# Talk Some Sense into It #MTBoS30 12 of 30

People joke with me and say, “You must do calculations in your head for fun.” My instinct is to deny this accusation, but it probably is true. Math educators run through numbers all the time. We play numbers games in our head like would you rather, without prompt or encouragement to do so. I see a fraction that *could* be written in different way, so I try. I see a complex shape that *can* be broken into parts, so I do. These instincts were fueled by years or even decades of me talking numbers in my head.

I should never assume that this is normal, even though it is one of my life’s missions to make it so for the general population. The CCSS Standards for Mathematical Practice says to make sense of problems and persevere in solving them (among other things). This idea of having a sense for numbers has been discussed by some awesome educators like Fawn Nguyen, and Sadie Estrella, and it’s been the hot topic of journalists and bloggers alike. I’m in the camp that says, “Feeling comfortable with manipulation of numbers directly supports comfort with manipulating the abstract.”

So today we were talking circles, and proportions of circles. Instead of just giving a formula…

…we developed our own.

It started with a spreadsheet.

then a geogebra applet

then into paper practice.

Students shouldn’t be given a formula, and then be expected to make sense of the abstract values without having developed a sense for the concrete values.

Math educators need to talk numbers with students daily. It just makes sense.

Posted on May 21, 2014, in Common Core, Good Teaching, MTBoS30 and tagged Common Core, MTBoS30, number sense. Bookmark the permalink. 1 Comment.

So true. There’s an interesting tension here: we also want students, eventually, to “leave the problem in variables as long as possible.” That way, they write more general expressions and can see relationships in the symbols. But you’re right—it will all get lost without starting with concrete numbers.

I recently did a calculus lesson about Mercator projections (see bestcase.wordpress.com) in which students had to think about the shape of 1°-square regions on the Earth’s surface. I decided (wisely, it turned out) to go all concrete on them and ask how many kilometers a degree of latitude was. Unsurprisingly, they had never heard that the meter was designed so that there are 10,000,000 meters from the equator to the pole, so it was rewarding for them to take 10,000 km and divide it by 90° to get 111 km per degree. More importantly, we could then talk about everything in terms of that 111 km. As in, at what latitude is that 1° box 111 km by 55.5 km? That is, where does it have a 2:1 aspect ratio?

In this case, the concrete numbers helped them reason their way to the symbolic representation. But in other cases, I see students substituting in numbers when it’s unnecessary, too early, so that the expression loses its meaning. I wonder how our inner mathematician decides when being concrete is more efficient and enlightening, and when it’s better to be abstract.