Apart from, of course, me.

Prompted by the What Should We Write About post, and in particular by a comment from newcomer Mr. Texas, it seems worthwhile setting up a few “resources” post. This is the first such post: who, or what, should people read on mathematics education?

People are free to suggest whatever they wish, however they wish, although anyone suggesting a podcast will be summarily shot: “Read” means read. A writer might be suggested generally, or just for a specific article. The suggestions can broaden into education more generally, and even further. Perhaps more focussed suggestions may be more helpful, but we’ll see what people throw up. Monographs and paywalled suggestions are also worthwhile: sometimes there are ways …

Note that the suggested writings should be *about* mathematics education, rather than textbooks and expositions of mathematics and the like. Of course there is not a clear dividing line, but I plan to publish a separate, similar post for teaching resources.

As people comment, I’ll update the post, with the suggestions and links somewhat cleaned and sorted. (I also plan to post an organising front page to the blog in the very near future, so that posts such as this don’t get lost in the blog jungle.)

********************************************************

**UPDATE (12/02/23)**

Thanks to everyone for their suggestions, and I’m sorry to be so slow to compile them. Here they are, including a suggestion of my own. Any links are to free or substantially free versions/sites, unless otherwise indicated. (Non-linked books can be located and purchased easily enough.)

Although some of the references are standards and/or appear to be very good, I’m simply listing pretty much all that was suggested (as long as it was more or less on some aspect of maths ed). Their value has to be determined by the reader (and can always be discussed further below). Of course, further suggestions are always welcome.

**Tony Gardiner **

Gardiner, who occasionally graces this blog with his presence, has great books of problems (including this free one, co-authored with Alexandr Borovik and discussed here). Two of Gardiner’s works specifically on mathematics education are:

***) ***‘Problem-solving’? Or Problem solving?*

***) ***Teaching Mathematics at Secondary Level *

**George Pólya**

Polya is a legend, of course, which means most current maths teachers have never heard of him.

***) ***Mathematics and Plausible Reasoning, Volume 1 and Volume 2*

**Greg Ashman **

***) ***Filling the Pail *(semi-free substack) and *Old Filling the Pail *(blog prior to the substack)

***) ***The Truth About Teaching*

***) ***A Little Guide for Teachers: Cognitive Load Theory*

**Craig Barton **

***) ***How I wish I’d Taught Maths*

**Hung-Hsi Wu **

Wu is another legend, with a home page full of great stuff. For instance,

***) ***Learnable and Unlearnable Mathematics*

***)** *Teaching Fractions According to the Common Core Standard*

***) ***The Rôle of Open-Ended Problems in Mathematics Education*

***) ***Pre-Algebra*

**Alexander Zvonkin**

**Felix Klein **

***) ***Elementary Mathematics From an Advanced Standpoint*

**Barry Garelick**

***) ***Traditional Math *(substack)

### Fan Lianghuo (editor)

***) ***How Chinese Learn Mathematics: Perspectives From Insiders*** **

**Dylan Wiliam**

Wiliam has a website with many articles and presentations on assessment.

**Daniel Willingham**

Willingham has a website (many broken links) with articles on thinking and memory.

**Brian Butterworth**

***) ***Dyscalculia: From Science to Education*

***) **(with Dorian Yeo) *Dyscalculia Guidance*

**Stanislas Deheane **

***) ***The Number Sense *(10/04/23)

**Jaime Escalante **

***) ***The Jaime Escalante Math Program *(10/04/23)

**James Cargal **

***) ***The Reform Calculus Debate *(10/04/23)

***) ***On Teaching in the Mathematical Sciences *(10/04/23)

***) ***On Learning in the Mathematical Sciences *(10/04/23)

**Alexander Renkl**

*) *Learning from worked examples in mathematics* (12/08/23)

**Paul Kirschner**

Mr. Big of cognitive load theory.

*) *There is more variation within than across domains* (12/08/23)

**John Mighton**

The JUMP math guy.

*) *All Things Being Equal: Why Math Is the Key to a Better World*

**Jean Schmittau**

**) Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy *(12/08/23)

**Viktor Blåsjö**

Blåsjö has a very extensive website called Intellectual Mathematics.

Highly recommended:

Filling the Pail – Greg Ashman.

In fact, I was re-reading one of his older blog posts just today:

https://fillingthepail.substack.com/p/quality-teaching-rounds-is-not-the

Yeah, GA was always gonna come up. If only he didn’t agree to appear on manels.

* panels.

If not Greg, then who …? I’d rather have Greg than the usual suspects. The wider the audience that hears him, the better. The pity is that what he says is against a backdrop of white noise and static from the muppet majority on these panels.

I wrote “manels” and I meant manels. And of course I was being sarcastic.

I struggle at times with Greg getting all the very limited airspace for the intelligent opposition. It queers the message. But there is no doubt that Greg bats for the good guys and bats well.

I agree that it is annoying that I get so much of the limited airspace but this is due to a number of factors:

1. Most of what is written about education is nonsense. Some business leader will write about how they have never needed to know the surface area of a sphere as if this is some great insight or a journalist will breathlessly report on a new attempt to engage [insert marginalised group] in maths education by getting them to meditate, patronising them with a show performed by failed actors or sending them on a high ropes course.

2. Most sensible maths teachers are unable to speak about maths education – even if they want to – because they will fall afoul of department media and social media policies. I am able to do so because I work for an independent school.

3. Most education commentary is not about questioning assumptions, it is about fitting-in with the established narrative. This causes those with questions to hesitate to engage.

I would love more people to fight this fight. I want to know when they do so I can amplify the message. A crucial insight is that the airspace is not fixed. When Marty and I worked on our group letter together, we created space for a flurry of articles about maths education. We can do that again.

I am particularly concerned about the fact that I focus mainly on teaching methods. While important, we need more informed commentaries on curriculum. I am just a physicist and not a proper mathematician so I am lacking here.

Thanks, Greg. Of course I wasn’t blaming you for monopolising the intelligent air space. You are known, you get asked and you respond. That’s the way it works. But, without really disagreeing with your 1-3, I think there is more to it and less to it than that.

In brief, most education (and other) reporters and editors are pretty bad. They are harried and have no time to breathe, but they are also not nearly as smart or as knowledgable as they imagine. The reporters are another version of your 3, where they are simply unwilling to question the educational system at a sufficiently foundational level. Which is why they will happily act as stenographers for ACARA and whichever institutional clown will hand them an easy story; who cares if the story is fundamentally nonsense? Of course Urban was different, and Carey is better than most, but he’s not great, and Bita and Baker can be appalling: they are either willing to knowingly write utter nonsense or they simply don’t have the ability to determine what is or is not nonsense.

You are evidently friends of, or at least friendly with, a number of these reporters. So, they go to you for a quote, and you give them good quotes. And they have their list of other go-to people, most of whom are dumber than rocks. So, it’s the same stuff over and over and over again: you say the same smart stuff and the same three clowns say the same stupid stuff. It’s all lazy and soporifically predictable.

But it is not just the reporters, and your last two paragraphs are key.

Of coursewe need more people to be outspoken, and in particular we need prominent mathematicians to be outspoken, to proactively raise the roof. And they don’t. In the past it would have been Tony Guttmann and Garth Gaudry, but Tony is a hundred years old (approx), and Garth is dead. The current leaders at AMSI and AustMS and AMT and AAS are simply unwilling to incite a revolution, or are actively undermining any such attempts.There are a few books that I’ve found useful, so I’ll tentatively suggest them.

* Craig Barton (2018) How I Wish I’d Taught Maths

(I like the bit about sequencing questions and minimally different examples.)

* G. Polya (1945): How to Solve It. I think it’s worth reading the first chapter about asking questions that help students learn.

* Hung-Hsi Wu (https://math.berkeley.edu/~wu/) has written several books about the mathematics that teachers should know. I haven’t finished reading any of them, but they’re on my list.

Polya should be compulsory reading in schools.

Just

How to Solve it, or also other works?is certainly a classic. I also liked . Not sure how available that one is though.

Thanks, wst. Obviously Polya and Wu were gonna be mentioned by someone. (The last time I looked at

HTSI, I wasn’t so impressed, but perhaps I was drunk.) I don’t know Barton but I’ll look.HTSI needs time to digest and really think about what Polya is saying. The headings of “read”, “plan”, “do”, “reflect” are nothing special in and of themselves, but when you drill down into it, there is some powerful stuff there that a lot of really good teachers probably do without thinking; good students likely do the same.

The power of Polya’s HTSI is in helping the not-quite-so-brilliant problem solvers get to a better standard.

Incredible book

Thanks, Dr. M. It looks very interesting.

Very interesting book. I am pretty sure any math educator would enjoy it.

Marty, also, there is a pdf out there.

Yes, I know.

At present I am reading Felix Klein (1908/1945). “Elementary mathematics from an advanced standpoint”. It deals with how mathematics should be presented in secondary schools.

I bought a new copy for a modest price, but there is a pdf version out there.

If a university mathematics department wanted students to think about teaching, this would be a useful starting point.

Thanks, Terry. You’ve mentioned it before. Impressive book.

Ministers of Education, or members of their staff, should read this blog which is a free source of interesting ideas.

Kind of you to say it, and weird of you to say it.

It’s got no more garbage than anything the fools they’re listening to are pedelling. And one might argue it’s got a helluva lot less.

I’m not arguing against the merits of the blog, which are, um, obvious. I’m arguing against the idea that institutionalised clowns would give such a blog any proper consideration.

Yeah, you need to change your name to Eddie Woo.

(It’s revealing that the institutionalised clowns keep listening to the same fools that caused the problems in Australian education. Things will obviously keep getting worse until that changes. Even then, there’s decades of damage to undo, and that doesn’t happen in a couple of years).

Smilin’ Marty’s Friendly Maths Blog.

Uh huh.

(Smilin’ and Friendly might be works in progress …)

Barry Garelick

https://barrygarelick.substack.com/

Thanks, SRK. His name has arisen at times, but i haven’t read much.

Please can anyone recommend any useful maths specific readings about cultures of thinking and/or thinking routines? I’ve searched and searched. I could only stomach one chapter of Ron Ritchhart’s book…

Here are four better readings for consideration. I checked your rules and I think biographies and novels are okay.

The crest of the peacock – non European roots of mathematics by GG Joseph. My fav dip Ed lecturer recommended this classic to me and I found it fascinating and relevant to becoming a better teacher: https://press.princeton.edu/books/paperback/9780691135267/the-crest-of-the-peacock

About the size of it – the common sense approach to measuring things by W Cairns. The first lines are: “This is a serious book about weights and measures. You can take that as a warning, or else you can take it as an invitation.”

In code, by S Flanagan. This is a biography about an award winning female code breaker. I’ve recommended to students. https://www.goodreads.com/en/book/show/653291.In_Code

October sky by HH Hickam. An inspirational true story about a group of school kids who joined the space race and had to learn and apply kinematics. I’ve recommended this to good students from year 7 upwards.

Hi, Tilbot. I’m not sure I can help. I don’t know what “cultures of thinking” means. (And I will avoid finding out what a Ron Ritchie might be.)

With your recommendations, I’m not sure whether they belong here, or on the “teaching resources” post to come, but they belong somewhere. I don’t like

Crest. The other three I’ll check out.Please can anyone recommend any useful maths specific readings about cultures of thinking and/or thinking routines?

Cultural foundations of learning: East and West by Jin Li.

Thanks very much, Anonymous. The book looks interesting, although I wonder how much practical it offers to teachers. The main message for maths teaching appears to be that Chinese students practice a hell of a lot more than Western students, which will come as a great shock to absolutely no one. Jin Li’s book also refers much to and reminded me of another book that looks interesting:

How Chinese Learn Mathematics: Perspectives from Insiders.Not strictly maths, but…

Among the readings I’ve done in my courses, an author that stood out is Dylan Wiliam. He was probably the most sensible of anything I had to read. I think his main bag is formative assessment, but his background is maths.

E.g. http://www.cms.sd23.bc.ca/resources/assessmentandreporting/Documents/Assessment%20-%20The%20Bridge%20Between%20Teaching%20and%20Learning.pdf

Another name I’ve come across is Daniel Willingham, a cognitive scientist, who has written on critical thinking.

E.g. https://people.bath.ac.uk/edspd/Weblinks/MA_ULL/Resources/Learning%20to%20Learn/Willingham%202008%20AEPR.pdf

Probably not new names to most readers of this blog, but I think worth pointing out.

Thanks very much, Anita. Definitely worth pointing them out. May I ask, what did you learn from these guys?

What did I learn.. hmm…

What I’ve found is that a lot of the stuff written on education (and maybe anything, really) seems to get very complicated, very messy, and very hard to make sense of. It’s as if most people, most of the time, simply regurgitate the stuff they heard other people say, and very rarely think clearly about what they’re saying. When I have on occasion found an author who seems to be able to sweep away the guff* and drill down to the core of the topic, it’s like a breath of fresh air. It helps me make sense of the topic and recognise the guff for what it is.

In the case of Dylan Wiliam, the topic was assessment. He made it clear that there is only one purpose of assessment: to find out where a student is at. There might be more to talk about, in terms of what you do with that information, and how to go about getting the information. But any other purpose put to it is guff.

Then I read more of what he had to say and he seemed to consistently be pretty sensible.

In the case of Daniel Willingham, it was about critical thinking, which seems to be hard to define and harder to find sensible stuff written about. I’m not sure if he did manage to define it, but a lot of things he said made sense. The main point I got was that critical thinking relies heavily on domain knowledge and experience, that it’s not a set of skills that can be taught independently of a context. So anyone who says students should be taught “critical thinking skills” is talking guff.

* Guff is a great word, by the way, introduced to me by my husband. Not sure if it’s in the dictionary.

So, from Wiliam you learned that the purpose of assessment is to assess, and from Willingham you learned that thinking is difficult. Rushing to the bookstore as we speak …

Well, no. I was finding it hard to answer your exact question, so I dodged. I might be able to do a better job now. What I learnt was:

* That there are actually papers out there that are readable and sensible

* How to think about these topics in a way that helps me make sense of what I read

* Maybe more importantly, how to recognise the guff in these topics

You might argue I should have known these last two already. But if you read enough crap, your brain can end up a quagmire. I’m not great at sorting the quag out for myself, I find it helps a lot to read something that clarifies a topic. Even if there’s no single new piece of information in it.

Not to say there weren’t new pieces of information in these papers, but it was a few years ago now, and what I wrote above is what stuck.

Yeah, ok, i was being flippant. Sort of. It still seems to me to that the worth of such papers/authors is mostly a reminder to not be dumb. I think I have a post worth writing on this. Will try to do soon.

You’re not wrong, but they can also provide a good and clear way of thinking about something to those of us who are not so good at coming up with one on our own.

And I think the reminder itself is necessary sometimes. “First, don’t be dumb.”

It requires a book?

I think the value of Willingham’s work is that he uses his background in and knowledge of cognitive psychology to address various common educational tropes – eg. why trying to teach a generic skill of “critical thinking” is futile, why relying on student self-assessment (however honest it may be) is highly misleading, why spaced and distributed practice (rather than just a couple of exposures to “get the gist”) is required for mastery, etc.

Obviously a lot of these results have existed for a while, at least in the cog-psych literature, but they haven’t always been explicitly connected to education, or connected in a way that is useful for teachers.

Thanks, SRK (and wst).

Wiliam “made it clear that there is only one purpose of assessment: to find out where a student is at”. This is a useful basis for thinking about assessments. I often return to it when setting questions or assignments. For example, if the student gives the answer as (C), what does that tell me about the student’s learning?

I don’t know. What did people answer the last ten times you asked this question?

A student’s answer to a multiple choice question tells you nothing about the student’s learning.

However, you can ask similar questions about many questions on VCE examinations, other than multiple choice questions. For example, questions about probability density functions on past papers were in fact questions about calculus dressed up as being about probability.

I wonder – has anyone found a book particularly useful for learning to teach maths to students with dyscalculia? There are a bunch of books. For example, Dyscalculia: From Science to Education by Brian Butterworth and Dyscalculia Guidance by Brian Butterworth and Dorian Yeo. However, all the books on this topic seem kind of expensive.

Is dyscaculia a real thing??

Well, I’m hardly an expert but yes? Apparently some people have specific difficulties with things like subitising, counting backwards, estimating, and telling quickly which of two numbers is larger. I’m curious to learn what methods of teaching/learning help them overcome that.

Yeah, but is “dyscalculia” supposedly analogous to being deaf, or to being hard of hearing? Yeah, some kids have a stronger or weaker sense of number, and some kids find it easier or more difficult to attain such a number sense, but is there more to it than that? Does the term do anything other than label a pretty arbitrary category of kid?

Even if so, it probably doesn’t affect the substance of your request: good and affordable references to understand help kids with poor number sense.

Psychologist reports are being given to teachers that list dyscalculia and a lot of other words I had to research. Apparently it can come down to a genuine FEAR of numbers, not just an inability to work with them.

Are they anything more than a word to describe a set of observations? Maybe, I’m not a psychologist. Nor a doctor.

My father was a psychologist and was smart as hell. I know exactly what such reports are generally worth.

I’ve finally updated the post, with a compilation of (most of) people’s suggestions.

Greg Ashman of course (note, his schtick gets pretty repetitive so just watch a few and you get the gist. Not meant as a slam–I actually cleave to his viewpoint more than Marty’s. But Marty does more true different content. (He has a blog, substack and a fair amount of videos…just Google.)

James Cargal:

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

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

https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1109&context=hmnj

Jaime Escalante:

Click to access ED345942.pdf

Dehaene:

How does one read a YouTube video?

Turn the speed to 2x and turn the closed captions on and the volume down? 😉 Not completely joking.

He has a book and several research articles also, but some are dated, so I really think the YT might be best first view of his views:

https://www.newyorker.com/magazine/2008/03/03/numbers-guy

https://scholar.google.com/citations?user=2Dd5uoIAAAAJ&hl=en

Sorry. I deleted my comment and will start again. (Hadn’t had my morning coffee and thought I was responding to a different suggestion to watch something.)

Deheane is a reasonable suggestion, and I’ll add it to the list, although: (a) as noted in the post, “read” means read, and suggesting a video got up my nose; (b) As with Butterworth, these neuro guys don’t thrill me.

As to your other suggestions:

Cargal seems good, and I’ll add.

Escalante is an interesting choice. He was a legend, of course, as captured in the great movie. But he was a teacher, not a writer. Matthews book is very interesting, and tells a much clearer story than the movie, but is kind of off the point for this post. But the article by Escalante that you linked is very interesting, and I’ll add.

Maybe another is the Hotelling article on statistics and its place in American universities.

https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-11/issue-4/The-Teaching-of-Statistics/10.1214/aoms/1177731833.full

I actually don’t agree with all of his points, but it is an interesting, classic take. And it does highlight some of the issues in that field, which continue to this day (competition with other departments on teaching of courses such as “statistics for nurses”, difficulty/need to have both capabilities to do pure research and also the need for some in the trenches applied survey exposure).

It is very reassuring to me as essentially a beginning teacher that I had stumbled across many of these people myself and they resonated and also very useful to get some more leads. Thank you – this is an excellent page and an excellent blog!

Thanks, JJ. Unfortunately, navigating the blog is very much like playing Adventure. I hope to put up a front page soon, so new guys can find their way around.

Alexander Renkl (on worked examples), e.g.

Renkl, A. (2017). Learning from worked-examples in mathematics: students relate procedures to principles. ZDM, 49(4), 571–584. doi:10.1007/s11858-017-0859-3

https://sci-hub.ru/https://doi.org/10.1007/s11858-017-0859-3

Paul Kirschner (on cognitive psychology of learning, including mathematics), e.g.

Kirschner, P. A., Verschaffel, L., Star, J., & Van Dooren, W. (2017). There is more variation within than across domains: an interview with Paul A. Kirschner about applying cognitive psychology-based instructional design principles in mathematics teaching and learning. ZDM, 49(4), 637–643. doi:10.1007/s11858-017-0875-3

https://sci-hub.ru/https://doi.org/10.1007/s11858-017-0875-3

John Mighton’s books on mathemaics education and his founding JUMP Math are also excellent, e.g.:

_All Things Being Equal: Why Math Is the Key to a Better World_ (Penguin, 2020)

Jean Schmittau (on the Davydov-Elkonin curriculum), e.g.:

Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, 19(1), 19–43. doi:10.1007/bf03173235

https://sci-hub.ru/https://doi.org/10.1007/BF03173235

All added, including Blåsjö.

Viktor Blasjo’s posts on mathematics teaching (and history of mathematics):