Monday, July 16, 2012

A pretty wonky post about how different curricula handle the properties of arithmetic in Algebra 1

I'm working on my Algebra 1 curriculum, and I came across two really different approaches to teaching the properties of arithmetic. Because I find them interesting, I want to share them, record how I plan to use them, and then ask a question at the end.

Here's how CME introduces the properties of arithmetic to 9th graders:

Students are asked to complete the above table and find several patterns in the table. They do this with the multiplication and addition tables. Students are given a big, arduous, concrete task and asked to find shortcuts. Students are given hints directing them towards particularly interesting shortcuts. Then kids share their work, and then we call it a day.*

* This table is actually from Day 4 of their sequence. Day 1 observes patterns in the addition and multiplication tables, restricted to non-negatives. Day 2 reviews arithmetic with negative numbers. Day 3 extends us to the full addition table. 

It's not until the next day that we have any sort of general expression of the properties of integers, and at this stage we still don't use any algebraic notation at all. Instead we articulate the "Any order" and "Any grouping" properties of arithmetic. Then we extend these principles to cover rational numbers and then the reals. 

Finally, students look at various algorithms and try to explain why they're true. But that's really hard, using English, and so after students struggle to justify the reliability of these algorithms using our common language and vocabulary, the students are presented with variables. Variables are ways of talking about arithmetic with generality. And, boom, then comes the rest of the Algebra 1.

Now, here's how Park Math introduces the properties of arithmetic to their kids:

From there they move to finding counterexamples to claims about this rule, motivating the importance of mathematical justification. From there they quickly move to the introduction of new rules and, then comes this problem:
In a sentence, CME eases kids into variable expressions via the need to justify arithmetic shortcuts. Park Math assumes that kids understand variable expressions (and equations) and asks them to justify algebraic rules.

The personal puzzle is how to hack these two approaches apart and come up with something coherent and useful for my classroom. I'm assuming the pace of CME is too slow for my kids but the abstraction of Park Math is too much to start with. My current plan is to start with CME and move on to Park.

But the more interesting question is why these two curricula assume such different things about the students who are beginning Algebra. Does anyone know why that is? Is there something in the water in Massachusetts/Maryland?


  1. Park looks to be a fairly high performing school (650 average SAT). CME says something about low barrier to entry for their books, so I suspect it is intended to be usable in low performing schools. That may be a lot of the reason for the difference in approach. I suspect CME is anticipating many students with very weak arithmetic skills and little to no ability to use order of operations.

    I wonder how many students will buy into looking at the underlying cause and using variables to confirm that the patterns they think they see will continue. There is certainly room for solid lessons if you can get the buy in. If not, you have spent a few days doing 5th grade lessons at best.

    My students are all over the place on order of operations. I think the Park stuff is a good vehicle for addressing that. They also are not strong at all on arithmetic with variables (like terms, distributive property, exponents). I am curious, but much less certain about using Park to address those deficiencies.

    Do you think giving and referring to the commutative and associative property is helpful?

    1. Your take on CME vs. Park sounds right. I'm thinking of just jumping to the "Day 4" activity on the first day of class -- there are some interesting patterns in the multiplication table, though less so in the addition table.

      Unless I'm looking at the wrong thing, I don't think Park will help much with like terms/distributive/exponents. In that first chapter they review these things lightly in the context of using operations to define new rules using non-standard symbols.

      I'm not sure what you mean by "giving and referring." In what context? Are you asking whether the properties have a place in the curriculum at all?

  2. In question #5 - 7 and some at the end one or both of the arguments for the new operation is a variable. You need to do some arithmetic with variables to answer the questions.

    I just mean do you state the associative and commutative properties and do you refer to them by name to justify steps in a problem - for example as you are working through a one variable linear equation. Do you expect the students to name the properties to justify their steps? I feel like emphasizing these properties makes something that is simple seem complicated. If students haven't interalized these properties through arithmetic, I don't see how formally defining them is going to help.

    1. I think that these principles allow us to give mathematical justifications and explanations of why various stuff either works or doesn't work in algebra. I'm less sure that knowledge of the properties directly helps students in solving a linear equation.

      At the same time, there's a really cool interplay between conceptual understanding, the ability to explain something and the ability to solve a math problem. So if kids are able to form coherent justifications about algebra, I'm relatively confident that they'll be better problem solvers.

      And that's the upshot of the CME unit, eventually: the patterns that you notice in the tables turn into principles, at first rough, then precise, and once precise they can be used to explain why various mathematical shortcuts work (and why others couldn't possibly work.) I think that's cool.

  3. Right, they require dist. prop, like terms, but really don't develop it. My students are very weak on this coming in - they need much more than a quick reminder. I don't want to get bogged down in the unit because I am throwing too much at them at once.

    1. One of the things that I liked most about what I did last year was that I taught baby factoring very early in the year. Basically, we just undistributed, but this helped a lot of kids go deeper into the distributive property and also set us up well for factoring a few months later.

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  4. (Accidentally posted as a response to someone else--I was wondering why it was indented...)

    First, I enjoy reading your blog. I have had a lot of opportunity to ruminate on stuff I probably wouldn't have considered on my own so far.

    Why do you say the Park text (the second image is from Park right?) "assumes that kids understand variable expressions (and equations) and asks them to justify algebraic rules "? I don't know the point of the exercise or what is supposed to happen but it doesn't seem to me that the student needs to know how to manipulate variables or equations. In problems 10 and 11 there only seem to be constants. That makes sense because if m and n are variables then what known quantities are we going to solve them in terms of? On top of that, m and n usually denote given quantities or when we consider the set of all m x n matrices we consider m and n as general quantities that don't vary with some other known quantities. The same goes for a, b, and c. So I am not sure what you guys mean for instance when mr bombastic says, "In question #5 - 7 and some at the end one or both of the arguments for the new operation is a variable. You need to do some arithmetic with variables to answer the questions," since the x \triangle 8 isn't equated to some known terms with respect to which it can be solved or another variable which can vary as changes. So I am inclined to agree with "I just mean do you state the associative and commutative properties and do you refer to them by name to justify steps in a problem - for example as you are working through a one variable linear equation. Do you expect the students to name the properties to justify their steps?" and agree with your statement that there is an interesting relation between explaining and solving which I think is probably the point of the whole exercise. I don't know. I am sort of confused about the point of the discussion. As for manipulating equations, I don't think the Park people expected students to manipulate the equations by canceling constants. It seems more likely they expected the students to chain together equations like:
    m \circ n = 3n - m \neq 3m - n = n \circ m to show that it isn't justified.

    Also, what do you mean by algebraic notation? Algebraic operations are the usual pemdas while arithmetic operation are pmdas.

    1. I didn't really mean that Park expects you to know how to manipulate variables and solve variable equations. I just meant that they expect you to understand what variables represent and what an equation, entirely full of variable expressions, would be saying.

      Check out #11. It says "is m*n = n*m for all inputs of m and n?" That's the full generality and abstraction of algebraic equations right there. CME introduces that on Day 20 or so. Park Math introduces it on Day 1.

  5. Late to the conversation, and really liking a lot of the stuff on this blog.

    CME Algebra 1, Chapter 1 exists for a lot of reasons, but you've hit on the big idea of that lesson: when extending an operation to a new number system, the rules need to stay consistent. (-3) x (-4) = 12 because of the rules of arithmetic for positive numbers, and it's not arbitrary. We want students to act the same way each time they're exposed to a new extension -- fractions, radicals, exponents, complex numbers. It's especially effective in working through zero, negative, and rational exponents, but it's a good habit in lots of situations.

    As for the slow introduction of variables, we did this because too many students do algebra without the sense that the underlying variable can represent a number. These are the kids who say "7-x" when you ask them for 7 less than a number. But the same kids won't say "-13" when you ask them for 7 less than 20. In my opinion, abstracting directly to algebraic notation as Park does, especially to "mystery operations", will leave some of these kids in the dust. I want kids in Chapter 1 to almost demand variables: saying things like "you take whatever number you've got and then you..." Doing this makes the transition to variables much smoother. And I agree that it's very dependent on your students: some groups of students may already have had this experience in a prior course.

    By and large, though, I find that too many students are rushed directly to algebraic notation and manipulation, and frequently have no idea what is really happening -- for example, a lot of kids see the equation "2x + 1 = 7" and think "x is 3" when x can actually be any number, and 3 happens to be the only number that makes the equation true. This seems like a nitpick but it explodes when equations like "x + y = 5" or "x^2 = 9" come along.

    Let me know what other questions you have, and thanks for all the useful thinking about math. Keep looking at those tables, there are lots of great patterns, including a really nice way of dealing with quadratic factoring!

    - Bowen (one of the CME Project authors)