I have started doing some tutoring on the side for one of the college students who helps at my children’s elementary school afterschool program. The young woman [I will call her Joanne] who needed help has impressed me over the years with her dedication to her work, and to the children. Upon seeing her talking to another parent about a hard math test coming up, I let her know I was a math teacher and I’d be happy to help her. We didn’t get together until recently, but working with her for just an hour was fascinating.

Joanne is taking what is essentially an Algebra I class at her college. She has attempted three other math classes and has either dropped them or failed them, and she needs to pass one math class to earn her degree. Her advisor suggested she try a self-paced course, which happened to also be computer-based, using the ALEKS program.

As we began to work on expanding polynomials, I shared a few techniques that seemed to help, at least at that moment. Based on some of the comments she made, I realized that she didn’t really understand what an expression like “2y^6″ meant. If students don’t understand the building blocks, and why they are useful, how can we expect them to make sense of multiplying one polynomial by another? Our assumptions can be risky, and mostly they are risky for the students in our classroom.

The other thing that has stuck with me that Joanne mentioned was about her former math teacher. She sees her high school teacher at her job, and that teacher has chided her about her math struggles, saying” Joanne, you know this stuff! We did all of this together in class.” Again, this is an issue of assumptions (which include mine as well). What did Joanne understand when she studied these ideas the first time? If she did really comprehend all aspects of her work at the time, what makes it so hard the second time around? If nothing else, it points out how rarely anyone is being asked to use some of the skills we emphasize in high school. How can we make the work that we do more relevant?

All of these ideas coalesced in my mind this weekend. I was out in the city, and overheard two people talking about the incredible sales going on. One person was accurately estimating what the sale price of something would be if it were 40% off. They talked through their technique, and reached a correct answer. Then, the person turned to their friend, joking and saying, “Math! It’s finally helping me!” This makes me sad. I’m not quite sure what to do with that feeling, beyond keeping it in my mind every day.