Slow Down...and Do the Math (Part 1)

The full text of the problem is missing, but you get the idea- it's a fairly typical seeming growing dot pattern.  Many of these patterns are deceptively simple- it's fairly easy to describe how it grows, to construct a table, and to come up with an additive relationship for how it grows.

From the Ontario curriculum, you might expect any student from even grade 2 and up to be able to say SOMETHING about this pattern.  The problem strongly fits explicit teaching of the math process "Representing"- tables, graphs, explanations, and algebraic representations all work.  The patterning/algebra curriculum is ideal for showing the variety of mathematical representations that we can use.

This problem worked very well for co-teaching in a grade 7 class.  After about 15 minutes of paired work, all student pairs had something down on the page.  They had made tables, extended the patterns, and made conjectures about how the pattern grew.

Math process of the day. How do we create representations of something so abstract as mathematics? pic.twitter.com/tUx71Rvw8r
— Matthew Oldridge (@MatthewOldridge) November 28, 2014

This is where I had a conversation with one of the teachers in the room about how the students "should" have been done 5 minutes prior. On the surface, that seemed like it might be true-there was something on every page, and the students' initial line of thinking was tapped out.

Having a bunch of experience seeing students work with growing patterns, I thought that many were "ready" to make the leap from additive to multiplicative reasoning, and quite possibly to generalizing using algebra for the nth term. Asking one or two key questions here would be the key to getting further with our thinking.

Here is an example of a pair that was ready to jump to generalizing:

The proof is the circling of the groups of 3 on the right.  Groups of 3 leads to the idea of multiplication by 3, which can get us to the ideas of variables and constants.

Another teacher suggested asking for the 15th term.  What typically happens here is that those students who need to keep adding to show the constant growth will mechanically extend the pattern, and get the correct answer.  But those who are ready to take the conceptual leap will often jump in with insights, like lightning out of a blue sky.

Here is where the differentiation happens- some students realize they understand the problem right down, all the way to the bottom.  They have broken it down to its mathematical elements, and can use those elements to construct their own explanations for what's happening.

Given an extra 5 minutes, here is what one pair had to say about the 15th term:

-it has 15+16+17 dots.

How did they know?  Term 1 has 1 +2 +3, term 2 has 2+3+4, etc. This was quite an efficient way of thinking about the pattern.  It would allow them to give the number of dots in the nth term with simple addition.

In the consolidation, some students revealed that they knew 3n + 3 would work for any term.  They  had little to no prior experience with algebra, but were able to make that generalization.  I think further work in this classroom (on other situations/patterns) could focus on generalizing through multiplying, and generalizing for any term.  Here is where precise explicit teaching through mini lessons, and purposeful practice work would come in.  The next week or two of classes could all be planned from what we learned that day. Such is the power of consolidating our thinking.

This is just the sort of insight that students will come up with, given more time.  Where we can, let's slow down, and the let the thinking happen.  Curriculum can be a rush, yes, but sometimes that extra 5 minutes is the difference between getting somewhere with our thinking, and really getting to a more mathematical place with our thinking.

Jo Boaler and others have written eloquently about the problem of speed in working with students on their number fluency.  Slower but deeper thinkers can often be turned off of mathematics by our constant rush-to get from topic to topic, strand to strand, report card to report card.  Let's slow down and do the math!