(I copy-pasted this comment from The Atlantic article on New Math. How mathematics is taught in school is a very important issue and I think one ignored by most people in the general public. FYI, our school district uses a horrible "new math" program but as it is taught with emphasis on knowing the algorithm as well as the memorization of addition/multiplication tables and the like, our district test scores are deceptively high. My take-away from the article and comments is that it is best to master the lower-level maths completely before moving on to fulfill some graduation requirement list.)

I've been a classroom teacher for 15 years and spent the 7 years ahead of that working in a Math Tutorial Lab in a state university, helping undergraduates as they struggled with math. I saw big time calculator dependence and continue to see that phenomenon in our high school students. I recently invited four faculty from our university to speak to our secondary (junior high and high school) math teachers about preparation for college math and STEM fields. They unanimously said, "Don't let your students become calculator dependent." "Don't let them have calculators all the time." They also said, "We don't care if they take calculus or not in high school, but please teach them algebra!"

Indeed. And trying to teach algebra to students who are not prepared for it is very difficult. In the recent decade (when students who learned from NSF-funded math programs reached secondary schools), we have noticed a drop in elementary math skills, including basic operations. We see students who cannot multiply without a lattice (they frequently can't multiply with it, placing decimals incorrectly or adding incorrectly) and we see students who cannot do long division. We see students who cannot multiply at all, and do repeated addition such as 1000+1000+1000+1000+1000+1000 instead of 6 x 1000.

They don't know when to multiply, either. When it comes to multiplying fractions, we see things like 2.666666... x 33 = ? because they can't manage the fractions, continuing their love affair with decimals (and calculators). They don't understand the concept of remainder and how to express it as a fraction, as one would in changing an improper fraction to a mixed number. They have to be taught long division at the algebra 2 level because it is needed for polynomial division. The structure of the standard algorithms (as those of us who learned math before the current reform era got here) is necessary for students to work with polynomials and rational functions.

Trying to teach students the difference between vertical asymptotes and holes in rational functions is extremely difficult when they are not in the habit of writing fractions in lowest terms or knowing how to convert improper fractions into mixed numbers. We cripple them for learning algebra (and hence, bar the door that leads to STEM careers) when we don't prepare them with the standard algorithms. I have seen it time after time -- students whose parents can afford tutors or Sylvan come prepared to learn algebra much more often than those who are dependent on whatever gets taught in school. I've taught honors classes and very low-level, slow-paced classes over the years. Those students in slow-paced algebra 1 are there mainly because they have absolutely nonexistet computational skills.

It's pretty tough to add -6x + 10x in your head if you don't know how to add -6 + 10. We see transfer students from states where nothing lower level than algebra 1 is taught in high schools. They will have passed algebra 1 and geometry from their previous schools, and they can barely pass prealgebra in ours.

It's an equity issue and we will not narrow the achievement gap until we have high expectations for ALL students. Watering down the curriculum will not do it. Pretending that everyone in a high school is taking algebra 1 or higher level math classes is such a disservice to those students who have not mastered prealgebra material. We can't pretend we are preparing students for STEM fields or college or even careers that do not require some vocational post-secondary education by not giving them a strong basic math education at the K-8 levels.

I've been a classroom teacher for 15 years and spent the 7 years ahead of that working in a Math Tutorial Lab in a state university, helping undergraduates as they struggled with math. I saw big time calculator dependence and continue to see that phenomenon in our high school students. I recently invited four faculty from our university to speak to our secondary (junior high and high school) math teachers about preparation for college math and STEM fields. They unanimously said, "Don't let your students become calculator dependent." "Don't let them have calculators all the time." They also said, "We don't care if they take calculus or not in high school, but please teach them algebra!"

Indeed. And trying to teach algebra to students who are not prepared for it is very difficult. In the recent decade (when students who learned from NSF-funded math programs reached secondary schools), we have noticed a drop in elementary math skills, including basic operations. We see students who cannot multiply without a lattice (they frequently can't multiply with it, placing decimals incorrectly or adding incorrectly) and we see students who cannot do long division. We see students who cannot multiply at all, and do repeated addition such as 1000+1000+1000+1000+1000+1000 instead of 6 x 1000.

They don't know when to multiply, either. When it comes to multiplying fractions, we see things like 2.666666... x 33 = ? because they can't manage the fractions, continuing their love affair with decimals (and calculators). They don't understand the concept of remainder and how to express it as a fraction, as one would in changing an improper fraction to a mixed number. They have to be taught long division at the algebra 2 level because it is needed for polynomial division. The structure of the standard algorithms (as those of us who learned math before the current reform era got here) is necessary for students to work with polynomials and rational functions.

Trying to teach students the difference between vertical asymptotes and holes in rational functions is extremely difficult when they are not in the habit of writing fractions in lowest terms or knowing how to convert improper fractions into mixed numbers. We cripple them for learning algebra (and hence, bar the door that leads to STEM careers) when we don't prepare them with the standard algorithms. I have seen it time after time -- students whose parents can afford tutors or Sylvan come prepared to learn algebra much more often than those who are dependent on whatever gets taught in school. I've taught honors classes and very low-level, slow-paced classes over the years. Those students in slow-paced algebra 1 are there mainly because they have absolutely nonexistet computational skills.

It's pretty tough to add -6x + 10x in your head if you don't know how to add -6 + 10. We see transfer students from states where nothing lower level than algebra 1 is taught in high schools. They will have passed algebra 1 and geometry from their previous schools, and they can barely pass prealgebra in ours.

It's an equity issue and we will not narrow the achievement gap until we have high expectations for ALL students. Watering down the curriculum will not do it. Pretending that everyone in a high school is taking algebra 1 or higher level math classes is such a disservice to those students who have not mastered prealgebra material. We can't pretend we are preparing students for STEM fields or college or even careers that do not require some vocational post-secondary education by not giving them a strong basic math education at the K-8 levels.

Ben qualifies as disabled in Math as he is at the 2nd-3rd grade level in Math facts. I've met many a kid who gets the higher order stuff, like trig, geometry, calculus, and still doesn't know how to add,subtract, multiply, or divide. He's like in the 90th percentile in math for the higher order stuff, the 10th percentile for +-*/. Figure that one out.

ReplyDeleteIt is "my opinion" that some kids aren't wired for math, just like some kids aren't wired for reading. It doesn't mean they aren't highly intelligent. Ask Ben's uncle, the lawyer, who hires secretaries and accountants to do his math for him.

Just saying.

You're absolutely right about the intelligences being not all as we think they are. But there is something to be said for giving average students the ability to add 5 + 7 without getting the calculator out. Handicaps are different than handicappING the student by lack of teaching, yk?

DeleteEmperor has a verbal IQ of 135 but still has difficulty deciphering sentences, so I get what you're saying. :)

Whenever I've encountered a kid with a problem in high school math (and have had an influence over them), I've taken them back and tutored them in basic math... specifically multiplication, division, fractions, decimals, and exponents. They are not going to get Algebra right without it. Granted, they may be able to grasp the concepts and use a calculator, but they will never-ever feel confident and so therefore they will HATE math.

ReplyDeleteIf it were up to me, each kid would use software that let him or her progress through math at their own pace. Some may end with Pre Algebra and maybe business and personal finance math. Others would end with college calculus. I'd rather them be know what they know than to flounder at subjects beyond their capability.

The thing is, too, how much "floundering" is related to a genuine problem like dyslexia and whathaveyou and how much is related to extreme dread of the subject and/or poor pedagogy over the years? And how often do teachers get the chance to test students on specific math processes and help them fill in those gaps? It's always on to the next thing with public school, unfortunately. It's almost all test prep now.

DeleteI would like to be snarky and just say everyone should homeschool but the reality is that not everyone can and these schools are training the majority of our future doctors and engineers. We should be concerned.

There's words in your post that I've never even heard of!

ReplyDeletePolynomial division? (Does that mean long division?)

Vertical asymptotes? Totally beyond my comprehension. (I finished school in 1967).

We didn't learn algebra until high school, but our primary years (elementary) were filled with addition, subtraction, multiplication and division, including long division which I was quite good at.

-6x +10x =4x.

Right?

I'm not always good with my "terms" either but I know what they mean when they are on the page. The vocabulary is more daunting than the mathematics sometimes. I believe -6x + 10x = 4x is a polynomial expression of some sort. And I also think you got the answer right. Math never really changes. :)

DeleteSadly many kids don't have the parents at home to help back up practicing the basic math facts as homework and perfecting them at an early age. I think a lot of kids hate math because of feeling insecure over the basics. :/

ReplyDeleteAnd I think schools have done a great job drilling into parents' heads that they are always the experts. And so parents are now not involved with homework or schoolwork of any kind. It is the school's job now.

DeleteLike it or not, this is the atmosphere that schools are reaping. Now they are looking for ways to "involve parents" and it will take a long time to turn that ship around.