Monday, April 6, 2009

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:]

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.

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Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.