January 02, 2010

How Good Are UW Students in Math?

UPDATE: A similar exam was given in another big lecture class: Earth and Space Sciences 102. Here is the exam and results: http://www.atmos.washington.edu/~cliff/ESS%20102%20Math%20Assessment%20Solutions-1.pdf Unfortunately, there were very similar results, with poor performance on many questions. Thanks to James Prager of ESS for sharing the exam with us. This is a big problem faced by all professors at the UW.

As many of you know, I have a strong interest in K-12 math education, motivated by the declining math skills of entering UW freshmen and the poor math educations given to my own children. Last quarter I taught Atmospheric Sciences 101, a large lecture class with a mix of students, and gave them a math diagnostic test as I have done in the past.

The results were stunning, in a very depressing way. This was an easy test, including elementary and middle school math problems. And these are students attending a science class at the State's flagship university--these should be the creme of the crop of our high school graduates with high GPAs. And yet most of them can't do essential basic math--operations needed for even the most essential problem solving.

A copy of the graded exam is below (click to enlarge) and a link to a pdf version is at:

If you want a blank version to give your kids and friends, it is at:

Consider these embarrassing statistics from the exam:

The overall grade was 58%

43% did not know the formula for the area of a circle
86% could not do a simple algebra problem (problem 4b)
75% could not do a simple scientific notation problem (1e)
52% could not deal with a negative exponent (2 to the -2)
43% could not do simple long division problem with no remainder!
47% did not know what a cosine was.

I could go on, but you get the message. If many of our state's best students are mathematically illiterate, as shown by this exam, can you imagine what is happening to the others--those going to community college or no college at all?

Quite simply, we are failing our children and crippling their ability to participate in an increasingly mathematical world. I have even heard from carpenters complaining they are finding it difficult to hire new apprentices that can do the calculations needed to build a house. When I buy something, cashiers have difficulty making change. During the past several years professors such as myself have seen a progressive decline in math skills. Math remediation rates have soared at community colleges to the 50% level. The math tutoring industry has exploded (over 300% in the past decade and a half). Foreign nationals are filling our technical positions here in the U.S. We are in deep trouble. And it has to be fixed or our state and nation will slip into second class status.

This is a national problem--WA state is not the only place where math ability has floundered. But I am amazed how major foundations (e.g., the Gates Foundation) and big-time politicians are so clueless about what is going on, and how quickly major progress could be made.

Now why have things gotten so bad and what can we do about it? As in any disaster the causes are multiple. And let me say at the outset, it can be fixed. A lot can be done quickly with only modest resources.

One problem is curriculum and textbooks. Starting in the mid-90s colleges of education and "curriculum specialists" in districts become enamored with a new way of teaching math--called reform or discovery math. Instead of teaching the basics --followed by practice to mastery, the idea was that students could only learn math they "discovered" themselves. Working on problem sets was considered "drill and kill." Direct instruction by teachers and equations in books were out. Long division was out. Integrated math books where all topics were swirled together were in. Group learning and playing with objects (manipulatives) were in. Describing one's through process was considered more important than getting the right answer. Most of this proved to be a disaster, but those pushing it--professors in education schools and district curriculum types--were believers, even though there was no empirical proof that it worked. Many felt that the new reform math could serve social and political goals--helping the disadvantaged or those that traditionally could not "get" math. (as if minority groups were not capable of learning the same math as the privileged!) And it didn't work, as the exam below shows. (Ironically, their new approach hurt the disadvantaged more than others who could afford tutoring and private schools.)

Another problem is that many teachers are ill-prepared by colleges of education to teach math. Many elementary schools teachers have weak math knowledge and large numbers of middle school and high school math teachers do not have degrees in math or technical subjects. And of course the ill-founded reform and discovery math approaches are pushed into their heads. Now these weak teachers could function marginally if they had good textbooks and curriculum, but they don't. And the curriculum and books are so poor that students can't take them home to learn the material on their own or enjoy the assistance of their parents. It would be hard to design a worse system.

And, of course, there are a host of other problems--too much video games and Facebook at home--as an example. A lack of a sense of student responsibility by some. An out of control "self esteem" movement among the education community and some parenting experts. Larger class sizes don't help (although Asian countries seem to be able to do a good job with big classes) And more.

Now, all this can be fixed...some of it very quickly. But before I talk about that I should acknowledge some positive changes going on right now.

Terry Bergeson, an enthusiastic support of reform math and the ill-fated (and poorly constructed) math WASL, was voted out of office (Superintendent of Public Instruction). Randy Dorn, who does not share her ideological baggage, was voted in, and has already made some positive changes. Recently, his office made a major public announcement saying that the standard algorithms for addition, multiplication, subtraction, and division WILL be required and tested on the new exam his office is creating. (It shows you have far we have gone into the dumps when something like this is a major advance!) He has also recommended some excellent textbooks on the elementary and high school levels for acquisition by state districts. And he has begun to restrain the antics of some holdovers from the Bergeson era.

With support by the State Legislature (who got sick of Bergeson's shenanigans) and the State Board of Education state math standards have been enhanced (improved, but still not as good as they should be) . And several districts, like Shoreline and Northshore, are moving to excellent curricula and textbooks.

Importantly, parents have begun to understand how the past system has really ruined their childrens' math education.

But the battle is hardly won. Some districts are going backwards, not forwards. The Seattle Public Schools, has poor discovery math books at all levels; the Seattle School Board voted in the Discovering Math series last spring (4-3 vote), even though the State Board of Education found it "unsound." The math coordinator for the city does not have a degree in math or science and is a real discovery advocate. Recently, the Seattle Board has voted to spend hundreds of thousands of dollars for tutoring from Sylvan Learning (talk about admission of failure). If your child is in this district you have a real problem in this failing district. The irony is that this district has a large proportion of minority students and they picked the worst possible curriculum for them.

But not all disaster districts are underprivileged. Take Issaquah. Issaquah has a Superintendent (Steve Rasmussen) and curriculum staff that are determined to railroad discovery math through the district. Currently, Issaquah is using one of the worst elementary school textbook series (Everyday Math) and their handpicked Math Adoption Committee has voted unanimously for the "unsound" Discovering Geometry and Algebra series. They even hired an outside contractor that supported their decision (you would not believe the reasoning. For example, Discovering Math would fit better with their current middle school math, Connected Math. So because the students are so weak for the current offerings, they need a weak math curriculum in high school!). If I was a parent in that district, I would let these people know that their ideological math decisions are simply not acceptable. High tech parents will have to pay to get their kids properly educated by Kumon or Sylvan or see them flip burgers for the imported high tech employees of the future.

Anyway, my 101 exam and other assessments given throughout the U.S. has shown that our nation and state have been running an unfortunate experiment on our children that has failed. It will not cost much to change the curricula and standards and only a modest amount to switch out the textbooks. These change will have a major impact on our childrens' math education.

Some people (like the Seattle Times) are fixated on testing our kids with high stakes tests, but this doesn't make sense (and is even cruel) if we don't have reasonable educational material in places taught by teachers who know their material. I would offer it makes more sense to test the teacher first for a competent knowledge of math! It is a very American thing to think you can change everything with a test, when the whole foundation is rotten. Amazingly, some business groups, like the Business Roundtable, don't understand this and are fixated on the exams and not the underlying curriculum issues.

So fix the curricula and books first. Properly educate the teachers second (and allow technical people to easily get into teaching). It will be much harder to change the mindset of the education lobby and particularly schools of education and "curriculum specialists" who tend to have an ideological approach without empirical support. I am embarrassed to admit some of the worst of the worst are at the UW. But the disaster that is our schools of education or K-12 education specialists in the math dept should be reserved for a future blog.

Finally, if you are interested to learn more about this subject go to the website of a statewide group interesting in fixing this situation:



  1. You got it, Cliff. I teach middle school math in SW Washington. I am appalled at the lack of preparation my students have when they enter 7th grade, and the attitudes of their previous teachers ("Oh, it's ok, I wasn't very good at math..." What?)
    I try hard to give my students what they deserve- knowledge and practice, but it seems like a losing battle at times. Thank you for leading the charge in Seattle.

  2. I cut off all the answers and printed out copies for me, my husband, and my 17 year old who is a junior. Rather happy to see that my son did best of all--100%. My husband got just one wrong. I couldn't remember how to do 1d, 1e, and 3b...but that still leaves me better than kids who took "math" 25 years more recently than me.

    The poor result with long division is the one that strikes me the most. That's a skill I use in day to day life. I've been working with a group of the brightest kids in my 12 year old's class, and I'm happy to say they are getting pretty good at it. Shouldn't take parent volunteers to make that happen!

  3. Argh, depressing. And it's not just math. I had an adjunct faculty gig at a large university in southern California in the '90s, teaching an Electrical Engineering class around one of my projects. An average of less than one senior per quarter had ever picked up a soldering iron, and written communication skills were horrific. I ended up spending a surprising amount of time just explaining what engineers do... and trying to get basic skills across that any self-respecting Maker (or electronics hobbyist, in the parlance of my day) would have down pat by the age of 12.

    But only two of the faculty members in this very large department had ever worked in industry, and one even confided that "undergraduate engineering education is not really a priority here."

    Tell that to the students, and the parents who pay the bills.

  4. I live in ISD. Who can I talk to?
    "Discovering" math is like discovering religion by praying to different gods and seeing which ones answer.

  5. Concerning, yes, but there are two unsupported assumptions here:

    1) That these students should have strong math skills. In reality, people get into UW based on aggregate grades to which math contributes *maybe* 25%, probably less. Many of those who scored poorly on this test would probably run circles around all of us in French or history or something else.

    2) That the test reflects students' genuine best efforts and the state of their knowledge. I know that, as a student many years ago, if I suspected the test wasn't related to my grade, I would have put zero effort into careful reading or double-checking answers. And if I suspected the teacher was expecting poor results, I certainly would have accommodated.

    It's still worth being concerned about, but it's probably smart to hedge a bit on accepting these numbers as accurate representations of math skills in the state's best-and-brightest.

  6. I interviewed some recent UW grads for an intern position at a local biotech company. The position entailed making solutions of small molecules (i.e. 250-500 g/mole molecular weight). I asked them to solve a simple problem where I gave them the MW and a mass that had been weighed out and asked what volume of solvent was needed to make a stock solution of a given concentration in moles/liter. Out of 4, one of them mostly got it correct, the other 3 couldn't solve it. This should be taught in 100-level general chemistry (or even in high school chemistry). Needless to say, I was disappointed.

  7. Keep up the good fight Cliff!

    I'm a 1983 U of W Civil Engr graduate (licensed PE) - please continue trying to save our math skills.

    It pains me to see so many of our graduates nearly incompetent in math and science as they leave high school.

    Thanks again!

    PS - occasionally I will 'test' the cashier and give them the change to my purchase to 'round it up' to a $1, $5 or $10 bill. Only about 10% get it. Some will just look at me, others will say, "I can't do that" as they had just entered in the bills that they saw in my hand, saying "my register won't let me change what I've already entered".

  8. There's another possibility here. This is one anonymous, non-graded test in a low-level class that is not math-heavy (I'm assuming a 101 class is an introductory, but having never taken it myself I could be completely wrong). The results could easily be explained by students simply not caring ("Why would I bother spending time on this test when it doesn't contribute to my grade?"). There could even be a malicious component here, since the top of the test says the outcome will determine how you develop quizzes and homeworks for the class. Students have an incentive to do poorly even if they do know the correct answers because it means an easier time for the rest of the class. If you believe they have sub-elementary math skills, you're probably not going to give them difficult math skills during the quarter.

    Thinking back ten years ago when I was in school, this sounds like a class I would've taken junior or senior year to fill credits (I actually took "Speech and Hearing Sciences 101" at my school, but AtmoSci 101 would've been just as good). With an over-full workload of CS and engineering courses, the last thing I'd want would be to have my "blowoff" easy credit course add even more to the load so I certainly would've thrown the test.

    That's not to say that you're wrong about the state of math education. "Discovery" math is complete BS and should be killed dead, but I don't think you can rely solely on this test as evidence of such.

  9. Cliff,
    Any more demographic data on the 202 students, in-state vs out, US vs students from other countries? Might be interesting.

  10. Excellent (and unfortunately dismaying) post. I have often thought that for reading, writing, arithmetic, and mathematics we should not grade. Rather simply note what level the student has mastered. i.e., reading, grade 12 month 6, Algebra - 3rd semester, Writing - college freshman level. Those functioning below their grade level would have some real information to assess themselves.

    And almost every student who attends classes, turns in homework in a timely fashion, behaves appropriately to teachers and fellow students should graduate - employees and higher education asks for a grade transcript to see what the person has accomplished academically, high school graduation itself is somewhat of a social thing.

  11. Hit the nail on the head Cliff. I graduated from Garfield High School last June and my math experience was very poor. The math program they have kids following there is hardly geared towards teaching kids math. . . my score on your test would have been around your class' average. I find this most embarrassing because I learned much of it in middle school but forgot most of it by the time I graduated.

  12. Ugh! Been stuck with discovery math (Intergrated Algebra and Geometray core I & II) for the past two years in Northshore School District. Our teacher gives most of what it wants you to discover because she knows discovery doesn't work. It is so mundane how many tasks it has you to discover how certain things work.

  13. Cliff, I see you ask for High School along with city and state. Do you have a breakdown of those numbers? I'd be curious if it is like our weather-- quite variable by location.

  14. Cliff,

    I graduated high school in 1985, from a school in Raleigh, NC. I took the quiz based on the PDF (without cheating) and missed ONE answer (1e). Given that I don't use much math in my technical writing work, I feel reasonably good about that score.

    My daughter is in fourth grade and we are constantly having to enrich the math education she's getting. I don't think it's the fault of the teachers; the old-school teachers would like to do more with many of the kids who are capable, but they don't have the time to get into deeper material, and they are discouraged from doing so.

  15. Oh my God, I completely bombed on that test. I have a high-school education from North Kitsap and it taught me English quite well but I don't recall ever even HEARING about such things as negative powers. I only got six answers correct on that test, including the "Name" and "High School" questions at the top. I admit, math was not my strong point but it was in part because of the way it was taught. I dig math now and wish I was more fluent in the language of numbers. Two to the negative power of two? Crazy.

  16. I'm a math TA at a similar institution and I can say that the results are not because the students simply don't care. Also these are the basics---I'm sure some of these students are great in history and french, but even if that is their focus they still should know this.

    Someone told me this theory once:
    The people pushing for reform math are the people who did poorly in math as a student. These people think that they had terrible teachers and they decided to change things. In reality, it is not that they had bad teachers it's just that they didn't keep working until they understood it. The result is that we have these people who have a fundamental misunderstanding about math who have changed the curriculum.

  17. Years of structuring teacher salaries based on seniority versus the value of the skills that the teacher possesses also has an effect. For example, a PE teacher with 20 years of experience will get paid more than a Math teacher with 5. The math teacher will probably get more compensation outside of education for their required level of knowledge, so why bother trying to break into the "educator mafia"?

  18. When I looked over the evaluation test, I was wishing this was a joke. Sadly, I knew that this is all too common. I currently attend a small university and work as a tutor for the resource center set up by the university. For those of you that say that the students were not taking the test seriously, and therefore, did more poorly than they are capable, I have tutored students entering first semester calculus and I have to say that the test given by this UW professor is right on the mark.

    Fortunately, by the end of the semester, most of the students that had regularly come in for math help had improved substantially. But the writer of the article is right in saying this phenomenon is not localized to Washington. This is an article written by a professor in Maryland who has noticed the same trend: http://www.aps.org/publications/apsnews/200910/backpage.cfm.

    To add to this, there is a professor in the physics department at my university that has given a similar test, (although a bit harder) for the past twenty years and has seen the same trend of declining scores on his pre-course test.

    I am glad there are still reasonable people out there who recognize this problem and want to fix it.

  19. I have to say that I disagree with your assumption that it is the current, in-vogue teaching methods are to blame for the poor results.
    When I was learning math in middle school we used mandibles, group work, discovery etc. Yet I scored 100% on this test.
    My mother and father grew up back when the more traditional methods were popular but neither of them would have been able to get even 50% on this test, and not just because they're quite rusty on the material.

    We've definitely got a problem with our math education, but I would not jump to blame the discovery technique of teaching. Done well it can be quite effective

  20. Very depressing, those are all problems that any high school sophomore should be able to do.

    If that wasn't bad enough, also consider that Prof. Mass's students have chosen to take a science class with the justified reputation for requiring math. Even for the science distribution requirements there are sciences that don't require quite so much math, and students who know they are weak in math are presumably taking them instead of Atmospheric Sciences.

  21. if I suspected the test wasn't related to my grade, I would have put zero effort into careful reading or double-checking answers.

    But this test should require practically no effort. It's been twenty years since I had high school math and just thinking up wrong answers would take me more effort than giving the correct ones. This test is dead easy, or at least it should be for any one who's had ONE year of high school math.

    The thing that depresses me the most is that this makes it clear that just helping my kids learn their math is not enough. If half their classmates in college can't handle these basics then they'll have a heck of time getting much of a college education.

  22. Hi there!
    I'm from Russia, Moscow... Old enough to know better... You see, never in my life could I imagine such statistics. May be in the deepest country... Tundra... Siberia... Bears...
    We used to know a very old quotation from the father of Russian science (Lomonosov, in the times of Catherine the Great, if you know): Mathematics is worth learning just because it's putting the brains into order.
    Who's going to fly to the Mars? Hindus?

  23. While I agree with your basic points about math education, your little quiz is grandstanding.

    Students who haven't done math for a bit (presumably at the beginning of a term) may not remember which sides of the triangle are part of cosine, for example. So I don't put much stake in your quiz.

    A better argument would be how students struggle with math in atmospheric science courses. What math is required there? Show some examples from your courses, so we can see what level of math competency is required to do meteorology.

  24. Where's The Math is a little too militant of a group. There needs to be skill and drill, but there also needs to be discovery based math which is closer to the real world.

    Skill and drill did not work for me in the long run. I need to be able to apply what I learned. Skill and Drill doesn't teach that.

  25. I am not impressed with the math text that my son has (3rd grade, Seattle Public Schools). The teacher is great but I don't like the textbook. We worked out a plan where my son takes an online math course instead. This is working much better for him (and he tested into 6th grade math!)

    The Everyday Math curriculum is boring. I can see it zapping any interest in math that a kid might have.

  26. ... declining math skills of entering UW freshmen...

    Why are you letting such underprepared children into your university? Is it just because of tuition? Maybe if the colleges and universities kept their standards higher, the K-12 establishment would follow suit. I realize it's elitist and undemocratic, but do you want excellence or not?

    Most of this proved to be a disaster, but those pushing it--professors in education schools and district curriculum types--were believers, even though there was no empirical proof that it worked.

    Not so. A little digging at your own university would reveal perhaps thousands of pages of empirical evidence of the discovery model's efficacy. Of course, it'd also reveal the efficacy of many of the other models, including direct instruction. Have you discussed such issues with the UW's Ed. college?

  27. Maybe if India did fly to Mars it would shake us out of our complacency, just like Sputnik did.

    Anna, my daughter is in 3rd grade in Seattle Public Schools too. What's the online math course you're talking about? Is it generally available? This is getting off-topic; feel free to e-mail me -- kkt at zipcon dot com

  28. Cliff does a great job of describing the problem here and some of the root causes of it. As a parent to two young children and a college student, I'd also like to add that a large part of what's causing this is the concept in many parents' mind that education is a process that begins and ends at school.

    Parents/guardians need to establish an environment at home that tells the kids that their education is of primary importance. Some of the important elements of this are; talking daily with the kids about the subject matter at school and their progress with it, limiting (or eliminating) TV time and video games or making their access contingent upon performance, keeping a channel of communication open to the teachers and staff at school and/or being involved at school to the extent that one's schedule allows. Most importantly, we as a society need to demonstrate a respect for education by compensating teachers in a way that attracts the best and requiring an administrative environment that recognizes their performance and allows for poor performers to be encouraged to find new fields to work in.

    Cliff mentioned comparisons to asian countries in his post, and you can be certain that some of the items I mentioned are inherent in those households.

  29. I'd also like to add that a large part of what's causing this is the concept in many parents' mind that education is a process that begins and ends at school.

    The idea that parents should stay involved with their kids, limit TV, get involved with the school are good ones. And, of course, it's best if schools can attract and adequately compensate the best instrutors.

    All the same, when I went public high school, my parents quite litterally paid no attention to what went on in my school. They didn't talk to me about my math classes, and my math teacher was very recently a PE teacher with no special background in math and who clearly was barely learning the material a week or so before he had to instruct the class. Nevertheless, I came out of public school perfectly capable of answering all questions on Cliff's quiz without trouble.

    The main difference I had in my favor? My textbooks.

    It didn't matter that my instructor was fairly incompetent. I read the text and did the problems and came away with the necessary basic skills. My parents didn't pay attention to my education. They trusted that I'd get the necessary preparation. Thankfully, I did.

    It shouldn't require super parenting to deliver students a satisfactory education. The part that makes the maddest about the decline of math education is that I feel like I have to watch my kid's schooling like a hawk ready to swoop down at the first sign of failure. My folks didn't have to do that. It doesn't have to be this way. Really.

  30. I just took the test and got 100%. But I did graduate HS in 1973...

  31. Right on, Cliff! Some things CAN be discovered, some cannot. Math is not one of those that can. Keep pushing, writing, pressing...esp. the Gates' maybe someone will hear...

  32. I am not afraid to supplement my kids' math education, as I do their reading and writing at home. The problem is that reading and writing come more naturally to me than math. I did make it through Geometry (which I loved) and TO trigonometry before I got bamboozled in high school in the '80s. I just don't know how to go about teaching my 2nd grader the things that are missing from Everyday Math.

    Can anyone steer me (us) to a great, accessible math curriculum to use at home, and not confuse the poor kiddoes?

  33. While I agree that this is shocking and dismaying, I don't think the blame lies entirely with Discovery Math textbooks. I just gave this test to my 14 year old daughter (whose K-8 school used Discovery Math series) and she scored a respectable 88%. She did have trouble with problem 4b, as did most of your students.

    Your points are well-taken - most resounding to me is this:

    "Many elementary schools teachers have weak math knowledge and large numbers of middle school and high school math teachers do not have degrees in math or technical subjects."

    If you do not have a basic understanding or admiration for your subject, how can you successfully teach it? I think this speaks to a larger societal problem in the US: we do not truly value mental effort in and of itself; we accept that the machine will do the work for us, and we want the fast, easy way. I am hoping, Cliff, that you are seeing the last of the fallout from "teaching to the WASL," but I fear it may not be so.

    For Shelley Pasco: check out Singapore Math or The Art of Problem Solving
    http://www.artofproblemsolving.com/ for some more challenging curricula.

  34. This is the most interesting blog. Also I like the weather thing, I was at UW unknown decades ago and I think I took a course from your dept (BSc Botany 1977).

    Anyway, another problem in University (and high school, I guess) is a bimodal distribution of math ability. Either they get it or they don't (maybe that means they got good schooling or they didn't). When I teach population genetics (the most math oriented class in biology) its always a tough pitch. So I mix math questions with the usual biology descriptive stuff. It doesn't really average out, the people who do math have high grades, but the people who don't have math ability don't fail.

    My 16 year old son got 100% tonight. It will make him confident when he goes to UBC.

  35. There isn't one answer to the way we should be teaching math. Some students can read a text book and learn the math. While others need direct instruction. Everyone needs a little drill and kill to learn basic facts and become fluent in numbers. But only teaching drill and kill doesn't give kids the necessary skills to understand the math. I am a second grade teacher using Math Expressions for the 2nd year in a row. I think this text book is working well for the majority of the students.

  36. Spot on Dr Mass. Our twins are in kindergarten. Went to your website and got the CA math test for 1st graders. Have been "testing" & teaching the twins since September. Am going to have to get the 2nd grade material soon. Always tell them "Math is Easy". Lately, have also told them "physics is easy and verrry interesting". They gobble it up, and it helps that they'll do anything for ice cream.

  37. After thinking about this post for awhile I can't resolve one issue. SAT Math scores for the middle 50% of incoming freshman at UW range from 570 to 680. It's been a long time since I took the SAT, but I wouldn't think that the students that bombed Cliff's quiz would be able to pull off those scores. Makes me think there are a few other variables out there that we are not talking about.

  38. 58% is pretty dismal. The UW physics department (11X series) seems to have similar grades (if not lower, from what I can recall) for their exams as well as the first year UW gen chem (pretty math intensive in the 1st two classes of the 3 quarter series). Anyone recall doing I.C.E. in gen chem? :)

    Although I'm by no means a 'math genius' and I can't crunch derivatives in my head while I fall asleep, I was really surprised when I took my physics 115 class at the UW last quarter and had to help a few students with their triangles and to manipulate numbers around to solve for X. I thought trig was something we all should know when taking a science class (esp if its the introductory year long series) and I was a bit dumbfounded that some students did not know the "SOH CAH TOA" rhyme (sine=opposite over hypotenuse, cosine=adjacent over hypotenuse, etc.) and did they did not know how find degrees in a triangle using the inverse. What was also sad was that many (if not all) text books for the basic science series at UW also have refreshers in the appendix or front/back cover - no one that I helped bothered to look in the book for this information. The gen chem, physics, and even the biochem books all have appendices that show these basic math refreshers. I struggled with math, but these were basics that I knew by heart and thought it was something science students would know as well.

    As a student from a horrendous math background (public school math in Hawaii? HA!), I do understand how frustrating it can be for everyone (students, faculty, etc.) because math was hard for me and all I needed was someone to explain the steps in math problems that textbooks or math professors do not write out because they assume students know each step in obtaining the solution. Another issue that I had in learning math after high school was the fact that many professors had sloppy handwriting and for many students, it can mean the difference between writing down letters and numbers that look similar (ie - 4, h, and 7s, etc.) that can make no sense when reviewing the material at home. This was a problem for me at UW and at the community college, where even the letters X would be written so messy that it was inadvertently re-written as a Y by many students. This is a problem I am having now in my Q SCI 291 class as well - messy handwriting. When the math or math intensive science class is 100+ students, it gets worse, with the small overhead and messy writing. (Sorry to vent, but these are math issues that have affected me in the university level, and I always mention this in prof reviews at the end of the quarter).

    I took the atms 101 class about a year ago with Prof Patoux, and I think he also mentioned that math-weakness was a huge issue in the atmsci dept as well. This blog is great, and I still keep up on reading it almost a year after my introductory weather class.

  39. New Seattle Mayor Mike McGinn has a website up to allow people to present their ideas to the community at large and to allow the community to vote for them, with the promise that the ideas that get the most votes will become priorities for his administration. You can find the Education Ideas portion of that website here:


    If someone with great familiarity with this issue puts together a "Better Math Curriculum" proposal here, I will vote for it.

    Many thanks, Dr. Mass, for your advocacy work on this issue.

  40. I easily scored 100% and graduated from HS in 1984. This was NOT a hard test.

    Thank you, Cliff, for keeping this in the fore - we have a second grader in SPS and math is our biggest concern - we can supplement fairly easily now but as each year progresses, the issue will get worse.

  41. I recently returned to college and was ranting to a good friend of mine about the absolute lack of critical thinking skills exhibited by the students of various ages. Subsequently, he linked me to this post and I have to say, I'm not as much shocked as I am disappointed.

    Math and Sciences have always been two problem areas in our educational system but the problems continue to compound and the disparity of the standards of education between our nation and others continues to deepen.

    I personally feel that a lot of problems stem from standardized testing, something which wasn't prevalent when I went to school. These days teachers have too much rigidity in what they teach and how they teach just so politicians can hold up the annual test results for the public. Nevermind the fact that such tests have such a low benchmark that they're utterly useless (here in Virginia, the 11th grade Standards of Learning exam is really only a 6th grade level test).

    In addition to your comment about teachers being ill prepared to teach math, Cliff, I would like to add that there are some teachers who are very qualified to teach math but are being shoved into other subjects instead. Such was the case with my former 7th grade Algebra I/U.S. History teacher with whom I remained friends until he retired and moved away last year. He held a Bachelor's Degree in Mathematics and loved teaching math. His students traditionally did better than other 7th Grade Algebra I students, school wide. However, two years after I was in his class, due to an influx of "new" teachers, many of whom claimed to be math teachers, he was taken out of his Algebra I class and forced to start teaching English and History instead. He never taught Algebra I, or any other math, again.

    As others have said, thank you for keeping this topic out there in the open.

  42. My son was enrolled in an International Baccalaureate (IB) school K-2nd grade. He disliked math despite "an aptitude" for it, and despite his penchant for solving problems "the wrong way" (his 1st grade teacher said! right answers but the "unexpected approach") He is now at a Waldorf school for 3rd grade where we were told in Sept that he was behind a grade level in math, despite the $$$ tuition etc spent at the IB school. However, now he loves math, talks about math, etc. walks around doing math problems etc. Perhaps ithe Waldorf approach is on to something after 75+ years--there is much recitation etc.

  43. I agree that the UW needs to look harder at the math skills of those entering.If this is the quality of the students entering the UW then I am even more convinced that my 15yo will be going to WSU!

  44. My husband, who teaches math at a Seattle area suburban high school, has been pressured by the administration to drop allegations of cheating against a student. (His evidence? He gives two different similar exams, an "A" and a "B", and the kid wrote the correct answers to the A exam on the B paper, with no supporting work). Why? Because the kids' parents are from a wealthy neighborhood, and the administration doesn't want do deal with them hassling them - it's easier for them to not stand by their teachers and fold on this issue.

    You could have perfect teachers and a perfect curriculum, but as long as kids from a certain social class know that their parents and their money talks, you're not going to get kids who take any aspect of education that seriously.

    What I always wonder - what will happen to these kids in the Real World? Will they continue to expect to be bailed out when they lie and cheat? By whom?

  45. I can't agree any more, I'm currently a Sophomore in high school, MPHS, and I know that if the kids in my junior level math class were to take the test most would fail. I was lucky to get teachers that covered negative and fraction exponents, as well as sine cosine and tangent.

    We changed just this year from the integrated or core program, and I have to say that I like the current program much better. It gives the equations point blank, none of the "you have some experiments to maybe figure out the equation" nonsense.

  46. The question is: how do we change people's perception of math? Just say "math" and 9 out of 10 Americans will roll their eyes. The most common response is: "oh I was never any good at math".

    And if the parents can't do math why would they encourage their kids? Or if they do, how will they be in a position to judge the quality of said education? Seems like a problem that will worsen with each generation.

    sorry for the following length

    After 8 years at a software job I returned to UW to study math, and finish a B.S. degree. I finally hit a wall at work that I couldn't get around due to the math and choose to go back and "get it over with" :-).

    My first class was precalc (math 120) and it was the hardest class I'd ever taken. I took me more than once to get through it. Frankly, without the support of the math study center I never would have gotten through it at all. 2.5 years later I'm still struggling with math but at least now it's more interesting (vector calc and complex analysis).

    I learned some important lessons on the way (that might have saved me the first time through school):

    1) It doesn't matter how smart you are, you will have to ask for help at some point...suck it up and do it.

    2) Math IS hard...but hard for everyone so stop whining...the **only** way to learn it is to keep working problems and asking for help until you get it.

    3) But while it is hard, the benefits are worth it...it changes the way you see the world (for the better)...it teaches you how to approach problems in a more structured way...and, importantly, it builds mental stamina...think of it as an investment that will pay huge dividends in anything else you do.

    4) Learn it early, because learning it in your 30's is a real pain.

    The one thing I hate about the way it's taught (even senior level math) is that the pure math teachers don't care about applications. The most common reply I get to "this is cool, but what is a practical application of it?" is "Uh...I never thought about it, let me get back to you on that". You often don't even know what it's for until the next qtr. when they use it to support the next level of math, and so on without apparent end.

    The problem is that it's taught exactly opposite to how you learn on the job, which is you start with a large problem and work backwards to figure out how to solve it. Here you're given a bunch of shiny tools with no idea what (real-world) problems to solve. After a while it gets truly frustrating. Though I presume if I were taking applied science courses I'd see more applications.

    I'd like to plug a free resource I've found very helpful with my math studies, the Kahn Academy. This guy has created close to 1000 10-min. youtube videos explaining math and science. It is fantastic!: http://www.khanacademy.org/


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  49. Other commenters raise good caveats regarding the interpretation of the grade data. But yes, Virginia, it really is bad. I TA'ed ATMS102 some time ago; that is the successor course, which should have more bias toward more math-literate students. But the story was the same. Arithmetic was a challenge, and algebra was too overwhelming for most.

    Personally, I would not have thrown a test like this in school -- I would have tried to make the quizzes appropriate to my level so I could get a leg up on the curve, and I would have been afraid that my handwriting would be recognized -- but I know other kids might think differently.

    It's not that the kids are dumb. They were capable of understanding and even debating key concepts. The problem is that they were failed very early on; math is like a beautiful 67 mustang, but they were shown el caminos and vespas.

  50. Cliff,

    Funny and pathetic at the same time. I'm going to have to start calling you "Cliff Math". Keep up the good work.


  51. Dr. Mass,
    First of all, thank-you so much for sharing these test results. It is indeed very worrying for most of us who have children and/or care about the future of our country.
    I am a senior research scientist and have received my education in Eastern Europe. Many things were done in a wrong way in the country where I grew up, but most likely, the majority of my generation would have passed your test with a 100% score at the end of the seventh grade.
    We were never taught or told that learning math, physics, and sciences is "FUN". Nowadays, here in the United States and in some other developed countries, teachers and educators make great efforts to teach math and sciences to be fun and easy.
    Stating in the first grade, students should be told and explained that learning is NOT FUN, it is PAINFUL, it is a SERIOUS matter, and HAS TO BE DONE: there are NO ALTERNATIVES. Offering many elective courses to young students is another reason we see this kind of disastrous test results.
    Again, thank you!

  52. I'm not going to disagree that many people are awful at math and that we need to have better math education, but let's put this in perspective. As a recent UW grad, I know that AS 101 has a reputation, whether true or not, as a somewhat easy class that people take to boost their GPA or to take a break from their major's classes. While I've never taken the class, I've heard it is actually harder than most people think. Nevertheless, mostly the kind of people that are taking this class are not the ones looking for an easy pass and are not necessarily the ones headed on an engineer or med school path. Not all UW students are super bright. I agree that we need to teach math better but this test is not conclusive evidence to me.

  53. Cliff: You ask for the High School on the top of the test. Did the students from traditional math districts score substantially better than those from discovery math districts? I'd love to see your data broken down that way.

  54. Some of those problems and concepts are not used commonly in most lives (cosines and negative exponents, for examples.) They involve more memorization than intuition, and if you haven't used them in a while, you forget. I have a PhD in chemistry and would have scratched my head about the cosine problem. My engineer husband would not. Although if it were an open book test, I'd expect any high school graduate to be able to quickly refresh his or her memory on these.

    I would focus also on basic statistics. Everyone needs basic statistics skills for personal finance and to assess political arguments based on numbers. High school should build a foundation of life skills.

    Written skills are also shockingly poor at my workplace (which happens to be at the UW.) And you all should, if looking for more to rile you up, check out the OSPI website. An English class could have a field day marking up the grammar and dangling modifiers there.

  55. Thank you Cliff for sharing. Looking at your test I just remembered how much fun that was. I graduated 1990 in Czech Republic and this was the basic thought in first and second year of high school (highs school was 4 years 14-18 years old) Everybody would know it even if you wake them up in the middle of the night. I am sure it was the same here.

    I do not know what is the situation there know, but I have heard it is getting worse as well.
    I remember we did have to work to learn it and to do lots of homework to understand it. Math teacher was very strict yet popular. When we started calcullus - last year of high school - he said "so far we had merchant counting, real math starts with calcullus".

    One needed to work and be prepared and thats what it takes - good teacher which will motivate the kids and students willing to work ... my 5 socialist cents ... :-)

  56. I printed the blank form and did the test, although I had to wing it on the triangle in (3b) because it isn't there in my copy. So I labelled one and then saw that I had it right when I substituted the labels on your solution page (and sin 0 = 0, cos 0 = 1 is enough to figure it out too).

    I was frustrated that the printed PDF with the answers was not the same test (but then I looked further down on the blog page).

    I missed two of them. I handled 1e) incorrectly and knew immediately what I did wrong when I saw your solution. I also couldn't do 3(a) closed-book. I knew the answer was less than 4r^2 and pi was in there, but didn't remember that it was pi/4 and don't know how to derive it though I must have seen it done once upon a time.

    Not too bad considering I graduated from Tacoma's Lincoln High School in 1957.

    So, an anecdotal experience. How does that relate to how mathematics is taught at present. I have three personal observations:

    1. The text books are awful and teachers who are slaves to them are ill-prepared and misled in terms of what is essential for building upon for mastery and what is incredibly irrelevant.

    2. There seems to be little understanding and teaching of arithmetic and high-school algebra/trigonometry as something that can be demonstrated and checked. Having the answer that satisfies the teacher seems to be the norm (and if this was a multiple choice test, I might well have done better too). That is very anecdotal and maybe it happened for kids in my time, but I can't recall it ever being tht way for me.

    3. Calculators are a problem. Not in themselves and I don't fault using them, but in the removal of learning by doing and the mechanical manipulation of numbers, simplifications, application of identities, simplifications, and back-checking and simplification. And using decimal calculations (with the help of a calculator) in place of working with rational fractions becomes a serious difficulty when there are variables and a calculator is not useful, as in 4(c). I wager that (1f) suffers from this condition as well.

  57. Lake Washington is choosing a new curriculum as well, apparently. http://www.lwsd.org/News/Elementary-Math-Adoption/Pages/Informational-Fact-Sheet.aspx I hope they pick wisely!

  58. As an old dinosaur from the HS class of 1957, I walked through this test in under ten minutes, writing down correct answers (except 4b, which required separating variables) out of my head.

    I admit that Alpheus Green was an exemplary math teacher, but part of his method was refusing to accept sniveling from students, rather insisting that anyone could do this stuff and that it would be expected. He was crystal clear in his logical expositions, and willing to stay long hours to extend help when asked by any student.

  59. I am a UW electrical engineering student and although these results are appalling, they in no way surprise me.

    Our education system is failing us in many ways:

    1) Lowered standards -- This spans all levels of the system, from grade school where children simply aren't challenged to any sort of threshold anymore (because we "can't leave someone's child behind") all the way up to the UW (and other universities) where anyone with a fistful of money can show up and waste a few years, many of whom do not even want to be here. A college education is something one must earn; it isn't an undeniable right! There's a reason a bachelor's degree has little value in today's job market.

    2) Lack of responsibility -- Most students establish their work ethic at a very young age. If a child's developing mind isn't stimulated, pushed and motivated, he or she is already at a disadvantage going into higher levels of education. Those who should be taking responsibility? Teachers and especially PARENTS! It's easy to blame everything on the student or the government but who honestly has the most control over how a child spends his idle time?

    3) Merit is rarely recognized -- I understand the good that has come from offering minorities and other sub-populations financial assistance. However, there is a serious problem when many recipients of such funding possess no desire to fulfill its intended purpose. I know dozens of bright, motivated people (with great math skills, by the way) who would make tremendous college students, researchers, lab techs, etc.

    Cliff, I am embarrassed for my generation and fear having to work alongside many of them in the future, but thank you for shedding some light on an issue that should be much more alarming to our nation than it is.

  60. How would our K-12 teachers do on this test? I'm sure they would not want us to know. This, I think, is the source of the problem. Our K-12 system is filled from top to bottom with people ignorant of mathematics. If the faculty have math amxiety, they will pick the textbooks that please them and be useless for teaching mathematics.

  61. I am a high school student with in the seattle public schools and The new discovery math textbook do not offer any help. looking in the textbook for explanations or rules doesn't even help, thank goodness our teacher gives us notes. Also, the new "working together" gives the one student who understand the opportunity to complete the worksheets, while the others copy off them or socialize. I find group work very detestable and usually have to wait until my teacher actually tells or teaches us how to do it. I actually miss integrated math, which offered at least some repetition.

  62. In my high school math class, the topics are a mess; completely jumbled together. For equations, we are given the formulas/ procedures, but the teacher never proves them. Or the reason is 'because the calculator said so'.

  63. Did you notice the discrepancy between the blank test and the solution? Look at problem 4b.

  64. I know this is an old post, but I want to post my thoughts about this whole disaster as well:

    1. It's obvious that students need to learn how to be curious about the world, how to think logically, and how to ask the right questions.

    2. A solid math education is a great way to learn how to do all of the above - I believe there should be earlier introduction of proofs and a bit of math analysis thrown in for good measure. A lot of people don't encounter proofs until MATH 300/310 at the UW, which is a damn shame.

    3. What people don't realize is that it's INCREDIBLY important to learn what I've enumerated in (1), no matter what. I've studied math and philosophy at the UW and am now working in a totally different field than what I studied (I work as a sysadmin these days), but you absolutely need to think logically in order to flourish in IT. I imagine it's the same case in many other careers as well.

    4. The impression of math that I see so far is that it's a damn boring field, where higher level math is reserved for masochists who only love algebraic manipulation of complicated equations; unfortunately, no one (aside from those who have to take higher level math) gets to the point where that perception is thoroughly smashed.

    5. Therefore, I think one way to go about resolving this discrepancy is to not only teach these algorithms and make sure they're memorized, but also have slight detours where we simply think about a problem and we explore ways to come up with a conjecture on how something behaves in general (not something as pedantic as what they teach these days in the Seattle Public School system; I'm thinking along the lines of what Vi Hart shows in her videos on Youtube.) The ease of calculation makes it easier to think about these other, more abstract problems in math, much like how a poet doesn't think about grammar or spelling (because he has mastered it a long time ago.)

  65. Your 'scientific notation' quiz question is dubious. You don't state how many significant figures your numbers are to. If the answer is 3, you may only have one sig-fig. That's not necessarily the case...

  66. I've been an online math tutor for now almost a decade and I couldn't agree more with you. The student really don't take the transition seriously and as a consequence they get stuck at the begining resulting in typical math anxiety.


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