Problem Solving
For me, Cognitive Load Theory (particularly the paper Cognitive Load during Problem Solving: Effects on Learning discussed in the Cognitive Load Theory section) 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? We have already seen some answers at the end of the Cognitive Load Theory section with the suggestion of a phased approach, and noting the importance of the expertise-reversal effect. Here we seek to see if there is such a thing as a set of problem solving skills that we can help our students develop.Research Paper Title: Learning to think mathematically: problem solving, metacognition and sense-making in mathematics
Author(s): Alan H. Schoenfeld
My Takeaway:
This is a fantastic summary of problem solving in mathematics. There were several points that caught my eye:
1) There are two rather contrasting definitions of a "problem":
Definition 1: "In mathematics, anything required to be done, or requiring the doing of something."
Definition 2: "A question... that is perplexing or difficult."
I had always thought of "problems" as being more of the second variety, but it is worth bearing in mind that when much of the literature refers to "problems", they mean the first type. And whilst we are on the subject, I think a useful way to think about problems is in terms of GCSE Maths Assessment Objectives:
AO1 - Use and apply standard techniques
AO2 - Reason, interpret and communicate mathematically
AO3 - Solve problems within mathematics and in other contexts
It is AO2 and AO3 that I think of when I hear the terms "problem" or "problem solving". It seems to me that AO1 style questions can be taught using the techniques of example-problem pairs, deliberate practice and careful use of desirable difficulties, whereas to help students to answer A02 and AO3 questions you maybe need something more.
2) The authors present a framework for thinking mathematically and solving problems. They argue that all of these categories - core knowledge, problem solving strategies, effective use of one's resources, having a mathematical perspective, and engagement in mathematical practices -- are fundamental aspects of thinking mathematically. I strongly recomend reading each of these sections.
3) The authors make clear the importance of the knowledge base for solving problems. They sum this up nicely as follows: In sum, the findings of work in domains such as chess and mathematics point strongly to the importance and influence of the knowledge base. First, it is argued that expertise in various domains depends of having access to some 50,000 chunks of knowledge in LTM. Since it takes some time (perhaps 10 seconds of rehearsal for the simplest items) for each chunk to become embedded in LTM, and longer for knowledge connections to be made, that is one reason expertise takes as long as it does to develop. Second, a lot of what appears to be strategy use is in fact reliance on well-developed knowledge chunks of the type "in this well-recognized situation, do the following.". For me it is the second point that is crucial - what is often seen as strategy is in fact knowledge. This will be a recurring theme throughout this section.
4) Strategy is still important: Nonetheless, it is important not to overplay the roles of these knowledge schemata, for they do play the role of vocabulary -- the basis for routine performance in familiar territory. Chess players, when playing at the limit of their own abilities, do rely automatically on their vocabularies of chess positions, but also do significant strategizing. Similarly, mathematicians have immediate access to large amounts of knowledge, but also employ a wide range of strategies when confronted with problems beyond the routine.
5) The acquisition of knowledge must come before any such problem solving strategies are developed, otherwise the following may happen: a reliance on schemata in crude form -- "when you see these features in a problem, use this procedure" -- may produce surface manifestations of competent behavior. However, that performance may, if not grounded in an understanding of the principles that led to the procedure, be error-prone and easily forgotten.
6) Teaching the classic problem solving strategies as suggested by Poyla may not be that easy: A substantial amount of effort has gone into attempts to find out what strategies students use in attempting to solve mathematical problems... No clear-cut directions for mathematics education are provided by the findings of these studies. In fact, there are enough indications that problem solving strategies are both problem- and student-specific often enough to suggest that finding one (or few) strategies which should be taught to all (or most) students are far too simplistic.
7) Novices and Experts approach problems in different ways. This finding is related to the work of Cognitive Load Theory on the issues of a means-end problem solving strategy. The author presents a fascinating graph of how studetns and mathematicians typically approach problems, showing how their time is split across the areas of Read, Analyze, Explore, Plan, Implement and Verify. Following a quick read, students typically spend the majority of time exploring. In Schoenfeld's collection of (more than a hundred) videotapes of college and high school students working unfamiliar problems, roughly sixty percent of the solution attempts are of the "read, make a decision quickly, and pursue that direction come hell or high water" variety. And that first, quick, wrong decision, if not reconsidered and reversed, guarantees failure. This is in stark contrast to the graph of the mathematician. The mathematician spent more than half of his allotted time trying to make sense of the problem. Rather than committing himself to any one particular direction, he did a significant amount of analyzing and (structured) exploring -- not spending time in unstructured exploration or moving into implementation until he was sure he was working in the right direction. Second, each of the small inverted triangles in Figure 4 represents an explicit comment on the state of his problem solution, for example "Hmm. I don't know exactly where to start here" (followed by two minutes of analyzing the problem) or "OK. All I need to be able to do is [a particular technique] and I'm done" (followed by the straightforward implementation of his problem solution).
8) Teaching students metacnognitvie strategies for problem solving is possible to help make the the students' graphs more closely resemble those of the expert. The author explains: However, it is the case that such skills such can be learned as a result of explicit instruction that focuses on metacognitive aspects of mathematical thinking. That instruction takes the form of "coaching," with active interventions as students work on problems. Roughly a third of the time in Schoenfeld's problem solving classes is spent with the students working problems in small groups. The class divides into groups of three or four students and works on problems that have been distributed, while the instructor circulates through the room as "roving consultant." As he moves through the room he reserves the right to ask the following three questions at any time:
What (exactly) are you doing? (Can you describe it precisely?)
Why are you doing it? (How does it fit into the solution?)
How does it help you? (What will you do with the outcome when you obtain it?)
He begins asking these questions early in the term. When he does so the students are generally at a loss regarding how to answer them. With the recognition that, despite their uncomfortableness, he is going to continue asking those questions, the students begin to defend themselves against them by discussing the answers to them in advance. By the end of the term this behavior has become habitual.
9) A related approach is suggested by Lester, Garofalo & Kroll on page 65 and is definitely worth reading. Three important conclusions come from this study in particular:
- Metacognition instruction is most effective when it takes place in a domain specific context.
- Problem-solving instruction, metacognition instruction in particular, is likely to be most effective when it is provided in a systematically organized manner under the direction of the teacher.
- It is difficult for the teacher to maintain the roles of monitor, facilitator, and model in the face of classroom reality, especially when the students are having trouble with basic subject matter.
Hence we see the importance of modelling and teacher-led instruction, together with the necessity for knowledge before problem solving.
My favourite quote:This chapter has focused on an emerging conceptualization of mathematical thinking based on an alternative epistemology in which the traditional conception of domain knowledge plays an altered and diminished role, even when it is expanded to include problem solving strategies. In this emerging view metacognition, belief, and mathematical practices are considered critical aspects of thinking mathematically. But there is more. The person who thinks mathematically has a particular way of seeing the world, of representing it, of analyzing it. Only within that overarching context do the pieces -- the knowledge base, strategies, control, beliefs, and practices -- fit together coherently
Research Paper Title: Critical thinking: why is it so hard to teach?
Author(s): Daniel T Willingham
My Takeaway:
This paper does a really good job of summarising Willingham's thoughts on problem solving. We all want our students to become better problem solvers, and to think like mathematicians, but the finding here is that unlike, say, how to factorise a quadratic equation or how to add two fractions together, problem solving per se cannot be taught - at least not to novice learners. Many times I have witnessed students able to solve one problem, but then being unable to transfer those skills to another problem that is clearly related. The key is that the two problems are only clearly related to me. Experts (and I am by no means saying I am an expert here, but relative to the students I am) and novices approach problems differently. When faced with, for example, a worded maths problem in context, typically students are focusing on the scenario that the word problem describes (its surface structure) instead of on the mathematics required to solve it (its deep structure). So even though students have been taught how to solve a particular type of word problem, when the scenario changes, students still struggle to apply the solution because they don’t recognise that the problems are mathematically the same. "Sir, you've not taught us about paint", is a particularly painful post GCSE exam memory for me. Anyway, for Willingham, there are two ways of addressing this issue, but both have important limitations:
1) Students need to be familiar with a problem's deep structure. Exposure to lots of variations of problems that have the same underlying deep structure certainly helps students to look beyond the surface elements of each problem and focus on the relevant mathematics underlying it. So, lots of problems that revolve around the concept of Lowest Common Multiple should help students better spot those. The problem? Well, think how many concepts there are in maths! And then, think about what happens when those concepts are combined! It soon becomes impractical to exposure students to variations of problems covering every single concept in maths.
2) Students need to know they are looking for a deep structure. This is known as meta-cognition. If students are given a problem, they should be prompted to think: "okay, this is a maths problem, it must be related to something I have done before, now can I figure out what it is?". Whilst this strategy undoubtedly helps students avoid the temptation to focus all their efforts on the surface structure, there is a big problem: if they then lack the knowledge to then answer the question, all their efforts are in vain.
This paper has had a profound effect on me. Firstly, when I am happy that students have a sound knowledge of a given topic, I then select a group of related problem solving question on that topic, but crucially I explicitly discuss how they are connected. I force myself and my students to do this, to really emphasise the cues and signals, to try and get around the undo attention give to the surface structure. Sure, I cannot do this for every sub-topic and every combination of topics, but it is a start. And what I no longer do is con myself into thinking that giving students lots of different problems with different underlying deep structures will somehow enable them to develop the skill of "problem solving". Without the knowledge of the topic in place, and without purposeful practice looking at connected problems, this is likely to be a futile and frustrating exercise for all involved.
My favourite quote:
What do all these studies boil down to? First, critical thinking (as well as scientific thinking and other domain-based thinking) is not a skill. There is not a set of critical thinking skills that can be acquired and deployed regardless of context. Second, there are metacognitive strategies that, once learned, make critical thinking more likely. Third, the ability to think critically (to actually do what the metacognitive strategies call for) depends on domain knowledge and practice. For teachers, the situation is not hopeless, but no one should underestimate the difficulty of teaching students to think critically.
Research Paper Title: Analogical Problem Solving and Schema Induction and Analogical Transfer
Author(s): Mary L Glick and Keith Holyoak
My Takeaway:
The use of analogies in conveying concepts is prevalent in subjects like History in English, but not much in maths (it is quite hard to come up with a story to explain how to find the turning point on a quadratic graph, for example). However, I have come to realise that analogies play a huge role in many of the problem-solving questions we give our students to do. Indeed, any in-context problem is surely an analogy, with the surface structure being used as a way to convey - or, as is often the case, disguise - the deep structure that lies below. Hence, work related to analogies and the issues of transfer are directly relevant to mathematics. These two classic papers clearly demonstrate the issues Willingham is describing above, as well as offering a solution. The first paper describes a famous example of a lack of transfer, whereby participants are told a story about a general trying to overtake a fortress ruled by a dictator and challenged to come up with solutions. They were then given a problem involving a doctor faced with a patient who has a malignant tumor in his stomach. The key point is that the two stories have identical solutions. In other words, they have different surface structures, but the same deep structures. However, very few participants made the connection. Indeed, the experimenters found that unless they prompted students to use the initial story, most ignored it and tried to come up with entirely new solutions to the tumor problem. Only 20% of students who attempted to solve the problem gave the correct answer without that prompt. This is clearly related to the finding in the Cognitive Science section that novices and experts think differently, with the former focusing on the surface structure of the problem. So, what is the solution? Well, the second paper provides a possibility. This time the researchers gave two different analogous problems to students (different surface structure but same deep structure) before presenting them with the problem to solve. When students were given these two example stories, with different surface details, and then given the problem, they were much more likely to give the analogous solution. Fifty-two percent of students – a substantial increase over the 20% from before – were able to give the analogous solution. How does this help us as maths teachers? Well, a simple takeaway is that two examples are better than one, especially when it comes to wordy problem-solving questions where the method to use is not entirely obvious. Notice that prompts were also important to help students notice the connection between the two problems. So, if we want students to go beyond the surface features of the problem, we need to present them with groups of problems with the same deep structure, together with our guided support, so that they can begin to make sense of the connections. The use of unrelated problems to solve here and there is unlikely to promote the ability to transfer knowledge to different situations that we need our students to have.
My favourite quote:
The experiments in Part I attempted to foster the abstraction of a problem schema from a single story analog by means of summarization instructions, or else either verbal or visual statements of the underlying principle. We found no evidence that any of these devices yielded more abstract representations of the story, nor did any consistently facilitate analogical transfer. In contrast, the results obtained in Part II were dramatically more positive. Once two prior analogs were given, subjects often derived an approximation to the convergence schema as the incidental product of describing the similarities of the analogs; furthermore, the quality of the induced schema was highly predictive of subsequent transfer performance. In addition, the same verbal statements and diagrams that had failed to influence transfer from a single analog proved highly beneficial when paired with two.
Research Paper Title: Domain-Specific Knowledge and Why Teaching Generic Skills Does not Work
Author(s): André Tricot and John Sweller
My Takeaway:
This brilliant paper offers a complementary view to the Willingham paper above and follows directly from the work we looked at concerning Cognitive Load Theory. The authors argue that there is an assumption that there exists such a thing as domain-general cognitive knowledge (i.e. generic skills such as problem solving and critical thinking), and this can be used to explain student achievement. This assumption then leads to the related assumption that we can teach these generic skills. The authors argue, however, that in fact domain-specific knowledge held in long-term memory provides a better explanation for the acquisition of skills. Moreover, they offer up several explanations of why we as teachers are so keen to embrace the idea of a set of teachable generic skills: "At any given time, we are unaware of the huge amount of domain specific knowledge held in long-term memory. The only knowledge that we have direct access to and are conscious of must be held in working memory. Knowledge held in working memory tends to be an insignificant fraction of our total knowledge base. With access to so little of our knowledge base at any given time, it is easy to assume that domain-specific knowledge is relatively unimportant to performance. It may be difficult to comprehend the unimaginable amounts of organised information that can be held in long-term memory precisely because such a large amount of information is unimaginable. If we are unaware of the large amounts of information held in long-term memory, we are likely to search for alternative explanations of knowledge-based performance"
My favourite quote:
We have argued that expertise based on biologically secondary, domain-specific knowledge held in long-term memory is by far the best explanation of performance in any cognitive area. Furthermore, in contrast to domain-general cognitive knowledge, there is no dispute that domain-specific knowledge and expertise can be readily taught and learned. Indeed, providing novice learners with knowledge is the main role of schools. We might guess that most school teachers in most schools continue to emphasise the domain-specific knowledge that always has been central, making little attempt to teach domain-general knowledge. Based on our argument, they should continue to do so. At school, children acquire knowledge that overcomes the need to engage in inefficient problem solving search and other cognitive processes. That knowledge allows people to function in a wide variety of tasks outside of school.
Research Paper Title: Classroom Cognitive and Meta-Cognitive Strategies for Teachers
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
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?
Research Paper Title: The Subgoal Learning Model: Creating Better Examples So That Students Can Solve Novel Problems
Author(s): Richard Catrambone
My Takeaway:
This paper is important for problem solving, Making the most of Examples and Self Explanations. The author points out that learners have great difficulty solving problems requiring changes to solutions demonstrated in examples. This is obviously incredibly important because it is impossible to construct examples that cover every eventuality, despite how careful we can be with concepts such as Boundary Examples. However, the authors found that if the worked example learners study are organised by sub-goals (in other words, a meaningful conceptual piece of an overall solution procedure), then the learners are more successful. In other words for multi-step procedures, teachers can encourage students to identify and label the substeps required for solving a problem. This practice makes students more likely to recognise the underlying structure of the problem and to apply the problem-solving steps to other problems. Sub-goal learning is hypothesised to be aided by cues in example solutions that indicate that certain steps go together. These cues may induce a learner to attempt to self-explain the purpose of the steps, resulting in the formation of a sub-goal. Across 4 experiments it was found that a label for a group of steps in examples helped participants form sub-goals as assessed by measures such as problem-solving performance and talk aloud protocols. That is no real surprise - labeling parts of worked solutions to more complex problems seems a sensible idea to draw students' attention to the features of that problem and why it is important in the grand scheme of things. As the authors point out, and as we know from our look at Student Self Explanations, compelling students to pause and consider the purpose of individual steps in a solution is likely to be important for their overall understanding. However, what I found particularly interesting is that not all labels were equally as effective. Abstract labels (in this case a symbol to stand for the total when calculating the mean) were more likely than superficial labels (labels tied to the surface structure of the particular problem, such as "total number of suitcases") to lead participants to form sub-goals with fewer ties to surface features. The issue, of course, with focusing on the surface structure is that students are less likely to be able to transfer such solutions to contexts with the same deep structure but different surface structures. However, before we are tempted to jump to the conclusion "abstract labels are best for everyone", there is a problem as explained in the quote below. Hence, we once again see the importance of knowledge, not only in allowing learners to circumvent the distracting features of the surface structure in the first place, but from fully benefiting from the power of sub-goals and labels.
My favourite quote:
Unfortunately, a label that is related to surface features of a problem will be more likely to lead a learner to form a solution procedure that is tied to those features. An abstract label is less likely to lead a learner to make this mistake, although the learner must have relevant background knowledge in order to take advantage of an abstract label. These results suggest that cues such as labels can play a strong role in the formation of solution procedures. Because of this, care must be taken to construct cues in a way to aid the formation of structured solution procedures. For learners with weaker backgrounds these cues might need to be tied at least partially to example features despite the danger that this may lead the learner to form representations that have erroneous surface ties. However, for learners with stronger backgrounds, the cues can be constructed more abstractly, thus helping them to form appropriate subgoals.
Research Paper 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 and tries 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.