And, for that matter, what are the bad mathematics curricula? Given that California has just shot itself in the everything it would be rude to ignore them.

We’ve been meaning to post this for a while, but simply haven’t gotten around to it. Mysterious commenter texas asked a number of “Where do I look?” questions. For one, on who should people read on maths ed, we put up a discussion point, which seemed to garner plenty of interest. So, finally, here is post on the second question (of three).

What are the good mathematics curricula, and what are the bad mathematics curricula, of which people are aware? And, why are they good, or bad? Of course people may differ in their opinions, and we’re happy and keen to have people debate the (de)merits. We’ll start the ball rolling with two curricula, about which there is probably little debate, at least amongst frequenters of this blog.

## GOOD?

**SINGAPORE: **Website

** Primary** (PDFs): Primary 2013 version (link fixed), Primary 2020 version (to year 3)

**Secondary** (PDFs): Express/Normal 1-4, Additional Material 3-4, Technical 1-4

**MOROCCO: Website (with linked exams)**

## BAD?

**AUSTRALIA: **Website

**Primary:** Website (F-10 version 9.1)

**Secondary**: Website (version 8.4)

**Germany (Baden-Württemburg): **Website and PDF

## TONY GARDINER’s “UNIVERSAL CURRICULUM”

**(25/07/23) **This is a bit different from what we had intended, but it seems well worth adding in, if only as a guide to what approved curricula do, and don’t do. As an appendix to his guide to a proposed English curriculum, Tony Gardiner included a suggested “universal curriculum”. Here it is:

Not a full curriculum, but I like the tenor (from Hungary):

Click to access hungarian-pedagogy-flyer.pdf

Thanks, Terry. If there’s anything like a full curriculum, even if not in English, I’ll add a link above.

Singapore says so much more with so much less

They do. They know when to shut up.

Here’s something from Germany. It is in German. It seems to have links also to final exams, some of which look pretty interesting.

http://www.zum.de/mathematik-digital/themenuebersicht.php?schulart=5&key=1818211450

Thanks, Glen. I’m not sure what our German correspondents might have to say, but I’ve added a link above.

The heading says ‘Topic Overview North Rhein Westphalia High School’.

See hjm’s comment.

Thanks, hjm. Deleted.

“Germany” does not link to the german curriculum. It’s a private organization that provides teaching material financed by donations.

Here is a link to the math curriculum of Lower Saxony, 60+ pages full of nonsense about „competences“ with an absurd level of instruction details and truisms, and that’s only for years 5 to 10 for one of three school types.

https://cuvo.nibis.de/index.php?p=download&upload=63

Thank, hjm. That sounds more like the Germany that Franz Lemmermeyer has enjoyed telling us about.

Alas, from Australia the link appears to just hang, though it was grabbable using a VPN.

I’m reading Franz’s blog and yours, and came to the conclusion that there is almost no difference between German and Australian when it comes to high school math, almost as if it is managed by the same insane people, writing the same stupid textbooks, based on the same fundamental misunderstanding of what math is about.

The only difference seems to be that we do not have multiple choice exams in Germany. Instead we have complete nonsense “context based” exams like: “A coastline has the shape of a polynomial of degree four…”.

Thanks, hjm. Yes, that’s the impression Franz gives when he occasionally pops up here. Note that we also have plenty of quartic coastlines, and the like.

Education in Germany is in the hands of the 16 states; each state has its own system. I think that some states do have a couple of multiple choice questions; since many of them keep their exams secret I cannot say which – so far, there were no multiple choice questions in the south, and there will be none in 2024. I expect no dramatic deterioration until 2027, when there will be problems in science exams asking the students for example to evaluate legalizing cannabis (in biology) or whether its is better to have anti-noise system in planes or night flight bans around airports (in physics); these evaluations must not be based on science but on what is better for society, whatever that means. They’re really trying hard to draw girls into STEM over here.

For those who can read some French I suggest having a look at the bac in Morocco:

https://www.sigmaths.net/bac2/bacMaroc.php

I discussed four problems from 2021 on my blog (in German). I just might translate an exam or two into English if anyone’s interested.

My impression is that Germany is behind Australia in Math – we do not cover integration by parts, substitution, complex numbers and similar “advanced” stuff. And what little we do teach is over the heads of most of our students, with a dramatic decline in the near future due to the corona years and the introduction if Ipads.

Hi Franz!

I’m wondering if you could clear something up for me. When I worked in Germany (Sachsen-Anhalt and Berlin, 2000s) the students we received for the BMath were consistently far better than those in Australia. To the point where one of our top Australian students would be roughly equivalent to middle of the pack in terms of Day One at university.

They all certainly knew integration by parts and had seen complex numbers.

Do you have an explanation for this?

I should add the disclaimer that I am not trying to discredit you (I would not be capable of that anyway). Rather I’m trying to marry my own experience with what you’re saying here (and elsewhere).

The eastern states had excellent mathematics teachers for quite a few years after 1989. These differences have vanished to the point that in Mecklenburg-Vorpommern, the grades of the final exams had to be adjusted twice in three years. I know an excellent teacher in Sachsen who had taught numerous winners in International Olympiads (mathematics and physics); he quit last year and told me the stiuation has become unbearable during the last 10 years, and that he will never ever enter the school building for the rest of his life.

At least they could calculate whether the trains were running on time.

Thanks very much, Franz. The Morocco exams look mighty serious, and I’ve added the link above. Any idea why they are the way they are? (I’m not sure it’d be worth your work to translate the exams, but I’d be interested.)

Thanks also for the depressing report on the ground from Germany. I’m less concerned with whether the topics are advanced per se, than whether whatever is done is done in deep enough fashion. Of course, within reason, more advanced topics permit more advanced mathematical reasoning, but these topics are not automatically chosen with any reason. For examples, the stats nonsense in Australian senior mathematics, and the relativity and quantum nonsense in Australian senior physics.

But, it could be worse, as your conjecturing suggests. At least we’re not doing the Māori nonsense in New Zealand (yet).

Hi hjm. As you certainly know, there is no “the German curriculum” so of course it doesn’t link to that. I was providing a link to a curriculum in use at some high schools in one of Germany’s states. I know it isn’t the same in every place in Germany (I lived and worked there in a math department for several years), but as it is indeed a curriculum in use in Germany it seemed to be relevant.

If we are going to have an honest comparison, it seems to me that any curriculum in use across a state (like we have in Australia with e.g. NSW, VIC) should be valid. Your one, and the one I linked, included.

Glen, what is the status of the curriculum to which you linked? hjm gave what I assume is a province-level curriculum, and one could similarly give a state-level curriculum for an Australian state. But what is yours?

It is, I believe, the curriculum for the Nordrhein-Westfalen Gymnasium. There are three kinds of high schools in Germany (this is true in any state). Practical, central, and theoretical. The Gymnasium is the theoretical one. It is not as selective/low enrolment as our most advanced mathematics subjects. However some preselection has taken place (earlier on). When I taught there (not in NW, I was in Sachsen-Anhalt), most students at university went to Gymnasium (and my university was not a “top” German university) for high school.

Thanks, Glen. So the link you provided earlier was the N-W provincial government guide to their Gymnasium curriculum?

Not the government guide, it is a non-government guide. It might also be out of date since I just learnt that they changed in 2022/2023.

The other website is the official one, it also has Gymnasium on it. Here’s an example (beispiel) for Gymnasium.

Click to access SiLP_Gym_G8_Mathematik.pdf

Jahrgangsstufe = year level. It doesn’t seem complete, and I’m not sure they actually prescribe a full curriculum.

It seems as though they give rules and regulations for the curricula, and then high schools may be able to choose a curriculum that meets the rules? Probably with a complicated process. Perhaps that would explain why the first link isn’t a government link. It could be one of many possible curricula that Gymnasia in NW can choose for their students. Take this with a grain of salt however, as I’m very unsure of myself.

Sure. I’m not going to try to use my ACARA-level German to decipher this. If you think there’s a link worth including I’ll do so. Then, if the Germans object again, I’ll remove again, and we can have a Thomson’s lamp-style link.

I think the language barrier might make including such a link not incredibly helpful. If these comments stick around, interested readers have the links anyway.

OK, sure.

For the central school in the same state, I found this:

https://www.schulentwicklung.nrw.de/lehrplaene/lehrplannavigator-s-i/hauptschule/mathematik/index.html

However a lot of it is rules about curricula (meta curriculum instead of actual curriculum). I found an “example” curriculum for their new rules (it apparently changed for 2022/2023) which is on that site. Everything auf Deutsch naturally.

The rules are … as you’d expect. The example is just bullet points in tables of material. It is kind of difficult to judge.

OK thanks. If there’s something clear enough that is central and authoritative (not a school’s interpretation), I’m happy to add the link.

The “Bildungsplan” for my state is here:

https://www.bildungsplaene-bw.de/,Lde/LS/BP2016BW/ALLG/GYM/M

with a pdf at the bottom. Each school is free to move stuff within the years 5-6, 7-8, 9-10 and 11-12.

In practice, all the young teachers stick faithfully to their books; all of them have a digital version, so they don’t even have to use the blackboard (or whiteboard). The results are predictable.

The books are rubbish, and I don’t use this word lightly. The proof of the similarity of triangles, which was correct up to the 1990s, is now full of errors and gaps and circular reasoning. The only correct proof up to grade 10 is probably that of the theorem of Pythagoras (if you accept the usual “Look!”-proof as a proof – I do not think much of proofs that do not make it clear where you are using something equivalent to the parallel axiom).

Thanks again, Franz. I’ve added links to B-W.

There’s also essentially zero proof in Australia K-10 mathematics (and no shortage of clunkiness beyond that.) I’m less concerned about axiomatic geometry, and I really like the “Look!” proof, as discussed here, but I recognise that I’m an outlier.

I’m pleased to hear you’re not trolling, but fewer colloquialisms might help.

The context is school curricula, either primary or secondary.

An upvote for Singapore Math, a version of which is approved for use in California, but more commonly used by homeschoolers.

Art of Problem Solving, of course, used by a few private schools in California, but mostly for after-schooling and homeschooling.

Thanks, VJ. People have talked highly of Art of PS, but I thought mainly as a source of good problems (for all levels). Are you saying it is a more complete curriculum thing as well? Or are you noting Art as a separate, related thing?

AoPS indeed is a legit curriculum, pre-algebra through calculus + extras you don’t get in a US curriculum like C&P and NT. AoPS also has an elementary math curriculum.

Thanks, A. Do you mean “legit” as in “sensible” or “legit” as in “formally approved”? Are there clear and easy links.

I’m not sure what you mean by “formally approved.” I don’t place much weight on accreditations, but here are some listed by AoPS: https://artofproblemsolving.com/school/handbook/prospective/accreditation .

It’s definitely a sensible choice for US students who want to arrive at college overwhelmingly prepared for any quantitative field. Or if you want to compete in the IMO, lol.

Thanks, again, A. By “formally approved” I mean a maths curriculum that is approved by a state of country for the schoolchildren of that region.

I’m not trying to be nitpicky or gatekeepy. I’m trying to determine regions of substantial sizer that do it right, or at least have the curriculum to suggest they’re in the ballpark.

You are probably aware that education policy is… complicated in the US. Policies, curricula, grade level standards vary by state, school district, and even by the individual school.

With that in mind, you can see at the bottom of this page a list of California public schools that allow the use of AoPS for students.

https://artofproblemsolving.com/school/handbook/prospective/working-with-institutions#vendor

But that’s a bit misleading because these schools actually represent hundreds of homeschoolers…not actual brick and mortar schools. Still, they are public schools and they do receive funding from the state, so.

Thanks, A. Yes, I’m less interested (here) in the kind of materials tolerated for homeschoolers than the stuff sanctioned for the masses.

I actually understand the Singapore curriculum! (have a remedial maths class so have been looking through the Oz one and it’s just so complex). It is a bit of a sign if the curriculum is so complex it’s hard for teachers to understand. (I’m no genius but the website has taken many goes just to get some idea of what’s going on). As an ex-bureaucrat, a rule of thumb of mine was that anything I couldn’t understand was probably mainly waffle and too complex to be all that useful in the real world. btw the link to the 2013 primary one seems to go to the 2021 one.

Thanks, JJ. Link fixed. And of course I agree entirely with your comments.

Israel: https://timssandpirls.bc.edu/timss2019/encyclopedia/pdf/Israel.pdf

This is an overview of the mathematics curriculum in Israel. The tables in the paper give more details. The source documents in the references are in Hebrew.

Thanks, Terry. I couldn’t make a lot of sense of the overview (and didn’t bother with the primary documents. The revisions smell suspiciously of PISA.

I’m not sure how this will fare when subjected to JJ’s “waffle test”.

Most national curricula fail the “waffle test”. So, like Alex (on Singapore) and Terry (on Hungary), I prefer the ones that are specific, and that focus on content (suitably described).

I have had opportunity to critique, and try to improve, several versions of our national curriculum. Such curricula are curiously hard to draft; but as part of my attempt to critique the proposed 2013-14 England curriculum:

Click to access obp.0071.pdf

I felt obliged to end with a draft to illustrate the points that had been made (Part IV, pages 271-316), using the overall structure with which local teachers and teacher-educators were familiar. But it was intended to be a “universal curriculum” – so I would be interested whether it makes sense to this mixed group or falls short.

Thanks very much, Tony. The entire books is very interesting: a little awkward to read, because it’s tied to a particular curriculum, but with most of the content much more applicable. I’ve embedded just the “universal curriculum” part in an update to the post. (I trust that was ok with you to do.)

The worst curriculum that I’ve ever seen (my older daughter was subjected to it in US grades 6 and 7):

Illustrative Mathematics

It contains almost zero math and very few problems; it wastes students’ time in fruitless discussions (“inquiry”); and it is even incorrect. For example, Unit 1 in the 6th-grade textbook (age 11) starts by having students discuss and figure out a definition of area. If I remember right, the teacher manual recommends a good chunk of a class period for this worthless activity. In my daughter’s class, they spent most of the period on it. The book doesn’t ever tell them a correct definition — which, as a physicist and not a mathematician, I’m not even sure about myself — and the glossary offers a wrong definition.

The same textbook then takes 56 pages to get to the entitled “Formula for the area of a triangle.” And that section doesn’t even state the formula! Students have to “discover” it from a few tables of triangle measurements and areas.

I showed the book to a friend with a math PhD from Harvard who is also involved with math education in schools. She said, “Sanjoy, it’s not that bad” — until I pointed out that the giant “1” on the cover referred to Unit 1, not grade 1.

Because it’s so terrible, it’s becoming one of the most popular curricula in the US. At first, it was available only for grades 6-8, but now it’s available for K-5 and even high school (9-11), ensuring that students will hardly ever study calculus in high school.

Thanks, Sanjoy, I take your word that IM is bad, but can you clarify its status? Is IM a curriculum that certain states have adopted, or is it a set of curriculum materials that certain states have approved, or what?