I was able to be in the class four days this week: every day but Wednesday (because they had a field trip that afternoon).
Much of what students worked on this week continued to be review of multiplicative comparison, and on Friday they moved on to “multiplying by 10s, 100s, and 1000s.
On Tuesday, I led the Teacher Time center, based on a multiplicative comparison worksheet. My teacher mentioned she would like it if we had a chance to use “n” in place of the unknown number. The first two questions on the worksheet gave students a multiplicative comparison situation, that ended with students doing an additive comparison. This shift seemed to be difficult for many students! It was interesting to see that many students “got” that the first two questions were multiplication – but most also struggled to flexibly move between multiplicative comparison and additive comparison.
The first question asked something along the lines of: A fire dragon burned 8 times as many huts as a fire beetle. The fire beetle only burned 3 huts. How many more huts did the fire dragon burn than the fire beetle?
First Thought. A few struggled with understanding that the first question started with 9 groups of 3. These students struggled to set the model up correctly and so I tried to give them opportunities to use tiles as physical manipulatives to understand the situation. Some students interchanged 9 groups of 3 with 3 groups of 9, which makes sense because multiplication is commutative. However, in the situation, (1 group of 3 + 8 groups of 3) will result in a different final answer (24 huts – 3 huts = 21 huts) than (1 group of 9 + 2 groups of 9) which results in (18 huts – 9 huts = 9 huts).
Second Thought. Most students found their answer by multiplying 9×3 = 27 huts. Only a few students really noticed or understood that 27 huts was not the answer, but that, instead, “how many more huts did the dragon burn than the beetle” meant to subtract. It seemed that shifting from multiplicative comparison to additive comparison was difficult. I tried to support the students by focusing their attention on the final question and asking them what they thought it meant, but many students seemed to really be mentally blocked from using subtraction.
Third Thought. I tried to work in the “n” with the first two groups, but I think even in a small group there is a wide range of understanding of what the tiles meant or what the squares in the model meant, so moving to calling the number “n” seemed to be just one other thing at the moment. Only one student among the twelve in those first two groups really seemed to respond to that.
Fourth Thought. I will keep working to communicate manipulative expectations. The groups really varied in their reactions to the availability of physical tiles. Some students immediately began grabbing tiles without using them as mathematical tools, but instead began creating patterns with them. I tried to communicate clearly, but again this communication is something I will have to continue working on.