The Common Core math quiz that has everyone outraged isn't about Common Core -- it's worse

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Why 5+5+5=15 Is Wrong Under the Common Core

A bafflingly graded third grade math quiz caused a firestorm on Reddit last week, and has spread across the Internet, causing several people to question Common Core math standards and the teacher's implementation of them.

The quiz is on some of the most fundamental aspects of basic whole number multiplication and how students encountering multiplication for the first time can think about problems.

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The first question asks the student to calculate 5 x 3 using repeated addition. The student wrote 5 + 5 + 5 = 15, and was marked wrong, with the teacher writing in the "correct" solution of 3 + 3 + 3 + 3 + 3 = 15.

The second question prompts the student to calculate 4 x 6 using an array. The student drew an array with six rows and four columns, getting the answer that 4 x 6 =24. The teacher again marked the question wrong, and drew in a nearly identical array of four rows and six columns:

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This, naturally, has a lot of people pretty riled up. Here's what's going on.

This is bad math

It is obvious to anyone who looks at this problem that the student did precisely what was asked. For the first question, they interpreted multiplication as repeated addition. For the second question, they went with a graphic interpretation of multiplication, looking at a stylized version of finding the area of a rectangle.

One of the most basic properties of whole number (and integer, rational, real, and complex number, along with many more) multiplication is commutativity: for any two numbers A and B, A x B = B x A. Order does not matter in multiplication; adding five together three times is exactly the same as adding three together five times.

To a third grader just encountering arithmetic for the first time, that might not be immediately obvious. This actually gives one of the strengths of looking at more visual illustrations of multiplication like the arrays in the second problem: Many students will very quickly see that an array with four rows of six columns has the same size as an array with six rows of four columns.

One possible rationale for the grading scheme could be a formalistic issue: The curriculum or teacher might have formally defined multiplying together two whole numbers A and B as the total number of objects in a collection of A groups of B objects each. In that case, 5 x 3 would be defined officially to be 5 groups of 3, or 3 + 3 + 3 + 3 + 3.

It still makes no sense to penalize that student, even in this case. Commutativity is one of the first properties that emerges from that definition, and the student is still, in both problems, capturing the essence of what multiplication is.

This is NOT a Common Core standard

This example, along with so many other viral math problems that baffle students and parents (like this subtraction problem, or this check mocking a first grade counting exercise), is being used as an example of Common Core math being unduly confusing or frustrating.

While this worksheet does present a frustrating situation, it has nothing to do with Common Core. Common Core lays out a set of objectives for what students should be learning in each grade level. It's still up to individual states, districts, and teachers to come up with the specific curricula and lesson plans to achieve those objectives.

As New York City high school math and physics teacher Frank Noschese told Tech Insider's Madison Malone Kircher, "The standards just lay out what kids should know and be able to do, not actual lessons. Nothing in Common Core forces the specific interpretation these teachers used."

The two Common Core standards listed at the top of the quiz ask students to "interpret products of whole numbers" and to "use multiplication and division within 100 to solve word problems." Absent from those standards is an insistence on slavish devotion to a pedantic hyper-formal definition with no particular mathematical meaning.

Indeed, penalizing the student for their recognition that 5 x 3 = 5 + 5 + 5 just as much as it does 3 + 3 + 3 + 3 + 3, or that a grid with six rows and four columns is the same size as one with four rows and six columns, goes against the deeper spirit of much of the Common Core math standards: to reinforce a fundamental understanding of what numbers and operations are and how they interact with each other to provide a solid foundation for further mathematical study.

However this grading mishap happened, it wasn't Common Core's fault.

See photos of Common Core testing and those who oppose and support it:

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The Common Core math quiz that has everyone outraged isn't about Common Core -- it's worse
FILE - In this Jan. 6, 2015 file photo, Common Core opponents wave signs and cheer at a rally opposing Mississippi's continued use of the Common Core academic standards on the steps of the Capitol in Jackson, Miss. Results for some of the states that participated in Common Core-aligned testing for the first time this spring are out, with overall scores higher than expected though still below what many parents may be accustomed to seeing. (AP Photo/Rogelio V. Solis,File)
FILE - In this July 21, 2014 file photo, students at a summer reading academy at Buchanan elementary school work in the computer lab at the school in Oklahoma City. Results for some of the states that participated in Common Core-aligned testing for the first time this spring are out, with overall scores higher than expected though still below what many parents may be accustomed to seeing. (AP Photo/Sue Ogrocki,File)
Chart shows which tests states use to test students on Common Core standards; 2c x 4 inches; 96.3 mm x 101 mm;
FILE - In this April 30, 2015 file photo, Leticia Fonseca, 16, left, and her twin sister, Sylvia Fonseca, right, work in the computer lab at Cuyama Valley High School after taking the new Common Core-aligned standardized tests, in New Cuyama, Calif. Test results for this yearâs Common Core-aligned exams are starting to be released, and while several states report higher-than-expected scores, vast numbers of students didnât test proficient in math or reading.(AP Photo/Christine Armario, File)
FILE-This March 2, 2015 file photo hundreds of Albuquerque High School students stage a walkout in Albuquerque, N.M. A legal challenge by a Washington-based testing company may halt a controversial assessment exam in New Mexico that has sparked school walkouts and anger from parents and teachers. A Santa Fe judge is scheduled to hear arguments Tuesday, March 9, 2015, in an appeal by the American Institutes for Research. The company is fighting the state's granting of a potentially multi-year contract for Common Core testing to Pearson. (AP Photo/Russell Contreras,File)
Dolores Ramos, 16, right, joins dozens of Highland High School students in Albuquerque, N.M., as students staged a walkout Monday March 2, 2015, to protest a new standardized test they say isn't an accurate measurement of their education. Students frustrated over the new exam walked out of schools across the state Monday in protest as the new exam was being given. The backlash came as millions of U.S. students start taking more rigorous exams aligned with Common Core standards. (AP Photo/Russell Contreras)
Graphic shows hypothetical sample math questions for standardized tests under Common Core standards; 2c x 4 inches; 96.3 mm x 101 mm;
Graphic shows sample questions for tests measuring Common Core standards; 2c x 5 inches; 96.3 mm x 127 mm;
In this Feb. 12, 2015 photo, Maguire Ballard, age 13, takes part in a trial run of a new state assessment test on a laptop computer at Annapolis Middle School in Annapolis, Md. The new test, which is scheduled to go into use March 2, 2015, is linked to the Common Core standards, which Maryland adopted in 2010 under the federal No Child Left Behind law, and serves as criteria for students in math and reading. (AP Photo/Patrick Semansky)
In this photo taken Feb. 12, 2015, sixth grader Alex Greuey, 11, reads through a problem in the English Language Arts section of the PARCC (Partnership for Assessment of Readiness for College and Careers) test as he and his classmates practice for the Common Core State Standards Exams at Morgan Elementary School South in Stockport, Ohio. On Tuesday, Ohio becomes the first state to administer one of two tests in English language arts and math based on the Common Core standards developed by two separate groups of states. By the end of the year, about 12 million children in 28 states and the District of Columbia will take exams that are expected to be harder than traditional spring standardized state tests they replace. In some states, they'll require hours of additional testing time students will have to do more than just fill in the bubble. The goal is to test students on critical thinking skills, requiring them to describe their reasoning and solve problems. (AP Photo/Ty Wright)
In this Feb. 12, 2015 photo, Yamarko Brown, age 12, works on math problems as part of a trial run of a new state assessment test at Annapolis Middle School in Annapolis, Md. The new test, which is scheduled to go into use March 2, 2015, is linked to the Common Core standards, which Maryland adopted in 2010 under the federal No Child Left Behind law, and serves as criteria for students in math and reading. (AP Photo/Patrick Semansky)
In this photo taken Feb. 12, 2015, sixth grade teacher Carrie Young, back center, answers questions from her students about an exercise on their laptops as they practice for the the Common Core State Standards Test in her classroom at Morgan Elementary School South in Stockport, Ohio. On Tuesday, Ohio becomes the first state to administer one of two tests in English language arts and math based on the Common Core standards developed by two separate groups of states. By the end of the year, about 12 million children in 28 states and the District of Columbia will take exams that are expected to be harder than traditional spring standardized state tests they replace. In some states, they'll require hours of additional testing time students will have to do more than just fill in the bubble. The goal is to test students on critical thinking skills, requiring them to describe their reasoning and solve problems. (AP Photo/Ty Wright)
In this photo taken Feb. 12, 2015, practice test books sit on a table in the Sixth grade English Language Arts and Social Studies classroom at Morgan Elementary School South in Stockport, Ohio. On Tuesday, Ohio becomes the first state to administer one of two tests in English language arts and math based on the Common Core standards developed by two separate groups of states. By the end of the year, about 12 million children in 28 states and the District of Columbia will take exams that are expected to be harder than traditional spring standardized state tests they replace. In some states, they'll require hours of additional testing time students will have to do more than just fill in the bubble. The goal is to test students on critical thinking skills, requiring them to describe their reasoning and solve problems. (AP Photo/Ty Wright)
File - In this May 6, 2015 file photo, Colorado Republican State senators, top left to right, Mark Scheffel and Randy Baumgardner have a discussion on the closing day of the 2015 Colorado legislative session, at the Capitol, in Denver, Wednesday May 6, 2015. A dozen rural schools in Colorado who want to opt out of Common Core linked tests can pilot their own assessments under a proposal passed in the waning hours of the state legislative session. (AP Photo/Brennan Linsley, file)
House Education Committee Rep. Mark Baker, R-Brandon, presents House Bill 385, which bans use of a Common Core-related test, wiping out high school exit exams in biology and U.S. history, and pushes the state Board of Education to adopt standardized tests published by the ACT organization in House Chambers, Thursday, Jan. 29, 2015 at the Capitol in Jackson, Miss. The House bill passed 116-3 and moves to the Senate for more debate. (AP Photo/Rogelio V. Solis)
FILE - In this Oct. 24, 2013 file photo, State Education Commissioner John King, Jr., talks to reporters before a forum on Common Core learning reforms at the Stephen and Harriet Myers Middle School in Albany, N.Y. A newly appointed Regents task force this month will begin reviewing feedback from the 20 statewide forums, with instructions to come back with ideas for smoothing the way forward for the ongoing shift to more demanding K-12 learning standards, student testing and teacher evaluations. (AP Photo/Mike Groll, File)
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This has nothing to do with higher order math

One explanation that was floated by several Redditors (ninjakiti here is one example) and blogger Hemant Mehta revolves around the fact that for arrays in computer science and matrices in linear algebra, order does matter.

The idea is that, in the second problem, an array or matrix with four rows and six columns is in fact a very different thing than an array or matrix with six rows and four columns. It's conventional to describe the dimensions of a matrix by putting the number of rows first and the number of columns second. In that case, the 4 x 6 array asked for on the quiz would be different than the 6 x 4 array the student drew.

I do not think this is the solution here for two reasons.

First, this is a problem involving basic single-digit whole number multiplication. It is likely one of the first times students are encountering multiplication. Students are unlikely to encounter matrices until their later high school years, and it would seem a rather odd pedagogical choice to enforce a convention in linear algebra and computer programming that students aren't going to see for nearly a decade, without any explanation of why that convention matters, at the same time students are expected to grasp the most basic properties of integer multiplication, like commutativity.

Second, commutativity is a bedrock mathematical fact, while matrix dimension notation is an arbitrary convention. One of the very few things I am willing to accept as an absolute, immutable, universal truth that holds in all times and places is that the order in which two whole numbers are multiplied together does not matter. 5 x 3 = 3 x 5 is written into the fundamental fabric of reality in a way very few other things are.

Meanwhile, describing matrices as rows by columns is essentially arbitrary. We could have just as easily chosen to write them as columns by rows. As it happens, modern mathematics settled on the former rather than the latter. This is similar to how English is read from left to right, while Arabic is read from right to left. Neither of those are fundamental, inevitable aspects of these languages; they are just how things worked out.

The idea that a student should be punished for recognizing and applying the fundamental truth of commutative multiplication in service of drilling in a completely arbitrary convention that they can easily learn when they need it ten years later strikes me as borderline insane.

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