Incorrect Past Practices can result from incorrect Pedagogical Reasoning

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

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?

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The logic being used by many teachers to defend Multiplication is drawn from ratios. A far more advanced concept than division by simple fractions (whose numerator is always 1). In Year 3, (8-9 years old) almost all kids would understand there are 3 thirds in a whole, 4 fourths/quarters in a whole, and so on. So dividing by a third (to just find how many thirds in a whole) is the easiest concept to introduce as plain common sense. It is intuitive. Equality of ratios is totally different concept from equality of fractions.

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?

4. sparkleghirlStar commenter

I'd say divided into 2. Not divided by half.

I've never ever heard anyone say 'increased' in this context.

What age group(s) do you teach and where? Are you perhaps in a different education system where things are done this way?

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5. hert0677New commenter

I don't draw on any such false assumption (i.e that 3 fourths in some sense increases to 9 twelfths). So your thesis is not true.

How can you know what is going on in "ALL" classrooms everywhere in the country? It seems incredible that you could possibly know what every teacher is doing everywhere, as you claim above.

I agree that visualisation is important when teaching equivalent fractions. Weaker students can often see that cutting a cake into four equal slices and eating 3 of them gives you as much cake as cutting a cake into 12 equal slices and eating 9.

You seem to think that division by fractions is intuitive to the sorts of less able learners for whom pedagogy matters most. I strongly disagree. They may know that there are 3 thirds in a whole one, but they do not readily relate that to the operation of "1 divided by a third". Weak students often have a poor grasp of what division is, including the fact that it is non commutative. Division where the divisor is greater than the dividend is bad enough for them. When the divisor is less than 1, it is an even greater mystery.

So I would join the others here in suggesting that trying to teach division by fractions before equivalence of fractions is a mistake. Do you know of many people with actual experience of teaching who have found such ideas useful (especially for the less able)?

The main point of agreement I have, though, is that visualisation methods (i.e. the basic concepts involved) often get disposed of too quickly for the less able, and get replaced with methods and rules that mean nothing to them and so are quickly forgotten.

Let's stick to Mathematics. All other issues are extraneous to the point at hand. This is what you are saying :
1 divided into 2 = Half or 2 ?
So 1 whole bar divided into 2 = half or 2?
Therefore: 2 whole bars divided into 2 = ???

Re: 1: I do not make any such claim. I know from my own experience and from the text-books and curricular material in use on this topic, worksheets and workshops, videos on youtube on the topic....not just here but the world over...that multiplication of the numerator and denominator is the norm. I will find it difficult to be convinced otherwise.

Re 2: You yourself are confusing equal ratios with equal fractions e.g. 1 out of 2 is the "same as" 4 out of 8" has two meanings:

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.

B: The condition for Fractions to be Equal is that the Equal Fraction actually has to have the same quantity. The amounts in equal fractions 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.

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.

Which is why the numerator and the denominator have to be SPLIT INTO SMALLER EQUAL PARTS to maintain the sameness of quantity. Splitting into smaller parts is a Division function, not multiplication. When 8 is divided by 2 it means: (a) 8 gets split equally into 2 smaller equal groups of 4 (b) or 4 smaller groups of 2 make 8.

Re 3. If a 9-year old (who has been taught the right way) is asked "How many halves make 1 whole?" s/he should say: "2". This is intuitive. This experiential insight, combined with learning of Fraction concepts makes it almost intuitive for children to answer the question: "How many thirds make 1 whole?" correctly. Within seconds. Takes very little to stretch this same inquiry to 4ths and 5ths...and so on until they get the idea. If fractions are not taught the way minds learn them, then yes, "Division where the divisor is greater than the dividend is bad enough for them." and other similarly "difficult" things will turn into a nightmare for them.

Re 4: So when you "join the others here" in supporting the practice of multiplying the numerator and the denominator, you actually prove the point you are refuting in 1. I think 90% of teachers right now, agree with you and will defend this practice. I don't blame them. Those who are supposed to know better, did not explain it to them. It's not their fault. Right now whether the ideas that I am proposing are useful or not have yet to be proven. The fact is: I am not proposing any NEW ideas. I am simply explaining the Math. Correcting it where it goes awry. Nothing more.

Re 5: Thank you. I think if those reading this thread now go see the videos I uploaded, they will be in better position to understand what this discussion is about.

8. BG54New commenter

Re the underlined sentence, equivalent fractions do not have to lead to the same absolute quantities in different examples, they just have to have the same value as fractions. The fact that in your example Anne has a smaller quantity of apples available to her than Joe is down to your arbitrary choice of quantity and has nothing to do with the equivalent fractions, each person gets half the available number of apples. Try looking at the equivalent fractions purely from the point of view of the unsimplified equivalent fraction having common factors in both numerator and denominator that can be cancelled to give the simplified equivalent fraction - I wouldn't bring "increase" or "decrease" into the discussion.

Evidently, according the UK Curriculum, "equivalent fractions have the same value". Universally, this is understood to mean that equal fractions must represent the same quantity, not the same ratio. Regretfully, your argument of common factors and cancelling to give simplified equivalent fraction is not an argument. It is a description of a practice that yields the original "reduced" fraction. The issue here is whether "value" suggest quantity or "ratio". Linguistically, and in terms of common sense, ratio cannot be considered a "value" as integers and fractions that represent actual quantities, do.

Moreover, the evidence is in the very text-books and worksheets schools are using. They show equal fractions in terms of equal quantities i.e. 3-fourths are shown as 9-twelfths. And correctly so. If what you say is true, then equivalent fractions would be depicted visually as:
3 apples out of 4 apples (shown visually) is the same as 9 apples out of 12 apples (shown visually) ! Such illustrations are NEVER shown to represent equal value.

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

11. brighton56Occasional commenter

To be honest, I got bored of reading everything but just to clarify - Year 3 children are 7 and 8 year olds.

12. colpeeStar commenter

Yes it’s all starting to sound a bit trollish. Perhaps we have all been had

13. sparkleghirlStar commenter

It seems that the OP is pushing an 'education system' promoted online and is based in Canada.

It could be that he knows little about how maths is taught in the UK and that in Canada it's taught completely differently.
I suspect however, that Canadian teachers don't talk about 'increasing' in this sense either. In fact, if I suggested this was an increase, the children themselves would correct me, because they know what 'equivalent' means.

I tried to give him a fair hearing and to be patient but it seems that the OP, despite apparently knowing little about how maths is taught in our schools. decided to come here and tell us all we're doing it wrong, believing we would be grateful for the enlightenment. I originally took his posts to be well meaning, now I see them as ignorant.

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14. hert0677New commenter

You said "When teaching Equal Fractions, teachers...ALL, without exception draw upon an implausible assumption." To know that, you would have to know many things you could not possibly know. Why not just withdraw it? You are more likely to get a sympathetic hearing from people if you make yourself seem reasonable.

No native speaker of English in my acquaintance thinks that equivalent (or equal) fractions are merely ones with equal numerators, or ever uses the phrase "equal fractions" in that sense. So Joe gets a different amount of apples but the same fraction of apples available to him as Anne. Who would ever suggest that Joe gets a bigger fraction of the apples available to him because he gets more apples (except, perhaps, you)?

You say that if one increases the numerator and denominator of a fraction (by, say, a factor of three), one increases the quantity. The quantity of WHAT? Certainly not the quantity the fraction represents. We would be increasing the quantities on the numerator and denominator, true, but so what? It does not follow that the fraction is any bigger. You have said that teachers are making some sort of error with their standard practices but thus far you have not succeeded in communicating what that might possibly be.

I agree. It doesn't follow that she can answer the question "What is 1 divided by a half?" The proportion of students who can answer your question correctly is significantly higher than the proportion who can answer my question correctly, especially among weak students. Anyone with any experience of teaching children would know that division by numbers less than 1 is conceptually very difficult for weak students. If you have limited experience in a classroom, I can see how that might be hard to accept.

Not so. You said that I (and every teacher "without exception" assumes that "3-fourths increase to 9 twelfths" I don't believe that three quarters increases to 9 twelfths. I believe they are equal, and there is no sense (in inverted commas or otherwise) that one is greater than the other. Why you think anyone believes or assumes such a thing is a total mystery to me and everyone else. You have pointed out that the numerator and denominator are both greater in 9 twelfths. So what? How would it follow that anyone (let alone everyone) assumes that one fraction is greater than the other fraction?

Can I ask what your experience of teaching in the UK is? You have had a fairly unsympathetic hearing from a bunch of people who, I suspect, have hundreds of years of teaching experience between them (in this country and abroad). That's not to say experience guarantees being correct, but when one's arguments appeal to what we expect students would find intuitive or difficult, experience counts for a lot.

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15. colpeeStar commenter

Thanks for the link @sparkleghirl , although ‘Education System’ might be a tadge strong for a couple of people with little teaching experience flogging some on-line content. The website itself is enough warning to be careful with one’s money - no irony that such an amateurishly bad effort is produced by a company purporting to be
committed to the use of visual communication via the use of dynamic imagery

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