P1a - Teaching THROUGH problem solving

P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving.

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.

P1 - Technology and manipulatives

P1. Students will solve problems (using appropriate technology).

This process standard includes the parenthetical statement, "using appropriate technology." Although the use of computers and other technologies are generally accepted by teachers and general public, the use of hand-held calculators, particularly in elementary classrooms, continue to be controversial. On the other hand, most people seem to endorse the use of concrete materials (manipulatives) in elementary school classrooms. So, what's the difference between technology (not just calculators) and manipulatives? Although some people may say both technology and manipulatives are both simply learning tools, I believe there is a fundamental difference in their nature.

Let's consider how long division algorithm may be taught using a very commonly found manipulatives, base-10 blocks. A typical instructional sequence will start with problems where students are asked to solve sharing problems using base-10 blocks. As students continue to solve these problems using base-10 blocks, teachers may encourage students to start drawing the picture of blocks and modify the picture as blocks are manipulated. Eventually, teachers will ask students to simply draw pictures of what they would do with base-10 blocks without actually working with the blocks. As students continue solving problems by drawing pictures, teachers will encourage students to use numerals to record the process - instead of drawing 4 flats (hundreds), students can simply write "4" under the heading of "flats (or hundreds)." Eventually students can organize the record using the familiar long-division notation. [See, for example, two activities Sharing Base-10 Blocks and Doing and Recording on my university web page: http://science.kennesaw.edu/~twatanab/.]

Now, here is a calculator game that may help students to develop mathematical thinking. It is called "NIM with Calculator." It is a 2-player game. Clear the calculator so that "0" is shown in the display. Players take turn adding 1, 2, or 3. The winner is the player who gets the sum of 21 after his/her turn. It is a very simple game and children do not have problem remembering the rules. When students become comfortable with the game, you may want to ask if there is a "winning strategy" for either player - the player who goes first or the player who goes second. It turns out there is a winning strategy for the player who goes first - that is, if you know the strategy, you can be 100 % sure that you will win if you go first. I encourage you to figure out the strategy.

Once you figure out the strategy, an interesting extension question is how you may be able to figure out the winning strategy if you change the goal number - for example, you can make the player who gets the sum of 24 to be the winner. You can then determine the relationship between the goal number and the winning strategy (in particular, the first number you must enter).

All of these activities - base-10 block division activities and NIM with Calculator - may be appropriate in elementary classrooms at the appropriate time. However, the roles these tools (base-10 blocks and calculators) play are very different in nature. With base-10 blocks, teachers ultimate goal is to help their students go beyond base-10 blocks. That's the reason teachers will start asking children to draw base-10 blocks or imagine what they might do with base-10 blocks. It is possible with base-10 blocks, and other manipulatives, for children to imagine what they would do and what the results of their actions might look like. Thus, they can examine the effects of their actions without having to use manipulatives. Of course, it is essential that children have opportunities to physically manipulate the blocks BEFORE they start imagining what they might do or what the results of their actions might look like.

On the other hand, calculators and other technological tools are often suited for such an instructional step. Children might be able to imagine which calculator keys to push, but it isn't always possible for students to imagine what the results of their actions may look like - in other words, they don't always know what they will see after they hit the "=" key. Similar point can be made with graphing calculators, dynamic geometry software, or productivity software like spreadsheet.

Teachers should be aware of this difference in the nature of these tools. Manipulatives are useful tools for students to make sense of different processes so that they don't have to use manipulatives to figure out the results. On the other hand, technological tools are to be used to help students think about mathematical relationships that might exist among different numbers and shapes in the problem context. We are not interested in "weaning" students from technology - we want our students to become better at using technology. Judicious use of technology requires us to pay attention to this difference.