What do we know about the science of teaching math?

[mathmarm]

11 hours ago, wendyroo said:

Then I guess I'm not clear on what you mean by linear progression.

Obviously, kids can't learn 4 digit addition with regrouping until they have learned 3 digit addition with regrouping. But I would really count those as the same skill (despite many curriculum holding out for a long time before teaching the 4 digit version like it is a much more difficult skill), and argue that if a child has truly mastered 3 digit addition with regrouping then they are fully capable of any number of digits.

I guess I don't see a lot of strict prerequisites in arithmetic. I do more so in upper level math - it is hard to factor polynomials if you don't understand exponents (though I think a large percentage of students try to do just that by rotely following algorithms) - but elementary math feels more to me like a web of complexly interconnected skills rather than neat skill lists that need to be learned in order.

I will disagree with you on the bolded. This is a matter of teaching sequence and the amount of "microconcepts" and "microskills" taught and mastered before hand. A student that has mastered the appropriate Place-value  "microskills" of combining mathfacts and regrouping skills will not need to learn to add/subtract 2-digit, then 3-digit, then 4-digit, etc problems. Instead, they can apply their understanding of "microconcepts" and "microskills" to calculating any amount of digits on the same day that they learn the vertical addition/subtraction format type.

 

[maize]

9 hours ago, UHP said:

Did you make much math progress as an adult, or does this still trouble you?

I avoided math in college (majored in anthropology). I have filled in gaps going through math with my kids. I took a college algebra class a few years ago and am currently taking calculus for the first time. 

[wendyroo]

8 hours ago, UHP said:

If a kid has mastered the prerequisites, she can learn one skill a day in a five-minute session, and practice the previous skills she's learned over another five minutes, and thereby obtain "expertise with negative numbers" over the course of a little more than two hours spread out over a little less than three school weeks. (Of course if she doesn't put that expertise to use right away, she is likely to forget... this is part of my objection to the "spiral curriculum" that I might get on a high horse about later.)

I think this mindset seems foreign to me because this pedagogy and these goals are so far from mine.

I expect "expertise with negative numbers" to take years, starting with a light introduction when kids are ~5 and slowly growing and solidifying through their study of algebra and beyond.

Our local public schools definitely take a "cram it into three weeks" approach, and I see nothing to recommend it. They don't introduce negative numbers at all until 6th grade. In my experience that is about 5 years of missed opportunities when the kids could have been building their mental model through real life experiences. They do put the skills to use right away, but only for a short while, only on a rote algorithmic way, and only at a simple, surface level. They do have a separate unit of "problem solving" each year, but in the rest of the units they progress just as you outlined, and spend almost no time asking challenging questions about the new topic that would force the students to truly understand it.

My goal, with all math concepts, it to introduce the topic early and often to the point that when it comes time to formalize it symbolically or algorithmically, the child already has a very intuitive understanding of how and why the numbers work that way. For example, with my 7 year old, just as I have with all of my kids, I very consciously ask questions, discuss problems, and introduce terminology that is preparing her for long division, for multiplying fractions, for finding volumes of solid shapes, etc. At this point it is just very basic conceptual ideas - you could split the M+Ms between your siblings by first splitting the reds, then the oranges, etc; if the recipe needs 1/3 cup oil, how much will we need if we triple it; if lines are measured in cm and areas of squares are measured in cm^2, what might we measure the volume of this box in? - but I firmly believe that this immersion approach will lead to stronger math students in the long run than waiting until they can tackle the whole idea at once and covering it as quickly as possible.

[UHP]

24 minutes ago, wendyroo said:

I think this mindset seems foreign to me because this pedagogy and these goals are so far from mine.

...

My goal, with all math concepts, it to introduce the topic early and often to the point that when it comes time to formalize it symbolically or algorithmically, the child already has a very intuitive understanding of how and why the numbers work that way.

It doesn't sound like you're describing a goal, it sounds like you're describing a process. I also have an aversion to "goals", especially long-term goals, in education or in parenting: I'm afraid of getting my hopes up. I would like the time I spend with my daughter to be beautiful and rewarding now, not to put lots of unpleasant pressure on her in pursuit of an exalted goal. I think of myself as executing a process, not pursuing a goal.

But part of that process is "spend n minutes per day in math lessons." We've been doing it for close to a thousand days and those minutes add up. How can she get the most out of them? To me, it seems like an urgent question. Does some portion of those minutes have to be devoted to negative numbers, for 5 years in advance of going through my 13 bullet points? My experience is no.

(To be fair, I did spend a week or so almost two years ago introducing negative numbers to my 6-year-old with hand-made manipulatives, inspired by @Not_a_Number's "poker chips." It was fun and she remembers something about it. Perhaps in a subtle way this was fundamental for what she learned since. But I was perplexed that, doing as much math as we do, we hardly ever discussed negative numbers in the intervening two years, not until a few weeks ago when Connecting Math Concepts got around to them.)

1 hour ago, wendyroo said:

They do put the skills to use right away, but only for a short while, only on a rote algorithmic way, and only at a simple, surface level.

I don't want to see the skills "put to use" in exercises that only keep the concept from sliding too quickly out of memory. I want to see them put to use (asap) in building something up, in teaching new concepts. My somewhat uncertain sense right now is that there's only one major use of negative numbers in this kind of building up: they allow for a much richer source of examples when introducing coordinate systems and graphs of functions. (That's something my daughter wasn't ready for two years ago, but is now.)

[daijobu]

I stumbled on this article about Australian math education, and I see a lot of similarity with US education.  The big difference is they seem to have systems that operate province-wide, so one uninformed bureaucrat can have a big impact.  The US system is highly fragmented, so you never know what you are going to get.  

Some excerpts:

One of its fundamental failings is the treatment of algebra, by which measure the new curriculum falls short by a country mile.

Algebra is the beating heart of mathematics. It is the naming of the quantity being hunted, setting the stage for its capture. It is how we signify pattern and how we express the relationship between quantities. You want to understand something else, probability or statistics or geometry? Algebra is essential. Algebra is how Descartes captured geometry, and how Newton and Leibniz captured calculus.

A fundamental insight is that algebra is simply arithmetic, just with numbers we don’t know. The current curriculum states this clearly, in a Year 7 instruction to ‘Extend and apply the laws and properties of arithmetic to algebraic terms and expressions’. By contrast, ACARA’s new curriculum instructs Year 7s to ‘generate tables of values from visually growing patterns or the rule of a function; describe and plot these relationships on the Cartesian plane’.

This is pointless, and it is not algebra, except in the most trivial sense. It is substituting data entry and graphical busywork for the critical practice of algebraic skills. The curriculum is choked with such nonsense, strangling the few stray instructions that might otherwise engender proper study. The term ‘algebra’ and the critical discipline of algebra have been distorted to meaninglessness.

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There is, to recoin a phrase, no royal road to algebra. A mastery of algebra requires practice and memorisation and struggle. And preparation. Prior to the arithmetic of numbers we don’t know comes the arithmetic of numbers we do. In order to understand a/b = c/d, we must first understand 4/6 = 6/9. This is the work of primary school.

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The treatment of mental skills is little better. Automatic recall of the multiplication tables is critical to all the arithmetic that follows: a student cannot apply 4 x 9 if they are struggling simultaneously to remember it or calculate it. The multiplication tables were mangled in the draft curriculum and then fixed, but only up to a point, where they are referred to as ‘multiplication facts’.9 This is weird and telling. Much worse, the tables (still) only reach 10 x 10, rather than the traditional 12 x 12. This matters. We have 60 minutes in an hour and 360 degrees in a circle for a reason. Having natural multiples and factors and fractions readily at hand is critical for the learning of arithmetic.

The ignorant decision to exclude 12 turns out to not matter, however, since few students even get to 10. The large majority of primary school students do not learn their multiplication tables. This is because, clear curriculum direction notwithstanding, the very large majority of primary school teachers do not consider the tables important, much less mandatory. Teachers have been fed other ideas.

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ACARA’s new curriculum is a large but entirely predictable step down. Modern education is steeped in grandiose perversion, with innovative but misconceived practices working not to improve but to undermine fundamental processes of understanding. There is Higher Order Learning, and 21st Century Skills, and Flipped Classrooms, and Child-Centred Learning, and Discovery Learning, and Inquiry Learning, and Play Based Learning, and on and on and on. Underlying almost all of this is the philosophy of constructivism.

Built upon the tautology that we only understand what we understand, constructivism claims that students must construct this understanding through their own experience. Constructivist approaches are to be contrasted with the boring old practice of directly teaching students, and in particular the teaching of clear facts and skills. Always lurking is the boogieman of Rote Learning. This boogieman in particular has frightened teachers away from orderly ‘tables’ and into embracing isolated ‘facts’.

There is a proper and important role for mathematical exploration, but that role in primary education is limited. It took thousands of years for civilisation to come up with the crystal concepts and truths and techniques of ‘elementary’ mathematics. Constructivism is the slowest and most painful and least successful method of mastering these fundamentals.

Not everyone, however, sees this as a drawback. A teacher focused on students’ Higher Order Thinking may have little concern for the lowly basics. Mistakenly. Rather than basic facts and skills being opposed to deeper thinking, the basics are the foundation for deeper thinking. Before twenty-first century skills, whatever these might be, there must come a mastery of seventeenth-century skills.

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Hand in hand with constructivism goes problem-solving, one of the great con jobs of mathematics education. Mathematicians love the idea of problem-solving, since it is a one-word definition of what they do. Mathematicians love to see children working on authentic, well-structured problems, with clear mathematical content and purpose. But school problems are different. School ‘problems’ are typically poorly defined, open-ended explorations with no measure of or concern for success, and with students ill-prepared for a venture of any significance. These problems are all the worse for almost invariably being about something other than mathematics, about a largely fictitious Real World. Thus, rather than a carefully crafted problem about prime numbers and factors, students ‘explore’ or ‘model’ the painting of walls and the graphing of mortgage rates. Students are presented with pseudo-problems that require little thought and inspire less.

ACARA and the education authorities regard the Real World as a great selling point. Hence the new curriculum has a stream on ‘Space’ rather than geometry. Hence NAPLAN has a test on ‘Numeracy’ rather than arithmetic. Hence the mandating of statistics way beyond its very limited pedagogical worth. Hence the unceasing focus on STEM, which reduces mathematics to an instrumentalist, utilitarian skill set. This all fits in well enough with the neoliberal notion of education as training, but it is otherwise reductive, and it erodes the basis for mathematical thinking. Real Worldness in school is almost invariably contrived, and thus as boring as dirt, because we simply need very little mathematics for our everyday lives.

Such Real Worlding also feeds back to devalue and to poison the teaching of mathematics. The critical point of mathematics, the source of its incredible power, is that it removes the distracting noise of the world. Mathematics abstracts from messy reality to create something much simpler, something that can be analysed and honed and generalised. And yes, mathematics then gives back, providing indispensable tools for the understanding of real-world phenomena. But the mastery of these tools is beyond the scope of school mathematics for any but the most banal of real-world situations. What results is simply the glorification of noise—the presentation of noise as the central topic of mathematics education. It is absurd, and disastrous.

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Mathematics education wastes untold time and energy and goodwill on electronic media: students watch videos instead of reading; they ‘move’ shapes on screens instead of shifting physical blocks; they push calculator buttons instead of computing on paper; they ‘prove’ statements by pressing Solve or Graph on their handheld computers.

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Plenty are fooled by constant references to ‘visual learners’ and ‘digital natives’, but there is no fooling reality. The perverting effect of these media is that students are not required to think or to reflect. They need never pay proper attention, to a teacher or even to their own thoughts. The electronic media stimulate and entertain, occupying the space where contemplation might have occurred. In his 1986 book Amusing Ourselves to Death, Neil Postman wrote on the effect of television on education: ‘The name we may properly give to an education without prerequisites, perplexity and exposition is entertainment’. That is where we are, except that now television, and much worse, are in and are intrinsic to the classroom.

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THE PAST IS A FOREIGN COUNTRY
And the name of that country is Singapore. For instance.

Asian countries dominate TIMSS, the international test of school mathematics. Asia even dominates PISA, the anti-algebraic non-test created specifically so that Western countries would feel better about themselves. Even if one wishes to concentrate upon ACARA’s snake-oil games, it turns out that attention to the basics is the way to go.

Australia once did much better, although that was long ago and is largely forgotten. The powerful forces of entertainmenting have been at work for many decades. Moreover, there are two intertwined forces, one political and one philosophical, which have directly perverted mathematics education.

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First, the political. Historically, for good and bad, Australian mathematics education was carefully controlled by education bureaucrats under the guidance of mathematicians.15 In the 1970s, teachers started to be given more autonomy.16 Also around that time, a fourth group, of education academics, was beginning to emerge as a force.17 Since then, and with varying overlaps and alliances, these four groups have tussled over the nature and control of mathematics education.18 All-out war broke out in the early 1990s, with the bureaucrats attempting to wrest control from the other three groups.19 The bureaucrats failed, but the downward slide was well underway. The power of education academics has continued to grow, and they are now much more closely wedded to the bureaucrats.

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With mathematics education academics now in possession of their own world, they are generally much less connected than they once were to the world of mathematics; they are less adept at and less interested in it. This lack of mathematical expertise encourages and necessitates an emphasis on other, non-mathematical concerns, laying the fertile ground for constructivist obfuscation. Much more time is spent in apologising for and avoiding the difficulty of mathematics than is ever spent addressing that difficulty, or in demonstrating the beauty and the power that can result from proper effort.

Edited January 2, 2023 by daijobu

[EmilyGF]

On 1/1/2023 at 8:28 AM, mathmarm said:

I will disagree with you on the bolded. This is a matter of teaching sequence and the amount of "microconcepts" and "microskills" taught and mastered before hand. A student that has mastered the appropriate Place-value  "microskills" of combining mathfacts and regrouping skills will not need to learn to add/subtract 2-digit, then 3-digit, then 4-digit, etc problems. Instead, they can apply their understanding of "microconcepts" and "microskills" to calculating any amount of digits on the same day that they learn the vertical addition/subtraction format type.

 

Agreeing with @mathmarm. RightStart teaches 4-digit addition with regrouping on the abacus in one step, skipping fewer digits. I just did it last week with dd6. It made sense to her rather quickly; she went from abacus to paper to "this is easy, why are we still doing this" in four 20-minute sessions.

Wrt Science of Math, I don't think there has been the same breakthrough as with reading. Reading dealt with the question of whether good readers sound out words or not. fMRIs showed that good readers used phonics and that perceptions that they did not (which is what all the Marie Clay stuff was built on) were wrong. I don't think there is a puzzle piece for math that big that science has uncovered. If I am wrong, let me know! 🙂

I came across this website: The Science of Math that has some short, research-based rebuttals to some math questions, but it isn't as deep as I'd like.

ETA: I work coordinating a study through psych lab currently that does research about young kids and mathematical interventions. Research is messy. I wish there was something as clean as an fMRI that could show what is really going on with some key part of math.

Emily

 

Edited January 3, 2023 by EmilyGF

[daijobu]

On 1/3/2023 at 5:05 AM, EmilyGF said:

I came across this website: The Science of Math that has some short, research-based rebuttals to some math questions, but it isn't as deep as I'd like.

 

 

Thank you for linking this; I hadn't heard of it before.  I was able to find a few podcast interviews, and I'm learning that the miseducation we see with reading is also happening in math.  

[Malam]

On 1/3/2023 at 8:05 AM, EmilyGF said:

RightStart teaches 4-digit addition with regrouping on the abacus in one step, skipping fewer digits

Which level is this at?

[EmilyGF]

5 hours ago, Malam said:

Which level is this at?

Level B, which is first grade.

[birchbark]

I just noticed this thread, and I wanted to post a video that addresses the OP's question as it regards math and the phonics analogy. This was made by the creator of Systematic Mathematics, who has since passed. (His business shut down but the site is still up if you want to poke around for more of his philosophy.) His point was that the issue isn't procedural vs. conceptual (he taught both), but a spiral vs. a systematic approach. I would describe it as more "mastery," but with an incremental, logical progression. 

 

 

Most of the vintage math books (pre-1965) will teach math in this way. Strayer Upton is one that is still in print.

[Malam]

How The Brain Learns Mathematics by David Sousa

[Helpdesk]

It doesn't really go to the OP's question--it's not research--but this article by Susan Wise Bauer might be helpful in understanding conceptual vs procedurals. It's part of what Well-Trained Mind is addressing in the Math With Confidence series by Kate Snow.  FAQ here.