Imagine that people who wanted to become a math teacher had to take a test to determine how well they understood mathematics. Sounds reasonable. However also imagine that this test was both ludicrously hard and assessed topics that had little to do with what teachers actually teach. Not so great.

Well unfortunately, this test exists and in California we call it the CSET (California Subject Examinations for Teachers). It’s supposed to ensure that math teachers have sufficient content knowledge but instead acts more like a misinformed gatekeeper. To show you an example of what I mean, check out these two problems below from the Algebra and Number Theory subtest:

Problem 1

Problem 2

Here’s your reality check: if you can’t solve problems like these, then you can’t pass the test and you won’t be able to teach math from 6th grade through calculus. I don’t know about you, but I know two things when I see these problems:

• I have no clue how to solve either of them.
• These problems are MUCH closer to something you’d see in a college level Linear Algebra class than anything in high school.

With this in mind, let me take a step back. The CSET has three subtests which have names that seem reasonable enough (click on the test names to see more practice problems):
1. Algebra and Number Theory
2. Geometry, Probability, and Statistics
3. Calculus and the History of Mathematics

If you want to teach middle school mathematics, you only need to pass the first and second tests. If you want to teach high school mathematics, you have to pass all three. I was a math major at the University of California, Los Angeles (UCLA) and I took the first and second tests about three years after I graduated. They were by FAR the hardest math tests I had ever taken. Most of the content was college level mathematics, and a lot of it I had never seen in high school or college. So this entire test hangs on the assumption that if you know how to do this math, you must know all the math that came before it. That would be like giving kids a Geometry final and having that grade represent all earlier classes.

When I first took the first practice test ahead of the actual test I got something like 4 out of 32 right on my first try! It took me two weeks with the answers AND explanations to fully understand how every problem was solved.

So think about what this means: there are potential educators who have solid content knowledge of middle and high school level mathematics yet are not able to pass these tests… and conversely, people could pass these tests yet still not know the mathematics they’d actually teach!

Conclusion
I’m fortunate. I studied and studied and passed the tests on my first try. However, I’m still bitter about it over 15 years later. How many people would make great secondary math teachers who deeply understand the content they would actually need to teach, yet can’t pass this test?

I’m all for ensuring that math teachers understand what they teach, but this test does not do that. Why are we not assessing the content knowledge teachers actually need? How do we go about revising these assessments?

If you’ve had similar experiences with these tests or one for where you live, please let me know in the comments. If you think I’m missing something, I’d also appreciate reading about that too. Thanks.

1. Jaimee says:

Yes! Thank you for posting this. Very difficult. I was fortunate that the District Math Coach was helping current teachers and substitute teachers study for this test, but it took me 3 times to pass subtest 1, and 2 times for subtest 2.
I took both tests together the first time and that was a huge mistake. Take them separately.

• Robert Kaplinsky says:

Yeah, I was very happy that I passed both of them, but I never took subtest 3 because I knew nothing about the history of math and was very rusty with my calculus.

• Barry Garelick says:

Test 3 is actually easier than the first 2. It’s limited only to calculus so it doesn’t cover the waterfront, and the history of math questions were not that daunting.

• kathy says:

Exam III only covers Calculus. The History of Mathematics was dropped in 2017.

Exam I covers Number Theory, Linear Algebra, and Abstract Algebra which are college level subjects. But high school teachers need to know more than their students so that teachers are able to know and relay the underlying reasoning of why the high school mathematics concepts are true.

Exam II covers Euclidean Geometry and Probability at a high school level (with a few higher level questions…but this test can be passed by a strong high school student.). Teachers need to be able to understand how and why these subjects are important and having a higher level of understanding helps the teacher know what to tell the students.

• Robert Kaplinsky says:

Two thoughts:
– Small miracle that they dropped history of math
– I think it’s all in the details of what “teachers need to know more than their students” means. I would MUCH rather teachers have deeper knowledge of content they might actually teach than broader knowledge on something they’ll never use. The reality is that they could pass the CSET AND STILL not know what they actually need.

2. Erika Berk says:

This. Is. Insane. I fully support the notion that anyone teaching math at any given grade level needs to be proficient 2 grade levels ahead of that, but in the case of this test, I would think the state could break it down a bit further than that. 6-12 is a big range. I grew up in California but moved to Texas where I went to middle & high school as well as college, earning my degree and 1-8 math certification (which allows you to teach Algebra 1 at the junior high level). I passed my HS (8-12) certification test in 2013 after 11 years of teaching 4th, 5th, 8th and Algebra I. I also tutored through Algebra 2 at the time. Back then the test was ~10% 8th grade material, 33% algebra I, 14% Geometry, and the other 43% of the test covered Algebra 2 through Calculus (and possibly beyond). Having never taken calculus, I knew I could most likely get at least a 60% with my every day math knowledge, but I needed to find another 20% within the rest of that test to pass with the required 80%. I did have to study in advance but not extensively and I passed on my first try.
Although I have that certification I would never accept a job teaching PreCalculus let alone Calculus. I don’t have a firm enough grasp of the material beyond those courses to do the students justice. Would junior high students benefit from having a teacher who can master the material through algebra or geometry, rather than all the way through…whatever these questions are asking about? I only got 9 right and I’m a darn good Algebra teacher & instructional coach.

• Robert Kaplinsky says:

What I think is worth reflecting on is that “Algebra” is being broadly defined. For example, the kind of algebra on this test is college level and not anything like what is taught in middle or high school.

3. MaryAnn says:

I had a similar experience with the Utah certification test. I entered teaching through an alternative route and, though I had taken all of the required math coursework, I was grossly unprepared for the certification test. I remember praying and telling God that he needed to work a miracle because if I didn’t pass, I was going to quit teaching. I passed by three points. The ironic thing was that I still didn’t really have the deep understanding of the content I was teaching that would have helped me and my students most. Things like why a negative a negative times a negative equals a positive, why anything to a zero power equals one, and the fact that the Pythagorean Theorem can be visualized as actual squares – these understandings transformed my teaching but came only after years of quality professional development after I earned my license. I think there is value in learning advanced math like Linear Algebra. However, I think it would be wise for math teacher education programs to require fewer of these courses and to offer and require more depth on the 6-12 content knowledge.

• Robert Kaplinsky says:

Exactly! When passing this out-of-touch test doesn’t actually measure the things you really need, there’s a problem.

4. Marcia says:

To avoid these tests, and because I have a California clear credential in biology, I spent nearly \$10,000 and took 48 units of math to earn a subject matter authorization in math (with a 4.0), and I am still limited in what I can teach .. essentially algebra, geometry. Pretty pathetic.

• Robert Kaplinsky says:

I feel ya Marcia. I found out that even though I was a math major in college, only one of my classes would count towards subject matter competency. I did not want to spend that kind of money or time to go that route and would have probably changed professions instead.

5. Josh says:

It’s actually college-level abstract algebra. Usually a third-year course for math majors with strong preparation.

• Robert Kaplinsky says:

Totally. I remember doing this in my upper division linear algebra class in college.

6. Mary says:

I live in Illinois. I didn’t major in math and got an additional endorsement. Its the only test I’ve ever thought I failed, and contained almost nothing that I would ultimately teach. I took it before finishing my course work so I knew I was a little unprepared but the questions above brought back the feeling. I studied with my friend who did engineering at Northwestern and he found them hard with a higher level of math than I had. There were few sample questions so I mostly knew I couldn’t do things but wasn’t even clear what they were. I didn’t even read answer choices for many questions because there was no point–I couldn’t make anything of the question. I finished first in a room of several hundred people (most taking other tests; there were 1-2 others taking math content) because there were so many things I guessed on. Ultimately, I did pass on my first try…by 1 point.

I wish the test had more to do with subject matter we’d teach. They could even have some sort of specialist test for people who will teach especially advanced math. Oh, and there was no statistics whatsoever (guess what I teach). It felt like the point was to make me feel like I wasn’t good enough honestly.

• Robert Kaplinsky says:

Right?! It’s like what’s the point of making an absurdly hard test that doesn’t actually measure what we’d need. We’d have a riot if we did that to our students but somehow it’s ok to do to potential teachers?

7. Cindy says:

Aspiring math teacher here. I have learned so much from your blog and videos — thank you!

For some of the non-California readers, maybe worth clarifying that with a Foundational level authorization (“middle school math”) one can teach through and including Algebra 2/Elementary Stat at any K-12 grade level (including at high schools). The “high school math” authorization allows the holder to teach courses beyond that.

Some Alg 2 content is not too far off from the CSET example Problem 2 in your post.

In my opinion, the big disconnect arises from the fact that the Foundational authorization doesn’t align with current middle school curriculum. Few (hardly any?) public middle schools in California offer Alg 2 or Statistics!

What would you think of something like the following:

Adjust the Foundational authorization to cover only, say, through Alg I/Geometry.
Pre-existing Foundational-level authorization holders could continue to teach through Alg 2/Stat.
CSET I and II would cover algebra (including basic linear algebra)/number theory/probability+basic stat/geometry/basic trig.
CSET III would cover calculus and the abstract algebra/higher-level linear algebra/higher level statistics topics, etc.

• Robert Kaplinsky says:

I think that the bigger concern for me would be to better align the test so that passing the test meant that you had the kind of deep conceptual understanding of the content you’d teach. As it is, you could pass this test and still be wildly underprepared to teach your math classes.

8. Brian Goebel says:

I took all 3 math CSET’s in the summer of 2015, and was actually very relieved that subtest 3 had been recently redesigned to remove the history of math, so it only includes calculus. As far as subtest 1 algebra and number theory, I remember my study book said there would only be a maximum of something like 3-4 questions regarding rings and fields (like the sample questions you posted), so unless you had recently studied it in college, your time would be better spent studying the other topics, as one could still pass the subtest while getting all of the ring and field questions wrong assuming you did very well on the other topics. What great advice! *sarcasm*
While I do believe that including these ring/field questions is unreasonable, this does bring to mind two issues. First is the lack of K-12 teachers that studied math in college. Considering the last two high schools I taught at, I would say only ~25% of the math teachers had any STEM related degree (let alone pure math). I’ve heard multiple teachers tell students that they “loved math in grade school but college math was too hard.” What kind of a growth mindset does that demonstrate to our students, if the teachers in role model positions gave up when things got tough? That honestly frustrates me even more than parents normalizing how it’s okay to be “bad at math!”
The second issue is how to redesign the current CSET, or some other form of subject matter fluency, so that it is more closely aligned with the actual content. Personally, I would happily forsake some of the CSET multiple choice, and start making teachers work on some problem based lessons or open middle problems! A lot of those problems are just as challenging, but relate a lot more than asking about the properties of a ring or field!

• Robert Kaplinsky says:

There’s a lot to unpack here. First, when I took the CSET, it was 2004 or 2005 and there were literally zero test prep books at the time. It was just after the switch away from the MSAT and I was on my own. Glad to see that the history of math section is gone. No tears will be shed for that.

I don’t share your take on math teachers without math degrees. I think this is really complex but there could be lots of reasons at play. I have used essentially NOTHING from my UCLA college math classes in life or even in teaching. So, while I’m grateful that having a math degree has opened doors for me, I kinda wished I had studied something more useful. Maybe we should be looking at the reality that many college math professors have no formal training as educators and that this could be a bigger issue.

Robert,

Your problem 1 (actually problem 9 from the sample materials at http://www.ctcexams.nesinc.com/TestView.aspx?f=HTML_FRAG/CA_CSET211_PrepMaterials.html) has all the issues you mention, and more:
Answer A is correct because it shows that multiplication in this setting is not commutative.
Answer C is ALSO correct because it shows that there is no zero element.
Answer D needs revision. Is it introducing matrix A twice? Is there a typo? Is it intentionally written that way, so as to assess the test-taker’s ability to ignore answers that are part gibberish? Is it part of a ploy to condition test takers to ignore questions that are part gibberish, so that they’ll miss a later answer that is correct by merit of being part gibberish? It’s unnecessarily exhausting. By my count, at least half the answers to this item need to be changed.

They did it again with #17 on the same prep materials, with answers A and D both being correct.

When I took the CSET–it must have been about the same time you did, give or take a couple years–I left with the feeling that some questions on the real test had the same issues.

Maybe your statement “It took me two weeks with the answers AND explanations to fully understand how every problem was solved” is testament to the quality of your math preparation at UCLA; I know people who’ve studied a lot longer for the CSET with less to show for it. Still, I would like to see a test with more relevant content on it–perhaps, a question that asks the test-taker to draw conclusions about student thinking. Maybe take excerpts from Error Patterns in Computation and ask what the “student” is doing. Tough to do this in a multiple choice format, though.

I’ve raised my concerns to the CSET folks, who thanked me for my input. If anything comes of it, I likely won’t find out for exam security reasons.

• You have a great point Brad about answer C. I suspect they included that because it is close to being the distinguishing property that you need to have a field, i.e. that inverses exist. I eliminated it personally because I thought “well that would make it more like a field rather than less.”

The only defense of this distractor, and I don’t really buy this myself, is that statement C does not REFUTE the claim that GL_R(3) is a field because statement C is FALSE in itself (take any matrix A, including the zero matrix, which has det = 0). Statement A is a true statement, and contradicts the definition of a field, so can be used to REFUTE. But that is a really narrow way to read this question, and a terrible way to distinguish who has mathematical knowledge for teaching 6-12 mathematics.

I totally agree that answer choice D is garbled. They implicitly give you a universal quantification and existential quantification on matrix A. So I guess you eliminate it by virtue of its nonsensicality. But it’s also eliminated because it says det (A)≠ 1 not det(A)=0