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.

]]>PS, I’ve liked other posts – “problems” in Math Memos. My approach is “jump into a problem and try to swim” – so it’s great to have access to some doozies. Thank you and keep ’em coming!

Cheers,

Richard

]]>Eric

]]>The Name Race is an icebreaker and name activity I’ve been using with students the first time I meet them. It doesn’t require students to memorize each other’s names, so it’s low stakes, but it’s energizing and fun: http://www.collectedny.org/frameworkposts/name-race/

I didn’t get a chance to do the Handshake Problem with my class. It’s been a struggle to decide what I should do and what I don’t have time for, with only one class a week. I did give the class a visual pattern (http://www.collectedny.org/frameworkposts/a-collection-of-visual-pattern-handouts/) with a growth pattern that is the same as the handshake problem (1, 3, 6, 10, 15, 21, etc.) but we haven’t had a chance to go over it as a group and only one students figured out a generalization for it on his own. Similar to you, my goal at the time was for them to recognize patterns and not necessarily make a generalization. I think it takes a fair amount of guidance for most of our students to get to a generalization for the handshake problem.

Best,

Eric