Tools: Materials or resources used to represent mathematical ideas and problem situations, such as paper & pencil, manipulatives, calculators, computers, symbols, metaphors or stories or examples. Students use tools to solve or explore mathematical tasks.


The main tools used in this lesson were graph paper, markers, and the arrays themselves. I gave each students one marker, but in reality, I should have given them at least two markers each so they could color code the place value components on their grids. Realizing this part way through the lesson, I had students switch markers if they wanted so they could at least write the multiplication problems in a different color. Also, markers may not have been the best choice because the students were unable to erase their mistakes.

The 1/2 inch graph paper we used to draw one-digit by two-digit arrays
The 1/2 inch graph paper we used to draw one-digit by two-digit arrays

The purpose of the graph paper was to represent the multiplication problem in a transparent way by showing students that they were actually finding how many boxes were contained within the rectangle. For the most part, students seemed comfortable with using the graph paper. However, a couple students did not position their arrays in the top corner of the paper and thus ran out of room. Although I did not do this during the lesson, it would have been a good idea to ask these students what their array now represented. In doing this, I would have been able to see if they understood the concepts even though they were unable to represent the problem I wanted.

Furthermore, several of the students drew their arrays incorrectly because they numbered the boxes inside the rectangle but then did not count the boxes with numbers as part of the array. Thus, they had one extra row and one extra column.

Because K drew the place value line on the incorrect side of the 10th column, the ones rectangle became 4 x 3 instead of 3 x 3.
Because K drew the place value line on the incorrect side of the 10th column, the ones rectangle became 4 x 3 instead of 3 x 3.

Others drew the place value line on the incorrect side of the tenth column, which resulted in the incorrect number of ones and the tens place being a group of nine rather than ten. With these students, I was not sure if their incorrect arrays meant they did not understand the task and the concepts behind it or that they just drew the arrays incorrectly. However, these errors often affected future calculations. Unfortunately, we did not have enough time to fully examine these students’ responses.

In general, I wish I had given students more time to explore using the arrays. As Hiebert (1997) explains, “tools play a kind of intermediary role in the development of meaning. In order for students to use tools wisely, they need to develop meaning for the tools. Once some meaning is established, the tools can be used to solve problems and to help students develop meaning for others things” (p. 55). Because using arrays was a new concept for students, I do not think any of them fully developed “meaning for the tools.” As a result, using arrays was a much less effective strategy than it would have been if students had been able to develop a relational understanding.

References:
Class notes.
Hiebert, et al (1997). Mathematical Tools as Learning Supports. In Making Sense: Teaching and learning mathematics with understanding. Portsmouth, ME: Heinemann. Chapters 5.
Skemp, R.R. (1976). Relational Understanding and Instrumental Understanding. Mathematics Teaching, 77, 20–26.

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