Incorrect Teaching Practices = Weak Pedagogical Reasoning?

Discussion in 'Independent' started by karisshad17, Apr 8, 2019.

When teaching Equal Fractions, teachers, textbooks, curricula, teacher-training, student-learning, visual demonstrations...ALL, without exception draw upon an implausible assumption: i.e. when multiplying the numerator and denominator by the same number, it produces the "increase" in the the number of equal parts ( 3-fourths "increase" to 9 twelfths!). This is patently absurd and fails the test of all four levels of Mathematical reasoning:
1. Visual Reasoning 2. Verbal Reasoning 3. Numerical Reasoning and 4. Pedagogical Reasoning.

In order to demonstrate this, I have uploaded 2 videos on My Resources for downloading for free. In addition, the videos may also suggest that Primary-Middle Math CAN be taught without the use of spoken language. We use verbal mediation (without visual mediation) to convert printed words to sounds, then to meaning. Likewise, it can be proved (as I do in the videos) that we use "visual mediation" to convert visuals directly to meaning without any recourse to verbal mediation. I remain open to any comments, inputs. And also to correction, of course!

2. frustumStar commenter

I've never heard/seen "increase" used in that way.

Although I think there's much to be said for visuals, a pitfall of avoiding verbal mediation altogether is that the correct use of the language is not modelled, and so pupils' explanations will sometimes use language incorrectly as they struggle for the right words to use.

I think I was a bit hasty. I agree with you. In can't teach without using language. I may be over-thinking about developing and designing videos: too focused on doing that without any voiceovers. But that means it gives the teacher complete freedom to wrap his/her language and communication around the visuals provided. Pl do see the videos. I think one can produce videos without a voice-over, not use them without language. I think that's what I should have been saying. Appreciate your comment on this.

4. caterpillartobutterflyStar commenter

I've also never, ever heard any teacher suggest that 3/4 would increase to 9/12.
I have seen lots of visual representations, on the IWB and using practical equipment.
I'm sure your video is lovely, but I think you are starting from the assumption that primary and middle school teachers are teaching incorrect maths. This isn't, in general, true. Certainly in upper KS2 in independent schools, maths tends to be taught by specialists who would definitely not teach that 3/4 increases to 9/12.

Mind you, no primary school teacher I know would say 'three fourths' in the classroom either.

If you show students that you "divide by half" in order to split wholes into smaller equal parts, (and show that division operation by half), before "flipping it to X 2" then that would be the correct thing to do. But if you begin by multiplying both the top and the bottom by 2 and say: "...so this way it breaks into 4 smaller parts....." then the choice of the multiplication operation to explain what you are doing, would (in the children's minds) suggest an "increase in number" from 2 (wholes) to 4 (parts). This is a logical and conceptual inconsistency. It doesn't square up. From what I have seen, this is the normal practice in schools.

Since you mention it: I have always used "Three-fourths" because I want students to associate the number 4 below with an ordinal placement. The original reasoning behind the use of an ordinal number to represent parts ensures consistency in reasoning. That doesn't disqualify "Quarters". It is the perfect choice for starting with Fractions (halves and quarters). But there comes a time when students need to understand WHY they are referred to as 3rds, 4ths and fifths, and nor 3, 4 and 5.

6. caterpillartobutterflyStar commenter

I don't think you've seen many primaries in action then!

That is hardly an argument. I think you would be able to argue the case from a mathematical and pedagogical point of view after you've seen the videos. Without it, discussions degenerate to adopting defensive positions of "I am right you're wrong"........ without even knowing what the other is talking about. Kills all discourse that way. Please do watch the videos.

8. caterpillartobutterflyStar commenter

My argument followed on from yours.

You are basing the usefulness of your videos on an incorrect assumption. In general primary schools do NOT teach that 3/4 increases to 9/12 and they do NOT use the term 'fourths' unless to point out to a child that we are not in America and we use the term quarters here.

I'm sure your videos are lovely, but they have no use if they are correcting something that isn't incorrect.

I assume, your response is "arguing" (in a sense) that equal fractions are indeed taught in classrooms by dividing the numerator and denominator by the same fraction (and I... "it seems"... am blissfully unaware of this common practice...). You are also implying that this is standard practice. Therefore, all text-books, worksheets, sample test-sheets, all curricular material show the division of the numerator and denominator by a fraction before proceeding to multiply by an integer. Consequently, the videos I uploaded do not shed any light on any problem relating to prevailing practices regarding the teaching of Equal Fractions. Is this what you are really saying?

I'm sure your videos are lovely, but they have no use if they are correcting something that isn't incorrect.

Well, thank you. But I am not into making lovely/nice videos, but useful ones. I can't argue with you over it because you don't appear to have seen them. But I think they do correct practices that are widely prevalent and that need to be questioned/revisited. Perhaps you can support your claim with evidence which would be quite visible in curricular materials and resources (text-books, worksheets) that teach and test this specific concept. I would be very interested to see them if you can point them out specifically. Thanks.

Also: for all who have seen the videos and are interested in this thread of discussion, I invite your mathematical arguments/reasoning offered in defense of the current prevailing practice. As far as I know (and have seen), it is about "multiplying the top and bottom numerals" of a fraction by an integer to obtain equal fractions. And it's done without any reference to the original division by a fraction. All other arguments sound quite extraneous to the actual issue on hand: Multiply by an integer or divide by a fraction first?

13. caterpillartobutterflyStar commenter

They do not and you have provided no evidence that such practices are widely prevalent. It is for you to demonstrate a need, not for me to demonstrate there is none.

sabrinakat likes this.

Wow! You sound combative and angry. Sorry. (I can provide PLENTY of proof!!!!!! ). But I shall stop aggravating you with my comments. Last thing I want to do is end up hurting people.

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Let me put this problem as mathematically as possible without meaning to arouse unnecessary hostility and anger. Chill. So here goes:

Place a bar on the table of a certain length (A). Then below it place an identical bar split into two halves (B) as below:

A. 1 whole bar
B. 2 half bars

In doing this you performed one of the following two mathematical operations:

C: 1 whole (equally split into ) half = 2 halves

[“equally split into” is the real-life version of “divide into” , or of the abstract version “divide by”]

D: 1 whole (increased 2 times) = 2 wholes

[“increased 2 times” is the real-life version of “multiply by 2” ]

Which one would you chose to explain what you did: C or D?

16. sparkleghirlStar commenter

I don't teach dividing top and bottom by same fraction. I don't directly teach multiplying either.

I (KS3 here) show them it visually, eg same piece of pie or cheese or whatever (pizza is our favourite) and show them that eg 2/3 of the first one is the same as 4/6 of the second one. When they've seen a few of those they soon work out how to change from one to another, usually without any help from me.

I'm not really sure what your point is.
If I were to say to the children 'divide each of those pieces by a half' I don't think they'd initially understand. So then I'd have to say 'imagine cutting each of those pieces into two' and they would see that they now have twice as many pieces. They would see that as multiplying the total number of pieces by two, and it would serve them very well both in understanding and it application.

Do you think that means they don't understand propery and if so, why do you think that?

17. caterpillartobutterflyStar commenter

Which is also the usual way for KS2 teaching.

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18. caterpillartobutterflyStar commenter

Nopes, neither...just bemused why someone would spend so much time on something that isn't needed.

Odd that the question I asked remains answered!? My point is that ones' theoretical understanding/belief/premise guides one's practices. I am trying to understand what that theoretical understanding/belief is that's guiding your choices. So far what everyone seems to be doing is describing how they display or arrange manipulative/visuals in such a way that "it appears obvious that a pair of fractions are equal". Then the children do the rest.....? What underlying mathematical operation expresses such equality? The obvious answer is "equal splitting". Which is division. 8 divided by 2 means 8 splits into 2 equal groups of 4. Likewise 1 whole splitting into two halves means it is being "divided into" two halves (NOT multiplied).
Children understand very well and very easily the concept of division by half if they're explained the concept visually (as shown in the video, though I really don't want to make this grate upon your ears). In fact it takes at most a couple of minutes to draw that knowledge (pre-cognition) out of them and give it a mathematical garment ( 1 divided by half). Perhaps you should try it. The video (part 2) actually shows how. The other point I am making is that children should understand where the "multiplying by 2" comes from. It is a follow-through-operation that is stitched into the concept (and operation) of "division by half". It is not a "doubling of the whole". If this is not done, then children grow up with a very ambiguous/confused notion of what multiplication means. e.g. "sometimes it means to double 1 whole to make 2 wholes. And sometimes it means to BREAK 1 into 2 equal parts".