Do Students really Understand the Operation they are Performing?

I recently read this great article from NCTM called “Capturing Children’s Multiplication and Division Stories: Learn the value of having students create their own stories and pictures to represent number sentences as classroom assessments.” Here is link to the article:

It is definitely worth a read and really inspired me to go out and try it in a class. I asked one of my grade 3 teachers I work with if we could try it in her class and give them a diagnostic assessment to see how well they understood multiplication. In the article the study that was done had them do a story and a drawing for a multiplication prompt and a division prompt. We only had the students complete a multiplication prompt. This class has done some work with multiplying but not much this year so far. I have added the curriculum expectations for grade 2 in Ontario below:


This what students in grade 3 have learned previously in grade 2 about multiplication and division. We also know students develop along a continuum for operations so the students in the class have a wide range of where they could be in their understanding of early multiplication.

Here is the prompt we gave them:

Make up a story and a picture about marbles for this number sentence: 4 x 3=

(We said they could change the topic from marbles if they wanted to.)

Here are some of the student work samples:


As you can see all three of these students knew the answer but were unable to make up a story that describes a multiplication situation. The two that were able to make a story up described an addition situation to arrive at 12.

The first pic shows the students understood it can be repeated addition but was unable to draw a picture that models 4 x 3 or make a story up about the situation. The second pic down knew the answer was 12 but drew an equal group picture that represents 6 x 2 not 4 x 3 and also described a addition situation in their story. The bottom pic was interesting in that this student knew the answer immediately. This student said work is done on times tables at home. This is a great example of students memorizing answers to times tables questions without conceptual understanding. The student may know answers but will struggle to be multiplicative if this is not addressed. The pic shows an addition situation as the story and an addition picture not even an equal groups for the picture.

I wanted to also show this piece of student work because I thought it was neat they tried to model it with a number line as their picture of the situation.


There is a few simple errors but they understood its three equal groups of four. The number line doesn’t start at zero and there are two groups that are 3 not four but they tried! I like that they tried to use the number line.

Overall  this assessment showed a clear lack of understanding of multiplication but was very informative for the teacher, she now knows she has a lot of work to do! I am going to show this too my grade 1 and 2 teachers and continue the conversation that multiplicative thinking needs to be developed simultaneously with addition and subtraction. I feel sometimes there is a thought that it needs to start after addition and subtraction and that it is something looked at more in grade 3. The foundations have to start in FDK to grade 2 with learning to unitize (seeing groups of as one unit) and starting explore what multiplication and division mean. Start counting equal groups, looking at equal group situations and solving basic multiplying/division prompts by direct modelling and fair sharing. Students in these grades don’t need to hear the word “times” or multiplication but that doesn’t mean they can’t start learning it by exploring what I listed above. I like the language Kathy Richardson uses in her book “How Children Learn Number Concepts”. She suggests using:  groups of, piles of,  rows of, stacks of, cups of instead of times.

I plan on trying this in some junior classes too, I suspect we will find some students with very similar misconceptions to this grade 3 class. Try it out in your classes, I think you will be very pleased with the information you find out about your students.

About stamp36

Instructional Leader for Trillium Lakelands District School Board
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