I received the following email from a parent recently. (I have deleted some of the email to focus on the main issue, and have lightly edited the part included here.)

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Dear Professor Devlin,

I am the father of a 9 years old boy. Recently, I stumbled upon a few questions set by his school and I noticed that the answer provided by my son is wrong. I was really puzzled by it, and I looked for more information pertaining to the topic of Multiplication, Multiplier and Multiplicand. I found an article written by you in Devlin’s Angle, dated January 2011

https://www.maa.org/external_archive/devlin/devlin_01_11.html,

which talks about multiplication. Having read that, I wish to seek your opinion on how you would perceive the answer to the question that was posted to my son and if it is wrong in any way?

It is a worded question:

A box of pencils consists of 12 pieces of pencils in them. The stationary shop decided to buy 165 boxes of pencils. Each pencil is set at $5 each for sale. The shop sold all the pencils. How much money did the stationary shop receive at the end of selling them all?

The answer:

The amount of money to receive at the end of selling all the pencils is:

165 x 12 x 5

= $9,900

The teacher mark my son wrong for the second line, as it should be

5 x 165 x 12

My point is that the first elementary rule in multiplication state that a x b = b x a, as in 3 x 2 = 2 x 3

Regardless of the placement of integer or whole number, there is no way you could deduct marks and mark it wrong, since no unit was written in the second line of answer unless you have written

165 boxes x 12pencils/box x $5

Since no unit was written, how could this answer be wrong, regardless if it is a worded question, as it might break the elementary rule of multiplication? If the elementary rule is to be broken, could the discussion of multiplier and multiplicand stand in this circumstance?

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The part of the email I have left out led me to suspect that the father was fairly knowledgeable about (at least) basic mathematics, and could even be, say, an engineer who makes regular use of mathematics but is (aware that he is) not an expert in elementary mathematics education.

As I was writing my reply, it occurred to me there is sufficient value in this brief exchange to warrant a post in this blog, which due to a variety of circumstances has lain idle for some months. And by “value” I mean value to all parties here, the student, the teacher, the parent, and perhaps also the person or entity that created the assessment, if other than the teacher. Here is what I wrote.

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Dear XXX,

Hmm. I don’t see any problem with writing the three numbers in a particular order. In fact, given how the question is phrased, I myself would express the calculation as 165 x 12 x 5. But, whatever the order, the student providing the answer clearly understands what is required, and their final answer is numerically correct. To mark an answer as wrong because of the order is idiotic, and really has nothing to do with mathematics.

There is a reason to mark the answer *as written* as incorrect, because the second line is nonsense. You can’t simply write

165 x 12 x 5

= $9,900

That’s improper use of the equal sign. If I were grading that test, I would note down for the student that the answer is numerically correct but add that what is written is mathematically wrong (in fact, as I just noted, it’s nonsense, but that’s not a good term to use with a student learning the subject), since it says that an integer is equal to a monetary amount. For instance, it would be okay write

165 x 12 x 5 = 9,900

Hence the answer is $9,900.

There are important issues involving the roles of the different values in multiplication, as you allude to in mentioning units. It’s possible the teacher was focusing on those, but simply marking an answer as wrong does not help the student learn to understand those roles. The **order** in which the numbers are written is not a mathematical issue, though some mathematical cultures probably have preferred conventions. (That’s all they are, however: **conventions**.) In contrast, knowing the different roles they play **is** a mathematical issue, and an important one.

So, I am in agreement with you regarding your comments on this particular question and answer. The problem I see is that a question designed to help students learn about multiplication to the extent of including units (i.e., not just a test of basic computational skills), as this one presumably was, cannot be effectively presented as a CORRECT/INCORRECT question. In particular, in this instance, the order in which the three values are written is not important, but knowing and understanding their roles is.

I think both student and teacher mis-stepped here. (In possible defense of the teacher, it may be that they were constrained by a systemic assessment framework that provided insufficient freedom to give this issue the degree of attention it required.) Analysis of the episode does, however, provide an excellent learning experience for all parties. And good learning is what education should be about.

I hope this helps. I will probably write a short post about this for my SUMOP blog (absent mention of any names or location), since I think it raises issues worth circulating. Thanks for writing. Best wishes to you and your son, KD.

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That was the exchange. Multiplication is a far more complex issue than is popularly assumed, even when restricted to the natural numbers. Indeed, it is the topic I have returned to more than any other in my monthly Devlin’s Angle blog for the Mathematical Association of America. As you will discover if you follow that thread (below), I stumbled onto the topic by accident, when a throwaway remark I made at the end of my September 2007 post turned out to open a hornet’s nest of misunderstandings.

**WHAT IS MULTIPLICATION? Posts on Devlin’s Angle**

September 2007, What is conceptual understanding?

http://www.maa.org/external_archive/devlin/devlin_09_07.html

June 2008, It Ain’t No Repeated Addition

http://www.maa.org/external_archive/devlin/devlin_06_08.html

July-August 2008, It’s Still Not Repeated Addition

http://www.maa.org/external_archive/devlin/devlin_0708_08.html

September 2008, Multiplication and Those Pesky British Spellings

http://www.maa.org/external_archive/devlin/devlin_09_08.html

December 2008, How Do We Learn Math?

http://www.maa.org/external_archive/devlin/devlin_12_08.html

January 2009, Should Children Learn Math by Starting with Counting?

http://www.maa.org/external_archive/devlin/devlin_01_09.html

January 2010, Repeated Addition – One More Spin

http://www.maa.org/external_archive/devlin/devlin_01_10.html

January 2011, What Exactly is Multiplication?

http://www.maa.org/external_archive/devlin/devlin_01_11.html

November 2011, How multiplication is really defined in Peano arithmetic