What Every Student Needs to Know for Multiplication (Part 2)

This post continues the Constructing Units team’s discussion about developing composite units with the goal of building children’s multiplicative reasoning. You can read part one here.

In the Towers Task, the teacher uses the child’s understanding of composites as a starting point, and then provides modifications to the original task which encourage opportunities for the child to incorporate different representations and apply reasoning and problem solving. It is important to know that once a child is using composite units, that means they are able to track the containing unit as well as the units that are within the containing unit. For example, Max, when he first began working on the Towers Task, was not able to track both the individual units and the containing unit (the tower). When he was asked for the total number of cubes used to build 6 towers of 3, he counted 1-2-3, 4-5-6, 7-8-9, 10-11-12, but then said he was “stuck,” unable to remember how many sets of 3 he had counted. With experience, he developed the ability to track both individual units and containing units.

When a child has developed the ability to track both the individual and containing units, the teacher can then modify the task for the student. Up to this point, students have been building towers one at a time and bringing them back to the sender, allowing time to reflect on what was built. For the adjustment, the teacher now asks the child to bring back multiple towers during a single trip. For example, they may be asked to bring 5 towers with 3 cubes in each for their first trip. The reason for the adjustment is that the child no longer needs individual trips for them to reflect on the fact that each containing unit (tower) is countable.

Once they have completed this adjusted task, the towers are then hidden. The act of hiding the towers eliminates visible objects that the child may count, prompting them to form mental images of the towers. The child is then asked to build additional towers. For example, 3 towers with 3 cubes each. The teacher should keep the amount of towers in this second group to 2, 3, or 4 when initially presenting the task. The second group of towers can be partially or completely hidden from the child. The choice to hide all, some, or none of the towers in the second group is made based on the teacher’s understanding of the child’s mathematics.

Having the child make two trips provides opportunities for additional goals. The three questions posed in Part 1 (How many towers were built? How many cubes in each tower? How many cubes altogether?)  remain the same but the child’s goals may change. The teacher still asks the question, “How many towers were built?” However, for the student to set the goal of determining total towers they must begin to see towers as countable units themselves and that they can be added with other similar units. Where do we go from here? Click here to go on and read part 3 of this series.

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3 Responses to What Every Student Needs to Know for Multiplication (Part 2)

  1. Kristin, Thanks for the inquiry. We learned the range of towers used varied by the student knowledge upon entering the task with a correlation to their grade level. For the first and second grade students without a developed composite unit, we began with four or five towers of 3 cubes. Some would have a memorized a skip count of 3, 6, 9 (maybe even 12). Our attempt was to provide situations that were beyond any memorized skip count they may have. In doing so they would need to construct a way of counting 3 more while keeping track of the number of towers they had counted. This promoted an understanding for the underlying structure which creates the skip count sequence. Usually, we saw this manifested with the child using three fingers on one hand for the cubes and the fingers on the other hand for the towers. Occasionally we saw students count the segments of their fingers. For example, if given four towers of 3 cubes they would extend four fingers to represent the towers and then count each segment of the finger as substitutes for the cubes.
    Moving to using 6 through 9 towers creates new challenges for students specifically if they are using one hand to count cubes and the other to track towers. They also may experience difficulties when moving beyond 10 towers. Each new difficulty requires a reorganization from the child in the way they count and track cubes and towers.

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