One of the most complex issues in teaching mathematics is how to handle examples. On the one hand, examples are important in mathematics because their construction is usually how we make sense of abstract ideas. On the other hand, students can get the wrong idea about examples. They can think that just by seeing enough examples done by a teacher, they will gain understanding of the subject; or that course assessments will be about completing examples just like the teacher did, and so they focus on replicating the teachers' examples instead of learning from them.
Last Fall semester I really felt this struggle as I taught my Modern Algebra 1 class. It's an upper-level course focused on number theory, ring theory, and fields. So there's a lot of abstraction and the only way to truly grasp the subject is to work with a lot of examples. Where I fell short in this class, and what I learned from the experience, is something obvious: When I'm the one doing all the examples, the students aren't learning the math as well as they could. Or as a colleague of mine put it, the one doing the math is the one learning the math. Whenever students ask to "see" more examples, I work them out, and students watch. But watching someone do a thing is not the same as learning the thing.
Why don't we have students generating their own examples more often? And how might such a strategy of student-generated examples look in practice? After my Fall teaching experience, I set out to see what research has been done on these questions, and I found this paper that I wanted to break open here today:
Anne Watson & John Mason (2002) Student‐generated examples in the learning of mathematics, Canadian Journal of Math, Science & Technology Education, 2:2, 237-249, DOI: 10.1080/14926150209556516
Link to paper: https://bit.ly/2SZ5CtX
This paper is a little different than other research articles I've written about here, in that it's a qualitative study. Qualitative research focuses not so much (or at all, in this case) on numerical measures of variables and their statistical differences, as it does on making careful observations of phenomena and then systematically analyzing what's observed. You'll often see qualitative research aimed at exploring questions that are difficult or impossible to operationalize, through anthropological-style observations, interviews, surveys, with the agenda of simply asking questions and making sense of the answers.
Some people think this makes qualitative research less rigorous than quantitative research. That's not the case. In my own research experience, doing qualitative research well is a lot harder than doing quantitative research; and doing quantitative research poorly is just as easy as doing qualitative research poorly. They're just complementary practices (and you'll often see them mixed together) and sometimes one is simply a better tool for the job.
Back to this study: The authors worked with kids (the study mentions 11- and 12-year olds in a few of the observations) on in-class exercises where students were asked to generate examples of five different kinds:
- Experiencing structure: Examples that involve executing and reversing processes, and "doing and undoing".
- Experiencing and extending the range of variation: Examples that elicit different kinds of examples of the same concept from different learners, or multiple representations of the same idea, or different questions that give the same answer.
- Experiencing generality: Examples that result in seeing a pattern in the examples that are produced.
- Experiencing the constraints and meanings of conventions: Examples asking learners to illustrate new concepts and invent notation or terminology to explain a phenomenon and then compare to standard mathematical notation and conventions.
- Extending example-spaces and exploring boundaries: Examples that satisfy some conditions but not others, or those that exemplify "what is and what is not" or what cannot be done within specified constraints.
Each kind of example accesses different cognitive aspects of the example-making process. It's appropriate to give examples of each kind of example here.
The experiencing structure example given was about solving linear equations. They were asked to start with a value of $x$ stated as an equation (like $x = 5$) and then build up a linear equation by repeatedly doing the same operation to both sides. So start with $x=5$, then add 4 to both sides to get $x + 4 = 9$, then subtract $10x$ from both sides to get $x + 4 - 10x = 9 - 10x$, and so on. Then at some point they stopped and presented their equations and the rest of the class asked to solve them. The question came up --- if you were given this final equation and asked to solve for $x$, how would you do it? Well, for the group that created the example, it was easy --- just reverse all the steps that were used to build up the equation in the first place. For the rest of the class, the process was about figuring out what those steps were. Both groups experienced a structural process that generalizes to solving other linear equations.
The experiencing and extending the range of variation task had students give examples of multiplying multi-digit numbers together and making visible their thought processes for how this might work. I've seen this example myself in my own kids' school work. When asked to compute $89 \times 4$, some will compute $80 \times 4 + 9 \times 4$. Others will compute $90 \times 4$ and subtract another $1 \times 4$. By letting kids make the rules and then making their work visible, the entire group is exposed to a multiplicity of examples, some of which might "click" with a student who wouldn't have thought of it otherwise.
In experiencing generality tasks the researchers used a method they called "particular-peculiar-general". The specific task they gave students was:
- Write down a particular number that leaves a remainder of 1 when divided by 7.
- Write down a number that leaves a remainder of 1 when divided by 7 which is peculiar in some way.
- Write down a general form of a number that leaves a remainder of 1 when divided by 7.
Students mostly contributed small numbers for the first task like 8 or 15, then weirder ones like 700001 and 1 for the second task. The authors said that discussion ensued about how to handle negative numbers, and that "the third request followed easily from these contributions". (The latter I have to admit I'm skeptical about, because forming the right generalization for these numbers isn't easy.) This is an instance of using examples in a different direction --- not constructing particular instances from general concepts but using particular instances to arrive at the general concept, which is something that would be right at home in my Modern Algebra class.
In experiencing the constraints and meanings of conventions tasks, the teacher gave students a general idea --- in this case, to represent a function whose output is equal to the input plus 3 --- and have students generate their own representation of the idea. Students came up with some wildly different ways to think about these functions; the paper shows one result involving a sequence of nested boxes that eventually results in the graph of the function $y = x + 3$. After looking at student examples, the idea is to debrief the activity by comparing representations, discussing their pros and cones, and then comparing student representations to mathematically standard representations. The idea is that
If students have had to develop notations for themselves, and compared their usefulness, they are more likely to understand and accept the strengths and idiosyncracies of conventional notations.
And who among us math teachers has not had to deal with students struggling to understand the "idiosyncracies of conventional notations"? I'm looking at you, inverse functions and logarithms.
Finally, the extending example-spaces and exploring boundaries tasks are what I've had my students do in the past: Build a sequence of examples that satisfy increasingly strict constrains. In the paper, students were asked to draw a quadrilateral, then a quadrilateral with a pair of equal sides, then a quadrilateral with a pair of equal sides and a pair of parallel sides, then a quadrilateral with all these features and a pair of equal opposite angles. The idea with this kind of example is to explore the space of possible examples of a concept and discern what's possible and what's not possible.
So, what did the researchers find when they gave these kids all these example-generation tasks? Again, while no quantitative data were collected, the researchers uniformly observed that
Students were actively, noisily, and verbally struggling with attempts to reorganize what they knew to fit the kind of example the teacher was seeking. Students were led away from limited perceptions of concepts and towards wider ranges of objects. They restructured their ways of seeing and experienced the creation of mathematical objects and notations.
Some caveats are in order here: This is great, but it is a long way from a systematic analysis of student observations, and unfortunately it's pretty much the only general conclusion that the researchers draw. They also tend to align their observations with their own experiences as mathematicians: In our own mathematical training, example-generation helped us, and look! It made these kids better too --- which sounds to me like confirmation bias. I'd like to see this kind of study done again with tighter controls on the observations and analyses, and in fact this has been done --- actually Watson and Mason went on to write an entire book about this subject.
For me, the main importance of this article is that it validates the idea that while instructors may need to give examples to students at times --- and there is some reason to believe that instructor-led examples can be helpful in reducing cognitive load for students --- there is also great value in placing the main work of example generation into the hands of the students. It also sparks ideas for how we might do this on a regular basis in our teaching. Watch this space for some future posts on specific activities for different classes; and leave your own ideas in the comments.