Evidence of Student Learning: Understanding is not an all or nothing phenomenon, but rather develops over time. Rather than thinking about whether a student does or does not understand something, or has specific knowledge, look for evidence of the kind of mental activity that they are engaging in as a result of the activity.
My goal in teaching the array method was to help the students see why other methods of multiplication made sense. Moreover, I hoped that students would be able to add this method to their repertoire of strategies for solving multiplication problems. Unfortunately, due to time, I do not feel that any of the students were able to fully develop a relational understand of the array method, which also hindered their ability to extend and apply this mathematical knowledge. Furthermore, because I was teaching them a new method for multiplication, they were mostly following my instructions rather than solving the problem in ways that made sense to them.
On the other hand, each student seemed to have some understanding of how to connect arrays and place value to multiplication problems and was able to articulate this information in conversations with their peers. I believe that with a follow up lesson, students understanding of the array method would greatly improve and they might even begin connecting it to other multiplication techniques. Below, I have further analyzed each student‘s learning individually:
A: Although A initially seemed reluctant to try a new approach and was set on finding the answer, by the end of the lesson, he did seem to understand the steps we took to solve multiplication problems using the array method. At times, A seemed to get frustrated when classmates were not initially understanding a concept and as a result would check out of the lesson. I struggled to keep him fully engaged, but based on the work I collected, he seemed to take in more than I originally thought. Early in the lesson, A drew a 13 by 6 array to represent 13 x 3. When comparing with his partners during a turn and talk, J pointed out his array was too large, but A said he wanted to keep it that way. However, a little later in the conversation, A adjusted his array to match the multiplication problem. This makes me wonder if giving A more to explore the array tool would have allowed him to understand the method better. At the end of the lesson, A seemed to be struggling to independently compute the last step of the array method. I worked with him one-on-one for a bit, and after talking through the problem together, he seemed to understand. From his work, it also seems that A has difficulty differentiating between addition and multiplication strategies. He first wrote, “40 + 20 = 600,” presumably using the multiplication strategy of adding on the zeros. He then crossed that out and wrote, “40+20=80,” where it seems that he multiplied four by two instead of adding them. Finally, he realized that 40 + 20 = 60. If we had more time, I would have like to give A an additional problem to see if he would be able to do it entirely on his own.
Z: Z appeared to have a working understanding of how to divide a number into its place value components, how to set up the original array, and how to find the area of a rectangle, but he had difficulty putting these pieces together. When he originally drew an array to represent 13 x 3, he placed the numbers inside the array but still correctly boxed off the correct number of blocks. However, after working with J and seeing that his numbers were outside the box, Z re-drew his array and ended up boxing off a 2 x 12 array. After examining his original array with him, he saw that it was correct, but I may have guided him to this point. Instead, I wish I had given Z more time to explore using arrays on his own so that he was able to make sense of them. Later in the lesson, Z re-set up an array to solve 13 x 3, but he drew the line separating the place value before, instead of after, the tenth column. However, this mistake actually showed a lot about his thought process. In the first rectangle, which should have been 3 x 10, Z still wrote 3 x 10 = 30. This presumably shows that Z understood that the first block represented three multiplied by the groups of tens in 13, even though his array did not actually represent this. In the ones place rectangle, which should have been 3 x 3, Z wrote 4 x 3 = 7. Here, instead of realizing that the ones place of 13 was three, he saw that there were four blocks in each row of the array. Although this was not the correct calculation, it shows that Z was using the array to solve the multiplication problem and that he understood how to find the number of blocks in the rectangle. Furthermore, even though Z wrote “x”, he actually added the numbers, which shows that he needs more review in the differences and relationships between multiplication and addition.
J: J originally struggled in setting up the array, specifically with where the numbers should be labeled and how that affects the array, which I think caused him to be frustrated with the rest of the method. Because he is not the only student who struggled with this, I wonder what would have happened if I did not number my boxes but instead just counted them out loud, emphasizing the number of grid blocks I was boxing off. In terms of place value understanding, J appeared to understand the meaning of place value and how to divide a number into its place value components. Furthermore, when working through the guided example of 15 x 5, J explained where these numbers were coming from in the 5 x 5 box. Unfortunately, J messed up drawing his arrays numerous times and I ran out of extra graph paper. I had graph paper that was formatted differently that I gave to him, but I think the difference in formatting threw him off. As a result, J seemed to check out towards the end of the lesson.
K: K was eager to participate throughout the whole lesson, and seemed to have a decent understanding of the array method. Although she helped in originally setting up the 15 x 5 array, she then struggled in understanding how that array represented the Thanksgiving problem. After talking with her peers as well as with me, it seemed K had a better understanding, but I was not convinced she fully understood. Although she was able to correctly solve 15 x 4 using the array method, she struggled with 13 x 3. Like Z (I think they may have worked together), K also drew the place value line on the 13 x 3 array on the left of the tenth column, even though she correctly divided 13 into the tens and the ones place below the array. Like Z, she also wrote 3 x 10 = 30, showing that she understood that three must first be multiplied by ten. In the ones box, she wrote 4 x 3 = 12. [She originally wrote 7, but changed it to 12.] She made the same mistake as Z in that her incorrect array led her to use four instead of three. Under the array she wrote, 30 x 7 = 107 [or it might have been 30 + 7 = 107]. Here, it looks like she combined multiplication and addition strategies and added seven to zero and then seven to three. Even though she did not do the math correctly, K understood that she had to do something with the two numbers to get the number of blocks in the entire rectangle. Below this, she then used the traditional method to solve the problem and arrive at 39, the correct answer. However, she did not do any further work after seeing her answers for the two methods were not the same.
C: C was able to set up the array and divide it into its place value components, but he seemed to not know what to do after that. In terms of setting up the array, C appeared to understand how the array represented the multiplication problem and even explain it to K when she was confused. Furthermore, with the later problems, he made up his own stories to represent the multiplication problem I gave. This shows that C has a pretty good understanding of what real world situations require multiplication (although I think it distracted him from other parts of the lesson.) Initially, C was confused by the smaller multiplication problems we solved (5 x 10 and 5 x 5 for the 5 x 15 problem), specifically where each set of numbers came from, but after K explained it to him, he was able to explain it back to the rest of the group. In the 13 x 3 problem, he set up the array, but then solved the multiplication problem using the standard algorithm. For the 13 x 4 problem (he ran out of room on the graph paper to fit 15 x 4), he set it up correctly and found the area of the smaller rectangles. However, instead of adding these numbers together, he then did 13 x 4 using the standard method.
Burns, M. (2005). Looking at how students reason. Educational Leadership, November.