If you want someone to stop doing something, it's helpful to train an incompatible behavior (this I learned from Karen Pryor). However, sometimes it's hard to know what that incompatible behavior should be.
Of course, you also have to see the existing behavior as a problem in the first place in order to want to change it.
I've noticed that many students who struggle in math are finger-counters. In other words, they use their fingers to add and subtract numbers. This does not seem to me to be a coincidence. In order to figure out a solution, I had to delve into why the finger-counting was happening and why it was a problem.
Happily, as usually happens, the root of the problem was also the source of the solution. The incompatible behavior had the added benefit of being hands-on.
This post will explain my reasoning and help you take the first step to help someone overcome the habit of finger-counting when performing addition and subtraction tasks.
Getting beyond ones
We all begin counting by naming numbers. At first these numbers are just words, and then we associate them with objects. After awhile, we understand that the sequence of numbers exists even without the objects.
The next hurdle is to realize that a given number itself exists without the objects. A number (say, 4) is an idea. Each number has its own unique and unchanging properties. You don't have to have four rubber duckies or coins to have the idea of four.
All you need is the idea of the number in order to manipulate it in various ways.
Most of us pick up this concept intuitively. However, for some students, this concept does not develop on its own.
Below, on the left, is a four-block from a Math-U-See set. On the right are 4 one-blocks.
Of course, these are equivalent. They are equal. But they are not the same.
There is a reason that the idea of four exists. You cannot go around counting out ones and expect to build higher math concepts. At a certain point, you will hit a wall.
And that is exactly what happens to many math students somewhere in the middle grades.
To truly fix this issue, you have to go back to single-digit addition. How many math tutors go back that far?
The root of the problem is that the student doesn't understand a given number except as a collection of ones. So, if we can show the student the numbers as separate things (like that four-block versus the one block, or a dollar bill versus a five dollar bill, or a ruler versus a yardstick), we can build a more mature number sense.
The good news is that it takes less than five minutes to establish this new awareness. Ready?
Note: I based this lesson on a purchased block set that I had in my possession. Homemade resources can work, too.
First, I hold a four-block in one hand and four green one-blocks in the other. I point out that these are equivalent but not the same. "After all," I say, "This is yellow, and these are green." This isn't a joke - it gets them questioning basic assumptions.
Next, I place a ten-block in the top groove of the board. Below, this is shown from the student's perspective:
Next, I offer a number sentence, like, "Six plus blank equals ten." I place a six-block on the left side of the lower groove, under the ten-block:
Next, the student finds the block that completes the number sentence. This is a self-correcting activity - they will know when they have the wrong block, because it will not match ten. You generally will not have to point out when something is amiss.
I gently remind them, "Do not count. Just try different blocks until you find the right one."
Generally, this is an intuitive process, and often you will see the student reach for the correct block without conscious thought.
We continue going through various combinations of addends that make ten. I look for increased confidence and speed in the student's movements as the first sign of improvement.
I also look for buy-in - it's a lot of fun when the student looks up grinning and says, "I was going to count on my fingers!" and then reaches for a block instead. We have trained an incompatible activity, because searching for a matching block means that the student cannot count on her fingers. In fact, it makes it hard to count by ones internally, too.
Once the student is confident with this activity, I have them complete a timed drill (contact me if you would like a copy of the worksheet). Most students demonstrate dramatic improvement on these timed drills over the course of a week, even with just five minutes a day invested.
Best of all, the students report that the rest of their math work gets easier immediately. As they perform math operations, they find themselves visualizing the blocks instead of counting. This means that their number sense is finally moving beyond the first-grade level.
Some things to keep in mind
- If ten is too daunting, start with making five. You might want to start with this anyway, so that your student has some quick wins.
- You will notice that the blocks are paired with their addends in the tray (for example, the seven-block and the three-block are adjacent). Your student may not notice this for days. I think it is best if you wait for him to come to this realization himself.
- Do not be surprised if students do not memorize these facts right away, or if memorized facts seem to disappear when the students are using the blocks. Once the concept is solid, the memorization will happen naturally.
- Even if students understand the commutative property intellectually, they may not be able to put it to use right away. For example, they will be very comfortable with 7 + _, but 3 + _ throws them for a loop every time. This will pass as they repeat the exercise and gain more clarity each time.
The next step beyond making tens is making teen numbers. I hope to address that in a future blog post. In the meantime, let me know if you have any questions about this activity.
Many of my best ideas have come from questioning basic assumptions in order to uncover unexpected or uncomfortable truths. It's a creative process, and a collaborative one. I am grateful to my students for being willing to expose their weaknesses and try a new way. The results have been remarkable. I hope we can help someone else out there!