Adult numeracy and HSE math students often think doing math is just memorizing procedures and formulas and getting the right answer. This activity is designed to explore and expand the definition of what it means to do math, while looking at measures of central tendency.

- Ask students what it means to do math. Record their answers on the board or on a sheet of newsprint/butcher paper.
- Have students work on the first problem, where they have to calculate the mean median, and mode for the given set of numbers.

- Let students work together – collectively your students will probably be able to come up with the procedures for calculating the mean, median and mode. And if they are not, you can share the procedure.
- Don’t give too much time for the problem – students either know it or they don’t.
- Before sharing the answers, have students define the procedures for finding all three measures of central tendency and write them on the board.
- Have students share their answers. For each one, ask the class if anyone else got the same answer. (Note: Depending on whether they use calculators, and how they round off, the mean will likely be given in different, though equivalent, terms)
- Have students flip the page and work on the other problem, asking for a set of numbers, given the mean, median, mode and number of integers in the set.

- Give students a few minutes of time on their own to work on it. Then you can have them share their thinking so far and work in pairs. If many students are struggling, you might stop after two minutes and facilitate a class brainstorm by asking students, “How did you get started?” (This question can help jump start stuck students, without taking too much of the problem-solving out of their hands)
- If some students finish quickly, you can direct them to the second question (“Is there more than one way to do it?”) and challenge them to see how many different ways they can find.
- After the class has processed the second problem, ask them how the problems are similar. Students may say things like:
*They are both about mean/median/and mode**You need to know the procedures for finding the mean/median/mode**We all found the mean in the same way (and the median and the mode)*

- Then ask the class how the problems are different. They may say things like:
*The second one took a lot more time to answer. It was much harder.**You need to understand the procedure to answer the second one, but the procedure is not enough.**There is only one right answer in the first one. The second problem has many (infinite?) possible answers that work.**It felt so good once I found an answer to the second one.**For the second one, you gave us the answers.**I knew the procedure, but I still wasn’t sure what to do at first. (for the second one)**You can solve the second one in different ways. There is just one way to answer each thing in the first problem.**You have to work backwards for the second problem.**For the second one, you have to find the median of an even number of integers. You need another step of the procedure for finding the median.**You have to make sure you meet all of the conditions in the second problem.*

- Return to the newsprint/butcher paper with their ideas about what it means to do math. With a different color marker, ask them if there is anything that would like to add after their experience with these two math problems.

There are lots of ways to use this activity. I like to do it towards the beginning of my time working with a group of students. As I wrote above, I often frame it in the context of expanding what it means to do math. I also use it to tell students that we are going to be working on more problems like the second one – problems that they will use to develop their own toolbox of problem-solving strategies. Our students need to develop perseverance and independence. We can help them through problems that require more of them; problems that are accessible to a wider range of approaches and therefore to a wider range of students.

Teachers can find problems like this throughout this site and especially under the Math Memos tab at the top of this page.

For more on helping students develop mathematical problem-solving strategies, teachers might be interested in watching Metacognition in Math: Developing Problem-Solving Strategies