Mathless teachers

Virginia will drop a basic skills test for would-be teachers which measures high-school-level reading, writing and math performance. Instead, the state will develop its own test of college-level reading and writing skills. Only math teachers will be tested on math knowledge.

Here are “advanced math” test prep questions for Praxis I, which is being abandoned. Thirty-five years out of high school, I can do these problems in my head. It’s hard to believe there are people smart enough to teach who can’t pass a basic math test. How are they going to average students’ grades?

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  1. SuperSub says:

    Does that mean as a Science teacher I don’t need to worry about English? I’ve always wanted to teach a class in Klingon.

  2. P. Abel says:

    As a high school math teacher I’m truly aghast at the VA decision. I’m assuming them every K-8 math class will be taught not by the mathless teacher who pass the new test but rather by K-8 math specialists.

    Otherwise can we look forward to incoming 9th graders arriving with zero knowledge of math?

    Will VA pay those then very valuable math specialists merit pay for doing what the rank & file can’t/won’t do?

    As it is incoming 9th graders arrive with very little math content knowledge. They can’t do multiplication, subtraction with 1000s or more, division, fractions, not to mention they have no concept of what a basic algorithm is nor how to handle them. Math to them is “fuzzy stuff” with no real basis in either content or meaning, and CERTAINLY no concept of mathematic’s beauty or rigor.

  3. Richard Brandshaft says:

    Congratulations in being able to do those in your head. I would need a pencil and paper for some of them.

    I might add that this test should use some debugging:

    “If Sam can do a job in 4 days that Lisa can do in 6 days and Tom can do in 2 days, how long would the job take if Sam, Lisa, and Tom worked together to complete it?”

    At least a week, since they’ll be getting in each other’s way.

    In the fraction 3/© , © may not be substituted by which of the following sets?
    {1, 2, 4}
    {1, 3, 7}
    {0, 10, 20}
    {1.8, 4.3}

    I presume they meant the last one, although there is no rule against writing a fraction with a non-integer in the denominator (or the numerator).

    Of the following units which would be more likely used to measure the amount of water in a bathtub?
    Either kilograms or liters works for me.

    “If a match box is 0.17 feet long, what is its length in inches the most closely comparable to the following?” Does that question make sense to anybody?

    Of the following units, which would be most likely to measure the amount of sugar needed in a recipe for 2 dozen cookies?
    degrees Celsius
    Gender discrimination against men. Only someone who cooks would know whether one is more likely to weigh the sugar or measure the volume.

    If 8x + 5x + 2x + 4x = 114, the 5x + 3 =

    If two planes leave the same airport at 1:00 PM, how many miles apart will they be at 3:00 PM if one travels directly north at 150 mph and the other travels directly west at 200 mph?

    I can almost do those two in my sleep, but I do wonder if they are fair to non-techies 20 years after their last math class.

  4. KateCoe says:

    If Tom can do the job in 2 days, who needs Sam and Lisa? Oh, right, they’re members of the teachers’ union.

  5. nicksmama says:

    P. Able, yes, Virginia is hiring math specialists (at least here in Northern VA).

    I really question how anyone going into the teaching profession cannot study or teach themselves how to do these problems. If they can’t learn, how are they going to teach?

    BTW, I don’t have a bachelor’s and have been out of high school for 20 years and yet I could still figure out the answers to these problems with a little help from some references such as Math on Call. I home school and have a large library of math references.

    I know that if my career depended on it, I would be doing a bit of brushing up.

  6. I would think – and maybe I’m making a big presumption here – but I would think that if one had a certain amount of pride in one’s job, and one didn’t know something, one would make whatever effort necessary to learn that? You know, so as not to look stupid or something?

    I mean, I’m a pretty narrow specialist in some ways but if I know I need a particular piece of knowledge to be able to teach decently, by golly, I’m going to get that knowledge – by reading, by doing online coursework, by asking someone who knows it to tutor me, whatever.

    I will say I teach at the college level in the sciences and my colleagues and I are sometimes shocked at the lack of basic arithmetic knowledge – and sometimes, the lack of desire to learn said basic knowledge – of some of our students. (I often get complaints on my evals that my ecology class involves “too much math.” Actually, it only involves about half the math I COULD put in…)

  7. Walter E. Wallis says:

    Where can I get 6%?

  8. “Thirty-five years out of high school, I can do these problems in my head. It’s hard to believe there are people smart enough to teach who can’t pass a basic math test. How are they going to average students’ grades?”

    Why assume that these math-phobic teachers will be assigning objective, numerically-based grades? My guess is anyone this terrified of Praxis math items will be giving “holistic” and subjective letter grades.

    That’s much more “personal” and “caring” than those nasty ol’ averages, you know.

  9. LibraryGryffon says:

    The answer to the fraction set is that the one which can NOT be used is the fourth option, the one with “0” in it.

    I spent about 10 minutes on the test, only used a pencil for 6 or 7 problems, and got 37 of 40 right. I haven’t had to use math for anything other than my checkbook for almost 20 years now. Of the three I got wrong, one was close, and I know that I was estimating to figure out which option to choose. If I had used the pencil on that one, I would have been right. I probably wouldn’t have missed the other two either if I’d spent a minute or two more on them or else done them on paper. If my daughters’ elementary school teachers can’t do this I don’t want them trying to teach children.

  10. “Grace has 16 jellybeans in her pocket. She has 8 red ones, 4 green ones, and 4 blue ones. What is the minimum number of jellybeans she must take out of her pocket to ensure that she has one of each color?”

    The correct answer is three, which is not one of the choices int he test. Or did they mean she is choosing the jellybeans at random with her eyes closed? If so, they should say so. Choosing three differently-colored jellybeans from a pocket (let’s say an apron pocket, since that is also not specified) is a simple matter of looking into the pocket and choosing three that appear to be three different colors.

  11. I looked at the question “which would be more likely used to measure the amount of water in a bathtub?”

    I imagine they wanted “liters” as an answer, but it’s actually very difficult to make a direct measure of the number of liters in a tub. Most people would do it by measuring the tub to get the volume, which isn’t one of the answers.

  12. Richard,

    Your post was painful to read. You are either being sarcastic or you just made the best case ever for keeping those questions on the exam.

    You cannot divide by zero.

    Volumes of water are *more often* measured in liters, not units of weight (kilos).

    Your third question is a unit conversion question 0.17feet = x inches.

    In the US, we use cups to measure sugar. They use weight in Canada and Europe.

    I took the CBEST in 1996. One of the questions asked that I measure a paperclip (a picture of the paperclip was placed right ablove a picture of a ruler). Another (multiple choice!) was “7 + 3 = ?” If teachers can’t do that sort of math they have no business in the classroom.

  13. Michelle Dulak Thomson says:

    Hmm . . . just took the test & got 40 out of 40, but must admit to have been guessing once or twice. I’d forgotten what “mode” means, for one thing. And I was getting tired of long division (heh!) by the end, so guesstimated a few, which ought to be good enough (obviously it was). But I wouldn’t have liked to do this without pencil & paper. It’s possible, but takes more time and effort to do it all in your head.

    Three questions: Do the teachers get to use calculators? How long do they have to complete the test? And is the proofreading on the real tests as bad as it is on this one? Jeez, these people are testing potential educators and they can’t bother to spell correctly or write grammatical sentences.

  14. Michelle Dulak Thomson says:


    I think people have answered most of your queries about the questions. I’d just add that the one with the two airplanes is purely an “ever heard of a Pythagorean triangle?” question. If they hadn’t put that aircraft toupee on it, it’d’ve been quite bald. (OK, that’s a weird image. Sorry.)

  15. While I do believe that any adult should be able to handle most of the problems without much difficulty, I’m not concerned about how this plays out in the middle and high school classrooms. The NCLB Act requires that anyone teaching math to middle and high school students must be a math specialist. NCLB mandates the equivalent of a BA/BS in any subject taught at those grade levels or a passing score on the single subject Praxis. In short, the math and science teachers are going to need to know a lot of math or they will never be considered Highly Qualified and the school district will be out of compliance with the law.

    Should an art teacher be able to handle the questions on this test? Of course they should from a practical sense. According to the article, 97 percent of the candidates do pass the test. So why’s this a big deal? The negative impact on kids in the classroom will be negligible at best.

    That being said, the questions on this test were mostly very easy. I also only missed three and none were because I didn’t know the math, but rather because I was speeding through the problems and misread a crucial fact. Something I always caution students against doing. We call them speeding ticket errors.

    Most of these questions should be answerable for the average California sixth grader. These skills are almost all covered in the grades 4-6 California math standards. Which btw are a lot more difficult than what was expected of me in the sixth grade in 1966. The standards are definitely not dumbed down as many claim. Fuzzy, constructivist math teaching methods that rely primarily upon boxes of Legos and games are a cruel joke and I think are the real reason many kids don’t ever learn the math skills that they need. Rote drilling for skill, direct math instruction, and limited use of constructivist tools for reinforcement of skills already taught, work better for most students in my experience.

    The airplane problem would confound most sixth graders unless they recognized that this is really a description of a 3-4-5 triangle or that they could also solve this using the Pythagorean Theorem.

    Most teachers that I know no longer calculate grades and averages from handwritten gradebooks. Computer gradebooks do it all for them in a fraction of the time. All they have to do is enter a description of the lesson, the points earned and the points possible. The software does the rest. They should of course know how to do it the old fashoined way just same. (I wrote my own spreadsheet to do the same thing and saved myself the cost of the software.)

  16. Mike in Texas says:

    There are some really interesting quotes in this discussion

    Otherwise can we look forward to incoming 9th graders arriving with zero knowledge of math?

    If Tom can do the job in 2 days, who needs Sam and Lisa? Oh, right, they’re members of the teachers’ union.

    I really question how anyone going into the teaching profession cannot study or teach themselves how to do these problems. If they can’t learn, how are they going to teach?

    Its as if some of you believe teachers are stupid and ignorant. Oh wait, I forgot what website this is, many of you do.

    However, if you read the article it clearly states it is the State Board of Education making this decision, not teachers. I don’t know how it is in Va. but here in Texas the Board is made up of politicians. If you read even further into the article (as Mr.E pointed out) it clearly states that 97% of the teachers pass it anyway.

  17. Michelle Dulak Thomson says:

    Mike in Texas,

    The article also says that

    For years, would-be teachers have complained that parts of the Praxis I were irrelevant to their qualifications for working in the classroom.

    So how did this test that 97% passed anyway become such a confounded burden? (We aren’t told what’s the passing score, for what it’s worth. In my high school, it was 65%. I’d be ashamed to have gotten more than thirteen answers wrong on a multiple-choice quiz with such easy questions, and I’m not offering myself as a purveyor of knowledge to youth or anything.)

    Anyone smart enough to be teaching should be able to handle sixth-grade math, which is what we’re talking about here. I mean, presumably they’ve already taken and passed sixth-grade math, along with 7th, 8th, 9th, 10th, 11th, and 12th-grade math, plus whatever their college distribution requirements forced on them. And even taking this rudimentary quiz is too much?

  18. ragnarok says:

    Mike in Texas, some things are indefensible, and a college graduate failing this test is one of them. No ifs, ands or buts. If in fact 97% pass it, (on the first try?), then it should be easy to fire the failing 3%, and presumably the union would support it, correct?

    The politicians on the State Board presumably are responding to pressure by killing off this test. Who do you think pressured them?

    Not sure, but I think Mr. E is being optimistic in thinking that most California 6-th graders could do this test. It’s at roughly the level of Saxon Math 76, and I doubt that most California public schools could use Saxon 76 in 6th-grade.

  19. nicksmama says:

    Mike in Texas,

    “I really question how anyone going into the teaching profession cannot study or teach themselves how to do these problems. If they can’t learn, how are they going to teach?” nicksmama

    “Its as if some of you believe teachers are stupid and ignorant. Oh wait, I forgot what website this is, many of you do.” MiT

    Mike, I don’t think teachers are stupid or ignorant. My mother in law was a brilliant math teacher for 30 years! I just wonder why they (current teachers) aren’t upset that people who want to become teachers aren’t held to standards like passing the Praxis I. Why are they allowing their profession to be dumbed down?

  20. As a high school math teacher myself, I believe that every adult who steps foot in a classroom should be able to pass that test – it is practical math.

    The problem that I see with kids coming to the high school (especially those who had elementary teachers who were not strong in math) is that they go through the motions but have absolutely no clue why they are doing what they are doing!!!

    Elementary teachers who have no grasp of how numbers work do not pass this knowledge on to their students, and they come to me knowing nothing of patterns in math and why things work the way they do! My stronger kids can see how I change one thing in an old method and extrapolate how it applies to the new type of problem. This is why I believe many kids fall apart when they hit algebra. They cannot get by with just knowing the steps of solving a problem – they have to know WHY. You can’t understand algebra unless you know why number operations work.

  21. Mike in Texas says:


    I am an elementary school teacher and there is nothing in that test elementary school children will be learning. The state of Va. has deemed other things or more impt and I actually disagree with that opinion. What concerns be is how quickly this discussion turned to bashing teachers in general. Perhaps some people fail it b/c the questions are poorly written, as in the bathtub example above. You can measure how much water a bathtub will hold in kilograms. Just like the military measures jet fuel in pounds, not gallons. Would you? Who knows and I probably wouldn’t, but that doesn’t make kilograms wrong.

  22. Mike in Texas says:

    BTW, I didn’t learn the Pythagorean Theorem until I was in high school.

  23. Michelle Dulak Thomson says:

    Mike in Texas,

    The point is that this is middle-school math, tops; and anyone applying for a teaching position has taken middle-school math, four years of high-school math, and probably more math in college. Is it too much to ask that people who’ve passed half a dozen courses in this material bone up enough on it to pass a test that any ninth-grader ought to pass with ease?

    I remember the controversy over CBEST in CA some years ago, when the test was deemed racist because there were different failure rates in different racial groups. The SF Chron ran sample questions from the exam — which was quite a bit easier than this one, by the way. Then there were sample passing and failing essays. Suffice it to say that the person who wrote the passing one was fit to teach, and the person who wrote the failing one was not.

  24. Keith Whitescarver says:

    Jane, There are now so many tests that teachers have to take in Virginia, that you miss the key element. Elementary teachers (and special education teachers) still have to take Praxis II in Virginia. That is essentially Praxis I on steroids for elementary teachers. Ending Praxis I is just eliminating a redundancy. All secondary math teacher must take the Math Praxis II exam. That is a test of higher level skills.

    I don’t think we have to worry about some poor teacher candidate not getting a chance to pass enough tests

  25. carpeicthus says:

    The relentless negativity on this site can be ridiculous. It looks like they’re reallocating resources and, what do you know, they’re making the tests relevant to someone’s field HARDER. Yes, I want every teacher to be a Golden God from On High, but in real life I care more that a high school English teacher can pass college English than knows how to divide fractions.

    But please, don’t get off your high horses, you might hurt yourselves.

  26. P. Abel says:

    Dear carpeicthus,

    Nice attention to detail 🙂

    I for one have a problem with K-6 teachers who TEACH math not knowing any math content.

    Isn’t that an issue worth discussing?

    Isn’t asking the following question legit: What math content will they be required to know?

    Shouldn’t science teachers be required to demonstrate mathematical knowledge as well?

    Social studies? or oh no, maybe econ/stats aren’t part of “that” subject either.

    Or should we insist that “ONLY” math teachers know math? Fine, pay us more if it is so onerous that “NO ONE” else wants to be responsible for knowing very basic math concepts. And whoops there go cross-curriculum projects…they might contain “math”.

    Tell me, how exactly does one calculate interest on their mortgage or loan? Doublecheck their bill at WalMart? Figure out a tip? Check a stock portfolio? And as a good citizen, it’s nice to know the meaning of statistics in an article or how to figure a dose of medicine for my child based on weight. Many of those concepts meet or exceed the math on the Praxis I yet all are concepts we as adults should have mastered.

    Or is it easier to just gripe about the negativity rather than look at the broader social implications of “mathless” teachers.

  27. Engineer-Poet says:

    Mike in Texas:

    Its as if some of you believe teachers are stupid and ignorant.

    Maybe some of us have direct, personal experience with teachers who were stupid, ignorant and also arrogant as all get out.  Maybe we want them kept away from the front of the classroom.

    FWIW…. 40/40, never touched a pencil.  That set question had me for a second until I noticed the zero in #4, and the substitution of “then” for “and” made me hope that the copy editor for that test isn’t working there any more.  (If you were looking for a reason why basic skills are so important, that typo brings it home.  Worse than HS grads who haven’t learned such things are teachers who can’t even begin to teach them.)

  28. Carpeicthus –

    Sing a couple of verses of “We shall overcome”. I’m sure you’ll feel better.

  29. Some teachers are brilliant, some are stupid, most are inbetween. But the system discourages knowledge-based teaching, especially at the early grades, and persists in Prussian regimentation under the guise of socialization. Consider that someone with the knowledge and passion to excel in math, or physics, or history usually becomes a mathematician, physicist, or historian – in industry or post-secondary academia – rather than a teacher. There are economic reasons for that, which means a shortage of specialized teachers (especially science), and an overabundance of educators depending on an ill-defined “emotional intelligence”, which usually means social dominance.

  30. Electroniceric says:

    I have to agree that if this test is representative it needs some serious debugging. I am a physicist, and I spent 10 minutes trying to figure how to use the “points” described in question 16, did some definite guessing on what “profit over” meant in question 6, and question 21 has a grammar error. So all this caterwauling about “they’d better pass this math test” seems pretty overstated until the test itself is decent.

    Conflating the drop in the mathematics requirement for VA teachers with the poor math performance by students is basically trying to assign causality to events that are merely correlated. Relaxing the requirements on teachers has to do with difficulties in teacher recruitment and more stringent requirements for hiring teachers, not necessarily the mathematical proficiency of teachers.

    In an ideal world, I’d like to see everyone be proficient enough to pass this “advanced” math test. On the other hand, I’d hate to throw out an excellent English teacher who simply happens to be terrible at math. And I certainly don’t think that this proves anything about anyone’s teaching. It’s basically saying, “boy, it’d sure be good if more people, including teachers, knew more math.” Yeah, sure that’d be good.

  31. Is this PRAXIS I test taken by elementary and high school teachers? ELED teachers normally teach in self-contained classrooms (at least where I live), meaning they must be conversant with all subjects. This includes math. I don’t see how ELED teachers can be exempted from a math test.

  32. ragnarok says:


    The point here is that the bar is set so low for this test that *anyone* who has trouble with it should be looked at very sceptically. If someone cannot do things that the vast majority take for granted (see P. Abel’s post above), their basic competence is open to question.

    It’s quite true that the test is sloppily worded, but I can’t believe that you seriously think that it’s so bad that you might fail. It’s this kind of rationalisation that lowers expectations and hurts the children.

    Since you’re a physicist, let me mention a physics high-school textbook (don’t remember the name) which asked students to add weight to mass. This is absolute nonsense – wouldn’t you want the teacher to know enough to disregard this crap? Similarly for math.

  33. I just wanted to point out to everyone who are hyper-analyizing the “Advanced Test” that this is not the actual math portion from the Praxis I. This is a product produced by A company who’s vested interest is in selling test prep services. So I suggest that we take these paprticular questions a little less seriously.

    I can’t speak for you, but I know that I’ve never taken a standardized test (PSAT, SAT, ASVAB, LSAT, California Bar, MSAT, CBEST, RICA, & more) that I didn’t have something to quibble about. It’s inherent to these tests.

    Part of the trick in performing well on these things is in being able to deduce the intent of the question’s author over whatever interpretations that you may be able to parse out from them. If you enjoy difficult tests then I suggest the multiple choice section of any Bar exam in the US. Many questions are supplied with all of the answers being either all correct or all wrong. The trick is finding the best choice among what are often very similar answers.

  34. BTW, I didn’t learn the Pythagorean Theorem until I was in high school.

    Posted by: Mike in Texas at June 28, 2005 03:22 PM

    I also didn’t learn the Pythagoreum Theorum until high school, but that was then. Today it is expected that a California middle school student know how to use it. No reason why not to. We introduce exponents and one step equations at fifth grade and they begin area and perimeter before that. We expect much more of kids today than we used to, regardless of how ready they are to learn it.

  35. Ragnarok,

    “Not sure, but I think Mr. E is being optimistic in thinking that most California 6-th graders could do this test. It’s at roughly the level of Saxon Math 76, and I doubt that most California public schools could use Saxon 76 in 6th-grade.”

    I don’t think that most California sixth graders would be able to pass the test. Many of the problems are still to abstract for most twelve years olds to conceptualize. However, California does require that the concepts be taught at those early grade levels. California is very big on spiraling math material so that the kids are introduced to concepts at a young age and see it again and again, year after year. Some get it very early, and others never seem too.

    My 14 year old took the test this morning. It took him about 30 minutes to complete it. 31/40, a passing score.

    The airplane/triangle problem was one of the ones that tripped him up. His other errors pretty much paralelled the problematic questions that the above posters questioned. When we reviewed his answers, it was evident that his problems were in his understanding of the question’s wording, not in the underlying math skills. With minor prompting from me he solved all of the others except for the g.p.a. calculation, which is a pretty abstract problem for a kid.

    As for the triangle, I suggested to him that he draw a picture of what the airplanes were doing. He quickly figured out the rest. Pythagorus wasn’t an obstacle, just a still developing ability to think abstractly.

  36. Richard Brandshaft says:

    Ouch! You got me on not noticing one of the choices for the denominator had a zero. I wish I could say that was my most embarrassing mistake ever, but alas no. It does go on my lifetime list, near the top.

    Yes, I know about right triangles and the Pythagorean Theorem. The question was whether a non-techie 20 years out of school would instantly see the underlying problem (or remember the Pythagorean Theorem). I had a demoralizing experience a few years ago: I saw Maxwell’s equations on a T-shirt and realized I could no longer read them. True, I hadn’t used them in nearly 40 years, but they once were so basic to me. I find stuff I learned in school often falls into two opposite categories: some become so much part of me I no longer have to think about them. Some I never used or thought about since and have forgotten. It’s easy to say “everybody” should be able to answer questions you find easy. But is it justified?