Yesterday’s post recounted a conversation with Tabitha (5) in which she asked for a “math class” problem.

I focused the initial discussion on where she learned what constitutes a “math class” problem.

But there’s lots more in there that’s interesting.

It wasn’t just her affective response (rejecting the driving in a car context, asking for a naked number problem) that matters here. Notice the way she engaged with the two problem types.

When I relented and posed the question, *What is two plus three?* she guessed. I know that she guessed because she (a) took no time to process, (b) asked rather than told me her answer, and (c) it was wrong despite being within her grasp.

When I posed the exact same problem in a situation she could imagine (she and Griffy had been to the arcade just that day), she engaged quite differently. Her body position changed. She paused. Her fingers moved. Each of these is an indication that she was *thinking*. And when she had an answer, she stated it; she did not ask.

The central tenet of CGI and an important belief underlying the IMAP work is that children can use contexts to solve problems that they cannot solve abstractly. Here it is in action. 2+3 is meaningless to Tabitha right now. But her 2 tickets combined with Griffin’s 3 tickets? That’s got meaning.

The conclusion here is obvious, right? We start with contexts kids understand and can reason about (here, combining tickets). We move to the abstract mathematical representations (2+3). We don’t save arcade tickets for *after* the kid understands addition. We don’t wait for symbolic mastery before doing some applications.

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This is exactly how I’ve been working with my son. We started with contexts that he understands, and we have since moved into more abstract ideas. This is essentially what Seymour Papert has recommended – the contextual work first (like the engineering, the designing, the building, the making, etc…) which motivates the abstract layer.

Tabitha in a bunny outfit solving math problems is great context for any math teacher educator! I like the way you phrased that, Christopher; children can utilize contexts to solve problems they cannot solve abstractly. However, I need reinforcement when I sell the value of contexts for students who CAN solve CGI (addition, subtraction, multiplication, division) problems abstractly. Here’s my current off-the-top-of-my-head list of reasons why accessible contexts, and making students reference the context throughout problem solving, are good (in no particular order): 1. they open the door to non-traditional and very insightful alternative strategies, 2. they engage thinking rather than mindless rule following, and 3. they remove the meaninglessness of math associated with repeated algorithmic practice. What am I missing here? In general, I’m grappling with how to best put context back into the mathematical experience of well trained symbolic manipulators. I’m open to suggestions there too.

Beautiful analysis of Tabitha’s thinking–I love this example.

“The conclusion here is obvious, right? We start with contexts kids understand and can reason about (here, combining tickets). We move to the abstract mathematical representations (2+3). We don’t save arcade tickets for after the kid understands addition. We don’t wait for symbolic mastery before doing some applications.”

But what is “understands addition” in context? I’d say that she understands addition. What she doesn’t yet have is “does addition on demand without any connection to [her] reality.”

The work from Math in the City (Mathematics in Context) from Cathy Fosnot is quite excellent and preaches the same message as shared here. It also provides opportunity for “abstraction” and computational strategy development.

I am instantly reminded of this. I see it so often kids can solve so much intuitively that as soon as abstraction (like x’s and y’s) that just through them off.

I’ve been toying around with the idea of teaching kids that they are just mathematical objects, and we should think of them just like we think as concrete things. Easy to say, but seems like it won’t work out like that. How do we bridge the gap.

I agree for a five-year old putting math in context is important. There seems to be an arch re: “memorization / equations are bad” that this may fit into. If so, I would not agree with that conclusion. It is important starting around entry to first grade, IMHO, for children to: a) have memorized the patterns of basic arithmetic equations, and to b) understand how more complex problems can break down into simpler step-by-step arithmetic. Doing this is a way that promotes right-minded learning (pattern recognition) is vital. I’ve done my best attempting this for my daughter with my app – mathflashapp – and her performance through the third grade suggest this is the right track. Many paths to a common end, though, so whatever works best. Overall, the single greatest factor for success is a parent who take the time to promote the value of learning math, and who suggests it is easy. Some hand-holding will be needed for many early concepts, but it is that hand-holding that is vital.

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So if I read you right,

Steve, you are suggesting that a strong instructional move with Tabitha (a five-year old) when she cited “3+3=4” as a math class question, would have been to make sure she knew that 3+3 isn’t 4, it’s 6? And that moving to an iPad app would have been just as good as shifting the conversation to a familiar context (and would perhaps have been a better one)? Do I state your argument correctly?Hi Christopher,

No, I poorly stated my point I guess. First, I do agree that for a child of five, context is likely better than equations. Our school curriculum in Kindergarten (5 yrs) uses visual grouping for add concepts. I think your example of the relevance to T. of the tickets example underscores why this works. My point was more toward folks who argue points like “finger-counting in fifth grade” is good because, well I guess it is because memorizing equations is bad (that is very poorly stated). Was not sure if you might be implying this.

I have been reading some of these “equations are bad” arguments, and these trouble me. Around 6-7, I think it is important for children to first internalize basic arithmetic equations as memorized, right-brain pattern recall. Once they do this, their minds are free to think about other aspects of the math problem in front of them. Once basic one-digit equations have been internalized, the next pattern needed is the simple process of stepping through more complex problems. For addition/subtraction, this the simplification of stepping through the number positions using ither carry forwards or borrowing as needed. Also being able to using chunking to solve multi-digit multiplication fits well here (once single-digit multiplication tables are memorized). When a child has these techniques right-brained, 356-275= or 35*12= is solved with confidence and ease.

Based on my daughter’s experience with Washington State’s math curriculum through third grade, she would never had done enough equations from first to third to have these patterns memorized (I prefer right-brained, since memorization seems to have bad connotations). So, I wrote mathflashapp, and she did 5,000 additional equations. The format and approach of the app IMHO helps quiet the left-brain creating an optimal mental state for right-brain patterns to be created. Many of her friends are currently beginning the slow decline that seems to affect many in the U.S. and it makes me mad. Ask any one of them what 6+9 is, and you will never get an immediate response. Fractions, of course, get pretty hard when the child does immediately visualize 1/3 + 2/4 is the same at 4/12 + 6/12. Now wonder fractions are the beginning of the erious “I am bad at math” syndrome.

I will end by noting the case of Ladd Campbell’s sister. Very bright girl, who because she was home schooled never did much math. Finger-counting still in fifth grade. Hated math. Believed she was terrible at it. After just over 1,000 flashes, she “GOT IT”. She was not bad at math, but had never done enough to internalize the basics. Now she loves math (her favorite subject), and is beginning to excel at it.

So, as Tabitha gets older, if you were to think equations, reps, and memorization is bad, I guess I would ask you to reconsider. However, yours is undoubtedly a corner-case because you clearly have a lot of interest in how/why T. will excel at math. Most of the kids my daughter goes to school with (we live in a relatively poor area – 25% of students are Title 1 math) have parents who think that the school is going to do everything needed, and, as I noted above, fractions are the beginning of the decline of Americans skills at math. Why?

aplogies for the typos above. hope you can read through them. –steve

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“SHE GOT IT!”

What did she “get,” exactly?

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