# SMP #2: Reason Abstractly and Quantitatively

by Pam Harris (@pwharris), author of Lessons and Activities for Building Powerful Numeracy

The second Standard for Mathematical Practice states: “Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. 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.”

If we encourage students to reason quantitatively (attending to the relationships between the numbers and using properties flexibly), they can abstract to general principles that can help solve other seemingly unrelated problems. Here’s one example.

Consider these two different problems:

The first is an example of what is called quotative division. The problem can be thought of as “how many 12’s in 252?” or “12 times what is 252?” That thinking might be recorded as 12 x ____ = 252.

A solution modeled with equations:

12 x 10 = 120
12 x 20 = 240
12 x 1 = 12
12 x 21 = 252
So 252 ÷ 12 = 21

Modeled in a ratio table:

After finding the answer, we need to re-contextualize: 252 minutes divided into 12 minute chunks per students means that the counselor has time to meet with 21 students.

This quotative problem feels multiplicative—you have the total (number of minutes) and the number in each group (how long the meetings are) and you need to find the number of groups (how many students). Many students (and adults!) feel like solving this type of problem using a multiplication strategy.

The second problem is an example of partitive division; it can be thought of as breaking 252 into 12 chunks and finding the size of the chunk. Many beginning students do not feel like this situation is multiplicative at all. Instead they feel like parsing the 252 minutes into 12 chunks (students) and finding out how many minutes are in each chunk. Students might draw 12 circles and deal out tally marks until they have dealt out all of the 252 minutes and then ask how many minutes ended up in each group. This can be tedious and very inefficient.

What if, instead, we encourage students to use the more efficient thinking for problems where it makes sense to do so? We could reason that if it takes 252 minutes for 12 students, then it would take half as long, 126 minutes, for half as many students, 6. It would also take half that time, 63 minutes for half the students, 3. Then 63 minutes parsed by 3 students is 21 minutes per student. This reasoning can be recorded in the following ratio table, with no need to recontextualize because we stayed in context the whole time:

Traditionally, teachers have told students that, since both problems are given as division problems, they should use the same method to solve them: long division. The long division algorithm suggests quotative reasoning. But partitive thinking can be more efficient in some cases.

This time, reasoning about the partitive question as ratios is not particularly helpful, as 92/23 does not simplify to a unit rate easily. Thinking about the problem multiplicatively could look like this:

Or, for the quotative problem, we could find equivalent ratios, reasoning that 92 ¸ 4 = 92/4 = 46/2 = 23. Returning to the context, we reason that 92 minutes divided by 4 minute acts results in 23 acts possible.

The reasoning in this set of problems illustrates what it can mean to “abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents.” In this case, reasoning that since each problem represents either 92 ¸ 23 or 92 ¸ 4 respectively, one can solve each problem using the mathematical properties of division.

Students who can use both strategies (dividing by multiplication and by finding equivalent ratios) can then generalize them to other situations. Look what happens with fraction division!

A problem like 5/6 ¸ 1/6 can be thought of quotatively as “how many 1/6’s are there in 5/6?” or 1/6 ´ ____ = 5/6.

How can we as teachers encourage the flexibility and generalized use of strategies discussed above so that students can reason abstractly and quantitatively? One way is to have students solve questions like the school counselor problems, where the numbers are the same even though the situations suggest different thinking. Finding the same solution but using different strategies can cause disequilibrium for students. The teacher can purposefully have students share strategies to solve the different problems and help the students make connections between the solutions. Also teachers can give students problems like those from the talent show, where decontextualizing is helpful because the numbers lend themselves to using properties not suggested by the given contexts.

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Pam Harris is the author of Building Powerful Numeracy for Middle and High School Students, which looks carefully at what research says about developing an instructional approach focused on developing number sense and understanding in higher math. She is also the author of Lessons and Activities for Building Powerful Numeracy, which includes detailed lessons and practical activities that promote strategies for teaching as much mathematics as possible with as little memorization as possible. Follow her on Twitter @pwharris.

## 16 thoughts on “SMP #2: Reason Abstractly and Quantitatively”

1. Joye W

Pam, I think you are having computation issues.  The last time I checked, 3/8 divided by 2/5 was 15/16, not 15/4.  And I have never in my entire career seen more convoluted reasoning for what should be simple division!  Dividing 252 by 12 is a piece of cake using long division, and I cannot imagine why anyone would want to make it into the complicated mess described above.  Knowing when to divide is very important, but applying some sort of bizarre reasoning is completely unnecessary.  No wonder our children are struggling so with computation!  Sorry, but in my book, quantitative reasoning in the talent show problem is knowing that, given a 92-minute show, finding out how many 4-minute acts could be performed is done using simple division.  It's simple and it's right and it doesn't cloud minds with unnecessary clutter.  I can't think of a better way to slam and lock the door to learning higher math than to use the methods described above.  This is making sense of mathematics?  The only part of it that makes sense to me is that it is not a surprise that our children in the US do not perform well in math.

1. Anne

I don't see where 3/8 divided by 2/5 is showing up as a problem. I like the quotative method in that it connects to proportional reasoning. As an adult this is how I solve mental division problems. Regarding the conceptual understanding of division of fractions, sooooooooooo many kids have no idea why the invert and multiply method works. They learn that before skill before practicing with visuals and then learning one of the several algorithms that work. One of the problems with Common Core that I find is the time pressure. How do we fit in the concepts before quickly teaching a skill and moving on?

1. Heather

The blog post seems to have been revised.  The example with 3/8 divided by 2/5 is not there anymore.

2. Barry Garelick

I think a better set of exercises would be what I had in my old arithmetic book, in which a scenario is described and the student is asked whether multiplication or division is necessary to solve the problem.  I agree with Joye that the methods you have outlined here only add confusion to what should be a relatively simple process.  Division is not hard to understand or make sense of.  One can still talk about problems that are partitive and quotative since that helps determine what the units are.  But the method for solving the problems are exactly the same: long division.

3. Heather

Joye, I do agree that 3/8 divided by 2/5 is 15/16.  I, too, am having difficulty following Pam's reasoning.  However, I think one intent of these standards is to have procedural fluency along with conceptual understanding.  This comes across in Pam's blog – we want students who understand procedures and link them to the concepts.  Other than the last example, I agree with her post and support the use of these strategies.

4. Heather

I see that Pam's book is for teachers of grade 6-12.   As someone in elementary math, here are some thoughts on how this problem could be solved with conceptual understanding that goes back to primary grades in terms of "doubling/halving".  I envisioned a wall divided into fifths.  I shaded two-fifths of it one way and wrote 3/8 gallon paint.  I shaded the next two-fifths another way and wrote 3/8 gallon paint.  I added 3/8+3/8=6/8, or doubled the 3/8.  Then, I knew the remaining one-fifth was half of two-fifths, so it would have half the paint, 1/2 of 3/8 is 3/16.  Lastly, I added 6/8 and 3/16 using common denominators 12/16+3/16=15/16.  The foundation for this type of reasoning is built in elementary grades with mental computation of doubles and halves.  For example, students in second grade understand that if they double 9, it is 18 and inversely, if they take half of 18, it is 9.  Students in fourth reason to make common denominators. Students in middle school divide and multiply fractions as well as apply concepts of ratios.  It all builds together keeping conceptual understanding alongside procedural fluency.

5. Candace

I do see what everyone is thinking but I must disagree with the  concepts you all are thinking . We must get students to think differently and not one way. Theses concepts are not what our thinking was back when we were younger. It's taking me time to grasp the thinking also. We as educators must began to look at change differently and not just one way. Many of our  children are having an issue with  not thinking outside the box (cognitively)and this hinders their true understanding. This type of math lets students realize that their thinking in a way that's letting them understand it their way that there isn't just one way to get to an answer.  I'm teaching second grade and they are feeling better because they understanding that they can get the answer in different ways. They are taking ownership of their learning. This math is getting them ready for the real world. I do understand your pain because I'm learning this also and it's all about how you  approach the concepts and to change our thinking.  I don't think Pam is saying to completely stop with the typical algorithms but to give students a change to use their own thinking to what's more efficient for them .

1. Barry Garelick

To think outside the box, one must have mastered inside the box thinking.  Teaching students to apply such techniques before mastery of the concepts of division is adding an unnecessary layer of complexity.  After mastery, many students do what Pam is advising on their own.  It can be introduced as an estimation technique later, not when they should be mastering when division is necessary.

Also, the insistence that students MUST show more than one way to solve a problem is misguided. The thinking is that a student has achieved "understanding" if a student can solve a problem in multiple ways. Insisting on having students come up with more than one way to solve a problem is confusing cause and effect. That is, forcing students to think of multiple ways does not in and of itself cause understanding. It is tantamount to saying: "If we can just get them to do things that LOOK like what we imagine a mathematician does … then they will be real mathematicians."

1. Eric D.

Well I think the problem with that is that you are making yourself the arbiter of what is inside and outside-of-the-box thinking. Maybe long division is much more outside-of-the-box for some people than we realize. Or maybe another approach is. Maybe a different approach is what helps connect the long division dots that didn't make sense before. I don't think we have to mandate different approaches, but it sure is good practice to explore more than one.

6. Eric D.

I really agree with Candance. To me, this represents flexible thinking and an ability to approach a problem differently. I think long division would be a fine way to tackle some of the examples, but if that's the only strategy students have then they get stuck when they are presented with something that isn't set up or presented like they've seen it in a textbook. I don't think Pam is saying this is the only way to solve these problems or that this is how you must solve these problems. To me, it's just wise to take a problem and look at many different ways to approach it and then compare and consider those approaches.

1. Katharine Beals

Pam asks, "How can we as teachers encourage the flexibility and generalized use of strategies discussed above so that students can reason abstractly and quantitatively?"

Perhaps the best way to encourage flexibility and abstraction in mathematical thinking is to move beyond the tyranny of the Base 10 number system, which, after all, is simply an anthropocentric artifact of our having 10 fingers. I propose Base 2 for starters; then Hexadecimal. These, after all, are the number systems of our 20th and 21st century computers, and, therefore, of 21st century math skills.

Of course, once students have done these problems in Base 2 and Hexicecimal, they can related their solutions to the anthropocentric counterparts in Bse 10. As Eric points out, "it's just wise to take a problem and look at many different ways to approach it and then compare and consider those approaches."

7. SteveH

I think you are making this process too vague. One should first start with mastery of basic skills and then tackle "word" problems. However, there are specific classes of word problems that students have to identify, like rate and D=RT, work, and mixture. These are "governing equations" and each has a certain number of variables, and most word problems require you to fill in numbers for all but one variable, for which you solve. Students have to identify the governing equation and fill in the values from the words in the problem statement. This is a generic problem solving technique that stays the same even if they ever get to solving Bernoulli's equation. It takes time and practice to master it, and the key ingredient is to see word problems where each value, in turn, is missing. This is just the start. You have to slowly and carefully work towards variation problems where, say, two people are painting a house at different rates. Students have to see that it is a rate problem and know how to determine a variation of the governing equation where there are two or more rates defined. In your problem, you could have two counselors who work at different rates.

A big tool that helps students, even in the earliest grades, are units. You can't add or subtract apples and oranges, but you can multiply and divide units just like factors and fractions. You can have units of students per minute. Technically, your sample equation:

(number of students)(12 minutes) = 252 minutes

is wrong. It's not 12 minutes, but 12 minutes/student. When that gets multiplied by number of students, you are left with a number with the units of minutes. Students loved it when I explained that units are treated just like factors or numbers – you can multiply and divide them and cancel them. Units can be used to determine if you have an equation set up correctly. If one term is minutes per student and you want to add it to a term that is students per minute, then something is not right. You can only add or subtract terms with the same units.

Your problem is a "rate" type problem.  Students have to learn to see that and understand that they should expect a value that talks about the rate – the amount of something per unit of time and that it will probably be multiplied by a time number so that the time variables cancel out: number of students/minute * minutes = number of students * (minutes/minutes) = number of students * 1. Units behave just like numbers. This was an epiphany for many of my students. It also simplified word problems that tried to trick them with different units, like a time in hours, but a rate in minutes. If you want to convert a rate from 10 feet per minute to the number of feet per hour, you would multiply 10 feet/minute * 60 minutes/hour = (10*60) feet/hour * (minutes/minutes). The units multiply and divide like fractions and the minutes cancel out.

Specific words also matter in word problems. When you give one of the equations as:

(12 students)(length of each meeting) = 252 minutes

it's not correct. (length of each meeting) should be given as (minutes per student). Students need to know what words like "per" and "of" mean. In your example, you are dealing with a simple rate problem where:

Rate * Time = Number

You have to get the student to first see when this governing equation applies to a word problem, and then see each variation where one of the variables is missing. This is just the start.

I find your use of "quotative" and "partitive" to be confusing and unhelpful. That is only two of the three basic forms that word problem can take, but there are so many others, as when two counselors work at different rates. What happens when the governing equation has 5 or 10 variables?

8. Gretchen

I am assuming the problem that was erroneous was taken out of the blog as I can't find it?  If there was a problem that had the wrong computation but the reasoning (not the answer) was paramount, then I hope those students learning math in this manner will not end up being the engineers, pharmacists and scientists who need exact and precise answers.  Can you just IMAGINE the lawsuits arising from the wrong answer?

I also agree with this statement:

"To think outside the box, one must have mastered inside the box thinking.  Teaching students to apply such techniques before mastery of the concepts of division is adding an unnecessary layer of complexity.  After mastery, many students do what Pam is advising on their own."

As a mother, I taught my children to master certain skills before they moved on to a new task.  As they gained confidence, they were indeed able to attempt more complex tasks because they were ready to do so.  Had I asked them for complex reasoning before they had mastered the basics, I would have seen the meltdown many children are experiencing nationwide via the developmentally NGOed crafted assessments and standards.

Whether or not some educational professionals believe THIS is the right way of instruction and another group believes it's THAT way to instruct children, the children will let you know if they are learning…or not.  What I resent as a parent is that the current group of educational professionals/NGOs in charge of math instruction don't really care about how children are struggling, crying, hating school because of the forced implementation of curriculum, standards, assessments.  They are insistent to prove a point with our children that in my eyes, ranks right up there with child abuse.

I would counsel any parent with a young child in public school following common standards, assessments and curriculum, to leave a system filled with educational professionals who are insisting one methodology is appropriate for all children.  They need to find the door marked "EXIT" as quickly as possible.  And then parents need to file class action lawsuits against those educators who refused to adapt learning to what is appropriate to children and instead, tried to pass off educational theory as research/data.

9. Bryan Pefound

Those students who do not know how to properly answer a question tend to either leave it blank, or write down as much garbage as possible. From what I am seeing above, it looks like the author has no idea how to utilize the long division algorithm properly. Did no one show you the beauty of greedy algorithms in your educational career? If you don't have a Msc or PhD in mathematics then I can understand why you haven't seen greedy algorithms. However, not having an MSc or PhD in math then begs the question: why are you giving advice (that is not even founded on proper research, mind you) on how to educate our children? From what I can see your only qualifications are having written a book, and that somehow correlates to being able to ultimately decide the fate of our children?

If you had continued in mathematics you would have seen many uses of greedy algorithms. Greedy algorithms are an integral part of computer science and I can't see a better way of giving a student a "content rich" problem than showing them the algorithms used by our electronic devices. However, since your scope of mathematics is so limited, your advice on how to properly teach children is also quite limited.

This is a major problem in math education today: non-qualified individuals who utilize bogus "research" and use our children as guinea pigs to justify their careers.

10. Eric D.

Bryan Penfound, are you honestly saying that if you "don't have a Msc or PhD in mathematics" you aren't qualified to be "giving advice on how to educate our children"? I find that to be unbelievably offensive and very ivory tower-ish. I'm a principal at a K-5 elementary school, but don't have an Msc or PhD in mathematics. In fact, I don't have a PhD at all. So should I stop coaching my teachers? Stop working with them to become practitioners? Stop learning more about what current research says about best-practice? Stop reading and thinking? Or am I unqualified?

Bryan, I suggest you include you contact information here so that we can all reach out to you. Perhaps you can begin traveling to schools across the country to share your fount of knowledge. Clearly you feel you are smarter than the rest of us. Bryan, you have an opinion. Fine. Discussion and debate is good. Let's do that. But I find it unbelievably rude that you would spend time to 1) tear down the author, and 2) call out what you perceive as nonqualified educators who are just trying to "justify their careers".