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I suspect that; just as one must "do" mathematics to learn mathematics, one must have practice teaching mathematics to become a great mathematics instructor.

Still, a good book on mathematical pedagogy could only help! I am looking for book recommendations related to mathematical pedagogy.

I am generally interested in teaching math to university students, but my strongest preference is for books that reference research in math education.

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In The Teaching Gap, James Stigler and James Hiebert analyze research from the TIMSS studies. Part of their conclusion is that much of our experiences of classroom culture are deeply imprinted in us, as what school is. “The scripts for teaching in each country appear to rest on a relatively small and tacit set of core beliefs about the nature of the subject, about how students learn, and about the role that a teacher should play in the classroom.” (p.87) So it's hard to change how we teach math if we are working within the school systems.

In Mindstorms: Children, Computers, and Powerful Ideas, Seymour Papert's research shows us some alternative learning environments that could make a big difference in how we learn things like math. (I loved this book, and highly recommend it.)

In What's Math Got To Do With It?, Jo Boaler discusses her research in what she calls complex instruction. Groupwork is part of it. Also the use of what educators might call low floor, high ceiling problems - problems that are engaging, easy to get started on, and offer challenging aspects. (I wrote about some of this here.)

I can also highly recommend Mathematical Problem Solving, by Alan Schoenfeld.

I reviewed 11 articles on math education in this blog post, including two by Schoenfeld, and one by Uri Treisman, whose UC Berkeley Calculus education research is very important to consider.

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My favored text that I consider essential for "getting started" as a college math instructor is:

Krantz, Steven George. How to teach mathematics. American Mathematical Soc., 1999.

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"Building thinking classrooms" by Liljedahl is a summary of effective teaching practices distilled from Liljedahl's research. TLDR: get students working on questions which require thinking independently before introducing new techniques. Have them work standing at whiteboards. Use visible randomized grouping to create the groups. Encourage knowledge to circulate through the classroom. Summarize the outputs of student work using their work. Assess students using transparent standards. Allow students to demonstrate their mastery in a variety of ways.

"Cheating Lessons" by Lang argues that effective teaching practices are also our best defense against cheating. TLDR: give frequent low stakes assessments rather than few high stakes assessments.

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  • $\begingroup$ Thank you! I knew about some of these techniques and was looking for the research behind them. $\endgroup$
    – Sue VanHattum
    Sep 2, 2023 at 14:21
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This isn't a book, but I recommend (along with looking at the texts suggested here), to read the article by Jaime Escalante.

https://files.eric.ed.gov/fulltext/ED345942.pdf

This is not to say his words are gospel. One of the key issues in math pedagogical research is people thinking there are single proven/research solutions versus competing approaches and a murky topic.

A couple more good articles to read, by James Cargal (On Teaching in the Mathematical Sciences; The Reform Calculus Debate and the Psychology of Learning Mathematics). With the main messages being DON'T be too "hard"...DON'T assume your kids are as smart as you, or headed to R1 math grad school...and DON'T assume that you can give them a more efficient path to wisdom by doing real analysis before calculus, for example.)

https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1431&context=hmnj

https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1234&context=hmnj


Note: Education research tends to be very shoddy. E.g. low sample sizes, confounding variables, teacher/student dropout effects, non representative samples, poor metrics, failures to replicate, even political bias on top of the statistical bias. You are better off skimming such things with a very open/critical mind (hmm...maybe this might work...looks like BS...etc.) versus a "this is research, it was printed, must be right" approach. In particular, you may be BETTER off having anecdotal discussions with teachers (including those with opposing views) than looking for "the answer" in print, when it doesn't definitively exist.

Unfortunately, pedagogical research is also weak on tools for in depth observation such as ethnographic studies, detailed case histories, wide ranging interviews, etc. Probably because they are more work to do...and education schools prefer the shiny appearance of scholarship from the shite statistical-looking studies they churn out, decade after decade.

To make my point further, consider the counter to the "answer" (really an opinion, because that is how this forum especially, but really much of non hard science/math SE functions best) above advocating Thinking Classrooms by Lilejedahl. See for example this criticism of the (facade of) research backing it up:

https://pershmail.substack.com/p/the-evidence-for-building-thinking

"We could imagine a version of this book that doesn’t say anything about research at all. “Here are my thoughts on what makes for good teaching,” it would say. “It’s based on my own teaching, observations, and the experiences of the people I’ve worked with. But don’t take my word for it—try it for yourself, and you’ll see that it works wonders.” Would that be OK?

I think the answer is, that would be very OK. That’s just telling people what you think. You have to be allowed to do that.

Now, how differently should we think about this system with the research support that Peter has given us? I’d argue, not much differently. It’s not that there’s no evidence, and it’s not that he’s playing loose with the facts. It’s just that the evidence is weak. It doesn’t support big generalizations. You wouldn’t want to bet the bank on it."

See also, the criticism by Greg Ashman (paywalled, but free 7 day trial):

https://fillingthepail.substack.com/p/peter-liljedahl-wants-to-make-kids

[Excerpting the main thesis of his pro-DI and anti-IBL view:

"It was Alfred North Whitehead, a mathematician and philosopher, who made the following observation:

'It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.'

Whitehead’s view is now strongly supported by cognitive science. It is widely accepted that working memory — roughly, the thoughts we are conscious we are having — is extremely limited, capable of processing about only four items at any one time. However, these constraints fall away entirely when dealing with organised knowledge held in long-term memory. Educational psychologists call these webs of knowledge that are related to each other by meaning, ‘schemas’. We can effectively activate an entire schema and solve problems without much conscious effort, provided we have a schema in long-term memory to activate."

[None of this is to say Lilejedahl is wrong and his critics are right...go ahead and look at it...just buyer beware of one more buzzworded edfad.]

P.s. If you really want "the research", you are better off starting with a lit search of your own, to include visiting a university library and scanning/skimming the books on the shelf. Then asking more focused questions. Expecting a dispositive lit search from a forum, as if there were a librarian or a grad student here to prepare it for you, is not the right tool for the job. (Not only is it not there, but you are missing the grounding you'd get from some search yourself.) You are better off with focused questions, after some minimal self education, and with accepting and being interested in anecdotes...versus "tha litrachure".

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    $\begingroup$ Completely agree! I personally found the practices Lilejedahl advocates to be beneficial in my classroom. I am not sure I even really believe that education could be a science: it is an art. People have their own unique goals, values, and styles. There are no universal truths. It is similar to other human relationships. No research will uncover the optimal way to be a brother, a friend, or a lover. Same with being an educator. I suppose once we have readily available mind/machine interfaces it will start to be more of a science. $\endgroup$ Sep 5, 2023 at 14:03
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    $\begingroup$ Shake your manly hand, Steven. Thanks for not crushing me with your BJJ. $\endgroup$ Sep 5, 2023 at 14:31
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One of my interests is East Asian mathematics education. A very good book that succinctly describes some of its philosophies is “Principles of Mathematics Education” published by the Association of Mathematical Instruction (of Japan) and written by Kô Ginbayashi in 1984. It's a very small booklet (52 pages, 5"x7" in size).

Its emphasis is on applied mathematics (and not on pure mathematics). The basic concept used is quantity (and not number). Quantities are either discrete (magnitude of a countable set; its unit of measurement is clear at the start) or continuous (magnitude of a continuum; its unit of measurement has not been determined a priori). For example, the number of children (how many) has only one possible unit of measurement (children). The amount of water (how much) has many possible units of measurement (gallons, liters, etc.)

Addition and subtraction only apply to quantities that have the same units of measurement, and multiplication and division involve quantities that can have different units of measurement.

Some continuous quantities can be added (like area or weight); some cannot (like density or speed). For example, you add the weights of two bodies to get their total weight, but you don't add the speeds of two bodies to get their "total speed."

The concept of a decimal is not a particular case of the concept of a fraction; the concepts are different. Decimals involve smaller units that are known in advance; fractions involve smaller units that are known only after the quantity to be measured is given. For example, I can measure the height of a person using decimeters, and if there is a remainder, then I use centimeters, and so on. But if I measure the height using, say, eighths of a meter, and there is a remainder, then I need to change the units.

There are a lot more concepts in the book. I wrote about a few of them in this blog post.

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    $\begingroup$ I'm nitpicking, for clarification. "Continuous quantities like speed cannot be added: the speed of two bodies is not the sum of their speeds." I posit that two speeds can always be added; what does the speed of two bodies mean? $\qquad$ "I can measure a person's height using decimal meters, but if I want to say what fraction of a person's height is another person's height, then I need to know both heights first." I can measure a person's height using fractional meters too; and if I want to know what fraction of B's height A's height is, I can do so using either decimals or fractions. $\endgroup$
    – ryang
    Sep 4, 2023 at 12:01
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    $\begingroup$ @JRN The operations model many different situations. In particular $A + B$ be be thought of as the answer to "How many units do I have when I combine A units and B units" or "If I have A units and you have B more units than I do, how many units do you have?" or "I had some units and lost B of them. I now have A. How many did I have?". The same operation answers all of these distinct questions, and more! So while your question about speed doesn't make sense, there are others that do. "I am jogging 1 m/s. My friend is running 2 m/s faster than me. How fast is my friend running?" $\endgroup$ Sep 4, 2023 at 14:15
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    $\begingroup$ @JRN It is interesting that velocities do not actually add this way in our universe, because of relativity, though. $\endgroup$ Sep 4, 2023 at 14:19
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    $\begingroup$ @StevenGubkin, I agree with the points you make. $\endgroup$
    – JRN
    Sep 4, 2023 at 14:25
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    $\begingroup$ @JRN Thanks for editing! $\quad$ "If two speeds can always be added, then why are you asking what the sum means?" Nono, I meant to nudge you to rephrase that paragraph, since you evidently intended "the speed of two bodies" not to mean the sum of the speeds of two bodies, nor, say, their average speed, etc. $\endgroup$
    – ryang
    Sep 4, 2023 at 16:28
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I find the MAA's Instructional Practices Guide to be thought-provoking.

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Perhaps Proofs and Refutations by Lakatos might give you a good perspective.

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