The Purpose of Math Activities

Let’s dive in to another conversation I had with a frustrated colleague today. What was she frustrated about?

Subtraction.

Darn it.

Now, here is where we ALL agree. No, seriously, I haven’t met a single person who has ever taught 2nd-4th grade not say this - subtraction is stupid hard. Double digit subtraction is a ridiculous crazy beast that has everyone banging their head against the wall by the end of the unit. I fully believe that this “unit” in math is the beginning of the end for many kids in their beautiful relationship with math.

But, me being me, I don’t accept it. In fact, I never accept that a concept should ever be this horrifyingly frustrating to students or teachers (we will talk multiplication later). It’s mathematics. There must be a way that makes sense and is less frustrating.

And I think there is…but before we can try it, we have to have a deep philosophical conversation (don’t you love me?) about the very purpose of math, specifically the purpose of math activities.

But before I even dive into that, some personal anecdotes. In my third grade classroom, I began experimenting with “different ways” to help students approach subtraction. My criterion was:

1. Needs to maintain place value - this is a whole other blog post - but I’ll summarize by saying place value should remain in tact and explored thoroughly throughout 2nd-4th grade and quick algorithms sometimes take away from that.
2. Should not be complicated and unconnected steps - this is the problem with the current way. We teach it, teacher says “the kids forget it unless we do it every day!” and then they forget it anyway. They hate, they hate math, and we are toast. There needs to be a way that actually infuses number sense and connectedness - not just arbitrary steps.

This led me to the 2 different visual approaches to math - partial “sums” and difference between.

I’ve done a blog post on these before(click here) - but this is a deeper conversation into how I got there. Both of these approaches are visual, maintain place value, use the full scale of double digit numbers (building the mental number line and mental number connectedness) and both went over with students so well.

In fact, every student started using one of these strategies by the end of the year instead of trying to remember the traditional steps that always frustrated them before. (and look, I’ve said it before and many times, I’m not staunchly anti-traditional-algorithm - but I think it’s valid to say it’s not always clicking with kids).

So, in moving to my new position, I began introducing these strategies to other kids. And that is what led up to the conversation in question.

Another teacher with the same anecdote - kids forgot double digit subtraction. They did that unit months ago, she gave a review today, they bombed and they were frustrated. The advice (not from me)…? Review it every single day. Drill it. This is fine advice, and certainly might work, but it begs the question - if we have to drill everything, at what point is it too much to keep in your head? And at what point is it just not fun any more? (I know the answer to the latter - the double digit subtraction unit -ha!)

My approach was a new strategy. But I knew what the pushback would be - it takes too long. It looks like (but isn’t) “more steps”, it looks different, it’s not “the way”. This led me to wonder…is getting the answer quickly always the purpose?
And I know I’ve asked this before but it was slightly different this time. When choosing an activity or resource, is exploring number and place value through computation ever a valuable goal in and of itself? Can thinking ever be the goal, with computing as an outcome? And is this the shift that we might just need?

When my students are working on partial" “sum” subtraction with negative numbers - it’s actually a lot more about them exploring numbers. It’s less about subtraction, and more about place value, counting backwards, tens vs. ones, etc. Students are doing so much thinking in that problem. And (in my experience, so yes anecdotal) they really enjoy it. Because students are people - and people actually love thinking. It’s fun!

With the traditional algorithm, students:
get a solution
subtract numbers under 10

With other strategies, like my examples, students:
get a solution
count backwards by ones, tens, hundreds
think about hundreds vs. tens vs. ones and what that means when subtracting off a number
think about negative numbers
subtract large amounts

While some view this as “too much work!” I view it as “so much thinking”. And there in lies the value - but it all depends on your purpose.

So I ask once again…

When picking an activity - what is our purpose? What mental load are students carrying and how is it going to stretch their mathematical minds?

It’s start to a thought. I’m gonna keep thinking.