# Incorrect Teaching Practices = Weak Pedagogical Reasoning?

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

1. ### sparkleghirlStar commenter

They see quite easily that if you divide something in half, you have twice as many pieces.

Not in my experience they don't.

What age groups do you teach and where?

2. ### caterpillartobutterflyStar commenter

You have started with an incorrect assumption about teaching methods.
Then you make an incorrect assumption about children's understanding and further knowledge.
Then you insist that, if we watch your videos we will learn how to avoid the mistakes in teaching we aren't making, which will mean the misunderstandings, which aren't being misunderstood, will removed.

If it wasn't the school holidays and we didn't have time on our hands, I doubt you'd have any replies.

sabrinakat, Pomza and digoryvenn like this.

Well said caterpillar!

What utter nonsense from the OP. They must have too much time on their hands.

Pomza and caterpillartobutterfly like this.

Honestly, with SATS just 4 and a half weeks away, I'm overjoyed if they remember what a fraction is, let alone what to do with one

5. ### drvsStar commenter

What country's curricula are you basing your assertions on?

In England, we teach equivalent fractions, not equal fractions, and we understand that equivalence and equality are not the same thing. I suspect that this is why a lot of our English members don't identify with the problem you are trying to address.

Your first video begins with the assertion that "equal fractions must represent the same quantity" and you base the rest of your presentation on why that is the wrong approach. This assertion doesn't apply to the English National Curriculum where we might more accurately state the basic assertion as: "equivalent fractions represent the same proportion of a whole". Note proportion, not quantity. This is a fundamentally different approach to yours.

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Thank you for placing this whole debate and discussion in the right context. The switch from "equivalent" to "equal" was made at least 15 years ago in Canada, USA, Singapore and I believe it is the same in Australia, but I am not sure. The distinction between equal ratios and equal fractions had to be made since 1 out of 2 is the "same as" 4 out of 8 has two meanings in real life applications:

A: As equal ratio (which leads to the concept of proportionality) it means that the numerator is always twice the denominator. In other words the amount being numerated (or taken, given whatever) is always twice the amount being denominated. This rule makes 1 out of 2 "the same as" 16 out of 32. In such a situation, equal ratios are generated by multiplying the numerator and the denominator (i.e. increasing them) by the same number. It leads up to the idea of proportionality.

B: However, Equal Fractions do exist in real life. A Fraction CAN indeed be perfectly Equal to another Fraction i.e. it can represent exactly the SAME quantity. The amounts in equal fractions CAN remain UNCHANGED. This is not possible to do if you multiply the amount being numerated and denominated by the same number. Because by doing so you are INCREASING the quantity (permissible in the rule of "equivalence" as you correctly point out).

I can give Anne 1 out of 2 apples. I can give Joe 16 out of 32 apples. I gave them apples in the same ratio: Half. But I did NOT give the SAME amounts. Anne gets just 1 apple. Joe gets 16! Each received DIFFERENT amounts.

However I CAN give them the same amounts and represent that as an Equal Fraction:

I can do that by SPLITTING the numerator and the denominator INTO SMALLER EQUAL PARTS to maintain the sameness of quantity. Splitting into smaller parts is a Division function, not multiplication. 1 Whole "splits" into "2 halves" if we DIVIDE it by half. It doesn't, of we multiply a whole by 2.

As far as I know, in the UK, schools and text-books have adopted "Equal Fractions" as a norm. The word "equivalent" is no more in use. Or maybe both are in use. In which case there are even more pressing reasons to seek clarity on this. Perhaps the UK Curriculum has not been updated? After all it's not written in stone. If you google "equal fractions" the correct definition is everywhere to be seen. BUT the conversion (by multiplication) to Equal fractions is following the older tradition of Equivalence, which is one of Ratios (Proportionality) and not of sameness in quantity as currently defined.

Also, text-books, and all curricular materials visually display/demonstrate Equal Fractions in terms of the
sameness of quantity: Half is visually depicted as 2 fourths or 3 6ixths and so on. And yet, the mathematical operation being performed to produce this "sameness in quantity" is incorrectly being taught as "multiply both the top and the bottom by the same integer".

I just uploaded a Video from Youtube which serves as an excellent example of how Equivalent/Equal Fractions is being taught in classrooms today.

9. ### drvsStar commenter

This is largely incorrect. The English National Curriculum was revised in 2014. It repeatedly references equivalent fractions, never equal fractions. The same can be said for the text and web based teaching resources which the major English publishing houses base on the National Curriculum. Given that the National Curriculum is the basis on which external examinations are set, it very much is set in stone.

Also, any google search for equal fractions, even when set to only return results for "equal fractions", returns exclusively hits for equivalent fractions.

Your pedagogical arguments are based in pure maths. Have you taught younger children? They do not have the cognitive capacity to deal with pure maths when first meeting concepts. The teaching of Science is similar - imperfect models which are accessible to young minds have to be used in order to build foundational knowledge and skills. A small number of students will build on those foundations in secondary and tertiary education to become pure mathematicians or scientists.

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That's a good note to end this discussion thread. Thanks to all who participated. And my sincere apologies to all who I may (very likely) have offended with my abrasive combativeness. I don't like myself for it and sooner or later it gets to me. People respond less to what is being said and more to how it is being said. And that continues to be my weakness.
Please do have a Happy Easter weekend ! My best wishes and Godbless!

I wouldn't worry, we quite like having a good scrap every now and again!

caterpillartobutterfly and Pomza like this.

13. ### caterpillartobutterflyStar commenter

I don't imagine anyone is offended.
More incredulous that anyone can be so naive and yet so arrogant.

A colleague pointed out that there are a lot of interested people (over 600) following this thread (I would request them to see the videos I uploaded so they may establish the context). The objections (that sound pretty angry) are coming from those who reject my arguments for whatever reasons (none mathematical).

Commentator drvs and yourself ( from among a clutch of 4-6 others) have been most patient and I am very grateful. I shall take an Easter break and return to pick up the threads of my argument until I .....and, perhaps, all those following this thread.....obtain a better understanding of the reasons that support the defensive stand re: prevailing practices on Equal Fractions.

Perhaps there exists an argument that could serve as an excellent defense of current practices. I am as keen as my objectors are, to be proven wrong (mathematically) just so that it may restore my credibility as an educator and practitioner. To conclude: Thanks so much for your kind words and your sentiments.

[QUOTE: I am as keen as my objectors are, to be proven wrong (mathematically) just so that it may restore my credibility as an educator and practitioner. [/QUOTE]

CLARIFICATION: " I am as keen as my objectors are, to be proven wrong (mathematically) just so that it may restore my credibility as an educator and practitioner IN THEIR EYES".

16. ### drvsStar commenter

You have made an assumption about prevailing practice in England which is incorrect, as has been made clear above. Have you accepted this? There is no defense to be made here.

Yes, re: the curriculum and its use of the word "equivalence" (not "equal") which clarification you offered, I have accepted as correct. However: the of the word "equivalent" does not suggest the equivalence of "ratios" but of exactly "equal quantities" as parts split into smaller equal parts. The proof of it is in how "equivalence" is demonstrated visually and numerically in all text-books and teaching resources. Which means that it is not multiplication of numerator and denominator that produces such equivalence, but division. (You have in a sense acknowledged this to be "an imperfect Model". (You evidently understand the point I am making.) And yet, prevalent practice clearly assumes the correctness of multiplication to produce an equivalence. (This is applicable to equal ratios only). So I remain firm on my advocacy for mathematically correct practices.

However, what I take from it all is that prevailing practices in the teaching of Maths in the UK (and elsewhere) are immersed in problems and issues that are considered far graver, more important, significant and relevant to teacher's professional lives, than the issue I am raising. I am obviously raising a pitifully "ridiculous" point of mathematical correctness (at a granular level) that has no place in the thinking and practices of UK teachers. And few if any have an appetite for it. This aspect of prevailing practice I have accepted (and respect) as important since it prevails over all else.

In pursuing this, I have likely failed to bring any value. I was quite mistaken in my belief that my point re: the generating of Equal Fractions might do just that. It didn't.

18. ### caterpillartobutterflyStar commenter

As do I, and indeed most teachers I know.
WOW! Another totally incorrect assumption. We look forward to some videos addressing some of these problems and issues.
No you are assuming teachers teach equivalent fractions incorrectly, which is untrue. And, when this is pointed out, you arrogantly suggest we simply have no appetite for the correct teaching.

Mathematical correctness is vital in maths teaching at all levels, however the vast majority of teachers manage this without too much hassle.