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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.


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15 Comments

  1. 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.

    • 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.

  2. 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.

    • 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. 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.

  4. 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.

    • 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. 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.

    • 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?

  6. 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.

    • 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.

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