This is the third follow-up post to a previous one aimed at parents and other non-math teachers, The Real Power of Studying Mathematics, about the significance of the Standards for Mathematical practice.
MP 1, Part 1: Making Sense of Problems
MP 1, Part 2: Persevering in Solving Problems
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In our last two 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? And what can parents do to support students?
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.
Reason Quantitatively
Reasoning quantitatively is one avenue of making sense. (Many thanks to Grace Kelemanik and Amy Lucenta for codifying this framework, and to my 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?)
At a recent 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.
Reason Abstractly
What does it mean to reason abstractly? In contrast to thinking carefully about the various quantities and units and how they are related given the context of the problem, reasoning abstractly means stripping away the quantities and units and other referents and working purely symbolically.
If I give kids a bunch of "naked" calculations to perform such as 24 x 7, without any context or real-world story behind it, I'm asking them to reason abstractly. The same goes for solving algebra equations like 7x + 9 = 30. There are no goats or apples or radians to think about, and all the relationships are spelled out for us symbolically; it's all just numbers and operations.
Reasoning abstractly does play an important role in solving real-world / contextual problems like the one above, though. Say for example that we give a kid a problem like:
- "There are twice as many kids wearing hats as there are kids with no hats, and there are 27 kids in all. How many are wearing hats and how many are not?"
The quantitative part is being able to say, "Ah, yes, I'm adding together two groups of kids where one is twice as big, so I need to compute 1/3 x 27 (or) 27 / 3 (or) 2/3 x 27 (or) solve 3x = 27, etc." The quantitative part is determining what calculations you need to do or what equation you need to set up or what have you, based on the quantities and relationships involved.
The abstract part is actually doing those calculations or solving that equation. It's reasoning through the symbolic manipulations in a way that has nothing to do with what the numbers or symbols actually refer to in the problem. The abstract part is, once I know I can find the answer to my problem by solving 3x = 27, being able to reason my way through with the numbers and symbols and operations and actually get an answer:
- "Ah, okay, so three multiplied by something is 27. So that means I can find the 'something' by going backwards and dividing 27 by three, which is 9."
(OR)
"Right, three times blank equals 27. That's a fact family I know, the missing number is 9, because 3 x 9 = 27."
Now, don't get me wrong; using the context to solve a problem is an important skill. In fact, pedagogically, having kids use a real-world context to reason through a problem is a critical tool for helping them to notice mathematical patterns and relationships and start to make sense of shortcuts they can use to solve problems more efficiently. One of the biggest problems in all of US K-12 math is that kids get pushed too quickly from solving problems intuitively in ways that make sense to them into abstract algorithms and procedures, before they've had time to see the connections between the two and transfer their understanding from the concrete to the abstract.
HOWEVER, it is still a skill we want them to have once they're ready.
Take this problem, for example:
- "I know I need to buy 4 packages of hot dogs and 5 packages of buns and that hot dogs cost $8.50 a package and buns cost $4.75, and then also there is 8.75% sales tax. How much is that going to cost me?"
It's completely feasible and logical to solve this problem by thinking:
- "Okay, I need 4 packages of hot dogs, and hot dogs cost $8.50 per package, so that's four groups of packages at $8.50, which is $34 for hot dogs. And then 5 packages of buns which cost $4.75 each, so 5 groups of buns at $4.75 each, so that's $23.75 for buns. Then $8.50 for buns combined together with $23.75 for hot dogs, that's $57.75 for the food. Then 8.75% sales tax, so if $32.25 is 100%, then 1% is $.3225. 8 of those is $2.58 for tax, and then there's the .75%..."
In fact, as kids first start to work with decimals and percents and performing various operations with them, this is 100% how I would want them to do it. I want them to take their time and think through the various quantities and relationships and use their common sense to figure out what should be added and what should be multiplied and with what and when and why. This is making sense.
However, I don't want kids solving that problem that way forever. I don't want them to have to constantly refer back to dollars and hot dogs at every intermediary step. Eventually, I want them to feel secure enough in their quantitative reasoning to read this problem a few times and then go, "Right, I can set up this calculation to find the total cost:"
As they carry out these calculations, they are NOT thinking about dollars and hot dogs; they're just doing calculations with numbers. This is abstract reasoning.
In our Parkview Elementary problem above, if we wanted to find the total number of students, the reasoning abstractly might look like:
159 * 3 = 477
The "reasoning abstractly" part is the part that a lot of people tend to think of as "the math part," but in reality, it's only part of solving a math problem, and we can't get there without doing the sense-making part first (even if for some types of problems we did it the sense-making a long time ago and figuring out what calculations to do is now second nature).
Reason Quantitatively...Again!
Even once we've translated everything into symbols and reasoned abstractly according to what we know about mathematical operations and equations, we're not quite done. Once I've got my answer, I still need to go back to the context of the problem and do a little quantitative reasoning in order to figure out what the answer really means.
Not quite sure what I mean by that? Take a minute to consider the following four division problems:
- Candace has 7.44 meters of ribbon from which to make 60 bows. How much ribbon can she use for each bow?
- The lunch budget is $744 and there are 60 students on the field trip. How much can each student spend on lunch?
- There are 744 students going on the field trip and 60 students fit on one bus. How many buses are needed?
- You have 744 grams of chocolate and it takes 60 grams of chocolate to make one cake. How many cakes can you make?
For all four problems, we can reason quantitatively and determine that we need to divide 744 by 60; then we can reason abstractly by actually performing the calculation 744 ÷ 60 and getting 12.4.
But I hope we can all agree that the answer to all three problems is not 12.4.
In order to determine what the answer is, we have to go back to the problem and revisit our quantitative reasoning. In order to answer question, should we round up, round down, leave it how it is, or adjust the format of the number slightly? (And then, of course, there is the question of units.)
Teachers often say to students, "When you finish the problem, always be sure to go back and make sure you've answered the question that was asked." That's SMP 2, right there!
What Can Parents Do?
(**Caveat**: It's always a little tricky to make suggestions to parents because a) not all parents have the luxury of being available regularly to support their kids with homework; b) even for those who do, or can shuffle things around to make the time, it often comes at great opportunity costs; c) parents have a limited amount of control over what and how much work kids are being assigned, how it's being presented at school, etc.; and d) when kids are struggling with school work, emotions between parents & kids can run high & parents trying to help can start to feel like torture for both parties.
So I don't want to come across as if I'm suggesting you're responsible for doing your kid's homework, or teaching them what they didn't learn in class for whatever reason, or even sitting patiently next to them night after night until they're done. But if you do have interactions with your kids around their math work, particularly if they're struggling with something, here are a few things to keep in mind that help support students' ability to persevere in solving math problems.)
- Say it with me now: Focus on sense-making. Everything in this post applies, because reasoning quantitatively is one specific avenue of making sense of problems. Doing calculations is a waste of everyone's time unless and until kids understand WHY those calculations make sense in the context of the problem.
I think there is a lot of power in the "Capturing Quantities" routine created by Grace Kelemanik and Amy Lucenta, and if you want to borrow from it to help your student with word problems at home, you might try asking them
- What are the quantities in this problem (that is, what can I count or measure)?
- How are those quantities related to each other?
- Can you draw a picture or diagram that shows those relationships? (Or even just one relationship at a time.)
- Go back to the question. I don't know if you have any idea how many times math teachers look at student work where s/he's done everything right *except* go back and answer the actual question, which includes interpreting the numerical answer correctly. Sometimes kids need a reminder to not just do the calculations and stop.
- See if they can explain what they've done so far. This is particularly useful with abstract reasoning, if a student is struggling with the symbolic process of doing a calculation or solving an equation. Questions like, "Can you tell me where this expression/calculation/equation came from? What do the numbers mean? Why are you adding/dividing/squaring?" Sometimes when students struggle with abstract reasoning, it's because they've lost sight of the meaning behind the symbols or where they're trying to go. On the other hand, these questions might also uncover deeper gaps in the meaning behind particular operations or relationships or symbols, and that is also useful information. Identifying the problem is a crucial first step in solving it.
Next time: We take a look at Practice #3, "Construct viable arguments and critique the reasoning of others"!
(Thank you for reading this far! I consider these posts to be living works in progress, so please do feel free to share any thoughts, suggestions, or questions. I want to make them as accurate and useful as possible!
