Author: Bill McCallum

Misconceptions about Multiple Methods

/ Bill McCallum / 12 Comments

You may have noticed that I am back to publishing regular blog posts! My goal for now is a blog post every second Wednesday. I am now also trying to answer forum questions promptly. I want to thank the readers who took up the slack for the last year and a half in answering questions in the forums. In particular, I’d like to call out abieniek, Alexei Kassymov, and Lane Walker, whose answers were always spot on.

Now to the topic of this post. There has been a lot of talk since the standards came out about what they say about multiple methods for arithmetic operations, and I’d like to clear up a couple of points.

First, the standards do encourage that students have access to multiple methods as they learn to add, subtract, multiply and divide. But this does not mean that you have to solve every problem in multiple ways. Having different methods available is like having different means of transportation available to get to work; flexibility is good, but it doesn’t mean you have to go to school by car, then by bus, then walk, then bike—every single day! The point of having multiple methods available is to encourage students to think strategically about what might be the best method for a given problem, not force them to solve every problem four times.

Second, the different methods are not unrelated; they form a progression, with the ultimate goal being the standard algorithm. For example, when students are first learning to multiply two digit numbers, they might use a rectangle to represent a product such as $42 \times 71$.

This shows the fundamental role of the distributive property in multiplying multi-digit numbers. You have to multiply each base ten component into each other one. Indeed, the same rectangle representation provides a visual proof of the distributive property itself.

At some later point students might just start writing down all the partial products, without using the rectangle to derive them.

Note the correspondence between the rectangle method and the partial product method, indicated by the colors. The first row of the rectangle and shows all the products by the 2 in 42 (in red); the second row shows all the products by the 40 (in blue). The products in the partial product method are grouped in the same way. There are many ways you can order the partial products, but if you group them as I have here, going from right to left in each two-digit number, as in the standard algorithm, you make an amazing discovery: you can add up all the partial products in each group (blue group or red group) in your head as you go along. That’s because, in each case, adding the 2 to the 140 or the 40 to the 2800, there are enough zeroes in the second addend to accommodate the first, so it is easy to write down the sum right away, without writing the addends separately.

OK, so it’s not always quite this easy, because every now and then you will have to keep in mind a bundled unit from the previous step (aka carrying), but you will never have to remember that for more than one step at a time, because each bundled unit gets used up at the next step. So if you invent a notation for remembering the bundled unit (what we used to call “little 1 in the corner” when I was growing up) then you can still avoid writing down all the partial products, and just compute the sum within each group as you go along. You have just created the standard algorithm.

The different methods are not isolated different ways of doing the same thing; they are steps towards fluency with the standard algorithm, fluency that is not fragile because it is supported by understanding.

Curricular Coherence Part 3: Using Deep Structures to Make Connections

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In this post I’d like to describe the third aspect of coherent content that Al Cuoco and I have been thinking about. In my fourth and final post on this subject I will talk about coherence of practice.

A difficult question in designing a curriculum is to decide which topics go together. The logical and evolutionary considerations described in my previous two posts help, in that they provide guidance on the ordering of topics. But that still leaves many decisions to be made. My goal this post is to show some examples of how deep structures can guide these decisions. (See my previous post for what we mean by a deep structure.)

CCSSM in 6th grade has the following standard about percents in the Ratio and Proportional Reasoning domain:

6.RP.A.3c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

One approach to implementing this standard in a curriculum would be to have a section on percents that covers everything in the standard. But there is another possibility which attends to the difference between the parts of this sentence before and after the semicolon. The first part introduces the concept of percent. The second half involves solving problems that are tantamount to solving the equation $px = q$, where $p$ and $q$ are constants. This is related to a standard in the Expressions and Equations domain:

6.EE.B.7. Solve real-world and mathematical problems by writing and solving equations of the form $x + p = q$ and $px = q$ for cases in which $p$, $q$ and $x$ are all non-negative rational numbers.

Thus another possibility might be to split the treatment of the percent standard into two places in the curriculum, with the introduction to percents occurring as a type of rate, in the section where ratios and rates are studied, and percent problems occurring in the section where solving equations is studied. If percents are regarded as a deep structure, one might choose the first arrangement; if rates and equations are regarded as deep structures, then one might choose the second. The second approach is the one we have taken in our soon-to-be-released middle school curriculum.

Another example of a deep structure is the profound connection between geometry and algebra. Imagine a 12 by 16 rectangle. Experiments with geometry software suggest that a square of side 14 maximizes area for the perimeter of this rectangle. If this is so, it should be possible to dissect the rectangle and fit the pieces into the square with something left over.

Trying several other rectangles of perimeter 56, a regularity emerges. Expressing this regularity in precise language leads to an algebraic identity that captures the dissection. Using an $a\times b$ rectangle, one has
$$
\label{eqagm} \left(\frac{a+b} 2\right)^2
-\left(\frac{a-b} 2\right)^2=ab
$$
This identity, inspired by geometric reasoning, can, of course, be verified in an algebra course. But its roots in geometry give it some extra meaning. And, it can be used to show how far off the rectangle is from the square.

Rather than separating the parts of this connection into two chapters or lessons, a coherent curriculum could use one story to develop both the necessary algebra and geometry, making it explicit that the main point is the connectivity of the ideas.

Curricular Coherence Part 2: Evolution from Particulars to Deep Structures

/ Bill McCallum / 2 Comments

In my previous post on curricular coherence I talked about how the principle of logical sequencing can determine the ordering of a set of topics. Since time is one-dimensional, and curriculum occurs over time, some principle for ordering is necessary. However, mathematics is not a linearly ordered set of topics; it is better viewed as a network. In this post I’d like to talk about deep structures. A deep structure is, roughly speaking, a node in the network of mathematical knowledge with many connections. Of course, this is not a precise definition; the organization of the subject into a network is to a certain extent a matter of judgment and preference, although some connections are dictated by the principle of logical sequencing. However, this will serve for a start in describing the principle of evolution from particulars to deep structures.

I’ll talk about two ways in which the such evolution occurs: extension and encapsulation. Extension is a process by which a particular principle is repeatedly applied to ever broader systems, thus revealing its nature as a deep structure. Encapsulation is a process by which a related array of concepts and skills becomes encapsulated into a single compound concept or skill.

Extension is exemplified in the way that arithmetic with whole numbers is extended to fractions, integers, and rational numbers through a program of preserving the properties of operations. The fact that $(-3)\times (-5)$ is 15 is a definition, rather than a theorem—it has to be that way if we want arithmetic with integers to obey the distributive property. The properties of operations start from observation of particular instances, and evolve into powerful deeper structures under-girding the number system.

A good example of encapsulation is the development of fractions. The standard 3.NF.A.1 expects students to “Understand a fraction $1/b$ as the quantity formed by 1 part when a whole is partitioned into $b$ equal parts; understand a fraction $a/b$ as the quantity formed by $a$ parts of size $1/b$” and 3.NF.A.2 expects students to use this understanding to represent fractions on a number line. These two standards encapsulate many prior ideas and activities: dividing a physical object into halves or thirds; recognizing a geometric figure as a fraction of a larger figure representing the whole; moving from area representations to linear measurement representations; understanding the number line as marked off in unit lengths; subdividing those lengths into $n$ equal parts and thinking of those parts as a new sort of unit, an $n$th, and measuring out distances in those new units; correlating all these activities to the numerator and the denominator of the fraction.

Encapsulation builds coherence by tying what were previously disparate ideas and actions into a tightly connected structured bundle which becomes viewed as an object in its own right.

Proportional relationships

An important type of encapsulation is the evolution of representations. Mature representations are a form of encapsulation, and should be developed through a sequence of intermediate representations whose structural features preserve information about the object being represented. In early grades students might start with pictorial representations; but even then the picture should be more than a picture: it should carry information about the situation. Over time, such pictures evolve into more abstract diagrammatic representations, and eventually these diagrams are replaced by even more abstract representations such as tables and equations. The figure shows such an evolution for representations of proportional relationships in middle school. That final equation $y=kx$ is very dense with meaning, or at least it should be so for students. By the way, this is the sequence of representations for proportional relationships that we use in our new middle school curriculum coming out in July.

What Does It Mean for a Curriculum to Be Coherent?

/ Bill McCallum / 3 Comments

Al Cuoco and I have been thinking about this question and have developed some ideas. I want to write about the first and most obvious one today, the principle of logical sequencing. I’ll write about others in the weeks to come.

Remember the distinction between standards and curriculum. While standards might remain fixed—a mountain we aim to help our students climb—different curricula designed to achieve those standards might make different choices about how to get there. Whatever the choices, a coherent curriculum, focused on how to get students up the mountain, would make sense of the journey and single out key landmarks and stretches of trail—a long path through the woods, or a steep climb up a ridge.

By the same token, mathematics has its landscape. CCSSM pays attention to this landscape by laying out pathways, or progressions, that span across grade levels and between topics, so that a third grade teacher understands why she is teaching a particular topic, because it will help students with some other topic in the next grade and build on what they already know.

This leads us to the first property of a coherent curriculum: it makes clear a logical sequence of mathematical concepts.

Consider, for example, the concepts of similarity and congruence. It is quite common in school curricula for similarity to be introduced before congruence. This comes out of an informal notion of similarity as meaning “same shape” and congruence as meaning “same shape and same size.” However, the fact that the informal phrase for similarity is a part of the informal phrase for congruence is deceptive about the mathematical precedence of the concepts. For what does it mean for two shapes to be the same shape (that is, to be similar)? It means that you can scale one of them so that the resulting shape is both the same size and the same shape as the other (that is, congruent). Thus the concept of similarity depends on the concept of congruence, not the other way around. This suggests that the latter should be introduced first.

This is not to say you can never teach topics out of order; after all, it is a common narrative device to start a story at the end and then go back to the beginning, and it is reasonable to suppose that a corresponding pedagogical device might be useful in certain situations. But the curriculum should be designed so that the learner is made aware of the prolepsis. (Really, I just wrote this blog post so I could use that word.)

Although the progressions help identify the logical sequencing of topics, there is more work to do on that when you are writing curriculum. For example, the standards separate the domain of Number and Operations in Base Ten and the domain of Operations and Algebraic Thinking, in order to clearly identify these two important threads leading to algebra. But these two threads are logically interwoven, and it would not make sense to teach all the NBT standards in a grade level separately from all the OA standards.

In the next few blog posts, I will talk about three other aspects of coherent curriculum: the evolution from particulars to deeper structures, using deep structures to make connections between topics, and coherence of mathematical practice.

Catching up

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The site was down for a few hours today because of a malware attack, but I think we have it fixed now.

I took the opportunity to catch up on comments in the forums; I was way behind! Thanks to all those who responded to readers’ questions. I will try to stay more on top of it. One of the things that has been keeping me busy is our work on grades 6–8 curriculum for Open Up Resources. It is being piloted this year, so that link is still password protected, but stay tuned!

Also, I am close to finishing up the Quantity Progression, the last one not yet done.