Hold the marbles: Abstract approach best for math

[flashl]

CHICAGO (Reuters) - Frustrated math students may have a good excuse -- some of the teaching methods meant to make math more relevant may in fact be making it harder to understand, U.S. researchers said on Thursday.

They said students who were taught abstract math concepts fared better in experiments than those taught with real-world examples, such as story problems.

Adding extraneous details makes it hard for students to extract the basic mathematical concepts and apply them to new problems, they said.

"We're really making it difficult for students because we are distracting them from the underlying math," said Jennifer Kaminski, a research scientist at Ohio State University, whose study appears in the journal Science.

Reuters

[liberal N proud]

1. But many teachers demand student all solve problems with same methods

My daughter learned a method in one school when she was in 3rd grade. When she got to 4th grade the teacher failed her on every paper and exam because she didn't use the teachers method for solving problems. The answers were correct, the work was shown, but the teacher failed her.

She has been screwed up on math ever since.

[DrDan]

3. well . . . what else would you expect

classrooms are flooded with non-math people They know one way to solve problems - and have a difficult time recognizing valid math concepts.

Until we own-up to the problem, and begin to pay certain subject-matter teachers according to their relative worth, I doubt it will go away. I would certainly put math/science/reading teachers at the top of the pay scale.

Having state legslatures getting involved with education is a further problem. Their concern is whether religious dogma is offered as part of the science curriculum instead of recognizing and dealing with our dwindling math/science resources.

In Florida, cuts continue. There is nothing left to cut - except more teachers. We are in a downward spiral overall in education. The repigs will stop at nothing to get their voucher programs - even if it means ruining public education along the way.

Let's face it - the repugs are taking this country down the tubes.

[ayeshahaqqiqa]

5. the failings of teachers are often to blame for kids not liking math

When I taught school, I tried to show that there were many ways to come to an answer. One of my favorite things to do was to show the patterns that are found in the multiplication table by making a colorful 100s chart--the numbers 1 to 100 in rows of ten--with the multiples of 2 red, 3 green, 5 brown, etc.

I also had my third graders create checking accounts for points that they were awarded every week for their work, and with which they could "buy" products like stickers, erasers, etc. They had to balance their checkbooks every week to determine if they had enough points to get what they wanted. Yes, it was a lot of extra work on my part, but it showed the kids WHY they needed to learn to add and subtract--and I showed them the different ways they could make sure their math was correct.

For a teacher to insist on only one way shows a lack of imagination and also a deficiency in understanding that different people learn in different ways.

[madeline_con]

12. There are those who aren't capable.

A lot of ExEd kids will never learn basic math, no matter how hard we try. :(

[Android3.14]

2. Old news, but still nice

Mathematicians and cognitive scientists have been demonstrating this for decades. All New Math has done is add impetus to the movement of employers offshore.

I taught mathematics to students earning their elementary school education certification (future teachers) and I can say, based on my evaluation testing at the beginning of each semester, that at least 20 percent of our college students cannot multiply single-digit numbers with consistency, that half cannot answer simple questions about mileage, distance, and tank capacity, and that most of them are earning certifications anyway.

Less than five percent can manipulate fractions.

If we want a progressive population, we must teach subjects such as science, math, and reading with "conservative" methodologies. We've been teaching our children for hundreds of thousands of years; it's unlikely we finally got it right in just the last 50 years.

[ayeshahaqqiqa]

6. Don't you think that it actually should be a requirement

that a student solve a math problem in at least two different ways? That means that no one can rely on one method, which very well may have been learned strictly by rote. It would show a true understanding of the concept behind the problem.

I was taught new math by college professors when I was in high school (my school was one of the first testing out new math), and what I found extremely aggrivating was the fact that these professors could never really explain the concepts behind the formulas and theories and why it was important for a person to know them. They apparently simply loved doing the math, sort of like I love doing cryptograms. A nice way to pass the time, but what use is it?

When I taught school, I made my kids understand the need to know fractions by having us cook. Practical, kept their attention, and gave them a challenge when I took away the cup measure and they had to figure out how to get 1 1/2 cups flour. Once this was done in a "hands on" way, it was a fun challenge for the kids to work out such problems on paper.

Math has been so "mis taught" in this country it is ridiculous.

[Android3.14]

7. Nope.

I also use cooking to teach my kids, and when I taught math in school I would find concrete examples as well. But first you teach them the theory and the process.

Cognitive psychology and large population studies show that a student should first learn and use the accepted algorithm before employing another method, that learning theory first followed by application works far better, and that drill-and-practice creates "automaticity" in calculations so that a student can use higher brain function to evaluate a solution rather than struggling to remember how to perform a basic arithmetic operation.

Think of how we learn music. In order for a student learn a scale, for example, the teacher shows them the scale, explains what it is, has the child demonstrate he or she can strike the keys for the scale, and then have them play a tune that features that scale. If we applied current math idiocy to teaching music, a person would hear a song, try to copy it until they did, and then the teacher might explain that this song is in the key of C, and ask the child to try to find the other notes in the key, and so on and so on.

Yet by learning traditional scales, and theory, a music student develops an 'ear' that allows him or her to creatively break the rules and develop their own music.

In mathematics and music, the basics and the advanced stuff take explaining, modeling, and drill-and-practice before a student can effectively demonstrate mastery of a skill.

[ayeshahaqqiqa]

8. You can tell I was taught new math, can't you?

Because that is why I have this notion.

As for music--an interesting example. I am a musician, and was shown all the things you described. And still, for the life of me, I can't read music. I can sit down at any instrument and play any song I've heard, usually after one listen--but reading music is beyond me. I thought I was reading it until I realized one day long before I could have memorized the piece that I was listening and not looking at the music. I've done tests that show I have what is called a "musical" brain and I am an auditory learner, which makes sense since I have very poor vision.

[Android3.14]

9. Exactly

I reference music because the tasks (math and music) share many of the same parts of the brain.

When someone learns a new piece of music, it is like learning how to solve a new type of math problem (especially when a person is first learning) he or she first tries to imitate what they hear (in math that would be memorizing the algorithm)and then, once they have the tune by heart (automaticity), then they can explore all the different ways to interpret the song, apply the melodies to other schema, and all that stuff.

[ayeshahaqqiqa]

10. Thanks for making this clear to me

I asked my husband about math on the drive to work, and what he said finally made things click. I'm still disgusted that I was basically short-changed when it came to math because of the way it was taught.

[ayeshahaqqiqa]

4. I was taught the "new math" back in the '60s

by the man who invented it (who also admitted it was a failure just before his death in the early 1970s). Max Beeberman and his associates came to my high school, which was a university lab school, to teach us all these wonderful concepts about math. And I never understood geometry or trigonometry, because, frankly, I couldn't see the point. How does one apply all these wonderful theories to the real world? The professors, who all loved math in and of itself, loftily told me that practicality didn't matter, and could not give me one concrete example of how to actually use the math they were supposedly teaching us. It took a carpenter, years later, to explain why it was a good idea to know and understand about angles, but by that time I'd forgotten all that I had supposedly "learned".

All I can say is that those mathematicians turned me off of mathematics entirely. You can teach the concepts any way you like, but for this person at least you have to tie it to something practical for it to make it worthwhile to remember.

[LWolf]

11. You can't use college students for a study

and then conclude that use of abstracts works better than concretes for elementary instruction.

<snip>

To find out the best methods of teaching basic math concepts, the researchers conducted several experiments using college students in which some students were taught concepts using basic symbols, while others were taught with concrete examples.

Children begin to be able to understand abstract concepts as they enter their teen years, according to Piaget.

http://chiron.valdosta.edu/whuitt/col/cogsys/piaget.html

The Handbook of Developmental Psychology agrees, noting that the ability to work with abstracts begins to develop between the ages of 9 and 15, and reaches optimum levels at around 19 or 20.

http://books.google.com/books?id=4d7FOANFv00C&pg=PA498&lpg=PA498&dq=at+what+age+can+a+child+understand+abstract+concepts+&source=web&ots=LdRxgLFeNF&sig=wxAso8Id2kjEEPQskQdxrjfak7g&hl=en

I don't teach math any more, but I loved it when I did. I found most traditional texts and curricula to focus on rote memorization of facts and procedure, leaving the understanding of concepts out of the picture and simply teaching children to choose an operation to solve a problem based on the appearance of certain key terms.

I also found that many math programs that use manipulatives and concrete examples do so without any real focus on the underlying concepts, which makes them just as ineffective.

In my experience, teaching the concepts as the first priority, and then teaching facts and procedures, produces young mathematicians that can solve problems correctly in a variety of ways without trouble.