# Solving America’s Math Problem?

My recent posting on homeschooling led to an interesting exchange on teaching math . . . so I thought I’d post a recent article about math education, from the Stanford University publication *Education Next*.

I’ve blogged about Duke professor Jacob Vigdor’s research before. His basic thesis is that efforts to expand enrollment in higher math – surely an admirable goal – coupled with a push to teach algebra earlier to all students, have led schools to dumb down math curriculum and have undermined the progress of America’s most gifted math students.

Concern about our students’ math achievement is nothing new, and debates about the mathematical training of our nation’s youth date back a century or more. In the early 20th century, American high-school students were starkly divided, with rigorous math courses restricted to a college-bound elite. At midcentury, the “new math” movement sought, unsuccessfully, to bring rigor to the masses, and subsequent egalitarian impulses led to new reforms that promised to improve the skills of lower-performing students. While reformers assumed that higher-performing students would not be harmed in the process, evidence suggests that the dramatic watering down of curricular standards since that time has made our top performers worse-off. Even promised improvements in the lower part of the distribution have at times proved elusive, a point illustrated below by the disappointing results of a recent initiative to accelerate algebra instruction in the Charlotte-Mecklenburg school district.

America’s lagging mathematics performance reflects a basic failure to understand the benefits of adapting the curriculum to meet the varying instructional needs of students. Recently published results from policies such as Chicago’s “double dose” of algebra, which groups students homogeneously and increases instructional time for lower-skilled math students (see “A Double Dose of Algebra,”

research, Winter 2013), support differentiation as the best way to promote higher achievement among all students.

http://educationnext.org/solving-america%E2%80%99s-math-problem/

What follows is a data-rich analysis of American students’ math performance. (I’ve inserted a couple of graphs from the article here.) Professor Vigdor’s bottom line is that pushing early algebra hurts struggling students while subjecting mathematically-talented students to courses that fail to challenge them or prepare them for more advanced work. Basically, this is the old argument for math tracking, combined for a plug for longer and more intensive math classes for students at the lower end of the curve. But since I’m a paleo-teacher, old doesn’t necessarily mean bad.

What do you think?

I think the math thing is real simple in its solution but difficult to implement and overcome.

Most parents seem to be against “too much homework” but for most people learning math takes hard work, learning math means practice, practice, practice–things many students and even parents aren’t willing to do.

Many secondary schools have block schedules with 90 minute periods. This is bad for students in most classes. It is especially in math. Instructional time is lost in this setting. Teachers would have to present their material for double the length of time. Most teachers don’t have the energy to do this nor should they try as students can’t sit and listen to math instruction for 80-85 minutes. We need to shorten classes or double block for math and allow lab time.

Finally, math teachers need to change how they teach. Students learn in different modalities but math teachers seem to be the most resistant, especially at the secondary/college level to recognized this. They need to show the practicality of math. I taught my own children a lot of math through having them play yahtzee, chess, monopoly, card games, having them build things, cook recipes, track the weather etc. Have them find ways to use their math, teach them about interest and the national debt for example, show how actuaries use statistics to set insurance rates. I know teachers are creative and can find ways. But just lecturing and then practice and lecture and practice and lecture and practice isn’t working. Students learn in different modalities to lecture, hands-on, to visuals etc. etc. But I hate to see it, math teachers generally use few teaching modalities and certainly modalities which are not as effective with today’s children as they might be in the past.

Techniques to flip the classroom have merit in teaching math if not overused and technology for students is available. Like them or not, most of those Kahn lessons are well done.

Finally, I think some of the teachers want to find ways to teach math more effectively but are hamstrung by policies and the core curriculum.

Hey, we have LOTS of points of agreement here. I don’t think block schedules work well for math (or foreign language.) It’s a shame, because they work very well for my own subjects: history, government, economics. I also agree that we should be teaching more math application. It always astonished me how many of my economics students – even students enrolled in pre-calculus or calculus – struggled with basic graph interpretation and arithmetic problems involving percentage change.

Our schools have tried for so long to be “all things for all people” that as a result are “nothings for nobody” (for want of a better thing). As an educator I am firm believer in tracking students based on ability and performance. We are so afraid of hurting a student’s self esteem that we are afraid to acknowledge that some students have gifts in areas that not every student has a gift in, especially in the elementary grades. Why? Because if Little Timmy gets acknowledgement for accomplishing something then we should also acknowledge Little Billy for something as well, even if Little Billy really has not accomplished anything at all. Then when our students get out into the world and they figure out that not everyone is “special” (and to tell you the truth most have figured it out long before that) it comes as a slap in the face.

With that said, helping every student begin to find their niche (not saying place, but niche) in the world should be a primary goal of education. Not everyone should go to a four year college. But every student should have the opportunity to pursue their dreams, while at the same time being taught that nothing is given to you for free. All great accomplishments require time and effort, which at the high school level we begin to see much more of a separation with those students who want to achieve (in the arts, in career and tech education, in math and science and the liberal arts), and those who want things given to them without effort. That changing of the mindset in our society is not going to be easy but I believe must take place.

All students are in need of basic math instruction. We all need to know how to balance a checkbooks and other financial skills that require some basic math skills, and some understanding of concepts like compounded interest. But when it comes to higher math, I think the need, desire, perhaps even innate ability, simply isn’t there for many.

As an analogy: Every student benefits from some basic physical education. Every student benefits from some basic exposure to music and/or art. Every student benefits from learning spelling, grammar, and basic composition. Every student benefits from some basic home maintenance skills that can be taught in an introductory “shop” class.

But we don’t expect every student to be on the Varsity football or basketball teams. We don’t try to persuade every student to be part of the performance band or choir or to try out of the lead in the school play. Not every student is going to become an author. And many students would not be well served by taking advanced industrial arts classes, at least not if doing so means time diverted from other areas of instruction.

Holding back the intellectually best and brightest in a vain attempt to get those with different skill sets to try to perform on the same level as the former makes no more sense hobbling the varsity football team with a bunch of players who have trouble catching a ball.

That all said, it would be useful to see the effects if we could get math specialists to teach math in early grade school. Ditto if we could drop a few of the new fangled and mostly failing attempts to “teach” math without requiring repetition, or drills, or actual work, and instead would go back to tried and true methods that worked for a couple of hundred years at least. Repetition and work are both fundamental doctrines of learning. That is true whether we are learning physical skills like shooting baskets, whether we are seeking spiritual knowledge, whether we developing artistic talents like playing piano, or whether we are learning math or science. But ultimately, no grade school teacher should ever stand up and say about fractions (or any other basic math concept), “This is really hard….” It is hard to teach what you don’t know and far too many grade school teachers are just not comfortable with the math concepts they are tasked with teaching. Too many don’t enjoy math.

The changes we need to make need to happen well before grades 5-7.

I do worry that elementary education attracts too many math phobes. Ditto for social studies education. One vital undertaught skill is graph interpretation (critical for success in some of the AP courses I’ve taught.) I wonder if some teachers shy away from graphs because they hated them back in math class.

Agreed, Mary. Just as calculators should not be permitted until a concept is well learned, graphing calculators should not be used until after students demonstrate a proficiency in manual graph interpretation.

Of course, MatLab, Mathematica, and other such graphing programs can be a great boon to teaching wherein a teacher doesn’t need great art skills in order to show students a perfect graph that is easy to decipher. But these teaching aids should not become a substitute for students learning to graph by hand and to interpret graphs manually.

The sad fact is, those with math skills generally have a lot of career options instead of, or at least in addition to, teaching grade school. If nothing else, someone good at math can more easily teach secondary level classes rather than the younger grades. Not to mention much higher paying (though different lifestyle) jobs in engineering, accounting, etc.

I don’t know to fix that exactly. I think differentiated pay for different specialties is needed (pay a premium for a few math specialists in each grade school) might be part of the solution. Part of it might be find some way to encourage those who have worked a career in another field to take their math expertise into the classroom (perhaps as part time math specialists). Maybe secondary grade math teachers could be persuaded to spend some time each day/week providing instruction to the lower grades.

But I suspect it is pretty tough to fix in 7th to 12th grades what wasn’t taught about math, or phobias developed about math, in grades K through 6.

I’m not fond of Vigdor’s article.

He argues that the achievement gap and generally dwindling math performance of US students has been addressed by making the math curriculum “more accessible” (i.e., it has been dumbed down). He then argues that it need not be dumbed down if the curriculum were differentiated between low and high performing students.

In fact, this is pretty much how it was in the 50′s and 60′s. Students did not need the 3 or 4 years of math in high school to get admitted into colleges. What he leaves out, however, is the quality of math education in the lower grades and how this has affected the number of students who might otherwise be high performing students.

There’s no disagreement that some kids are smarter than others. Most people know that you can’t just set a standard (like algebra in 8th grade) and do nothing else. But Vigdor overlooks overlooks that issue and then claims that the failed initiative defines some IQ/algebra correlation. There are many other variables to consider–which he doesn’t.

The “Math Wars” are about curriculum and teaching methods, but this article skips over that analysis. Most schools separate kids starting in 7th grade. In affluent areas, since “enough” students get onto the top math track in high school, (often due to tutors, learning centers, or help from parents), educators will not look for any fundamental issues in K-6. They only assume that it’s a relative problem.

Why not interview parents to see what is done (or not) at home and try to find out how the best students got there? There may see an IQ connection, but it’s not that simple. There are things one can do to separate the variables. But too many authors of the recent spate of articles about math, algebra and its need, either can’t or won’t.

In his report, he pooh poohs the idea of introducing Singapore Math into classrooms, citing the usual cultural differences argument which is specious. (Teachers in Singapore have better math background; students go to school all year round, so there’s no forgetting concepts during the summer; the culture promotes education and hard work, etc). He neglects the fact that Singapore’s texts present the material clearly and succinctly and that there have been successes in schools in the US that have used it).

Thanks for the thoughtful comment. I like Singapore Math, too, and used it when I was homeschooling. And I certainly agree that we need teachers with better math preparation.

Should math tracking begin earlier? Maybe. I share what I assume, from your comment, is your worry that affluent kids will be tracked into more demanding math. On the other hand, giving struggling students extended hours of math early on – which Vigdor recommends – makes a lot of sense to me.

I don’t disagree that extended hours of math for struggling students makes sense. I question what math instruction students are getting in K-6 that are producing such quantities of students deficient in math skills, compared with those in the 60′s and 70′s. Vigdor does not address this as I indicated in my initial comment.

And yes, flexible ability grouping does not automatically result in “tracking”. “Tracking” is a no-no word in edu-circles, because it once referred to placing students in trade/vocational curricula vs college prep, without student or parent input. Now we have full inclusion classrooms, in which students with huge differences of abilities must co-exist. Ed schools and the ed-establishment proclaim that this is managed through differentiated instruction. This really amounts to differentiated assignments, and also to “just in time” learning, which I won’t go into here, but if you want an inkling, see http://www.educationnews.org/commentaries/book_reviews/99791.html

IQ is a nonsensical concept. Aptitude for mathematics is a recognizable attribute just as that for art or music.

As a secondary math tutor I work with students with very varying preparation and interest in mathematics. I am still reflecting on Professor Vigdor’s research. I do think that the math curriculum has been watered down for many reasons over the years. However, I disagree with him that Professor Chrystal’s math text is a fair comparison to modern Algebra 1 high school texts. In the free PDF of this work the full title is: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges. Professor Chrystal writes in the Preface that the text is NOT intended for use by absolute beginners. He suggests that students have already been exposed to algebra by gradual generalization of arithmetic (A powerful approach abandoned once Sputnik flew over the horizon-my tutees love it.) through simple equations and possibly quadratic equations. He also expects students to be familiar with construction of literal formulae, such, for example, as that for the sum of money during a given term at simple interest.

He notes that the problem sets for the book were collected from final exam questions at Cambridge University. Good heavens!!

I agree that introducing algebraic concepts into arithmetic (generalizing arithmetic) makes great sense. I did this when I homeschooled, and felt that it helped prepare my kids to handle higher math.

I agree with a lot of what Chuckie is saying. I think we don’t need everyone to have enough math to be a CPA or a mechanical engineer, probably getting most students through basic algebra and/or consumer math is good, much beyond that is debatable if ability and/or interest isn’t there.

I think block is bad for most subjects. There are good teachers like Mary that can make it work in Social Studies, English etc. but most teachers aren’t that skilled and instructional time is lost as many teachers aren’t effective with transitions or maybe changing directions and modalities.

Plus, I think even in AP classes some teachers would love to meet with their students every single day. With block days if a student misses a day it is like missing two days. If multiple absences occur this effects learning. With extended weekends sometimes a teacher won’t see students for several days. This makes learning and teaching skills difficult. Students often procrastinate with the block schedule. Block schedules have their merit but I would use block type scheduling within a traditional schedule, again thinking out of the box. I am not even against modified block where schools have block scheduling for two days and traditional schedule for three days. This way in a regular week of school teachers are meeting with their students four times a week vs. sometimes two which can occur on a block schedule. There you can have the best of both worlds for most subjects and allow some teachers to do extended learning activities while in some subjects like math can benefit from shorting times but more meeting times.

Thanks for the “good teacher” comment. What I liked about the block was that I could use the first ten minutes of class for a quick assessment on the assigned readings (students could use their study guides, if they filled one out – basically this was an effort to get students to prepare a little before class), and still have time for other activities. My government classes often began with a news clip, for example. These worthwhile activities nevertheless eat up time.

I like the modified schedule as a compromise, though it can get complicated.

Absences are tough. I became an attendance witch over time. Students could miss up to two of my opening quizzes per quarter – excused OR unexcused absence. If they knew they were going to be absent they could take the quiz in advance. If they missed more than two, for whatever reason, they could only make up the grade by writing an essay during lunch or after school.

Of course I made exceptions in the event of a serious illness or accident. But I found that some of my students accumulated lots of excused absences, presumably with parent (or sometimes coach) cooperation. What I would tell them was that in the real world, 8 “free” absences (16, really, since we had a block schedule) would be enough to get them fired from most jobs . . . however many times their parents called in!

Countries that track students the latest and the least have students with higher achievement in mathematics.

That’s interesting. I’d love to post a source.

Growing up in Moscow, Russia I learned a parable about four non-musical beasts – bear, donkey, monkey, and goat – deciding to perform a musical number. Each of their several attempts was a complete disaster. They tried to fix it by rearranging their seating order several times with no success. Finally a finch came and told them that reordering the seats will not make musicians out of them – a more drastic change was required.

I believe when it comes to teaching math and other technical disciplines we often fail to recognize the root of the problem. It is not so much how we teach as it is in who teaches and whom he teaches. One problem is that our culture of perpetual low-aptitude entertainment always rushing around after things of superficial nature creates bears, goats, monkeys, and donkeys out of our children with respect to analytical ability. The bureaucratic government-dominated non-competitive process-rather-than-result-oriented educational system is more likely to bring a bear, a goat, a monkey, or a donkey for a teacher than a finch.

I do not have a practical solution for the problem on a government level as what it takes to solve it will go against the very nature of our culture and it is not realistic to legislate such a change. However, for those parents that want to produce a math finch, I do have some advice. Become a math finch yourself or as close as you can get and home school or at least spend enough time with your kid exploring the subject. If that is not an option, find a kid in your neighborhood that is a math finch and just have him hang out at your house a lot with the understanding that you expect him to share what he knows. The best way to develop a musical ear is to be exposed to a lot of good music. The principle holds for math and other technical fields.

I was one of the kids in school with flat feet, allergies, and a deep rooted dislike of PE classes. If the school sports program had been run with the same logic as some of the math programs, the government would be telling schools that kids like me needed to be on the football team. We would have spent our practices lifting five pound weights and doing mild calesthenics. Playbooks would have been simplified to things kids who hated sports could remember. The good atheletes would either come down to our level or find something else to do.

And, when we got creamed playing teams that taught and trained by the old rules, the government would come in and tell us it was because we weren’t applying the new principles correctly.