Math Practice At Home: Finding and Making Use of Structure

The Common Core State Standards (CCSS) include eight “Mathematical Practices” that teachers and students are encouraged to use throughout their mathematical lives. These are different from the grade-level content standards in a few ways. Firstly, these Mathematical Practices are not grade-specific. Rather, they remain constant practices throughout the child’s education, with the hopes that they become habit. Further, as the words practice and habit imply, these are not mathematical procedures or rules, but are more akin to mindsets and broad strategies.

For a broad look at all eight Math Practices, check out the post What Are the Math Practices and What Do They Have to Do With Me? For a deeper dive into the other seven practices, check out the list at the bottom of this post.

The seventh principle is ultimately about making life (and math) easier, more manageable, and more connected by observing common patterns and using them to simplify or better understand problems. Officially, the statement reads that “mathematically proficient students look closely to discern a pattern or structure,” but the examples that follow clarify the uses and purposes of these observed structures. The examples vary from making an arithmetic problem simpler to compute, to recognizing common properties within arithmetic problems, and being able to view both the larger problem as well as individual pieces of the problem.

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Using structure to simplify a problem

Robert Kaplinsky, a math coach for teachers and prolific content creator, gives two great examples of using mathematical structure to simplify a problem in his post on this standard. He takes the view that this is the primary purpose of the standard. He’s not wrong. We’ll see in the conversations below that ultimately these two purposes swing back to making both math tasks and future math learning manageable.

Some additional examples of using structure to simplify a problem include

Adding (or subtracting) the same constant to both parts of a subtraction problem to avoid regrouping (or borrowing in previous terminology). Howie Hua, a teacher of math teachers, provides an example in his short video. This uses the structure of subtraction as a means of finding the distance between two numbers combined with the structure of moving/changing two numbers together so that their distance remains the same.
Understanding subtracting a negative as adding a positive. Most ways of introducing this also use the structure of subtraction as a means of finding the distance between two numbers on a number line.

In algebra, moving between different equivalent forms (such as factored form, graph, standard form, vertex form, etc) of a polynomial to easily see different characteristics and patterns. This uses a wide variety of different structures as well as the structure (properties) of equivalency.

Using structure to recognize (and use) common properties

There are many reasons to recognize and use common properties. There are whole fields of math research dedicated to this (number theory comes to mind), and in fact it’s an underlying theme of mathematics to discover these properties and see explore how they apply in a variety of situations.

However, the one main purpose of these properties that most of the audience reading this will understand and accept is to make the problem simpler to do. In that way, this part of the standard is really a continuation of the first part.

Multiplying by 11 ($\times 11$) by multiplying by 10 ($\times 10$) and multiplying by 1 ($\times 1$) and then adding them together. This uses the structure of the distributive property* to simplify computations.
Adding $239+52$ by instead adding $240+51$ uses the associative property of addition (by thinking of the $52=1+51$ and “associating” the $1$ with the $239$ instead).
Multiplying $2 \times 32 \times 5$ by instead multiplying $2 \times 5 \times 32=10\times 32$ uses the commutative property of multiplication.

*Technically the distributive property of multiplication over addition, though since it’s the most commonly used of the distributive properties, we usually leave out the last four words.

Viewing the problem at different zoom levels

Being able to view both the parts and the whole of a problem is useful throughout life. Math is a great place to practice that skill, as well as to see the payoff.

Break an image, such as a complicated geometric figure, into smaller parts that are easier-to-deal-with shapes. (In this, math is a lot like drawing!)

In using the order of operations, looking at the larger problem to recognize which operations are present is important to determine where to start. Then focusing in on the individual operations is important to making progress through the steps.
Simplifying an expression or solving an equation in algebra uses the same skills of zooming out and in as the order of operations problems. It also uses the bird’s eye view of knowing what the problem is asking for and what the original structure actually is (simplifying or solving; expression or equation).

How to practice recognizing structure at home

Sort groups of objects

The standard itself suggests a powerful way to begin recognizing structure in different forms: have your child sort objects according to whatever characteristics they see. Hand them a pile of objects and ask them to sort them into groups. Maybe take turns sorting them with your child, using different characteristics you see to model how they might sort in different ways and by different characteristics (or structures). If you have a variety of paper clips and binder clips of varying sizes and colors, you might give them a handful and have them sort. They might sort into paper clips vs binder clips. They might sort into colors. They might sort by size. They might do all three in turn. Hand them a pile of crayons and they might sort by color, by pointiness, by length, by brand. Sorting piles of coins, shapes cut out of construction paper, cookies, and more can help them learn to recognize structure.

Play Which One Doesn’t Belong?

The idea behind this “game” is to examine the structure of 4 pictures and recognize similarities and difference among them. In most cases, there is at least one reason that each image “doesn’t belong” or is different than the others. They each belong when looking at certain characteristics and each don’t belong when considering other characteristics. I always thought of this as an analogy to humanness and our desire to belong and tendency to “other” each other. Thus, I also see this activity as a good way to introduce ideas of acknowledging how we are all the same and yet all different and how that’s okay (and even good!)

There are a variety of sources to get you started with this activity:

The website wodb.ca has a wide variety of images to use for this activity. Simply choose an image and ask “Which one is different?” or “Which one doesn’t belong?” (your preference; increasing numbers of people prefer the first).

The book Which One Doesn’t Belong? (affiliate link) by Christopher Danielson uses images accessible to children of all ages (including very young) in a format they can touch and point to.
The hashtag #wodb has been used to share user games throughout social media.

Pick an image and take turns explaining which one is different than the others and why. Eventually see if you can create your own!

Ask “How is this [situation] like [that situation]?”

When you notice something happening in your family life that reminds you of a different event, ask your child if they can see the resemblance too. This may not be directly mathematical, but it is very much structural. It can become a highly useful skill in noticing underlying structure in word problems in particular. This develops the skill of noting underlying structure despite outward context, which is the basic essence of determining how to solve a problem. Model drawing these connections and noting these similarities and your child will follow.

Practice finding “similar problems” in a homework set

If there are multiple problems in a homework set or a worksheet, spend time looking at how they are the same and how they are different. For example: Do they all use the same multiplication process? If so, how do you know? Are they all addition, or are some subtraction? How do you know? These types of analysis set your child up for structural thinking that takes them beyond the sometimes-overwhelm of context and allows them to solve problems correctly even when they aren’t familiar with the context.

Ask “How Many?”

I’ve used this example many times in this blog and on my youtube channel. It’s so incredibly multipurpose that I can not overemphasize its utility across the skills spectrum. It’s a lighthearted, multi-age, multi-skill game that supports this particular skill in ways similar to the Which One Doesn’t Belong game discussed above. It requires no special images, skills, or equipment and can happen anywhere and anytime just going about your day and through the world.

Even though it does not require anything special, if you do want a source to get started, consider