A study of students in Israel by Victor Lavy and Edith Sand has discovered a surprising result…or maybe not so surprising to you, but I was rather shocked. Math teachers score girls’ performance lower when they know their identities.

In math, the girls outscored the boys in the exam graded anonymously, but the boys outscored the girls when graded by teachers who knew their names. The effect was not the same for tests on other subjects, like English and Hebrew. The researchers concluded that in math and science, the teachers overestimated the boys’ abilities and underestimated the girls’, and that this had long-term effects on students’ attitudes toward the subjects.

For example, when the same students reached junior high and high school, the economists analyzed their performance on national exams. The boys who had been encouraged when they were younger performed significantly better.

They also tracked the advanced math and science courses that students chose to take in high school. After controlling for other factors that might affect their choices, they concluded that the girls who had been discouraged by their elementary schoolteachers were much less likely than the boys to take advanced courses.

But…*math*. Isn’t that one of those incredibly objective disciplines in which questions all have a right answer and a best method, and there’s no wiggle room for adjusting a score? Just like all of science — there’s no subjectivity here at all.

No, when you’re evaluating how well students think, there’s always lots of room for taking student knowledge into account. I teach genetics, and it’s a good example: I grade exams with my nice brief key by my side, and when students come up with the same answers I do, it’s easy and fast. But when they don’t, I have to look much more closely. Did they just make an arithmetic error in the last step? Did they understand the basic concepts, but just fail to integrate them all? Did they demonstrate a complete lack of comprehension of basic Mendelian principles? I have to see some sign of understanding in the work to make the effort to track through the problem more carefully, and it would be tempting to, for instance, know that this student did poorly on their last exam, so it’s not worth the effort to try and figure out what dumb mistake they made this time.

(I take steps to avoid that trap: I grade papers anonymously, not looking at the name on the first page.)

But I have a hard time imagining taking a negative attitude towards a math problem on the basis of the solver’s sex. Apparently it’s common enough that it actively skews assessments downward, though.

I’m familiar with the Swedish study that showed a pervasive bias against women scientists on the job market, but it’s clear the problem goes much deeper: women are being discouraged from going into math as early as middle school.

The paper also tried to puzzle out what was going on with these teachers, and found some other interesting correlations.

Older and single teachers seem to favor boys over girls: the coefficient of a dummy indicator of being older than 50 years old is positive and significant (0.206, SE=0.104), and so is the estimate of the indicator for single teachers (0.315, SE=0.202). The estimated coefficient for teachers from Europe-North America origin is negatively and significantly correlated with teachers’ biases (-0.204, SE=0.113). The other individual characteristics that we examined are being married (positive but insignificant) and the number of children and the proportion of daughters, both of which have negative coefficient but not significantly different from zero.

So older teachers are more biased in favor of boys; there’s hope that that effect will diminish as a newer generation of teachers takes over. I don’t think we can insist that teachers get married.

I also wondered about the effect of the teacher’s sex on this problem. Buried deep in the paper is an interesting revelation: they couldn’t look at that because *all of the teachers in their sample were women*. We learn two things from that, of course: that women can propagate sexist attitudes (no surprise), and that teaching is a deeply gendered profession. The gender distribution in the teaching profession also has to be sending a message to girls and boys.

Here’s the authors’ conclusion.

We also find that favoritism of boys among math and science teachers has an especially large and positive effect on boys math test score and on their successfully completion of advance math and science studies in high school; the respective effect on girls is negative and statistically significant. The estimates of the direct-subject effect in math are of special interest because of the considerable gender gap in math achievements and its impact on future labor market outcomes. Moreover, since this gap in math achievement partly results from teachers’ stereotypical biases against girls in mathematics, eliminating these biases will go a long way toward reducing the math achievements gender gap, and it will also decrease the gender gap in enrollment in advanced math studies. The impact on the various end of high school matriculation outcomes carries meaningful economic consequences because these high stakes outcomes affect sharply the quantity and quality of postsecondary schooling and impact earnings at adulthood as well.

Another message we should take away from this: **teaching is important**. All you primary school teachers out there are shaping society as a whole.

SQB says

I’ve noticed that the handwriting-equivalent of pink, is rounder letters.

PZ, are you able to discern the gender of the students whose exams you’re grading by looking at the handwriting? If so, how do you try to counter that?

Giliell, professional cynic -Ilk- says

No. It’s something maths teachers like to believe, but many studies have shown that maths scores tend not to be any more objective than those in languages.

The good news is that around here teachers get taught how to make and grade more objectively.

Which is significant because it’s one of the MRAs favourite talking points being disproved: They claim that the “feminized” atmosphere especially in primary schools are discrimination against boys.

Zeno says

Most of the time I don’t know whose exam I am grading because I grade multi-page tests one page at a time. After page one, there is no name to identify the test-taker. My observation is that the girls in my calculus classes can hold their own versus the boys. In three of my last four calculus classes, the top student was female. I hope that my experience is not an outlier.

Giliell, professional cynic -Ilk- says

SQBMy personal observation is that it’s individualised fonts. Especially in middle school girls invest time in handwriting (sometimes to the point that their notes are so elaborate and tidy that half is missing cause they didn’t have timt ot finish) while boys tend to have a shoddy version of the official primary school style (and if you’re really good you can tell who their primary school teacher was)

Beatrice, an amateur cynic looking for a happy thought says

Maybe schools over here are different, or maybe the reason is that my high school was math-oriented, but I never got the impression that girls’ knowledge in math was treated differently than boys’ or that any less was expected from us in maths.

I did encounter that kind of thinking in physics classes.

anbheal says

When I was in high school there was also what I guess you’d call a Winnowing Effect, or Pruning Effect. In freshman year, those who had been better performers in junior high, and so went straight into geometry or algebra 2, were about 60/40, boys to girl ratio. And there was a clear drumbeat in the background that this was a Boy’s Path. The girls were also steered toward biology for their mandatory science requirements, rather than chemistry and physics. By 10th grade there was only one girl in my physics class. Who doubtless saw all the other girls wandering around campus with their brood of ducklings waddling behind them, and being told how cute it was and how much more fun biology was. By the end of calculus 1 there was only one girl left in the class, and when the high school’s curriculum ran out and we were bussed to MIT for our final semester’s coursework, the head of the math department decided it wasn’t really worth including her. We boys also teased her relentlessly from the end of 11th grade through graduation, though we all liked her and admired her. I put myself in her shoes now, and remember her crying on the days when she’d enter the class to find her desk inverted or some equation on the board tweaked to make fun of her. I chanced across her in Manhattan in the 90s, as a thirty-something, and she clearly went on to have a terrific life, but she was in marketing, not science. A pity, for she was brilliant. But the teachers and the boys conspired to make math unpleasant for her, and at some level or another we had done so to every other girl in my class by the time senior year had arrived, leaving her the last standing. It’s rather awful, in retrospect.

twas brillig (stevem) says

IDK, but speculating from my own experience, grading some math can be a little more lax than only strict. For example, I often had math problems where you were to derive a particular equation from another, to then throw in a particular set of numbers and calculate the answer. Often I would put the wrong sign in the equation, or (also common) be off by a factor of 2 [pi vs 2*pi, often being the conundrum]. These could be graded as either, -1, -2,(i.e. partial credit) … or simply “WRONG” (-10, or -20) (i.e. full credit loss) … so … I could imagine them being lenient with the boys, with the single point deductions presuming typo errors, while girls lose the entire problems worth for a single mistake.

Beatrice, an amateur cynic looking for a happy thought says

re: grading math objectively

Math isn’t just about written tests. At least a third of my grade came from solving problems on the blackboard. That can include a lot of explaining why you did this or that or how you would solve something else, so your knowledge can be tested better than on a written exam, but the teacher can also be less objective in grading the pupil.

Rowan vet-tech says

Girls have a distinct writing style compared to boys?

Pardon me, but my initial reaction to *that* was lolwut.

Maybe it’s the field I’m in, we all have crappy handwriting except for a single coworker who writes in miniscule but perfectly formed letters at incredible speed so she’s both neat, fast, and highly legible.

Maybe it’s the schools I went to, the girls didn’t practice writing anymore than the boys did in my elementary, middle, and high school.

Maybe it’s my age, where even then cursive was becoming less important and pretty much no one used it after we learned it unless we were forced to.

Maybe it’s all and none of those, but while sitting with a friend who is a teacher, there was definitely NO noticeable distinct things between the girls in AP biology and the boys with regards to hand writing other than typical individual differences. Neat boys, neat girls, messy boys, messy girls, illegible scrawls for both…

wcorvi says

All of my early teachers were women The prevelant attitude was that girls were supposed to be good at academics – math, spelling; boys were supposed to be good at lunch and recess. Since I was more academically oriented, I was kind of an outcast.

Until High School, that is; then everything changed. Girls were interested in make-up, clothes, and boys who were good at academics (had career possibilities). This got even stronger in college.

Beatrice, an amateur cynic looking for a happy thought says

re. Differences in writing style between boys and girls.

I think it started disappearing somewhere at the beginning of Uni, but I definitely noticed it before.

In my time, yes, rounded style was usually favored by girls. Of course, none of this was set in stone and there were exceptions but girls usually did have a neater handwriting, with more rounded letters. Boys kinda didn’t move past the cursive we learned in first grade while girls soon started using some variation or combination of print letters and cursive as well as playing with handwriting (looking at old note books, I noticed that a couple of years all my letters were leaning to the left, then to the right and then they pretty much steadied themselves somewhere in the middle where they still rest when I write).

PZ Myers says

I see a lot of student handwriting, and you’d be surprised. Women turn in essays with crabbed, spiky handwriting, and men turn in clear, flowing, rounded letters, and everything in between. There is a tendency for women’s handwriting to be better…but it’s a really poor proxy for gender.

PZ Myers says

Oh, yeah: The tiny writers, whose answers have me reaching for a magnifying glass. I’ve seen no correlation with sex on that at all. Every semester I have one or two students like that, and they’re equally split between men & women.

latveriandiplomat says

@2:

I’d be very interested in links or references to one or more such studies. I’m genuinely interested, this is not a rhetorical request.

Island Adolescent says

That’s clearly a rhetorical question y’alls…

Obviously PZ is saying grading math has an element of subjectivity just like the rest of science.

latveriandiplomat says

That can happen, or less extreme versions of it, which is why it’s important for teachers to have a uniform approach to partial credit. This includes anticipating the most likely errors and how they’ll be scored, and also double checking to make sure similar answers got the same amount of partial credit. Students can always surprise you, of course, especially at the secondary level.

This is why partial credit is a huge PITA, but teachers do it because the alternative (insisting on perfection) is very discouraging for students, and feels unjust.

In most primary school material, the most likely mistakes are obvious, and the partial credit for those should be worked out in advance (and part of the rubric). So, I have a heard time believing this is the source of the difference in this study.

PZ Myers says

Partial credit is also important so you can figure out where the student is going awry, so that you (the teacher) can learn what needs to be emphasized/repeated in class and in problem sets.

I’ve been doing this crazy thing of putting lots of really basic questions on genetics exams — like, what’s the ratio of progeny phenotypes in a cross of two heterozygotes? — because I’m really trying to dissect student understanding and find out where they’re going wrong. Typically at the end of the semester, there’s a bimodal distribution of grades, and I’m tired of it. I think the only way to fix it is to make sure the bottom hump in the distribution gets lots of remedial instruction. The top hump I don’t have to worry about.

A. Noyd says

Rowan(#9)I got teased by the other girls in sixth grade for using small block letters rather than big, loopy handwriting with circle- or heart-dotted i’s like them.

victorbogado says

What if the student got to right answer by using a completely wrong logic and pure luck?

Giliell, professional cynic -Ilk- says

Rowan VTIt’s a tendency, not an absolute. When I was still in primary school we got grades for our handwriting. I usually scored a D. C- if the teacher was generous.

Re: gradingCurrent state of the art in grading, at least in Germany:

-When setting up an exam, already formulate a solution (this also helps in eliminating confusing questions)

-Also assign points for parts of the answer, say if the question is “describe process X”, how many points does each step get, what are core terms you want to hear.

-When grading, don’t grade exam by exam but question by question. that way you can make sure to stay consistent and also more objective.

-Also, in case you noticed that

youfucked up a question, feel free to delete it retroactively. (Maybe not possible for college level teaching)latveriandiplomatSmart as I am I cannot remeber the password to the lecture scripts, but Ingenkamp is the guy you’re looking for. At least in Germany he did groundbreaking work on how the same exam is graded very differently by different teachers, also depending on the background knowledge about the student.

neuroturtle says

A very quick browse through PsycInfo indicates very mixed results regarding gender and handwriting… people tend to not be able to differentiate. Usually small sample sizes. What they *do* find is that people will see handwriting differences when gender of the writer is identified.

This may become a hands-on (lol) demo in research methods for my intro class.

Anne Fenwick says

For hand writing differentiation, age may be a factor and of course it’s not an absolute. Now we all type, my handwriting (which even I haven’t seen for a while) is a piece of work.

Regarding grading – that’s another pro to the switch back to exams in the UK, and away from the touchy feely assessment stuff. I homeschooled my kid till age 11. After she entered state secondary school, the ‘profiling’, time wastage, and domination of the classroom by boys kicked in almost immediately. It’s so bad, I’m quite likely to arrange for her to sit her exams privately, after a final homeschool year.

leerudolph says

victorbogado:“Right answer” on a math test (in every university mathematics department I taught in from 1974 to 2012) is a property of the entire path, not just of its endpoint; as it is for PZ’s genetics exams.

Giliell, professional cynic -Ilk-:Your description of “Current state of the art in grading, at least in Germany” could have been written about the state of our art (at ditto). I’m not sure any Germans were involved.

sugarfrosted says

As far as grading math I do know a bit, (I’m a grader at UW.)

(Agreeing with the OP) It’s way more subjective than may people would think. Say there are two people, who both make a mistake in their reason. Both are minor mistakes, but one trivializes the problem and the other makes a mistake that doesn’t trivialize the problem and in fact makes it a lot harder and they end up not finishing it. Which person gets a better grade it’s tough to say. In situations where they didn’t do a canonical clear and correct proof it’s tough to decide on a grade. At that point is when bias can enter.

@20

I have no idea how (2) could translate over to grading homework or tests with proofs. This method only really works if a student gave a canonical proof. Often times there are tons of ways to prove a statement, so this doesn’t seem like a one size fits all. There was a class were I proved something using transfinite induction and I was the only student who did and it ended up being quite different

On (4). For any test with a time limit this isn’t really a good solution. If a student wasted their time on a problem that was harder than it was meant to be, they end up getting penalized simply for attempting it.

(1) and (3) are really good advice in any grading situation, though.

Also as far as handwriting, the differentiation between male and female isn’t really there. Cursive tends to be associated with women’s writing, but I’m one of the few grad students here that writes in Cursive, I want to say the ones that write in cursive are in fact male.

mnb0 says

“(I take steps to avoid that trap: I grade papers anonymously, not looking at the name on the first page.)”

As a math and physics teacher I do the same. I do two more things:

1) I look for things done right and grade them, ie I don’t “punish” mistakes;

2) I compare papers to make sure I grade the same way for every candidate. The latter is initially time consuming, but I have quickly developed a good memory for this.

Yup, girls on average perform better than boys with me. I am male, 51, but not single.

mnb0 says

@19: “What if the student got to right answer by using a completely wrong logic and pure luck?”

Zero points. Actually one boy got very angry for me doing so.

I see Giliell uses basically the same method for grading as me. That’s not surprising – I’m Dutch.

Dalillama, Schmott Guy says

Giliell#2Well, that’s just stupid. Not just because of this study, but teaching jobs, especially in primary schools, have been principally filled by women for at least the last century and a half (in the U.S., anyway).

garnetstar says

I also grade in the ways that you all have described above–PZ and mnb0 and Gilliel. I’ve never noticed any gender-correlation to anything, though, whether handwriting or quality of prose or correct answers.

Only in the case mnb0 @26 mentions: I have had male students whine for points if they happen to get a correct answer for no good reason. The females who have done that haven’t complained about the zero.

latveriandiplomat says

@20:

These are standard practices in the US as well, I think, at least at the secondary school level. One of the things that makes partial credit a lot of work (in math at least) is that if a student makes a mistake early on, you want to follow through the rest of there work and see that it is correct. There is some subjectivity here, because in some cases an early mistake can make working out the rest significantly easier, and you don’t want to give too much partial credit for that.

Thanks for the Ingenkamp refererence.

@19

It depends on how the rubric is set up. A typical algebra problem might be worth two points: one for setting up an equation and one for solving it successfully. In such a rubric, the student has done neither, and would get no points. It’s important to state “Show your work” as part of the question or for the test as a whole.

If a student writes down a bad set up and the answer pops out of nowhere based on nothing else they’ve written down, that can be a sign of looking over someone else’s shoulder.

@24: Nobody is doing proofs in primary school.

It’s still a mystery to me how much subjectivity was creeping into primary school level assessment as described in the article. It seems to indicate a serious lack of professionalism on the part of the teachers.

sugarfrosted says

@29, I was responding to a more general case that the con.. fuck it. I’ll just never comment here.

anat says

latveriandiplomat, even in primary school, when solving ‘word problems’ there is much room for subjectivity. Did the student understand the problem? Did they translate it correctly to mathematical terms? Did they deduce correctly which information they needed to find out in their intermediate steps?

anat says

anbheal, over the course of my school years (1970s to 1980s) the advanced level of biology (roughly equivalent to AP Biology in the US) went from being a boy-majority subject to a girl-majority subject. Though in Israel (where I grew up) students who take advanced biology often also take an advanced class in chemistry or physics. In my school the classes that were blatantly majority male were the ones with both advanced math and advanced physics. Math+chemistry was more or less balanced, as was biology+physics. (OTOH the advanced literature and languages class was 100% female).

In my high school math teachers were roughly an even gender mix, physics teachers were 100% men (mostly immigrants from the Soviet Union), chemistry mostly women and biology teachers 100% women. In 3 different elementary schools I attended all teachers who taught math (mostly the general class teachers, but one school had a specialist math teacher for the 5th and 6th grades) were women.

futurechemist says

I don’t remember my own elementary/middle school math classes, but from looking at some of my youngest brother’s math tests, it seems his teachers favored the “no partial credit” approach for simplicity and/or efficiency. For instance there was a question where he made a fairly simple math error in part a and got the wrong answer, so no points for part a. He completely correctly used that wrong answer in parts b and c, but got no points there either because wrong input –> wrong output. If I was grading that exam, I’d probably have given half credit on part a, full credit on parts b/c. But from teaching at the college level, I’ve become more and more convinced that primary/secondary schoolers are being taught getting the answer is the most important part, rather than learning the method.

I teach chemistry at a fairly large university. I give lots of short answer questions and insist on partial credit for nearly all questions, and I’m sure my TAs are not fans because of it. In gen chem, where there might be 2000+ students/year, they’ve given up on that approach and just give multiple choice exams. Those are 100% objective.

latveriandiplomat says

@30: I wasn’t trying to be a jerk or drive you away. I’m sorry that my response was so harsh. I didn’t intend to be, but clearly I was and I apologize.

The thrust of the article about primary school education specifically. Which to me makes the problem more mysterious than the general question of subjectivity in mathematics grading at all levels. So, I kind of wanted to focus on the primary school aspect, but I didn’t mean to become the topic police.

Any situation where a test is graded differently depending on who is doing the grading is a problem. While perfect objectivity is perhaps unobtainable it is definitely a goal of a properly written test and the accompanying rubric to be as objective as possible. We’re not talking about a few points randomly different here or there. We talking about a systemic bias against one group of students. Alarm bells should be going off.

@31: But the goal should be to minimize subjectivity. A well written test question is assessing a method taught in class. So there should be a correct method to solving the problem, with multiple steps, each step can be judged correct or incorrect. And most student responses to a well written question will be objectively gradable against that rubric. There will always be a few student responses that surprise and are difficult to grade, and various teachers would grade those responses differently. But that would not account for a systemic bias against a certain group of students.

Any teacher has unconscious biases, that’s precisely why objective assessments are so important. You don’t want your feelings about a student (even if you’re not particularly aware of those feelings) to influence the assessment process.

Zeno says

This is one of the main reasons I refuse to give Scantron-graded multiple-choice exams. The answer is not the be-all end-all. Of course, sometimes students get huffy and want to argue when they get few or no points (depending on how much of the set-up they got right) because their work does not support their accidentally correct answer. As one student sententiously informed me, “The last I heard, math is all about right answers.” “You heard wrong,” I told him.

anat says

latveriandiplomat, #34, No, even at the elementary level there are multiple ways to approach math problems. The goal isn’t to get everyone to use one method, the goal is to have everyone understand the problem and find at least one way to solve it.

The Vicar (via Freethoughtblogs) says

@36, anat:

Um, much though I hate to disagree: at the elementary level, and even at lower post-elementary levels, the goal usually

isto get everyone to use the same method — to make sure that everyoneunderstandsthat method, or at least knows how to turn the crank on the machine. It’s only later, when everyone has learned a toolbox of methods, that flexibility tends to be rewarded. (And this makes sense. If you’re teaching — for example — the quadratic equation, and one of the kids solves all the homework problems by factoring the polynomials, it may be clever but it tells you fuck-all about whether the kid understands the quadratic equation, which is the point of the exercise.)anat says

Well, neither my teachers nor my child’s teachers agree with you. When my child was in 2nd grade the teacher had posters with the children’s methods for solving problems. They had multiple ways to solve even things like adding two digit numbers. In my elementary school days the teacher would solve an example, and not infrequently a student or two would say ‘I solved it a different way’ and they would be invited to the board to show their solution.

Similarly, none of my secondary teachers ever insisted on a specific method. Any method was accepted if the steps were mathematically valid. (Also, we were expected to develop the formula for solving quadratic equations ourselves, it wasn’t given to us.)

Grewgills says

Having a clear rubric and grading a problem at a time or a page at a time (if not too many problems on the page) not only makes grading more fair, it makes it MUCH faster. An additional advantage to this is that common errors and omissions become obvious and are thus easier to address. Having a clear rubric also helps in the rare instance when a student tries to argue that their answer was graded unfairly. It is easy to point to the rubric and walk them through exactly why they got the points they did (or did not).

Giliell, professional cynic -Ilk- says

sugarfrostedAdvanced maths is a bit out of my depth, but I’m pretty sure the maths didacts have worked on that problem as well.

The question is, what would be a better solution? Make the whole class repeat the test because you fucked up and that one question now affects your reliability?

DalillamaWe’re talking MRAs. Stupid is the implied default.

latveriandiplomatWell, at least for German highschools patial credit is written into the law. Because especially in maths you would really fuck over students who make a simple mistake while doing calculus. As I demonstarted in highschool by getting 9+7 wrong (it would have been so much easier to go on using the square root of 16 instead of the square root of 15) early on but solving the whole long problem correctly, only with the wrong number. My teacher hated me.

Having a kid in primary school I can easily imagine how. One of the things they have to do is simple calculations with money. My kid is really good at it, but she’s also prone to not bothering with doing things neatly, so she often forgets to put the units into her calculations.

So she’ll write 50ct+20+10ct+5ct+2=87

How do you grade that. Partial credit? How much? No deduction?

There’s a lot of subjectivity in this unless you take steps to make your grading more objective. Of course, thinking that your subject is totally unbiased helps with fucking up. The more people are convinced of their objectivity, the more biased they act.

robinjohnson says

The luckiest objective result I ever got for an objective physics question was when I’d started it, had no idea where to go, left the rest of the page blank, and moved on. When I got it back, I was surprised to see I’d got full marks for that question. The lecturer had filled in the rest of the working in red biro to show me what I should have done, then apparently got distracted by something, looked at the page again later, seen the complete answer, and marked it correct.

leerudolph says

(1) If the polynomials in question, which presumably have integer (or maybe rational) coefficients, can all be factored over the rationals, then the test is badly made; if they’re not factorizable over the rationals, and the kid factors them correctly anyway (inevitably using quadratic surds in the factors), then as a teacher I’d be ecstatic. (2) As someone who’s taught calculus to at least ten thousand college students over the years, I can testify that essentially none of them “understands the quadratic equation” by the time they get to college, which makes me skeptical that very many of them ever did understand it (

viceshow some capacity, once, to treat it as a black box and do that job tolerably well).anat says

Giliell, my eighth grade science teacher was very harsh about units. Any time a student did not write the units where they were supposed to, the teacher would draw a picture of a cat, claim the answer was given in units of cats and deduct 10 points. And yes, people ended up with negative marks on some of his tests.

anat says

leerudolph, the only way to *understand* the quadratic equation is to derive it from first principles on one’s own.

We started with simple problems : x^2 -4X +4 = 0 —> Hey, that’s (x-2)^2 =0

We went to things like: x^2 -4X +3 = 0 —-> Well, let’s see, (x-2)^2 -4 +3 = 0 Yes, we know how to do that.

So after a few examples of that we generalized to x^2 + bx + c = 0

And once we had the above it wasn’t that hard to generalize further to cases when x^2 was multiplied by something other than 1. I understood the formula to solve quadratic equations because I knew where the various parts came from.

throwaway, never proofreads, every post a gamble says

anat @44:

Shouldn’t that be (x-3)(x-1) = 0; x = 3 | x = 1?

throwaway, never proofreads, every post a gamble says

Oh, I see what you’re doing there. Gotta have the A, B, C.

leerudolph says

Oy. I repeatedly read (and typed!!!) “quadratic formula” where you (and I!!!) typed “quadratic equation”. Sorry. My reply to what you didn’t write would have been absolutely correct, though (well, except for my writing “formula” for “equation”).

neilschipper says

leerudolph, I hit on a way of presenting the proof of the QF that set a few students’ eyes aglow: Using “completing the squares” to solve a quadratic, and then something much bigger

It’ll be slow going for you as it’s intended for students seeing it for the very first time.

I doubt it’s original (‘it’ being the presentation, not the derivation, which is conventional) but I’ve not seen in done quite that way in any text or youtube video.

Giliell, professional cynic -Ilk- says

anatThat’s just plain abuse of power. I’m often wondering what teachers think a test is supposed to do. Hint: It’s not supposed to be comic relief for sadistically inclined teachers. A test is supposed to give as accurately a picture of a student’s knowledge on the matter as possible. Obviously, somebody who solved 80% of the tasks correctly but forgot 100% of units is not somebody who has a negative knowledge about the subject matter. Deducting points: yes Deducting more points than possible to achieve: bullshit. And what’s more: teachers only do that shit on students whose parents won’t object. It’s probably not something a teacher will do in a class where 2 students are the children of teachers and one the kid of a lawyer.

anat says

I’m pretty sure some of us had lawyers as parents. The school supported policies of deducting points for procedural things – writing in the wrong kind of notebook could lose you 20 points. So my teacher’s policies would have been backed by the school. Anyway harsh grading would be a source of pride for school and students alike – it’s cult indoctrination basics.

Giliell, professional cynic -Ilk- says

anatI’m glad that over here such things are regulated by the eeeeevil government. I will admit to having deducted a point for lack of name, which is within the “up to 2 points (in the 15 points grades system) for violation of form”

mithrandir says

Re: handwriting – even if handwriting lacks any

generalgender-specific trends, it still tends toindividuallyidentify the student. And once the grader knows which student goes with the handwriting, they know the gender, and the unconscious biases can creep in. It may be possible to “blind” against that, too, but I think that is the question that was being asked of PZ, not assuming he could guess the gender of the handwriter from just the handwriting shapes.PZ Myers says

Doesn’t matter.

My procedure is to ignore the name at the top of the first page. Grade page 1 for all students, then page 2, then page 3, etc. When I’m at the end, add up the scores for each page, write the total on the first page. Then sit down with the gradebook and transcribe each score by name — and that’s the only time I link name to score.

I don’t have any association between the page by page performance and the individual, so I don’t have the opportunity to learn individual students’ handwriting.

David Marjanović says

Then it probably wasn’t luck, but (incompetent) cheating: the kid copied the correct answer from someone else and believed to be done.

chigau (違う) says

I looked upthread but couldn’t find this link

http://m.xkcd.com/385/

Michael Kimmitt says

I’m not sure why this is bad. What if that accurately reflects the students’ motivations entering class? Classes are not taken via a random selection process with no correlation between students…

I always felt like a bimodal distribution was way easier to score; I definitely knew which group to pass and which to, um, not pass. It was also generally pretty obvious by Week 3 which mode a given student was likely to fall into, though not where in that mode.

Dr Marcus Hill Ph.D. (arguing from his own authority) says

I’ll apologise right now, since I know this is liable to be a totally TL:DR comment. I’ve been a mathematics educator for decades, and taught 11 year olds to (currently) undergraduate mathematicians, and I’ve also trained teachers and am an active mathematics education researcher, so I’m afraid I’ll ramble uncontrollably on this topic.

Marking mathematics can be objective to an extent. I’ve spent many years marking (and supervising teams of markers) for A-Level maths. For the benefit of colonials and other assorted foreigners, these are public exams taken by thousands of 18 year olds across the UK (and, indeed, internationally). One of the most important parts of the process of marking is putting together a mark scheme that generates an extremely high rate of inter-marker reliability. The scheme is refined by the senior team looking at a bunch of student responses (they’re all scanned in by the exam boards question by question and thus anonymous to markers) and getting examples of the errors actually being made, so the scheme can then be made to work consistently for common errors. Only when this has been done do the markers get the revised scheme, and then they have their reliability regularly monitored. In short, two students producing the same answer will (to a high degree of certainty) get the same mark. However, the ability to have such an objective mark scheme relies almost entirely on the fact that the person writing the exam has one eye on how easy it will be to generate a consistently applicable mark scheme. Questions that require the students to demonstrate insight or to think beyond a procedural application of algorithms are invariably the ones that cause reliability problems in the marking stages.

When writing and marking assessment items for undergraduates, I want to test their understanding on a deeper level. Luckily, my classes are small enough that inter-rater reliability isn’t an issue (it’s just me doing the marking), but marking proofs, as others have mentioned, is inherently inexact, maybe as much as marking essays (which I also do).

Younger kids can (and do) use loads of different methods to solve problems. On the one hand, I always try to ensure the primary teachers I train understand this and look out for potential variants. On the other, the short-sighted education secretaries of the current government in the UK have seen fit to mandate that in public assessments taken at the end of primary education, only the “correct” methods (read: the ones politicians learned) will get partial marks, and since schools get judged on the outcomes of those tests, there is considerable pressure on teachers to get kids to use a method they don’t understand (and thus are more likely to fuck up) than one they do understand and can use reliably.

However, to get to the topic of the original post, it’s a very well known and documented phenomenon that the under-representation of women in mathematics is pretty much entirely attributed to social factors. It’s a cycle that affects girls from a very young age, when the attitudes of their parents and teachers that maths is a “male” domain start to be felt – usually unconsciously. When I trained teachers, one of the early exercises I’d do with trainees would be to ask them to draw a picture of a maths teacher. Despite the fact that, statistically, around 60% of their secondary maths teachers and 90% of their primary teachers would have been women, and the fact that they were sitting in a room full of trainees which was usually hitting around the one testicle per person mean, they still almost all drew pictures of men. The research also shows that both teachers and pupils are more likely to attribute good marks in mathematics to hard work for girls and to talent for boys. In short, all sorts of social expectations lead adults and, specifically, teachers to subconsciously have lower expectations of girls and treat them as such, and this in turn leads the girls to internalise these expectations in their self-images as mathematicians, and this in turn leads to many of them making choices that steer their education away from mathematics.

Unfortunately, as with any problem which is socially created and driven, it’s not one with an easy or quick solution. Even practitioners who are aware of the research and actively aim to counteract the effect can have a limited effect even in their own teaching – the problem with subconscious actions is that it’s hard to avoid them consistently. This doesn’t mean we shouldn’t try, of course, but it does mean that the fact is that it will be an uphill struggle.

OK, I’ll shut up now. Apologies again for the vast rambling rant…

opposablethumbs says

Marcus, I only wish more educators were as aware as you are. A little while ago I was involved in checking the adaptation for European use of a pre-school-to-end-of-primary-school maths scheme originally developed in Japan. I particularly noticed the (probably completely unconscious) gendering of the illustrations and the examples, and sent my employer links to reputable studies of the deleterious effect on girls and offered to balance these out more evenly – hell, after the first dismissal I even offered to put in the extra time unpaid – but was told not to bother, as the client would not be interested and there would be too much extra trouble making sure that everything matched across the question booklets, the answer sheets and the teachers’ booklet …

talk about disappointing. It would have taken really very little extra effort, and the result would have been a significantly better product. It still rankles :-(

Dr Marcus Hill Ph.D. (arguing from his own authority) says

It’s as bad for ethnic stereotyping. Things have improved a bit lately, with pictures and names in textbook examples showing more of a gender and cultural mix. However, we still don’t do enough to show kids that maths isn’t just for people like me (white middle class men, that is). Ask someone to name as many mathematicians from history as they can, and see if any come up with Germain, Noether, al-Khwarizmi or Ramanujan.

L E says

Late to the discussion, but I find it interesting that this bias seems to kick in about middle school, which is pretty much when subjectivity in math evaluation kicks in. I have plenty of experience grading math papers for the younger grades (my mother was a 4th grade teacher and we all got roped into grading at exam time) and there’s simply not enough complexity in math for much subjectivity at that point since it’s mostly basic arithmetic – they don’t even really get much in the way of word problems until fifth or sixth grade.

Middle school is also when pre-puberty starts to kick in and peer pressure to fit in starts to have a major effect. I have a distinct memory of having a discussion with a teacher about something I didn’t understand (in math, actually) and she told me not to worry about why it worked, just use the method she’d taught us. Not only was that really dismissive (although I realize now that she probably didn’t have particularly good understanding of the underlying concept herself) of me as the student, but afterwords all my friends accosted me in the hallway about how they couldn’t believe I’d started a “fight” with Sister G (a fight which consisted of me saying “but why does it work?” and her saying “just do it the way I told you and let’s move on.”), something that no one would have batted an eye at any of the boys in the class doing.

I was lucky in that that sort of gender largely disappeared in High School because I went to an all girl’s school. There was probably more emphasis on “girls” subjects like art and history, and we certainly had more available courses in biology than chemistry or physics, but the idea that math was somehow a male topic wasn’t a huge thing (helped that our Calculus teacher, who was female, had a PhD in math from the Univ. of Michigan).