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I will be teaching Euclidean geometry to future teachers, and I am feeling a bit lost (I know geometry, but I am not that familiar with mathematics education).

Is there some recent (as concise as possible and maybe classic) mathematics education book that focuses on geometry? I am reading Freudenthal's relevant chapter in "Mathematics as an Educational Task", which is very interesting, but it does not contain any practical instructions on how to teach geometry (it is also a bit old).

Also, is there some short enough mathematics education book that provides some necessary background to the different schools of thought in mathematics education?

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  • $\begingroup$ Based on the tags you used, am I correct in assuming that you are teaching undergraduate students, who in turn will be teaching high school students in the future? $\endgroup$ – Joel Reyes Noche Feb 4 at 15:47
  • $\begingroup$ @JoelReyesNoche Yes, you are correct. $\endgroup$ – Sumac Feb 4 at 16:37
  • $\begingroup$ I'd say that any decent textbook that has sections with definitions, theory, proofs and exercises automatically teaches how to teach the subject. I have never respected the overly scripted books designed for teachers, although I understand that different subjects have specific pedagogical peculiarities. Also, I suggest looking at high school curriculum: modern math programs have been decimated by NCTM standards and Common Core, there are very few proofs in them and the sequencing is not often logical. Whether you want to build your course aligned with these programs is an important decision. $\endgroup$ – Rusty Core Feb 4 at 18:43
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    $\begingroup$ I took a course in geometry which was required for future high school teachers at my university. It was out of "Pure And Applied Undergraduate Texts: Geometry for Undergraduates." (bookstore.ams.org/amstext-8), with some extra noneuclidean readings. It is aimed at students with less proof-writing skill & explicitly mentions future teachers as part of its target audience. $\endgroup$ – Opal E Feb 5 at 0:44
  • $\begingroup$ Polya wrote several books that could be understood as about how to teach mathematics (for example, both volumes translated as Mathematics and Plausible Reasoning and How to Solve It?). Although he does not address "different schools of thought in mathematics education" in those terms, he takes an ecumenical and flexible viewpoint and offers a lot of useful ideas illustrated with concrete and realistic examples appropriate to real teaching settings. His book are not focused on geometry, but they do include some geometric examples. How he attacks a geometric problem is interesting to see. $\endgroup$ – Dan Fox Feb 5 at 16:56
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Not to blow my own (and my co-authors') horn, but this may be exactly what you're looking for:

Herbst, A., Fujita, T., Halverscheid, S., and Weiss, M. (2017). The Learning and Teaching of Geometry in Secondary Schools (IMPACT: Interweaving Mathematics Pedagogy and Content for Teaching), 1st Edition. London: Routledge.

(Amazon link, if you prefer.)

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From the publisher's website:

The Learning and Teaching of Geometry in Secondary Schools reviews past and present research on the teaching and learning of geometry in secondary schools and proposes an approach for design research on secondary geometry instruction.

Areas covered include:

  • teaching and learning secondary geometry through history;
  • the representations of geometric figures;
  • students’ cognition in geometry;
  • teacher knowledge, practice and, beliefs;
  • teaching strategies, instructional improvement, and classroom interventions;
  • research designs and problems for secondary geometry.

Drawing on a team of international authors, this new text will be essential reading for experienced teachers of mathematics, graduate students, curriculum developers, researchers, and all those interested in exploring students’ study of geometry in secondary schools.

To be clear, this is not a "how-to" handbook, but it does provide a pretty comprehensive summary and synthesis of what the mathematics education research community knows about the teaching and learning of Geometry, including a historical survey of how that has changed over the last 150 years.

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  • $\begingroup$ Thank you for your suggestion. I read the first chapter and browsed through the whole book. I think this is exactly what I was looking for. It is well organized and full of relevant references and quotes. I will definitely add it to the reading material for my class. $\endgroup$ – Sumac Feb 13 at 10:13
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In addition to whatever book you choose, I recommend getting the students / future teachers to play with geometry.

Two sites I love:

And I have started finding great basic geometry puzzles which I am sharing with my college students in hopes of improving their visualization skills:

  • Geometry Snacks, by Ed Southall and Vincent Pantaloni
  • More Geometry Snacks, by Ed Southall and Vincent Pantaloni
  • Puzzles by Catriona Shearer (@catrionashearer)
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  • $\begingroup$ These seem really interesting. Thank you. $\endgroup$ – Sumac Feb 5 at 19:46
  • $\begingroup$ Software that allows you to perform geometric constructions such as geogebra or Geometer's Sketchpad can also be very useful. $\endgroup$ – Brian Borchers Feb 11 at 4:44
  • $\begingroup$ @BrianBorchers Yes, I agree. It is my intention to cover some things in geogebra. $\endgroup$ – Sumac Feb 13 at 10:14
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This is likely not exactly what you have in mind, but a recent book by an author that is very critical of how geometry is traditionally taught and who offers a starting point for how it should be taught is Measurement, by Paul Lockhart. While it may not be the book you want to use as your reference, I think it is a very worthwhile read to understand how one particularly opinionated math educator thinks geometry should be taught. But that mathematician's viewpoint does represent what a lot of mathematicians think about geometry education (and math education more broadly), even if most of us aren't quite as passionate about it as Dr. Lockhart. (You can also read his famous A Mathematician's Lament online here to get a sense of his criticisms of the traditional approach to teaching mathematics. He addresses geometry specifically in his critique.)

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  • $\begingroup$ Thank you for your suggestion. I looked at Lockhart's book. It is certainly a very interesting read, but it is not what I am looking for. $\endgroup$ – Sumac Feb 13 at 10:17
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Alfred Posamentir (granted, a generation ago) was big in Math Ed and big in Geometry.

If I were teaching undergraduate Geometry to future teachers, I’d look for Ed Moise’s book Elementary Geometry from an Advanced Standpoint. He wrote the Geometry book back in the “New Math” days—perhaps more rigor than needed for high school Geometry—but it’s perfect for undergraduates.

His book builds up Geometry as far as you can without the parallel postulate. I think he called that “Synthetic Geometry” (although it’s been since Fall ‘88 when I took the course). He then goes into the various cases where the parallel postulate gives us unique and tasty geometrical flavors.

Since then, I discovered Taxicab Geometry and wonder if it wouldn’t be a nice way to ease the undergoos into a non-Euclidean idea with surprising results (e.g., circles appear as squares in Taxicab Geometry).

One thing about Moise is that he holds out on Non-Euclidean Geometry until the very end, and doesn’t do a helluva lot with it. The title says exactly what the book is, and it also expresses what I think the course ought to be. If you’re looking for a serious Non-Euclidean Geometry book for undergraduates, I can make some recommendations, but you’ll have to tell me you at least glanced at Moise.

NB: His high school Geometry text is available on the Internet. It started as a series of typed notes (search for SMSG, the cool kids way of saying “new math”) and you’ll find them. Also, if you open an account on the library section of archive.org, a relatively mature 1990s-era version of the book (co-authored by Floyd Downs) is available for checkout in Encrypted ePub and Encrypted PDF formats, accessible with Adobe Digital Editions software.

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  • $\begingroup$ Thank you for your suggestions. I really like your idea about Taxicab geometry (I did not know anything about it). I will try to incorporate it into my course. $\endgroup$ – Sumac Feb 13 at 10:19
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I recommend to avoid the "interesting" suggestions of books that include high school geometry but with some advanced insights. These suggestions likely come from the commenters here (many were research mathematicians at least in grad school) versus the needs and enjoyment of high school teachers.

Instead assign a popular, straightforward example text actually used in schools. For simplicity, I would pick one a bit more to the side of old fashioned. This also gives you sort of a base case to compare different ed fads to.

In addition I recommend to assign a text on the different main methods of teaching geometry. If you can't find one, then assign readings on each major "school": articles, etc.

It is not critical to be encyclopedic. Just give them some flavor of a few different approaches (Common Core, traditional, Saxon, etc.) That way when they go teach, wherever they get a job they feel a bit confident that there are several ways of skinning a cat. That there is not one correct school of geometry education (i.e. there is no Euclidean proof of how to teach Euclid). Do NOT feel like you need to be an expert on every school of math ed (too harsh on yourself). Just give them an honest exposure to at least three methods--you will learn something also. [However I would NOT emphasize your own lack of knowledge--students find that a turnoff when the teacher prostrates himself like that...and it reduces confidence.]

A quick Amazon search found this for a math ed text: https://www.amazon.com/Learning-Teaching-Geometry-Secondary-Schools/dp/0415856914#reader_0415856914 I actually like the modeling approach but would prefer it had a bit more of a review of different methods (has some in a historical context) versus just pushing their approach. But take a look at it. And do a Google/Amazon search yourself.

In addition, there is a very active blogosphere of math ed (secondary school edu refomers as well as practitioners). It is NOT a well connected graph (haha) to this MESE community. Below are a few links. You can look through and find others as well. But I think having the students read some blogs will give them excellent insights from real practitioners and also help them understand the whole multi cat skin idea. Also I think they will enjoy that versus just articles and books (less dry). Plus there is the possibility they just play the link game and wander onto other things that interest them on the topic.

http://talesofahighschoolmathteacher.blogspot.com/

https://crazymathteacherlady.wordpress.com/

https://thegeometryteacher.wordpress.com/

https://blog.feedspot.com/high_school_math_blogs/

https://educationrealist.wordpress.com/?s=geometry

https://larrycuban.wordpress.com/

https://garyrubinstein.wordpress.com/

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  • $\begingroup$ "there is not one correct school of geometry education" - this is true in relation to good programs, the sequencing and even base postulates can be different. But in relation to Common Core, Saxon, etc. it can be either mediocre, bad or truly awful. $\endgroup$ – Rusty Core Feb 8 at 1:22
  • $\begingroup$ I agree with most of what you have written. Nevertheless, assigning some text that is actually used in schools does not seem like a very good idea (I will of course comment on some parts of them in class). I would like my students to get an idea of more advanced concepts in euclidean geometry even if they never use them in teaching. $\endgroup$ – Sumac Feb 13 at 10:24
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Not what @Sumac seeks, but it would be good to keep this book handy so you can add historical context at appropriate junctures.

The first five chapters show how the ancient Greeks established geometry, together with its numerous practical applications, while more recent findings on Euclidean geometry are discussed as well.


         


Ostermann, Alexander, and Gerhard Wanner. Geometry by its history. Springer, 2012. Springer link.

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  • $\begingroup$ I really liked this book (I just browsed the material covered). It can complement the material I will cover in class. $\endgroup$ – Sumac Feb 13 at 10:25

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