You have probably noticed that your child’s math classes are unlike the ones you attended as a child. Of course, the world in which we live has changed too, and the changes in math class are reflective of these larger changes. When I was a grade 12 student, my classmates and I found it hilarious that our math teacher kept his calculator in the breast pocket of his suit. What could be nerdier than that? Nowadays, students as young as twelve have more powerful calculators in their pockets all the time (and would probably sleep with them under their pillows if we didn’t insist on phone-free time at night). Today, devices that are capable of complex computation are ubiquitous, and we must change how and what we teach in response.
Problem-solving has always been a feature of math instruction, but in order to understand how and why math instruction has changed, we need to think of problem-solving as a process, broken down into its constituent parts. The first stage is to formulate the question. Once the question has been posed, there comes the stage in which students select the appropriate operation(s) or model(s) for the problem. Students then have to do the computations. The final stage is to reflect on what the answer means and whether the final answer makes sense within the context of the problem.
These steps were evident in my high school math classes, although I wasn’t necessarily engaged in each phase. In most of the textbooks I encountered as a student, the question had always been formulated for me. Similarly, I was generally told what kind of math to apply to the situation. For example, the textbook might feature a picture with a graph superimposed on it, or a diagram labelled with the relevant details of the problem. Once I had completed the computations, reflecting on my answer consisted of checking the answer in the back of the book and the only real reflection happened during tests when there wasn’t an answer key available. Because so much of the instruction was geared towards computation, I was never explicitly taught how to recognize situations to which math could be applied, nor how to determine how reasonable my findings were in the absence of an answer key. Now that the computation phase of problem-solving is more easily outsourced than ever before, we can devote more time developing skills in formulating questions and reflecting on answers. Dan Meyer explains this very well in his 2010 Ted Talk.
When problem-solving skills take precedence over computation skills, there is more space to use math for inquiry. Rather than learning math skills apart from their work in other subjects, students can begin to employ their math skills in authentic and interesting situations, using it to further their inquiry into other areas. For example, rather than giving my students a series of word problems involving addition, subtraction, multiplication, and division, I asked my students, “Can Canada’s forests be the lungs of the world?”, which connected to their science unit. In order to formulate the question, they had to brainstorm what kinds of data or facts they would need, and they then set out to find them. In so doing, students learned how to formulate questions, and search for and identify relevant information. Once they had data about how many trees there are in Canada and how much oxygen a tree can produce, they began doing calculations. At this point, students who needed practice with the computation took a break from the larger task to work on some more basic exercises. Not only were they more motivated to develop the skills, but the skills fit within an authentic context that made them more meaningful.
Shifting the focus from computation to the other phases of problem-solving also creates more space for learning math by inquiry. Rather than teaching students rules or algorithms, students can investigate patterns in order to discover the rules for themselves. Instead of learning the rule by rote, students discover the reasons behind the rule, resulting in a deeper level of understanding that can be transferred to novel situations. For example, knowing math facts, such as 2 x 3 = 6, is useful; however, knowing how 2 and 3 combine to make 6 and knowing that there are other ways to combine numbers to make 6, offers students more options when they get to more sophisticated exercises, from equivalent ratios and fractions to factoring polynomials in order to solve quadratic equations. I recently saw a great video in which a math educator explains this very well.
Computation will always be an important element in the problem-solving process and, consequently, in math curricula; however, computation skills are no longer the primary goal. By placing computational skills within the broader context of problem-solving, and by devoting time to developing the skills that precede and follow computation in the problem-solving process, students will develop a deeper and richer knowledge of math. Similarly, their problem-solving skills will be broader, stronger, and more transferrable between disciplines.