I have students in an enriched class that demand for me to give them more practice. I tell them that we practice mathematics with daily class activities. They don’t want practice, they want repeated practice. They are accustomed to receiving repeatable drills to cement understandings.
I have learned to compromise with this demand. I do believe there is a place for basic skills training in mathematics, and would raise an eyebrow at anyone who claims these unnecessary. I do, however, also believe that the heart of mathematics is problem posing, problem framing, and problem solving.
Here is how I’ve infused an ounce of conceptualization into regular drills. (I use this for both practice in a large group discussion, small group rotation format, take home work, as well as unit exams.)
The work begins like many math classrooms with a set of problems to do. In this post, the topic at hand is solving equations (at the Grade 9 level).
I’ll give ten or so to show the possible variety in structures, and then begin to ask questions that allow students to think deeper about the rules they just employed. Most of these questions focus on flexible use and mathematical communication.
Here’s a question from my most recent unit exam on solving equations:
The question then reads:
Fill in the blank with the number that makes this equation as simple as possible. Explain your choice.
Once you’ve explained your choice, go ahead and solve the equation. Show all work.
The results were fantastic. It was excellent for me, as a teacher worried in skill development as well as deep, conceptual growth, to see that these students were grappling on a deep level with the content when probed to do so. I was assuming that many students might choose “3” to match the denominators on the left-hand side. This scared me, because I felt that I was baiting my students into mistakes. Turns out, not a single student responded with “3”. The most popular responses were “2” (foreseeing the first inverse operation), “6” (choosing a LCM of all denominators present), “5” (fully simplifying the fifths), and “1” (assuming that eliminating fractions is always easiest).
The exam questions are a nice break from traditional assessment while still affording the convenience and balance of a pencil and paper test. My favourite format for these conceptual drills is a small group jigsaw where each group answers, explains, and rationalizes their actions to the larger group. It sparks great discussion.
Here’s a question used in the unit on solving equations:
The question then reads:
Change a single digit from the equation above to make the problem as simple as possible. Explain why you made the choice, and then proceed to solve the equation. Show all work.
Popular choices include changing the “2” to a “1” and shifting the “5” to a “6”. These moves both have ample justification and spark great conversations. Eventually the topic of fractions came up, and students said that they would like to avoid them altogether. That led me to the natural extension:
Is there a number that we can replace “2” with to avoid fractions altogether? How many of these numbers exist? How can we find them?
The discussion skyrocketed from there.
It causes me pause to think about why discussions like these don’t happen more often. Is it a time issue? Do teachers see them as wastes of time? Do teachers struggle with the dimensions of problem posing necessary to see beautiful math staring them right in the face? Is it downright confusion of the purpose of mathematics?
**TANGENT: I think teachers don’t practice looking for mathematics. We waste our time trying to appear mathematical by partaking in various stereotypical mathematical whimsies such as an undue infatuation with Pi day and the obligatory kudos to binary clocks. There is more to mathematics than surface niceties.
It is one thing to preach balance but to continually teach at the poles. One day we work on a task and “construct” mathematical knowledge, and the next we “lecture” and “practice”. Learning doesn’t operate on this notion of average–flip-flopping will only confuse students. We need to develop a curriculum and supporting pedagogy that lives between the two worlds at the same time. Procedural and conceptual are not nearly as mutually exclusive as they are mutually dependent.
PS. For another foray into this conceptualizing of drills see David Coffey’s worksheet adaptation.