Fluency with Maths Facts
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.
Research Paper Title: Assisting Students Struggling with Mathematics
Author(s): Institute of Education Sciences
My Takeaway:
The second appearance of this brilliant paper which is full of research-based evidence and practical strategies for assisting students who struggle with mathematics. This time we turn to a recommendation that is directly relevant to this section:
Recommendation 6 - Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.
The paper then goes on to suggest three practical strategies to aid with this:
1. Provide about 10 minutes per session of instruction to build quick retrieval of basic arithmetic facts. Consider using technology, flash cards, and other materials for extensive practice to facilitate automatic retrieval.
2. For students in kindergarten through grade 2, explicitly teach strategies for efficient counting to improve the retrieval of mathematics facts.
3. Teach students in grades 2 through 8 how to use their knowledge of properties, such as commutative, associative, and distributive law, to derive facts in their heads.
For me, strategies 1 and 2 are all about drilling, whereas strategy 3 lends itself well to the concepts of Number Talks which will be covered later on in this session. The overall conclusion is clear - a knowledge of basic mathematical facts is necessary in order for students to achieve success in mathematics, and as teachers we should help students gain this by making a focus on fluency a regular part of each lesson.
My favourite quote:
These studies reveal a series of small but positive effects on measures of fact fluency128 and procedural knowledge for diverse student populations in the elementary grades.In some cases, fact fluency instruction was one of several components in the intervention, and it is difficult to judge the impact of the fact fluency component alone. However, because numerous research teams independently produced similar findings, we consider this practice worthy of serious consideration. Although the research is limited to the elementary school grades, in the panel’s view, building fact fluency is also important for middle school students when used appropriately.
Research Paper Title: Improving Basic Multiplication Fact Recall for Primary School Students
Author(s): Monica Wong and David Evans
My Takeaway:
This is a great overview into the importance of times tables, and the strategies students employ. There are a number of key points addressed:
1) Whilst conceptual understanding of the times tables is important, it is not sufficient. Automaticity must be achieved in orer to free up space in working memory to solve more complex problems (I love this quote: The importance of automaticity becomes apparent when it is absent. Lessons may stall as students look up facts they should recall from memory. Thus conceptual understanding is necessary, but insufficient for mathematical proficiency)
2) Basic multiplication facts are considered to be foundational for further advancement in mathematics. They form the basis for learning multi-digit multiplication, fractions, ratios, division, and decimals Many tasks across all domains of mathematics and across many subject areas call upon the recall of basic multiplication facts as a lower-order component of the overall task. To enable students to focus on more sophisticated tasks such as problem solving, proficiency in basic facts and skills is an advantage. Without procedural fluency and the ability to recall facts from memory, the student’s focus during problem solving will be on basic skills rather than the task at hand, thus drawing attention away from the learning objectives of the task. If the student cannot perform these basic calculations without the need to use calculators or other aids, higher-order processing in problem solving will be impeded
3) When answers are predominantly recalled from memory, the student should be able to answer approximately 40 basic mathematics questions correctly in one minute
4) The order in which facts are introduced and sequenced can assist students to become proficient in learning and recalling basic multiplication facts. Facts that can be learned easily should be presented first during practice (e.g., 0, 1, 10, 2, 5, 9), then they should be followed by the more demanding multiplication sequences (e.g., 4, 7, 3, 8, and 6). Students should also be proficient at counting from 1 to 100 and be able to skip count.
5) To improve speed of fact recall, students should be given a specific time to respond to a question or a constant time delay, typically starting at five seconds and gradually reducing to one and a half seconds. Reducing the response time forces the student to abandon inefficient counting strategies and attempt to retrieve the answer from memory
6) Practice on computers is said to afford some advantages over more traditional delivery modes. Students can progress at their own rate and practise using varying representations (horizontal or vertical). Feedback is immediate and scoring systems automatically monitor progress. Students who used computers as part of their usual instruction generally learn more in less time and retain the information for longer
7) A constant time delay approach is effective for developing fluency in students with learning difficulties. for The students were then taught 15 unknown multiplication facts using computer-assisted instruction based on a five second constant time delay procedure. The results indicated that the constant time delay procedure was an effective method of teaching multiplication facts to those students.
8) Interspersion of known and unknown facts in each practice session increases the speed at which facts are committed to, maintained in, and retrieved from long- term memory. It also assists in the remediation of errors from previous sessions and improves the speed of retrieval of known facts from long-term memory.
9) Procedural and conceptual misunderstandings need addressing. Some students consistently calculated the A x 0 multiplication fact incorrectly by writing the value of the A, whereas its commuted counterpart, 0 x A was answered as 0. The revision of concepts and procedures needs to be included in the practice sessions to ensure students possess the necessary pre-skills to answer questions accurately. This can be achieved through building conceptual knowledge though the use of manipulatives, through building ‘generalisable’ rules, and through providing practice in the use of virtual manipulatives. Virtual manipulatives allow the introduction of concepts, and provide practice and remediation
10) While proficiency in multiplication facts is important, there are also other basic facts that require practice to maintain ongoing development of mathematical proficiency, such as addition and subtraction facts. Therefore practising other basic arithmetic skills could be included in practice sessions like those used in this study. Initially, multiplication facts may be practised separately to promote proficiency; later they could be mixed with other facts to allow students to become more proficient in selecting from and discriminating between operations. Although it is a more
My favourite quote:
In summary, the belief that the development of basic multiplication fact recall is enhanced by practice has been supported. Results generalised to the study group have shown that a systematic practice of basic multiplication facts by interspersing known and unknown facts improved students’ recall of these facts for all but a few. For these few students, revisiting multiplication concepts may be necessary. Without this improved recall of basic multiplication facts, working memory is consumed by the most fundamental of problems. Releasing working memory capacity allows students to tackle more difficult tasks such as multi-step problems or questions demanding higher-order thinking.
Research Paper Title: Developing Multiplication Fact Fluency
Author(s): Jonathan Brendefur, S. Strother, K. Thiede and S. Appleton
My Takeaway:
This paper investigates if presenting times tables with different representations can help students develop fluency in them more effectively than drilling. The researchers used third, fourth and fifth grade students, splitting them up into two groups. the Strategy group received instruction based on cognitive and social-interactional framework for fluency development; whereas, the Drill group received fluency instruction for basic multiplication facts using an approach emphasizing memorization and rehearsal techniques typically practiced in schools. The techniques used in the Strategy group were Based around Bruner's theory of Modes of Representation, and used the following procedure:
1. Strategy group students began by building arrays with physical models (e.g. tiles) and finding arrays in pictures as well as the surrounding environment. Students then drew diagrams of the arrays on either grid paper (to structure the drawings) or freehand.
2. Students then transitioned from arrays to using a 12x12 blank grid as a multiplication table as both a way to list facts they knew but also as an example of an array. Students overlaid their derived facts strategies (e.g. 8x5 + 8x1 to recall 8x6) on these 12x12 multiplication grids.
3. Eventually, students' materials were removed and they were engaged in what was called "fluency-talks". Students sat on the floor with no writing materials or manipulatives available and were presented with various facts. They had to discuss as a
class how they might use related facts to solve the unknown facts presented on the board.
4. To culminate the strategy group's fact development, pairs of students created sets of strategy cards, which were essentially multiplication flash cards (with a fact on the front of the card), but strategy cards included derived facts strategies the pair preferred written on the back of the card. The pairs would alternate describing two or three facts strategies for each card's fact.
The results were that sstudents receiving instruction through drill and rehearsal gained on average 0.79 facts per minute over the five weeks, with positive gains of 2 facts per minute in fourth grade 2.36 facts per minute in fifth grade. Students receiving instruction grounded in the social-interactional approach demonstrated an average gain of 6.08 facts per minute, with the highest gains in fourth grade of 6.65 facts per minute. This suggests a different approach is needed to introduce students to times tables at a younger age, or possibly to help those older students who do not have a firm grasp of times tables develop the fluency they need.
My favourite quote:
The evidence from this study also demonstrates that students receiving instruction grounded in a framework built upon Bruner's Modes of Representation combined with social- interactional elements significantly outperform students who receive instruction grounded in a behavioristic theory of learning. These instructional activities designed for the Strategy group emphasized strategic thinking and mathematical relationships between multiplication facts and created greater and more consistent gains in fact fluency than activities emphasizing memorization and repetition. Although Brownell (1935) had similar conjectures and findings, more recently Russell (2000) found that students build an understanding of multiplication facts through problem solving and, then, sharing and examining their own strategies. One of the implications to this finding involves memory: one might presume that students will remember any piece of information (including multiplication facts) more accurately if the information is connected to already easily-remembered information (Hiebert & Carpenter, 1992). In the case of multiplication facts, this would mean that students who know 8x5 would ideally spend instructional time learning to use that knowledge to solve 8x4, 8x6, and 8x7 and so forth. Time spent building flexibility with facts would in turn produce fluency with facts as students would have related strategies to refer to should memory fail them. This would contrast with a similar amount of time being spent by students trying to commit 8x4, 8x6, 8x7, etc. to memory.
Research Paper 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).
Research Paper 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.
Research Paper 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.
Research Paper 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 (see Anxiety section for more) - 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.
Research Paper 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.
Research Paper 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.