Individualized Math Strategies

Problem being examined: Join Change Unknown:

student name finds 12 rocks on the playground. Everyone comes and wants to play with his rocks. How many more rocks does he need to find to have 21 rocks altogether?

This problem was presented to my small group of math students that I am working with this semester in my internship. It was one of two problems they were to solve during their independent practice. The problem was glued on a large sheet of construction paper, that would act as their, “works space,” providing them plenty of room to demonstrate a strategy that works best for them. One or two of my students occupy just a small area to solve their word problems, but for some, this piece of construction paper becomes an incredible space for them to break the problem down, or a map to represent the numbers using multiple strategies and trials, and more specifically, a space to solve!

One little girl, “Madeline,” amazed me by turning in a true treasure map of strategies that would eventually lead her to the gold (correct answer). I was sitting next to her watching this work unfold. Her first stop lead her to a traditional multiplication algorithm (which just happened to be something they were working on in class). Instead of letting her continue to solve the word problem this way, I asked her why she was choosing to solve the problem using the, “window” multiplication method. She responded that she thought that is how I wanted her to solve the problem. I gave her a huge smile, and reassured her that this is her work, and SHE should choose a way to solve the word problem that made sense to the problem itself and to her. She quickly crossed this algorithm out, and moved to an “inventive” direct model strategy. She began by drawing one stick person (which I assumed to be the student in the problem), and then started drawing five more people, stopped, looked at the problem again and made a self-prompted self-correction. She crossed out the

sentence, “Everyone comes and wants to play with his rocks.” Then crossed out her picture. She moved right along her treasure map, and started drawing circles. Madeline again did not resort to the algorithm, which I at that time noticed one of her peers was writing (21 – 12 = 9). She drew almost a diagram of circles, drawing lines to separate sections, and then writing the numbers each group represented beside them. I noticed then that she got frustrated when she realized she drew too many circles, but I at that time did not realize what that diagram was being drawn to represent. She crossed out that picture and moved to her final picture. She first drew 12 rocks and wrote the number beside them. She drew a line (similar to her diagram above) and counted up to twenty-one drawing one rock as she said each number. She finished by counting the rocks below the line and writing, “9 rocks.” This again represents a direct model utilizing counting up by one to solve for the answer. While she did not use the first diagram of circles, after looking at her work, I realized that incidentally she had indeed solved for the correct answer using that

diagram as well!

While this student is still in the first stage of solving word problems using direct modeling, as most of my students are, I feel through her self-corrections, and her ability to pair the whole numbers in the word problem with her pictures, she is on the brink of utilizing more inventive strategies to solve whole-number computation. She also drew her objects within a diagram-like formation, which is not demonstrated by her other peers, and is outside the norm of direct modeling. Her counters were drawn a way that represents the steps and what she needs to accomplish in order to solve for the answer. I feel when her confidence in solving increases, she will begin to develop strategies that break the problem down and demonstrate a solid understanding of the operation and its properties.

One method Madeline could have used to solve for this word problem, utilizing an invented strategy, would be to use her knowledge of whole numbers and subtraction. The student could take away the one from 21, which gives me 20 (starting with the largest whole number) and then mentally recall that 20 – 12 is 8, adding on that missing one gives me 9 so I know the answer is 9 rocks. This combines mental math with whole number knowledge. Another inventive strategy that would combine her use of mental math, while also utilizing a more manipulative-based way of solving, (using her direct model background knowledge) would be to break the numbers down using base-ten concepts. Starting again with the larger number, 21, the student would represent the number using two ten sticks and one unit square. The smaller number, 12, would be represented with one ten stick and two unit squares. This usesthe direct model concept, but instead of manipulatives representing the objects and the action, they represent the numbers. If the student does not have base-ten manipulatives, a picture can be draw instead. The student would first draw each whole number using the base-ten concept. The student would draw the two sides as separate, but then solve by combining, and taking away evenly to see how many were left. First a ten stick from each side is taken away (the take away has to be even), and then break apart the second ten stick into units. Now there are two units left from the 12 side and eleven units left from the 21 side. I take away my even sets again, removing two units from each side. The answer will be the number of units that are left.