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I'm a retired university math prof and I now have a retirement job teaching at a small private high school. This is my 4th year. This school teaches Algebra 1 to 8th graders. Geometry to 9th graders. Algebra 2 to 10th graders. Precalculus to 11th graders. Calculus to 12th graders. I have come to the opinion that for many students, these subjects are coming a year too soon. I'm convinced that if we'd let most students get another year through or past puberty, and develop their ability to think abstractly more, then they'd have less trouble in math all through high school.

I found this study:

https://www.nber.org/papers/w21610

which tangentially supports my opinion.

My question is: Is there other research that more directly addresses when to teach the various subjects of middle school and high school math?

My observations are: I've listened to the middle school teachers complain that they can't teach Prealgebra to 7th graders because they can't add fractions. And the 5 and 6th grade teachers complain that they can't teach adding fractions because the students don't know the multiplication table. I used to complain that college freshmen who took Calculus in high school were usually worse off than the freshmen who hadn't. By taking Calculus in high school, many students seemed to be permanently warped in a way that made Calc 2 much more difficult. My suspicion is that by ever trying to cram more stuff in earlier, things get a lighter treatment. So that students are never quite prepared for the next class. And this effect snowballs.

I don't know how many times I had a sophomore in Differential Equations in my office who had never seen Pascal's triangle or had no idea how to factor a quadratic. Or he sort of knew how, but couldn't execute. I would complain, "I wish the high schools would teach the high school math solidly, so that I can teach Calculus to students who are ready." And I'm observing the same effect in this high school.

Here in Texas, they have the STARR test. It's a monster. By studying it, we can tell that they're having 5th graders do stem-and-leaf plots and find quartiles of date. Sure, they can do it, but it's just one more thing crammed into the curriculum that distracts from learning to add fractions. And these easy things, like finding quartiles, can be learned by a 10th grader in one day. Why are we torturing 5th graders with it for a week? A typical STARR test problem is to show a polygon in the shape of Utah, and assign lengths to some of the sides. These lengths are all mixed numeral. The student is supposed to find the area of the polygon. So first he has to add and subtract mixed numerals to find the lengths of the other sides. Then he has to multiply some of these to get the area. So this is a multistep problem testing more than one thing. Not only is more material being ever shoved into the lower grade curriculum, but also a higher expectation of problem solving skills. I don't think most 5th graders are ready for the Utah problem.

So my question is, again, does anyone know of real research that would support my opinion? (Or negate my opinion?)

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    $\begingroup$ finding quartiles, and be learned by a 10th grader in one day. Why are we torturing 5th graders with it for a week? I winced while reading this. $\endgroup$
    – ryang
    Feb 10, 2023 at 15:22
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    $\begingroup$ "I wish the schools would teach the [lower] math solidly, so that I can teach [the higher math]" - this is a completely separate hypothesis from "they're not old enough", and I suspect it's the correct explanation. See also this TED talk from Sal Khan about how we need to put more emphasis on the fundamentals. $\endgroup$ Feb 11, 2023 at 1:01
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    $\begingroup$ Just an anecdote, but: this is about the same track I took through math education when I was in high school, and personally, I've always felt it went too slow. I'm probably not a typical student, but I was bored out of my mind through every math class until finally reaching calculus in 12th grade and starting to really understand the reasoning behind all the stuff I'd previously been taught as "it's like this because that's how it is, stop asking". I hated math for all its rote memorization until I got to calculus, and now I love it. $\endgroup$
    – Hearth
    Feb 11, 2023 at 21:07
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    $\begingroup$ "they can't teach Prealgebra to 7th graders because they can't add fractions" I sincerely doubt this is a child-developmental issue. I was taught to add fractions in 3rd grade (primary/elementary, so around age 8-9) and it was considered commonplace in the classroom to be able to do so by 4th grade. I know anecdotal evidence is not what you're asking for but I'm hard-pressed to think that a 7th grader is not developmentally able to add fractions when taught properly, which if true would frame challenge the basis of your question as would mean it's being taught too late and/or badly. $\endgroup$
    – Flater
    Feb 12, 2023 at 22:30
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    $\begingroup$ [..] To put it differently, the fact that I could immediately recall that 7 times 8 is 56 when I was in middle school is precisely because I was taught multiplication tables early on in primary/elementary school. It is not proof that I should have only been taught multiplication tables in middle school. $\endgroup$
    – Flater
    Feb 12, 2023 at 22:40

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I think you have to consider tracking. The answer for the average kid is not the same as for the top or bottom ends. Exeter and Thomas Jefferson do fine accelerating their entire cohorts by a year (if not more). Conversely, we don't live in Lake Woebegone where the 100% of the kids are above average...and California has struggled with early algebra for all.

As far as "research", start with this:

https://www.maa.org/external_archive/columns/launchings/launchings_06_09.html

The referenced studies Morgan and Claric and Morgan and Ramist are good ones to look at. In essence, the takeaway from Bressuod's work:

  1. For the fraction of kids that accelerate, and get a 4 or 5 on the AP (AB or BC), they do fine in follow on classes in calculus, outperforming kids who did not accelerate. For kids with a 3, it's more of a wash (and many schools don't accept a 3 and more kids with 3 will self select out of college advanced placement).

  2. The vast majority of kids who experience calculus in high school do NOT get a 4 or 5 on the AP. And it is unclear if their acceleration mildly helps them, is a wash, or is detrimental. But arguably, it may not be worth the effort expended if no college credit ends up being earned, even if a wash. And may be really hurting the kids who are lower tier and could benefit from a year of increasing smarts (it is well known from IQ literature that we get smarter from age 12 to 18) or even maturity.

I don't have research on the amount of time spent on fractions and how it correlates to later performance. I suspect it is a hard nut to crack regardless. Upper level teachers have been complaining for decades about prior prep. But there is no magic switch. Not everyone is going to achieve high scholastic performance.

I personally sort of like the baby stats exposure. A lot of elementary school is phenomenological learning, vice axiomatic. So why not let them play with it. And "analysis" (casual meaning, not math) is creeping into business life with even secretaries playing with Excel now.

I am anti calculator in primary school math though and think more time with paper and pencil is better. The kid who needs calculator help, needs to not have it. The kid who doesn't need it, can pick it up easily later in science classes (7+) where doing calculations of inconvenient numbers. I mean, do we really think they won't know how to push the buttons, unless shown in a class? There is research in both directions on this, though and it is politicized. (But of course I'm fine with calculator use for the baby stats as the numbers can be a hassle.)

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    $\begingroup$ Lake Wobegon. . $\endgroup$ Feb 13, 2023 at 2:13
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    $\begingroup$ I agree. My spontaneous answer to the title question was "you cannot start early enough". But that is probably only true for some. Good point. $\endgroup$ Feb 13, 2023 at 8:56
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You will find two schools of thought about teaching algebra 1 in grade 8. One can be characterized by the Robert Moses "Algebra for All" philosophy. See this article by Dubinsky and Moses:

Philosophy, Math Research, Math Ed Research, K–16 Education, and the Civil Rights Movement: A Synthesis

In particular, see the section towards the end of the paper titled The Algebra Project and the Civil Rights Movement and the conclusion. In this school of thought, having all students take algebra 1 in grade 8 is most equitable. Of course this hinges on infrastructure to provide adequate foundations before grade 8, as well as infrastructure to teach algebra 1 in grade 8 to all students.

The other school of thought is that algebra 1 should "never" be taught in grade 8. Deciding who has access to algebra 1 in grade 8 and then providing this selective access is framed by institutional societal inequities, and motivated by the so called "rush to take calculus in grade 12." This was a source of debate as California revises it math framework. See for example this article:

California revises new math framework to keep backlash at bay-- Offering algebra in eighth grade remains a local choice

The work of the Dana Center Launch Years Initiative is relevant to this discussion. See

Launch Years Initiative Course Frameworks

Read at least the Executive Summary. This initiative is not without its own controversy, but I think it brings valuable ideas to the discussion.

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    $\begingroup$ But I suppose "equitable" would apply if every student failed. $\endgroup$
    – B. Goddard
    Feb 10, 2023 at 17:58
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    $\begingroup$ @B.Goddard Leave it to the Woke to find a way to inject the Civil Rights Movement into a simple question about whether 7th graders should learn Pre-algebra. The second I saw the phrase "Algebra for All" I was suspicious that this answer was Woke ("all" being a Woke watch word meaning "some"), and sure enough, "equitable" (socialism) and "access" (means of production) are right there in the next paragraph. Lastly, "brings valuable ideas to the discussion" means "advances the dialectic". We are on to your game, Marxists. See newdiscourses.com/2023/02/woke-all-means-some for more info. $\endgroup$ Feb 13, 2023 at 18:57
  • $\begingroup$ @B.Goddard In Woke, "all" (as in, "we will educate all children," e.g.) means "some". Because Woke assumes power dynamics like "systemic racism" make the existing situation unfair, the emphasis on "all" implies redistribution schemes to make up for it for "equity." $\endgroup$ Feb 13, 2023 at 19:08
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Supposing for a moment that the answer to that question is a resounding, inexorable "Yes!" - these subjects are taught a year too soon. I think it's very important to ask the next question, which is no less important, why is it too soon? and here I think it is important to distinguish two totally different options:

  1. There is something inherent in the age and intellectual development that happens up to that age, that prevents students from dealing with the subjects adequately.
  2. There is something inherently wrong in the mathematical education that the students go through before they reach that age.

Now, I have my own opinion about which one is more likely - but that's not the point. I just want to point out that it's important to approach this question in an unprejudiced way.

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  • $\begingroup$ In the answer by "guest" above, it's asserted that "it is well known from IQ literature that we get smarter from age 12 to 18." If this is true, then that pretty much puts science on the side of your option 1. In fact, the thing that might be "inherently wrong in the math education" could be the attempt to teach more and more abstract topics and ever younger and younger ages. $\endgroup$
    – B. Goddard
    Feb 11, 2023 at 19:58
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    $\begingroup$ @B.Goddard - maybe it's true and maybe not. The thing is, the primary education of math that I personally had first hand experience with, had so much wrong with it, that it's difficult for me to know what is the part that abstract thinking capabilities etc. of the students play in this. It may be that we don't know how to teach/encourage young students to think abstractly. I really don't know. $\endgroup$
    – Amit
    Feb 11, 2023 at 20:05
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    $\begingroup$ I went through during the "new math" so I also have first hand experience. I'm pretty sure we suck at teaching math. Partly because most of the people teaching math hate it. The Texas legislature has pretty much destroyed elementary math education here with their poor implementation of "no child left behind." That was the beginning of this philosophy to cover every topic in every year. Somehow they got the idea that if you introduced data handling/stats early and every year, students would find it easier. I think the opposite has happened. $\endgroup$
    – B. Goddard
    Feb 11, 2023 at 20:45
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    $\begingroup$ @B.Goddard For me the deepest problem is something that isn't math related at all. It's a problem of lack of encouragement, and intimidation. Again and again, I see young students lose their interest, and get discouraged, because they get "burned" for not understanding something "elementary". It can happen to even the most talented ones! And when that happens at a young age, it is very difficult to reverse that sentiment later. It did happen for me because I came across the correct mentors, one could say. But there's no guarantee... $\endgroup$
    – Amit
    Feb 11, 2023 at 20:52
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My (Math) student teaching hours were split between two extreme polar opposite high school classes: The AP Calculus class, and the General Math, which was very basic. The basic class was doing tax forms. The AP Calculus class was doing derivatives of tangents (or something). I was in the AP track. I never learned taxes at school. My mom showed me early. I just kept hoping those kids moms like mine. Summary: Flawed system. They need to ensure the kids have what they need before they overfill them with other concepts that yes- they may not be just yet ready for.

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A bit of a tangent from a different school system, and with no studies to back it up...

My suspicion is that by ever trying to cram more stuff in earlier, things get a lighter treatment.

My experience is not as teacher, but as parent in the German school system which switched from a 13-year education in my state to 12-years (and back to a voluntary 13-years). I myself had the 13 years, and my state school system was considered the toughest in Germany, back then (all pre-Pisa, when states had individual power over their system).

In the 12 years, it was excessively obvious that cramming more stuff in is bad. Speaking bluntly, pupils quickly learned to forget everything after a test, to make space for the next "content", to be forgotten after the next test, and so on. Teachers obviously had a very hard time stuffing their teachings into those short years. There was very bad filtering between 4th and 5th grade (where pupils choose one of three higher school types).

As far as I know, this was one of the main motivations to give schools the option to offer 13-years again, next to 12-years. We'd rather have a little older children enter the workforce or uni, than totally burned out younglings (who were not even adults when entering uni, often!).

So yes, cramming more stuff seems to be bad.

In contrast, if I recall my own time in school, I clearly remember that I never had to try hard (especially in maths); I had good marks in maths, not great, mainly because I was very lazy and basically never prepared anything at home at all. I found the stuff interesting, which is the main reason I attribute the good marks to; the secondary reason being that there always was more than enough time for everything.

Comparing that to my children's maths (in the 12-years curriculum), everything seems relatively unconnected and compressed. I find it very hard to see how someone (especially if they are not high-flyers in maths) would take anything much of interest from it, or even come along for the ride, easily.

So if I were to fight the good fight, my main focus would be to reduce the amount of mandatory "stuff", and give everything more time to sink in (maybe offer optional knowledge for high achievers). Also, there are many low-hanging fruits in the teaching itself (you don't write anything about that, but for my kids they occasionally had multiple-choice-tests in maths?! You got to be kidding me). Motivation trumps everything.

The issue now is, does switching everything one year later help with providing more space? Or are you then just concentrating more content into the latter years? Even, or especially in the second half of adolescence, pupils have other stuff on their mind than school, and even some younger 2x-year-olds are quite child-like.

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I don't have a research based answer to add, however, I have some anecdotal evidence to offer. My wife has used Singapore math curriculum for the most part with my children. This curriculum integrates algebra in stages from the earliest grade levels. In fact, such is the case with all branches of mathematics. It integrates geometry, proof techniques, symbolic algebra, trigonometry etc.. which is appropriate to the grade level each year. In this way they revisit basic algebra probably 5 to 6 years. In the end it gives students a much deeper understanding of algebra and numbers if you stick with it. Anyway, by 10th or 11th grade the student is ready for calculus. I'm currently teaching my 14 and 16 year old children second semester university Calculus. You might complain, my kids aren't typical. That is true, but what is also true is that there are other kids like my kids who ought to have the opportunity to excel in highschool mathematics which includes calculus (and more for the truly interested kids...) Is calculus best for all students, certainly not. Just like university is a place where a lot less people should go. The ubiquity of university education is the larger problem.

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This is another tangential remark, but perhaps (tangentially) relevant.

A decade or more ago I was invited to give a math talk to middle-school students in the U.S. (6th, 7th, 8th grades). I talked about Art Gallery Theorems. Understanding required the notion of a polygon, and the notion of visibility within a polygon, of course both new to them. I walked them through many slides of examples of polygons with few vertices: triangles, quadrilaterals, pentagons (already getting complicated). And interactively we figured out how many guards were sufficient, and how many necessary (one-third $n$).

I was surprised and delighted that not only could at least some of the students grasp these abstractions (the "necessary and sufficient" concept in particular is not easy to grok), but they immediately tried to work on the elementary open problem I posed (concerning "edge guards").

My point is that I was underestimating their capabilities. They knew some algebra and a little geometry, but guards in art galleries does not need much machinery. It just needs basic logic, geometric intuition, and imagination. And they could handle that, and (at least a subset) were excited to explore a question whose answer was not known (and is still not known).

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    $\begingroup$ 9th, 10th, 11th graders in the U.S. are high school students. Perhaps you mean ages 9, 10, 11 (roughly 4th, 5th, 6th grades)? $\endgroup$
    – shoover
    Feb 18, 2023 at 2:35
  • $\begingroup$ Oh, you are right. They were 6th, 7th, 8th grade students. Will correct. $\endgroup$ Feb 18, 2023 at 12:08
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As others have noted, there are two distinct hypotheses being presented. The first is that younger students are developmentally incapable of learning algebra. This is generally not supported by education research.

You may be interested in the research summarized in this chapter from Algebra in the Early Grades titled "Early Algebra Is Not The Same As Algebra Early." The authors worked with students in an ethnically diverse school in Greater Boston. They conducted weekly early algebra activities, following the students from second grade to fourth grade. Here's an excerpt that I think is relevant:

We witnessed a dramatic shift in students’ thinking over one and one-half years. At the beginning of grade three, students interpreted a story with indeterminate quantities as a single tale about particular people and amounts. By the middle of grade four most students construed this sort of situation as entailing many possible stories involving variable quantities in an invariant functional relationship. A good many of the students made use of algebraic notation to convey the variations and invariance across the stories.

The decisive changes in their thinking surprised us. Several years ago, when invited to assess the evidence regarding whether young students could ‘do algebra’, we downplayed the discontinuity between arithmetic (as generally taught in K-8) and algebra. Our initial findings showed that young students could make mathematical generalizations and express them in algebraic notation...

Our present findings have convinced us that there is indeed a large leap from thinking in terms of particular numbers and instances to thinking about functional relations. But the fact that most students throughout the United States do not make this transition easily, nor early, may well say more about our failure to offer suitable conditions for them to learn algebra as an integral part of elementary mathematics than it does about the limitations of their mental structures.

Of course, the success of early algebra programs depends on a variety of factors, and it's not a given that earlier is always better. But according to the authors of this RAND working paper who studied the effects of California's push for early algebra:

Our analyses indicate that 8th grade Algebra enrollment has a substantial and lasting positive effect on students’ advanced math course enrollment, and more modest positive effects on students’ mathematics and ELA test scores. Importantly, however, we find evidence to suggest that the effects of 8th grade Algebra vary substantially across schools and placement strategies. In some settings, early Algebra has large and statistically significant positive effects on student achievement and attainment; in other settings, these effects are negative. Descriptive analyses suggest that the effects of early Algebra placement are more positive in schools with larger shares of relatively high-achieving and socioeconomically advantaged students, as well as schools that set high test score thresholds for enrolling in 8th grade Algebra.

The second hypothesis is that students are not adequately prepared for algebra in eighth grade due to bad curriculum design choices. This is significantly harder to address and research, because students' algebra readiness depends on a host of factors besides the design of the intended curriculum. Additionally, there are very few opportunities for comparative longitudinal studies on curriculum.

Since quantitative studies are hard to come by, another way to gain insight on this question is to study the history of the Common Core State Standards for Mathematics (CCSSM.) The new standards were motivated in part by the perception that existing U.S. mathematics curricula were inadequate, especially when viewed in international comparison. R. James Milgram and Hung-Hsi Wu, both mathematicians who have had a strong presence in curriculum reform efforts, wrote this in the introduction to a proposed K-7 mathematics intervention program, prior to the release of the CCSSM:

Analysis of the results from TIMSS suggests that the U.S. school mathematics curriculum is a mile wide and an inch deep. It covers too many topics and each topic is treated superficially. By contrast, the structure of mathematics instruction in countries which outperformed the U.S. follows a strikingly different pattern. In all cases, only a few carefully selected focus topics are taught and learned to mastery by students in the early grades. At the fourth grade level, since the students in these countries have not been exposed to as broad a curriculum as U.S. students, it sometimes appears on standardized tests such as TIMSS that they perform at a comparable level to U.S. students, but by grade eight the students in the leading countries are far outperforming our students. In fact, key test items already show serious weaknesses in our fourth grade student performances. This difference becomes even greater by the end of high school, where even our top students do not match up well with the average achievement levels of students in these countries.

Wu would go on to become a staunch proponent of the CCSSM. This is what he said in a piece for the AFT journal American Educator:

the very purpose of mathematics standards (prior to the CCSMS) seems to be to establish in which grade topics are to be taught. Often, standards are then judged by how early topics are introduced; thus, getting addition and subtraction of fractions done in the fifth grade is taken as a good sign. By the same ridiculous token, if a set of standards asks that the multiplication table be memorized at the beginning of the third grade or that Algebra I be taught in the eighth grade, then it is considered to be rigorous.

The CCSMS challenge this dogma. Importantly, the CCSMS do not engage in the senseless game of acceleration—to teach every topic as early as possible—even though refusing to do so has been a source of consternation in some quarters. For example, the CCSMS do not complete all the topics of Algebra I in grade 8 because much of the time in that grade is devoted to the geometry that is needed for understanding the algebra of linear equations.

There is general support among mathematicians and mathematics educators for the idea that, in the past several decades, U.S. mathematics curriculum standards have been unfocused and shallow. In my estimation, the predominant attitude towards the CCSSM is that the K-8 standards have improved in this regard, though there have been problems with uptake and implementation. However, a group of prominent mathematics educators called for the high school standards to be revised in an open letter last year, stating:

Over the past ten years, mathematics educators have recognized that the K-8 Common Core State Standards, while not perfect and certainly in need of tweaking, are internationally benchmarked, coherently based on research-affirmed learning progressions, fewer and deeper by design, and, most importantly, achievable. In stark contrast, the high school standards are cluttered, outdated, and, because of their scope, unteachable and unachievable for many. They are not based on coherent progressions, they do not accommodate the impact of widely available technology (including computer algebra systems), nor do they articulate equally rigorous differentiated pathways that reflect the broad range of post-secondary mathematical needs. By clinging to increasingly obsolete content, they leave little room for statistics and modeling, and most problematically, they reinforce serious inequities and limit opportunities.

Based on my personal experience teaching the high school standards, I agree.

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  • $\begingroup$ Thanks for the hard work. I'll digest for awhile. $\endgroup$
    – B. Goddard
    Feb 17, 2023 at 16:23

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