Jacob (not his real name) is a 6th grader who struggles with his multiplication facts. Recently I sat one-on-one with him to capture his thinking as he completed Jo Boaler’s math cards multiplication matching activity. He had one prior experience with the activity when he worked with a small group during homeroom math support, but this was the first time he was completing it with me.
I shuffled the cards and handed Jacob the deck. I did not mention any strategies to get him started. I wanted to see how he would get himself started. He began by keeping the cards in the deck–faced up. He removed the top card, looked at it, then placed it at bottom of the deck, as one would do if you were working with flash cards. He went through three cards before stopping at 9×7, where he spent about 30 seconds mentally calculating. “Sixty-three,” he said.
“How did you figure that out?” I asked.
“I knew 5×7 was 35, and I kept adding 7s.” I noted he is using a derived fact, using a fact he knows to find out the answer to one he doesn’t.
Jacob placed 9×7 on the counter top then immediately drew 7^2 then 8^2. “I don’t remember what those are,” he said placing the cards to the side.
The next card he drew was 9×4. It took him about 25 seconds to complete the mental math computation. “36,” he said, then created a new column for 9×4.
“How did you calculate that?” I asked.
“I knew 9×5 was 45 then took away nine.” It’s becoming obvious he is using landmarks or friendly numbers, then adds or subtracts accordingly.
The next card he drew was a 7 by 9 array. He knew to place it in the 9×7 column he just started, but when I asked him the product of 7 and 9 he had forgotten. He counted the squares, but he counted each square in an inward spiral motion, working his way from the outside in.
“Sixty-four,” he said.
I suggested he count again. He counted each square in spiral form again and said, “Sixty-two.”
I asked, “If there were 10 columns of 7, what would that be?”
“Seventy. (PAUSE) Oh, it’s 63.”
“Because I need to take away a seven.”
Before we moved on, I wanted Jacob to think about a strategy that would help him complete the activity efficiently while at the same time revealing his mathematical thinking.
“What do you think we can do to help us with this activity before we forget what 9×7 is? I’m pretty sure there will be another card that represents 63.”
At this point he spread out the unmatched cards, found 63, then placed it at the top of the column. Then he began looking for another card.
“What are you looking for?” I asked. He found the 4 by 9 array to place it in the 9×4 column.
He looked at the array for about 15 seconds then said, “9 times 4 is 36.”
“How do you know?”
“I made it 10 times 4, which is 40 then I took away 4.”
I didn’t ask him but I assume he was searching for the 4 by 9 array to give him a visual representation to support the same friendly number strategy he previously used. He then proceeded to look for 36 and placed it at the top of the column.
We spent the remaining 15 minutes completing the activity.
At the end of the period I mentioned to Jacob that I was pleased to learn he was utilizing a strategy to compensate for an overall weakness in his multiplication facts.
To my pleasant surprise Jacob actually has strong number sense. He is decomposing numbers and using derived facts to think about multiplication. He is utilizing that strength to compensate for a lack of fluency with other math facts. In addition, he is using visual models to construct his math arguments and uses them to defend his derived fact strategy.
I’ll be working with Jacob next year and it is our goal to improve this fact fluency. This observation gave me insight into Jacob’s mathematical thinking. He has more number sense than I previously gave him credit.
Special thanks to Capri Paluch (@CapriPaluchD95) for helping me process this observation.