I accidentally created an online argument between myself and other passionate educators on a Jo Boaler dedicated Facebook page. After countless hours of endless research and sending messages to math educators I admire, I still had a question that could not seem to be answered. In my perpetual attempt to improve my craft, I innocently asked how to make those “procedure teaching days” meaningful for students who had prior knowledge. So, for example, if we have students who go to Kumon, Russian Math, or even have access to a private tutor and are taught lots of procedures, how can we challenge them on days we are having the rest of the class engage with them? Many of these students feel the discovery piece is a waste of their time since they already know the end result. As often as I remind students that procedures without a solid conceptual foundation are weak, there is still push back and I was looking for a way to engage them, not for my ego, but for their betterment. Furthermore, if a teaching goal is still for students to be able to perform algorithms to help them “Look for and make use of structure,” what is the best way to make those days of instruction inspirational for everyone?
My post got a lot of comments, and not all were kind. Disgusted responses such as, “You should never teach your students procedures, what are you even doing as a member on this Facebook page” were peppered in along with, “Following, if you find out, please tell me.” Ok, I am exaggerating the first statement, but only a little. The message was underlying and clear. Although my post was not about the straight teaching of procedures, it got me wondering:
Is it true, are procedures a dirty word in our current math classrooms?
Now listen, I am all about trying to have students discover and conceptualize before utilizing any procedure. A circle unit my students recently completed as a review is a great example. Students measured the distance around a circular object and then across the diameter of it to provide them an opportunity to discover the pi ratio. Before the lesson, I asked them what they knew about pi. They responded that it was 3.145…and used in circles. Many commented on “pi day” celebrations of the past. None knew it was a ratio. This was a small example of procedure without substance.
For the area of the circle, students studied images of a circle being sliced up into wedges and slowly transforming into a rectangle, which they were then able to compare to the area formula for a rectangle, hence, discovering the formula for the area of a circle.
For circumference, students engaged in Rolling Tires 3-Act Lesson by Andrew Stadel.
However, after these periods of discovery are over, students still apply the formulas in and out of context with the procedure. The conceptual piece is enhanced by the algorithm, and vice versa. My critics questioned my inclusion of this piece of teaching, so I have to question it too. However, how will isolated discovery without practice allow concepts to enter the long-term storage area of the brain? Aren’t both needed? I think yes.
I have researched (through the printed word, professional development courses, and on-line videos of educators I admire). The same message delivered in different ways is that the algorithms of the past are not meant to magically disappear. “There is no new math” is a popular phrase. In my current state of teaching, I agree. Algorithms are wonderful, important, and revolutionary methods. It is not wrong to know how to use algorithms, but it is not productive to use them and have no idea why they work. Shouldn’t we have both?
I really want to know.