sharing a whole number of objects with a whole number of people (5÷7)

the equivalence of a division problem and a fraction (7/8 = 7÷8)

finding a fractional group of a whole number (2/3 of 12)

finding a whole number group of a fraction ( 5 groups of 3/7)

how many groups of a fraction go into a whole number ( 12 ÷ 1/4)

how do you divide a fraction into smaller units (1/4 ÷ 5)

I put this up on the board (without the colors or the words, those came out of the discussion):

S: All the numbers are even. Oops, no they're not.

S: They all have fractions in them...except for 4 · 2

S: A lot of the fractions have 1's.

S: Wait a second... aren't 1/3 ÷ 4 and 1/3 · 1/4 equal to the same thing? OHHHH, I get it!

At this point, I realized, once again, that their minds just don't work like my mind. They don't see what I think they should (didn't we just spend 20 minutes figuring out that these problems were equivalent?), and more importantly, they don't think the way I want them to... yet. The pattern we are looking for here is not adding squares to a figure, or a decreasing sequence. We are not comparing obtuse triangles and acute triangles. We are looking for a pattern much more abstract, involving operations and numerical relationships. And learning to think about this kind of pattern is one of the biggest reasons we teach mathematics (or should be!). I was pretty stoked.

So we continued to discuss, and we found that the labels, especially the word "divisor", came in handy here, and one student suggested that you could get the answer to a division problem by "multiplying by the reciprocal of the divisor." I was thrilled. We had arrived at the ageless trope, bane of anyone who dreads top-down, teacher driven math instruction, through the work of the students. I wanted them to get there; it seems a useful tool, especially in later algebra classes. But I wanted to reap the fruit of having them think it out as well. And I will do it the same way next year.

HOWEVER..

The next day I gave them some problems to practice the skill.

S: Can you remind me how to do this again?

S: Yeah, do you flip the front one or the back one?

S: Don't you flip both?

REALLY?! And of course, if your classroom is like mine, and we are completely honest, this happens time and again year after year. I want the students to understand math and always grow their mathematical thinking, their map of the math landscape. And I want them to continually add to their mathematical toolbox, have computational competence. I want both. And I have always believed as many of us do, that introducing the computational procedures at the start ends the thinking. So we explore first, and the computation evolves from what we discover. THEREFORE, the exploration and understanding must enhance the student's competence with computation. If they understand it, they will remember it better and use it with more facility.

I am not so sure anymore. Why does it keep happening that we derive the formula for the area of triangle over 3 days of work, and half the class needs a reminder the next day?

WHAT'S GOING ON

**1) The connection between Understanding and Recall is not straightforward.**We assume that understanding why will reinforce remembering how, but I am not sure it is that direct a relationship. I suspect those two parts of the brain communicate only loosely with each other. And I suspect that we make important, but perhaps erroneous and misconstrued assumptions about how these work, like assuming that building understanding is enough to give students lasting skills.

**2) The symbols are different than the math.**That means the students often don't see the same relationships in the symbols that they do in the math or the models. Consequently, the students often aren't thinking of the symbols the way we think they are. We want students to grow their mathematical thinking and understanding and experience with mathematical ideas. We also want them to gain operational knowledge for using mathematical symbols, and again I think the connection is much less direct than we imagine. I know that if I ask my students to show me 3/5 of 60, I get this:

So I think we need to give more nuanced thought to the connection between mathematical thinking and remembered procedures, the purpose and role of each and the nature of how they interact with and reinforce each other... or don't.