Let’s look together at a line of people at the checkout counter of a supermarket. It may seem surprising at first sight, but no one seems to panic as they approach the salesclerk! No one even seems to fret! They have no worry at all about the order in which their purchases are scanned or whether their discount coupons are scanned each after the discounted item or all together at the end or somewhere in between. They understand that it doesn’t make any difference if they pay for the sandwiches and the chicken wings at the deli and for the prescriptions at the pharmacy instead of paying for everything at the checkout counter along with their other purchases. This lady is leaving the checkout counter. Let’s ask her a few questions.
Me: So your name is Mary. You seem to clip coupons quite efficiently. How much did you save today with those coupons?
Mary: Let’s see. I had a $2.00 coupon on a pound of cheese and three $1.00 coupons on soft drinks. I saved $5.00. No, more than that. My sales slip reminds me here that I had a $3.00 coupon on a box of cereal. I also used a $2.00 coupon at the pharmacy on some vitamins that I bought along with my prescriptions. In all, I saved $10.00. My coupons are paying for this nice big piece of meat.
Me: Congratulations. I seem to remember that the second item you scanned was a jar of pickles. I saw you give it back at the end. Why did you change your mind?
Mary: I just had two twenty-dollar bills. I didn’t want to spend more than $40.00. At the end, I noticed my bill was going to be almost $43.00. So I handed back that $4.00 jar of pickles. See, I just had to pay $38.95.
Me: Thank you, Mary. You’ve been very helpful.
The scene and the imaginary conversation demonstrate a good amount of mathematical understanding on the part of those shoppers. We all agree, I guess, that it is intuitive knowledge, not the result of math classes. If the shoppers don’t panic as they stand in line, it is not because they suddenly remember that the commutative property of addition tells them that the total cost of their purchases is not affected by the order in which their items are scanned. They don’t need deep reflections on grouping to make the choice of paying their deli purchases at the deli or at the checkout counter. In fact, those shoppers know more than the commutative and associative properties could teach them, as these apply to addition only, not subtraction.
Focusing on these shoppers helps us stare intuitive mathematical knowledge in the eyes and convinces us of its concrete, practical reality. We can then ask ourselves how, on this and on entirely different topics, we can ground our teaching strategies on establishing a strong connection between that preexisting knowledge and the topics we want our students to learn. This may imply not just a recasting of the topics but a different understanding of what it means to know. The Common Core State Standards seek to be an antidote to knowledge that is “a mile wide and an inch deep.” We hope Teaching to Intuition can make a contribution to that vitally important goal.