# Fuzzy maths jumps the shark

By September 1977, Happy Days was an enormously popular TV Show. Initially a supporting character, Fonzie had began to dominate and, in an infamous plot line, he is seeing performing a water-skiing jump over a confined shark. It is debatable as to whether this really was the point at which the show began its long decline; it continued in production for another seven years. But it did demonstrate a high point of absurdity for a program that was originally about a romanticised version of 1950s American family life. And so the phrase “jump the shark” has entered the lexicon to represent such a tipping point.

I wonder whether we are now reaching the high water mark of fuzzy maths; that movement launched in 1989 by the National Council of Teachers of Mathematics (NCTM) in the U.S.; a movement that eschews what it sees as ‘rote’ memorisation of maths facts and procedures in favour of prioritising understanding. Fuzzy maths seems to have taken over much of North American maths education. Despite efforts to make the new Common Core State Standards pedagogically neutral, there is evidence that they are being used to pursue a fuzzy maths agenda. In Canada, large-scale implementation of fuzzy maths is associated with a parallel decline in test scores.

Given the current popularity of fuzzy maths, let me nominate a candidate for the jump-the-shark moment.

The Telegraph is reporting an image taken of a Year 3 maths test that has been posted on Reddit. It shows a question:

“Use the repeated addition strategy to solve : 5 x 3″

The answer given by the student is correct, 15, but it is marked as incorrect due to the way the child has worked it out. He or she has written “5 + 5 + 5” and the teacher indicates that it should instead be “3 + 3 + 3 + 3 + 3”.

This is obviously barking mad. But it demonstrates the kind of hole that fuzzy maths sucks us into. Clearly 5 + 5 + 5 = 3 + 3 + 3 + 3 + 3 =15. It is all the same. This demonstrates a key property of multiplication; that it is commutative i.e. 5 x 3 = 3 x 5.

However, it seems as if the teacher does not want the student to know this yet and that the student is meant to strictly interpret “5 x 3” to mean “five lots of three.” This is a reasonable interpretation. However, the commutivity of multiplication has so suffused our culture that allowing only this interpretation is quite unreasonable. If I gave you a shopping list that had “tin of beans x 3” on it, you would not interpret this to mean “tin of beans lots of 3” you would interpret it to mean “3 tins of beans”.

Similarly, it is quite legitimate to interpret “5 x 3” to mean “five, three times”.

And so, in the name of ‘understanding’ we head in to being both confusing and wrong. Never mind the fact that we really don’t want students to have to work out 5 x 3 using a repeated addition strategy. It is essential that such basic maths facts are memorised so that precious working memory resources may be devoted to higher level aspects of problem solving. The correct answer should be sufficient in this case. And what message is this poor students getting about maths?

The next question on the paper is equally bizarre. Asked to draw an array, the student draws it the wrong way around. Yes, the array might have 24 elements and the answer might be 24 but, for some strange reason, the teacher wants 4 rows of 6 columns and not 6 rows of 4 columns.

You might just put this down to one teacher being idiosyncratic. You may suggest that fuzzy maths does not really require this sort of thing and this particular teacher is operating under a misconception. You may think that nobody would defend this, even those who are committed to fuzzy maths.

Not so.

The NCTM defended the marking of the paper. Diane Briars of the NCTM commented, “We want students to understand what they’re doing, not just get the right answer.”

Funnily enough, this would seem to achieve the precise opposite.

UPDATE: It has been brought to my attention in the comments below that the defence that is attributed by The Telegraph to Diane Briars is identical to statements that she is reported to have made in this news report from May 2014. So perhaps she did not defend it after all. Can anyone shed light on this?

### 20 Comments on “Fuzzy maths jumps the shark”

1. gregashman says:

Reblogged this on The Echo Chamber.

2. How I wish that child’s parents went in and asked some difficult questions!

3. howardat58 says:

‘The NCTM defended the marking of the paper. Diane Briars of the NCTM commented, “We want students to understand what they’re doing, not just get the right answer.” ‘

This is so sad. The kid understood perfectly. The whole idea is that kids explore, find, and try out ways of seeing calculations, “doing sums” and so on. Treating an alternative way of seeing something as a “method” or “strategy” to be learned defeats one of the main objectives of the Common Core. Looks like the NCTM hasn’t got this yet.
As for the second example who says that 4×6 cannot be represented by an array of dots or ones in ANY orientation. If I am standing at the side of the kid who has done 4 rows of 6 columns then I see 6 rows of 4 columns. What changed?
I think that the teacher doesn’t get it.

4. Josh Fisher says:

Painful. Here’s a strained defense having to do with *matrices*, for corn’s sake: https://goo.gl/97K4Ao.

5. ben says:

I’m going to point out this hullabaloo is from a week or so ago and Briars quote is from an NPR story last May as far as I can tell. So yes she was defending the common core standards but as far as I know she still believes in the commutative principle.

• gregashman says:

So is the Telegraph story a misrepresentation?

It states:

The National Council of Teachers of Mathematics (NCTM) in the US defended how the paper was marked, saying it gives students a better understanding of the problems they are solving.
“Part of what we are trying to teach children is to become problem solvers and thinkers,” said Diane Briars, president of the NCTM.
“We want students to understand what they’re doing, not just get the right answer.”

• Paul Hartzer says:

See the date.

Do the math.

Unless Briars has invented a time machine and deliberately gone back in time. The only person capable of such actions isn’t involved in merely fuzzy maths, he’s involved in wibbly wobbly timey wimey math.

So I would say that The Telegraph might be fibbing just a wee bit.

Also, I distinctly remember a hullaballo like this about a year ago, except it wasn’t Common Core or America, it was Korea or Japan or something and it was at the high school or college level and it was to show the intransigence of the traditional teaching, in praise of alternatives. I swear it was exactly the same question. So I question whether this test really happened.

• gregashman says:

You have a point about the comment – I will update the post.

6. barryg99 says:

Part of this idiocy may have its roots in an essay that Keith Devlin wrote in which he asserted that multiplication was not repeated addition and should not be taught that way. While it is true that as one goes upward in math, one learns that multiplication is an operation and can mean more than repeated addition. But for 1st or 2nd graders it is an appropriate and logical place to start–and probably where Devlin himself started on his climb up the math ladder before he kicked it down and declared it unnecessary. Stating that 5 x 3 is 5 groups of 3 supposedly gets students prepared for the day when they see 1/2 x 6 and can then interpret it as half of a group of 6. But insisting on such interpretation is a ludicrous development of a “habit of mind” that is largely unnecessary. It is what the fuzzies seem to think represents “deep understanding”, and really represents a very shallow and faulty understanding–and unfortunately one that serves to confuse more than enlighten students in lower grades.

7. CriticalNumeracy says:

This is the most conservative traditionalist way to interpret an “understanding” approach.

All the downsides of rote memorising/give us the answer we want, with none of the upsides of actually getting students to understand or flexibly use proper skills. A fascinating if horrifying case study in change implementation.

• gregashman says:

I’m not sure it’s fair to implicate traditionalists in such nonsense!

8. Paul Hartzer says:

The similar issue in Asia I was trying to remember was also at the elementary level, but the direction was backwards: The issue was whether 4 + 4 + 4 + 4 + 4 + 4 = 4 x 6, or only 6 x 4.

Even so, the folderol here and thirteen months ago strikes me as the same in a different way, and makes me suspicious that the US version isn’t a hoax-ified version of the Indonesian one, being used to make Common Core look bad.

Common Core does have an oddly phrased standard that seems to support this inanity: CCSS.MATH.CONTENT.3.OA.A.1
Interpret products of whole numbers, e.g., interpret 5 × 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 × 7.

Some people appear to be interpreting this as meaning that 5 x 7 = 7 + 7 + 7 + 7 + 7 + 7 + 7 and not 5 + 5 + 5 + 5 + 5. But it’s important to note that the phrase “repeated addition”, which appears all over the place on “Common Core-aligned” worksheets, appears exactly zero times in the standards themselves.

… in which a former high school maths teacher claims that abusing children with this inanity is actually good for them, because it prepares them for the realization in high school that multiplication is not always commutative (it isn’t, for instance, with arrays).

9. Tunya Audain says:

The Same Conditions Are Ushering In Even More Education Malpractice

Some say it’s government failure behind poor performing schools — failing to bring in quality control regulations . Some say it’s the education system itself that rules the roost, a job security haven for its workers and devil take the hindmost.

At any rate, fads prevail in public education that drive logically minded people nuts, be they parents or concerned teachers. Whole language and fuzzy math should be movements that are way past their shelf life as research findings do not produce good ratings; but they persist.

It was Dr Seuss who called-out the “whole language” movement. Years after publishing his “Cat in the Hat” (1957) he said: “I did it for a textbook house and they sent me a word list. That was due to the Dewey revolt in the Twenties, in which they threw out phonic reading and went to word recognition . . . I think killing phonics was one of the greatest causes of illiteracy in the country . . . there were two hundred and twenty-three words to use in this book . . . I read the list three times and I almost went out of my head. I said, I’ll read it once more and if I can find two words that rhyme that’ll be the title of my book . . . I found ‘cat’ and “hat”, , , “

This photo should be a cautionary tale where fuzzy math prevails and where 2 + 2 = 5 if the student can show his work http://i.stack.imgur.com/hl9F8.jpg

Now, these two movements are just dress rehearsal for what’s going on today. “Transformations” are the name of the game currently — Common Core, Personalized Learning, 21st Century Learning, etc. Hard skills as reading, writing, arithmetic and knowledge are being downplayed in favor of soft competences as creativity, collaboration, critical thinking, etc. These have been deemed THE essentials that employers will demand in the future. Does anyone REALLY know what the future holds?

Of course, a cynical observer would say these new competencies are not measurable by objective standards so deliberately avoid accountability. It’s time to adopt hardheaded policies and practices that support proven evidence instead of pie-in-the-sky philosophies.

• gregashman says:

That’s a great quote. Thanks. I found the source and Tweeted it.

• I have said this before in my blog: Educationalists seem to exist in a world where they imagine the ‘workplace’ to be some kind of open place office staffed entirely by poloneck-wearing hipsters on flexitime, all sipping frappucinos, googling on their iPads and chatting (collaborating). Essentially, this is a dreamworld.

Can you teach soft competencies anyway? It seems these are really only possible with a foundation of knowledge and with the kind of social skills that come with maturity, wisdom and experience.

10. Stan says:

Regarding the 21st century work. That would be where everyone today works. Students parents, teachers, teachers spouses and so on. It is not an unknowable place.

Regarding knowing what someone in the grade you are teaching them 5×3 needs to know I ask the following. If teachers in each grade from 1 to 11 simply asked what is most important to have their students learn and asked this of those teaching the next grade why wouldn’t that lead to the correct answers? What’s wrong with walking 50 steps to the class where the next grade is being taught to find out what needs improvement in the current grade?

11. howardat58 says:

Most of the difficulties in 5×3 arise from what the symbols mean.
On the one hand this reads as five times three, which in common parlance is five lots of three.
On the other hand this reads as five multiplied by three, which means three layers of five in each, as in 3-ply, 4-ply, plywood. So how you see 5×3 does depend on your reading, rather dramatically in this case.
This has nothing to do with “commutative”.

12. […] “Use the repeated addition strategy to solve : 5 x 3” (Reddit, via The Telegraph, via Greg Ashman); the student answered “5 + 5 + 5” and was marked wrong because the answer ought to be […]