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.
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?