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The National Mathematics Advisory Panel final report states that algebra is the gateway to higher math, to a college degree, and higher earnings from employment. It also states that success in algebra is dependent on understanding multiplicative reasoning, including proficiency with rational numbers and proportional relationships. Furthermore, there is ample evidence that difficulty with multiplicative reasoning is pervasive and failure rates in algebra classes are unacceptably high. One of the report's recommendation is:

The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percent, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected. (p. 18)

If improvement in the teaching of fractions is necessary for improvement in student achievement in algebra, then isn’t multiplicative reasoning the gatekeeper to higher math rather than algebra?

Does anyone know of any research that sheds light on this question? For example, math placements exams for entering college students could be analyzed for proficiency with multiplicative reasoning and with algebra, and related to math courses taken in college. The same could be done for 8th grade and 12th grade standardized math tests or other tests like the PISA, and related to college attendance as well as math courses taken. The difficulty, I expect, would be access to all the necessary the data for individual students.

In a blog post, krikii says “Research has shown that the extent to which elementary school students master rational numbers is a strong predictor of future success in mathematics.” There is ample research showing poor performance on rational number tasks, but what research is there that uses rational number knowledge to predict future success in math?

Another way to research this is to ask, "When do students start to dislike math and not want to take any more math courses?"

UPDATE: littleO’s comment and Tom Au’s answer got me thinking about defining more clearly what I mean by “gatekeeper to higher math.” Students are having difficulty in making the change from additive reasoning to multiplicative reasoning, and in making the change from working with numbers to working with variables. Which adjustment in thinking is a greater separator between those who continue on to higher math and those who don’t? An important research question would be, “What percent of those who are not proficient in algebra are proficient in multiplicative reasoning?”

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    $\begingroup$ For now, just a comment: You may find some interesting material in this book; check out the first couple of chapters, by Zazkis and Campbell (who also served as the editors). $\endgroup$ – Benjamin Dickman Jan 15 '15 at 22:29
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    $\begingroup$ I think the term "gatekeeper to higher math" is not sufficiently well defined for there to be a "correct" answer to this question. But I think "when do students start to dislike math" is a very clear question. I would guess it's usually at the moment they start memorizing rather than understanding math, which for many students I think is when they learn to perform algorithms for addition, subtraction, multiplication, and division in elementary school. $\endgroup$ – littleO Jan 16 '15 at 0:12
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    $\begingroup$ I wonder if difficulties with rational numbers and difficulties with algebra frequently have the same source: that students who don't realize fractions denote numbers also end up not understanding that letters are used to denote numbers. Without the key realization that mathematical symbols have meanings that determine their properties, one would be forced to resort to rote memorization of arbitrary-seeming rules. $\endgroup$ – Daniel Hast Jan 16 '15 at 2:22
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    $\begingroup$ @Benjamin - Thanks for the reference. It doesn’t answer the question but it is related, and it is interesting. I am still thinking about the distinction between integer arithmetic and rational number arithmetic in relation to quotative division and partative division. $\endgroup$ – Burt Furuta Jan 16 '15 at 9:27
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    $\begingroup$ But here's a thought on my mind at the moment, which is not well-developed and for which I cite nothing: An important step in getting to higher mathematics (roughly: more conceptual than procedural) is understanding and schematizing (mathematical) structure. Additive structure involves breaking down to sums - even a sum of all 1s. Multiplicative structure involves factoring - even a factorization of all primes, which (the FTArith) is what I think of as an early nontrivial fact ("theorem") not discussed/understood well even by many math teachers. This separator is what I might focus on... $\endgroup$ – Benjamin Dickman Jan 16 '15 at 19:49
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First, a disclaimer: I am a mathematician, and not a math educator (at least, not beyond tutoring, and teaching algebra, statistics and some calculus as a grad student); thus, my answer is going to be colored by the experience of someone who has learned a lot more math than I have taught.

The answer depends on what you mean by "higher math". If by higher math, you mean "calculus", then all I can say is that it's known that someone who isn't good at algebra is going to struggle in calculus; indeed, I have personally noticed that concepts of calculus can be pretty simple to understand, but individual problems can, algebraically speaking, get pretty scary, particularly if you make a mistake along the way. I am not familiar enough with the differences between multiplicative reasoning, and algebraic reasoning, to comment on which is more important to understanding calculus.

On the other hand, if by higher math you mean topology, abstract algebra, real and complex analysis, and so forth, then I would propose that the true gatekeeper is being able to prove things. It is a well-known observation in mathematics that a lot of students who think they love math because they are good at calculus, algebra, differential equations, and so forth, will hit a brick wall when they hit proof-heavy classes, and many don't make the transition. Indeed, occasionally someone who isn't good at equations will somehow find themselves in a position to learn higher math, and find that they are pretty good at proofs!

Now, when I say "proving things", I should point out that geometric reasoning, with its formal "proof tables", isn't sufficient, and may even hurt interest in proofs. Because I had an interest in learning everything I could, in my first year of college (and perhaps even before), I was trying to learn non-Euclidean geometry, graph theory, and number theory, and this gave me a foundation for linear and abstract algebra and other classes later on, and was certainly helpful in graduate school, where proofs are so central to everything.

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    $\begingroup$ Being well rounded in anything is generally the key to success. :) $\endgroup$ – Simply Beautiful Art Sep 26 '16 at 0:24
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This is not a true answer to the question, rather a criticism of its premises.

If by "multiplicative reasoning" you mean, at least in part, the ability to work with linear dependence between variables or numbers, then there are indications that insisting on this as much as is done during the primary education may in fact prevents students to succeed in math, and that this reflects way beyond the school and university walls.

I learned about this in the last issue of the EMS Newsletter in the article by the Education Committee of the EMS, "Solid Findings: Students’ Over-reliance on Linearity" (page 51).

In short, students are taught into assuming linearity for the sake of giving them proficiency in "multiplicative reasoning" (change of term is mine, hope I did not misinterpreted it) through "real-life" exercises, but the side effect is that they may become unable to even think about a non-linear relation between the numbers at hand.

The following very clear example is given in the above article: "Farmer Carl needs approximately 8 hours to fertilise a square pasture with a side of 200 m. How many hours would he need to fertilise a square pasture with a side of 600 m?" I let you read the article to see how much students can fail to solve this exercise.

In conclusion, of course multiplicative reasoning is not by itself bad for math, but making it important and teaching it in some (currently common) way can do a lot of harm.

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Why do kids hate maths? In my view, performance anxiety and the abstract nonsense that often passes for algebra.

Edit: these are not a main stream views as I believe curriculum change is necessary. This is intended to be thought provoking rather than "the answer" and is primarily based on personal teaching experience, and seeing alternative teaching techniques in practice.

Timed tests

There is research that has been mentioned before (I'll look for it, but if someone remembers, please add it) that timed tests common in maths education leads to anxiety which makes kids hate maths. They are also expected to have high levels of accuracy, so that it is more about avoiding mistakes than constructing an answer.

Students need to avoid mistakes at a rapid pace (often a question every 10 seconds), in an academic version of dodge ball, but a dodge ball that goes on your record and is reported to parents. If you do badly you may need to have make up academic dodge ball ordeals which are repeated until you are good enough at it. I feel stressed just imagining it.

In my teaching experience, I've seen a very mathematically gifted kid who hated maths because the teacher's measure of maths success was speed tests in times tables. Recent "back to basics" movements have just made it worse.

Algebra

As a former student of pure maths, I find the next section difficult to write.

I would suggest that rational numbers, algebra, and analytic calculus encourage a thinking model that is largely at odds with decimal numbers, algorithms, and numeric methods.

[edited] Though I often hear the sentiment that mathematics is all about proof (algebraic thinking) this is not in line with many mathematicians' and engineers' experience with mathematics.

The reason that so many people hate school maths is that it is largely dominated by algebraic thinking models that most students find unintuitive and irrelevant. In other words, to them it is just abstract nonsense. [end edit]

It is no surprise that comfort with algebraic thinking models, which have largely dominated high school curricula, is highly correlated with success in current curricula (we needed a survey to find this out?). It does not investigate whether this style of mathematics deserves its current dominant position in curricula.

Decimals are often taught as a special case of rational numbers, but I have heard arguments that suggests students would be better served by dropping rational numbers from curricula until high school algebra. I am not sure if there is research on this.

At lest schools currently teach decimals, unlike much other practical mathematics. Most mathematics curricula do not even include algorithms. It is mostly relegated to an optional subset of technology, along with cooking and fashion design.

Though numeric calculus is easier to understand and totally dominates real world applications of calculus, current curricula only briefly include it as an intro to analytic calculus.

From my own experience, replacing algebra and calculus with algorithms and numeric methods allowed a middle school student to complete much of a calculus based Physics course. This student had virtually no algebra beyond substitution (the basis of algorithms) and he used memorised heuristics for common patterns of equation rearrangement.

Multiplicative Reasoning

In my experience of teaching how to combine ratios and proportions, I have caught myself teaching formal algebraic methods to students that can do it intuitively. I eventually realised that by enforcing the formal algebraic methods, I was working against and possibly undermining students' intuition.

I personally wonder if the cause of some problems in multiplicative reasoning is caused by current teaching methods undermining intuition.

In order to fit problems into the formal algebraic language we are used to, we are forced to express problems in an unintuitive way that most students rightly react against, in an attempt to preserve their often algorithmic mathematical intuition. Similarly, generations of programmers have mostly rejected mathematically correct "functional" programming and algebraic type theory.

Papert, Resnick, and others at MIT have researched the use of algorithmic style thinking to allow students to develop abstract thinking and mathematical skills, however, it has been many years since I have researched this, so if anyone has any references, please add them. I don't think that they looked much at multiplicative reasoning or number specific skills.

As I have also been an IT teacher, I can confirm that teaching algorithmic mathematics allows students to develop mathematical skills much more quickly and without the cognitive dissonance often associated with teaching algebra. Students naturally solve mathematical problems that they may not even recognise as "maths".

Sorry I don't have links to solid research yet, but it has been a long time since I researched it, and hopefully others can fill in my blanks.

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    $\begingroup$ You write "I would suggest that rational numbers, algebra, and analytic calculus encourage a thinking model that is largely at odds with decimal numbers, algorithms, and numeric methods." I think that is utterly wrong, in particular a lot of science you call "how engineers and programmers have used practical mathematics to build the modern world" would not have been invented without algebra and analytic calculus. Perhaps what you are referring to, among others, is a tendency of people who learned analytic calculus to dislike the "dirty" numerical solutions. Cont. $\endgroup$ – dtldarek Jan 16 '15 at 14:21
  • $\begingroup$ Cont. However, that is not the fault of rational numbers, but rather our attitude, how we had learned them or how we were taught. For example, there are tasks "prove this no-variable inequality". Ordinary a simple use of calculator would be sufficient, but that is often considered not an acceptable solution. Why not? An appropriate approximation with error-bounds check is a proper proof. The fault is with us (the teachers, the students), not in the tools themselves (analytic calculus doesn't forbid you performing numerical computation). Fin. $\endgroup$ – dtldarek Jan 16 '15 at 14:22
  • $\begingroup$ @dtldarek I largely agree that it is the fault of us as teachers. However, sometimes a language encourages modes of thought. The mode of thinking encouraged by algebraic language is simply different to algorithmic thinking, and is unfortunately often at odds with students' mathematical intuition. It is amazingly beautiful and, as you note, has important applications. But those are not the majority of applications and there are now very powerful alternatives that are also more intuitive for students. Algebra is currently the gatekeeper for maths, but maybe it shouldn't be. $\endgroup$ – Richard Jan 16 '15 at 15:12
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    $\begingroup$ The supposed distinction between "algebra" and "algorithms" is hard to parse. E.g., how is an algorithm described but "by algebra", in addition to numerical examples? A description of "iteration" is "by algebra". The often-rumored schism between allegedly proof-obsessed mathematicians and results-oriented scientists is almost entirely a "pop" concept, rather than substantiated among mathematicians and scientists. What is more often manifest is a non-mathematician's interpretation of "what math is/ought to be", which is most often a severe caricature... but possibly foisted upon students. $\endgroup$ – paul garrett Jan 16 '15 at 20:41
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    $\begingroup$ Richard, thanks for your concern... It's not so much "offensive", but perhaps missing a certain mark. That is, "school math" is almost universally (to my observation, and given other accounts) disconnected from genuine mathematics, and also from the applications and general utility of mathematics. Perhaps this is inevitable, but/and the difficulties students find should not be attributed to the nature of the thing itself, but to the stylized, often-benighted, often-zealous-for-rules version of "mathematics" they are subjected to... and the immediate irrelevance of that ... [cont'd] $\endgroup$ – paul garrett Jan 16 '15 at 23:49
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I would say that algebra is the gateway to higher "pure" math, while multiplicative reasoning is more like the gateway to higher applied math.

Multiplicative reasoning is about algorithms and numerical methods that "engineers" use to solve problems on a day-to-day basis. In that regard, it is the "gateway" for these kinds of workers.

But algebra is the gateway to higher pure math, precisely because it forces students to stop thinking in terms of numbers, and start thinking in terms of symbols. These "symbols" are found more and more at progressively higher levels of pure math, which gives algebra its importance.

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  • $\begingroup$ You make an interesting link between multiplicative reasoning and applied math. Is it related at all to the engineer's former ubiquitous use of slide rules? They are just about multiplicative reasoning in physical form! $\endgroup$ – Richard Jan 16 '15 at 16:13
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    $\begingroup$ @Tom Au – You make a good point in that algebra requires thinking in terms of variables not just numbers. However, higher math uses multiplicative reasoning as well, and it is that change from additive reasoning to multiplicative reasoning that many students are not getting. So is the change from additive to multiplicative reasoning the main separator between those who go on to higher math and those who don’t, or is it the change from numbers to variables? $\endgroup$ – Burt Furuta Jan 16 '15 at 18:00
  • $\begingroup$ @BurtFuruta great question. May I comment that in my experience there is also a difference between the shift to variables as unknowns (which I don't think presents difficulties) and manipulating a variable as a function or a pure mathematical object (proper algebra). The introduction of variables with exponents (advanced multiplicative reasoning) often happens at the same time as the second shift - it would be fascinating to untangle the changes. $\endgroup$ – Richard Jan 16 '15 at 19:40

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