# Measures of Central Tendency (and Expanding What it Means to Do Math)

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

## One thought on “Measures of Central Tendency (and Expanding What it Means to Do Math)”

1. On 13 December 2017, my intermediate HSE math class at the Lehman ALC focused on concepts and practice with mean, median and mode.

Following the protocol found on the CUNY HSE Math Curriculum Framework, I first asked students to take a few moment to reflect on the question, “What does it mean to do math?” And then to write their thoughts out. Here are some of the responses:

Crystal: “What math mean to me is different. It’s hard work, you have to think before answering any question. It plays a part in every day life, for example: money, paying bills, buying clothes, food, car note and also school loan etc.”

Monique: “Doing math is a part of everyday life. Without math it would be kind of difficult go through a day without knowing math. For example with doing math if you go to the store and you buy something and you give your money you would know instantly how much change you got back.”

Samantha: “Math is a part of everyday life whether we like it or not. Math teaches us the basics like how to count, shapes, and numbers. I personally don’t like math and on some aspects its useless and unnecessary. But its a tool we need in life therefore we have to know it.”

Meshia: “What does it mean to do math? To solve a problem even though it can be frustrated at times, but at the end of it you get the understanding of doing it.”

Yohansel: “To do math means everything because it helps how to deal with personal finances. Sizes and time scientifically. Doing math helps to show the improvements of life.”

Gladys: “What it mean to do math is that the focus is on students actively figuring things out, and testing ideas, making conjectures, developing reasons. Sometimes students work in groups, some of them like to work in pair or by themselves, but they always sharing and discussing the result.”

Marvin: “Doing math mean getting a problem working on it until you come up with an answer and understand the problem you just did.”

Quite a few students volunteered to read aloud their responses. This led to a lively class discussion about math – its usefulness, frustrations, dislike of – and, in several case the perceived uselessness of the more complicated math (“Who does algebra in real life?” “I hate word problems!”).

This opening gambit (if that’s the word), warmed up the Wednesday math class, and we plunged in.

I wrote on the board: MEAN, MEDIAN, MODE, and asked the class if they knew what these terms meant. I thought mean and median would be familiar – perhaps not mode. Because we’d done mean and median problems (I thought) on a semi-regular basis. But, wow – not so. Some sort of knew – remembered. There then a followed a discussion/clarification – examples on the board, etc. Until I felt everyone was clear about those terms. MODE is easy (if not such a common term) – simply a repeating number.

With that accomplished I handed out the first worksheet: Find the mean, median, and mode of the following set of numbers: 2, 34, 5, 37, 45, 5, 16. I asked students to first work alone, and then, if they wished, share results with a neighbor. After perhaps 15 minutes, we came together as a class and compared answers. I thought this would be pretty easy after our previous discussion of the 3 terms. BUT, some did not understand to find median you first had to put the numbers in order small to large. This led to a consideration of odd number of terms (7 in this case) vs even number of terms when seeking the middle (median) number. There isn’t a middle term if the numbers are even. How do you find it? It’s halfway between the 2 numbers closest to the middle. I put several examples on the board – you have to average the 2 numbers. Interesting to me that students struggled a bit with this concept. (And it would provide a difficulty in working on the next evolution of this exercise!)

So, by doing this straightforward practice, students got further (experiential!) experience with these terms. NOW it seemed they knew what was to be known about M,M, and M!

I handed out Worksheet 2: Construct a set of numbers, with a mean of 7, a median of 6 and a mode of 3. Could there be more than one way to do it?

I read the problem aloud and we discussed it to be sure everyone understood the problem before setting off. There was consternation! Groans! This was the “headache”! They worked for a while diligently at it. How to get into this problem? Was there a key? I kind of steered them toward the notion of “mean”, a mean of 7, with 10 numbers . . . A light went on over Marvin’s head! 70! It’s gotta be 70! And the elated Marvin explained to the rest of the class how he got that: “10 divided into 70 gives you 7. The mean is 7.” I helped him by writing the calculation on the board. I could see people starting to get it. Oh, yeah . . . That seemed to me – to us – the key to the problem! Now we had a way of working on it. 10 numbers whose sum is 70. YES!

I could see it unlocking students: Now they could start to play with 10 numbers, which they had to add up to 70. Moreover, at the same they knew they had to repeat the mode of 3. How many times, I asked. “At least twice”, someone shouted. And then someone else said, “Could be three times!” And then someone else said. “It’s gotta be at least two times!” This was like learning on steroids!

Now began a period of intense experimentation. Conferencing with neighbors. Calling me over to see if they’d gotten it. Or were on the right track. Check, I would say: your series of numbers has to satisfy the three conditions of the problem. They would get their numbers to total 70, including at least 2, 3s. But that damn median! I would try and steer them back to our discussion of “How do you find the median in an even string of numbers?” How many number in your string? That seemed a tough nut to crack, oddly enough. Then a few began getting it: those two middle numbers had to add up to 12: half of 12 is the magic median number 6. But then some would write 6 in – and have 11 numbers! Oh, delirious confusion! Until, gradually, students began getting it – a trickle, then more.

I asked those who correctly solved this problem to put their string of numbers on the board (with their names). Four students complied. As a class we tested their results: 10 numbers, mean of 7, median of 6, mode of 3. YES – for all 4 students. But, we realized – NONE WERE THE SAME. 4 different- correct – solutions to the same problem. One student’s mode repeated twice, while another’s went 4 times; the middle numbers were different (3 were the same), but they each totaled 12 – to give the median of 6. I asked the class they thought perhaps there could be more solutions. Yes, they said, probably. We played by expanding the mode to 5 – that didn’t seem to work.

What surprised me about working on this exercise: what I assumed they knew about mean and median – they didn’t. Which resulted in a mini-lesson. I began to wonder if this exercise was going to go nowhere . . .

But what amazed me was how they focused then and worked on the more – abstract – problem – using their knowledge – and working method of “trial and error” – to figure out solutions. I could practically see thinking dancing in the air above their heads.

I would like to have returned to their original writings about math – and see how their (positive) experience with this problem may have changed their thinking about math. But . . . we ran out of time.

I did record on board student work on Part 2 of the problem – see it here – https://drive.google.com/open?id=1xOJnAP7x4X7SLJUXmQSeUJeKEWxslTPI.

As for me – I felt high after this class. A most rewarding experience.

So, thanks for a great math thinking exercise – keep ‘em coming.