# Noticing number strategies

Recently I ran a math class for a few younger students, including my son. The objective of this class were to start making connections between how students visualize numbers and early arithmetic strategies.

Here are some of the various ways students visualized a group of dots to figure out how many total dots there were.

Here are some of the strategies students used when they doubled numbers in their pattern.

I asked students to look at the visualizations for the dot patterns and the arithmetic patterns and see if they noticed anything in common between the dot visualizations and the doubling strategies. They didn’t.

So I told them to find a partner, talk to their partner about the same questions, and then told them we would discuss it again. Two minutes later, I asked students to sit back down in the group after their discussions and re-asked the same question: “What is common to the strategies you (as a group) used for the dot visualizations and the doubling strategies? What is the same about what you did?”

One student said that in both cases, the students counted in order to find the answer, which was true. In both the dot visualizations and the doubling strategies, one strategy at least some students used was counting.

Another student said that in the dot pattern and the doubling strategies, we were trying to duplicate something. Not everyone knew what he meant by duplicating, so he explained it again, and I revoiced him and used the word doubling, which everyone understood.

Finally, another student had an epiphany. “When we did the dot pattern and when we did the doubling, something in common is that we figured out a big group and left some over to figure out at the end.”

Here are the actual two strategies he linked together with his observation:

In the first one, in order to figure out 23 x 2, first we do 20 x 2 and we save the 3 to figure out later. In the second one, first we counted the perimeter and saved the inside shapes for later.

I’m excited by this observation as I had never thought of thinking of the distributive property as “saving part of the calculation” for later (although I have often seen other visualizations of the distributive properties).

# Thinking about rates and times

As we were traveling back from track practice tonight, my son and I had a fairly natural conversation about driving on the highway.

Son: “How fast are we going right now, Daddy?”

Me: “According to my speedometer, we are going 2 miles over the speed limit. Oops!”

Son: “So if we were traveling for a whole hour, we would only go 2 miles??” Here my son is double-checking his reasoning with me and thinks I might be a bit crazy to suggest that we are going only 2 miles an hour. He has a strong sense that the speed he can observe outside the car does not match the calculation he has done.

Me: “No, I mean we are traveling 52 miles an hour and the speed limit is 50 miles an hour, so we are going 2 miles an hour too fast right now.”

Son: “Oh. Okay.”

Me: “I have a question for you. If we were traveling at 60 miles an hour, how far would we get in ten minutes?” Now, if we were not in a car and I was not trying to strike up a conversation with my son, I probably would not have asked this question. In the context of driving a car down a highway, I think this makes sense as a conversation starter – at least to a parent who is a mathematician and a mathematics teacher.

Son: “10.” (nearly instantly)

Me: “How do you know that?”

Son: “Well, there are 60 minutes in an hour and we are traveling at 60 miles an hour, so ten minutes must be ten miles. I don’t know how to explain this, Daddy!” Actually, you did explain it just fine son, but somehow you feel your explanation is inadequate.

Me: “I think I understand. The miles and the minutes have to be the same.” My son did not articulate his thinking in this way. I don’t know if this was helpful or not, but I felt that he wanted me to revoice what he had said using different language.

Son: “Yeah.”

Me: “I have a harder question.”

Son: “Okay, but I want it to be just slightly harder not a lot harder, okay?”

Me: “Okay. If we were traveling at 50 miles an hour, how far would we travel in ten minutes?” This question comes straight from a video I watched of Magdalene Lampert teaching during a workshop she ran over the summer. I was curious, how might my son approach this problem?

Son: “Hrmm. Maybe 9 miles an hour?” I love that my son’s first instinct is to estimate.

Me: “How do you know that?” I always ask this question, so this is normal for my son, rather than being like a quiz. I’m not quizzing him to find out how he understands, I am genuinely curious.

Son: “Well, it has to be less than 10 miles an hour. 50 is less than 60 and 9 is less than 10.”

Me: “How could we double-check that?” This is the teacher in me. I’m never satisfied with the first explanation.

Son: “We need to find nine times six. Hrmm. Nine plus nine plus nine plus nine plus nine plus nine.” My son was probably counting these nines in his head but he would have had to keep track of how many nines he was counting a couple of years ago.

Me: “What is that?”

Son: “Nine plus nine. Eighteen. And nine. Uh… I don’t know.”

Me: “Can you work it out? Also, I recommend paying attention to the answer.” This comes from MP7 in the Common Core and George Polya’s heuristics for problem solving — pay attention to the form and structure of your answer to problems as it can make other problems easier.

Son: “Okay. 27. Oooooohhh. Plus 9. 36. Plus 9. 45. I see the pattern. Plus 9. 54. Plus 9. Oh I can stop.” Meanwhile, while my son added on the nines, for each addition I pointed out the associated multiplication fact.

Me: “So does nine work?”

Son: “No, it’s too big. Let’s try six.”

Son: “6 and 6 is 12. 12 plus 6 is eighteen. Eighteen plus six is 22. No, 24. 24 plus 6 is 30. 30 plus 6. No, it doesn’t work. It’s too small.” I continue repeating the associated multiplication facts for each multiple.

Me: “What’s next?”

Son: “Let’s try 7. No, that will probably be too small. Let’s try 8. 8 plus 8 is 16. 16 plus 8 is 24.” At this point, I said that 8 times 3 is 24. My son stopped adding, “Really? Hunh.”

Me: “Yes.”

Son: “24 plus 8 is 32. 32 plus 8 is 40. 40 plus 8 is 50, no 49, no 48! Too small. It must be between 8 and 9.” Do kids that understand numbers only as counting objects recognize that there are other types of numbers in between the ones we use for counting? It seems to me that kids need plenty of experience with situations that require other types of numbers to motivate an intellectual need for those types of numbers. We don’t go look for tools, even thinking tools, arbitrarily; without some clear kind of need for the tool.

Me: “Okay.”

Son: “Hrmm. Maybe it’s 8 and a half. No, that’s too big because 6 halves is 3.”

Me: “Right, and then we would have 48 and 3 which is 51. Too big.”

Son: “It can’t be 8 and a quarter either. That would be 49 and a half.”

Me: “So it must be between 8 and a quarter and 8 and a half.” This is a key idea that I am sorry I did not let my son develop himself.

Son: “What about 8 and a third? Let’s see. YES!! 6 thirds is two, so it works! I got it! Daddy, that problem was not a ‘little bit harder’ than the other problem. It was WAY harder.”

I find it hard not to turn off my teacher self when my son and I are discussing mathematics. I find myself really curious about how he will approach problems and I often find myself straddling the line between pushing him to articulate his thinking and just being a bit annoying.

On the other hand… the teacher part of me is now curious; which mathematical standards and practices did my son and I uncover in our conversation together? I remember Magdalene Lampert telling me once that when she teaches, she tries to hold all of the standards for the year in her head while working with students because she is never really working on one specific thing at once; she is trying to connect all of the ideas of the year together.

# Understanding place value

Last night I had an interesting realization about my son’s understanding of place value. It is clearly incomplete.

We were continuing our made-up story about Max, the 701 hundred year old 7 year old. He was cursed at the age of 7 to never age and to never die, and now he is 701 years old and working hard with his friends to try and break the curse.

At one point Max gives an explanation of curse to the sea elf king in order to ask the king for help and Max says how many years old he is. I left the actual number of years Max had been alive out of the story though and so my son filled it in.

“Max has been alive for six hundred and four years,” my son said.

“Oh, how did you get that?” I asked.

“Six hundred and three plus seven is seven hundred, so six hundred and four plus seven is seven hundred and one,” my son replied.

He spoke so confidently and assuredly that I did not correct him. Also, I wasn’t totally clear at that time exactly at that time what he was thinking, and it was late.

I think that he was regrouping the ten and confusing a regrouped ten as a one hundred. He did the same thing when he first tried counting on from 100 at age four. I remember quite clearly him counting 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200. I also remember how stubbornly he continued with this understanding.

I know of some strategies that I can use to continue building his understanding of place value. I am not sharing this story to look for help or a diagnosis but just to point out that a child who the previous night confidently and accurately multiplied 400 by 21 in his head still has unresolved issues with place value.

I think my take-away is that I need to be cautious about what assumptions of knowledge I make about my son (and of course the same is true of my students).

# “It’s 8400”

I had a fascinating exchange with my son last night. I was telling him a made up story, and at one point this character who is cursed and very old said that his age was “Four hundred times a score and one.” My son asked what a score was, and I told him twenty.

He then tried to figure out the person’s age. He started by asking what is four times twenty, and I told him he could figure that out. He counted up by fours to get eighty, and then said that four hundred times twenty is eight hundred. He thought for a moment and said that no, it must be eight thousand, which meant the final answer is eight thousand four hundred.

This makes me curious about how he understands the number twenty. He knows apparently that four hundred times something is one hundred times whatever four times the something is, although I am not clear he would explain it like that. However, he apparently did not use the fact that twenty times something is the same as ten times two times the something.

Thoughts?

My son and I were walking back from the park, and he was trying to convince me that I had time to play Clash of Clans, and that I really should start playing it.

He said, “Daddy, if you played one third of one quarter of your time, that would not be very much. See look, if you had one minute, one quarter would be 15 seconds, and one third of that would be five seconds. And if you only had 1/3 of that time, that would be five seconds divided by three. Daddy, what is five divided by three?”

I asked him, “What do you think it is?”

He replied, “One and one third, I think.”

I asked him, “So let’s check. What would one and one third times three be?”

He said, “Let’s see. One times three is easy. That’s three. One third times three is one. Oh, that would be four total, not five. Maybe it’s one and one half?”

I asked him, “We can check that. What would one and one half times three be?”

He said, “Hrmm. Four and a half. That’s not right either. It has to be bigger. What about one and three-quarters? If I multiply that by three, that’s six and one quarter. That’s too big. So it has to be between one and one-half and one and three quarters. Is two-thirds between one half and three quarters? Let me check. Two thirds times three is the same as two thirds plus two thirds plus two thirds. That’s one and one third plus two thirds is two. So it works! Five divided by three is one and two thirds!”

I then pushed my luck and asked, “What’s half of one third?”

My son said, “One quarter! See,” while showing me his hands, “here’s one quarter and here’s half of one third. They are the same size.”

A woman passed by at this point and overheard our conversation and my question to my son. She stopped to listen.

“Okay, so how many halves of one third fit into a whole?” I responded to my son. The woman at this point started to speak, but I said to my son, “And remember the point of this is for you to figure this out yourself.” The woman looked at me and nodded and then stopped speaking.

After some thinking my son said, “Six, because three thirds is a whole. Oh, but four one-fourths fit into a whole, so they aren’t the same size.” My son then said, “One half of one third is a sixth, right? Because it fits six times into a whole.”

I push my luck again, “What about a half of a sixth?”

My son said, “One twelfth, then one twenty-fourth, then one forty-eighth. Oh those are the same numbers from Threes! Except instead of getting larger, they keep getting smaller.”

We stopped our conversation about fractions at this point as my son started talking about how in the games of Threes, the numbers aren’t actually multiples of three and the game is misnamed.

This is the first time I have heard my son talk about fractions other than thirds and halves. I saw my role here  was to find the right questions to ask at the moment that my son has an intellectual need for other fractions. These kinds of conversations are common-place between me and my son, but still every time I am amazed by how much thinking about this he must be doing when I’m not around.

# I eat five pancake!

This morning we had mini-pancakes for breakfast. I gave my youngest son (who is two) three mini-pancakes. When he was still hungry, I gave him two more.

Half-way through his final pancake, he said:

There is interesting research that suggests, through an ingenious experiment, that we are literally born with some knowledge of numbers. Not symbols, like the word “five” my son is using above, but numbers. It seems to me that while my son is developing his language, it is an excellent time to develop his innate understanding of number and connect it to the language he is learning.

So I pay attention to what my son says and even at two years old, when he is first developing language, we count together, we group objects when we play, and I help him give language to the thoughts he is already having.

# What’s 192 plus 192?

My son, “Daddy, what’s 192 plus 192?”

Me, “What’s 200 plus 200?”

My son, “400.”

Me, “What’s 192 plus 192 then?”

My son, “I don’t know.”

Me, “Try breaking it down like this. 100 plus 100 plus 90 plus 90 plus 2 plus 2. Does that help?”

My son, “Hrmm. Okay, it’s 284. No, I mean 384. Do you want to know why I want to know?”

Me, “Absolutely. Why do you want to know what 192 plus 192 is?”

My son, “I started with 1 plus 1, and now I’m at 8192, and I want to know what 8192 plus 8192 is.”

Me, “Do you mean you added 1 plus 1 to get 2, and then 2 plus 2 to get 4, and so on?”

My son, “Yes.”

What makes this exchange especially interesting to me is that I remember attempting to do exactly the same calculations my son is doing, at about the same age.