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LostInParadise's avatar

What do you think of this way of teaching fractions?

Asked by LostInParadise (27708points) May 12th, 2012

From all that I have heard, it seems that the introduction of fractions marks the point where a great many students turn off to math. There is therefore good reason to concentrate on how they are taught.

Doing arithmetic with fractions is something that is completely alien to the young students first seeing them. I do not ordinarily advocate mathematical formalism in K-12 education, but I think this is one case where it may work.
Then show how numbers can be represented as arithmetic operations. It is not just that you have the equation 2+2=4, the = means identical. 2+2 is another way of representing 4.

Secondly, introduce the fraction bar as simply meaning division. Start working with fractional representation of whole numbers, like 2/1 and 6/2. Do arithmetic with whole numbers represented in fractional form. The advantage of this is that students can check their work. To add 6/2 and 8/4 in fractional form, find the common denominator to get 12/4 + 8/4 = 20/4 = 5, which is what we would expect.

Finally, say that we are going to represent one half as 1 divided by 2, or ½. We are going to extend our arithmetic of whole numbers to include these new types of numbers, using the same rules. Then show how they behave as expected. 2*½ = 2/2 = 1. 3 + ½ = 7/2, which is between 6/2 and 8/2, that is between 3 and 4, as we would expect. I think that something along these lines would make students more comfortable with working with fractions.

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20 Answers

Dutchess_III's avatar

I believe consistency and memorization is the biggest key to math. However you decide to teach it, stay with that.

Something that I hate that they do today is as a kid is working to understand one way of doing a problem, they tend to show them another way of doing the same problem. WAIT until they have the first way down cold before you show them a new way.

Mariah's avatar

Your method would be a lot better than the method I was taught with. My teacher made it seem like fractions were special, different types of numbers. We were taught the rules to add and multiply them, and the whole ordeal seemed to be a completely separate situation from “normal” numbers.

I remember I was having trouble with a problem at one point, and the teacher came over to help me. He said the fraction bar was just a division sign, and my reaction was “wait, WHAT? Why didn’t you tell me this sooner!” After this moment it all fell into place and made sense.

It seemed that the teacher didn’t realize that I didn’t know that fractions are just division. I think this can be part of the problem with teaching – something seems so obvious to the teacher that he just doesn’t even mention it, not realizing that it’s not so obvious to the students.

JLeslie's avatar

Well this is very interesting. I think partof the reason I did well at math, and I have said this before on other Q’s, was because when I was very little my grandpa drew pies in the sand on summer vacation and taught me fractions. It was confusing to me. Then a year or two later I began learning fractions. Because of what my grandpa had taught me, it was pretty easy for me to grasp, even though I had not understood it well before. It built up my confidence in math.

As to the method you describe, I don’t like it, I find it very confusing for very young children. I think part of the key to math is children being able to feel confident in math skills, and see themselves as “math people.” I am also against wordy/word problems in large numbers at young ages. A child should not fail math because their reading comprehension is not very good. Then they will suck at math and reading.

I don’t know how it is for other math people, but when I was young many times I did not understand the math, I just did the work with logic and just applying the rules. Later chapters that had been difficult from earlier in the semester would begin to make sense. People frustrated with math seem to want to understand it right away from what I can tell, instead of just doing it. It’s different when you are older and doing much more advanced problems like Calclus, then you need and want to understand, but if you are in calculus probably you do have a very good concept of what is being solved and how.

I would be interested to know if the theory of this new teaching method for math has been successful where it was tried out?

LostInParadise's avatar

@JLeslie , What I am trying to say is that we should look at math as a creative endeavor. We are going to introduce these new numbers 1/n and then we are going to define the arithmetic on them by blindly following the rules we established for whole numbers There is nothing additional to understand. This gives a sense of excitement and discovery. Working with fractions is important, because they extend arithmetic from the discrete to the continuous. It is the movement from counting to measuring.

Later on, irrational numbers can be introduced in the same way. We define a number x = square root of 2 that satisfies x^2 = 2. Again we define the arithmetic using these numbers as the same arithmetic we used for whole numbers and then rationals. Using the Pythagorean Theorem, it is easy to show that a right triangle whose sides are unit length has a hypotenuse of length square root of two. More advanced students can be shown a proof that square root of 2 can’t be represented as a ratio. Other students can work out decimal approximations and see that no matter how many places they go, there does not seem to be any repeating decimal. Most of us, however, can live our lives without giving serious consideration to irrationals. Understanding fractions is much more important.

JLeslie's avatar

@LostInParadise I just think it is confusing for a young mind. Verbally are we still going to say, “two plus two equals four?” I think what we say has to match what we write for small children. I guess if everything is replaced, if we all write all the time 2/1 symbolizes 2 then maybe it will work? There is no way an average 6 year old is going to understand 12/4 = 3/1 I don’t think.

I was arguing to not be so creative in math, the opposite of what you are arguing for.

I sent your Q to some people who are math oriented, or have had opinions on math questions, let’s see what they say.

LostInParadise's avatar

Are fractions being taught to 6 year olds? Maybe they can be taught what a fraction is, but certainly not the arithmetic on them. At that age, I would not even teach them how to write fractions. It is enough that they can talk of one half or three fifths.

Fly's avatar

I think that the idea behind this method is good, but I have to agree with @JLeslie that the actual method that you are suggesting would have been very confusing to me. Really, it’s too much at once. I can’t remember the specifics, but the way that I was taught did result in my understanding that they are just division in the end, though they were not introduced to me that way. It was more of a gradual introduction, and as I understood more of the concept, more of it was introduced to me. This worked really well for me and I never had much of a problem with fractions.

I think you are forgetting that children do not have a very good understanding of division at the age that they are taught fractions. I learned fractions in third grade (I was 7) and I had just learned division that year. While you and I understand division very well now and can easily make these connections, I would not have had a strong enough grasp on division to understand the method that you are suggesting at that age; it would have just left me confused and frustrated. Sorry!

JLeslie's avatar

@Fly I still don’t think of fractions as division. I know I can divide to get it rewritten in decimal forms of course, and how to move the numbers around on either side of the equation, but when I think of ¾ths, I am just thinking 3 pieces out of a possible four to make a whole.

LostInParadise's avatar

@Fly, I can’t remember when I learned fractions. My gut feeling is that it is okay to introduce them at a young age, but that arithmetic with fractions should be postponed until the age of at least 10 or 11. The brain is constantly developing in children. There is no point in introducing concepts before the students are prepared for them.

@JLeslie , If you think of fractions as divisions, then it follows easily from the rules that 3 pieces of size ¼ is the same 3 pieces divided into 4 equal divisions. Whether you are consciously aware of it or not, that is the assumption you are making when you divide 3 by 4 to get the decimal representation.

JLeslie's avatar

@LostInParadise No, but for a child .75 is not easily understood as 3 pieces out of 4, and that is the answer to the division.

FYI I took Business Calculus in College because I wanted to, not because it was required, and I am a business major, so I took a year of finance, a year of statistics, and I tutored Algebra to Jr High kids. I know math, I’m very good at it. Not more advanced than what I stated, but certantly k-12 I am competent. I don’t need fractions and division explained to me for the debate we are having.

LostInParadise's avatar

@JLeslie , I meant no offense. The nomenclature we use of “three fourths” encourages the image of ¾ as dividing by 4 and then multiplying by 3. In a similar way, I think the use of a minus sign to both denote subtraction and to indicate that a number is negative, causes people to wonder why minus a minus is plus. If we used n and p in front of numbers to indicate negative and positive then we would have subtracting n is the same as adding p and subtracting p is the same as adding n, completely symmetric. But I digress.

The point I am trying to make is that if students get used to working with fractions that are whole numbers and thinking of them as division problems, verifying the results on their own, then it should be relatively easy to view fractions in general as the division of one number by another.

JLeslie's avatar

@LostInParadise No offense taken :) just letting you know I get the explanation, just disagree.

I understand why minus a negative is positive is confusing, but then is when I am back to just leanr the rule not the why. Why comes later in math. As @Fly kind of pointed out as well, the full understand came later for her.

I do wonder, were you always a math person and interesting in changing the teaching and learning on the subject? Or, were you not a math person, and maybe have insight I don’t get about people who have trouble with math? It would be very interesting if you are the latter, because I come from the standpoint of let the people inclined towards math excell early. I do think there are many people who might even be so inclined, but psyche themselves out for various reasons.

Fly's avatar

@LostInParadise I should add the I was in the Magnet Program, so I likely learned earlier and more quickly than most kids. We covered fractions for about a week or so, and I would say that by the end of that time I did fully understand fractions. I have also always been very good at math. I don’t know at what exact age fractions are normally taught, but I do know that it is in elementary school and it is early on in fifth grade at the latest for the average student, and earlier for most, (at least in our school system). It would not be practical to postpone learning arithmetic fractions to the age that you suggest (again, using my school system as an example) because all math classes past elementary school require a general understanding of the functions of fractions as the lowest level of math is pre-algebra, and I do not personally think that this is at all unreasonable.

I can understand why you think that this method would be successful, but you are thinking of this method in hindsight, from the point of view of a fully comprehensive adult. I can tell you that, having fairly recently been a kid of that age myself and having always been good at math, I did not think about things even close to the way that you are approaching this, and I would not have been able to comprehend this method. As I said before, it is just far too much information to process at once, and I would not have been able to make those connections.

JLeslie's avatar

Sorry for so many typos in my last post.

SpatzieLover's avatar

While I understand what point you are trying to make @LostInParadise, I agree with @Fly & @JLeslie.

This information is above the average math student.

Could fractions be taught this way? Certainly, but to an older child, or a gifted math student. IMO, this will not do anything but create a math phobia to a younger child upon the introduction of fractions.

What I see is the necessity to teach and make certain small children (under age 8) understand and can recite their math facts. They should have memorized their addition and subtraction tables prior to moving on to multiplication. Then, the multiplication table should be memorized. Then, division and fractions.

The real issue I’ve seen all along is that children are moved along prior to fully understanding the basics. Then confusion builds to the point of frustration.

I homeschool (and recall my elementary math curriculum well). As part of our math curriculum we won’t begin fractions until the equivalent of 5th grade. My son advanced in math so for us it will be approx. one year away…approximately age 7, late in second grade

@JLeslie said I was arguing to not be so creative in math, the opposite of what you are arguing for. I agree.

LostInParadise's avatar

I hear what you guys are saying and I respect your opinions. However, I am not convinced that students would be afraid of the idea of math as invention. I know from personal experience way back when I did substitute teaching, that students like math puzzles, much more than I had expected. I have heard from some teachers that they give math puzzles as a reward for completing the rest of the work, which always strikes me as a little funny. The reward for doing tedious math drills is the chance to use one’s creativity to tackle a challenging recreational problem.

Fly's avatar

@LostInParadise Yes, but these math puzzles are more about the puzzle, not the math. The math portion tests only one, simple concept and is relatively easy. And, these puzzles are given after the students have fully learned a concept.

To be perfectly honest, I don’t see how your idea is particularly creative. It just seems like a roundabout, overworked way to teach fractions. There is a difference between being creative and unnecessarily complicating a method that really needs to be simple, step-by-step, and straightforward as opposed to abstract. I am telling you, as a current student who has always been advanced in math and in general, that the method that you describe is just not practical. Though I fully comprehend fractions, I find it difficult to follow your logic even now, and I certainly don’t see how young children who are being introduced to fractions for the first time would understand it.

JLeslie's avatar

What I still wonder is if @LostInParadise took to math easily and was advanced all the way through school? Most of us on here are math people and don’t like the idea proposed. But, what about the student who always felt challenged or afraid of math? Does this new way suddenly make sense? I have my doubts, but you never know. I like the idea of attracting more people to math, but my biggest fear is confusing, frustrating, or leaving those behind who will really excell and pursue math and math related careers.

It’s like how everyone thinks iPads are fantastic and so intuitive. I have one, I use it all the time, but I don’t find it particularly intuitive. I am not sure if it is because I have used PC for so long, or if my brain just works differently? It seems very creative people like Apple products generally, people who were not very good on computers previously like the iPad, and Apple has been cited as helping wake up autistic kids who seemingly take right to the ipad.

LostInParadise's avatar

I did take to math easily. It was my major in college. I am particularly interested in the best ways of teaching it, so I may come up with unorthodox ideas. One thing I know is that what is currently being done seems to be in need of improvement.

JLeslie's avatar

@LostInParadise I am in favor of finding better ways of teaching math. What I worry is that in the last 25 years we have moved away from a system that was working to one that doesn’t. That math has been muddied with mixing it together with needing an a strong reading comprehension at a young age. How old are you out of curiousity. I am just wondering if you learned math similar to how I did, I am 44. Although, since ever state can do something different it would not necessarily mean anything even if we were the same age. When I moved to MD from NY I was way ahead in math. Thank goodness we had been tested in school, honestly, the testing is what said I was on an 8th grade level in 5th grade and that is the only reason I internalized I excelled at math, and was ahead of my grade. I knew I liked it, but didn’t know I was better than average.

I moved in 5th grade. For some reason in 6th grade I was put into the lowest math group, and I told my dad after a couple of months after school started. He told me to tell the teacher I wanted to move up. I did, he asked me to do two problems the kids in the more advanced class were doing, and I did. I answered those problems having not been taught that unit during 6th grade, it was out of memory from maybe 4th? Funny it was a fractions problem, I remember having to find a common denominator. I still have no idea why he had put me in the slower group initially. If I had just gone passively along I would have been basically doing nothing in math that year.

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