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Worked Examples: The Importance of Choice

Cognitive Load Theory really brought home to me the importance of examples in teaching. This view was reinforced in my podcast interview with Daisy Christodoulou who described a lesson from her past that didn't go as planned because students had not understood her explanation of a concept. This made me realise I had been making a mistake in my teaching - I put the emphasis of my planning on the explanations I would give my students, with the examples I chose playing a secondary role. I have now come to realise that the choice of example are more important than anything else. This section is my attempt to explain why.

Research Paper Title: Exemplification in Mathematics Education
Author(s): Liz, Bills, Tommy Dreyfus, John Mason, Pessia Tsamir, Anne Watson, Orit Zaslavsky
My Takeaway:
This is a brilliant summary of relevant research into the use of examples in the teaching and learning of mathematics. There are many things I found fascinating, but here are a few of my key takeaways:
1) The concept of "varied examples" discussed by Matron can be an effective way to encounter concepts. The authors notes that what is needed is variation in a few different aspects closely juxtaposed in time so that the learner is aware of that variation as variation. This for me is similar to the concept of minimally different examples favoured by the likes of Bruno Reddy and Kris Boulton, whereby just one aspect of a particular example is changed each time. This should mean the student is more likely to notice key features of examples, and allows the teacher to have more control over the discussion. Bruno explained it to me in terms of a science experiment, whereby you are looking to isolate the effect of one variable, therefore you only change one thing at a time.
2) A teacher's poor choice of examples can have a detrimental effect on learning by making it more likely students will jump to the wrong conclusions. A study by Rowland documents this for novice teachers in a primary setting, where the unintentionally ‘special’ nature of an example can mislead learners. I have been there myself. When teaching the effect of squaring a number, I have been known to choose "2", hence potentially reinforcing the major confusion between squaring and doubling. Or asking which is bigger, 2.7 or 2.85, which learners may get right without understanding place value (asking them to choose between 2.7 and 2.65 would be a better example).
3) A study by Wilson reports that learners can be distracted by irrelevant aspects of examples, so the presence of non-examples provides more information about what is, and is not, included in a definition. This is crucial. I used to assume that learners would understand the definitions I gave. However, you can argue that to understand a definition you need to understand the concept that is being defined, and hence definitions are perhaps not the best way to introduce new concepts. This is where examples come into play, and crucially the explicit use of non-examples to illustrate to students what does and does not fit into the definition. Something as simple as showing students carefully selected images of quadrilaterals and non-quadrilaterals will enable them to build up their own understanding (and subsequent definition of what a quadrilateral is far more effectively than presenting them with a definition of something they know little about.
My favourite quote:
Examples play a crucial role in learning about mathematical concepts, techniques, reasoning, and in the development of mathematical competence. However, learners may not perceive and use examples in the ways intended by teachers or textbooks especially if underlying generalities and reasoning are not made explicit. The relationship between examples, pedagogy and learning is under-researched, but it is known that learners can make inappropriate generalisations from sets of examples, or fail to make any conceptual inferences at all if the focus is only on performance of techniques. The nature and sequence of examples, non-examples and counterexamples has a critical influence of what opportunities learners are afforded, but even more critical are the practices into which learners are inducted for working with and on examples.

Research Paper Title: The "Curse of Knowledge" or Why Intuition About Teaching Often Fails
Author(s): Carl Wieman
My Takeaway:
The Curse of Knowledge refers to the idea that when you know something, it is extremely difficult to think about it from the perspective of someone who does not know it. It is a finding that has been demonstrated across a number of studies, several of which are referenced in this paper. The Curse of Knowledge clearly has major implications for teaching, because teachers (we hope anyway) know more than our students. It brings into focus the important distinction between subject knowledge and pedagogical knowledge - you can be the best mathematician in the world, but unless you can find ways to communicate that knowledge effectively to students, then you are unlikely to be a good teacher. That is where knowledge of misconceptions is crucial, and will be discussed further in the Formative Assessment section. However, this paper made me consider another implication of the Curse of Knowledge for teachers - the importance of our choice of examples. If the suitability of our explanations is called into question by the Curse of Knowledge (perhaps we pitch them at too high a level, go too fast, use sophisticated language, or refer to concepts students do not have a complete grasp of yet), then perhaps we can avoid some of the damage of this via an increased emphasis on explanations. A running theme through this section is that explanations may be more important than examples, and for me the Curse of Knowledge provides another justification for that argument.   
My favourite quote:
This “curse of knowledge" means is that it is dangerous, and often profoundly incorrect to think about student learning based on what appears best to faculty members, as opposed to what has been verified with students. However, the former approach tends to dominate discussions on how to improve physics education. There are great debates in faculty meetings as to what order to present material, or different approaches for introducing quantum mechanics or other topics, all based on how the faculty now think about the subject. Evaluations of teaching are often based upon how a senior faculty member perceives the organization, complexity, and pace of a junior faculty member's lecture

Research Paper Title:
Shedding Light On and With Example Spaces (and a very useful summary version here)
Author(s): Paul Goldenberg & John Mason
My Takeaway:
This paper (and the excellent summary) builds nicely on the previous paper, and contains some beautifully written points that have really influenced the examples I now give my students:
1) Well chosen examples can make-up for limited vocabulary and conceptual understanding of students that can render definitions alone pretty useless. To quote: Examples might be thought of as bits of context—ways to give information other than “saying what the word means”—allowing vocabulary-learning in school to grow at least slightly closer to the natural language learning at which children are so adept. Examples allow teachers to use a word communicatively until students are able to use it as well. Teachers can use the word rather than explaining it because the example provides the context and carries the meaning. Only then, when the students already have a rough meaning from communicative use in context can one effectively clarify the meaning formally with other words, through discussion and/or definition.
2) Including non-examples can give a less ambiguous signal of understanding. To quote: Asking a student to circle all the parallelograms in a collection of figures that includes non-parallelograms, prototype parallelograms, and various special-case parallelograms that are often thought not to be parallelograms because they have their own special names, we take each special-case figure that the student does not circle as evidence that the student’s understanding of parallelogram is incomplete. But if we ask a student to draw a parallelogram, we expect not to see the special cases that we’d hope the student would circle in the previous example. And, in fact, if we do get them as responses, we might well take that as evidence of error or incomplete understanding.
3) The definition is still important for clarity and confirmation. This avoids you having to present a potentially infinite number of examples and non-examples. This definition and needs to come after exposure to carefully chosen examples and non-examples once students have a rough idea of what is going on. This avoids you having to present The lovely section on "smaglings" at the end of the summary paper conveys this point brilliantly.
My favourite quote:
Without belaboring it, here’s the point: just as definitions without examples are generally insufficient to convey meaning, so are examples without definitions. No matter how numerous and varied our examples and non-examples are, unless they are exhaustive (i.e., the set of smanglings is finite, and we have encountered every one of them as either an example or non-example), examples alone are insufficient to allow us to decide all cases, because they provide no way of knowing whether or not some perverse exception lurks among the cases that have not been seen. But the examples—and especially the task of trying to choose among the unknowns and then defend that choice—make it much easier to perceive the dimensions of possible variation and the range of permissible change. One advantage for students of encountering this meta-mathematical idea is that it helps motivate what otherwise often seems like bizarre over-particularity in the wording of definitions. There is a lot we must say to define a smangling in a way that allows us to decide, definitively and without question, which of the unknowns is and isn’t a smangling.

Research Paper Title:
Getting Students to Create Boundary Examples
Author(s): Anne Watson and John Mason
My Takeaway:
Building on the previous papers, here we are introduced to the interesting concept of Boundary Examples. Boundary examples distinguish between having and not having a specified property. The authors assert that if students are only offered well-behaved examples, or examples which have additional, but irrelevant, features, then the reason for careful statements of conditions to a theorem or definition might pass them by, and they may well develop the idea that it is possible to have ambiguous or undecided cases. The authors offer the example of sequences. If students are only shown increasing or decreasing linear sequences, they may focus on the fact that sequences are either increasing or decreasing, and be oblivious to the fact that some sequences go up by different amounts (e.g. 1, 1, 2, 3, 5, 8...), some have limiting values (e.g. 8, 4, 2, 1, 0.5...), and some have the same terms throughout (e.g. 1, 1, 1, 1...). If students leave a lesson thinking that all sequences have constant differences and the only thing that distinguishes them is whether they go up and down, then their understanding is incomplete. Elsewhere the authors explain "We use the word ‘boundary’ because we see students’ experiences of examples in terms of spaces: families of related objects which collectively satisfy a particular situation, or answer a particular mathematics question, or deserve the same label. Such spaces appear to cluster around dominant central images". The point is, that by not explicitly addressing examples "on the boundary" there is the danger that key features will be lost at the expense of others. The authors make the bold claim: "If you cannot construct boundary examples for a theorem or a technique, then you do not fully appreciate or understand it". My big takeaway from all this is to ensure that I do not always provide the "obvious" worked examples in class - instead being aware of unusual examples that still fit into the topic that I am teaching. Polygons with convex angles, straight lines in the form x =, linear equations like 4 - 2x = 10, the mean of algebraic terms, and quadratic expressions that do not factorise, are all possibilities that spring to mind. The key point is that if these types of examples are left out, it may be possible for students to appear as though they have understood a topic, whereas in fact they only have a surface level of understanding. Furthermore, challenging students to construct a particular example, then a peculiar example (eg. one which no-one else in the class is likely to think of), and then (if appropriate) a general, or at least maximally general example, seems a really useful practice to develop, all whilst considering the burden on students' working memories. The authors acknowledge that the process of creating boundary examples is likely to be tricky for students at first. Their advice is to give students time, work with them over a period of weeks and months, using the concept of boundary examples wherever appropriate. Eventually students will get better at it, and their understanding of mathematics will improve. As the authors say: "we cannot afford not to invest the time needed in order to enable students to appreciate the ideas to which they are being exposed". I discuss the importance of examples in my podcast interview with Daisy Christodoulou.
My favourite quote:
We have found that many students do not appreciate the range or scope of choice of objects which are permitted by a theorem. Most theorems can be seen as a description of something which is invariant-amidst-change, and the theorem states the scope and range of change that are permitted. But if students have not tried to construct examples for themselves, have not probed the role of various conditions in making a theorem or technique work, then they are unlikely to use it appropriately, and probably unlikely to think of using it at all!

Research Paper Title: Basic skills versus conceptual understanding: A Bogus Dichotomy in Mathematics Education
Author(s): H. Wu
My Takeaway:
This is a wonderful paper which attempts to tear down the myth that you can either teach students basic skills with little understanding, or give them conceptual understanding but without the basic skills underpinning this. For Wu, it is possible to do both, and an important component of that is the examples we choose. Wu makes the bold statement that: we should not make students feel that the only problems they can do are those they can visualize. To illustrate this point, he takes a common way of developing conceptual understanding of how to divide fractions, which is to use nice fractions that students can visualise. For example, when attempting to explain 2 divided by 1/4, we can either show a 2 litre jug filling up cups with a capacity of 1/4 litre, or ask "how many quarters in two wholes". According to Wu, this is all well and good, but then how does that help when dealing with numbers that are not quite so nice, such as 3/7 divided by 2/5, not to mention to introduction of mixed number fractions? Are we to ask students to take a leap of faith, saying "well, you saw how you could conceptualise the nice numbers, and how the written algorithm worked for them, right? So now, just trust me that it will also work for these numbers that are not so nice?". Wu argues that a natural consequence of such an approach is that children develop a sense of extreme insecurity upon the sight of any fraction other than the simplest possible. For Wu the answer is to introduce meaning behind standard algorithms. So, in this case students would be shown exactly why the method of "keep, flip change", or "inverted multiplication" works for dividing fractions, which they can then apply to any problem, no matter how complex. I fully agree with the notion that only using nice examples can lead to big issues when things get more complicated. However, I have a slight issue with the proposed solution, and it is related to my belief that sometimes it is better to teach the how before the why. You only need to read the paper to see how complex it can be to explain why the division of fractions algorithm works. How much conceptual knowledge do students need to have in place about the multiplication of fractions, algebra and generalisation in order to understand it? And are they likely to have this in place at the stage of their mathematical development when they first encounter the division of a fraction? The issue is highlighted even more starkly with the use of the algorithm for written addition later in the paper. I 100% agree that teaching the understanding behind why a robust algorithm works is essential, but I am just not convinced that this needs to come before students have used it. After all, what do we do with the student who cannot understand why the algorithm for dividing fractions or adding number works? Ban them from using it? Is there anything wrong with teaching students the algorithm well, getting them confident with it, and then revisiting the algorithm later in their mathematical development when they are in a better position to understand exactly why it works?
There is a second point in this paper that I feel is key to our discussion - the idea of creating a need or a purpose. This is something Dan Meyer discusses in his wonderful headache-aspirin series. Wu does a fantastic job of creating a purpose for the written algorithm for adding and multiplying by showing how painstakingly slow life would be without them. For me, this creation of need is far more powerful than any contrived real-life context would ever be.
My favourite quote:
Finally, we call attention to the breathtaking simplicity of the multiplication algorithm itself despite the tediousness of its derivation. The conceptual understanding hidden in the algorithm is the kind that students eventually need in order to prepare for algebra. In short, this algorithm is a shining example of elementary mathematics at its finest and is fully deserving to be learned by every student. If there is any so-called harmful effect in learning the algorithms, it could only be because they are not taught properly.

Research Paper Title: Relational Understanding and Instrumental Understanding
Author(s): Richard R. Skemp
My Takeaway:
I wasn't too sure whereabouts on this page to include this classic article by Richard Skemp, but I have opted for the section on the importance of the choice of examples, for a reason that will hopefully become clear shortly. This paper discusses an important distinction between two different types of understanding:
1) Relational understanding - knowing both what to do and why it is done that way
2) Instrumental understanding - the ability to be able to do something without really understanding why (rules without reasons)
Obviously the former is the most desirable, but the author argues that most of school mathematics involves the first, with students being taught maths via a set of rules which enable them to successfully answer questions across a narrow domain (usually with an exam in mind) without really having a clue what is going on. In an attempt to support his view of the importance of relational understanding, the author plays devil's advocate and lists what he sees as the most commonly stated benefits of instrumental understanding:
1) Within its own context, instrumental mathematics is usually easier to understand; sometimes much easier
2) So the rewards are more immediate, and more apparent. It is nice to get a page of right answers, and we must not underrate the importance of the feeling of success which pupils get from this
3) Just because less knowledge is involved, one can often get the right answer more quickly and reliably by instrumental thinking than relational
Then he discusses what he sees as the benefits of relational understanding:
1) It is more adaptable to new tasks
2) It is easier to remember
3) Relational knowledge can be effective as a goal in itself.
4) Relational schemas are organic in quality
But here's my question: who is to say that relational understanding needs to come before instrumental understanding? The author makes the point several times that relational understanding is harder than instrumental (often requiring knowledge of other areas of maths, like the example of circumference of a circle), and that we should not underestimate the importance of students' feelings of success and achievement. Why not teach instrumental understanding really well, via example-problem pairs using examples that cover the entire domain of the topic (hence negating the argument that students with instrumental understanding can only answer a narrow range of questions, and once again highlighting the importance of the choice of examples), and then return to relational understanding once students have tasted success and developed more mathematically? This could be at the end of the topic, or more likely the following term or year, bringing in the related topics necessary to achieve relational understanding, hence tapping into the positive effects of spacing and interleaving. Students revisiting the topic will do so with a feeling of success, and perhaps even have their interests piqued as to why the methods they have learned and used successfully work, as opposed to attempting to teach the why first, when students may lack the skills to understand it, and have no context in which to understand it either. Now, I am not saying that the How should always come first, nor indeed that relational understanding should not be the ultimate goal of teaching. But for topics where relational understanding is likely to be difficult, then it is seriously worth questioning whether we would not serve our students better by teaching the how first. 
My favourite quote:
Suppose that a teacher reminds a class that the area of a rectangle is given by A = L x B. A pupil who has been away says he does not understand, so the teacher gives him an explanation along these lines. “The formula tells you that to get the area of a rectangle, you multiply the length by the breadth.” “Oh, I see,” says the child, and gets on with the exercise. If we were now to say to him (in effect) “You may think you understand, but you don’t really,” he would not agree. “Of course I do. Look; I’ve got all these answers right.” Nor would he be pleased at our devaluing of his achievement. And with his meaning of the word, he does understand.