Ways of thinking and ways of doing

Somewhere back in days of Facebook fury about the Common Core there was a post from an outraged parent whose child had been marked wrong for something like this:
$$
6 \times 3 = 6 + 6 + 6 = 18.
$$
Apparently the child was supposed to do
$$
6 \times 3 = 3 + 3 + 3 + 3 + 3 +3 = 18
$$
because of this standard:

3.OA.A.1. Interpret products of whole numbers, e.g., interpret $5 \times 7$ as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as $5 \times 7$.

The parent had every right to be upset: a correct answer is a correct answer. Comments on the post correctly pointed out that, since multiplication is commutative, it shouldn’t matter in what order the calculation interpreted the product. But hang on, I hear you ask, doesn’t that contradict 3.OA.A.1, which clearly states that $6 \times 3$ should be interpreted as 6 groups of 3?

The fundamental problem here is a confusion between ways of thinking and ways of doing. 3.OA.A.1 proposes a way of thinking about $a \times b$, as $a$ groups of $b$. In other words, it proposes a definition of multiplication. It could have proposed the other definition: $a \times b$ is $b$ groups of $a$. The choice is arbitrary, so why make it? Well, there’s an interesting discovery to me made here: the two definitions are equivalent. That’s how you prove that multiplication is indeed commutative. It’s not obvious that $a$ groups of $b$ things each amounts to the same number of things as $b$ groups of $a$ things each. At least, not until you prove it, for example by arranging the things into an array:

You can see this as 3 groups of 6 by looking at the rows,

and as 6 groups of 3 things each by looking at the columns,

Since it’s the same number things no matter how you look at it, and using our definition of multiplication, we see that $3 \times 6 = 6 \times 3$. (We leave it as an exercise to the reader to generalize this proof.)

None of this dictates the way of doing $6 \times 3$, that is, the method of computing it. In fact, it expands the possibilities, including deciding to work with the more efficient $3 \times 6$, as this child did. The way of thinking does not constrain the way of doing. If you want to test whether a child understands 3.OA.A.1, you will have to come up with a different task than computation of a product. There are some good ideas from Student Achievement Partners here.

20 thoughts on “Ways of thinking and ways of doing

  1. Welcome to the party. This post is about 5 years too late. Districts like ours have already reconciled this and the hundreds of other issues that were set in motion by the standards.

  2. Good for your district! I’m sure others would be happy to hear about those “hundreds of other issues.”

    • If you are serious, let’s start with the debacle in the appendix that became Algebra 2 and is now part of college and career readiness in my state. The issue simply is: how can one expect to teach that to the typical student in one school year?

  3. I am always up for a serious and civil discussion about issues in math education K–12. To your question, I agree that forcing everybody to accomplish the standards in 3 years is a bad idea. There are four years in high school, and some students need all four. Many states, districts, and schools struggle to handle that problem in a humane and even-handed way. And because we live in a local-control system, where implementation of the standards is up to each individual state that adopts them (and in some states devolves to districts or even schools), there are many different solutions out there. The standards themselves do not specify an arrangement into courses and do not require that all standards be covered in three years. The near universal agreement that end-of-high-school testing should be required in grade 11, rather than grade 12, has always struck me as strange. That’s not the way it is in other countries, for example Australia, the country I grew up in.

    All that aside, because they were worried that states might want guidance on arranging the standards into courses in high school, Achieve created Appendix A. It was not intended to be taken as a mandate but rather as a model, as is stated clearly on page 2 (http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf). However, understandably I suppose, many policy makers took it as gospel, and that has resulted in the situation you decry here.

    I would point out that the PARCC and Smarter Balanced frameworks did not follow Appendix A to the letter, and put some important limits on the complexity of items for certain standards. However, as I say, I think the real problem here is the assumption that testing happens in grade 11. That should be an option for students who are ready for it, of course, but not the norm. I’ve been saying this for years, but I don’t know of any efforts to change it.

    By the way, if you want to continue this thread, it should probably go over in the forum on arranging the standards into courses. http://commoncoretools.me/forums/forum/public/arranging-the-high-school-standards-into-courses/. I’ll repost it over there.

    • I appreciate the response. Those efforts to change what is occurring around the country could come from you and other leaders, especially since graduation, college and career readiness, and many other factors affecting kids are in play. The common core standards put this issue in motion, and, although some of the consequences may have unintended, they are still the responsibility of those who set this course in motion, especially that ESSA, PARCC, and high-stakes testing are in play.

      As far as PARCC goes, I would disagree that they put limits on the tested standards. In fact, didn’t PARCC “invent” standards to be tested? These are some of the integrated, C, and D standards. One standard even asks high school students to “use reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity.” PARCC has interpreted this to mean that students can use right triangle trig on non-right angles, for example. PARCC has revised and re-revised the PLDs several times. Algebra 1 students are also tested on “securely held knowledge.” The common core standards put this chain of issues in motion.

      And, from my experience, asking elementary students to take a high stakes test that takes 4 hours is a bit much. The high school test is 4.5 hours long. Way too much testing. The common core standards put this chain of issues in motion.

      Finally, I do appreciate an honest and candid dialogue. Many of my colleagues and I have been frustrated as we find the balance between content and mastery, especially in Algebra 2. We have seen high failure rates on PARCC, as defined by 3 or lower, and then told that high failure rates mean that PARCC and the standards are so rigorous. So rigorous as to be unattainable. We have even been told that unless a student is in the 60th percentile in math (on MAP tests, for example), they cannot get a 4 or higher on PARCC. We are frustrated because it is not possible for all students to be greater than the median. The common core standards put this in motion.

      We are champions for our students and want them to succeed. We have high standards for ourselves and our students. We hope that those who set this in motion can see what has transpired and help us in the efforts to truly help every child succeed in math.

  4. Ohio recently handled this issue by rewording the standard as follows:

    3.OA.1 Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. (Note: These standards are written with the convention that a x b means a groups of b objects each; however, because of the commutative property, students may also interpret 5 x 7 as the total number of objects in 7 groups of 5 objects each).

    Commutativity needs to be revisited regarding fraction multiplication. For example,

    5 x (1/3), interpreted as “5 copies of 1/3” [4.NF.4a]
    (1/3) x 5, interpreted as “1/3 of a copy of 5” . [5.NF.4a]

    The second is harder than the first, and the important work is showing through pictures that they are equivalent. A robust understanding of fraction multiplication involves both of these ways of thinking.

  5. I have to question, though, whether the standards should be defining operations (or any mathematical concepts) in the first place. After all, as Bradford Findell remarks, there is a limit to how far students can stretch this concept of multiplication once the count of groups becomes a fraction, a decimal, an irrational number or a variable.

    To my reading, this standard does not define multiplication at all, but gives a student a way to make sense of multiplication given the scope that they will need to apply it in Grade 3, but perhaps under-sells the fact that this is indeed an interpretation and not a definition. That there are other interpretations (see, 4.OA.1 for starters), and none of the interpretations is truly what multiplication is in and of itself. This subtlety is not something that 8- and 9-year-olds should necessarily recognize, but it is something that the skillful teacher should be mindful of in order to keep students from getting too locked in to any particular interpretation of a concept that demands flexibility.

    • Well, I could quibble and point out that 4.OA.1 is about interpreting a multiplication equation, not about interpreting the operation itself, but I won’t 🙂 You are quite correct that 3.OA.1 does not come right out and say it is defining the operation. That’s just my interpretation of the standard. Your interpretation of it as giving students a way to make sense of multiplication is equally valid. And I think for the main point I was trying to make in this post your interpretation would work equally well. That is, there are two versions of this “way to make sense” (a groups of b or b groups of a) and you need to pick one and then see how it is equivalent to the other in order to make sense of the commutative property.

      As for your general point about what standards should be doing, I agree that there is a line to draw between standards and curriculum, and reasonable people can disagree about where that line should be. Wherever it is, discussions about how to cross that line are a large part of the purpose of this blog.

      • I agree and appreciate the work that you’ve been doing to help make meaning of the standards and help bring them to life in the classroom!

  6. Bill, I need some help with something and I cannot for the life of me log into WordPress (I’ve tried two usernames, password resets, nothing is happening!) I am looking at dividing decimals in 5th grade. I know from your previous posts that it should be decimals divided by whole numbers, or whole numbers divided by decimals. My questions is should the students be given problems with or without remainders? Any response would be much appreciated, again, I’m sorry I wasn’t able to do it in the forums!

    • Carrie, I don’t know why it took so long for this to come through! In answer to your question, I would say that finding quotients with remainder is purely a whole number activity, so once you are working with dividing fractions or decimals, you are just finding the quotient. So, for example, $2\div 0.8 = 2.5$.

      • Perfect, that is exactly what I was thinking reading over the standards and the progression. Thanks so much, Bill! I really appreciate the response!

      • So sorry to bother you again, Bill. I meant to ask you one more question! Since in 5th Grade Base Ten Operations students learn about multiplying and dividing by Powers of Ten, should these ever be included in expressions? Or strictly using digits (no exponents). Thanks for any input, again, WordPress is not allowing me to log in, so I apologize for not being able to utilize the forums! Cheers!

      • Well, I was looking at the algebraic expressions or equations, so I didn’t know if something like 4 x (10² + 5) = b would be acceptable for 5th grade since they should know the order of operations and have been exposed to (exponents) powers of ten in the NBT standards. Just didn’t know if they were to be included? (No other exponents, just with 10) Thanks again!

      • There’s nothing wrong with the $10^2$ there, but expressions and equations don’t really begin in earnest until grade 6.

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