In previous posts, we talked about MP 1--what it means to make sense of a mathematics problem (part 1) and persevere in solving it (part 2). Today, we'll continue Math Practice #2: Reasoning abstractly and quantitatively. What does it mean to reason abstractly and quantitatively? What does it look like? What does it look like to struggle with either or both?
Like MP 1, MP 2 has two distinct parts:
1) Reason abstractly
2) Reason quantitatively.
To understand the real significance of the second practice, we need to understand each of these two pieces, as well as why they are presented together in a single practice. After thinking about this for a bit, though, I find it easiest to explain if I turn it around, and we talk first about reasoning quantitatively, and THEN about reasoning abstractly.
Reasoning quantitatively is one avenue of making sense. (Many thanks to Grace Kelemanik and Amy Lucenta for codifying this framework, and to my former colleague at WestEd, Cathy Carroll, for bringing it to my attention.) When we reason quantitatively, we're thinking about the quantities involved (goats, apples, meters, seconds, miles per hour, the x-coordinate, degrees, radians, the difference between functions f and g, etc.--basically, anything we can count or measure) and considering their relationships to one another (How does the number of goats compare to the number of apples? What is the relationship between the number of minutes elapsed and number of feet traveled? How is the x-coordinate related to parameter t?)
In a previous Math PD Seminar, we spent some time working with teachers on ways to support students with MP 2. The problem stem we used for a lot of our discussion was the following:
Two thirds of the students in Parkview Elementary School wear something red during the last School Spirit Day. Of the students wearing something red, half of them were wearing red hats. Of the students wearing red hats, two thirds of them are boys. 53 girls were wearing red hats.
Source: Fostering Math Practices
We call it a problem stem because, as you may have noticed, there's no question to answer at the end. The purpose of leaving off the question was to focus on making sense of the problem, rather than immediately gunning for the answer. We wanted everyone to really focus on reasoning quantitatively as they thought about the problem stem, about what quantities are involved, and how they are related.
If we were to peer into the mind of, say, a 5th grade student who is reasoning quantitatively about this problem, we might witness an inner monologue that goes something like this:
- "Okay, two-thirds of kids in the school wear red [Identifying a quantity explicitly mentioned in the problem]. So I'm going to draw a pie chart, divide it into thirds, and outline 2/3 in red. So I can see that 1/3 of kids are NOT wearing red [Identifying an implicit quantity not explicitly mentioned in the problem stem], which is only half the amount of kids wearing red [Identifying a relationship between two quantities, kids wearing red and kids not].
"Of the students wearing something red, half of them are wearing red hats.' Okay, so students wearing red, that's my 2/3 that I outlined. [Identifying a relationship] And half of them are wearing red hats. Well, I can see from my diagram that half that amount is 1/3 [Relationship]. So 1/3 of the kids in the school are wearing a red hat [Relationship]. And this other 1/3 is kids who are wearing red, but the red isn't a hat [Implicit quantity].
"Let's see...2/3 of the kids wearing red hats are boys. So where's 'kids wearing red hats' [Quantity]? Oh yeah, it's this 1/3 I labeled here [Relationship]. So I need to divide that part into thirds...And let me shade two of those parts and label them 'boys in red hats.' The leftover 1/3 there has to be girls then [Quantity AND relationship], and the problem tells me there are 53 of them, so I'll label that too."
Thus far in their thinking, this student has not performed any calculations--but they've done a heck of a lot of math! All this business above is an example of reasoning quantitatively--making sense of the problem by identifying explicit quantities of interest, using those to identify implicit quantities of interest, and then determining how those quantities are related to one another. As the "official" definition of SMP 2 puts it,
- "Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects."
Reasoning quantitatively is one way to making sense of a problem, and if that's the approach I take, I can't really do anything calculation-wise until I've understood what quantities are important and how they are related.
Next time we'll look at reasoning abstractly, and how these two skills complement each other and help us solve problems that are presented verbally or narratively rather than just symbolically.