Today we're going to unpack one of the arguably least understood of all the Mathematical Practices: MP8!
This is the third post in our Mathematical Practices series. To see the previous two posts, check out the links below.
Defining MP8
Mathematical Practice #8 (MP8) states that mathematically proficient students are able to “look for and express regularity in repeated reasoning”. What does this mean? MP8 is the ability to notice patterns (elements that are repeating) when doing similar calculations in order to find a general approach. MP8 often requires investigating a set of related problems instead of solving a single, standalone problem.
This can look like:
Generalizing a relationship, pattern, or formula
Extending a pattern to a different case
Using past problems as a model for current problems
Trying several numbers and observing the process
Identifying parts of a process that are repeating
Using inductive reasoning to make conjectures
Attending to repetition in counting, constructing, and calculating
Being aware of one’s reasoning in a problem
Evaluating throughout if answers make sense
Students using MP8 ask themselves:
What was my process? Was it the same every time?
What about this process is repeating?
Do I keep doing the same thing over and over again?
How can I generalize the repetition?
Can I predict what will happen in this problem based on my work in previous problems?
What seems to be the rule here?
Will this method always work?
MP8 is less about noticing patterns in answers but about noticing patterns in the processes that generated those answers. This can be a tool for “shortcutting” the process to include only its most essential elements.
For example, suppose a student is given that the 2nd and 5th terms of a geometric sequence are 8 and 27, respectively, and asked to find the common ratio. A student might begin by drawing a diagram like the one below. They could then write an equation involving the common ratio, r, that could be used to solve for r.
Doing a few problems of this category should illuminate the repetition in the process. I can shortcut the process by noticing that each time I solve for the common ratio, I investigate the ratio of the given terms which represents some power of the common ratio. This power is determined by how many terms are between the two given terms. I can undo that power to find the ratio between consecutive terms, i.e. the common ratio.
Later, when I am given the 4th term and 11th term of a geometric sequence and asked to find the common ratio, I can “shortcut” the process and use the expression below to find the common ratio.
Warning: it is a huge disservice to offer this generalization to students ahead of time (and think about all the notation that would be needed to communicate this precisely!). We often think that giving students the shortcut up front will help them be successful (especially our "struggling" students) but it actually does the opposite. It short-circuits their mathematical thinking and reasoning by avoiding the productive struggle!
The complexity of MP8 is this:
You cannot prescribe how many times students will need to “repeat the reasoning” before they are able to generalize! We have to avoid the rush to the generalization that forces students to plug and chug using an algorithm they don’t understand.
Giving students the rule or method ahead of time and then just asking them to apply it to 20 practice questions does not foster their mathematical sense-making. They need time to experience that repetition so that they can own the method.
Examples of MP8
In this section we'll look at EFFL lessons from all of our Math Medic courses that help students put MP8 into action.
Algebra 1: Solving quadratics using the zero product property
After repeated problems of a similar type, students notice that if two quantities multiply to zero, then one of those quantities must be zero. This then offers a generalized approach for solving for the zeros of a quadratic—write the quadratic as a product of two factors and then find the values of x that make each factor equal to zero. See the full lesson.
Geometry: Finding the sum of the interior angles of any polygon
In this lesson, students use an applet to determine the sum of the interior angles of many different polygons. Specifically, they note that there is a predictable structure to these polygons related to the number of triangles that can be drawn from one vertex. After looking at multiple cases, students generalize that the number of triangles is always two fewer than the number of sides in the polygon and that since the sum of the angles in each triangle is 180˚, there is a pattern to how the sums of all the interior angles of polygon are generated.
Algebra 2/Precalculus: Writing equations of lines in point-slope form
Students start with the cost of a 4-topping ice-cream and are asked to find the cost of ice cream with more toppings (5 and 7 toppings in question 3). By doing repeated calculations of this form, students see that each time they start with what they already know (the cost of a 4-topping ice cream) and add on the additional cost which is determined by the number of additional toppings and the cost per topping. The margin notes make explicit where the numbers 1 and 3 come from in their explanations and what they represent (MP2), which is crucial for determining the (x-4) portion of the general equation. See the full lesson.
Precalculus: Writing an inverse function
In this lesson, students start by being given a function that gives the weight of an adult dog (dependent variable) based on the number of kilocalories in its diet (independent variable). They are then asked about the reverse: if they know the ideal dog weight, can they determine how many kilocalories they should feed the dog to attain that weight? By repeatedly going through the process of being given an adult dog weight and undoing the equation to solve for the number of kilocalories, students note that there is a predictable pattern to these calculations. The ideal adult dog weight is raised to the ¾ power and the result is multiplied by 130. Thus, this pattern can be generalized to find the number of kilocalories needed to attain an adult dog weight of any number of kilograms, k. They have just found the inverse function!
What’s the difference between MP7 and MP8?
The final two mathematical practices are often grouped together. They both focus on strategies mathematicians use to make sense of and solve problems. They even use very similar language!
MP7: Look for and make use of structure.
MP8: Look for and express regularity in repeated reasoning.
While many problems or tasks do require students to use a combination of these mathematical practices, it is worthwhile to clarify their differences. Why? It's not because it's important for students to identify whether they're using MP7 or MP8. It's because by being explicit about the different strategies that mathematicians use, we give students greater access into problems. We are helping them add tools to their toolbelt. Clarifying definitions helps us clearly articulate our instructional goals and find the right kinds of tasks to use with our students. The table below summarizes the major differences.
The long and short of it?
Through whatever content we're teaching, we should support students in identifying repetition in calculations or processes so that they can develop their own shortcuts or generalized approaches. The goal is to get students to think and reason for themselves, not to blindly adopt the traditional algorithm or our own preferred method.