Math Relearning/Progressions/K-5 Number and Operations in Base Ten

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These are my comments on the Common Core Math Progressions document Number & Operations in Base Ten: Grades K-5 (PDF).


"Intertwined" - Yes, that was what was so annoying about trying to group it with number sense and even measurement before getting to arithmetic. It's a hard concept to explain without bringing in those operations.

"Base-ten units" - I was using "unit" the way they're using "one," but I can agree to their usage.

"number as composed" - What does composing and decomposing base-ten units tell us? Why do we spend so much time on it? Reducing to one-digit calculations is one important consequence that the progression covers. It is a new perspective to think of the standard algorithm that way.

I think to convince people Common Core is a good idea, you have to be detailed and specific about the benefits of the new methods in addition to explaining how the methods work. What problems do students need to solve that the standard algorithms can't accomplish? What mistakes do students make with the standard algorithms?


"essentially arbitrary marks" - Aha! Take that, Jeremy.

Grade 1

"two-digit subtraction with and without decomposing" - What does this mean? How does it prevent the mistake of subtracting the smaller digit?

Grade 2

It would be good to see a chart of the multi-digit operation algorithms used before the standard algorithms to make it easy to study the problem solving options for this stage.

I think composing and decomposing units could still seem like a trick to get the digits to turn out right. Maybe what I want is a way to see that this is all equivalent to working with ones.

These documents get hard to read, and I think it's because details are left for me to fill in. I think this is common in math texts, so I should just get used to it. But I do wonder what the right combination of explanation, equations, diagrams, and so on is that would make math discussions fairly effortless (for me) to follow.

Right now I know I'm more excited about the idea of math than about learning math itself, because I've hardly done any of it yet. I also know that there's a drudgerous, discipline period of any subject you're studying after the romantic, honeymoon period ( It probably corresponds to the conscious incompetence phase of skill development. I know math will be no different. But that's something that's explicit in the Common Core, the practice of persevering through solving problems.

Grade 4

Starting fractions with 1/10 and 1/100--good idea! So is associating them with money.

"'oneths' place" - This is an example of the fact that many features of math (such as the decimal) have very specific meanings and purposes that aren't obvious at first and can easily be missed or confused with other possibilities, which can lead to mistakes. That's why I'm using conceptual math for this relearning project. It also illustrates the usefulness of asking "why," "why not," and "what if" for getting at these reasons and meanings.

New to me (NTM): "area models" - This idea was introduced before grade 4, but even though I knew the formula for area, before this project I hadn't thought of using area as a general model for multiplication and certainly not as a way of illustrating the properties of multiplication. This is a good example of the many connections among mathematical ideas. Finding these is part of the fun of math, in my opinion.

"recording the carries below" - This is even more compact than the standard method. It illustrates that the conceptual math people really do think through the methods they teach. They pay attention to things like efficiency and clarity, how those values can conflict, and the conceptual and procedural mistakes learners make that need to be corrected or that can be avoided. The place-value symmetry around the ones place and decimal point is another example. Parents don't typically think of all this. And this illustrates that specialists have a deeper and more extensive understanding of their domain than nonspecialists, which often leads them to surprising results that nonspecialists don't understand. If the nonspecialists don't think of the field as a deep one that can hold such surprises, they might think the specialists' conclusions make no sense and unnecessarily complicate things.

"shifting the result to the left" - I read somewhere that you shouldn't teach the trick of adding zeros to multiply by multiples of ten. I was puzzled by why this would be a problem until the article made the point that it leads to the mistake of adding zeros to the right of decimals. My first reaction was to dismiss this concern because to me, the trick of adding zeros includes nuances like that. I already understand the principles involved. But yes, if you're teaching it to people who don't know the concepts yet, you can't just tell them to add zeros to the right. It's better to speak in terms of shifting, which matches the idea of place value and results in the effect of adding zeros in the relevant cases but also takes the decimal cases into account.

"Products of 5 and even numbers" - This example brings up several issues for me:

"violate the patterns" - This confusion could be interpreted a few ways. Maybe the students have simply thought they'd identified a pattern, and now they have other cases to account for. That's normal math activity. But maybe they've established an intuition about zeros that's wrong, and 5 is surprising them, and maybe they think the answer's wrong. When this happens, I always want to explore the implications of the intuition: if there are any cases where the intuition works, what would be the implications in a universe in which it was somehow correct, what's the confusion that leads to the intuition, what incorrect answers we'd get if we followed the intuition, what's the correction to the intuition.

Or maybe the students think the rule of adding zeros is just one of those magical rules in math that no one can explain, and 5 is just a magical exception to it. Magical thinking in math violates its whole spirit. As far as I can tell, the idea of math is that it's completely logical, aside from the human factors that go into discovering the logic. One of the most valuable things I've learned so far is that everything we do in math has been worked out in detail by someone, and so yes, we do know how it works; and with patience, research, and careful thought, I can find out all the logical steps and concepts that make it work.

I was surprised that the authors associated all the products that have "extra" zeros with 5. Are there really no ways to get a multiple of 10 that don't involve 5? My preliminary examples suggest not.

As I go along, I want to try to inform the math tricks I came up with as I grew up, like the one about multiplying an even number by 5 by dividing by 2 and tacking on a zero. Or an odd by dividing one less by 2 and tacking on a 5.

NTM: I'd never thought of the standard multiplication algorithm as being an application of the distributive property.

Those division methods are a little different from the long division I learned, so I'm not sure of the nuances. That'll be something to pay more attention to as I'm studying the curriculum.

Division is an operation I'm less clear about on its meaning. It would take some time to interpret long division in terms of place value. I imagine it'd be that way for many people. It seems to be one of the harder standard algorithms for people to grasp and remember. Maybe they'd welcome another method in that case. This progression isn't giving me hope that there's something easier though, but maybe there's a clearer explanation of the usual methods.

Grade 5

NTM: "Recording division after an underestimate" - Interesting! I like how a slight adjustment to the standard algorithm can be used for this other purpose to get at the answer from a somewhat different angle. It seems elegant.

"placement of the decimal point" - I'll have to wait till the fractions progression to really get this paragraph. But it's interesting that fractions are given as only one way to interpret these decimals. Of course, not every decimal can be represented as a fraction.

This isn't attached to anything specific in the text, but I've been thinking one way to represent my takeaways from the material would be to condense the concepts into diagrams that show their relationships, such as showing the multiplicative and exponential relationships in place-value. Since I'm not having to build up the concepts myself now, I can afford to put them together in other interesting ways.


And now I've gotten completely sidetracked by Keith Devlin's articles on multiplication:

He's pleading with teachers not to teach multiplication as repeated addition, because it creates problems when they introduce negative numbers, fractions, and calculus. He doesn't tell them how to teach it, since he's a mathematician rather than a K-12 math educator, but he says however they do it shouldn't contradict the modern understanding of multiplication, which turns out to be from abstract algebra, so that's a good reason for me to learn it.

He also says it's pointless to ask what multiplication is. The only way to deal with it is to axiomatically describe its properties. He mentions scaling, but he says this is only an application of multiplication rather than the abstract thing of multiplication itself. In the final article, though, he does say his mental concept of multiplication is centered on scaling. In any case, he and I seem to have different ideas of what the question "What is it?" means. To me listing its properties answers the question, I think, depending on the properties.

But I do have a problem with the way he talks about it. He says, "Unfortunately, trying to find an answer holds back mastery of mathematics, which largely depends on getting beyond the concrete and into the realm of the abstract - on recognizing that the 'What is it?' question is simply not appropriate for the basic objects and operations of mathematics. 'It' is what 'it' is. What is important is what 'it' does."

What I want to know is whether multiplication has a single definition that can predict its effects on different kinds of numbers. If not, why do we use one term for these different operations? But apparently it does have one. "In particular, there is just one kind of number, real numbers, one addition operation, one multiplication operation, and one exponentiation operator (where the exponent may itself be any real number). You get everything else by restricting to particular subsets of numbers."

He also gives this intriguing quote from Adding It Up: Helping Children Learn Mathematics, which is available as a free PDF from the National Academies Press:

"The number systems that have emerged over the centuries can be seen as being built on one another, with each new system subsuming an old one. This remarkable consistency helps unify arithmetic. In school, however, each number system is introduced with distinct symbolic notations: negation signs, fractions, decimal points, radical signs, and so on. These multiple representations can obscure the fact that the numbers used in grades pre-K through 8 all reside in a very coherent and unified mathematical structure - the number line."

I'd been thinking the number line was a good general representation of numbers, but I don't know how to generalize confidently about math unless someone who knows a lot more math tells me. It's also interesting to see that point about the unity of arithmetic. Something to look forward to learning about. And yes, I've also been looking forward to making sense of all the notation. Anyway, Devlin's endorsement of that book makes me want to read it.

I like articles like this series by Devlin because they connect higher level math with people who only understand and deal with the lower level. We need lots of that kind of bridge content. I think that's a goal of the Numberphile channel on YouTube.

Also the discussion on the post he links to looks fascinating: One of the commenters thinks multiplication encompasses several models, including repeated addition, and this commenter seems fairly thoughtful. So maybe I can't just take Devlin's word for everything even though he's a mathematician. One thing is clear: Multiplication isn't as simple as I thought it was.

At the abstract level, why do we have a multiplication operation that's defined the way it is? I gather we could define any operation we wanted with any properties.

Back to the progression

"how many tenths are in 7" - This is a helpful way to think of division by a fraction/decimal.

Ah, 5.NF.5 has students interpreting multiplication as scaling.

Extending beyond Grade 5

NTM: It's interesting to split up two-digit multiplication like a polynomial. I, of course, had never thought to do that because I don't remember anything about polynomials.