One of the most valuable achievements of the Common Core State Standards, in my opinion, was the identification of the 8 Standards for Mathematical Practice. Here we get a glimpse of what it means to actually do mathematics, rather than just a set of statements about what students should know. Regardless of what standards your state has adopted, it seems pretty universally agreeable that as developing mathematicians, students should be able to
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
But these are way more than just skills that need to be taught. These are habits of mind that need to be developed. How do we do this?
The first step is the very non-trivial task of understanding what these statements actually mean. Even more than a decade after the launch of these standards, some of these still seem pretty fuzzy. Practices 1, 3, and 6 seem to get all the air time and I think make the most intuitive sense. But what about the others? What exactly is the difference between practices 7 and 8? What do they mean by “regularity in repeated reasoning”?
We’ll tackle these questions in upcoming blog posts but for today, we’re going to unpack what might be the most misunderstood of all the practices: Reason abstractly and quantitatively. Now, you might be thinking that this one is actually one of the most clear. Reasoning is a good math-y word, as is abstractly (that describes 90% of upper-level math!), and quantitatively just seems like the bread-and-butter of mathematics. We love numbers!
It wasn’t until about 2 months ago that I truly understood what it means to “reason quantitatively”–and turns out it doesn’t have anything to do with being able to deal with numbers!
Definitions
A quantity is an attribute that can be measured or counted. When we think of quantities, we think of marbles, money, or M&Ms, not abstract concepts like the golden ratio or the number pi. Math gives us a shorthand for referring to those quantities, using symbols like m to represent the number of marbles. The symbol m isn’t just a letter or a label, it represents a quantity of real-life marbles. Now that we have this shorthand we can use expressions like 0.25 m to represent how much money you earn for selling m marbles, if you sell each marble for $0.25. You can use the symbolic mumbo-jumbo of m/25 to represent the number of bags you need for your m marbles if you can fit 25 marbles into one bag. In this case m/25 isn’t just the division problem of m divided by 25, it represents the number of bags needed (a quantity).
Quantitative reasoning is not numerical reasoning. It is not being able to flexibly manipulate numbers, or even algebraic expressions.
Quantitative reasoning is the ability to interpret numeric and algebraic symbols as representing quantities and the relationship between them. Oftentimes it is the ability to use symbols to capture the relationships in a contextual situation.
Once I learned this, I started seeing quantitative reasoning EVERYWHERE in our curriculum. And noting that the CCSS writers deemed it worthy of being its very own mathematical practice, I finally realized why students sometimes struggled to make meaning of the expressions and equations in the contexts we introduced. Quantitative reasoning is a complex practice that needs to be developed over time, in developmentally appropriate ways.
Let’s look at some additional examples so we can make this more concrete.
Patty's Driveway
Suppose there are 2 inches of snow on Patty’s driveway. At midnight it starts snowing at a constant rate of 1.5 inches per hour. And suppose now we wanted to write an equation for S, the height of snow on Patty’s driveway, t hours after midnight.
Well, that’s easy.
We might ask students what S represents (the height of snow), and what t represents (the number of hours after midnight). We might ask them about the y-intercept and what that means (y-intercept of 2, represents the inches of snow already on the ground at midnight).
Okay, but what about this one: what does 1.5t represent?
If you said the rate at which snow is falling, I think you would be in the vast majority of people. But the rate at which snow is falling is given by the slope, which is just 1.5, not 1.5t. The quantity 1.5t represents the number of inches of snow that have fallen in t hours. At 4 AM, 1.5(4)=6 inches of snow have fallen, and since there were already 2 inches of snow on the ground, Patty’s driveway must now have 8 inches of snow on it. This may seem obvious, but it is not to students! The most common wrong answer students come up with for their equation is S=2t+1.5. They know the variable goes with one of the numbers, but besides just following the pattern of y=mx+b, what good reason can we provide for why it “goes with” the 1.5, and not the 2? Even if we say that “the variable goes with the slope”, why are we attaching the number of hours to the rate at which snow is falling rather than the initial amount of snow?
A student who is reasoning quantitatively would be able to answer this question. They would see that the equation S=2+1.5t is really saying
The amount of snow on the driveway at a certain time =(is) the amount of snow that was already on the driveway + the additional snow that came down since midnight.
How do we measure that “additional snow”? If it snows 1.5 inches every hour, for t hours, then we have a multiplicative relationship. The additional inches of snow is given by 1.5*t. I could go on about the reasoning necessary to interpret 1.5t as the product of two quantities which represents a new quantity, rather than just a number and a variable that are positionally next to each other. But I won’t.
A student reasoning quantitatively would then not be too troubled when a few days later they encounter an alternate equation to represent the same scenario:
What quantities do those symbols represent? S is still the number of inches of snow on Patty’s driveway after t hours (thank goodness!). But our “initial value” doesn’t represent the amount of snow on the ground at the start (t=0, or midnight) but rather the amount of snow on the ground at 2 AM. It’s still a “starting value” of sorts, but as it turns out “starting values” or “anchor points” as we like to call them, can be any point on our line (or parabola, or exponential function). It is simply a reference point. The expression (t-2) represents the number of hours that have passed since 2 AM, or the horizontal distance we are away from our anchor point (not exactly quantitative reasoning but a helpful graphical interpretation nonetheless). So what does 1.5(t-2) represent? It represents the number of inches of snow that have fallen since 2 AM. Note that the underlying relationship is exactly the same!
The amount of snow on the driveway at a certain time =(is) the amount of snow that was already on the driveway + the additional snow that came down.
Okay, but what if that certain time is 1 AM? Our equation doesn’t make sense now! Or does it?
The amount of snow on the ground at 1 AM is the amount of snow on the ground at 2 AM, but we have to remove that final hour of snowfall. We are undoing, or reversing, the accumulation of snow that happened during 1 AM and 2 AM, which can be represented symbolically as 5+1.5(-1) or 5 inches-1.5 inches=3.5 inches.
And for all you Calculus teachers out there, consider this problem.
Suppose there is some amount of snow on Patty’s driveway at midnight. At midnight it starts snowing at a rate given by the function f, measured in inches per hour. How could we write an equation for the height of snow on Patty’s driveway, 7 hours after midnight?
How would a student who is reasoning quantitatively make sense of this? The answer is left as an exercise for the reader.
Now that we know the meaning of reasoning quantitatively and are convinced of its importance, what are some ways we can develop this kind of reasoning in our students? Check out our related post with 4 tips for helping students reason quantitatively.
Interested in learning more about the Mathematical Practices? Check out our post on MP7: Look for and make use of structure and MP8: Look for and express regularity in repeated reasoning.