Minimally Different Problems

Meanings are acquired from experiencing differences across a background of sameness, rather than from experiencing sameness against a background of difference.

Ference Marton & Ming Fai Pang

One common thing I do in class is have students practice something. Some students get bored quickly, some work happily along, others struggle. This post is an attempt to design practice in a way that supports the learning of all of these students.

Students have been introduced to arithmetic series and need to practice. Here are two sets of problems:

Which sequence of problems better helps all students?

I’m going to argue the first. Three reasons:

  1. First, each problem only varies in small ways from the previous problem. Students’ attention is then focused on these small changes, and they are more likely to make sense of the components of an arithmetic series problem, rather than having to start from scratch for each problem. When the problem changes in only one way, students can better understand the impact of that change on the mathematics.
  2. There is more potential for extension. The structure of the problems means that students can find shortcuts, using one answer to more easily solve another. Then, we can return to those ideas to review as a class, providing more opportunities for discussion than typical practice.
  3. There is more opportunity to scaffold success. A student who is struggling might have trouble at first, but varying only one element of the next problem makes it more likely that they can use what they figured out right away and better consolidate their understanding.

This idea comes from Variation Theory, which Craig Barton talks about in How I Wish I’d Taught Maths. He writes:

By working through carefully chosen sequences of questions, students have to carry out procedural operations, thus engaging in vital practice. But through connected calculations, they also have the opportunity to consider the deeper structure. Such variation allows students to anticipate, notice and then generalise, instead of permanently playing catch-up (249).

I think there is more potential in these sequences of problems both for students who already have strong skills and have the opportunity to notice new connections, and to students who are struggling with the concept and can benefit from only focusing on the essential differences between problems. But the problems above are only one very narrow type of question. What about when students need to distinguish between similar problems?

If you do not know what English is and you hear 100 people speaking English, you will have no better idea of the meaning of “a language”. If you do not know what “a lively style of writing” is, and you read 100 articles, all of them written in the same lively style, you will still not know what “a lively style of writing” means.

Mun Ling Lo

Let’s say I want to help students distinguish between arithmetic and geometric series, and as a secondary goal practice identifying the common ratio of a geometric series. The above sequence of problems strips away any unnecessary ideas, and gives me a great chance to see exactly where student thinking breaks down, and to address those breakdowns. I don’t think all practice should be structured as variations of one problem; after these six problems, I might start with a new sequence focused on different ideas and asked in a different way. But by only varying a single element of a problem at a time, I get more precise information about what students know and don’t know, and can facilitate a more fruitful discussion of the problems. 

I think there is one more possible use for this type of minimally different problems. Let’s say I want to introduce students to sigma notation. I often struggle to explain sigma notation concisely, and a few examples can go a long way. I might give students a few examples of sigma notation to notice and wonder about. But with too much variation, it just looks like Greek alphabet soup. By only minimally varying problems, I give students more to latch onto, and make it more likely they notice what I would like them to notice:

I really like these sequences of minimally different problems. They still serve goals I had before. But now, students’ attention is more focused on the essential ideas of a topic, sequences of problems scaffold success for more students, and I open up natural opportunities for differentiation as students can make new connections and generalizations. While I’m only starting to experiment with minimally different problems, I also think that over time these problems could help students to see that math can make sense and isn’t just a collection of disconnected ideas. As students see more sequences of problems like these, they might start to believe that they can find shortcuts and new strategies for problems, and develop a disposition to look for patterns where they might not have before.

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