Posts
Math by analogy is when teachers substitute ideas completely unrelated to math in order to make some concept "easier". Usually, this is because they themselves do not understand the meaning behind what they are teaching, so they cannot explain it accurately. Math by analogy substitutes presumed common context for reasoning. Yet most young students don't share enough common context to build the analo
You see math by analogy in both big and little examples, from the use of it to "explain" greater than and less than to its use in teaching place value. The most common analogy I see used by teachers and their books is that "a fraction is part of a whole".This analogy has devastating results. I routinely (in 100% of classrooms not using Singapore math, in more than 50% of the students) hear:
- 1. "there's no such thing as ten ninths." that's the majority response in classroom after classroom. Why? Because a fraction is PART of a whole. How can a part of a whole be bigger than the whole? What's the whole then?
- 1b. therefore, they believe no fraction can be bigger than 1.
- 2. "You can't divide 6 things among 7 people." 6 things isn't one whole. It's 6.
- 3. "three thirds is A Whole." Not one.
- 3b. Therefore, they don't know 3 divided by 3, written as a fraction, is 1. I often hear of students who ask "is this a division problem or a fraction problem?"
These problems are so severe because these students have teachers who manage not to notice these errors. No problems in their books, no lesson script in the teachers guides illuminates this to the teacher. They only see the most trivial of problems. 10/9 is beyond the pale.
The correct explanation is that a fraction is a number. What number? A number defined on the number line as follows:
1/3 is the point on the number line when you break the unit length into 3 equal length parts, and take 1 part. the endpoint of that part is 1/3.
4/3 is the point on the number line when you break each unit length into 3 equal length parts, and take 4 parts. the endpoint of those parts is 4/3.
Yes, teachers will need to build up to this. They should do so.
27 comments:
Well that's depressing. I've spent a good fraction of this year (sorry) teaching my 3rd grader those ideas in her math course. It does take a lot of practice and repetition and building!
But practice, repetition and building are inauthentic and just plain BAD! Why does the ed world accept the necessity for same in the arts and athletics while rejecting it for academics? Could it possibly be because (1)mastering something takes effort and (2) some kids will learn more and faster? (very bad in a world that demands equal outcomes)
In my basic math class in community college, I see this problem all the time. It is very pervasive and rears its ugly head in unexpected places. (At least, I don't expect it.) For example, to solve the problem: what is 1.2% of 12, we teach them to convert 1.2% from a percent to the decimal equivalent. But, they say, 1.2% is already a decimal!!
I don't recall having that problem, but I had others. The goal is not necessarily to know all of them and get it right the first time. There is a lot to be said for practice, practice, practice. Drill and understand.
When you make the transition between simple fractions with numbers to rational expressions, you run into lots of subtle understanding problems. I found that students couldn't tell me what a factor was. They couldn't believe that you could move a factor up or down in a fraction just by changing the sign of the exponent. They saw an experssion and thought that there was only one required thing to do to it; simplify. It was somehow illegal to have an expression with a negative exponent.
The laugh is on me--this morning I asked my 3rd grader if you could have ten ninths and she said no, you can only have nine ninths. We went over it again and she remembers now...until tomorrow! (Every day she has at least one math question involving 11/7 or 7/3 or something.)
I guess this shows that these concepts need a *lot* of practice and repetition in order to sink in.
Also, my experience has been that students are taught that ten ninths is not simplified: they must change it to one and one ninth.
If anyone is interested: I got fed up with the lack of practice problems and poor quality of practice problems in my textbook so my colleague and I created a website where students can practice basic math. The problems are randomly generated using mathematical algorithms so the program can generate any number of problems at each level. If you want to check it out, you can register as a student. Once you log in, enroll in apdwyer11003. This is my 'play' class where anyone can try the problems.
If you are more interested, contact me and you can register as an instructor and set up your own class.
Sorry, I forgot to give the website. It is http://vorce.info/mathbooster/login.html
It is often necessary to teach "half-truths". I think it is reasonable to describe a fraction as parts of a whole, and when it comes time to discuss 10/9, one can explain that 10/9 is an improper fraction equal to 1 + 1/9 and that it is actually "proper" fractions such as 1/9 that are parts of a whole.
How do you ask this question? "Is 10/9 a fraction?" seems awfully leading.
I asked my kid this: "What is *this*," I said while pointing to a sheet with 10/9.
"1 and 1/9," he said.
Hmmm ... time for a followup question. "What type of number is this?" I was hoping for "a rational."
"Mixed number," he said.
So ... does he know that 10/9 is a fraction? Sorta ... and he is quite comfortable changing improper fraction into mixed numbers and vice versa, but he may not have totally nailed that 10/9 is a fraction, just an improper one [we *have* gone over this, but mostly in terms of *using* them, not in naming them ...].
He certainly knows that it is a *number*.
So ... how was "is 10/9 a fraction" asked?
-Mark Roulo
Jean, I wouldn't despair, and I wouldn't decide it's just an issue of practice.
If you have the wrong ideas in your head, practicing things that fit the right ideas just doesn't stick. It's like the practice works fine in a Very narrow range of problems but as soon as you need to do problem solving outside of that, you default to intuition.
Showing, for example, 2 pies and 3/4 of another makes it easy to just think the fraction is 3/4.
I think using a number line is a very good way to show the reality of the improper fractions. You show all of the fractions with denominator 3, including 4/3, 5/3, 6/3, etc. all the way to 300/3. You show the equivalent fractions too, the infinite ability to write a fraction as equal to others. this may help the light bulb to go off.
Mark, the question wasn't asked that way. here's an example: a 6th grade class was being shown the number line for the first time, the day after having significant trouble handling problems like 5 2/3 - 3 1/2.
The teacher began showing the definition: break the write length into n equal parts. 1/9,2/9, etc. He got to 10/9, and the class shouted "no such thing!"
Individual students do it all the time. Asked to write "1.2 as a fraction" using the rule they used to write .35 as a fraction, they couldn't. why? "You can't have 12/10" is the standard refrain.
No, you cannot teach children half truths in math. Half truths in math are falsehoods. You don't know what they will learn and what damage they will do when they wrongly equate true things with false ones. At best, they learn that teachers can't be trusted, at worst, they believe they can't reason to the right answer in math, but teachers decide what is right and wrong.
I'll try the number line, thanks.
This is a kid who has made pretty slow progress with math this year--she can grasp the ideas, but she has a permanent visual condition that made vision therapy necessary this year. Problems were most evident in math (and that's what clued me in). We've seen a lot of improvement lately though!
"Mark, the question wasn't asked that way. here's an example: a 6th grade class was being shown the number line for the first time, the day after having significant trouble handling problems like 5 2/3 - 3 1/2."
VERY interesting.
It seems that the kids had not been introduced to improper fractions. Is that the case?
If they hadn't seen improper fractions, when would they get to problems like 5 1/2 - 3 2/3?
-Mark Roulo
The concept I'm struggling with teaching my 4th grader is why when dividing a decimal by a decimal (e.g. 1.2 / 2.4) one multiplies by a power of ten to get rid of the decimal. If I write the problem as a fraction, she can easily solve it so clearly she understands the concept of equivalent fractions. But if I write it as a traditional division problem with the )---- she gets all messed up.
What can I do to get her to understand this concept?
CrimsonWife, Maria Miller of Math Mammoth has made some very good videos on that. You can find the videos at her website, or go to youtube and find the mathmammoth channel.
oh, they HAD been introduced to improper fractions, and to converting between them and mixed numerals.
This teacher was reviewing this material when I suggested using the number line. His students had been shown it at least twice before in his class. He said his students had done poorly on an assessment covering this, so he was reviewing it again.
By state standards, these kids had also seen improper fractions and mixed numerals in grades 5 and 4 as well.
But this example could have come from many other classrooms in countless other schools. I routinely meet students who say the same thing, and hear teachers say their kids say the same when introduced to the number line.
I think what is happening is this:
Since a fraction is part of a whole, they take "improper fraction" to mean illegal. the emphasis on converting to a mixed numeral is then the procedure to get rid of that illegal object that isn't allowed. the fraction is only a fraction then when it's back to a part of a whole.
The idea that these are equally valid mathematical objects does not exist in their heads.
CW, I don't know what you are saying she is struggling with. here is where I would start. (Maybe this is what you mean she can do)
Decimals are fractions. They are the special case where the denominator is a power of ten. The decimal point is shorthand for writing that denominator. You place the point so that the number of decimal digits matches the power of ten.
So .5 is shorthand for 5/10, 1.2 is shorthand for 12/10, .0300 is shorthand for 300/10000.
So to divide decimals, you must already know how to divide fractions. The simplest way is then to convert the decimals back into fractions and divide. 1.2 / 2.4 is then 12/10 divided by 24/10. You then turn that into multiplication of fractions:12/10 * 10/24. the tens cancel. You are left with 12 divided by 24.
For whatever division problem, yon can always separate the fractions into the powers of ten and the old numerators. So that's where the "multiplying by the power of ten" comes from--you are literally multiplying one fraction by another, and one is a power of ten.
So, does she know all of that?
Big push from CCSS and at NCTM was the understanding of a unit fraction. SO:
5 x 3/4
5 x (3 x 1/4)
(5 x 3) x 1/4
Powerpoint from a session by Tad Watanabe discussing unit fractions:
Challenges and Opportunities of Common Core: Fraction Teaching and Learning
CW, Your student may benefit from setting up the traditional long division algorithm for
120/240, then 12/24, then 1.2/2.4
and comparing the ease of solving as well as the solution. Moving the decimal is a convention.
She would also benefit from drawing the pictorial representations of those fractions and seeing that they are three different names for the same thing, in certain contexts. We humans like to make life easy, so we rename to make our computations easy.
If I write 1.2/2.4 in fraction form, she can easily solve the problem, but if I write it as a traditional division problem with the 2.4 outside of the division symbol and the 1.2 inside, she gets all messed up.
We've gone over numerous times that the line separating the numerator and the denominator in a fraction represents division. So I'm not understanding how she can solve the problem easily if it's written in one fashion but get stymied by it if it's written differently.
I don't remember this being a particularly difficult concept to grasp when I learned it myself in elementary or middle school.
you mean literally you write a complex fraction 1.2/2.4 with no explanation for why that is legitimate?
Is it clear to you why "complex fractions are fractions" is necessarily legitimate? really?
I'm going to gently suggest that your explanations are Very procedural and she lacks the understanding yon have in Your brain to make sense of what division and fractions mean.
Saying that 5/6ths of a pie is really "the same" as doing long division on 5 divided by 6 doesn't really explain why. But the powers of ten explanation is a bit easier to see if you can Factor them out of the division problem.
Is 12/24 as 24|---12 a problem for her too? or just with the decimals?
Show her what I wrote before:
12/10 divided by 24/10 works down to 12/24 x 10/10. Show her 12/100 divided by 24/10, becoming 12/24 x 10/100, and a dozen more. don't skip any steps. show where the powers of ten go.
I see a tradeoff between not causing future problems and being pedantic to the point where students don't get it anyway. There are many levels of understanding, and it's difficult to find the right balance. There are many things I didn't really understand until much later. Could they have been avoided in the first place? Maybe.
I would love to have all K-6 teachers read Wu's book. Better yet, require them to take a college course based on the book. That's the real problem. How do you get people to do something they don't believe in.
1.2/2.4 is easy for a kid to see because the numbers are obvious multiples. Try it with nonobvious numbers.
The student needs more experience with the pictorial representation of fractions and decimals to understand.
She understands that one can multiply the numerator and the denominator by the same number because that is equivalent to multiplying the whole fraction by one. So if I write a complex fraction with decimals in both the numerator and denominator, she knows to multiply both by the same power of ten to get rid of the decimals.
However, if I write the exact same division problem not as a fraction but by using the )--- she doesn't solve it correctly. I don't understand why the format of the problem makes such a difference to her when there are decimals involved.
when I see a student or hear from a teacher of a student who has trouble with a procedure it is always because they don't
understand why they are supposed to do whatever they think they are supposed to do.
In those cases, the students misperform the procedure because the procedure
has no meaning or the wrong meaning to them.
Lgm is correct: a student struggling with fractions and decimals needs a great deal of pictorial representation. they need that before they attack any procedure at all. And then, they need it to explain the procedure, too.
the pictorial representation to start with is the x-axis number line and y-axis number line.
the unit square is the best place to explain decimals. Break the unit length on the x-axis into tenths, and the unit length on the y-axis into tenths and show all of the equivalences between Various decimal notations.
A great way to connect division to fractions is to show the array model for multiplication + division, and then show the area model for fractions.
so a 5 by 6 array shows how 5 groups of 6 and 6 groups of 5 yield 30.
an area model for fractions then shows 1/5 of the 30 sq unit area is 6 and vice versa.
It is also the best way to explain multiplication of fractions, which is different than equivalent fractions.
A beautiful way to explain multiplication of fractions is as follows:
1/2 x 1/3 is the area of a little rectangle whose sides are 1/2 and 1/3. So you draw the halves on the x-axis number line and you draw the thirds on the y-axis number line. Drawing out all of those lines across the unit square, you and up with 6 congruent rectangles.
Since the whole area is 1, and the 6 congruent rectangles all have the same area, each must have area 1/6.
the argument generalizes to a/b x c/d.
both number line and area models are good for division of fractions.
You said some thing worrisome though. You said she knows
2/3=20/30 because 2/3 * 1 = 2/3 * 10/10 =2 * 10 / 3 * 10
but why does she know
2/3 * 10/10 =2 * 10 /3 * 10?
the above statement says without proof or reasoning
multiplication of fractions means you multiply the tops and then you multiply the bottoms.
It doesn't say anything about fractions being equivalent.
Your statement invokes multiplication of fractions to show equivalence but then what did you hang multiplication on?
that is not what the rule of equivalent fractions says. Equivalent fractions says 2/3 = 2*2 / 3*2 = 2*3 / 3*3 = 2*4 / 3*4 etc,
and is shown by drawing the pictorial representions that these really are the same fraction.
In other words, how can you go from fraction x fraction to numerator x numerator /denominator x denominator?
it's just a rule you already take for granted and your student has no basis for.
Steve H calls this pedantic, but the point is that she can't be correctly reasoning from what you said. so somewhere, your student's understanding will fall down.
It falls down not where you expect it, but falls down nonetheless.
I'd suggest that since these procedures for complex fraction division and long division of decimals have no meaning to her,
she can't make them have any common meaning.
All she has got is that you "multiply the top and bottom" to get rid of decimals in Fractions,
but no one taught her how to multiply a decimal and another decimal by things when they are in a long division problem.
if your first response is "what do you mean? she knows multiplying by ten moves the decimal point over" i gently suggest you
force yourself to answer "why" to a depth of 3 for every such procedure you've taught.
her proceduralism might look like understanding to you but that's because you have vast quantities more knowledge and context to go with those procedures than your student does.
"no one taught her how to multiply a decimal and another decimal by things when they are in a long division problem."
She has been shown this numerous times. For whatever reason, it's going in one ear and out the other. I go over it with her and then a week or two later she'll be doing a mixed review problem set and get it wrong. Fraction problems she can solve easy-peasy. It's just these decimals divided by decimals that are tripping her up.
>>She has been shown this numerous times. For whatever reason, it's going in one ear and out the other.
This is the phrase I hear from some of my child's teachers. It is my tip off that the kid missed a pre-req or the concept for the lesson. Kid cannot process, that's why it's going straight through.
Doing it fine today, then not being able to a week or two later indicates that a procedure was memorized rather than understanding developed. There are some older posts about the repeatability of procedural memory on this site.
If kid can't divide decimal by decimal, pictorial representation of decimal as a fraction is not there and fundamental understanding of division is not there.
I had a similar experience this week in one of my son's classes. He has a teacher who uses procedural too much....teacher couldn't even get fundamental concept of what a radian is across. 30 minutes of frustration tying to use calculator to convert degrees to radian and back showed he hadn't nailed the procedure in class. 2 minutes reading the paragraph in the textbook explaining and showing concept made the 'aha' moment happen and the homework become a piece of cake.
Post a Comment