Ahh, such beautiful work is shown above- or is it? Today I put up a basic addition problem (basic for my older students) and this is what I saw. These are all examples of great work correct? Early in my career I would say yes, now I take pause to consider other things.

The original problem looked like this:

I know I didn’t need to put in commas or align it horizontally, but I wanted to see what my students would do when I simply stated “Here is your warm-up problem.” No questions were asked other than “Do I need to write that down?” After 2 minutes of work time (and even that was too long, but I gave it anyway) I asked students what I needed to do. “Put 6831 on top, and 1897 on bottom Mr. A.” Students took my horizontal representation and immediately changed it into a vertical one to get down and dirty with the math.

My board now had this on it:

“Now what do I do?” “One and seven is eight, three and nine is twelve.” So I wrote:

“**NO Mr. A! You have to carry the one.**” “Carry the one? What? Where am I taking the one?” They look at me weird first, but then realize that I am trying to get them to explain their process and thinking. The answer I get is “Put it above the 8.” Throughout this whole process, there is no mention of place value or why they start with the ones.

So I show them a few different ways to also do the problem:

And… silence. No shining light from the heavens or angels singing. They sit there and look at what I did. This is the hardest part for me, being quiet. I stand there and let them think about the problem. Here is the catch, I have to be sure that they are thinking about the problem and not checking out mentally. Doing a walk-through while students are using this time, every student was analyzing my problems and trying to make sense of them (I was pretty happy about that). The place value was the easiest for students to understand, the equivalent expressions took a while (and some are still sketchy on it, but we’ll get there).

Like I tell my students, there is no “best way” to do math. Find the way that connects to you, makes sense and is explainable. Remember why your method work so you can be flexible and use it in a variety of ways. In this day and age of my Math teaching, there is no “standard algorithm” that fits every student.

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But do they know what they are doing with the numerals when using the standard algorithm, or doesn’t it really matter so long as they have faith in it? And even if they know what they are doing is it always necessary to be able to explain it to others? I am in a permanent two-minds state on this, but I remember well my trouble in following the fast fourier transform algorithm, reaching a state where I could explain it to my students, most of whom got it, but none of whom would be able to explain it. Since the thing was invented to enable comouters to calculate the fourier transform quickly that didn’t matter.

” Explain your reasoning” is quite different.