remedial math

Pattern Power

If you have little kids and you’ve been privy to an episode of Team Umizoomi, then perhaps the title of this post evoked a little jingle in your head. You’re welcome; I’m here all day.

My daughter, although she doesn’t choose Umizoomi over Mickey Mouse as often as I’d like, picked up on patterns relatively quickly after watching this show a couple of times.  She’s 3 years old, and she finds patterns all over the place.  Mostly color and shape patterns, but a string of alternating letters can usually get her attention as well.  These observations of hers made me realize that pattern seeking is something that is innate and our built-in desire for order seeks it out.

High school students search patterns out as well.  For example, I put the numbers 4, 4, 5, 5, 5, 6, 4 so that the custodian knew how many desks should be in each row after it was swept.  It drove students absolutely CRAZY trying to figure out what these numbers meant.  I almost didn’t want to tell them what it really was as I knew they’d be disappointed that it lacked any real mathematical structure.

I’m not as familiar with the elementary and middle school math standards as perhaps I should be, but I’m confident that patterns are almost completely absent from most high school curriculum.  Why are most high school math classes completely devoid of something that is so natural for us?

Dan Meyer tossed out some quotes from David Pimm’s Speaking Mathematically for us to ponder.  This one in particular sheds light on this absence of pattern working in high school mathematics:

Premature symbolization is a common feature of mathematics in schools, and has as much to do with questions of status as with those of need or advantage. (pg. 128)

In other words, we jump to an abstract version of mathematical ideas and see patterns as lacking the “sophistication” that higher-level math is known for.  To be completely honest, this mathematical snobbery is one of the reasons I discounted Visual Patterns at first.  Maybe it was Fawn Nguyen’s charisma that drew me back there, but those patterns have allowed for some pretty powerful interactions in my classroom.   I’ve used them in every class I teach, from remedial mathematics up to college algebra because they are so easy to  differentiate.

I think high school kids can gain a more conceptual understanding of algebraic functions with the use of patterns.  For example, this Nrich task asks students to maximize the area of a pen with a given perimeter.   The students were able to use their pattern-seeking skills to generalize the area of the pen much  more easily than if they had jumped right from the problem context to the abstract formula.  

I also notice that the great high school math textbooks include patterns as a foundation for their algebra curriculum.  For example, Discovering Advanced Algebra begins with recursively defined sequences.  IMP also starts with a unit titled Patterns.   I think these programs highlight what a lot of traditional math curriculums too quickly dismiss:  patterns need to be not only elementary noticings of young math learners but  also valued as an integral part of a rich high school classroom.

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Resilience leads the way

Intrigued by Matt Parker’s tweet yesterday, I decided to have a go at it.
Arrange the numbers 1-17 so that adjacent numbers sum to a perfect square.
It’s the kind of problem that makes students who struggle at math, or hate puzzles, shutter. I decided to take this problem on, in my lowest level remedial math class. This class has about 20 kids, 9th-12th grade, all who have failed a previous math class. These kids range in ability as there is a large variety of classes failed.
I started off by having them number little pieces of paper with the numbers 1-17. I think this helped in setting up the task in a low pressure way. Numbers 1-17, how hard could it be?
Then we talked about what numbers were square numbers. Those who thought they could begin were off.
Others who still couldn’t wrap their head around how the numbers could be arranged, together, came up with some examples of pairs of square numbers. Then, all students were able to make triples with adjacent square sums. They built on those smaller sets and began to come up with strings of 4 or 5 or 6 numbers. One student noticed that the highest square number that could be made from the numbers 1-17 was 25, so 16 and 17 needed to be on the ends of the arrangement since only one other number each would sum to 25 from the list.
Toward the middle of the task, I saw students getting frustrated that they only had a few left and they couldn’t seem to place them. We talked about how 1 could pair with multiple numbers on the list. That discovery seemed to re-energize them to rearranging more numbers and persevere in solving the problem.
The students had differentiated themselves at this point and some were working alone and some together. What I found very interesting is that when one of them solved it, they weren’t immediately drawn to that person to show them the answer. They wanted to figure it out on their own. They weren’t rushed either as students finished. Sometimes when students finish quickly, others become frustrated and just want the answer.
The students also developed some interesting strategies, like grouping pairs that totaled 16 and 25. By the end of the 30 minutes, every single student had arrived at the correct solution. I’m not sure if it was the physical manipulative or the puzzle-like feel of the task, but I was so proud of this group of kids. These are students who have already failed at math and have convinced themselves that they are inherently bad at it. Today they proved that not only is the latter completely false, but also that success is in math is achievable with perseverance and resilience.

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