Curricular Coherence Part 2: Evolution from Particulars to Deep Structures

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?

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

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.

Illustrative Mathematics Session at the Joint Mathematics Meeting

Illustrative Mathematics organized a special session at the Joint Mathematics Meeting on January 7, 2016 in Seattle, WA called Essential Mathematical Structures and Practices in K-12 Mathematics. Here is a description of the session:

The mathematics curriculum in the US has been shaped by myriad forces over the years, including the competition for market share among publishing companies, economic realities of school districts’ purchasing power, the ease with which teachers can deliver the material, traditional expectations of what mathematics classroom work should look like, and so on. Surprisingly absent from these forces is the nature of the discipline of mathematics itself. The focus of this special session was on identifying and describing the essential mathematical structures of the K-12 curriculum, as well as the key mathematical practices in the work of mathematicians that should be mirrored in the work of students in K-12 classrooms.