# Math Lessons with Two Parts and a Combination

This is fairly open ended, so I understand if people consider this to be off-topic.

I'm interested in creating math lessons where two groups each learn how to use a different simple math skill, and then the groups are combined and challenged with problem types that require both skills.

Here's a simple example:

Group A would learn the definition of factorial and run some easy factorial problems.

Group B would learn that if you have certain fractions (like $$\frac{8 \cdot 15}{5}$$) it's easier to cancel first and then multiply.

Combined Groups form with a mix of people from both groups A and B. They are given problems like evaluating $$\frac{10!}{8!}$$ where having both skills makes running the problem pretty easy.

Is there a specific name for this sort of collaborative learning? What are some fun/interesting problems to run in this way?

• This sounds something like a "jigsaw" activity. – Xander Henderson Feb 27 at 2:38
• I have done a few things like this in different ways. For example, in a combinatorics course I had one group work with Stirling numbers of the first kind and another with Stirling numbers of the second kind. The groups then swapped explanations and were assigned some identities to prove that had both kinds of Stirling numbers. – Jordan Feb 27 at 2:47
• @Jordan Please share your experiences in an answer. – Tommi Brander Feb 27 at 6:05
• This is called "project-based groupwork". Bad pedagogy. Also, by the time the factorials are explained, the process of simplifying fractions must have been licked out. @Jordan Your approach makes more sense. – Rusty Core Feb 27 at 19:06
• @RustyCore Can you please explain why you believe that "project-based groupwork" (which is a very broad category which contains the method suggested in the current question as an element) is "bad pedagogy"? The scholarly work that I have seen on the topic is all neutral to positive. – Xander Henderson Feb 28 at 13:38

## 2 Answers

This sounds something like a "Jigsaw" activity (1), (2), (3). The basic idea is as follows:

1. Divide your students into groups of equal size; say, four students in each group. For specificity, lets label these groups $$A$$, $$B$$, $$C$$, and so on. Don't actually have them get into these groups yet. In each group, assign each student to a different task, say tasks $$\alpha$$, $$\beta$$, $$\gamma$$, $$\delta$$.
2. Have the students work individually to learn the task which they were assigned. Give them some specific instructions and some relatively simple or straight-forward problems to complete in order to become familiar with their given task.
3. Have all of the students who have been assigned to the same task collaborate with each other in an "expert group" to master that particular task. That is, put all of the $$\alpha$$'s together into a group, and have them learn the thing that they need to learn. Same with the $$\beta$$'s, $$\gamma$$'s, $$\delta$$'s, etc. For example, you might have one group learn about Pascal's triangle, another group think about combinations, a third consider expressions of the type $$(a+b)^n$$, and so on. Give the group more complicated tasks (within the area that they are meant to master) to complete together.
4. After a reasonable amount of time has passed, have the students get into the original groups $$A$$, $$B$$, $$C$$, etc. Give these groups a task which requires the expert knowledge obtained by each member of the group. For example, ask them to prove the binomial theorem.

There is a nice diagram on the page linked as (3), above: The links I gave above give quite a bit more detail, but they seem to be more geared towards an elementary audience and reading comprehension-style exercises. However, I am sure that a reasonably clever instructor could figure out how to make it more mathy. ;)

• To prevent that all $\alpha$s rely on one good student of this expert group, it might be advisable to insert 1.5: Have everyone work on their own on the given task, and only then compare/merge their work in the expert groups. – Jasper Feb 27 at 11:48
• @Jasper Indeed. Honestly, the first thing that came to mind when I saw this question was "Think Pair Share," but that isn't quite the right idea (though it is another useful strategy). I'll revise my answer. – Xander Henderson Feb 27 at 13:18
• It is funny you mentioned the Binomial theorem. That is exactly the context in which I have done this in the past! It worked out quite well. Students were amazed to discover that such (on the surface) distinct questions lead to the same sequence of numbers. – Steven Gubkin Feb 28 at 16:57

I did something like a jigsaw activity in an undergraduate 300-level topics in combinatorics course. I split the class into two groups, and each was assigned a reading and several exercises related to Stirling numbers. One group worked with Stirling numbers of the first kind and the other group worked with Stirling numbers of the second kind. The next class, the two groups split in half and matched with half of the other group. Each of these mixed groups took turns presenting their material (within the mixed group), and then the two mixed groups were given another assignment including proofs of identities involving both kinds of Stirling numbers.

I also use similar strategies in an undergraduate exploration/discovery-based math course, but in a more informal way, as the course is completely student-driven. In that course, if I find a student struggling to prove something, instead of giving them a hint, I'll find another student not already busy, give them an idea (which is a hint for the first student), and then suggest they work together. In that class, I spent most of my in-class time looking for opportunities to facilitate things like this.