I'm going to write about the same process standard as the last entry; however, this time, I want to focus on the actual indicator, "build new mathematical knowledge through problem solving."
Teaching through problem solving has been a major emphasis in mathematics education over the last (at least) 2 decades - the emphasis on problem solving was there in the 1980 NCTM document. So, it is not necessarily a new idea, but it's not quite clear what this might actually look like in a real classroom. Some people have discussed the three related ideas:
* teaching for problem solving
* teaching about problem solving
* teaching through problem solving
Teaching for problem solving is exemplified by the common textbook organization where students are taught various rules and formulas in a unit, and at the end of the unit is the lesson(s) titled "applications." Students are taught necessary tools, so to speak, and they are given numerous problems for which those tools may be useful.
Teaching about problem solving typically means teaching various problem solving strategies such as guess and check, draw a diagram, look for a simpler problem, make a table, etc. Some textbooks will include a mini-unit on these strategies throughout their textbook, and students are asked to solve problems using the specified strategy.
However, neither approach really produces new mathematical knowledge by solving problems. Teaching through problem solving means students will solve a problem, using only what they have previously learned. Then, by examining their solution strategies, they will generate a new idea/rule/formula. Let's take a look at an example.
In the GPS, students are expected to learn how to determine the area of rectangles and squares by multiplying their dimensions in Grade 3 (M3M4c). Then in Grade 5, students are expected to derive the formulas for calculating the area of parallelograms and triangles (M5M1 b & c). Somewhere in between, students are often asked to find the area of L-shape like the one shown below.
If you ask students to find the area of this shape in many different ways, they may come up with solutions like the ones shown below.
All of these methods will determine the area of the L-shape. However, if you are teaching through problem solving, your real lesson starts once these solution strategies are shared because the goal of the lesson is NOT to determine the area of the L-shape. Rather, you may ask students, "What is common about ALL of these strategies?" One conclusion students may reach is that all of the strategies are somehow using rectangles and squares, shapes for which they already know how to calculate the area. Thus, by discussing that question, students may reach a new understanding that "when we are given an unfamiliar shape, we may still be able to calculate its area by somehow making a familiar shape (or a collection of familiar shapes)."
Your lesson may not stop there. You may want to ask students to sort these strategies - "which strategies are alike?" Often times, students will come up with the following three categories:
* divide the given shape up into several familiar shapes
* cut and re-arrange to make a familiar shape
* make-it-bigger
Thus, students can learn some specific strategies for creating familiar shapes by critically analyzing these strategies.
So, what can we say about teaching through problem solving? One important idea is that the discussion after various solution approaches are shared is the meat of the lesson. That means we must make sure that we leave sufficient amount of time for such discussion. Too often, we see lessons where very little time is left after the last solution is shared. Sometimes this happens because teachers lost track of time as they circulate around the classroom. Other times teachers feel that students need more time to solve the problem. However, I think it is very important for us to remember that the goal is not the answer to the problem. Rather, even if students have not completed their solution, perhaps their incomplete answer may still be sufficient for conducting productive discussion.
Teaching through problem solving is extremely challenging. It requires teachers to have deep understanding of mathematics they are teaching. It also requires teachers to understand their students' mathematical knowledge so that they can anticipate various solution strategies might come up. Furthermore, teachers must have a plan on how to orchestrate the discussion once strategies are shared. Few teachers, if any, can naturally do this; however, it is something teachers can learn, too. Japanese teachers continuously sharpen their craft of mathematics teaching through a process called lesson study. You can learn more about lesson study and also watch some interesting lessons by clicking here.
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