Open Set

Carl is on a hike in the woods. He has already walked 2 ¼ miles out of the 3 ½ -mile journey. What fraction of the way is that?

Here is a typical middle school fraction problem. When I teach material like this, my first advice to students is always to draw a picture. But my second piece of advice is this. Put in simple, familiar numbers instead of the fractions, then solve the problem using those numbers. Look back at what you did, and use it as a model for how to answer it with fractions. 

Here is what it might look like if the student follows my advice: 

Carl is on a hike in the woods. He has already walked 2 ¼ 10 miles out of the 3 ½ 20 -mile journey. What fraction of the way is that?     

That’s halfway, I know because 10/20 = ½. I divided the first number by the second. To answer the original problem, I would need to do 2 ¼ divided by 3 ½. 

I use similar strategies when teaching algebra too. 

Jeanette bought x postcards for $1.50 each. What was the total cost? 

Change it to … 

Jeanette bought x 10 postcards for $1.50 $2 each. That would be $20. I got that by doing 2×10. So, I need to multiply the number of postcards by the price.

Over the years, I’ve noticed this method helps a lot of students. The new problem is not as scary and confusing now that the numbers are within the realm of familiar. Students are more willing to think about what is happening in the situation when they think they will be able to find their way to the end. The confidence factor is major. 

But there is something else going on here. Students often know the answer to the simpler problem BEFORE they know what operation they used. They intuit the operation correctly when the numbers are familiar, even if they don’t realize what they are doing. That’s number sense showing up. The ability to come up with the answer is not always the same as knowing the math steps. There is a lot of sense-making and math reasoning that students do when the arithmetic is basic, when they can do it in their head. Familiarity with numbers and their connection to the real world opens a door to math problem solving.

This method, where they solve a problem with simple numbers, allows students to build on prior knowledge. They can connect this thing that makes little sense to them (2 ¼ divided by 3 ½, and 1.5x) to something that does make sense. They learn that these complicated operations are extensions of familiar ones, and they are empowered to make meaning out of them. Over time, they learn that these harder problems are accessible, even if they are less obvious. 

Now, give those same students a calculator too early, and you have helped nothing. In fact, you are getting in the way of their learning. If a student does not know that 10 is half of 20, then the easier numbers don’t help. The problem is still a bog of confusion; the student’s number sense cannot help them make sense of these problems. 

I use calculators in my classes. This is not a diatribe against calculators. I think Desmos and Geogebra have done amazing things for the learning of mathematics. (I still pine for Geometer’s Sketchpad, a void that Geogebra simply cannot fill.) And this is certainly not a call for “back to basics.”

However, when I see how students at my high school are simply given calculators all the time, I worry. They are not keeping their basic arithmetic skills sharp and fresh. The calculators mean that instead of honing their (limited) arithmetic knowledge through practice, students are just letting it atrophy and fade. Along with it, they are losing the ability to use their number sense to help them advance to the next level.

That’s what overuse of calculators will do to a math class. What does overuse of AI do to education in general? We are about to find out.