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Recommended Educational Research Papers for Teachers to Read

My interest in educational research only really began when I spoke to the likes of Dylan Wiliam, Will Emney, Mark McCourt, Greg Ashman and Tom Bennett for my Mr Barton Maths Podcast. Up until that point I guess had had been blindly going about my business, teaching based largely on gut instinct and the snippets of advice that had been popular throughout the first few years of my teaching career (learning styles, to name but one!). I had never really questioned whether what I was doing was the best for my students... until now (notice the pause for dramatic effect).

This page contains the academic research I have read to date that has influenced the way I plan and teach mathematics. It is meant as a complement to my Recommended Books for Teachers to Read page. It will be continually updated. I have divided it up into broad categories, but these are somewhat arbitrary due to the many interconnections between the papers. The recurring theme throughout everything reviewed on this page is trying to improve students' understanding and retention using evidence based research.

As well as linking to the original paper, I have also tried to summarise my main takeaway from it, explaining the practical ways it has changed the way I plan, teach and help my students.

I really hope you find this page useful.

Contents

Cognitive Sciencekeyboard_arrow_up
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Cognitive Science (along with Cognitive Psychology - often the two are hard to separate) is concerned with how people think, what motivates them to think, the importance of what they think about, and what conditions foster effective thinking. For Daniel Willingham, memory is the residue of thought (I love that phrase!), so if we can help students think, then we can help them remember and learn.

Article Title:
The Science of Learning
Author(s): Deans for Impact
My Takeaway:
This paper was a game-changer for me, and is a superb way to begin any voyage into the world of educational research. It is a summary of existing cognitive science research about how students learn. Each of the cognitive principles is stated clearly, and is presented alongside classroom implications, making it both practical and incredibly teacher-friendly. All of the principles identified are fascinating (and references are provided if you wish to dig into each one deeper), but what struck me in particular were the common misconceptions about how students think and learn: Students do not have different “learning styles”; Humans do not use only 10% of their brains; People are not preferentially “right-brained” or “left brained” in the use of their brains; Novices and experts cannot think in all the same ways; and Cognitive development does not progress via a fixed progression of age-related stages. How many of those have you heard? How many have you had training on? As I say, this was the paper that first opened my eyes.
My favourite quote:
The Science of Learning does not encompass everything that new teachers should know or be able to do, but we believe it is part of an important — and evidence-based — core of what educators should know about learning.

Article Title:
Why don't students like school? Because the mind is not designed for thinking.
Author(s): Daniel Willingham
My Takeaway:
This is an excellent summary of one of my all-time favourite books, Why Don't Students Like School. There is so much to takeaway from this, but the biggest thing for me is about how students think. Students (like all of us) are naturally curious and enjoy thinking, but that curiosity is fragile, and if the conditions are not right then students will avoid thinking. For Willingham, memory is the residue of thought, so unless students are thinking, there is little chance they will remember or learn. Willingham stresses that successful thinking requires three things: 1) information from the environment, 2) facts and procedures stored in long term memory, and 3) space in working memory. The problem is that working memory capacity is limited, and gets filled when students need to carry out basic procedures or search for facts. Hence, when faced with complex problems, if those facts and procedures are missing from long term memory, working memory gets filled up, thinking becomes hard, students stop doing it, and no learning takes place. Here is the key point: experts and novices think differently. Experts rely on retrieving whole schema (connected items of information) from long-term memory to get around the limits on our fragile working memories. Novices (i.e. most students) don’t have these memorised schema to rely on so attempt to hold too much information in their working memories which leads to cognitive overload. Therefore, we can make thinking easier for students by ensuring they have access to sufficient facts and procedures stored in long term memory, which frees up capacity in working memory, which can then be used for problem solving. As teachers we can help students acquire these facts via Explicit Instruction, emphasising the value of deliberate practice to improve, not taking study skills for granted, and praising effort not ability, and help students retain them using the work in the three Memory sections.
My favourite quote:
Teachers often seek to draw students in to a lesson by presenting a problem that they believe interests students, or by conducting a demonstration or presenting a fact that they think students will find surprising. In either case, the goal is to puzzle students, to make them curious. This is a useful technique, but it’s worth considering whether these strategies might also be used not at the beginning of a lesson, but after the basic concepts have been learned.

Article Title: A Simple Theory of Complex Cognition
Author(s): John R. Anderson
My Takeaway:
I am not entirely sure that "simple" is a good description of this theory, at least not for me anyway! But the main message is a crucial one: to be able to think, we need knowledge. Long-term memory is capable of storing thousands of facts, and when we have memorised thousands of facts on a specific topic, these facts together form what is known as a schema. When we think about that topic, we use that schema. When we meet new facts about that topic, we assimilate them into that schema – and if we already have a lot of facts in that particular schema, it is much easier for us to learn new facts about that topic. And how best to learn these facts? Well, the sections on Explicit Instruction and Memory should provide some answers.
My favourite quote:
All that there is to intelligence is the simple accrual and tuning of many small units of knowledge that in total produce complex cognition. The whole is no more than the sum of its parts, but it has a lot of parts.

Article Title: Self-Efficacy: An Essential Motive to Learn
Author(s): Barry J. Zimmerman
My Takeaway:
I think a lot about how to motivate my students. I have tried attempting to make the maths we study more relevant to their lives, tried the all-singing, all-dancing lessons, used videos, technology and more. These strategies have had mixed success, but even the best have not proved sustainable in the long run. This article has convinced me what I think I have always known - students are motivated by their own success. If you can convince students that they are successful at maths (self-efficacy is defined as "as one's belief in one's ability to succeed in specific situations or accomplish a task"), then they will be motivated to learn more. How you do that is up to you. For some it will be a Mindset approach, whereas for others it will be Explicit Instruction to ensure students are equipped with the fundamentals needed for more complex thinking, and hence further success. 
My favourite quote:
This empirical evidence of its role as a potent mediator of students’ learning and motivation confirms the historic wisdom of educators that students’ self-beliefs about academic capabilities do play an essential role in their motivation to achieve.

Article Title: When More Pain is preferred to Less: Adding a Better End
Author(s): Daniel Kahneman, Barbara L. Fredrickson, Charles A. Schreiber, and Donald A. Redelmeier
My Takeaway:
Often I would end my lessons with a tricky question - maybe a past exam question. This would be the most difficult question I had asked all lesson, and its purpose would for both myself and my students to see how far they had come. I would usually build it up: "okay, this is as tough as it gets. Can you do it?". And often the result was that some could and some couldn't, and I was okay with that as it was meant as an extension question. But having read this paper, I now look at it from the perspectives of the students who could not do the question. What is their  impression of that lesson? What will they remember? Will it be the 45 minutes of success they enjoyed at the start, or the 5 minutes of "failure" at the end. Cognitive psychology suggests the latter. This fascinating paper looks at how people's memory of an experience is often dominated by the feelings of pain and discomfort during the final moments, as opposed to what happened during the rest of the experience. Applying this to my students, when thinking about their maths lesson, many of them would have judged it as a failure because of that final, tricky problem. This negative emotion - potentially swirling around their heads the couple of days until their next lesson - could lead to a subsequent lack of confidence and a lack of engagement in mathematics, neither of which are conducive to learning. Hence, I now always end my lesson with a question that is of mid-range difficulty, or maybe even easier. The majority of the learning has happened in the first 45 minutes - my objective in those last 5 minutes is for my students to feel good about themselves ready for their next maths lesson.
My favourite quote:
The results add to other evidence suggesting that duration plays a small role in retrospective evaluations of aversive experiences; such evaluations are often dominated by the discomfort at the worst and at the final moments of episodes.

Article Title: Organizing Instruction and Study to Improve Student Learning
Author(s): Harold Pashler et al
My Takeaway:
Much like The Science of Learning above, this is an outstanding overview of key research findings from cognitive psychology that have direct implications for the classroom. A wonderful feature of this paper is that it also addresses potential roadblocks to implementing the finding in the classroom and suggests possible strategies to overcome them. The recommendations, each of which are covered in a comprehensive and yet easy to follow way, are: Space learning over time, Interleave worked example solutions and problem-solving exercises, Combine graphics with verbal descriptions, Connect and integrate abstract and concrete representations of concepts, Use quizzing to promote learning, Help students allocate study time efficiently, Help students build explanations by asking and answering deep questions. These recommendations have implications for planning, teaching, and helping students formulate effective revision strategies.
My favourite quote:
We recommend a set of actions that teachers can take that reflect the process of teaching and learning, and that recognizes the ways in which instruction must respond to the state of the learner. It also reflects our central organizing principle that learning depends upon memory, and that memory of skills and concepts can be strengthened by relatively concrete—and in some cases quite nonobvious - strategies

Explicit Instructionkeyboard_arrow_up
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Explicit instruction may be thought of as teacher-led instruction. It is more interactive than simply lecturing, involving questioning and responsive teaching, but a key characteristic is that the teacher dictates the content and structure of the lesson, in contrast to more student-centered approaches. A great deal of research suggests this is the most effective way to help children learn, and despite having moved away from it over the course of my career, it is now the central structure upon which I build my lessons.

Article Title:
Why Minimal Guidance During Instruction Does Not Work (and a shorter, easier to digest version Putting Students on the Path to Learning)
Author(s): Paul A. Kirschner and John Sweller
My Takeaway:
This article, more than any other, changed my way of thinking. It provides evidence that for everyone apart from experts, partial guidance during instruction (the kind used in inquiry-based learning or project work) is less effective than full guidance (direct or explicit instruction). This suggests that practice (often encouraged by Ofsted and senior management alike) of encouraging students to "discover" learning, and for the teacher to be the "facilitator" of learning and not the expert imparting their knowledge, is detrimental to students' learning and long term development. Without facts and procedures stored in long term memory, students cannot become the problem solvers we all want them to be. And crucially (and possibly counter-intuitively), they cannot become problem solvers by simply trying to solve problems! Daniel Willingham (see Cognitive Science section) emphasises that experts and novices learn differently, and this paper is also careful to point out that full guidance may not be the most suitable instructional technique for students once they reach mastery, or become experts. This point is discussed later when we look at problem solving.
My favourite quote:
If the learner has no relevant concepts in long-term memory, the only thing to do is blindly search for solutions. Novices can engage in problem solving for extended periods and learn almost nothing.

Article Title: Principles of Instruction
Author(s): Barak Rosenshine
My Takeaway:
I absolutely love this paper. It presents 10 research-based principles from cognitive science and studies of master teachers, together with practical strategies for classroom implantation. The 10 principles are all based around the model of explicit instruction, and sound so simple when you see them written down: 1) Begin a lesson with a short review of previous learning. 2) Present new material in small steps with student practice after each step 3) Ask a large number of questions and check the responses of all students 4) Provide models. 5) Guide students practice. 6) Check for student understanding. 7) Obtain a high success rate. 8) Provide scaffolds for difficult tasks. 9) Require and monitor independent practice. 10) Engage students in weekly and monthly review. For me, this is the structure of a very well planned lesson, encompassing aspects of spacing, retrieval, worked examples, modelling, formative assessment and problem solving , all of which will be discussed in greater detail further on this page. For each element there are links to further research to support the author's claim.
My favourite quote:
Education involves helping a novice develop strong, readily accessible background knowledge. It's important that background knowledge be readily accessible, and this occurs when knowledge is well rehearsed and tied to other knowledge. The most effective teachers ensured that their students efficiently acquired, rehearsed, and connected background knowledge by providing a good deal of instructional support. They provided this support by teaching new material in manageable amounts, modeling, guiding student practice, helping students when they made errors, and providing for sufficient practice and review. Many of these teachers also went on to experiential, hands-on activities, but they always did the experiential activities after, not before, the basic material was learned.

Article Title: The Role of Deliberate Practice in the Acquisition of Expert Performance
Author(s): K. Anders Ericsson, Ralf Th. Krampe, and Clemens Tesch-Romer
My Takeaway:
To become an expert at something, you must practice in the right way. This article (along with the excellent book, Peak), outlines a model of Deliberate Practice. A key feature of this model is that you isolate an individual skill and work on it, receiving regular and specific feedback so you can improve you performance. Crucially (and this was a game-changer for me), the skill you are practising may not look like the final thing you are working towards. It is why young Spanish footballers start on tiny pitches, playing 5-aside, working on specific drills - you learn far more from that than you ever would in a big 11-aside game where you hardly touch the ball, and there are so many other factors out of your control. It is why professional musicians practise scales over and over again instead of constantly playing full pieces. The implications for teaching are huge. When we teach a topic, say adding fractions, we need to break the skills down into their most minute components. If we want students to have success in tricky exam questions (e.g. those carrying several marks, or those of a more problem-solving nature)  then it might not be the case that we should give them practice of these questions. Instead we should isolate individual skills (such as basic numeracy, or identifying exactly what the question is asking), and practice these in isolation until students have mastered them before even going near an exam. The activities students do in practice may look nothing like what they will be expected to do in the final exam, but that is absolutely fine. Moreover - and for me this is just as important - students need immediate feedback on their practice. Think about a tennis player hitting hundreds of balls during a service drill. Each time they hit a serve, they immediately see the result and can adjust their technique to improve. How can we ensure students can do the same during their practice, so errors are identified and not compounded? It could be as simple as giving them immediate access to the answers. This is why two teachers I interviewed for my Mr Barton Maths Podcast always supply students with the answers - John Corbett during classwork, and Greg Ashman for homework.
My favourite quote:
During a 3-hr baseball game, a batter may get only 5-15 pitches (perhaps one or two relevant to a particular weakness), whereas during optimal practice of the same duration, a batter working with a dedicated pitcher has several hundred batting opportunities, where this weakness can be systematically explored

Article Title: The Emotional Dog and Its Rational Tail: A Social Intuitionist Approach to Moral Judgment
Author(s): Jonathan Haidt
My Takeaway:
Something a bit different to end this section. I was introduced to the concept of belief systems, together with the work of Joathan Haidt, by Neil Atkin. This fascinating paper emphasises the importance of intuition over rational thought when it comes to moral judgements, and argues that an individual's intuitions often remain despite the existence of evidence to contradict them. Applying this to education suggests a potential limitation to the model of explicit instruction is that it assumes students will believe what they are told. When that involves something like how to calculate the percentage of an amount, or how to use trigonometry to find missing sides in right-angled triangles, this is probably fine as students are unlikely to hold any intuitive contradictory beliefs. However, I am not convinced that is true for all topics. Take something like probability. Probability is filled with so many counter-intuitive results (the birthday paradox, the Monty Hall problem, etc) which may directly contradict the belief systems held by students, that simply being told how to do something and what the answer is may not be enough. For topics such as these, students may well need to be convinced of the answer - for it is only when they are convinced that their belief systems can be changed. Is this also true of misconceptions? Take a skill such as adding two fractions together. By instinct, students are likely to simply add the numerators and denominators. Now, this is perhaps not representative of a deep belief system, but such misconceptions are so prevalent among students (how many Year 11s have you seen making that mistake despite 6 years of being taught it?), that once again they may need convincing as opposed to just being instructed. How do we convince them? Well, we have traditional tools such as mathematical proof, and more modern models such as Dan Meyer's 3 Act Math structure.
My favourite quote:
Moral reasoning is an effortful process, engaged in after a moral judgment is made, in which a person searches for arguments that will support an already-made judgment.



Cognitive Load Theory keyboard_arrow_up
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Dylan Wiliam described Cognitive Load Theory as "the single most important thing for teachers to know", and alongside Explicit Instruction it has transformed the way I teach. Here we look at key aspects of the theory relating to worked examples, presentation of information, and the development of problem solving skills.

Article Title: Cognitive Load Theory and the Format of Instruction (and a shorter, easier to digest summary Story of a Research Programme)
Author(s): Paul Chandler and John Sweller
My Takeaway:
Cognitive Load Theory explains so much of the behaviour I have seen in my students over the years. Cognitive Load Theory is concerned with how cognitive resources are focussed and used during problem solving and learning. Many problem solving tasks encourage students to engage in cognitive activities that are removed from the goal of the task - processing redundant information and listening to lengthy sets of instructions are just two of these. These cognitive activities fill up working memory capacity and impede learning. Indeed, it was a real eye-opener for me to discover that students could be working hard, and yet not actually learning anything. The authors suggest well-constructed worked examples and being careful with the presentation of information can help reduce this cognitive load, and they are two simple things that I feel have significantly increased the effectiveness of my own teaching recently. For an in-depth discussion with a maths teacher who regularly practices the lessons from Cognitive Load Theory, you can listen to my interview with Greg Ashman. And the papers that follow in this section look at some specific "effects" identified by Cognitive Load Theory and their implications for explicit instruction.
My favourite quote:
Cognitive load theory has been used to explain why studying worked examples can facilitate learning compared with problem solving. In essence, searching for suitable problem-solving operators is cognitively demanding and directs attention away from aspects of the problem important to learning. In contrast, many worked examples are far easier to process than the equivalent problems and direct attention more appropriately.

Article Title: Learning from Examples: Instructional Principles from the Worked Examples Research
Author(s): Robert K. Atkinson , Sharon J. Derry , Alexander Renkl , Donald Wortham
My Takeaway:
A key component of Cognitive Load Theory is the "worked example effect", whereby learners who study worked examples perform better on test problems than learners who solve the same problems themselves. There are so many fascinating concepts discussed throughout this paper extolling the virtues of worked examples, but I am going to focus on one that has directly changed the way I teach. I used to do a load of worked examples at the start of the lesson, and then give the students a set of problems to work on for the remainder of the lesson. The problem (and it seems so obvious now) was that by the time I had done the 3rd worked example, students had forgotten the first as they had not had the opportunity to practice it for themselves. Hence, they would start working on the questions, get stuck, and I would need to go through it all again. This paper recommends interleaving worked examples with related questions for students to solve alone. Something as simple as doing worked examples for all the odd numbered questions whilst students solve the even ones can be beneficial. Greg Ashman discussed this strategy when describing how he plans his lessons when I interviewed him for my podcast.
My favourite quote:
First, transfer is enhanced when there are at least two examples presented for each type of problem taught. Second, varying problem sub-types within an instructional sequence is beneficial, but only if that lesson is designed using worked examples or another format that minimizes cognitive load. Third, lessons involving multiple problem types should be written so that each problem type is represented by examples with a finite set of different cover stories and that this same set of cover stories should be used across the various problem types. Finally, lessons that pair each worked example with a practice problem and intersperse examples throughout practice will produce better outcomes than lessons in which a blocked series of examples is followed by a blocked series of practice problems.

Article Title: Learning from Worked Examples: What happens if mistakes are included?
Author(s): Cornelia S Grosse and Alexander Renkl
My Takeaway:
I am a huge fan of "spot the mistake" style activities, whereby I show my students a complete solution to a problem and they must identify, explain and correct the mistakes. However, following my research into Cognitive Load Theory, I wanted to ensure that evidence suggested this was a sound technique. This paper provides the answer. Learning from worked examples where errors are included can enhance learning and transfer, but only if students have good prior knowledge of the topic. This makes perfect sense. Firstly, if students do not understand the topic, then how are they to spot the mistakes? Secondly, if their understanding is not secure, then their working memories are likely to become overloaded whilst searching for the right and wrong answers simultaneously. As a result, I will be more careful with my use of Spot the Mistake in future.
My favourite quote:
Learning with incorrect examples poses challenging demands on the learners. They have to represent not only the correct solution in their working memory, but also the incorrect step with an explanation why it is wrong. Learners with low prior knowledge who cannot form larger chunks for information coding can easily be overtaxed.

Article Title: The Split-Attention Effect as a Factor in the Design of Instruction
Author(s): Paul Chandler and John Sweller
My Takeaway:
Understanding the possibility of the Split-Attention effect has really changed my teaching. This suggests that when presenting students with worked examples where a diagram is involved (for example, most geometry topics), the text should be carefully integrated within the diagram, and not separate from it. If not, there is a danger that students' working memories will deal with the text description and the diagrams separately and become overloaded. Similarly, when students are presented with written information, talking over the top of it can also bring about this split-attention effect. So, the simple practice of putting up a slide of text and allowing the students the opportunity to read it in silence BEFORE reading it aloud or discussing it can again reduce the split-attention effect. But, as the next paper shows, things are not quite so straight-forward.
My favourite quote:
While cognitive load theory was used to predict that studying worked examples could be superior to solving the equivalent problems, the theory also was used to predict that not all worked examples would be effective. Many worked examples, particularly in mathematics and science, consist of two or more sources of mutually referring information. Diagrams with an accompanying textual explanation are a common example. For instance, worked examples in areas such as geometry and trigonometry consist of both a diagram and a set of textual statements. Usually, neither source of information is intelligible by itself, and meaning can be extracted only by mentally integrating the text with the diagram. Mental integration requires searching and matching each statement in the text with its appropriate entity on the diagram. According to cognitive load theory, this preliminary process of searching and matching text with diagram has the same consequences for learning as searching for operators to solve a problem through means-ends analysis. In both cases, cognitive effort is directed to a search process that is unrelated to learning. Attention is misdirected and cognitive resources are inappropriately allocated to an activity that is only engaged in because of the way the material is structured. Different structures can eliminate the search process freeing resources for learning.

Article Title: The Instructive Animation: Helping Students Build Connections Between Words and Pictures in Multimedia Learning
Author(s): Richard E. Mayer and Richard B. Anderson
My Takeaway:
The findings of this paper may at first glance seem at odds with the Split Attention Effect, but I think there is a key message running through them both. The authors conducted an experiment in which students studied an animation depicting the operation of a bicycle tire pump or an automobile braking system, along with concurrent oral narration of the steps in the process, and tested their performance on both a retention test and a problem solving test against groups who had the animation alone, narration alone, or no instruction. On both tests, the group who had the animation alongside the oral narration performed the best. The authors conclude that these results are consistent with a dual-coding model in which retention requires the construction of "representational connections" and problem solving requires the construction of "representational and referential connections". The obvious implication for teaching from this paper is that pictures and words together can be more effective than pictures or words alone. What are the implications for the Split-Attention Effect which postulates that students' working memories are in danger of cognitive overload if faced with too many contrasting mediums? Well, I think the key is in the relevance of the medium. We discussed in the paper above how if text was carefully integrated into geometrical diagrams then it aids the students. I think the same is true here. If the oral description is there to support the comprehension of the animation, then it should benefit students by easing the strain placed on their working memories. But it is obviously a fine balance. In the Real Life Maths section, I discuss a paper where the use of video to support maths comprehension had a negative effect on performance. For what it is worth, having carefully read both papers, my advice is as follows. For occasions where either text alone or a diagram alone will not adequately describe the situation to students (such as a geometric problem), then carefully integrate the two together. Where one or the other may suffice, I would advise presenting them separately and then together. For example, if I was demonstrating the lines of symmetry of a rectangle, I would show the diagram, give the students chance to process it, and then provide an oral description whilst the diagram was still on the screen. Hopefully that would have the effect of recusing the cognitive load on working memory, whilst also taking advantage of the concurrent presentation of information outlined in this paper. 
My favourite quote:
What makes an instructive animation? The results presented in this article demonstrate that animation per se does not necessarily improve students' understanding of how a pump or a brake works, as measured by creative problem solving
performance. For example, in both experiments, students who received animation before or after narration were able to solve transfer problems no better than students who had received no instruction. In contrast, when animation was presented concurrently with narration, students demonstrated large improvements in problem-solving transfer over the no-instruction
group. We conclude that one important characteristic of an instructive animation is temporal contiguity between animation and narration. We hypothesize that contiguity of words and pictures during instruction encourages learners to build connections between their verbal and visual representations of incoming information, which in turn supports problem-solving transfer.
 
Article Title: Cognitive Load during Problem Solving: Effects on Learning
Author(s): John Sweller
My Takeaway:
One huge question from all we have seen so far on Explicit Instruction is: "how do we get our students to become good problem solvers?". This paper offers the first clue. For students to become good problem solvers they need to form mental schemas from domain-specific knowledge which they can then apply to different situations. Unlike experts, novices lack the appropriate schema to recognise and memorise problem configurations. They set about solving problems by focusing on the detail and ignoring structure. However, problem solving itself is not an effective way of developing these mental schema. Why? Well, because developing these schema whilst also trying to solve problems (problem-solving search) overloads working memory, and hence the crucial mental schema are not developed. My key takeaway from this is that students do not learn from problem solving. Problem solving must come at the end of the process. Taking this further, I believe there is little point going through lots of difficult exam questions in the hope students understand them and make connections. If the basic skills are not in place, then problem-solving search suggests that there will simply not be enough capacity in working memory for students to develop the mental schema necessary to learn.
My favourite quote:
Most mathematics and mathematics-based curricula place a heavy emphasis on conventional problem solving as a learning device. Once basic principles have been explained and a limited number of worked examples demonstrated, students are normally required to solve substantial numbers of problems. Much time tends to be devoted to problem solving and as a consequence, considerable learning probably occurs during this period. The emphasis on problem solving is nevertheless, based more on tradition than on research findings. There seems to be no clear evidence that conventional problem solving is an efficient learning device and considerable evidence that it is not. If, as suggested here, conventional problems impose a heavy cognitive load which does not assist in learning, they may be better replaced by nonspecific goal problems or worked examples. The use of conventional problems should be reserved for tests and perhaps as a motivational device.

Article Title: The Expertise Reversal Effect
Author(s): Alexander Renkl and Robert K. Atkinson
My Takeaway:
So far all the talk has been of using Explicit Instruction as the best method of teaching students to become fluent in the facts and procedures they will need to learn more complex skills. Specifically, in early skill acquisition, learning from worked examples is very advantageous, and learning from problem solving is not. However, as this paper describes, instructional techniques that are highly effective with inexperienced learners (novices) can lose their effectiveness and even have negative consequences when used with more experienced learners (experts), hence "the Expertise Reversal Effect". The argument is that worked examples contain information that is easily determined by the more experienced learners themselves and, therefore, can be considered redundant. Devoting working memory to redundant information effectively takes away a portion of the learners’ limited cognitive capacity that could be devoted to germane load. Moreover, redundant information may even interfere with the schemas constructed by experienced learners, preventing them from seeing the deeper connections in problems that are essential for transfer. Of course, one major difficulty of this is recognising when students have made the transition from novice to expert and hence can start to be exposed to more complex problems. It is a delicate balancing act!
My favourite quote:
When a problem can be solved relatively effortlessly, analyzing a redundant worked example and integrating it with previously acquired schemas in working memory may impose a greater cognitive load than problem solving. Under these circumstances, practice in problem solving may result in more effective learning than studying worked examples because solving problems may adequately facilitate further schema construction and automation

Article Title: Structuring the Transition From Example Study to Problem Solving in Cognitive Skill Acquisition: A Cognitive Load Perspective
Author(s): Alexander Renkl and Robert K. Atkinson
My Takeaway:
This paper offers a slightly different take on problem solving than purported via the Expertise Reversal Effect above, but still under the umbrella of Cognitive Load Theory. The authors propose a fading procedure, in which problem-solving elements are successively integrated into the study of worked examples until the learners are expected to solve problems on their own. The crucial difference here is that students do not need to become "experts" before being exposed to problem solving - it is something that is gradually introduced alongside the acquisition of knowledge. Just like the paper above, we do not start with the problem solving questions. But at the same time the authors caution against introducing problem solving too late.
My favourite quote:
First, a complete example is presented (model). Second, an example is given in which one single solution step is omitted (coached problem solving). Then, the number of blanks is increased step by step until just the problem formulation is left, that is, a to-be-solved problem (independent problem solving). In this way, a smooth transition from modeling (complete example) over coached problem solving (incomplete example) to independent problem solving is implemented.

Problem Solvingkeyboard_arrow_up
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For me, Cognitive Load Theory provides a sound argument for why students cannot learn to be problem solvers before they have sufficient knowledge and procedures stored in long term memory, and Explicit Instruction provides a model for students acquiring those knowledge and procedures. But how do we then help our students take that next step towards developing those key problem solving skills that we all want them to have?

Article Title:
Research-Based Strategies for Problem-Solving in Mathematics
Author(s): Florida Department of Education,
My Takeaway:
This is a good summary of various problem solving strategies, and their applications in the classroom. The paper is structured around Polya's classic four-step approach to problem solving: 1. understanding the problem, 2. devising a plan to solve the problem, 3. implementing the plan, and 4. reflecting on the problem. Each of these four stages in analysed in great detail, with helpful suggestions for strategies to use in the classroom, together with examples and links to relevant research. I particularly like the emphasis on the first part of this process - understanding the problem. What should be the simplest part of this process if often the most difficult, but suggested techniques such as The Paraphrasing Strategy and Visualisation may help. However, what struck me most about this paper is the sheer variety of strategies outlined. There is no single "problem solving strategy" that students can learn and then apply to any problem. Topic-specific knowledge, together with an understanding of deep structure of problems that only knowledge can bring, is needed to correctly select the most appropriate strategy for a given problem and apply it successfully.
My favourite quote:
The first step in the Polya model is to understand the problem. As simple as that sounds, this is often the most overlooked step in the problem-solving process. This may seem like an obvious step that doesn’t need mentioning, but in order for a problem-solver to find a solution, they must first understand what they are being asked to find out.
Polya suggested that teachers should ask students questions similar to the ones listed below:
    Do you understand all the words used in stating the problem?
    What are you asked to find or show?
    Can you restate the problem in your own words?
    Can you think of a picture or a diagram that might help you understand the problem?
    Is there enough information to enable you to find a solution?

Article Title: Problem-Solving Strategies: Research Findings from Mathematics Olympiads
Author(s): Cheung Pak-Hong
My Takeaway:
This paper takes Polya's approach to problem solving ties to apply it to tricky Mathematical Olympiad questions. A variety of problems are presented, together with a discussion around which problem solving strategies could be used to tackle them. Those sitting this paper are among the very best student mathematicians in the world, and hence their approach to solving problems should shed some valuable light onto the most effective strategies. The conclusion of the authors is of crucial importance: the most effective problem-solving strategies are topic-specific. There is no generic problem-solving strategy. To be successful at these most challenging of problems, students must know their topics inside and out. This once again gives credence to the view expressed by Cognitive Load Theory that you cannot teach problem solving merely by exposure to problems. Topic-specific knowledge must be acquired in order for appropriate mental schema to develop, which allows students to have the best chance of solving problems.
My favourite quote:
This pilot study revealed that, for problems at Olympiad level, while heuristics suggested by Polya are useful in analysing the problems and in exploring feasible solutions, most of the more effective strategies are topic oriented. Olympiad problems in geometry are almost excluded from this report because although common strategies for solving them do exist, such as expressing quantities in terms of areas of triangles, they are confined to geometry.



Inquiries keyboard_arrow_up
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With minimal guided instruction coming under fire in the Explicit Instruction section, you may not think there is a place for inquiries in mathematics. But if done carefully, I believe there still is, with potential benefits in terms of motivation and problem solving.

Article Title:
Inquiry Teaching
Author(s): Andrew Blair
My Takeaway:
Inquiry teaching is often misunderstood as "learning by discovery". This article provides a good summary of the key distinction, together with the perceived benefits of inquiry teaching. The amount of structure or guidance offered by the teacher is determined by students' ability to think independently and critically. A key feature of inquiry lessons is "negotiation" between the teacher and the students over the structure and direction of the lesson. It is worth noting, however, that proponents of the direct instruction approach will likely balk at statements such as "Students not only meet concepts and skills in a meaningful pursuit of answering their own or peers’ questions, they also participate in a debate about how to learn and why to learn in a certain way" and "(the teacher must) redistribute authority to students". But I think that if the inquiries are well structured, and students only participate in them when they are ready (i.e. they are not used to teach concepts, just to apply them), then they have a valid place in the maths classroom.
My favourite quote:
In teacher directed strategy lessons, there is little guarantee that a method permits students to reach an ‘ideal’ understanding. In the investigative classroom, the learning experience is rigidly structured, leaving students the unenviable responsibility to discover a conceptual relationship. It is only the inquiry classroom that offers students a mechanism to harmonise conceptual learning with the method of learning.

Memory: Forgetting, Testing and Interleavingkeyboard_arrow_up
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How many times have you seen students seemingly understand something one lesson, and then act as though they have never seen it before only a day later? Learning without the ability to retain and retrieve is not really learning at all. These papers look at the crucial, and fascinating, subject of memory, specifically with regard to forgetting, spacing and interleaving. The power of Testing blew me away so much that it is a separate section. There are also profound implications for students revision, which will also be dealt with explicitly in the Revision section. The underlying message running throughout these papers blows my mind - in order to improve learning, we need to make it more difficult. Bjork's paper on Desirable Difficulties brings this all together beautifully.

Article Title: Learning versus Performance
Author(s): Nicholas C. Soderstrom and Robert A. Bjork
My Takeaway:
This is one of those things that sounds so obvious when you say it out loud, but I had been missing it for 12 years of teaching: learning and performance are two different things. The goal of teaching is to facilitate learning, which can only be inferred at some point after instruction, whereas performance is something that can be observed and measured during instruction, via some form of assessment. The key point here is that current performance can be a highly unreliable guide to whether learning has happened. We are at risk of being fooled by current performance, which can lead us as teachers to choose less effective conditions of learning over more effective conditions - i.e. use strategies that lead to short-terms gains in performance, such as teaching to a particular test. Moreover, it can lead students to prefer certain revision techniques over others as they can see short-term improvements in performance. Here is the ultimate kicker - high performance may lead to lower learning! We will get into that more in the next paper. There is another part of this paper I love - the concept of "overlearning" - i.e. students continuing to practice after their performance (via an assessment) has seemingly peaked. If learning and performance are two different things, and performance is a poor indicator of learning, then it makes sense for students to continue to practice despite a good performance in a test. But I will make two further points on this. Firstly, students need to be informed of this concept and its benefits, otherwise they are likely to question why they are working on something that they can already do. Secondly, a maths-specific study, The Effects of Overlearning and Distributed Practise on the Retention of Mathematics Knowledge by Doug Rohrer and Kelli Taylor, discussed later in this section found that overlearning had no significant effect on retrieval, whereas distributed practice did. They do not dismiss the concept of overlearning, but suggest ways in which students' time could be better spent. Note: this wonderful paper also makes an appearance on the Marking and Feedback section with its important implications for the timing of feedback.
My favourite quote:
Students and instructors alike often fail to appreciate the distinction between current performance and long-term learning, which makes them susceptible to mistaking the former as reliable index of the latter.

Article Title:
On the Symbiosis of Learning and Forgetting
Author(s): Robert A. Bjork
My Takeaway:
The work of Robert Bjork and Elizabeth Bjork on the importance and benefits of forgetting has blown my mind. Their "Theory of Dissuse" explains that that human memory is characterised by a storage capacity that is essentially unlimited, coupled with a retrieval capacity that is severely limited. The theory distinguishes between the retrieval strength of a memory representation - how accessible it is at a given point in time, which is influenced by local conditions, such as recency and current cues - and the storage strength of that representation, which is an index of how entrenched or interassociated that representation is with related representations in memory. Bjork argues memories do not disappear, they simply become lesson accessible. But the theory goes further than that - it is actually beneficial to forget! Why? Well, because if something is highly accessible, no learning can happen. Think about when someone gives you a phone number to remember - you say it over and over again (high retrieval strength), but can you recall it the next day? Probably not (low storage strength). As we forget, retrieval strength dips, then when we practice that material again there is a noticeably larger increase in storage strength, and long-term recall improves, often significantly. The cycle continues, with each phrase of forgetting and recalling leading to greater storage strength. This has huge implications for both teaching and revision. Firstly, it sets up nicely the work on interleaving and spacing that we will cover later in this section. But it also calls into question the idea of giving students loads of the same type of question to do following the teaching of a new concept. Isn't that similar to asking them to remember a phone number? Sure, they can all do the questions by the end of the lesson (high retrieval strength), but then when asked to do a similar question next lesson, or next week, students often cannot (low storage strength). How do we rectify this? Well, maybe we give students less questions to answer on a given topic (to borrow a phrase from my past as an Economics student, there is a diminishing marginal utility to each one), allow them to forget it, and then cover the material again at a later date. This cycle of practising, forgetting, and then practicing again, as opposed to keep practising the same thing, allows the storage strength of these skills to increase, which is needed for long term learning. Of course, this will be a frustrating experience for students (and us!), so students may need informing of the power of forgetting.
My favourite quote:
Perhaps the prime example of forgetting enhancing learning is the spacing effect, one of the most robust and general effects from the entire history of experimental psychology When a second study opportunity is provided after a delay following a first study opportunity, rather than being presented with little or no delay, long-term recall is enhanced, often very significantly. Again, though, were the studied material to be tested following a short delay or a long delay, we would observe that the longer delay results in poorer recall of the studied material—that is, more forgetting. Similarly, interleaving, rather than blocking, the learning trials on separate to-be-learned tasks produces more forgetting between trials on a given task during the learning phase, but tends to enhance long-term retention and transfer.

Article Title: Making Things Hard on Yourself, But in a Good Way: Creating Desirable Difficulties to Enhance Learning
Author(s): Elizabeth L. Bjork and Robert Bjork
My Takeaway:
This is paper is one of the best things I have ever read. It is directly related to the issue of distinguishing between learning and performance, and argues the seemingly counter-intuitive point that you improve learning by making it more (desirably!) difficult. These deliberate difficulties will likely lead to a reduction in short-term performance, but an improvement in long-term learning. The implications for lessons (and of course lesson observations!) are huge. The authors suggest the following strategies to induce desirable difficulties, many of which have been discussed above: allowing students to forget some of the material covered before it’s reintroduced (spacing); mixing up different content in order to prevent students developing the illusion of knowledge (interleaving); asking questions about material which has already been covered rather than restudying it in order to prevent students developing a false sense of familiarity and fluency (retrieval practice); varying the conditions in which instruction takes place in order to prevent contextual cues from building up and making it harder for students to transfer what they’ve learned to new contexts (variation); and progressively reducing the frequency and quantity of feedback given in order to prevent students from becoming dependent on external sources of expertise. I fully agree with all of this, but I will add one thought - if things are always too difficult for students, then motivation quickly disappears, and no teaching strategy can be effective if your students have switched off and have zero desire to engage in the learning process. Indeed, this does seem in contrast to Dan Willingham's view of thinking discussed on the Cognitive Psychology section - if thinking is too hard, students stop thinking - as well as Cognitive Load Theory. Much like David Didau (in this wonderful blog post on the same paper), I feel students need to taste success first in order to be motivated (see the Zimmerman article on the importance of self-efficacy in the Cognitive Psychology section for more on this), then we must ensure key knowledge and processes are stored in their long term memory, before finally things can become deliberately difficult. Didau proposes what seems a very sensible three stage plan to achieve this: Encode success, Promote internalisation, Increase challenge.
My favourite quote:
The basic problem learners confront is that we can easily be misled as to whether we are learning effectively and have or have not achieved a level of learning and comprehension that will support our subsequent access to information or skills we are trying to learn. We can be misled by our subjective impressions. Rereading a chapter a second time, for example, can provide a sense of familiarity or perceptual fluency that we interpret as understanding or comprehension, but may actually be a product of low-level perceptual priming. Similarly, information coming readily to mind can be interpreted as evidence of learning, but could instead be a product of cues that are present in the study situation, but that are unlikely to be present at a later time. We can also be misled by our current performance. Conditions of learning that make performance improve rapidly often fail to support long-term retention and transfer, whereas conditions that create challenges and slow the rate of apparent learning often optimize long-term retention and transfer.

Article Title:
Spacing and Interleaving of Study and Practice
Author(s): Shana K. Carpenter
My Takeaway:
This is a really nice introduction to the concepts of spacing and interleaving, which are two of Bjork's "desirable difficulties" described above. Best of all, both of these concepts are relatively easy to implement. The "Spacing Effect" refers to the findings that learning is better when two or more exposures to information are separated in time (i.e. spaced apart) than when the same number of exposures occurs back-to-back in immediate succession. The "Interleaving Effect" contrasts a "blocking" approach, whereby students study the same type of material over and over again before moving on to a different type of material, against an "interleaving" approach, where students practice all of the problems in an order that is more random and less predictable. The general findings are that both spacing and interleaving can have produce significant benefits with regard to memory and learning. Spacing is relatively easy to implement. You just need to ensure you routinely revisit material at fixed intervals throughout the year - and a paper in this section provides guidance for what those fixed intervals should be. Interleaving is more difficult. This would require teaching something like fractions not in a block, but splitting it up. So, you might teach equivalent fractions first, then move onto mean, then solving equations, and then come back to adding fractions. That is a big shift from the traditional scheme of learning. However, as the studies in this section will show, you can reap some of the benefits of interleaving by teaching topics in the exact same order as you always have done, but by simply adjusting your homework and assessments. A key point that also needs raising is that blocked practice might be more appropriate when a skills is first being learned. After all, spacing and interleaving work by inducing students to retrieve concepts from long-term memory. If concepts are not there in the first place, then there is nothing to retrieve!
My favourite quote:
Students and instructors are faced with these decisions on a daily basis. Research on human cognition has shown that learning can be significantly affected by the way in which repetitions are scheduled. This research can help inform the decisions that students must make concerning when to study information in order to maximize learning.

Article Title:
Using Spacing to Enhance Diverse Forms of Learning
Author(s): Shana K. Carpenter, Nicholas J Cepeda,  Doug Rohrer, Sean H. K. Kang & Harold Pashler
My Takeaway:
The concept behind spacing is relatively simple, and its effects can be profound. This paper finds that performance on final tests of learning is improved if multiple study sessions are separated (spaced apart) in time rather than massed in immediate succession. The optimal length of the spacing depends on when the material is going to be tested. In general, longer spacings are more beneficial when the test is a long way away, whereas shorter spacing are more beneficial for tests in the near future. These longer periods may be weeks or even months. Three practical recommendations for teachers: 1) teachers should dedicate part of each lesson to reviewing concepts learned several weeks earlier 2) homework assignments should be used to re-expose students to important information they have learned previously 3) teachers should give exams and quizzes that are cumulative. This last point has an extra benefit - as well as re-exposing students to information that they have previously learned, cumulative exams and quizzes also provide students with a good reason to review information on their own.
My favourite quote:
The key criterion is that information should be reviewed after a period of time has passed since the initial learning. Particularly if the goal is long-term retention, the findings from Cepeda et al. (2008) suggest that the ideal time to review information may be several weeks or months after it was initially learned.

Article Title: Spacing effects in learning: A temporal ridgeline of optimal retention
Author(s): Nicholas J Cepeda,  Edward Vul, Doug Rohrer, John T Wixted & Harold Pashler
My Takeaway:
After the last paper you may be thinking: "wouldn't it be great if someone could tell me the optimal time to revisit content with my students depending on when their test is?". Well, your wish is my command. This paper offers a rough guide:
Time to test: 1 week, Optimal interval between study sessions: 1– 2 days
Time to test: 1 month, Optimal interval between study sessions: 1 week
Time to test: 2 months, Optimal interval between study sessions: 2 weeks
Time to test: 6 months, Optimal interval between study sessions: 3 weeks
Time to test: 1 year, Optimal interval between study sessions: 4 weeks
My favourite quote:
To put it simply, if you want to know the optimal distribution of study time, you need to decide how long you wish to remember something. Although this poses challenges for practical application, certain conclusions can nonetheless be drawn. If a person wishes to retain information for several years, a delayed review of at least several months seems likely to produce a highly favorable return on a time investment— potentially doubling the amount ultimately remembered, holding study time constant—as compared to less temporally distributed study.

Article Title:
Interleaved Practice Improves Mathematics Learning
Author(s): Doug Rohrer, Robert F. Dedrick, and Sandra Stershic
My Takeaway:
This study is fascinating. Previously I had thought interleaving had to involve completely hanging the order lessons are taught. However, that is not the case. Seventh-grade students saw their teachers’ usual maths lessons and received regular homework assignments. Every student received the exact same problems, but the scheduling of the problems was altered so that students received blocked or interleaved practice. Later, students received a review of all the content, followed 1 or 30 days later by an unannounced test. Students following the interleaving program performed significantly better on both tests, with a greater difference in performance found on the test taken 30 days later, suggesting a significant impact on retention. Now, the "blocked practice" approach is one I have used regularly. Teach students fractions, give them an assignment on fractions. Then teach them equations, given them an assignment on equations. Then it is time for angles and averages, then let's have a review of all four topics, and then a half term test. Our assignments do consist of short revision questions at the start which are taken from a variety of questions throughout the year, but the main focus of each assignment is undoubtedly on the topic that has just been covered. This paper suggest that approach is not quite good enough. The assignments should consist of a mixture of questions from previous topics, and not just the quick-fire revision ones. Moreover, each assignment should contain a majority of questions from other topics, not a minority. For example, the kind of fractions problems that would have appeared as questions 4, 6 and 9 on our first assignment should be included in the second assignment instead, alongside questions on equations and any other topics previously taught, and so on. I particularly like the fact that there are no implications for changing the way or order topics are taught, or the content of assignments - it is simply a case of varying the composition of the assignments themselves. This seems like a very quick win.
My favourite quote:
Benefits of interleaved practice have been consistently observed with a variety of mathematics skills and with students in elementary school, middle school, and college. As argued here, these benefits arise because interleaved practice provides students with an opportunity to learn how to choose an appropriate strategy (or learn that they cannot do it). In short, interleaved practice simply provides students with an opportunity to practice the very skill they are expected to learn.

Article Title: The Benefit of Interleaved Mathematics Practice is not limited to Superficially Similar kinds of Problems
Author(s): Doug Rohrer, Robert F. Dedrick & Kaleena Burgess
My Takeaway:
This paper discusses the point raised in the previous paper, but then takes it further. The correct solution to most mathematical problems involves two steps: identify the strategy needed to solve the problem, and then successfully carry out that strategy. They argue that most mathematical assignments deny students the opportunity to practice that first step - identifying the strategy. For example, if a lesson on the Pythagorean theorem is followed by a group of problems requiring the Pythagorean theorem, students know the appropriate strategy before they read each problem. An alternative approach is interleaving, where a majority of the problems within each assignment are  drawn from previous lessons, so that no two consecutive problems require the same strategy. With this approach, students must choose an appropriate strategy and not only execute it, just as they must choose an appropriate strategy when they encounter a problem during a cumulative exam or high-stakes test. Such interleaved practice also ensures that problems are spaced, which can improve retention. Unlike the previous paper, the authors do not attribute the benefit of interleaving to merely enabling students to better discriminate between problems. They argue solving a mathematics problem requires students not only to discriminate between different kinds of problems, but also to associate each kind of problem with an appropriate strategy. Blocked assignments often allow students to ignore the features of a problem that indicate which strategy is appropriate, which precludes the learning of the association between the problem and the strategy. Interleaved practice can help the development of both of these crucial skills.
My favourite quote:
Although it might seem surprising that a mere reordering of problems can nearly double test scores, it must be remembered that interleaving alters the pedagogical demand of a mathematics problem. As was detailed in the introduction, interleaved practice requires that students choose an appropriate strategy for each problem and not only execute the strategy, whereas
blocked practice allows students to safely assume that each problem will require the same strategy as the previous problem.


Article Title: The Effects of Overlearning and Distributed Practise on the Retention of Mathematics Knowledge
Author(s): Doug Rohrer and Kelli Taylor
My Takeaway:
The final paper on interleaving from my Doug Rohrer trilogy, and this has equally big implications. In two experiments, 216 college students learned to solve one kind of mathematics problem before completing one of various practice schedules. In Experiment 1, students either massed 10 problems in a single session or distributed these 10 problems across two sessions separated by 1 week. The benefit of distributed practice was nil among students who were tested 1 week later but extremely large among students tested 4 weeks later. In Experiment 2, students completed three or nine practice problems in one session. The additional six problems constituted a strategy known as overlearning, but this extra effort had no effect on test scores 1 or 4 weeks later. A few things struck me. Firstly, the improvement in performance from distributed practice was not seen immediately - this is important for both students and teachers to know. Secondly, as has been pointed out in the previous studies, most mathematics textbooks rely on a format that emphasises overlearning and minimises distributed practise, as do most of the worksheets and sets of questions I give my students to do in class. Finally, with regard to overlearning, the authors do not dismiss it (and indeed, with the problem of distinguishing between learning and performance discussed earlier in this section, it is dangerous to assume a student has learned something), but instead suggest a better use of students' time. The authors sum this up very nicely: "we suggest that assignments should err slightly in the direction of too much practise, perhaps by including three or four problems relating to each new concept in the most recent lesson (in addition to any examples given in the written lesson or class lecture). However, beyond these first three or four problems, the present data suggest that the completion of additional problems of the same type is a terribly inefficient use of study time. Instead, our findings suggest that the student should devote the remainder of the practise session to problems drawn from earlier lessons in order to reap the benefits of distributed practise"
My favourite quote:
With this distributed practise format, each lesson is followed by the usual number of practise problems, but only a few of these problems relate to the immediately preceding lesson. Additional problems of the same type might also appear once or twice in each of the next dozen assignments and once again after every fifth or tenth assignment thereafter. In brief, the number of practise problems relating to a given topic is no greater than that of typical mathematics textbooks, but the temporal distribution of these problems is increased dramatically.

Article Title: Why interleaving enhances inductive learning
Author(s):
Monica S. Birnbaum, Nate Kornell, Elizabeth Ligon Bjork & Robert A. Bjork
My Takeaway:
A fascinating paper to end our discussion on interleaving (for now!). An inductive approach to teaching involves showing examples of how a concept is used in the hope that students will ‘notice’ how the concept works, as opposed to a deductive approach, whereby the teacher provides the material pupils need to think about and reduces the quantity of information they are required to hold in working memory. It would seem sensible to think that a deductive approach works best, and indeed when learning basic content that does seem to be the case. However, this research suggests that students tend to understand and remember more when learning occurs inductively, but only if this is combined with an interleaved approach to studying content. In short - students best learn a rule by seeing examples containing the rule mixed up with examples that don’t. Why does this work? Well, frequent alternation of categories (interleaving) has the advantage of highlighting features that serve to distinguish categories. Conversely, infrequent alternation of categories (blocking) has the advantage of highlighting information that remains constant across the members within a category. For me, it depends what you want to achieve. If you wish students to become fluent in a particular skill or concept, then a blocking approach is likely to be best. However, if you want students to be able to distinguish between problems, recognising key features and selecting the most appropriate method to use (as is often needed for more wordy questions, questions where the required method is not obvious, or on high-stakes tests), then interleaving is necessary. The research we looked at in the Explicit Instruction section suggests that students need key knowledge and processes to be stored in long term memory before they can become effective problem solvers, so my takeaway is that blocked practice may be optimal when students first encounter a skill, but interleaving is needed for students to develop their learning further. On a related note - learners are likely to find the process of interleaving more difficult than blocking. This is largely because they have to think more. Crucially, they may feel they are not learning as much and begin to lose motivation. This relates directly to Bjork's "desirable difficulties" discussed later in this section.
My favourite quote:
The great majority of the  participants in the present study, as well as those in prior  studies, judged that they had learned  more effectively with blocked than with interleaved study. Thus, a bit of practical advice to learners and educators seems  warranted: If your intuition tells you to block, you should  probably interleave.


Memory: Testingkeyboard_arrow_up
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One of the golden rules of school life is that students hate tests. But I would like to refine this rule to say students hate "high-stakes" tests. The papers in this section will outline a finding that has possibly transformed my lessons more than any other: the power of regular, low-stakes testing. By low-stakes, I mean short tests (typically 3 to 5 questions) that the students take at the start of every single lesson based on content from earlier in the year (to take advantage of the Spacing and Interleaving effects), they mark them themselves, the marks are not recorded anywhere, I don't know their scores, students don't know each others' scores, and I go through the answers straight away so students have immediate feedback. You will not believe me when I say this, but the students love them. More importantly, the effects on their retention have been profound. In this section I try to get to the bottom of the power of testing.

Article Title: Ten Benefits of Testing and their Application to Educational Practice
Author(s): Henry L. Roediger III, Adam L. Putnam and Megan A. Smith
My Takeaway:
This paper is stunning. It details (as the title suggests) ten benefits of testing, citing research to support each claim. Each benefit is worth discussion here, but I am going to limit myself to just three. 1) The testing effect: retrieval aids later retention. A fascinating study described here explains that students who followed a pattern of study-study-study-test for a topic performed better on the final test than a group who followed study-test-test-test. But - and here is the key - when tested a week later, the exact opposite had occurred, even though the latter group had much less exposure to the material. Testing led to better long-term retention. 2) Testing causes students to learn more from the next learning episode. When students take a test and then restudy material, they learn more from the presentation than they would if they restudied without taking a test, hence it helps with revision. 3) Testing improves transfer of knowledge to new contexts. This is the holy grail! We have seen in the Cognitive Psychology section that the inability to transfer knowledge to new situation is one of the characteristics that defines novice learners and inhibits their learning. It seems that testing can help with that, possibly by enabling learners to form and strengthen schemas that they can later draw upon in new contexts. Add to all of this the benefit that testing can give you the teacher information about gaps in students' knowledge which you can resolve in class, and it promotes the benefits of testing to students so that they may use it in their revision, and you have possibly one of the most important teaching tools ever.
My favourite quote:
We have reviewed 10 reasons why increased testing in educational settings is beneficial to learning and memory, as a self-study strategy for students or as a classroom tactic. The benefits can be indirect—students study more and attend more fully if they expect a test – but we have emphasized the direct effects of testing. Retrieval practice from testing provides a potent boost to future retention. Retrieval practice provides a relatively straightforward method of enhancing learning and retention in educational settings.

Article Title: The Critical Importance of Retrieval for Learning
Author(s): Jeffrey D. Karpicke, et al
My Takeaway:
For years I have underestimated just how powerful actively practising retrieval, relative to other more common revision techniques, can be. Three findings in this paper particularly stood out to me. Firstly, repeated studying after learning had no effect on delayed recall, but repeated testing produced a large positive effect. I have seen this myself - students think they have cracked a topic like adding fractions because they have got questions correct in the past, and so during revision may merely glance over their notes on fractions (studying) instead of trying questions out (retrieval). Secondly, students’ predictions of their performance were uncorrelated with actual performance. The students in the study were not aware of the benefits of practising recall, even after they had done it! This is both fascinating and worrying, and suggests that students need to be convinced of the power of this strategy long before they start their revision. Regular low-stakes testing in the classroom seems a sensible way of achieving this. Finally - and I have saved the best to last - if retrieval is so important for learning (as you will see in the next paper, it can actually cause learning), then is it a good idea to just do a load of past papers for exam preparation, as I have done for many years? NO! Why? Well, exam papers are designed to cater to a wide variety of abilities. Hence, they do not always encourage retrieval from long term memory, but instead test elements of problem solving (specifically, elements not stored in long term memory). We have seen in the Cognitive Load Theory section that problem solving does not always lead to learning, and hence students can work through exam papers and not actually learn anything. I conclude that until students have covered the full course, regular, low-stakes tests on topics that students have already studied is the best way to profit from the positive effect of retrieval.
My favourite quote:
The conventional wisdom shared among students and educators is that if information can be recalled from memory, it has been learned and can be dropped from further practice, so students can focus their effort on other material. Research on students’ use of self-testing as a learning strategy shows that students do tend to drop facts from further practice once they can recall them. However, the present research shows that the conventional wisdom existing in education and expressed in many study guides is wrong. Even after items can be recalled from memory, eliminating those items from repeated retrieval practice greatly reduces long-term retention. Repeated retrieval induced through testing (and not repeated encoding during additional study) produces large positive effects on long-term retention.
 

Article Title:
Retrieval Practice Produces More Learning than Elaborative Studying with Concept Mapping
Author(s): Jeffrey D. Karpicke, et al
My Takeaway:
This paper is by the same team of authors as the previous one, but makes two key additional points. Firstly, it makes explicit the finding that practising retrieval is more effective on long term retention than attempting to encode information during the learning process - possibly using strategies such as mind-mapping, and making notes. Practising retrieval was found to be more effective than more traditional revision strategies when students were later tested on both factual recall and more problem solving questions. However, possibly the more interesting point is that the researchers found that retrieval practice can actually produce learning (as opposed to being neutral). This is sometimes called the Testing Effect (or the Retrieval Effect) and is possibly due to the cognitive strain experienced when trying to reconstruct knowledge, which is related to the fascinating paper on Desirable Difficulties discussed later in this section. Once again, for me this emphasises how important it is that students are aware of the power of self-testing during revision (and not necessarily of complete exam papers, as discussed in the paper above), and the importance of low-stakes tests in the classroom.
My favourite quote:
Research on retrieval practice suggests a view of how the human mind works that differs from everyday intuition. Retrieval is not merely a read out of the knowledge stored in one’s mind; the act of reconstructing knowledge itself enhances learning. This dynamic perspective on the human mind can pave the way for the design of new educational activities based on consideration of retrieval processes.

Article Title: Both Multiple-Choice and Short-Answer Quizzes Enhance Later Exam Performance in Middle and High School Classes
Author(s): Kathleen B. McDermott, Pooja K. Agarwal, Laura D’Antonio, Henry L. Roediger, III, and Mark A. McDaniel
My Takeaway:
This paper is key for proponents of low-stakes testing. There is little surprise that the authors found that practicing retrieval of recently studied information enhances the likelihood of the learner retrieving that information in the future, as we have seen the benefits of testing for retrieval throughout this paper. But the key difference with this paper is that  the format of the quiz (multiple-choice or short-answer) did not need to match the format of the critical test (e.g. end of unit exam) for this benefit to emerge. This supports the research related to deliberate practice, whereby the activities involved in practice do not need to exactly replicate the final performance for the practice to be effective. The authors also find that frequent classroom quizzing with feedback improves student learning and retention, and multiple-choice quizzing is as effective as short-answer quizzing for this purpose. There is more discussion of the merits of multiple choice questioning in the Formative Assessment section, but a key takeaway for me here is that both multiple-choice questioning and short form skill-based questions are ideal to use in the classroom for regular low-stakes testing.
My favourite quote:
First, we consistently observed that the format of the quizzes did not have to match the format of the unit exam for the quizzing benefits to occur. This is the most important finding of the present report, in that it is novel, was unanticipated from the laboratory literature, and is a critically important practical point for teachers. Even quick, easily administered multiple-choice quizzes aid student learning, as measured by unit exams (either in multiple-choice or short-answer format). Further, the benefits were long lasting: Robust effects were seen on the end-of-semester exams in Experiments 1a, 1b, and 2; that is, both multiple-choice and short-answer quizzing enhanced performance on end-of-semester class exams (again, in both multiple-choice and short-answer formats).

Memory: Revisionkeyboard_arrow_up
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A game-changer for me when reading research into revision was just how ineffective some of the most common revision strategies that students common use really are. Making students aware of this is crucial. Alongside the research presented in this section, I would strongly advise reading the papers in Memory: Forgetting, Spacing and Interleaving, and Memory: Testing especially with regard to the importance of testing and spacing. They are two very simple techniques that are likely to significantly improve a student's ability to store and recall information. Two final things I tell my students as a result of this research. Firstly, I emphasise the difference between learning and performance is key, because many of the most effective revision techniques will not be as effective in the short-term on performance, but are likely to lead to long-term learning. Secondly, I tell them that revision should not be easy. If it feels easy (like re-reading and highlighting does), then they are not learning. 

Article Title: Improving Students’ Learning With Effective Learning Techniques
Author(s): John Dunlosky, Katherine A. Rawson, Elizabeth J. Marsh, Mitchell J. Nathan, and Daniel T. Willingham
My Takeaway:
This paper outlines 10 learning techniques and discusses the relative merits of each one, in terms of their impact on learning and whether this is also true given different student characteristics. The two techniques that come out on top are Practice Testing - which can include something as simple as recalling information from flashcards, or working through questions from a textbook/worksheet, and Distributed Practice - spreading the learning/revision of a topic over different time periods instead of cramming. Interestingly, some of the techniques found to have little positive impact on learning include: highlighting, re-reading and use of imagery for text descriptions. This all fits in nicely with my mantra to students: the only way to get good at maths is to do maths. Note, that simply doing exam papers may not be as worthwhile as focused practice on a specific topic as it is often hard to isolate areas for improvement. The paper on Deliberate Practice in the Explicit Instruction section deals with this in more detail.
My favourite quote:
Concerning students’ commitment to ineffective strategies, recent surveys have indicated that students most often endorse the use of rereading and highlighting, two strategies that we found to have relatively low utility.

Article Title:
What makes distributed practice effective?
Author(s): Aaron S. Benjamin and Jonathan Tullis
My Takeaway:
This paper seeks to explain the benefits of distributed practice during students' revision in terms of a theory of reminding. As well as confirming the benefits of distributed practice that we have seen in the paper above, and in all the work on spacing in the first Memory section, the authors of this paper make three additional points that I found particularly interesting. Firstly, they point out that distributed practice is a bit of a balancing act. We have seen in the work of Robert Bjork the importance of forgetting for learning. Students must have the opportunity to forget in order that when they practice the material again the storage strength of their memories are increased. Therefore, the timing of distributed practice is crucial - too much forgetting leads to unlikely reminding, and too little forgetting leads to impotent reminding. This will differ from student to student and topic to topic, and the best way for students to tell is probably through repeated self-testing to find the perfect balance. Secondly, because reminding slow does forgetting, the authors suggest that expanding interval retrieval schedules should prove superior for long-term retention than constant-interval schedules. So, students may revisit (specifically, retest themselves) on fractions after one week, then a further two weeks later, then a further three weeks, and so on. This requires careful planning on the student's part. Finally, the authors explain who reminding enhances the memory of the initial learning of the event, not the subsequent retrievals of it, so it is of utmost importance that the initial encoding is sound. This is expanded on in the quote below, and obviously has implications for us teachers in terms of the importance of how we initial present ideas and concepts to students.
My favourite quote:
A second implication of the reminding view presented here is that the memory enhancement occurs for the original event, not for the reminding event. Applied to skill learning, this view suggests that it is the quality of the original encoding that is particularly important for successful acquisition. If one initially learns a poor golf swing, for example, then the many later practice opportunities will serve to reinforce those bad habits, not correct them. How feedback plays a role in such corrections remains unexplored: it may be that corrective feedback can decrease reminding on an undesirable original memory, or that it serves to “tune” that memory by enhancing reminding of the original event and the corrective information.

Article Title:
What will improve a student's memory?
Author(s): Daniel Willingham
My Takeaway:
This article (along with Willingham's outstanding book Why Don't Students Like School) has had a profound effect on how I plan my lessons. Memory is the residue of though. Quite simply: students will remember what they are thinking about, so plan your lessons accordingly. If they are revising quadratic equations by making a poster, will they be thinking about the subtleties of the algebra, or the colour of the highlighters they are using?  And in terms of revision advice for students, the following three principles are excellent: 1) If you want to remember what things mean, you must select a mental task that will ensure that you think about their meaning, and if the task has little meaning, then use a mnemonic. 2) Memories are lost mostly due to missing or ambiguous cues, so make your memories distinctive, distribute your studying over time, and plan for forgetting by continuing to study even after you know the material. 3) Individuals’ assessments of their own knowledge are fallible, so don’t use an internal feeling to gauge whether you have studied enough. Test yourself, and do so using the same type of test you’ll take in class.
My favourite quote:
Many of my students also tell me that they reviewed their notes and were quite surprised when they did not do well on the test. I’ve found that these students typically know little about how their memories work and, as a result, do not know how to study effectively.

Article Title: Expecting to teach enhances learning and organization of knowledge in free recall of text passages
Author(s): John F. Nestojko, Dung C. Bui, Nate Kornell & Elizabeth Ligon Bjork
My Takeaway:
This is a nice revision strategy to use that has been shown to be effective. Simply telling students that they will teach something to another student changes their mindset so much that even if they don’t actually teach the information, they remember it better later on when tested. The effect on their retention was even greater than if they were just told to learn the material for a test. This suggests that a pretty easy way of improving a student's revision programme is to regularly have them "teach" other students, as the preparation for this teaching could have significant benefits.
My favourite quote:
Expecting to teach appears to encourage effective learning strategies such as seeking out key points and organizing information into a coherent structure... Students seem to have a toolbox of effective study strategies that, unless prodded to do so, they do not use.


Formative Assessment keyboard_arrow_up
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Dylan Wiliam (interviewed on my podcast here) likes to think of formative assessment as "responsive teaching", and it is a fundamental part of every lesson I teach. When done well it allows me to get an accurate snap-shot of my class' understanding of a concept at a given time and adapt my teaching accordingly. Needless to say, I am slightly biased when it comes to advocating the use of multiple choice questions for formative assessment, having created Diagnostic Questions. This section will be my attempt to survey the literature for the most effective practices around.

Article Title:
Inside the Black Box. Raising Standards Through Classroom Assessment
Author(s): Paul Black and Dylan Wiliam
My Takeaway:
This is the bible of formative assessment, full of practical steps that are easy to implement and can make a huge difference. The key finding is that formative assessment raises standards of learning, but there is still plenty of room for improvement. Practical strategies that classroom teachers can use to improve students' learning include: feedback to any pupil should be about the particular qualities of his or her work, with advice on what he or she can do to improve, and should avoid comparisons with other pupils; for formative assessment to be productive, pupils should be trained in self-assessment so that they can understand the main purposes of their learning and thereby grasp what they need to do to achieve; and opportunities for pupils to express their understanding should be designed into any piece of teaching, for this will initiate the interaction whereby formative assessment aids learning. For me, that last point is crucial. Formative Assessment is (or at least should be) responsive teaching. It provides an amazing opportunity to find out what your students do, or do not understand, and react accordingly. As a result of reading this paper, together with the papers that follow, I ask three diagnostic questions at the start of every lesson, and adapt the lesson depending on the students' responses.
My favourite quote:
The main plank of our argument is that standards are raised only by changes which are put into direct effect by teachers and pupils in classrooms. There is a body of firm evidence that formative assessment is an essential feature of classroom work and that development of it can raise standards. We know of no other way of raising standards for which such a strong prima facie case can be made on the basis of evidence of such large learning gains.

Article Title: Is there value in just one?
Author(s): Caroline Wylie and Dylan Wiliam
My Takeaway:
This paper deals with answering a key question a teacher regularly needs to ask themselves: "when are my students ready for me to move on?", and was very influential when I was developing Diagnostic Questions. It makes the point that waiting for test data to confirm this is often unsuitable due to the time lag and the abundance of information, and relying on student judgement of their own readiness is flawed. With teachers needing to make this decision "on the fly" mid-lesson, the authors suggest that one well-written question, can do the job. The question should have three characteristics: 1) designed for easy collection of information (I use one finger for A, two for B, etc); 2) Incorrect responses assist the teacher diagnose what students do not understand, and, ideally, provide ideas about what to do about this (this is all comes down to the choice of good distractors); 3) Correct responses support a reasonable inference that students understand the concept being assessed (students should not be able to get the question correct whilst still holding key misconceptions). Whilst the information gleaned from asking this one question is not going to be perfect, there is no quicker, more accurate way of a teacher doing it mid-lesson. I ask three of these questions each lesson. If the responses of my students reveal misconceptions, then I respond accordingly - even if this takes up more of the lesson than I had intended. For my take on the use of such questions, see my Pedagogy videos.
My favourite quote:
The issue of “readiness to move on” is a familiar one to teachers, and the decision not to move on will initiate routines that are also familiar: to engage in whole class remediation, to pull aside a small group or individual students for additional assistance, to construct alternative learning opportunities that will assist students in their learning, and so forth, so that the teacher can later reassess the situation and proceed in the learning sequence.

Article Title: Using Diagnostic Classroom Assessment: One Question at a Time
Author(s): Joseph F. Ciofalo and Caroline Wylie
My Takeaway:
This paper also looks at the use of diagnostic questioning in the classroom. There is a nice focus on the characteristics of good questions with some nice examples provided that illustrate these characteristics clearly. My favourite part is the description of how these questions might be used in the classroom, providing a suggestion with how teachers might deal with different response scenarios from students. I have recorded a series of videos about how I use these questions in the classroom each day here.
My favourite quote:
The goal is to help teachers better utilize questioning and discussion to improve student learning, a key strategy within formative assessment. By starting with the instructional decisions that teachers make, the intention is to create both a resource and a habit of mind that is of obvious value to teachers. Lesson planning is a natural part of every teacher’s daily activities. Embedding a focus on questioning strategies—through carefully constructed diagnostic items—supports teachers in improving instruction by providing them with a natural extension to their practice, rather than requiring a major shift in mind-set.

Article Title: The memorial consequences of multiple-choice testing
Author(s): Elizabeth J Marsh, Henry L Roediger III, Robert A Bjork, Elizabeth L Bjork
My Takeaway:
We have seen in the section on Memory and Revision the existence of the Testing Effect (or Retrieval Effect), whereby testing can actually enhance learning. This paper attempts to show that whilst multiple choice questions are often used to measure understanding (via formative assessment), they can also have a similarly positive impact on learning as other forms of assessment via the Testing Effect. Interestingly, these benefits are not limited to simple definition or fact-recall multiple choice questions, but extend to those that promote higher-order thinking. The authors also address something a common criticism of multiple choice questions - the fear that the choice of wrong answers (the distractors) can cause students to learn false facts. The authors find that such persistence appears due to faulty reasoning rather than to an increase in the familiarity of the distractors. There is no doubt, however, that good distractors have the potential to cause students to learn faulty information. The authors put it like this: "multiple-choice lures may become integrated into subjects’ more general knowledge and lead to erroneous reasoning about concepts." However before we make all multiple choice questions illegal, two things need saying. Firstly, the benefits of using multiple choice questions in terms of eliciting information about students' understanding in an efficient way, and positive effects on their learning via the Testing Effect far outweigh the costs. Secondly, the issue of learning false facts can be overcome via immediate feedback. This could be a discussion in a lesson when using a multiple choice question for formative assessment purposes, automated marking when using a system such as Diagnostic Questions, or through task-specific written feedback.
My favourite quote:
More generally, the prevailing societal emphasis on testing as assessment is unfortunate, because it obscures the critical pedagogical aspects of testing. Tests, optimally constructed, can enhance later performance, provide feedback to the learner on what has and has not been learned, and potentiate the efficiency of subsequent study opportunities.

Article Title: Multiple-Choice Tests Exonerated, at Least of Some Charges: Fostering Test-Induced Learning and Avoiding Test-Induced Forgetting
Author(s): Jeri L. Little, Elizabeth Ligon Bjork, Robert A. Bjork, and Genna Angello
My Takeaway:
A common criticism of multiple choice questions is that because students know the correct answer is one of the listed options, they can get it correct by recognition as opposed to the retrieval process from long term memory that we know can aid learning. This paper addresses that criticism by testing whether multiple-choice tests could trigger productive retrieval processes—provided the alternatives (distractors) were made plausible enough to enable test takers to retrieve both why the correct alternatives were correct and why the incorrect alternatives were incorrect. In two experiments, they found not only that properly constructed multiple-choice tests can indeed trigger productive retrieval processes, but also that they had one potentially important advantage over cued-recall tests. Both testing formats fostered retention of previously tested information, but multiple-choice tests also facilitated recall of information pertaining to incorrect alternatives, whereas cued-recall tests did not. In other words, so long as the multiple choice question is a good one (see papers earlier in this section for the rules of a good multiple choice question), then when searching for the answer students thoughts turn to reasons why the distractors are incorrect, and hence they exercise the very retrieval processes they have been accused of bypassing. This is huge, because it suggests that whilst multiple choice questions have all the advantages of speed of information gathering discussed earlier, they have an extra advantage of encouraging the students to think harder than if the question was presented without any options to choose from.
My favourite quote:
The present findings vindicate multiple-choice tests, at least of charges regarding their use as practice tests. In fact, our findings suggest that when multiple-choice tests are used as practice tests, they can provide a win-win situation: Specifically, they can foster test-induced learning not only of previously tested information, but also of information pertaining to the initially incorrect alternatives. This latter advantage is especially important because, typically, few if any practice-test items are repeated verbatim on the subsequent real test. From that standpoint, the advantage of initial multiple-choice testing over initial cued-recall testing is a truly significant one.

Marking and Feedbackkeyboard_arrow_up
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Whether we like it or not, most of us have to mark and give feedback to students. For many (including me), it is one of the worst parts of being a teacher. The best advice I ever heard on feedback was from Dylan Wiliam, who said "the only good feedback is that which is acted on". You could write the best feedback in the world, but if the your students neither have the time nor the inclination to do it, or if they simply do not understand what you are asking them to do, then it is an absolute waste of time. Here we look at what the evidence has to say about marking and feedback, both in terms of the type and the immediacy, and even when feedback can have a negative effect!

Article Title:
A marked improvement. A review of the evidence on written marking
Author(s): Education Endowment Foundation and Oxford University
My Takeaway:
Lots of practical, easy to implement strategies to improve the effectiveness of marking. My favourites include: dealing with careless mistakes and misconceptions differently when marking; making feedback as specific and actionable as possible so the students actually understand it; "acknowledgement" marking (i.e. a tick to say you have seen the work) is a waste of time, and hence a good rule is mark less but mark better; specific time must be dedicated to students to respond to marking. And then the big one for me: awarding grades/levels for each piece of work means students focus upon these marks at the expense on the formative comments the teacher has written.
My favourite quote:
It also appears worthwhile to caution against elements of dialogic or triple impact marking that do not follow the wider principles of effective marking that are underpinned by relatively stronger evidence summarised elsewhere in this review. For example, there is no strong evidence that ‘acknowledgment’ steps in either dialogic or triple impact marking will promote learning.

Article Title: The Effects of Feedback Interventions on Performance: A Historical Review, a Meta-Analysis, and a Preliminary Feedback Intervention Theory
Author(s): Avraham N. Kluger and Angelo DeNisi
My Takeaway:
When I interviewed Dylan Wiliam, he cited this paper as the most surprising piece of research he had ever encountered. The authors found over 3,000 research studies published between 1905 and 1995, but found that only 131 of the studies were well-enough designed for their results to be taken seriously. The 131 studies reported 607 effect sizes, which showed that, on average, feedback did increase achievement. But—in what I think is one of the most counter-intuitive results in all of psychology—231 of the 607 reported effect sizes were negative. In almost two out of every five studies, feedback lowered performance. What are we to take from this? Well, the authors suggest that the further feedback moves away from the task itself and towards the individual student, the less effective it is - even going as far as to have a negative effect! If teachers are going to dedicate a significant proportion of their time to giving feedback, we must make it task-focused. The following papers in this section look at the specifics of what that might entail.
My favourite quote:
The central assumption of Feedback Intervention Theory is that Feedback Interventions change the locus of attention among 3 general and hierarchically organized levels of control: task learning, task motivation, and meta-tasks (including self-related) processes. The results suggest that feedback intervention effectiveness decreases as attention moves up the hierarchy closer to the self and away from the task.

Article Title: Formative Assessment: Practical Ideas for Improving the Efficiency and Effectiveness of Feedback to Students
Author(s): Geraldine O’Neill
My Takeaway:
A really good summary of effective feedback practice, together with the value of this idea to either the teacher or the students, all referenced to relevant research-based evidence. There are many interesting practices cited here, but the ones that stood out to me were: 1) Consider feedback in different media/formats, such as on-line, audio-feedback, verbal class feedback, use of ‘clickers’ in large class contexts. This may not only save the teacher time, but may make feedback more permanent and easily accessible to students. 2) Student Requested Feedback -  ask students to submit specific requests for areas for feedback. As this is student-focused feedback it is more inclined to motivate students to act. This also encourages students to take some responsibility in the process. 3) In class peer and self assess feedback activities in terms of discussing and giving feedback to annonymised work. This will help students engage in the feedback process and make them more aware of the desired standard in relation to their own work. 4) Comment in actionable language. This relates to the recurring theme throughout this section that feedback should be task-focussed, but also reminds us that if the students cannot understand what the feedback is on about, it is likely to be a complete waste of time.
My favourite quote:
One of the key themes emerging to address this dilemma is to develop students own self-monitoring skills in order to help them narrow the gap between their performance and the standards expected of them (Nichol, 2009; Clarke, 2001). The timing, type and specification of feedback can also improve student ability to self-monitor. In addition, good feedback should feed into some specific actions that can be used in the next assessment (Nichol & McFarlane-Dick, 2009). Feedback need not always be from the academic staff, students themselves are a good resource to each other when given guidance on how to do this. New technologies also open up some efficient feedback opportunities.

Article Title: Task-Involving and Ego-Involving Properties of Evaluation: Effects of Different Feedback Conditions on Motivational Perceptions, Interest, and Performance
Author(s): Ruth Butler
My Takeaway:
This is a really interesting study into the effectiveness, and indeed the effects, of different types of feedback. A total of 200 fifth- and sixth-grade students with high or low school achievement were given takes to complete. Individual comments, numerical grades, standardized praise, or no feedback were received after Sessions 1 and 2. The results following a post-test were that interest, performance, and attributions of effort were highest at both levels of achievement after receipt of comments. Indeed, post-test scores were one stand deviation higher for students in the Comments group, with no significant differences between socrers in the other groups. Similarly, ego-involved attributions were highest after receipt of grades and praise. In other words, grades and praise had no effect on performance, and served only to increase ego-involvement. It is perhaps of little surprise that comments were the most successful in generating the kid of response we would want from students. However, I was taken aback by the fact that grades had the same impact as praise in generating ego-involving responses. My conclusion, based on my own experiences with my students, is that grades cause students to think about themselves - eliciting emotions ranging from joy to despair - and this prevent them from focussing on the task itself. This will be discussed further in the next study.
My favourite quote:
The present results further confirmed that individual comments yielded higher task-involved perceptions and lower ego involved ones than either grades or praise and that no feedback yielded perceptions of both kinds of factors as being
relatively non-determinative of both effort and outcome. The similar and ego-involved perceptions induced by grades and
praise seem particularly significant. Both anecdotal evidence (Holt, 1964) and some research findings (Harter, 1978; Maehr & Stallings, 1972) suggest that grades are perceived as potent sources of control over learning.


Article Title: Enhancing and undermining Intrinsic motivation; the effects of task-involving and ego-involving evaluation on interest and performance
Author(s): Ruth Butler
My Takeaway:
The last paper showed that task-focussed comments were preferable to grades in terms of subsequent performance on tasks, as well as general involvement and interest in the task. This begs the question: what if comments and grades are combined? This paper provides the answer. 48 eleven-year old Israeli students were selected from the upper and lower quartiles of attainment from 12 classes in 4 schools and worked in pairs over three sessions on two tasks (one testing convergent thinking and the other, divergent). After each session, each student was given written feedback on the work they had done in the session in one of three forms: A) individualised comments on the extent of the match of their work with the assessment criteria that had been explained to each class at the beginning of the experiment; B) grades, based on the quality of their work in the previous session; C) both grades and comments. Students given comments showed a 30% increase in scores over the course of the experiment, and the interest of all students in the work was high. Students given only grades showed no overall improvement in their scores, and the interest of those who had high scores was positive, while those who had received low scores show low interest in the work. Perhaps most surprisingly, the students given both grades and comments performed similarly to those given grades alone—no overall improvement in scores, and interest strongly correlated with scores—and the researchers themselves describe how students given both grades and comments ignored the comments, and spent their time comparing their grades with those achieved by their peers. The message is simple: If you are going to grade or mark a piece of work, you are wasting your time writing careful diagnostic comments.
My favourite quote:
The results confirm the importance of distinguishing between task involvement and ego-involvement when investigating intrinsic tak motivation. As hypothesized, both high and low achievers who received comments continued to express high interest both on Session 2 when they anticipated further comments, and at post-test , when they did not.

Article Title: The Power of Feedback
Author(s): John Hattie and Helen Timperley
My Takeaway:
The first line of this paper hooked me in: "Feedback is one of the most powerful influences on learning and achievement, but this impact can be either positive or negative." This is a fascinating, comprehensive paper that addresses all the major issues surrounding feedback. One particularly interesting part of the paper for me was the four levels that feedback could be directed: 1) feedback can be about a task or product, such as whether work is correct or incorrect (e.g. "you need to show more working out". 2) feedback can be aimed at the process used to create a product or complete a task (e.g. "you need to remember to find the lowest common denominator before adding the fractions"). 3) feedback to students can be focused at the self-regulation level, including greater skill in self-evaluation or confidence to engage further on a task "you have started this problem off correctly, now check Q1 again and see if you can use this to help you finish the question". 4) feedback can be personal in the sense that it is directed to the “self (e.g. ”you are superb!”). The authors conclude that 4) is the least effective, 2) and 3) are powerful in terms of deep processing and mastery of tasks, and 1) is powerful when the task information subsequently is useful for improving strategy processing or enhancing self regulation (which it too rarely does). Once again we see the need for feedback to be task focused (the authors stress that praise is rarely effective), and also be actionable for the student. It is also worth reading the section on positive v negative feedback (p98) just to see how tricky a topic this is. My takeaway is that is very much depends on the student themselves. Some students need praise, but for others praise is either wasted or can lead to negative consequences. Likewise, some students (quite a few, in my experience) simply need telling that this work is not good enough - a metaphorical kick up the arse. Finally, I loved the part that discussed how feedback is likely to be ineffective if the student simply does not have the relevant knowledge to solve the problem in the first place. You can give all the task-focused prompts in the world, but if the students cannot do it, then feedback is likely to be pretty useless. In that instance, instruction and work-examples are likely to be far more valuable. Of course, determining whether a student could not do the work, or could not be bothered to do the work, can be difficult.
My favourite quote:
Feedback, however, is not “the answer”; rather, it is but one powerful answer. With inefficient learners, it is better for a teacher to provide elaborations through instruction than to provide feedback on poorly understood concepts. If feedback is directed at the right level, it can assist students to comprehend, engage, or develop effective strategies to process the information intended to be learned. To be effective, feedback needs to be clear, purposeful, meaningful, and compatible with students’ prior knowledge and to provide logical connections. It also needs to prompt active information processing on the part of learners, have low task complexity, relate to specific and clear goals, and provide little threat to the person at the self level. The major discriminator is whether it is clearly directed to the task, processes, and/or regulation and not to the self level. These conditions highlight the importance of classroom climates that foster peer and self-assessment and allow for learning from mistakes.

Article Title: Learning versus Performance
Author(s): Nicholas C. Soderstrom and Robert A. Bjork
My Takeaway:
This is the second appearance of this lovely paper following its debut in the Memory section. This paper seeks to question the common assumption that immediate feedback is the best kind of feedback. The logic makes perfect sense - if a learner gets something wrong, surely they need to know that they have got it wrong immediately, and why they have got it wrong, in order to correct their mistake? And indeed, during the early phases of knowledge acquisition, immediate feedback may be beneficial to ensure mistakes do not become embedded. However - and wait for this because it is a classic - research presented in this paper suggests that delaying or even reducing feedback can have a long term benefit to student's learning. Why? Well, because regular, immediate feedback can cause learners to become overly dependent upon it (almost seeing it as a crutch to their learning). To relay this to themes covered in other sections of this page, immediate feedback prevents students from thinking hard, and also having the opportunity to forget. Once again we have the issue of distinguishing between learning and performance. The authors suggest that delaying or reducing feedback is likely to have a determinantal effect on short-term performance, but a positive effect on long-term learning.  What are the implications for us teachers, especially given that the Hattie paper above stresses the potential positive benefits of feedback? Well, one idea I had reading this paper was when initially marking a piece of homework to give no feedback whatsoever, simply to provide ticks and crosses. Then give this work back to the students and see if they could identify the source of their errors and correct them. When I next take the books in, this is when I would give feedback. In my view, there are two advantages of such an approach. Firstly, it will give students time to think, and hence become less reliant on me. Secondly, it may enable me to better identify actual gaps in their knowledge. As discussed in the paper above, feedback when students do not have the knowledge to begin with is likely to be a waste of time. However, if indicating where students have gone wrong allows students to have a better stab at those questions, then I can then better distinguish between gaps in their knowledge versus the kind of mistakes that well constructed, task-focused feedback is likely to help resolve. The knowledge gaps may then need addressing again via instruction.
My favourite quote:
Feedback, however, is not “the answer”; rather, it is but one powerful answer. With inefficient learners, it is better for a teacher to provide elaborations through instruction than to provide feedback on poorly understood concepts. If feedback is directed at the right level, it can assist students to comprehend, engage, or develop effective strategies to process the information intended to be learned. To be effective, feedback needs to be clear, purposeful, meaningful, and compatible with students’ prior knowledge and to provide logical connections. It also needs to prompt active information processing on the part of learners, have low task complexity, relate to specific and clear goals, and provide little threat to the person at the self level. The major discriminator is whether it is clearly directed to the task, processes, and/or regulation and not to the self level. These conditions highlight the importance of classroom climates that foster peer and self-assessment and allow for learning from mistakes.



Group Work and Cooperative Learningkeyboard_arrow_up
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Group work has always been a thorny issue for me - probably not helps by a previous school's obsession with Kagan training. Whilst I see the benefits of having students work together, talk about mathematics, and learn from each other, I am also acutely aware of the possibilities of students being off-task, and relying on others in the group to do the work for them. I guess it comes down to one question: do students learn more being in a group, or working individually? And of course, cooperative learning extends to those occasions where you might say "discuss this with your partner for 10 seconds". Here is my attempt to use educational research to find out the best practices for group work and cooperative learning.

Article Title:
Cooperative Learning and Achievement: Theory and Research
Author(s): Robert E. Slavin, Eric A. Hurley and Anne Chmberlain
My Takeaway:
This is a fascinating review on the research into cooperative learning, which comes to a positive conclusion about its  potential effect on achievement. Alongside reviewing contrasting theories of cooperative learning, the paper identifies two key facts that are necessary for successful cooperative learning. The first is the importance of structuring group interactions. The concept of reciprocal teaching is discussed, with emphasis placed on the importance of the teacher modeling these interactions. Not assuming students instinctively know how to effectively work together will be a key theme of this section. Secondly, there is the issue of group goals and individual accountability. The key finding is this: cooperative learning is most consistently effective when groups are recognised or rewarded based on individual learning of their members. Without this, there is the danger that members of the group will free-ride on others' work. This is obviously tricky to do, and a suggestion from the authors is found in "my favourite quote" below. In short, groups work best when there are 1) group goals (so pupils are working as a group and not just in a group), and 2) individual accountability (so any pupil falling down on the job harms the entire group's work). One other part of this paper I found fascinating was the section on which ability of student benefits most from cooperative learning. You could argue that high achievers could be held back by having to explain material to their low-achieving group mates. However, it would be equally possible to argue that because students who give elaborated explanations typically learn more than do those who receive them, high achievers should be the students who benefit most from cooperative learning because they most frequently give elaborated explanations. Interestingly, research has failed to provide a conclusive answer to this, and the authors remain in support of cooperative learning for all.
My favourite quote:
If students can only do as well as the group and the group can succeed only by ensuring that all group members have learned the material, then group members will be motivated to teach each other. Studies of behaviors within groups that relate most to achievement gains consistently show that students who give each other explanations (and less consistently, those who receive such explanations) are the students who learn the most in cooperative learning. Giving or receiving answers without explanation has generally been found to reduce achievement (Webb, 1989, 1992). At least in theory, group goals and individual accountability should motivate students to engage in the behaviors that increase achievement and avoid those that reduce it. If a group member wants her group to be successful, she must teach her group mates (and learn the material herself). If she simply tells her group mates the answers, they will fail the quiz that they must take individually. If she ignores a group mate who does not understand the material, the group mate will fail, and the group will fail as well.

Article Title: Group Work for the Good
Author(s): Tom Bennett
My Takeaway:
Tom Bennett reviews the research in favour of group work and teaches a different conclusion. He argues that much of the research derives from the field that can be broadly termed Constructivism - the idea that students are active participants in the process of learning, and not passive recipients of experience and factual knowledge. This area has been covered in detail in the Explicit Instruction section of this page. Bennett argues that once you question the validity of constructivism, then the arguments in favour of group work start to lose their power. Bennett goes on to outline what he sees as four main drawbacks of group work: 1. Disguised inactivity - if you give a task to three or four people, one or two may realise it's time to freeze, because others will carry the burden of the task, and in the meantime, they can coast under the guise of "research" or "running the group." 2. Unequal loading - Related to this is the problem that while ever student might participate, the participation might be profoundly uneven. 3. Inappropriate Socialisation - students may end up competing to see who can discuss the task the least. 4. Unfair assessment - When a teachers praises a pupil, it's a clear one-to-one relationship, whereas in grading groups, we often must give collective grades. For me, the key point that the author raises is the opportunity cost of group work - what else could the students be doing with their time? If Explicit Instruction could achieve the same result in 5 minutes that a group could achieve in 30 minutes, then how can we justify it? But if the activity (and the class) lends itself well to a group work activity, then the benefits of having students share ideas, learn from each other, and even something as simple as to vary the type of classroom activity to reengage students, may well be worth it.
My favourite quote:
Here's my parting advice: use group work when you feel it is appropriate to the task you want your students to achieve, and at no other time. The irony of the advocates' position is that while it correctly identifies the many benefits to using group work, their error is made when group work is preferred over other strategies because of some imagined potency, or when it is fetishized as a method imbued with miraculous properties. It isn't dogma, it isn't a panacea, and it isn't the messiah. It's one strategy among many. And it's a perfectly reasonable part of a teacher's arsenal of strategies. Not because pseudo-research has settled the matter, but because the teacher feels it appropriate at that time, for that lesson, with those children. And not before.

Article Title: Group work and whole-class teaching with 11- to 14-year-olds compared
Author(s): Maurice Galtona, Linda Hargreaves and Tony Pell
My Takeaway:
This findings of this paper provide somewhat of a contrast to the one above. Here researchers compared the academic performance and classroom behaviour of pupils when taught new concepts or engaged in problem solving in sessions organised either as cooperative group work or whole class, teacher directed instruction. Comparisons of attainment were made in classes of pupils aged 11 to 14 years (Key Stage 3) in English, mathematics and science. The attainment results suggest that a grouping approach is as effective, and in some cases more effective, than when whole class teaching is used. Classroom observation indicated that there were more sustained, higher cognitive level interactions when pupils worked in groups than during whole class discussions. Crucially, the researchers argue that the group work results could be improved still further if teachers gave more attention to training pupils to work in groups and if more time was given to debriefing after group work. For me these last two points are crucial. We cannot assume students intrinsically know how to work in groups, and without training then the drawbacks that Bennett identifies in the paper above may well be realised. This possibly even extends to students talking in pairs. The next paper in this section provides a suggestion for a simple form of training. Secondly, a debrief may go some way towards providing accountability to group members. By discussing what went well and what didn't throughout the group work, and drawing attention explicitly to groups that worked well, the teacher can emphasise the point that group work is to be taken seriously, and offer directions for students to improve.
My favourite quote:
There are a number of reasons for claiming that the group work could have been more effective, to do mainly with the context in which teachers had to operate when taking part in this study. First, as part of training pupils to work effectively in groups it is vital that teachers brief and debrief the class so that they can begin to gain metacognitive awareness of what it means to be part of a group. Debriefing sessions therefore are particularly important because they not only evaluate how individuals responded in the groups but they also call for participants to make suggestions about suitable strategies for improving the situation on future occasions. After each session, observers completed a lesson overview schedule which recorded, amongst other things, whether or not briefing or debriefing had taken place. It was noticeable, however, particularly in science, that teachers rarely found time for these debriefing sessions. It was rare, for example, to observe a science lesson where the teacher with, say, five minutes of the period left preferred to keep discussion of the results over until the next lesson and instead engaged in a debriefing exercise. More often teachers preferred to use an evaluation sheet which they handed to pupils as they left the class. Thus the exercise tended to take the form of an additional homework task rather than generate a debate on the consequences of the previous classroom activity.

Article Title: Teaching children how to use language to solve maths problems
Author(s): Neil Mercer and Claire Sams
My Takeaway:
This paper cites observational research which suggests that primary school children often do not work productively in group-based classroom activities, with the implication that they lack the necessary skills to manage their joint activity. To counteract this, the authors explored the role of the teacher in guiding the development of children’s skills in using language as a tool for reasoning. It involved an interventional teaching programme called Thinking Together, designed to enable children to talk and reason together effectively. The results obtained indicate that children can be enabled to use talk more effectively as a tool for reasoning; and that talk-based group activities can help the development of individuals’ mathematical reasoning, understanding and problem-solving. The important part of this study for me is that the students were explicitly shown how to effectively work in groups, with special emphasis on the language they should use, instead of assuming that they would automatically know how to do so. The lessons preceding the group work were explicitly focussed on making students aware of the need and benefit to work together. They were encouraged to discuss things and ask questions, include everyone’s ideas , ask what people think and what their reasons are , listen to each other, and so on, all of which was supported by clear modelling from the teacher. If we are going to embark upon group work (or even paired work), then such a structured approach before the task itself may well be a sensible step. However, one be aware of the relatively small sample size used in the study before drawing any significant conclusions.
My favourite quote:
More generally, our results enhance the validity of a sociocultural theory of education by providing empirical support for the Vygotskian claim that language-based, social interaction (intermental activity) has a developmental influence on individual thinking (intramental activity). More precisely, we have shown how the quality of dialogue between teachers and learners, and amongst learners, is of crucial importance if it is to have a significant influence on learning and educational attainment.

Mindsetkeyboard_arrow_up
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Carol Dweck's work on mindsets has proven both popular and influential across schools over the last decade. The general thrust of her work is students with a growth mindset believe that their intelligence is not fixed, embrace mistakes as learning opportunities, and strive to improve through practice. This can be most readily seen in her book, Mindset. Whilst I see the advantage of having a growth mindset, I am yet to be fully convinced of how to practically develop such a way of thinking in my students. I share some more thoughts in my Takeaways below. 

Article Title:
Boosting Achievement with Messages that Motivate
Author(s): Carol S. Dweck
My Takeaway:
A great introduction to Dweck's work on Mindset. She outlines her belief that students with a fixed mindset follow the cardinal rule "look smart at all costs", which leads to their desire to not work hard, not make mistakes, and if you do make mistakes to not try to repair them. In contrast, those with a growth mindset follow the rule "it is much more important to learn than get good grades", which leads them to take on challenges, work hard, and correct any mistakes. Studies quoted show that teaching students a growth mindset results in increased motivation, better grades, and higher achievement test scores
My favourite quote:
Many teachers see evidence for a fixed mindset every year. The students who start out at the top of their class end up at the top, and the students who start out at the bottom end up there. Research by Falko Rheinberg shows that when teachers believe in fixed intelligence, this is exactly what happens. It is a self-fulfilling prophecy. However, when teachers hold a growth mindset, many students who start out lower in the class blossom during the year and join the higher achievers.

Article Title:
Academic Tenacity: Mindsets and Skills that Promote Long-Term Learning
Author(s): Carol S. Dweck, Gregory M. Walton, Geoffrey L. Cohen
My Takeaway:
This particular paper presents some compelling findings and practical strategies to help foster a growth mindset and develop academic tenacity. Two finding that stood out to me were: 1) Students who received effort praise chose challenging tasks that could help them learn, while students who received intelligence praise were more likely to choose tasks in their comfort zone that they could perform well on; 2) Classrooms that encourage competition and individualistic goals may be particularly ill suited to minority students, who are more likely to be reared in cultural contexts that emphasize the importance of communal and cooperative goals over individualistic or competitive goals.
My favourite quote:
Research shows that students’ belief in their ability to learn and perform well in school—their self-efficacy—can predict their level of academic performance above and beyond their measured level of ability and prior performance

Article Title:
Does mindset affect children’s ability, school achievement, or response to challenge? Three failures to replicate.
Author(s): Yue Li & Timothy C. Bates
My Takeaway:
I include this paper for balance as it was the most empirically rigorous one I could find that questions Dweck's findings on mindset. Specifically, the authors find praise for intelligence failed to harm cognitive performance and children’s mindsets had no relationship to their IQ or school grades. Finally, believing ability to be malleable was not linked to improvement of grades across the year. Two points I would raise here: the authors do not find that promoting mindset beliefs to have a negative effect, and more importantly work by the likes of Dan Willingham and others discussed above strongly believe that effort and practice can significantly increase achievement. Hence, I personally have no problem promoting a growth mindset, both in students and teachers. However, a growth mindset without the success that will most likely follow from an Explicit Instruction approach may not be sustainable in the long run, in my opinion.
My favourite quote:
We find no support for the idea that fixed beliefs about basic ability are harmful, or that implicit theories of intelligence play any significant role in development of cognitive ability, response to challenge, or educational attainment.


Fluency with Maths Factskeyboard_arrow_up
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Research from the Cognitive Science, Explicit Instruction and Cognitive Load Theory sections of this page stress the importance of students having key mathematical facts ready to be retrieved from long term memory so they do not take up valuable space in working memory. For want of a better expression, I will refer to this as developing fluency with maths facts. Over the last few years, I have tried to get students to develop this fluency via number talks, and more recently through drills/rote learning. This section is my attempt to find research-based evidence on the merits of these, as well as to address the issue of whether fluency is important when students have calculators and mobile phones!

For an overview of how Number Talks work, I would recommend this article, together with this video from Jo Boaler.

For an overview of drilling, I would recommend reading the following two blog posts by teachers at Michaela Community School: Dani Quinn and Hin-Tai Ting.

Article Title: Automaticity in Computation and Student Success in Introductory Physical Science Courses
Author(s): JudithAnn R. Hartman, Eric A. Nelson
My Takeaway:
Students (and parents!) often say to me something along the lines of: "why do I need to work that out when I can just bang it into my calculator/smart phone?". To be honest, my answers over the years have not been great. Fortunately, with the findings of this fascinating paper, I now have some evidence up my sleeve. This paper looks at the effect the move away from the practice of key mathematical skills in US high schools has had on students taking science degrees. The authors find that between 1984 and 2011, the percentage of US bachelor’s degrees awarded in physics declined by 25%, in chemistry declined by 33%, and overall in physical sciences and engineering fell 40%. Data suggest that these declines are correlated to a K-12 (kindergarten to the end of high school) de-emphasis in most states of practicing computation skills in mathematics. The authors cite recent studies in cognitive science that have found that to solve well-structured problems in the sciences, students must first memorize fundamental facts and procedures in mathematics and science until they can be recalled “with automaticity,” then practice applying those skills in a variety of distinctive contexts. Even with access to a calculator, students working memories can become overloaded, which can prevent them being able to solve more complex problem and hence inhibit learning. To explain this further, I can do no better than to quote from the paper itself:
My favourite quote:
As one example, if as part of a calculation “8 times 7” cannot be recalled, the calculator answer of 56 must be stored in working memory so that it can be transferred to where the calculation is being written. On a problem of any complexity, that storage may bump out of working memory an element that is needed to solve the problem. An answer from a calculator takes up limited working memory space; an answer recalled from long term memory does not.

If arithmetic and algebraic fundamentals are automated, when examples are based on simple ratios or equations, room is available in novel WM for the context that builds conceptual understanding, and problem solving builds an intuitive, fluent understanding of when to apply facts and procedures (Willingham 2006). Conversely, if a student lacks “mental math” automaticity, conceptual explanations based on proportional reasoning or “simple whole-number-mole ratios” will likely not be simple. If a student must slowly reason their way through steps of algebra that could be performed quickly if automated, the “30 seconds or less” limit on holding the goal, steps, and data elements of the problem in working memory ticks away.

In recent years, the internet has facilitated the finding of facts and procedures, but new information occupies the limited space in novel WM that is needed to process the unique elements of a problem. Unless new information is moved into long term memory by repeated practice at recall, during future problem solving that new information will again need to be sought, and when found, it will again restrict cognitive processing (Willingham 2004, 2006).


Article Title:
Fluency without Fear
Author(s): Jo Bolaer
My Takeaway:
Jo Boaler agrees on the importance of knowing mathematical facts, but argues that the memorisation of math facts through times table repetition, practice and timed testing is unnecessary and damaging. Boaler goes further to argue that when teachers emphasise the memorisation of facts, and give tests to measure number facts, students suffer in two important ways. Firstly, for about one third of students the onset of timed testing is the beginning of math anxiety, which can block working memory and prevent learning taking place. Secondly, they can put students off mathematics for life. Boaler advocates a move away from seeped and memorisation, and towards encouraging students to work with, explore and discuss numbers. This will allow them to commit important facts to memory, but in a fun and engaging context. She then goers on to describe Number Talks, as well as some other strategies for developing a sense of number. My concern with this is founded in personal experience of using and observing Number Talks extensively over the last three years - whilst it is supposed to be the lowest achieving students who gain the most from them, I have found they are often held back from fully participating by their lack of knowledge of key maths facts. It ends up being the teacher suggesting strategies, students copying them down and seemingly understanding then, only for them to be unable to transfer them to a new calculation in the next Number Talk. Contrast this to higher achieving students, who can happily break apart and put back together numbers in wonderfully efficient ways, and really seem to get a lot out of the Number Talks, sharing and discussing each others' approaches. I believe all students can (and should) get to this level, but I don't think this can be done through Number Talks alone. Students cannot develop fluency with numbers without these facts in long term memory. I have found that the more comfortable students get with their times tables and number bonds, the more readily they can think of, and successfully carry out the kind of efficient strategies that Number Talks promote. I draw the analogy with problem solving - you cannot teach problem solving by just showing students how to solve problems. Likewise, I believe you cannot teach fluency with numbers simply by showing students how to be fluent.
My favourite quote:
High achieving students use number sense and it is critical that lower achieving students, instead of working on drill and memorization, also learn to use numbers flexibly and conceptually. Memorization and timed testing stand in the way of number sense, giving students the impression that sense making is not important. We need to urgently reorient our teaching of early number and number sense in our mathematics teaching in the UK and the US. If we do not, then failure and drop out rates - already at record highs in both countries - will escalate.

Article Title: Developing Fluency with Basic Number Facts: Intervention for Students with Learning Disabilities
Author(s): Katherine Garnett
My Takeaway:
This paper provides a fascinating insight into attempts to develop fluency with students with learning difficulties. The authors cite that on timed assessments, 5th grade students with learning disabilities completed only one-third as many multiplication fact problems as their non disabled counterparts. Interestingly, the students with learning disabilities were very much slower, but not significantly less accurate, than their non-disabled peers. Additionally, they demonstrated basic conceptual understanding of the basic maths operations. Thus, many students with learning disabilities establish basic understanding of the number relations involved in basic facts, but continue using circuitous strategies long after their non-disabled peers have developed fluent performance. And as we have seen from Cognitive Load Theory, it is the inability to recall facts from long term memory that will hinder such students in solving more complex problems as their working memories will become overloaded. However, the author points out that becoming fluent is not simply a case of remembering a load of facts - it is about forming a well-developed network of number relationships, easily activated counting and linking strategies, and well-practiced navigational rules for when to apply which maneuver". The author goes on to argue that the only way to develop these skills is though several years of frequent and varied number experiences and practice, and drilling is not enough. The author recommends presenting Challenge Problems and discussing strategies, much in the same way as Number Talks, and a really useful collection of prompt questions is provided. I am still not convinced that strategy comes before knowledge, but I am convinced that simply knowing facts is not enough given the immeasurable number of combinations of facts that would be required to answer every single maths problem!
My favourite quote:
In investigating the effects of challenge problems, many of the guidelines offered here would be useful, especially the emphasis on interactive, oral work. Regularly including challenge problems in student/teacher interactive math work could well promote the "mental math" prowess needed by so many students with learning disabilities who cling to number lines and paper-pencil routines.

Article Title: Mastering Maths Facts: Research and Results
Author(s): Otter Creek Institute
My Takeaway:
This authors provide a really good summary of the key findings from this paper: "Learning math facts proceeds through three stages: I) procedural knowledge of  figuring out facts; 2) strategies for remembering facts based on relationships; 3) automaticity in maths facts—declarative knowledge. Students achieve automaticity with math facts when they can directly retrieve the correct answer, without any intervening thought process. The development of automaticity is critical so students can concentrate on higher order thinking in maths. Students who are automatic with math facts answer in less than one second, or write between 40 to 60 answers per minute, if they can write that quickly. Research shows that math facts practice that effectively moves students towards automaticity proceeds with small sets of no more than 2 —4 facts at a time. During practice, the answers must be remembered rather than derived. Practice must limit response times and give correct answers immediately if response time is slow. Automaticity must be achieved with each small set of facts, and maintained with the facts previously mastered, before more facts are introduced. Suggestions for doing this with flashcards or with worksheets are offered." So, this paper stresses the importance of the automaticity of maths facts, and advocates the use of timed drills to achieve this. I found the suggested order to be interesting: strategies comes before automaticity. This did not seem to fit in with my experiences described in my Takeaway on the Boaler article above. However, digging a little deeper, the authors suggest that strategy comes first for addition and subtraction facts, but memorisation is needed for multiplication facts. As most Number Talks require an element of times table knowledge, this certainly fits in with my experiences. The paper also makes one point that I feel is of paramount importance: if students are relying on a counting strategy to solve basic maths facts (such as counting on fingers for multiplication), then no amount of drilling will help them transfer these facts to long term memory. For students relying on these strategies, drills can become a painful process. Students need to be moved away from inefficient strategies as soon as possible, and the authors suggest that timed drills might be a way to achieve this. Of course, we need to bear in mind Boaler's important point about the dangers of maths anxiety - but there are plenty of ways to make this kind of drilling fun and non-threatening, as the Michaela blogs demonstrate.
My favourite quote:
What is required for students to develop automaticity is a particular kind of practice focused on small sets of facts, practiced under limited response times, where the focus is on remembering the answer quickly rather than figuring it out. The introduction of additional new facts should be withheld until students can demonstrate automaticity with all previously introduced facts. Under these circumstances students are successful and enjoy graphing their progress on regular timed tests. Using an efficient method for bringing math facts to automaticity has the added value of freeing up more class time to spend in higher level mathematical thinking.

Article Title: Mental calculation methods used by 11-year-olds in different attainment bands
Author(s): Derek Foxman
My Takeaway:
This is fascinating. A sample of 247 eleven year old children was divided into three bands of attainment as measured independently by their scores on a written test of concepts and skills.They were then given a series of mental arithmetic questions to answer during one-ton-one interviews, and crucially asked to explain how they arrived at their answer. An example of one of the questions children were given is "I buy fish and chips for £1.46. How much change should I get from £5 ?". Their responses were classified as either being Complete (e.g. £5 − £1 − 46p), or Split (e.g. £5 − £1; £1 − 46p). There were three main findings: 1) Complete number methods were far more successful than Split number methods, even more so in the two lower attainment bands than in the top band. 2) For all three questions, Complete number strategy use declined from the Top to the Bottom attainment band, while Split number strategy use increased from Top to Bottom. 3) Complete number strategies were used far more frequently than either Split methods or the Algorithm for working out the in context questions. I was surprised by these results - for me the Split strategy seems more efficient, and is how I approach the problems. But, then again, I can see how it is prone to error. Relating this to Cognitive Load Theory - if a students does not have facts and processes stored in long term memory, then think of the cognitive demands placed on working memory when trying to process a problem, split it up, work out the individual components, and then put it back together again. Without such facts and procedures in long term memory, the complete strategy was always going to be the most successful.
My favourite quote:
The main significance of these findings is that the two mental computation strategies represent different attitudes towards numbers. The Split strategies suggests that numbers up to 100 are viewed as consisting of tens and units and children using them attempt to deal with these values separately. Such strategies can frequently lead to the sort of errors that occur when using the written standard algorithm. By contrast, Complete number strategies treat numbers as wholes. Furthermore, the calculation steps are sequential so that subtotals are operated on as they occur and do not have to be stored separately in memory.

Article Title: Developing Automaticity in Multiplication Facts: Integrating Strategy Instruction with Timed Practice Drills
Author(s): John Woodward
My Takeaway:
This study seeks to establish whether it is better to use a strategy of timed drills to teach multiplication facts, or a combination of timed drills with activities that promote the use of strategies to teach these facts. The strategies involved were those similar found in Number Talks, for example the multiplication fact 6 x 7 was shown, through discussion, to be equivalent to 6 x 6 + 6. Groups of students were taught using either Drills or and Integrated approach for 4 weeks. They were then given three types of test: Computation, Extended Facts and Approximations, followed by an Attitude Towards Maths survey. Both groups performed equally well on Computations, with the Integrated group performing better on the Extended Facts and Approximations tests. Both groups reported the same level of happiness in the attitudes survey, which may surprise those who fear the Drill and Kill strategy. My only reservation about recommending an integrated approach based upon these findings is that we do not have longitudinal data to see the levels of retention. The students in the Drill approach got through far more computations, and hence there is a chance that their degree of retention will be higher than the Integrated group.This study supports my view that an integrated approach can work - but the key knowledge and facts must be in students' long term memories before attempting to develop and discuss these strategies with numbers, and drilling may be the best way to achieve that. Then, hopefully, you can end up with the best of both worlds.
My favourite quote:
Results from this study indicate that an integrated approach and timed practice drills are comparable in their effectiveness at helping students move toward automaticity in basic facts. If educators were only considering facts as a foundation for traditional algorithm proficiency, either method would probably suffice. Yet, the educationally significant differences between groups found on the extended facts and approximations tests should encourage special educators to consider how strategy instruction can benefit students’ development of number sense.


Real Life Mathskeyboard_arrow_up
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"Sir, when will we ever use this in real life?", is a question I never tire of hearing... And for years, I have not had a good reply. My response has often been to try to shoehorn a supposed "real-life" context into my lesson, and believe me "shoehorn" is the correct way of describing it. Did David Beckham really consider the properties of quadratic graphs when lining up a free kick? Advocates of real life problems argue that they have a motivating factor, but as we will see in some of the papers that follow, this must be weighed up against the potential for misinterpretation. More recently, I have given up. I simply tell the students we are doing mathematics for the love, challenge and beauty of doing mathematics, and everyone has gotten on a whole lot better ever since. So, it was with some interest that I sought to see what educational research had to say about the use of real life contexts in mathematics.

Article Title: Relevance as MacGuffin in Mathematics Education
Author(s): Dylan Wiliam
My Takeaway:
In this brilliant paper, Dylan Wiliam looks at the use of "real life" maths problems in the classroom. Wiliam shows examples of where the context adds nothing at all to the question: " Alan drank 5/8 , of his pint of beer. What fraction was left?", as well as examples where the context can actually lead students to get the "wrong" answer: "a disc jockey plays two records. The first lasts 2 5/8 minutes and the second lasts 3 1/4 minutes. Find the total time taken to play both records" - the point being that DJs are likely to talk or play jingles between songs, and hence students may (sensibly) choose to account for this in their answer. There is a fascinating insight into how, when presented with a real-life problem, many girls seek to relate the problem to their existing knowledge, supplying any information that they feel is missing from their own experience, while boys are often content to tackle the problem in isolation from their previous experience. Finally, Wiliam suggests three criteria that should be applied to ensure that the contexts used in problems are useful: commonality, match and range. My main takeaway from this was to think really carefully about what the context is adding to the question. If the effect is neutral (as in the pint of beer question), then why bother with it? If it is negative (such as with the DJ, or if it is likely to disadvantage some students), then remove it. But if a metaphor or a framing can be used to impose extra clarity on the problem, without altering its underlying structure, then this would be valid. I will certainly be checking my examples more carefully in the future.
My favourite quote:
In the vast majority of classrooms, relevance is a MacGuffin—a device to motivate learners; to convince them that the activities they are given are somehow of the real world, even though they do not appear to be connected to it. Students know this, and as a result their thinking in mathematics lessons becomes divorced even from their thinking elsewhere in school, let alone the world outside school. Mathematics lessons thus become literally mindless—an activity in which students come to believe that thinking is not helpful.

Article Title: The effect of using real world contexts in post-16 mathematics questions
Author(s): Chris Little and Keith Jones
My Takeaway:
Before we get into the study itself, the authors have a really nice way of summing up the dilemma of using real world contexts: "On the one hand, by making a connection between the abstract world of mathematics and  everyday, or scientific, contexts, we are reinforcing the utility of mathematics as a  language for explaining the patterns and symmetries of the ‘real’ world. On the other  hand, if we manipulate and ‘sanitise’ real-world experiences to enable them to be  modeled by a pre-ordained set of mathematical techniques, then the result can appear  to be artificial and contrived". I like that! Anyway, in this interesting study, alternative versions of the same questions (all on sequences) were presented in explicit, algebraic, word and pattern contexts, and set to  a sample of 594 Year 13 students (aged 17-18) in a one-hour test. The results showed that that setting sequence questions in real-world contexts does indeed add to the overall demand, though a context can on occasions provide ‘mental scaffolding’ to help the solver to use context-specific heuristic strategies. My key takeaway is that we need to think carefully what our aim is. If it is to ensure our students are comfortable with the basics, then there is little point setting questions in a real-world context. Interpreting the question and deciding what strategy to use all impose additional cognitive load to the student's working memory, and as we have seen in the sections on Cognitive Load Theory and Cognitive Science, that may mean that no learning actually takes place. However, the researchers did find that some real-world contexts provided additional support for students in guiding them towards the correct method, so it may not be as clear cut as we would like. Of course, if your aim is assess if students can apply their knowledge and skills to different contexts, then using real-world questions may be a way to do that. I guess we just need to be aware how we interpret students' results in that case - do they struggle because the fundamentals are not in place, because they do not understand what the question is asking, or because they have been misled by the context itself?
My favourite quote:
The potency of algebraic formulae lies in their universality and blindness to individual contexts, and, in resorting to context-specific thinking to solve these questions, students are avoiding the necessity to transfer and abstract from context to mathematical model, which is, arguably, the heuristic strategy intended by the questions.

Article Title: Do realistic contexts and graphical representations always have a beneficial impact on students’ performance?
Author(s): D. De Bock, L. Verschaffel, D. Janssens, W. Van Dooren, K. Claes
My Takeaway:
This is an amazing study which looks at the effect of using video to aid mathematical explanation. One hundred and fifty-two eighth graders (13–14-year olds) and 161 tenth graders (15– 16-year olds) participated. They were give a paper-and-pencil test about the relationships among the lengths, areas and volumes of different types of rectilinear and non-rectilinear figures. The problems were administered in or out of an authentic context and setting, and either with or without an integrated drawing instruction, leading to four combinations of groupings. The "authentic context" invovled students watching clips of a screen version of ‘Gulliver’s Travels’, a world in which all lengths are 12 times as small as in our (and Gulliver’s) world. In the drawing instruction groups, students were provided with a drawing of the (geometrical) object introduced in the problem, and they were asked to complete the drawing by making a reduced copy using the given scale factor. How did the groups perform on the test? Students who watched Gulliver’s Travels and who received the video-related items performed significantly worse than the students from the other groups. Surprisingly, students who had to make a drawing performed significantly worse than students from the non-drawing groups, although the authors admit this is probably due to the nature of the linear scale factor task itself. To explain the video results, one possibility suggested really resonated with me: students perceive video as a less difficult medium than written materials, and therefore are inclined to invest less mental effort in working with information transmitted by this easy medium as compared to media that are perceived as difficult. As we have seen in the Cognitive Science and Memory sections, students remember what they think about, and sometimes thinking needs to be difficult. When watching a video (or indeed any other form of novel media we might use in a lesson, such as a website, app, or music) are students thinking about the task itself or the form of media? Any increase in engagement and motivation must be weighed up against this.
My favourite quote:
Some of the qualitative findings indicate that the reason why these two new forms of help did not yield the expected positive effect, was that they conflicted with students’ implicit norms, expectations and beliefs about doing mathematics, especially about their appreciation of formal and informal strategies and of drawings as a valuable modelling tool. From that point of view, classroom interventions that are only partial and instantaneous, and that are unable to influence or alter these more fundamental attitudes and beliefs, have little chance of success. Most likely, only a long-term classroom intervention, not only acting upon students’ deep conceptual understanding of proportional reasoning in a modelling context, but also taking into account the social, cultural and emotional context for learning, can produce a positive effect in defeating the illusion of linearity.

Article Title: The role of context in linear equation questions: utility or futility?
Author(s): Chris Little
My Takeaway:
Four linear equations questions were presented, each set in a context. Two of these questions were : 1) In 18 years time, Halley will be five times as old as he was 2 years ago. How old is he now?  4) The largest angle of a triangle is six times as big as the smallest. The third angle is 75°. Find the size of the three angles. The author argues that these questions may be regarded as ‘applications’ of linear equations, in the sense that they involve formulating and then solving them. However, they have no practical utility value - none of the results provide significant information about the context. So, what is the point in the contexts? They are certainly not motivating. But they are unlikely to introduce any of the misinterpretations highlighted in previous papers in this section. The author makes a really interesting point: whilst the underlying algebraic structure of all four problems is the same, the contexts may actually encourage students to try different methods to solve them - trial and improvement, for example. I have seen this a lot with my students - often the context prevents students from spotting the topic that the question intends to test, and as a result the students end up trying something different. My takeaway here is similar to the paper on post-16 students - think about what we want our students to achieve. Do we want them to practise solving equations? If so, then maybe it is best to cut out the context and just have the equations. If we want them to practice recognising when it might be appropriate to formulate a question in terms of a linear equation, then we have seen in the sections on Cognitive Science and Cognitive Load Theory that students will only be in a position to do that - and crucially, to learn from it - when the basic knowledge of how to solve linear equations is secure in long term memory. Otherwise, the context may inhibit their learning.
My favourite quote:
Some researchers into word problems (Greer, 1997) have advocated introducing more elements of realism into classroom tasks, for example by  adding irrelevant information, which then has to be discounted by the solver.  However, expecting students to engage in genuine mathematical modelling activity  before they serve an apprenticeship in formulating algebraic equations, and learning  abstract, analytical methods for solving them, is perhaps itself unrealistic. Many  students find the process of translating real-life numerical concepts into algebraic variables demanding enough, without being deflected by realistic ‘noise’.