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In the comment thread of this academia.SE question, the following generated some strong reactions:

My very different (community-college) perspective is that the math discipline will end up as a filter no matter how much the institution desires otherwise. Most of our students will never grok 8th-grade algebra no matter how many times they try, nor under what circumstances. Other disciplines can be made easier in many ways; math has a central unavoidable core of intellectual honesty. Our institution may in fact get rid of any real math as a basic requirement for this reason. E.g., writings of Andrew Hacker.

Is there any meaningful, reasonably objective way to determine what fraction of the population is incapable of learning algebra, even with repeated effort and competent instruction? I'm looking for evidence-based answers that cite references, not anecdotes or personal impressions. It would seem obvious to me that the answer has to be greater than zero (e.g., there are developmentally disabled people) and less than one (because some people do learn algebra).

Success rates in high school algebra probably do not even provide an upper or lower bound on the answer. It seems to be common for high school algebra teachers here in the US to pass students who can't do the work but make an effort. And much instruction probably is not competent.

Edit: To make the discussion easier for non-U.S. SE participants to follow, note that "GED" is an alternative way of getting a high school diploma in the U.S. It apparently stands for "General Educational Development" -- although I always thought it stood for General Equivalency Diploma.

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    $\begingroup$ @JamesS.Cook: The GED seems like a weak method of trying to infer an answer to this. Plenty of people fail high school algebra, therefore would not pass the GED, and then learn algebra later on, e.g., in community college. $\endgroup$ – Ben Crowell Sep 10 '16 at 1:59
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    $\begingroup$ @BenCrowell there is no algebra in the GED. It's mostly middle school math. It's primarily arithmetic, basic formulas like area or perimeter etc. Perhaps it has changed since I took it, but, high school graduates who cannot pass the GED are the same as those who cannot do arithmetic. It seems to me inability to do arithmetic is a significant factor in folks inability to learn basic algebra. $\endgroup$ – James S. Cook Sep 10 '16 at 16:07
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    $\begingroup$ Can not learn algebra seems to be a much harder thing to be objective about than Do not learn algebra. The latter is answerable, I think. $\endgroup$ – mweiss Sep 11 '16 at 19:39
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    $\begingroup$ On the other hand, we have the anecdotal observation of a math instructor in one of New Jersey's most affluent districts thinking that 1/3 is "Near three, isn't it?", and needing to be corrected by her students. "It appears that the higher scores in the affluent districts are not due to superior teaching in school but to the supplementary informal 'home schooling' of children." (Patricia Clark Kenschaft, "Racial equity requires teaching elementary school teachers more mathematics", Notices of AMS 52.2 (2005): 208-212.) $\endgroup$ – Daniel R. Collins Sep 11 '16 at 23:34
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    $\begingroup$ @DanielR.Collins: OMG, that's pretty horrifying. But the "near three" teacher was teaching 5th grade. K-8 teachers in the US are often incredibly bad academically, and especially in math. High school jobs are a different deal, especially in affluent school districts. My impression is that when people who don't know algebra are teaching high school algebra, it's typically in low-SES schools that are desperate to hire math teachers. $\endgroup$ – Ben Crowell Sep 12 '16 at 1:55
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I spent some time looking for information that might provide some kind of reliable, evidence-based answer to my own question. I came up with the following, which is not perfect, but I thought it would be worth posting as an answer. I tried two methods of probing this: standardized test scores of students with high socioeconomic status (SES), and heritability of standardized test scores as measured by twin studies.

Standardized test scores of students with high SES

For this method I used data from the US.

We can't use success in high school algebra as a way of determining innate ability to learn algebra, for a variety of reasons. Students in some states are required to take algebra as early as 8th grade, but many are not developmentally ready at that age, so their failure doesn't necessarily indicate a permanent inability. Anecdotally, many US high school algebra teachers report that they pass students who haven't learned the material but who were punctual, obedient, and turned in homework. Because many high schools have algebra as a graduation requirement, there is intense political pressure to say that students have passed algebra, regardless of whether they really understand it. Many students fail algebra not because of a lack of innate ability but because of a lack of motivation or interest. Many public schools serving a student population with low SES do not provide competent instruction.

To get around these problems, I looked for data on SAT math scores among kids with high SES. Among the most affluent families in the US (the top decile of SES), about 80% of kids go to college (although, surprisingly, only about 2/3 of those get a degree). The vast majority of these kids were exposed to competent instruction in algebra during high school, and they were typically under intense social and family pressure to get into a good college. Nearly all of them took the SAT, which is required by almost all schools in the US that have selective admissions.

For students with the highest SES (more than $200,000 per year of family income), the average SAT math score is about 600. A score of about 600 is also used as a cut-off by many colleges in determining which of their students are allowed to take calculus without having to take a prerequisite course. The math section of the SAT includes a lot of algebra. Therefore it seems reasonable to me to take a score of about 600 as indicating pretty decent mastery of algebra. If the mean of the scores for this group roughly coincides with the median, then about 50% of people in this group have mastered algebra.

It seems reasonable to assume that there is no significant difference in innate mathematical ability between the most affluent people in the US and the rest of the population. Therefore the scores of this group can probably be interpreted as a rough measure of what level of achievement you get in math, at this age, given a favorable environment. That is, they may give a reasonable estimate of what nature permits, given the right nurture.

The mean score for Asians (regardless of SES) is also about 600. If we assume that there is no innate difference in mathematical ability between Asians and other ethnic groups, then this would seem to support the hypothesis that a mean score of about 600 represents a broad average of potential mathematical achievement, among college-bound kids at age 18, and that this potential is reached if these students receive competent instruction and grow up with cultural expectations that pressure them to do well in academics.

Since about 80% of high-SES students go to college, and about 50% of those get an SAT math score of 600 or higher, my estimate would be that something like 50-60% of high school kids in general are not capable of learning algebra. This seems to be in the right ballpark to support Daniel Collins's statement that:

Most of our students will never grok 8th-grade algebra no matter how many times they try, nor under what circumstances.

Many people who don't master algebra in high school nevertheless go to college, where they will typically be required to take remedial ("developmental") courses in math. Some succeed in learning algebra at that point, so we should correspondingly reduce our estimate of the percentage of the population unable to learn algebra. However, remediation in math appears to be extraordinarily ineffective.

The most questionable assumption in my estimate seems to me to be that high-SES kids who get less than 600 really tried to learn math in high school. They were probably under strong pressure to succeed in math, but they may have responded to that pressure by doing the minimum required in order to get a certain grade, and doing the minimum may have meant going through the motions of taking a math course without actually learning the math. If there were many students like this, then the percentage of the population lacking the innate ability to learn algebra may be much lower than the above estimate. The size of this bias is not likely to be huge, however, because then one would expect a significant number of these students to succeed when remediated in college, whereas in fact remediation almost never works.

Since there are a lot of shaky assumptions in the above estimate, it's interesting to see if we can place any more definitive upper and lower bounds on the figure.

As an upper bound, the percentage of high school kids in the US incapable of learning algebra must be no more than about 94%. This can be inferred because out of 4.3 M Americans aged 18, about 1 M take the SAT, and 25% score 590 or higher on the math portion. Thus about 6% of the population demonstrates mastery of algebra every year, when they reach this age.

As a lower bound, the percentage incapable of learning algebra seems sure to be at least 3%, since that's roughly the percentage of the population that is considered intellectually disabled.

Heritability of standardized test scores as measured by twin studies

Kids in the UK have to go to school until they're 16, and at that age they take a standardized exam called the GCSE, which includes a test of math. The GCSE's "higher tier" is 30% based on algebra. The level of the algebra questions is about at the level of a first high-school algebra course in the US, i.e., it overlaps with the math portion of the SAT, but doesn't include the most difficult material you would find on the SAT.

Twin studies show that GCSE math scores are 55% heritable. Only about half of this genetic influence appears to be due to intelligence; the other half seems to be because of other partially heritable personality traits such as "self-efficacy" and behavior problems.

I would like to study this approach more thoroughly, but these findings seem to show that we can't necessarily think of a hypothetical innate inability to learn algebra solely as an innate intellectual shortcoming. For example, a student who refuses to do his algebra homework may be doing so partly because the heritable component of his intelligence tends to make the activity hard and not enjoyable for him, but also partly because a partially heritable component of his personality makes him unwilling to stay on task in order to achieve a future goal.

If the twin studies obtained data on SES, then it would be interesting to combine the two approaches.

A counterintuitive part of the interpretation of these data is that high heritability of test scores goes hand in hand with a more socially equal, meritocratic society. Suppose we make some change in social policy that makes us more meritocratic, e.g., Head Start programs, or eliminating the practice of giving preferences for college admission to children of alumni. By doing this, we've reduced the amount of variability in the environmental factors affecting education. Therefore the amount of variation in educational achievement will go down. Of this smaller remaining amount of variation, the fixed amount due to genetics will now represent a greater fraction.

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    $\begingroup$ Excellent work at a very hard question and thanks for writing it up. Granted that the main estimate excluded low-SES students; I might hypothesize that low-SES cases have the capacity to inflict more permanent intellectual disability, emotional management disabilities, language skill problems, etc. which could possibly push the number unable to potentially master algebra higher than among high-SES students. $\endgroup$ – Daniel R. Collins Sep 11 '16 at 19:25
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    $\begingroup$ Example: Groce, et. al.: "Malnutrition in pregnant women, infants and children leading to developmental delays and physical, sensory and intellectual disabilities is well-documented in maternal and child health as well as disability-specific literature." unicef.org/disabilities/files/… $\endgroup$ – Daniel R. Collins Sep 11 '16 at 19:26
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    $\begingroup$ There's also the metacognition issue, at all levels. In my experience, very many (perhaps 1/3 or more) of students in lower-division undergrad math grossly misjudge the level of effort required, and similarly misjudge their own effort, and misjudge their own competence. Failing to appreciate that one must actually pay attention from beginning to end of an algebra computation consistently leads to random, needless mistakes, which give the appearance of incomprehension... especially when combined with lack of appreciation of what "checking one's work" might really mean. So... [cont'd] $\endgroup$ – paul garrett Sep 11 '16 at 21:07
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    $\begingroup$ ... [cont'd] I don't feel like I have any idea (despite having taught thousands of college kids) what the actual limitations might be, because I think it's masked by larger behavioral issues. Thus, similarly, I'd think that middle-school through high-school social/behavioral issues completely swamp actual capacity (or lack). $\endgroup$ – paul garrett Sep 11 '16 at 21:09
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    $\begingroup$ I appreciate the thought and hard work in this answer, but I have very very serious methodological bones to pick with almost every deduction in it. The two most salient to me are: (1) trying to extract meaningful info about innate underlying capacity from the amount of US students successful on the SAT assumes fundamentally that the learning conditions presented to the students, from birth until the SAT, are close to optimal. These conditions span 16-17 years and every dimension of their lived experience. We can't have any reasonable scientific idea of how close to optimal this is. Cont'd... $\endgroup$ – benblumsmith Sep 16 '16 at 23:30
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Say student X tries a dozen different teachers and tutors and learning methods to grok algebra and they all fail. Is X incapable or is there another approach that would work better?

The question basically cannot be answered decisively.

If the world's greatest math teachers all combined efforts to find new ways to teach X, would that work? Could any empirical study test them all? [One thing to keep in mind here is that most of us are unaware of how incredibly narrow our math education experiences are and that there are many radically different ways of teaching math.]

To know what share of the population is inherently incapable of learning math, we'd have to know ALL teaching methods across ALL time compare that with a PERFECT and COMPLETE understanding of the brain. Ain't gonna happen any time soon.

i.e. In the next thousand years, what improvements will math pedagogy undergo? Maybe X will just learn algebra Matrix-style in 2063.

I suggest other questions:

  1. What are the best known ways to teach algebra to diverse students?
  2. How can we scale up them up?
  3. How can improve them further?
  4. What else do we need to learn about algebra pedagogy?
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  • $\begingroup$ I think that according to philosophy.stackexchange.com/questions/16356/…, it is impossible for one to understand one's own brain. It might be possible for somebody who got a more complex brain at a later age to fully understand a simpler brain of a 6 year old but I highly doubt it. I believe that research on how to teach them in such a way that they will learn is actually an easier problem than studying how their brains work. If they cannot be made to understand fractions at that age, it might be $\endgroup$ – Timothy Dec 25 '18 at 23:42
  • $\begingroup$ because it's in fact impossible to teach them in such a way that they will learn. I wonder if in addition to that, attempting to teach them when they're too young can cause them to form a bad memory of struggling and decide later as an adult to give up and resist all attempts to learn what they are. $\endgroup$ – Timothy Dec 25 '18 at 23:44
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Your question will not have a research-based answer.

You assume that there is a determinable "fraction of the population" that "is incapable of learning algebra, even with repeated effort and competent instruction". Philosophically, I wonder if this fraction is well-defined. In your own answer, you mention the SES reflecting the society as an important factor. Would you assume the society as given in your mental model or potentially being a subject to change? What about neuroscientific methods that could be invented? The "fraction" you are looking for is very fictional.

Now, assuming you solve this problem, how could you practically determine this fraction? You may test all people of the world, but tests only give estimates of people's abilities, not their potential. What evidence could you give to make us believe that a person really cannot reach a given level? Would you assume, for example, that people from the 5th century were unable to learn complex numbers just because no one did?

Even assuming you will find a way to estimate peoples potential, what would the pedagogical implications be? Still you would try to help them learn as much as they can. The learning support, however, should rather be based on what they actually have learnt and not potentially could learn.

Finally, the question may evoke some substantial concerns. Both teachers' and students' thinking of people having fixed abilites has proven to hinder learning as the "dumb" ones may seem to deserve less care or effort. Moreover, what if someone is able to learn large parts of stochastics, geometry and calculus but not algebra? Wouldn't you want to know this? So, how much measuring would be enough? Shouldn't we rather use our ressources for something of more practical use? Thinking of fixed abilities, in addition, is likely to produce distinct worlds in the educational system. A world for the "good", a world for the "dumb", maybe another world for the "very gifted" students, disregarding many problematic effects of such separations (e.g., focussing on "good" mechanisms of evaluating their potential instead of focussing on learning; focussing on learning algebra at the cost of social learning). We know people have different abilites, but should this knowledge lead our decisions?

This is just to explain why I strongly believe that you won't find "evidence-based answers that cite references, not anecdotes or personal impressions".

I would like to add that from a philosphical point of view, I still find your question interesting. In some "applied research", however, progress may not only be given by answers to questions but also by changing questions and rethinking the situation and what you find interesting and why. This question is a candidate for this, I think.

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    $\begingroup$ "We know people have different abilites, but should this knowledge lead our decisions?" Absolutely it should. The idea that one size fits all is not just wrong, it's dangerous for our best students. $\endgroup$ – James S. Cook Feb 12 '17 at 18:04
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    $\begingroup$ I don't buy the claim that this question is not "philosophically...well-defined." About 1-2% of the population is profoundly mentally retarded. These folks may, for example, have had mothers who had toxoplasmosis. They may have difficulty with walking and talking. They are not people who are going to learn algebra. On the other hand, something like 20% of the population takes the SAT and gets a score of 600 or more, which indicates pretty strong mastery of algebra. Therefore if we let x be the percentage of the population capable of learning algebra, x>20 and x<99. $\endgroup$ – Ben Crowell Feb 13 '17 at 4:32
  • $\begingroup$ @JamesS.Cook: This is a misunderstanding. I did not want to say that one size fits all. My claim is that most of the researchers I know would find it more important to research how to best teach students based on their actual knowledge. And they think that teachers should rather care about helping the individual student to learn instead of determining their ability to reach a given level (in principle, not their actual ability and demands). Just a matter of ressources. There is nothing bad about identifying and helping gifted students, they also have the right to get teaching that fits them! $\endgroup$ – Anschewski Feb 13 '17 at 15:33
  • $\begingroup$ @BenCrowell: I absolutely agree, that some are definitely able to understand a given content and others are not. Being well-defined, however, is a matter of the boarder cases here. Did I get you wrong? $\endgroup$ – Anschewski Feb 13 '17 at 15:37
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    $\begingroup$ @BenCrowell: Besides practical determination with perfect precision, I think we would need more information on the model of "being able" you have in mind. My philosophical concerns include, for example, what kind of future treatment you would assume possible. Is a child that will die from cancer in 3 weeks in principle unable to learn algebra because it will die? Were all people from 10.000 b.c. unable to learn algebra because it had not yet been developed? The boarders between the clear yes-or-no cases are blurred, that's what I wanted to say. $\endgroup$ – Anschewski Feb 14 '17 at 9:48
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In Jean Piaget's theory of cognitive development, the "formal operational stage" was the last one identified, achieved between adolescence and adulthood (ages 11 to 15-20), in which "intelligence is demonstrated through the logical use of symbols related to abstract concepts" (Wikipedia).

One overview of research on this topic was presented by Mary Carol Day, "Thinking at Piaget's Stage of Formal Operations" (Educational Leadership, October 1981, link). She writes:

Piaget's position has often been used to predict that one who can think in a formal operational manner will always do so. However, only about 50 percent of those over 12 years of age who are presented with tests of formal operations perform in what would be considered a formal operational manner. This is true of college-educated adults as well as adolescents. In addition, even people who use formal operational skills on one task may not use them on another.

These unexpected results have prompted a variety of responses. One is that Piaget was wrong; formal operations is not a stage attained by everyone. A second response came from Piaget himself. He maintained that all individuals attain formal operations, but perhaps only in areas with which they have had much experience (Piaget, 1972). A third approach has been to reexamine the meaning of the term "stage." A stage can be viewed as a description of all cognitive activity that occurs during the stage (or cognitive performance) or it can be viewed as a description of the highest level cognitive activity of which one is capable (cognitive competence).

The available data tend to support the second and third alternatives rather than the first...

Note that the figure of "about 50%" of students who perform formal operations when tested is very much in line with Ben Crowell's answer here and other similar estimates.

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You're asking a big question. These kinds of things are not really settled yet to allow some pretty numerical answer. Instead of pushing so hard for a nice pretty answer, ALLOWING open discussion, opinions, debate, would be actually be more revealing.

For that matter the terms of the question are not precise (like how much intensive education are you willing to posit? 60 years of one on one tutoring?). I don't mean this in some neener, neener be precise manner. Just that even defining the terms is worthwhile and leads to a more manageable question.

I think to start with, try some easier questions: what percent of the population passes* 9th grade algebra on first try? (That's not the answer to your ultimate question, but it's informative to start thinking about the topic.)

Maybe as a second question, you could look at what percent of students eventually pass basic algebra. Restrict it to a state perhaps if that makes it easier to evaluate. (Make sure you account for dropouts of course.) That would tell you the impact of SOME higher intervention (repeating courses).

*ideally you define this with an achievement test, but if these are not nation-wide, use grades. Or you could look at some state(s) that do have an achievement test and consider them indicative of the overall population (maybe not precisely, but approximately).

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    $\begingroup$ -1 Does not serve as answer to question; more appropriate as a comment(s). $\endgroup$ – Daniel R. Collins Oct 30 '17 at 4:13
  • $\begingroup$ My point is that it is difficult to explicitly answer the question (as one would solve an algebra equation) but that one can bound the problem, can answer "easier questions" that still advance the understanding. A simple START to understanding the situation would just be to say "what percent of the population doesn't learn algebra 1 now". Obviously this is an upper bound of the percent incapable. But it's still relevant (in that considerable effort is put into teaching them). $\endgroup$ – guest Oct 14 '18 at 20:57
  • $\begingroup$ But you are right. I could have edited it down and made it a comment. ;-) Some of the best stuff is in comments. ;-) $\endgroup$ – guest Oct 14 '18 at 21:26

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