# Incorrect Teaching Practices = Weak Pedagogical Reasoning?

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

1. Oh dear: here we go again on mathematical correctness. Opinions, I respect, but not incorrect mathematical thinking which continues to be presented in defense of prevailing practice.
{Permit me to clarify that I am referring only to the prevailing practice around the teaching of Equivalent Fractions. I am not tarring ALL prevailing practices in teaching mathematics with the same brush. Just to be clear}

Case A: When 1 is divided by half the answer is "Two". Question: "Two" what? Answer: "Two Halves"

Case B: When 1 is multiplied by 2 the answer is also "Two". Question: "Two" what? Answer: "Two 1's"

If prevailing practices treat "two" (i.e. the numeral "2") to mean the SAME thing in A as in B, then what is being preached is the startling notion that "Two Halves" (in A) are the same as "Two One's" (in B).

THIS, in a nutshell, is what prevailing practice is advocating and defending. I wait for a mathematically reasoned argument to dispute the correctness of what I am stating. If none is forthcoming (opinions and recourse to prevailing practice notwithstanding), then I leave it to all who are following this thread to come to their own conclusions on this.

2. well, I'm not thick. I'm highly intelligent. You are making yourself about as clear as mud, to me.

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Oh dear: here we go again on mathematical correctness. Opinions, I respect, but not incorrect mathematical thinking which continues to be presented in defense of prevailing practice.
{Permit me to clarify that I am referring only to the prevailing practice around the teaching of Equivalent Fractions. I am not tarring ALL prevailing practices in teaching mathematics with the same brush. Just to be clear}

Case A: When 1 is divided by half the answer is "Two". Question: "Two" what? Answer: "Two Halves"

Case B: When 1 is multiplied by 2 the answer is also "Two". Question: "Two" what? Answer: "Two 1's"

If prevailing practices treat "two" (i.e. the numeral "2") to mean the SAME thing in A as in B, then what is being preached is the startling notion that "Two Halves" (in A) are the same as "Two One's" (in B).

THIS, in a nutshell, is what prevailing practice is advocating and defending. I wait for a mathematically reasoned argument to dispute the correctness of what I am stating. If none is forthcoming (opinions and recourse to prevailing practice notwithstanding), then I leave it to all who are following this thread to come to their own conclusions on this.

4. "following" might be a pretty strong word for what people are doing on this thread....

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No problem. Let me try verbally (it’s all explained in the videos I uploaded. Very, very easy to understand)
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When we say Half (as in 1 over 2) it could mean any one of the two things (X and Y below):

Case X: Half a bar of chocolate i.e. the whole bar is split into 2 equal parts and we take one of those parts.

It could also mean:

Case Y: I had 2 bars of chocolate, and I took one bar.

Equivalent Fractions is an extension of Case X, NOT of Case Y.

So, if you take half a bar of chocolate (as in Case X) it can be shown that you took 2 Quarters out of 4 Quarters of the SAME chocolate. So the amount you took, doesn’t change. Same amount. So in this case “1half over or out of 2 (halves)” = “2 (quarters) over or out of 4 (2 quarters)”.
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What is the mathematical operation that converts 1 half into 2 quarters? The only way to do is to split the1 half further equally into 2 smaller parts. They will turn into fourths. The mathematical operation for splitting into1 into 2 smaller equal parts is division, not multiplication. So this is what it looks like:

1…(half)……… …divide by (into) half ………. 2 quarters
—— = —-
2 …(halves)………divide by (into) half ……….. 4 quarters

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Prevailing practice suggests the contrary :

1…(half)……..multiplied by 2 ……………2 quarters (??????? not true!)
—— =
2 …(halves)….multiplied by 2 ……………4 quarters (??????? not true).

Because if you multiply 1 half by 2 you will get 1 Whole. So the numerator will be 1 Whole
And if you multiply the 2 halves below also by 2 , you’ll get 4 halves or 2 wholes.
So the answer will be 1 bar out of 2 bars. Here the original quantity (1 half out of 2 halves) changes (to 1 whole bar out of 2 bars). The original amount increases from Half a bar to 1 whole bar.

This is representative of Equal Ratios, not of Equivalent FRACTIONS. In Equivalent Fractions, the original quantity remains unchanged.

(Once again, all of the above is explained in the videos using just animation and NO words. Unless you see them, I can't guarantee that all these words/explanations will make one's head any less muddier. .
Sorry.

6.
7. sorry, I don't think you are very good at explaining things. Except I now want chocolate

8. No it isn't the prevailing practice. And this is the basis of everyone's argument with you.
None of us have any issue with the maths you have explained (in my quoted post above). It is entirely correct. However it is also what we teach in schools and is the prevailing practice.

The issue is that you arrogantly suggest that you are the only person to understand equivalent fractions and that teachers across the UK are teaching it incorrectly and so have absolute need of your videos.

We can only object to your assertion about incorrect teaching as this is how you entitled your thread and what you persist in posting, despite every single responder telling you differently.

9. It's Easter...chocolate is essential!

But be careful when dividing your eggs in half...you might think you'll get two eggs and then be disappointed that you actually end up with two halves!  10. --------------------------------------------------------------------------------------------------------------------

Actually, no responders so far have accepted that what I described as prevailing practice (multiplying the numerator and denominator) as being incorrect. Nor have they said that I am correct in my maths re: the generating of equivalent fractions. I have gone over the comments in this thread and I can't find anything confirming the correct practice by division as proposed by me. I have said they need to see my videos to understand my point, yes, and I still insist they do so that I don't have to spend so much time explaining why and how. The need to see them is not absolute but absolutely necessary, yes.

I never claimed that I am the only person who understands equivalent fractions. All those I have taught equivalent fractions, understand them, many as young as 8-9 years old! As well as teachers I have trained. And a few of my own peers, too.

As for the rest:

1. Please suggest to me the name of a textbook used in Year 3- 4 that uses division by fractions as a means to generate equal fractions. In the UK, or the US or anywhere else.

2. Point me to the worksheets that expects Year 3 -4 students to divide by a fraction to generate equal fractions.

3. Point me to any video being used as a supportive resource that demonstrates the division of the numerator and denominator by a fraction to generate equivalent fractions.

Perhaps they exist and that I am not aware of them. In fact I am sure there are a handful of teachers who use the correct approach. They would be exceptions that prove the rule. But if you insist that all schools in the UK teach Equivalent fractions the correct way (i.e. by the division of fractions), then it's hard to understand why all that I have seen on the internet, on google, the videos explaining the concept, and discussions with teachers and trainers here, who confirm it.....all, without exception, confirm that the standard prevailing practice is to teach by MULTIPLYING the numerator and the denominator by an integer.
If it was otherwise, why would we be discussing it?

On a lighter note: let's not get overwhelmed by attacks of "Brexit Moments" during Easter. Peace!

11. I try, for whatever it's worth, driven by sheer Hope.....right up to the bitter end. Tch! Tch! 12. You need to get out more. Maybe get a hobby or something?

13. I have never seen such an effort to justify an inaccurate video...

You fixed a problem which doesn't exist, and then get uppity about it when people tell you that! And in order to defend your mistake, you spend 3 pages trying to make maths harder.

What a fine maths teacher you must be.

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I am seeking/soliciting evidence that the specific problem I am identifying (multiplying the numerator and the denominator by an integer) "doesn't exist" in prevailing practices. I have received none so far. I myself would like to be spared the 3-page explanations if commentators raised mathematical issues specifically around the videos. In calling the videos "inaccurate" (mathematically, I assume), you un-intendedly explain a lot of un-discussible issues around maths-education. Mathematical concepts are known to become easier, not harder, when everything inside it connects logically, and draws upon sound mathematical reasoning....because learners find them much easier to understand, remember and recall.
I am sorry my videos...and I myself, as a commentator....make you feel the way you do. I wish I didn't...

15. Noted. Thank you! 16. Can’t you just add a zero or move the decimal point, or something easier...?

17. Now I am bit lost with all of this fascinating discussion and although I use real-life and visual examples to teach students .... the basis of equivalent fractions is (like many topics that use the scaling concept like in algebra, pie charts, ratios, purchasing etc) is simply x/x=1 for all x. This is why multiplying top and bottom by the same number is simply multiplying it by 1 and therefore the value doesn't change just the the way it is presented. Fundamental for students to appreciate that 6/6 = 1 (in my opinion) along with other things like what 100% actually means.

Now back to my chocolate bar and marking Flanks likes this.
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Extremely refreshing to be presented with a sample of mathematical reasoning. Impeccable. Nothing wrong or mistaken here. I am so grateful you raised this approach to tackling Equal fractions. Now you will have to bear with me regarding what follows because what you raise requires me to put it within the scope of mathematical perspectives. So I will give it shot and see if it makes sense to you or other interested readers. Because....and you may not believe this....but Maths is VERY VERY interesting!

FIRSTLY:

ANY number (x) multiplied by 1 retains its original identity (as x) . That includes rational numbers (or fractions). This is another aspect of mathematics quite unrelated to the concept of Equivalent Fractions.

1/2 x 3/3 = 1/2 is no different from 8 x 3/3 = 8. Why? Well, because if you multiply 8 x 3 and divide the answer by 3 you’ll get back the same 8.

Likewise, in 1/2 x 3/3, if you increase half 3 times, then decrease it back 3 times, you will end up with same 1/2 as before.

This is quite a pointless (almost meaningless exercise in abstraction for 8-9 year olds). Not because abstraction is meaningless. It is not. It is very powerful and packed with meaning. But if the connection between the abstract and the real-life learning that inspired the abstract in the first place is lost, then abstract numerical operations become soulless & hollow and yes, utterly meaningless.
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SECONDLY:

Let’s look at what you’re saying:

1/2 x 2/2 = 2/4 (But 2/2 = 1) so 1/2 x 1 = 1/2. Therefore, we can infer that 2/4 is the same as 1/2.

This is based on understanding 1/2 to mean “division as equal sharing” e.g. 1 pizza shared equally between 2 people : 1pizza divided by 2 people expressed as 1/2.

When you double the pizza to 2 pizzas (multiply 1 on top by 2) and also double the number of people to 4 ( multiply 2 at the bottom by 2 ) then you end up sharing 2 pizzas among 4 people.

That would result each person getting 1/2 a pizza.

So in this case: 1/2 = 2/4 means: to divide 1 pizza equally among 2 people is the same as dividing 2 pizzas among 4 people. Why? Because each gets 1/2.

There is no equivalence of Fractions involved here. One pizza is “increased” to 2 pizzas. So 2 people are increased to 4 people to maintain the sameness of the ratio 1:2

There are no halves and quarters involved to explain the equivalence of value of two equivalent fractions e.g. 1 half = 2 quarters = 3 sixths and so on.

(Incidentally, this argument is covered in the first of the two videos I uploaded).
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Hope I didn't ruin your experience with the chocolate-bar. If so, my sincere apologies.

19.
20. Good Lord Above!
Please reassure us @karisshad17 that you are not a teacher.

I'm a maths graduate and maths specialist teacher, however even I almost lost the will to live when wading through your latest 'explanation'.
I seriously hope you never try to teach anyone anything.
And you have absolutely convinced me never to bother watching any video where you 'explain' mathematical concepts.
Thought you were disappearing off for Easter at this point...clearly not.

Flanks likes this.