People that have lived in their city for a long time often know shortcuts that wouldn’t be suggested by Google Maps. Because they know the ins and outs of their town—which roads are always blocked by accidents, which light takes forever to turn green, and which back roads get you directly between Point A and Point B—they can take a more non-standard route that often saves them time in the end.
Similarly, mathematicians find efficient strategies that take advantage of the underlying patterns and relationships between the objects in the problem. They know the landscape of mathematics well enough to be able to anticipate which strategies will be more efficient than others. They can plan a solution “route” that leverages the best and most crucial features of the problem.
This is what it means to engage in Mathematical Practice 7 (MP7): Look for and make use of structure.
What is Mathematical Practice 7?
Looking for structure means recognizing something about the problem that makes it easy to solve. It is making use of a relationship embedded in the numbers or objects in a way that gives insight into the solution.
MP7 is the closest we get to problem solving. It’s being clever. It’s the positive aspect of laziness. Can I do this with less work? How can I take advantage of what I know, specifically mathematical relationships between the objects, to “crack” this problem?
A key observation is that a problem’s “underlying mathematical relationships” and “embedded properties” that we often refer to when talking about MP7 are essentially what we mean when we talk about conceptual understanding. When students look for and make use of structure, thinking is at the forefront.
The opposite of structural thinking, then, is doing the same procedure every time without attending to the features of the problem or underlying relationships with the objects in the problem.
This is the biggest difference between humans doing math and computers doing math. Computers must use algorithms, and because they can compute quickly and accurately, they can use an algorithmic approach to find an answer every time. Computers don’t analyze the underlying structure of the problem. They don’t look for patterns or shortcuts. Computers can only do plug-and-chug.
Humans can think. They can notice patterns and structure. They can detect nuances. They can learn from experience. They can use previous problems to solve new ones. They can find efficient strategies.
Examples of Structural Thinking
The easiest way to demonstrate the use of MP7, I believe, is showing how the same problem can be solved with and without the use of structure. You can also download a printable version of these examples to have as a classroom reference.
We’ll start with an example from elementary school.
Notice that the non-structural approach relies on a procedure that will work but is not generally associated with meaning and can often go haywire if even a tiny error is introduced.
In an Algebra 1 class, I like to ask a question like this:
Note that this structural approach takes advantage of a key property of linear equations—the constant rate of change!
And now for an Algebra 2 example:
Note that the non-structural approach is often longer because it requires going through a predetermined set of steps each time.
The same can be found in Geometry problems.
Structural thinking in this example is not about having memorized a rule (doubling a radius quadruples the area of the circle), it’s about identifying the area of a sector as being composed of two parts, the fraction of the circle we have and the area of the entire circle, identifying what changes, and being able to identify how that change affects the overall problem.
Without using structure, you often go backwards and then forwards. If you are given an area, you must go backwards and find the angle, then forwards to find the area. Making use of structure is the direct path, the shortcut. What’s the shortcut? Understanding the components involved in determining areas of circles and sectors!
In some cases, a problem is completely inaccessible without being able to make use of structure. Here's an example from Precalculus.
Operationalizing MP7
In the above examples, I tried to demonstrate a variety of ways of thinking and reasoning that are related to using structure. These can be summarized in the following ways.
Looking for and making use of structure means:
Recalling and using properties
Pausing to understand the problem instead of jumping right to an algorithm
Contemplating before calculating
Letting the numbers and objects in the problem guide the solution path
Using what you know and what you notice
Finding efficient strategies based on the features of the problem (the numbers, what is being asked for)
Chunking complicated objects into pieces you recognize
Changing the form of objects to make them easier to work with
Connecting mathematical ideas and representations to make meaning
Looking for and leveraging shortcuts
Recognizing something about the problem that makes it easy to solve
Breaking something complicated down into its component parts
People who use MP7 ask themselves:
What do I know about how this mathematical object (function, shape, number) behaves?
How is this problem similar to a problem I have solved before?
How can I avoid a calculation here?
What’s the easier version of this problem?
Can I rewrite or rearrange the parts to make this easier to work with?
Can I decompose the object into parts?
What can I safely ignore about this problem?
What do I know about this problem? What do I notice about this problem?
Let's see how a student might apply some of the habits of mind listed above to solve a problem from a quadratics unit.
Connecting ideas and representations: Finding the solutions to f(x)=0 means finding where the y-value is 0, which occurs at the x-intercepts of the graph.
Knowing and noticing: There will be two solutions, since there are exactly two x-intercepts. One x-intercept is already given. I only need to find one more. Usually writing the quadratic in factored form is an easy way to find the zeros and x-intercepts. The given x-intercept is not a “nice number” so I probably won’t be able to write the equation in factored form very easily. There must be something else I can use. The y-intercept seems irrelevant to the problem, but we’ll see.
Recall and use properties: A parabola is symmetric. Both zeros/x-intercepts must be equidistant from the axis of symmetry. I can find how far away each x-intercept is from x=4.
After this, the student can do the appropriate calculations. But notice how much of this problem didn’t have anything to do with calculations or algorithms. It was unpacking the structure of the problem and finding an efficient strategy for solving the problem.
How to detect when students are using MP7
This is where we often get things wrong, because students who are using MP7 often get their work critiqued as being “nonstandard” or “not how you’re supposed to do it”. I found that my students who were really leaning into using structure were the same ones that often refused to put a pencil to paper. They said they wanted to do the problem “in their head” but what I think they meant is that they wanted to do the problem “with their head”. They wanted to crack the problem!
When we present multiple solutions to a problem in class, oftentimes a student will present a solution done using the standard algorithm. An alternate solution presented by a student using MP7 often starts with “You could have just done …” or “I found an easier way to do it” or “I noticed that …so I could just…” Look for the word “just” in a student’s explanation. This is often a key indicator that a student is using MP7. Sometimes looking for structure feels like cheating or being lazy. But it is not! Seeing structure involves heuristic thinking which is a key skill of mathematics.
When other students respond to this alternate solution path they often say things like “Oh, that’s WAY easier” or “I didn’t think of that but that makes sense.” Being able to identify the structure of a problem often includes some kind of aha lightbulb moment and we know that these don’t always happen for all students at the same time.
Pitfalls when helping students develop MP7
Thinking of “structure” as a problem “type”; teachers will often teach their students to use certain “look-fors” to classify a problem as a certain type, in order to be able to apply a particular rule or algorithm. This is not what it means to make use of structure.
Teaching MP7 explicitly; there is no set of steps to follow to arrive at this skill. Students need multiple opportunities to solve rich problems and to be able to explain and compare their solution strategies. You cannot force or rush an aha moment.
Calling a student’s structural approach “advanced,” or “different”, or implying that it is for those students who have progressed beyond the standard algorithm. Instead, the teacher should help the student articulate what it was that they noticed about the problem that made it easy for them to solve.
Summary
MP7 is about being lazy, but in the best way. It is about finding efficient strategies and avoiding tedious calculations.
MP7 is about using what you know and what you notice. Students using MP7 will be able to identify similarities between a current problem and others they have solved in the past and they will notice when objects in the problem could be rewritten, rearranged, composed, or decomposed to make them easier to work with.
MP7 is about pausing, taking a step back from the problem to notice helpful features of the problem, before selecting a solution strategy. This is the opposite of how a computer would solve a problem, which is to execute the same algorithm regardless of what numbers, functions, shapes are at play.
The ability to look for and make use of structure is I believe one of the defining characteristics of good mathematical thinking. For this reason, it’s my favorite of the eight mathematical practices. When we teach for conceptual understanding we help students understand the innerworkings of a problem and detect its patterns!
Interested in learning more about the Mathematical Practices? Check out our post on MP2: Reasoning Abstractly and Quantitatively and MP8: Look for and express regularity in repeated reasoning.
Comments