Yes, algebra is necessary.

7/30/2012

When I first saw yesterday's New York Times op-ed, I mistook it for a joke. The title, "Is algebra necessary?" had the ring of Thurber's classic essay "Is sex necessary?" a send-up of psychological sex manuals of the 1920s.

Unfortunately, the author, Andrew Hacker, poses the question in earnest, and draws the conclusion that algebra should not be required of all students.

His arguments:

A lot of students find math really hard, and that prompts them to give up on school altogether. Think of what these otherwise terrific students might have achieved if math hadn't chased them away from school.
The math that's taught in school doesn't relate well to the mathematical reasoning people need outside of school.

His proposed solution is the teaching of quantitative skills that students can use, rather than a bunch of abstract formulas, and a better understanding of "where numbers actually come from and what they actually convey," e.g., how the consumer price index is calculated.

For most careers, Hacker believes that specialized training in the math necessary for that particular job will do the trick.

What's wrong with this vision?

The inability to cope with math is not the main reason that students drop out of high school. Yes, a low grade in math predicts dropping out, but no more so than a low grade in English. Furthermore, behavioral factors like motivation, self-regulation, social control (Casillas, Robbins, Allen & Kuo, 2012), as well as a feeling of connectedness and engagement at school (Archambault et al, 2009) are as important as GPA to dropout. So it's misleading to depict math as the chief villain in America's high dropout rate.

What of the other argument, that formal math mostly doesn't apply outside of the classroom anyway?

The difficulty students have in applying math to everyday problems they encounter is not particular to math. Transfer is hard. New learning tends to cling to the examples used to explain the concept. That's as true of literary forms, scientific method, and techniques of historical analysis as it is of mathematical formulas.

The problem is that if you try to meet this challenge by teaching the specific skills that people need, you had better be confident that you're going to cover all those skills. Because if you teach students the significance of the Consumer Price Index they are not going to know how to teach themselves the significance of projected inflation rates on their investment in CDs. Their practical knowledge will be specific to what you teach them, and won't transfer.

The best bet for knowledge that can apply to new situations is an abstract understanding--seeing that apparently different problems have a similar underlying structure. And the best bet for students to gain this abstract understanding is to teach it explicitly. (For a discussion of this point as it applies to math education in particular, see Anderson, Reder, & Simon, 1996).

But the explicit teaching of abstractions is not enough. You also need practice in putting the abstractions into concrete situations.

Hacker overlooks the need for practice, even for the everyday math he wants students to know. One of the important side benefits of higher math is that it makes you proficient at the other math that you had learned earlier, because those topics are embedded in the new stuff. (e.g., Bahrick & Hall, 1991).

So I think there are excellent reasons to doubt that Hacker's solution to the transfer problem will work out as he expects.

What of the contention that math doesn't do most people much good anyway?

Economic data directly contradict that suggestion. Economists have shown that cognitive skills--especially math and science--are robust predictors of individual income, of a country's economic growth, and of the distribution of income within a country (e.g. Hanushek & Kimko, 2000; Hanushek & Woessmann, 2008).

Why would cognitive skills (as measured by international benchmark tests) be a predictor of economic growth? Economic productivity does not spring solely from the creativity of engineers and inventors. The well-educated worker is more likely to (1) see the potential for applying an innovation in a new context; (2) understand the explanation for applying an innovation that someone else has spotted.

In other words, Hacker overlooks the possibility that the mathematics learned in school, even if seldom applied directly, makes students better able to learn new quantitative skills. The on-the-job training in mathematics that Hacker envisions will go a whole lot better with an employee who gained a solid footing in math in school.

Finally, there is the question of income distribution; countries with a better educated populace show smaller income disparity, and suggesting that not everyone needs to math raises the question of who will learn it. Who will learn higher math in Hacker's ideal world? He's not clear on this point. He says he's against tracking, but notes that MIT and Cal Tech clearly need their students to be proficient in math. Does this mean that everyone gets the same vocational-type math education, and some of those going on to college will get access to higher math?

If that were actually implemented, how long before private vendors offer after school courses in formal mathematics, to give kids an edge for entrance to MIT? Private courses that cost, and to which the poor will not have access.

There are not many people who are satisfied with the mathematical competence of the average US student. We need to do better. Promising ideas include devoting more time to mathematics in early grades, more exposure to premathematical concepts in preschool, and perhaps specialized math instructors beginning in earlier grades.

Hacker's suggestions sound like surrender.

Anderson, J. R., Reder, L. M., & Simon, H. A. (1996). Situated learning and education. Educational Researcher, 25, 5-11

Archambault, I., Janosz, M, Fallu, J.-S., & Pagani, L. S. (2009). Student engagement and its relationship with early high school dropout. Journal of Adolescence, 32, 651-670.

Bahrick, H. P. & Hall, L. K. (1991). Lifetime maintenance of high school mathematics content. Journal of Experimental Psychology: General, 120, 20-33.

Hanushek, E. A. & Kimko D. D. (2000). Schooling, labor-force quality, and the growth of nations. The American Economic Review, 90, 1184-1208.

Hanushek, E. A. & Woessmann, (2008). The role of cognitive skills in economic development. Journal of Economic Literature. 46, 607-668.

matthew link

7/30/2012 12:51:31 am

Your arguments (as ever) are compelling and comprehensive. It seems to me that getting instruction right in the elementary grades is a far better solution than tossing out higher level math in general.

As you note in closing, high SES families will inevitably position their progeny for UVa and other elite schools "by any means necessary." The fights over which children get into honors track math at our local middle school make Romney's attacks on 'Obamacare' look polite.

I suggest that the day Hacker pulls his grandchildren out of Algebra is the day his proposals will gain wider credibility among the Grey Lady's readership.

Der Alte

7/31/2012 05:47:34 am

An even better solution than "getting instruction right in the elementary grades" would be to restrict immigration to people whose children are likely to do well in school.

Take a look at this picture:

http://www.flickr.com/photos/maaorg/7362471620/

The children of immigrants are a clear majority of this select group.
None of the immigrant parents were from Mexico. I think it's very unlikely that there will be a USAMO winner in your lifetime whose parents came here from Mexico, and it's a virtual certainty that there will be no USAMO winner in your lifetime whose parents were illegal immigrants from Mexico.

Dan Willingham

7/31/2012 09:06:56 pm

The OECD analyzes PISA data by immigration status. In virtually every country, the children of native-born outperform the children of immigrants, but that difference varies by country. The US does a worse job than most industrialized countries in educating immigrant kids.

Der Alte

8/1/2012 06:01:35 am

"The US does a worse job than most industrialized countries in educating immigrant kids."

Have a look at the graphs here:

http://www.vdare.com/articles/pisa-scores-show-demography-is-destiny-in-education-too-but-washington-doesnt-want-you-to-k

Asians do very well in American schools, on average. And it's utterly amazing how dominant Asians are at the very highest levels of academic competition. For example, look at the names in the results of the 2012 MATHCOUNTS competition.

http://mathcounts.org/Page.aspx?pid=1872

Hispanics do very badly in American schools, though not as badly as blacks. And even though many, probably most, of our Mexican immigrants are from the bottom of Mexican society, it looks like Mexicans here do better on the PISA tests than Mexicans in Mexico. So, maybe the problem with Mexican performance in American schools is with the Mexicans, not with American schools.

If we want high levels of achievement in our schools, why don't we keep out immigrants whose children are unlikely to do well in our schools?

Kirsten Larson link

7/30/2012 01:19:13 am

Sadly, we've been having this discussion in California for some time now, and I don't buy it. I have students in my college courses who cannot convert percentages into decimals or solve a beak even analysis. And these are not students studying engineering or physics, but rather business management.

Tony Lima link

7/31/2012 12:44:21 pm

Kirsten, does beak even analysys only apply to birds? :)

Jose Vilson link

7/30/2012 01:27:57 am

This is exactly what I was looking for. Better you than me, because mine would have had a few more swears. Thank you!

Marshall link

7/30/2012 01:29:43 am

Thank you. You point out Hacker's asinine arguments succinctly. That was a poor piece.

I would assert that most math teachers will draw a distinction between Algebra 1 and Algebra 2. At what point have kids abstracted enough for transfer? It always seemed silly to me that all kids must know how to find the vertex of a parabola, for example.

Bottom line is though that we should be having this discussion on the merits, not the red herring arguments that Hacker presented.

Dan Willingham

7/31/2012 09:09:29 pm

when is enough enough? great question and I don't know the answer, but i'm sure it's more than "citizen math." I agree with you that the math curriculum could be reviewed.

Gigi

7/30/2012 02:50:37 am

I agree that having math skills are VERY important for life, but I have seen entirely too many students give up on school based on either/both Algebra and/or Geometry frustrations. The personality type who likes math as a subject doesn't always like to teach it to others (especially the little ones where - IMO - if it's not learned early (like reading and a foreign language), it's highly unlikely that it will be embraced as a subject of interest later. Our "one size fits all" educational system is not working. Our emphasis on the same math for all leaves a number of students out of the equation.

Gigi

7/30/2012 02:55:10 am

OK...I haven't had enough coffee to make my grammar skills apparent - please forgive.

dan willingham

7/31/2012 09:12:50 pm

I think this view puts too much emphasis on the innate abilities of the individual--if you don't have the personality type that likes math, then you're going to have a hard time. the education systems in other nations have done a much better job of making sure that most kids are competent in math, and I doubt their populations have a higher %age of the right personality types. They have devote more time to math and they teach it better.

Gigi

8/1/2012 02:49:24 am

Sorry for the confusion...yet another reason to not post something in the morning. I was referring to the personality of teachers attracted to teaching math. Those who love the subject enough to major in it in college often lean toward being ISTJ (introverted, sensing, thinking, judging). Most teachers in most other K-12 subject areas lean toward being ENFJ/P (the opposite). When I was working with Curry School graduates seeking teaching jobs, being an ISTJ was sometimes an issue for those who were in the program, especially once they got into their student teaching and realized that there is little time for them to get a break from the students which is needed for their personality type (if you believe in the MBTI). Combine the personality issue with the lack of money and respect in the field, it makes it really hard to find math teachers. I don't think that the children's personality impacts this learning, at least not in the lower levels of math (outside of the lack of good teaching making them not like the subject very much).

mommy

7/30/2012 03:03:01 am

It seems to me that a lot of the problem with math is not the more abstract concepts and applications, but rather the boring focus on timed math facts to the exclusion of any conceptual content during the formative years. I was teaching the other kids math in K and 1, but by grade 2 I was bored out of my skin and turned off to math.

dan willingham

7/31/2012 09:14:03 pm

the national math panel mostly agreed--most american kids have a terrible grasp of the conceptual side. but they also don't know their math facts as well as they should

Steph

7/30/2012 03:18:15 am

What are your thoughts on waiting until children are older (ready to learn, as the Finnish say)?
We lived in a very cold rural area for a few years when my son was in grade school, so I homeschooled, and I used the Singapore math books which did not cover long division at his level. When he returned to public school in 5th grade, he was dismayed to find the others had learned something he had not. I taught him long division in ten minutes that afternoon.

dan willingham

7/31/2012 09:17:00 pm

Steph, I don't see much evidence that we're starting too early. (evidence like crucial developmental changes that come after we typically start teaching certain concepts, so kids are butting their heads against a stone wall, trying to understand something their brains can't process yet).

callejohan

7/30/2012 04:54:21 am

This discussion reminds of the old arguments of why it is important to study Latin -- because it is so good for other reasons, even if it's a dead language. Bunkum to that. With respect to the present discussion, the argument should really be for two tracks. Some people really have a hard time with math, but are still capable of creative thought. Yes, algebra is an essential skill in today's world, but not everyone needs it. One size does not fit all! And that's a really, really original thought ..... don't you agree?

Emily

7/30/2012 10:33:47 am

I'm pro-tracking. Most high school students shouldn't be taking calculus. I'm open to the idea that a lot of kids don't need algebra II or trig. But a basic algebra class? The vast majority of students have the cognitive ability to handle algebra at a meaningful level. It's even on the GED. Now, for those who don't, I'm open to the idea of a more vocational math track, but you should have to be really pretty terrible at math before we give up on teaching you algebra, and that's not what this guy is arguing.
Also, getting a high school diploma should not be an indicator of being capable of creative thought, it should be an indicator that the person holding it has a basic level of knowledge in certain subjects. We can argue about what those should be, but "capable of creative thought" is an awfully low (and strange) bar.

callejohan

7/31/2012 09:20:44 pm

i see an important difference between latin and math. no one thought latin was useful in any context, but build mental discipline. most see algebra as the gateway course to success not only in higher math, but in all of the sciences. (Calculus is a prerequisite to major in psychology at uva)

dan willingham

7/31/2012 09:22:15 pm

haha that comments was from me, not callejohan--mistakenly types your name, sorry. It's pretty early :)

Gigi

8/2/2012 10:18:45 am

I was going to ignore the "calculus being a pre-req to major in psychology at UVA" but then I realized that it's yet another example of an unnecessary math being required for a particular pursuit.

IF all undergrads majoring in psychology were going to pursue a clinical psychology PhD program post graduation, THEN it MIGHT be a good reason to take calculus DURING the program (rather than as a prerequisite) BUT most psych majors do NOT go that route, partly b/c -- as Dan noted -- the math thwarts them in grad school for clin psych....thankfully there are many routes to working with people as a counselor (not with just getting a PHD in clinical psychology).

Thankfully the faculty in the psychology department had not come up with that when I was an undergrad at UVA. I just wonder if it's a way to keep the riff-raff out of the program..... I can tell you as someone who has been counseling others for over 16 years, I NEVER use calculus, much less algebra. Statistics yes....but I did not have to take calculus in order to take statistics.

PLEASE give students a break (and our wanna-be psychology majors....I agree with someone's comment that it's a very important field to study so it does NOT need this level of math as a gatekeeper to it).

Dave Barnes

7/30/2012 05:39:05 am

1. Lets start with the fact that Hacker self identifies as a "political scientist". The only place that politics has ever been scientific is in Asimov's Foundation series and that was fiction.

2. Algebra is trivial. Quantum physics is difficult. If you cannot master algebra then you deserve to not graduate from high school.

Darin

7/31/2012 01:35:23 pm

There is also a case for grammar. Split infinitives.

Chris Stephens

8/1/2012 09:47:36 am

Of course, split infinitives aren't ungrammatical. But I guess you're making a case for your point, but not in the way you intended.

Betty Peters

7/30/2012 08:28:20 am

It is not that algebra I is too difficult; it is that the students are not being prepared properly. Across America and in England, they come to algebra I lacking basic arithmetic skills such as automaticity in math facts up to "times 12," full understanding and ability to do long division using the traditional method without a calculator, and deep understanding of fractions along with percentages, ratios and decimals. See the article below about the new math report from Carnegie Mellon. It brings up the same conclusions as the National Math Panel's report in 2008 regarding fractions and long division. The Carnegie Mellon report noted our schools of education must begin to prepare elementary teachers much better for teaching arithmetic. You can't teach what you don't know!
http://www.psychologicalscience.org/index.php/news/releases/knowledge-of-fractions-and-long-division-predicts-long-term-math-success.html

Gigi

7/30/2012 10:40:12 am

Here here Betty! It does start with elementary school teachers not liking the subject so they don't have fun teaching it which then leads to a less than optimal start to learning the subject. I didn't get my first good math teacher until college (thanks Toni Wegner!) where I thought that a stats class was going to keep me from getting my college degree. Math as an obstacle to school success is a shame. There are too many subjects out there to have one in particular impact so many people (either high school or college).

@ Dave and the rest of society who agree with your point that if one can't master algebra, then the student should be a high school drop out. I agree with Dan that there are MANY factors that contribute to the drop out rate but math is a biggy. I have been working for the past three years trying to give students confidence that they can learn algebra and/or geometry. For some of these students, they would have dropped out if I could not offer them what is called in Virginia as the Modified Standard diploma (MSD) which still involves three math classes...just different levels (intro classes counted and a real world math class - personal finance). This year the DOE has decided that they are no longer allowing the MSD as a diploma option (it was only available to special education students). I predict that unless our society starts to value teachers (especially math teachers!), our dropout rate will go up among that population in particular but I see similarities with other populations of students as well, especially low SES.

In an ideal world (and yes..I know that our current budget situation impacts this), we need to find *many* ways to engage our students and not lose them to "standardized" learning/testing. The repercussions are huge for both the families and the communities in which these families live. When the parents can't help the students with their algebra homework (and the students can't afford tutoring and/or otherwise make bad choices about paying attention in class), students end up not caring at all about the subject since no one really likes the feeling of failing. To have students fail the same class 2-3 times is such a waste of time. There are too many better ways to use that time (apprenticeships, volunteering in the community, mastering other subject areas, etc.).

Thankfully, the doors of the ivory tower at the time of my admission were open since they did not require me to be a math or SAT-wonk (obviously that was a LONG time ago). Fortunately in college, you don't have to continue to pursue math as much as in high school. I found that if I didn't use the math, I lost it. A lot of jobs don't use algebra.

Although I agree that high expectations lead to results among students, not everyone has the same motivation for academic success (as currently defined by the DOE). It's a shame when someone feels "less of a person" when they can't master Algebra (or whatever subject area isn't their forte). Let's make "hoops" more manageable so that we have less of a welfare system or people finding that dealing drugs is the best job that they can get out there (which believe it or not I've had to use as an example of why they should stick with math so that they're not ripped off)......

It takes empathy to recognize that not everyone is cut from the same cloth and that they should not be condemned because their strength is not the same as what the DOE and politicians have decided is necessary for success in academics.

Obviously this touches a nerve for me. My heart goes out to the families who have children who, for whatever reason, can't master the subject(s) we think they should. The reward in my job comes from figuring out the "many" paths that the students/parents can take to meet their goals..... Quite often there isn't just one path (but a lot of people will tell you there is!).

Gigi

7/30/2012 10:44:41 am

Comment deleted

Mazungu

7/31/2012 07:42:31 am

BTW, the expression is not 'here, here', it is 'hear, hear', as in 'hear him, hear him!' http://en.wikipedia.org/wiki/Hear,_hear

Although I agree that Algebra is important and should be taught in high school, I don't think it's a bad idea to evaluate occasionally what is being taught. For instance, I personally think basic economics is far more important for the average high school graduate to grasp than chemistry or biology. Everyone needs to understand economics (how can you vote, otherwise?); not everyone needs to know chemistry.

Alton

7/30/2012 10:41:26 am

A student's skills matter. Kids earning C's in arithmetic are poorly positioned to have a good experience in Algebra; so we invented Algebra 1 & 2. Kid's earning C's in arithmetic were poorly positioned to have success in Algebra 1 & 2; so we invented Pre-algebra.

As a HS Algebra teacher, I find that the lowest half of my students can't enjoy math. The pace is too fast, the number of concepts too many and the higher performing kids get the answers faster, with considerable ease - then sit looking pained, while the wait for their slower colleagues.

If I had tracking, I'd teach the less skilled students half as many concepts, at a slower pace - and I think they would enjoy it. (But I suspect that too many of these would imagine themselves "good at math").

dan willingham

7/31/2012 09:28:54 pm

alton, did you mean "not good at math?" this is another significant problem, i think--the pervasive american belief that some people are good at math and some aren't, compared with the belief more common in east asia that these differences don't matter much, and that persistence and hard work are the key.

Sue Jones link

8/5/2012 11:25:11 am

Agree wholeheartedly -- but persistence and hard work are more likely to be learned if it's clear where to persist and how to work hard, which requires good instruction.

Clifton Chadwick

7/30/2012 05:36:25 pm

Conditions at real life work do change but to dismiss something as basic as algebra should only be done based on a serious study of the knowledge and skills requirements of major sections of the world of work. I have not seen any study of that sort. I do not know if the Dictionary of Occupational Titles has relevant information.

The idea that the brains of American children can't handle algebra would be ridiculous since children from places like India, South Korea, Singapore, Finland and many other countries obviously can cope with the subject.

My main suspicion is that poor teaching is to blame, possibly combined with changes in motivation and dispossition.

dan willingham

7/31/2012 09:30:14 pm

clifton, i agree, the international data show that it can be done. (as I noted in the comment above, american attitudes about what it takes to be good at math are a problem, not just the quality of teaching.

Cal

8/1/2012 12:51:51 pm

Actually, the international data doesn't show it can be done. We don't have any real evidence that students with IQs under 90 (to pick one metric) can master algebra. There's little evidence that Finland et al have done this.

Justin Ma

7/30/2012 05:50:43 pm

Thank you for this reply. I recently started pursuing a third of four degrees (including one in applied statistics) and have been questioning myself whether it's worth it. Maybe it's not, but you've reminded me about what I've always believed in. I just have had trouble explaining it to myself lately.

Betty Peters

7/30/2012 07:34:13 pm

Thanks to those who responded but I do not believe you have read the information from the Carnegie Mellon Univ. report . It is not long and the excellent video is very short. This is a well researched report that has given me hope that we can truly help the students of AL with their math preparation. Again, please read this:
http://www.psychologicalscience.org/index.php/news/releases/knowledge-of-fractions-and-long-division-predicts-long-term-math-success.html
Let's take this up across our various states. We can turn this around in one generation.

Gigi

7/31/2012 01:54:52 am

Thanks again Betty! Here's the main point - "Because mastery of fractions, ratios and proportions is necessary in a high percentage of contemporary occupations, we need to start making these improvements now.” I was fortunate to learn fractions and long division but I don't have to employ Algebra at all, ever, since taking the classes. I recognize the importance of having taken it once upon a long time ago.

I would love to give the Algebra I or II standardized test to a large segment of our adult population and see how they do. I suspect a number of them would not pass it (again...if you don't use it, you lose it). I hope that we, as a society, can come up with multiple math classes that are "real world" just in case a student isn't lucky to get a teacher who can teach Algebra I. I am not saying that students should not be taught Algebra I; I'm only highlighting that IF it doesn't go well after two times, that we need to find something else to teach them.

Having been in education recruiting for a number of years, finding math teachers is the hardest thing encountered by *any* school - whether private or public - since as another pointed out, there are so many other jobs that pay more and have less hassles than going into teaching would.

Algebra I should not be the "gatekeeper" to a high school education (even though I don't mind Organic Chemistry being the gatekeeper to med school - at least those students have lots of other options to pursue...high school dropouts do not).

Steve Owens link

7/30/2012 11:13:01 pm

"But the explicit teaching of abstractions is not enough. You also need practice in putting the abstractions into concrete situations."

True, but this is where things all too often break down. When students and teachers get in the weeds with the 500 examples its all too easy to lose track of the big ideas that motivate the enterprise. How do you maintain that connection, and create coherence (math should make sense)?

Connecting the abstract to the operational is a job for a teacher with genuine understanding of math. Where are we going to find enough people to educate 50 million public school students when mathematically adept people can find more lucrative (and easier) work outside education?

Der Alte

7/31/2012 05:30:56 am

Some students are easy to teach. Some students are very hard to teach.

On average, Hispanics in the 12th grade have academic skills at the level of non-Hispanic whites in junior high. I think it's safe to say that, on average, our schools find it hard to teach Hispanics.

Right now, about a quarter of babies born in America are Hispanic.

America's immigration policy over the last 50 years has been insane.

dan willingham

7/31/2012 09:34:45 pm

i mentioned this above, but it bears repeating. children of immigrants are outperformed by children of natives in every country--the u.s. does a poorer job of this than most. and the children who were born here are not, on average, succeeding.

Der Alte

8/1/2012 06:41:19 am

"children of immigrants are outperformed by children of natives in every country"

This claim is spectacularly false when it comes to the highest levels of academic achievement in America. Again, look at the names in the results of the 2012 MATHCOUNTS competition.

http://mathcounts.org/Page.aspx?pid=1872

And even with respect to averages, Asians (mostly immigrants and children of immigrants) do better than children of American natives on the PISA tests (as well as on other tests).

8/2/2012 05:01:47 am

@deralte Yes, let's exclude low performers aka cherry pick our data set to make our overall performance look better. Clearly some people are congenitally unable to do math. Is this the logic? I couldn't disagree more. I believe strongly that our curricula (especially elementary math prep leading up to algebra) and teacher preparation and training can and must help everyone at every level of ability, socioeconomic background and culture to master the math basics--and I include in that basic algebra. Look at JUMP Math's results in underprivileged schools and you'll see that it's doable jumpmath1DOTorg/research_reports

dan willingham

7/31/2012 09:32:11 pm

steve great question and i don't know the answer. the way this problem is often framed is: "the people doing the teaching are a product of the system that we think isn't working."

Sherwood Botsford link

7/31/2012 01:57:12 am

Overall I agree. Where possible, however, examples should be drawn from everyday situations.

Calculating the consumer price index is a good thing -- if it fits into your curriculum. Better: Calculate the cpi 3 different ways. Better yet: Give another price list almost identical, and get them to calculate it using the differences.

Calculate the present value of an annuity. The pay off period of a loan. The amount of money paid in interest maintaining a large monthly balance on a charge card. Calculate the worth of an air mile.

Find the point where building roof trusses on site is better than hauling them in, given a cost of transport, and cost of labour. (Differential calculus)

Whenever possible give them real problems. Real problems often have extraneous information. Real problems often have multiple solutions. Sometimes they are under spec'd, and you have to dig for the extra stuff, or make assumptions. (Does the cost of labour as you get further from the truss factory go up because they drive further, or down, because local people will work for less? A site built truss has a different design. Will the materials difference make much of a difference?)

The other thing is variety of problems. Give them one type of problem, and they memorize an algorithm for that type without understanding. Mix them up. At best they develop deeper understanding. At worst they develop a super algorithm to decide which algorithm to use.

A friend of mine taught high school physics and calculus as separate courses. I pointed out to him that calculus and first year physics go together like ham and eggs. Velocity is the first derivative of position with respect to time. Hadn't occurred to him.

I've actually used systems of equations twice in my life: Once on a canoe trip where I had maps with magnetic declination pruned off. Declination was one variable, the other was our position on an arbitrary line parallel to the shore. The other time was making bee hives. They aren't square, but rectangular. I had 16 foot long boards. There were several ways to cut a 16 foot board into sides that were efficient. I had to use a system of equations to figure out how many to cut each way to minimize waste. Saved almost 10% of the material.

dan willingham

7/31/2012 09:35:53 pm

i would like to take a math class from you!

Gigi

8/1/2012 02:58:17 am

Hear hear :)! Me too.....

Der Alte

7/31/2012 05:04:00 am

The book Real Education by Charles Murray is much more sensible on these issues than either Hacker or Willingham.

http://www.amazon.com/Real-Education-Bringing-Americas-Schools/dp/0307405389

QUOTE:

With four simple truths as his framework, Charles Murray, the bestselling coauthor of The Bell Curve, sweeps away the hypocrisy, wishful thinking, and upside-down priorities that grip America’s educational establishment.

Ability varies. Children differ in their ability to learn academic material. Doing our best for every child requires, above all else, that we embrace that simplest of truths. America’s educational system does its best to ignore it.

Half of the children are below average. Many children cannot learn more than rudimentary reading and math. Real Education reviews what we know about the limits of what schools can do and the results of four decades of policies that require schools to divert huge resources to unattainable goals.

Too many people are going to college. Almost everyone should get training beyond high school, but the number of students who want, need, or can profit from four years of residential education at the college level is a fraction of the number of young people who are struggling to get a degree. We have set up a standard known as the BA, stripped it of its traditional content, and made it an artificial job qualification. Then we stigmatize everyone who doesn’t get one. For most of America’s young people, today’s college system is a punishing anachronism.

America’s future depends on how we educate the academically gifted. An elite already runs the country, whether we like it or not. Since everything we watch, hear, and read is produced by that elite, and since every business and government department is run by that elite, it is time to start thinking about the kind of education needed by the young people who will run the country. The task is not to give them more advanced technical training, but to give them an education that will make them into wiser adults; not to pamper them, but to hold their feet to the fire.

Christopher Stephens

8/1/2012 10:14:26 am

The trouble with Murray's main argument in this book is that he doesn't have enough evidence to support his strong claims about who is and who isn't able to become more educated. He seems to think that most poor children, for example, are unable to benefit from a liberal arts education. He thinks that somewhere around first grade one can make a determination of our innate abilities, and then track some students to vo-tech, etc.

The simple fact is, his central claims and recommendations outstrip the evidence he has.

It is a common problem for those attracted to quasi-genetic determination arguments; we almost never know the norms of reaction for complex traits (or even for simple traits in humans).

Further, we have a poor track record of figuring out who should be tracked and who shouldn't (in part because of our ignorance about too many environmental variables).

Also: we do have a track record of some changes in education policy making dramatic improvements in education of the poor.

Der Alte

8/1/2012 03:51:56 pm

"He thinks that somewhere around first grade one can make a determination of our innate abilities, and then track some students to vo-tech, etc."

Can you quote Murray saying that we should track students into vo-tech somewhere around first grade?

"Also: we do have a track record of some changes in education policy making dramatic improvements in education of the poor."

Name some of those changes, and give evidence of dramatic improvement.

Christopher Stephens

8/2/2012 05:51:58 am

If you're interested, there are many examples of dramatic improvements in Karin Chenoweth's book, It's Being Done: Academic Success in Unexpected Schools

Tony Lima link

7/31/2012 12:48:20 pm

Really impressively argued. There may be a few tidbits in my post at http://gonzoecon.com/2012/07/algebra-is-hard-so-why-bother/ that you'll enjoy. And there is a difference between political science and politics when it comes to academic departments. Hacker is simply misplaced in Poli. Sci.

Darin

7/31/2012 01:39:36 pm

Let's make a turn. Check out Steven Pinker's work on violence. He (accurately) attributes increasing peace to the ability to abstractly think about human rights and not the ability to empathize with people who we will never meet. Algebra brings peace.

Brooklyn Parent

7/31/2012 02:24:15 pm

My anecdotal experience based on my child's passage though public school in Brooklyn confirms that in general elementary school teachers are not attuned to math teaching and vastly prefer to teach reading and writing. Early years of math consisted mostly of writing! Writing story problems and then solving them. For kids who are slow to read and write, but are fine with math, this is a real turn off. Fortunately in middle school and high school teachers can be found who like to teach math and are good at it.

As to Algebra 1, as taught in NYC is is really not a challenging subject! I think Andrew Hacker has really gone off the rails here. Deciding that high school students can't handle Algebra seems to be tantamount to shutting them off from a vast array of opportunities that might involve some basic math. Because that is what Algebra 1 is.

While I am at it, I want to say that my son's 9th grade Geometry teacher was one of the best teacher's he has ever had. Learning Euclidian geometry taught him as much about logic as geometry. Mastering geometry gave him a new perspective on future career choices. I shudder to think of the students who would be deprived of these experiences under Hacker's imagined math-free educational system.

By the way, his high school was Title 1, but the students were expected to take four years of math and graduate on time and most did.

dan willingham

7/31/2012 09:42:44 pm

brooklyn parent "tantamount to shutting them off from a vast array of opportunities that might involve some basic math" that's exactly the part that bothers me so much. people are often responding "I never use math." Point taken, but do you want to shut off a bunch of career opportunities in 9th grade?

When I was in graduate school I attended a session for undergraduates on "How to be admitted to graduate school." (I was there to tell them grad school life was like.) Someone asked the professors running the session what coursework an undergraduate should take if he wanted to go to clinical psychology graduate school, so as to end up a therapist. The professor's unhesitating reply "Math. The number one reason people fail out of our program is that they can't cope with the math, so you need to show admissions committees you're prepared."

Der Alte

8/1/2012 07:18:00 am

Mr. Willingham, do you think that maybe there are many people whose lack of thinking ability is "shutting them off from a vast array of opportunities that might involve some basic math" ?

For example, consider the American 8th graders who score in the bottom 10% on the NAEP test. Do you think that maybe most of these students are simply incapable of understanding basic calculus, no matter how hard they work and no matter how they are taught?

And how about you? Do you think there might be results in mathematics that are simply beyond your ability to understand, no matter how hard you try? Do you think that there might be math problems that can be solved by some people but not by you?

"Ability varies. Children differ in their ability to learn academic material. Doing our best for every child requires, above all else, that we embrace that simplest of truths. America’s educational system does its best to ignore it."

You seem be be one of those who does his best to ignore this simple truth.

8/1/2012 08:10:27 pm

Interesting discussion. I think that any discussion of readiness for algebra must invoke both Liping Ma and John Mighton. Liping Ma talks about the roots of the problem with a stunning and simple exposition of how Chinese and American teachers teach four different math concepts/skills of varying degrees of difficulty. Very readable especially for the layperson. John Mighton (of non-profit JUMP Math) has created a curriculum through years of testing, tweaking and testing again in the classroom. (the way all curricula should be developed) with stunning results. Philosophically he is aligned deeply with Liping Ma's findings, but importantly he has unlocked the means to deliver on dramatically better elementary math. He moves math-phobic, underprivileged kids in classrooms from 25th % performances to 90%+ -- bell curves go way right and even more importantly tighten up -- performance gaps close (data from 3rd party sources). HIs book "The End of Ignorance", along with Liping Ma's thesis-cum-book are must reads. Especially for those of us scratching our heads over the back and forth jargon and opinion giving. Where other people assert that math is for everyone (or should be), they pinpoint with evidence, HOW math could be for everyone.

Dan Willingham

8/1/2012 07:00:30 am

@Der Alte: the graph you're linking to doesn't depict a fair comparison. Every country has subgroups that perform better than others. So breaking down American kids and comparing them to the *average* in other countries is misleading. David Berliner has, for years, been using this technique to make a different point. He shows that wealthy American kids are doing very well by international comparisons and poor kids are not. He concludes that poverty is the problem.
Regarding the "children of immigrants" argument. . . I don't think that inspection of the last names of tops-scoring kids in a math competition is the way to go here. PISA data show that children of immigrants do worse, on average, in most countries.
All of this is not to deny demographic differences in the performance of US kids. (They are attenuated but don't disappear by most metrics when SES is accounted for.) My point is that it's doesn't make much sense to me peg US kids poor performance in math on immigrant kids. The notion that US kids are doing just fine in math (or would be doing just fine) if not for these kids strikes me as unsupported. Maybe that's not what you're saying--that's what I'm hearing.

Der Alte

8/1/2012 07:35:35 am

My main point is that we can easily do a pretty good job of distinguishing those immigrants whose children will do well in American schools from those immigrants whose children will do badly in American schools.

The children of Chinese who come here for graduate school in science and engineering, on average, do extremely well in our schools. Precisely these children constitute a large fraction of those on the MATHCOUNTS list.

The children of Mexican illegal immigrants seldom do well in American schools.

So, why not have an immigration policy that makes America smarter, rather than an immigration policy that swells the ranks of our underachievers?

Christopher Stephens

8/1/2012 10:16:44 am

Alternatively, why not have a policy to figure out if there's a way to improve the scores of other immigrants?

Der Alte

8/1/2012 03:41:58 pm

"Alternatively, why not have a policy to figure out if there's a way to improve the scores of other immigrants?"

Yes, why not import tens of millions of sows' ears and then try to figure out how to turn them into silk purses?

dan willingham

8/1/2012 09:29:18 pm

this blog entry is about whether or not most kids should be taught algebra. I don't know anything about immigration policy.

8/2/2012 04:58:24 am

Yes, let's exclude low performers aka cherry pick our data set to make our overall performance look better. Clearly some people are congenitally unable to do math. Is this the logic? I couldn't disagree more. I believe strongly that our curricula (especially elementary math prep leading up to algebra) and teacher preparation and training can and must help everyone at every level of ability, socioeconomic background and culture to master the math basics--and I include in that basic algebra. Look at JUMP Math's results in underprivileged schools and you'll see that it's doable. http://jumpmath1.org/research_reports

Der Alte

8/6/2012 03:24:13 am

"this blog entry is about whether or not most kids should be taught algebra. I don't know anything about immigration policy. "

Immigration policy is extremely relevant to the issue of "whether or not most kids should be taught algebra", since our Hispanic students, on average, do much worse in math than our non-Hispanic white students.

Willingham combines willful blindness on the micro-scale (ignoring individual differences in intelligence, as in his article "Teaching to What Students Have in Common") with willful blindness on the macro-scale (ignoring the effect of the demographic riptide that our educational system is swimming against).

What I would regard as the criminal irresponsibility of Willingham's micro-scale blindness is explained in chapter 4 of "The g Factor - General Intelligence and its Implications", by Chris Brand.

http://www.douance.org/qi/brandbook.htm

The catastrophic effects of our immigration policy are described in the following ETS report.

http://www.ets.org/perfect_storm

8/2/2012 05:00:27 am

Der Alte

8/2/2012 01:21:23 pm

"Yes, let's exclude low performers aka cherry pick our data set to make our overall performance look better."

Whose overall performance? Asians?
Are you referring to my observation that the top performers on MATHCOUNTS and USAMO are mostly Asians?
Only an idiot would say that this observation is supposed to imply that all Asians are math geniuses. Of course the distribution of Asian math competence is very wide. That distribution, however, has far fewer at the low end and far, far more at the high end than the distribution for most other groups, especially for Hispanics and blacks.

"Clearly some people are congenitally unable to do math. Is this the logic? I couldn't disagree more."

So you think that even people with congenital brain defects that render them unable to learn to speak can still "do math" ?

There is very wide variation in the ability to "do math". Some of that variation is genetic. The smartest children can easily learn math that the dumbest find, at best, very difficult, maybe impossible. You're not doing anyone any favors by refusing to recognize that fact.

Richard Rasiej

8/1/2012 11:13:01 am

I read Andrew Hacker's piece on Sunday. I have just retired from my "give-back" career, seven years of teaching high school math in Los Angeles, and many of the points raised in his piece resonated a bit with me, even though I had to force myself to take his argument seriously. Then, fortunately, I came across a link to your blog reply yesterday, and it straightened my thinking out.

Nevertheless, I think there is a question that we are all dancing around without really addressing: WHY is the pass rate for Algebra so low?

One of the reasons I am interested in this question is that I am staying involved in math education, although I will now be concentrating on professional development in mathematics for K-5 teachers. My underlying hypothesis is that students have difficulties with middle school and high school math (which are generally taught by specialists) because of misconceptions and outright errors acquired when they were being taught the basics by generalists, many of whom themselves were terrified of the math they had to "master" in order to become teachers.

Sue Jones link

8/5/2012 11:33:30 am

What I often see tutoring "underprepared" students at the college level is that many assume from the start of class that they can't really understand the math, so they focus *hard* on memorizing the procedures (these are folks who do have that "hard work is more important than ability" concept).
Unfortunately, the memory burden gets huge when you don't see the meaningful connections. Suddenly a plus sign doesn't mean add (negative numbers) ... students do things like generalize "okay, if it's got parentheses, then everything is raised to a power" so that (2x^3) is 8x^3 ... because they don't understand what parentheses mean.
This is not innate; they learned this (often from those terrified teachers Richard Rasiej mentions).

ezra abrams

8/1/2012 12:40:12 pm

1) It is an observable fact that algebra is required for a very, very small % of hte population
2) therefore, the only reason to do stuff like resolution of partial fractions is either as weed out, student challenge, or professorial self satisfaction (professors - we are smart, we passed algebra, therefor any smart person will pass algebra)
3) to see the truth of (2) imagine if many semesters of music were requried; all professors would be good at music, and those who couldn't pass music would be "dumb"
4) even if one ignores the observable fact that only a tiny, tiny fraction of the population needs algebra, one has to answer a fundamental question: we have so many hours in the school day, and 1,000s, perhaps 10s of 1,000 more information then the students - any student - can learn.
Therefore, we ahve to select the most important information.

My personal observation of life is that most of the misery in human existence, aside from abject privation, is due to lack of understanding of ones fellow humans; from interations with one's immediate family to interactions between nations
Therefore, anyting that could reduce this misery is more important then algebra; I think that psychology is could help, and therefore suggest that from kindergarten to senior year of college, every single student in America is required to take at least one full semester of pscychology per year

ezra abrams

8/1/2012 12:43:03 pm

Quote "Economic data directly contradict that suggestion. Economists have shown that cognitive skills--especially math and science--are robust predictors of individual income, of a country's economic growth, and of the distribution of income within a country (e.g. Hanushek & Kimko, 2000; Hanushek & Woessmann, 2008). "

Cause or effect ?
I'm not sure how you would answer this

ezra abrams

8/1/2012 12:47:42 pm

quote
What of the other argument, that formal math mostly doesn't apply outside of the classroom anyway?

The difficulty students have in applying math to everyday problems they encounter is not particular to math. Transfer is hard. New learning tends to cling to the examples used to explain the concept. That's as true of literary forms, scientific method, and techniques of historical analysis as it is of mathematical formulas

This doesn't really seem to address the point
Where, in ordinary life, does a cubic equation, or resolution by partial fractions, come into play ?
NO WHERES
Algebra actually obscures use of math in ordinary life because it trains you to think of exact solutions and exact equations, whereas in life it is more useful to be able to apply rules of thumb, as in general the data - from CPI to gas mileage is sort of one significant figure data.
Not only that, in most real life situations, the trick is to figure out how to get behind or around the misleading information that you are given; most news stories, and certainly most for profit and political information, is wrong or slanted, and you don't need algebra - you need basic math skills to get behind this stuff.

dan willingham

8/1/2012 09:35:27 pm

LL I'm a fan of jump math. and it's incredibly inexpensive.

Jim

9/5/2012 05:35:54 pm

ok, before i say anything, I need two things clarified. One: how is everyone defining "higher level mathematics" because I consider all classes available in high schools to be lower level since they are still method based. Two: Algebra is an extremely broad subject even in the high school curriculum ranging from finding an unknown quantity to introductory level group theory (although it is not called that). So what parts EXACTLY does one not use outside of high school.

ezra abrams

9/6/2012 12:52:29 am

what is "method based" and why is it lower and what is the alternative
Anything with a variable (x) is algebra, and algebra is un needed by 90% of the population
Also, aside from surveryors and engineers, I'm not sure anything with a sine or cosine is ever usful outside of high school, unless one is a crossword puzzle afficianado

I find it astonishing that people think that ordinary people - say80 % or so of the population ever, ever need algebra or calculus, particularly in with modern computers.
Not that long ago, celebrated mathemeticians like Tukey wrote whole books on how to graph data with pen and paper, and tricks to make the graphing easier, since (I've done this) it is darn hard to prepare a simple scattergram with, say , a rapidograph pen set.

Now, all that effort is useless (if you look at the books Tukey wrote on graphing and analyzing data, most of hte book is "tricks" to make it easier to aanalyze data with pen and paper; you can see something similar in the specialized field of enzymology where the initial linear rate of an enzymatic reaction is important; these reactions take the form of a rectangular hyperbola when plotted; with modern software it is easy to extrapolate to the plateau; in the old days you had to do all sorts of trick to linearize the data so you could fit a line by eye and get a slope..., see the book by Cleveland or the book Enzyme Kinetics, by K Plowman, McGraw-Hill, 1972)
another point in that argument is that you use to be able to buy semilog paper with decimal or time X axis, and 1,2,3.. cycles on the y axis,,all to make manual plotting easier...

ezra abrams

9/6/2012 12:52:42 am

ezra abrams

9/6/2012 12:53:06 am

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