I was reading Dan Meyer’s latest post Teaching for Tricks or Sense Making and felt compelled to post for a few reasons. One, I’ve tackled the topic of how to teach zero and negative exponents before and I’ve just taught this very topic in three classes. But two, it relates to why I haven’t posted in a year.

Last school year I started teaching a new-to-me course called Theory of Knowledge, which is basically an epistemology course that asks how we know the things we claim to know. This course has been very rewarding to teach but has also taken up a lot of time and thought-space for me, which is why I haven’t blogged at all.

The current area of knowledge we’re covering is mathematics and one of the questions that arose was what is the point of learning things beyond basic arithmetic. If the majority of the students will never again solve a quadratic equation, why do we spend so much time on factorization, completing the square, the quadratic formula, forms of a quadratic function, etc. In essence, what’s the point of schoolmath?

A student (bless her) said that we study it because it teaches us to think and to problem solve. And my response to her was, “But does it?” I think that is the idealized claim (and certainly that is the impetus behind the MTBoS’ Nix the Tricks), but I’m not 100% convinced that we’re achieving that. I think that despite our best efforts many students simply follow rules or procedures that they only half understand. I saw that play out recently.

This year in my advanced 8th grade classes, we covered the rules of exponents (in an algebra-only context) and scientific notation. In a discovery-style lesson on scientific notation, in one class the students came up with the idea that if 10 raised to the *n*th power was a 1 with *n* zeros behind it, then 10 to the 0th power was just a 1. They also managed to get to the idea that 10 to the -1 should be one-tenth. I felt like in that class that I had taught for sense-making for small numbers in scientific notation, whereas in the other class I had merely taught a trick for it (in the interest of time; they had also had the discovery-style lesson on scientific notation for large numbers).

On the test I put two sense-making questions. The first asked them to find (-1)^{97} (no calculator) and explain their answer. A majority were able to give a completely correct justification. But I also had a fair amount tell me that the answer was -97, two tell me it was -1 x 10 ^{97}, and one tell me it was (-1)^{100} – (-1)^{3}. What mistakes of thinking are here?

The last question asked the students to do a mini-investigation. The first part asked them to calculate 2^{5}, 2^{4}, 2^{3}, 2^{2}, 2^{1}, then give the pattern as you go down in exponents, then use the pattern to find 2^{0} and 2^{-1}. The number of students who could actually *use* their pattern was limited, across both the trick-learning and the sense-making class. From the sense-making class, despite having seen that 10^{0}=1, many students still thought that 2^{0}=0. Among students who did get that part, lots of them then thought that 2^{-1}=0.2! In one particularly egregious case, a student told me that 2^{-1}=0.2, 2^{0}=2, 2^{1}=20, 2^{2}=200, 2^{3}=2000, etc. In other words, there was no sense-making at all! In fact, our class-wide explanation of how the powers of 10 worked had primed some students to *not *think about how powers of 2 might work, or how any power actually works.

I’m still excited about this question and can’t wait to go over the test with them to talk about how we should deal with pattern finding and its application, how to critically think and get over our biases of what we *think* the answer should be, and above all to prove to them that yes, shockingly, any number raised to the power of 0 is 1, by asking them to recall the rules for multiplication and division of exponents.

Which brings me back to Theory of Knowledge, because knowing in mathematics starts with a conjecture (I recognize a pattern that it seems that any number to the power of 0 is 1), followed by a rigorous proof using deductive logic to verify that it’s always the case. Using tricks robs students of what my student says the actual purpose of math is — to learn to think logically!