Teaching with Problems at a Cafe

This is a story to support my belief when it comes to learning math. 

1. I believe that students need to have basic math facts stored in long-term memory. 
2. I believe students need to use these math facts to support logical arguments for mathematical conjectures.

I overheard two young girls (ending grades 4 and 2) practicing addition and multiplication facts in a cafe today. The older, Kate, asked her younger sister, Sara, "What is ten thousand plus one hundred?" to which Sara replied "Eleven thousand?"

Kate: No.
Sara: Eleven hundred?... eleven...
Kate: No, ten thousand one hundred.

Kate: How about 23 times 18?
Sara: uuummm, thirty- no forty one.
Kate: Times! silly
Sara: oh, hmm...

I felt compelled to save Sara from questions that were a bit beyond her apprehension, while challenging Kate with something more than just declarative knowledge. Since their mother was present, it might be an opportunity to model effective differentiation of a problem space to involve both of her children. I happened to be reading a book on "Teaching with Problems" and wondered how they would fare with one of Lampert's problems of the day. The focus of Lampert's teaching is to have students "reason from assumptions to their implied conclusions". I knew I could ask a question involving 2-digit addition and subtraction (a skill they both could perform), that would still be challenging for Kate to figure out.

I wrote the same question on two sheets of paper and gave the girls pencils to experiment with.

I. Problem of the day

I expected the girls only to attempt part a of the problem, but I included part b because I was curious how someone would go about using hundreds with such a problem format. This is something Lampert alludes to in her book but doesn't go into detail on. The girls did not end up making their own assumptions about what could fit in the boxes, but operated under an assumption that single positive digits would occupy each box.

The girls started by finding one correct subtraction problem, each declaring: "I found one!"

I asked them to share what they found,
Kate: 46 - 23 = 23
Sara: 33 - 10 = 23

At this point a friend of Sara's named Jane (also ending grade 2) joined her in solving the problem. I explained to their mother that the girls were now practicing their subtraction facts without being asked to do a worksheet, because they were necessary experiments to find a pattern that would lead to conjectures to count all of the subtraction pairs they could find.

Both solution paths started as lists of all subtraction pairs involving tens {33-10, 43-20, ... 93-70}:

II. Kate's initial experiments

III. Sara & Jane's initial experiments

The girls were satisfied that they had found multiple solutions to the problem, but I reminded them that the question wasn't to solve the subtraction problems, but to figure out how many there were. I also wondered if we could make an estimate of how many there were, without listing them all. As the girls began to lose interest in the problem, I realized they would need more guidance to come up with estimates. Kate's mother encouraged her children by saying, "I am curious to know how many there are! Aren't you curious?"

I pointed out to Kate that some of her solutions had too many digits, and she crossed out 103-80 and 113-90. She drew a bold line to indicate the maximum for subtracting by tens. I asked her if she had found all of the subtractions ending with the digit 0 and she said, "yes", although at this point she had not considered 23-00 (top left of fig IV. added to fig II).

I guided her with a plan that might help us come up with an estimate. If we had found all of the subtractions ending with the digit 0, could we do that for all of the ending digits? For example, I pointed out that she had found one subtraction ending with the digit 3, her first attempt which was 46-23 = 23. This spurred her on to find all subtractions ending with the digit 3.

IV. Kate's further experiments

With further guidance from me and her mother, Kate came to an initial estimate of 80 combinations. She had discovered 26-03 and realized 23-00 would also work, so she was making a conjecture that there were 8 possible combinations for each digit, and 10 total digits. I left Kate to work with her mother on the final details of her approach and checked on Sara & Jane.

I asked, "How many combinations do you think there are?", and Sara replied "We found 14!"

V. Sara & Jane's further experiments

Given the level of support Kate needed to make an estimate of 80 using the final digit approach and her higher level of mathematical ability, I was glad that there was the beginning of a different pattern on this paper. Namely, {23-0, 24-1, ... 29-6}. I pointed out that we could continue with this pattern, but when we repeated 33-10, Sara noticed and was like "uh oh! we already are counting this one!"

VI. Sara & Jane's new pattern

I started listing all of the subtractions in order without writing them to suggest that we would repeat all of the subtractions by ten she had listed earlier with this new pattern. I wondered out loud when this pattern would end and I asked, "what is the biggest 2 digit number that we can subtract from?" and Sara replied "99". I told her to leave space for all of the other pairs, so she wrote it very small in the bottom corner.

After all of this, I had lost Sara a bit and she was still guessing her estimate: "88?" I could see that having two different approaches on the same list was interfering with her ability to see the new pattern I was trying to extend. But she continued thinking about the problem, and before I left the shop she suggested to me, "My estimate is 76, because there will be 76 subtractions." I noted that she wasn't counting 23-0, and she revised her thinking and said "77!"

VIII. Sara & Jane's response

VII. Kate's response

Overall, I feel like I did too much leading in this exercise, and it uncovers some of the pitfalls that a teacher will encounter using open-ended problems. However, given the time constraint of meeting in a cafe, and the design constraint of not knowing the girls' prior knowledge; I was pleased that both girls had come to believe in their guesses, had consensus in their results, and had used different approaches.

More importantly, the discourse between the sisters shifted from Kate demonstrating her prowess in mathematics by asking her sister declarative questions that were beyond Sara's level of understanding, to an exploration in which both girls were engaged in productive struggle. There is no clear "winner" in a situation where two girls come to the same answer through different solution paths, both require help, and both make small contributions throughout the exploration.

We need to equip our students with a strong foundation of math facts, but let us not forget to ask questions that require them to use this knowledge to produce new knowledge.

I recently read Magdalene Lampert's excellent analysis of teaching in, Teaching Problems and the Problems of Teaching. Without which this experience would not exist, so thanks for that effort.