Intensive, well-designed training didn’t improve seventh-grade math teachers’ knowledge or their students achievement in a federally funded study by the American Institutes for Research and MDRC. From *Education Week*:

The program studied was “far more intensive and extensive—and better—than the typical professional development” that teachers receive, noted Elizabeth Warner, an economist at the federal Institute of Education Sciences’ National Center for Educational Evaluation and Regional Assistance, and the project officer for the study.

Over two years, teachers were supposed to get 114 “contact hours” of training on how to teach about rational numbers, including summer institutes, one-day follow-up seminars, and in-school coaching visits.

Teachers with one or more years of training did score higher on “knowing what types of graphic representations will best convey specific ideas clearly, and knowing the common student misunderstandings.”

But training didn’t lead to higher student achievement.

Teachers’ general math knowledge, which wasn’t affected by the training, correlated to significantly higher student achievement, the study found.

A similar study on early reading, completed in 2008, “showed no statistically significant impact on student achievement after teachers were exposed to one of two year-long staff development program,” notes *Ed Week*.

It would be interesting to see how the results break down in association with the teachers’ college-level experience with math. That is, what is the base line for math teachers who have a significant concentration or major in math as compared to those who don’t, and was there any difference in the effect of the training program? I’m recalling the worst math teacher I had, a gym teacher who fell back on math when she decided she was too old for gym – I don’t think any amount of training would have made her more effective.

The whole notion of concentration or major in math is bogus. I have an 800 on the GRE Quant, but majored in English (790 V, which is much harder to get). Never took a math class in college that I recall. Outlier, sure, but that score probably puts me in the 99th percentile of all math teachers nationwide. Similarly, I passed the California CSETs, and not all math majors manage it.

The GRE test score would be a much better tracking device metric than how much math was taken in college.

Correction–it’s not “bogus”; it’s just incomplete.

How about we start having teachers who teach math demonstrate that they have aptitude for math BEFORE they begin teaching math in schools? The average education major has, at best, average aptitude in math. Many teachers who teach math have no background in or aptitude for math, and many of those teachers dislike math intensely. At our little hole in the wall teacher licensing factory, the students who receive departmental math awards have SAT math scores of barely 500.

Very little is expected on the GRE beyond Algebra I and Geometry. One’s performance on that test is essentially a measure of how quickly and naturally one intuits the truths of those elementary classes.

So Cal’s right that it would probably be a good measure of how well a 7th grade math teacher understands these things, but I’d not use it for anything higher at all.

I suspect 114 hours of an actual math class, with homework and tests and such that had to be passed rather than merely encountered would show better results.

And to sum up what I really think of this “study” of this “intensive” training:

114 hours over two years? That’s not training. That’s a coffee break.The average education major has, at best, average aptitude in math.So what? The average education major doesn’t become a math teacher–in fact, arguably, the average education major doesn’t become a teacher without passing a much harder math test. Cite a stat with some meaning.

One’s performance on that test is essentially a measure of how quickly and naturally one intuits the truths of those elementary classes.Only 4% of GRE testers get 800. About 10% get 750 or over. Every elite economics and engineering grad program in the country uses it as the primary assessment mechanism, telling applicants that they probably shouldn’t bother applying if they can’t get above 750.

So it clearly provides a lot more than that, unless we are to assume Michael Lopez knows more than the elite econ and engineering programs in the country–all of which need more than algebra to do well in.

Besides, “anything higher” than algebra and geometry is algebra II, precalc, and calculus–comprising maybe 20% of all the math teaching positions in the country.

So even assuming your boneheaded assessment of the GRE is accurate, it covers the same subjects as taught by 80% of the math teachers in the country. Passing the CSETs or some other qualifying test covers the rest.

Focus hard: all new math teachers have to pass a qualifying test. Seventh grade math teachers will, over time, be entirely qualified for single subject math (high school math credential), so any notion about teaching math is not only unnecessary, but pretty ignorant.

Intensive, well-designed training didn’t improve seventh-grade math teachers’ knowledge or their students achievement…Then how do we know it was well-designed? Kind of by definition the training wasn’t well-designed.

I don’t think math teachers need to have majored in math… but certainly having had a few levels of math ABOVE what they will be teaching is necessary. An elementary school teacher who has absolutely mastered algebra/trigonometry/pre-calc/geometry would be sufficient… a middle school math teacher who has mastered at least one semester of calculus, and a high school teacher should have completed and mastered calc I, calc II, multivariable calculus, and linear algebra.

And when I say mastered, I mean mastered. Rigorous standardized testing, not just having taken the classes in college. After mastery has been established, a year of practical pedagogy is all that is really needed followed by an intensive on-the-job apprenticeship under a master teacher.

Part of the problem is that it is rare for school districts to pay teachers different amounts depending on subject matter. If you really want high quality math & science teachers, you are going to have to pay them more.

Oh. So now I’m boneheaded.

I’m not claiming that I know more than econ and engineering PhD’s about what makes a really good graduate-level economist or graduate-level engineer. That would be silly, because I don’t.

But I can, and will, say the following things:

First, I think I’ve been very patient with your insulting, snide, groundless attacks on me the last few months. I’m done being patient.

Second, I almost certainly know more about graduate schools in general than you, and I know that most respectable programs in ANY field don’t place much emphasis on the GRE at all. You’re just flat-out, dead wrong when you say it’s the “primary” assessment that they use. I frankly don’t even know what would possess you to think such a silly thing. Coursework, letters, your publication record, and where you went to school are going to be MUCH, MUCH more important.

(At this point you’re probably thinking you’ll backtrack and say something like, “But they do use it as their *first* assessment, which is what I meant when I said ‘primary’.” But that’s not what you meant at all: you said that the test “provides a lot more than that” and that the programs require more than algebra to be successful, and that I clearly couldn’t be right about the GRE because those economists and engineers use the test as their PRIMARY way of gauging that required advanced math.)

Now, they’ll cut you from admission for bad GRE’s, sure, but so what? That doesn’t prove you right at all. They’d also cut you if you didn’t graduate from second grade, and that doesn’t mean that graduating from second grade is somehow a measure of one’s skill in the higher-level mathematics needed for the program. If you can’t get a 750 on the GRE, that’s proof that you just don’t “speak mathematics” and you’ve no business trying to sort through economic models at Harvard, MIT, or Chicago.

Third, I also think I probably know more about what it takes to succeed on the GRE than you do. (At the very least, I know that my overall score was higher than yours.) There wasn’t a single thing on it that I didn’t learn by the end of 9th grade, when I finished geometry.; it’s all intro-level 8th grade statistics, basic “Chief Soh Cah Toa” kind of algebra, means and median, reading graphs, and other basic crap like that. I know there’s no calculus on it because I can barely do calculus; I never took the class and I learned all my calculus in Physics. I know there’s no real trig on the test, because I’ve never taken trigonometry, ever.

Fourth, so what if only 4% of testers get an 800? How does that in any way prove your point? “Oooh wow. We’re in the top 96% of the country.” Yeah, well, lucky us. That

mean that only 4% of the country knows how to do the math on the test, or that the math is of any level higher than I have stated. That just means we’re in the top 4% when it comes to doing itdoesn’treally fast and under pressure. Which is pretty much what I said earlier: it’s about fast and natural mathematical intuitions.Fifth, your pointing out that “anything higher… compris(es) maybe 20% of all the math teaching positions in the country” is silly and childish. I already admitted that you were probably

RIGHTabout using the GRE as a measure for the sort of middle-school math at issue in this study. I merely said that I wouldn’t use it as a measure for subjects that it doesn’t test.Sixth, I just want to reiterate how utterly bitchy you’ve been. I was

supportingyour point and you just had to jump in and start insulting me. Unbelievable.Seventh, I realize that you’re inordinately proud of your GRE score, and that you’d probably like to think that the test is more important and useful than it really is. (And on reflection, that’s probably what stirred you to attack my downplaying of the exam.) But speaking as someone who took it prepless and cold on a morning right before heading off to work and got an absolutely perfect score, I can say without a doubt that the test is a test of one’s

facilitywith basic concepts — mathematical, verbal, and written — and not of having actually learned anything substantive at the collegiate level at all.Finally, you want me to “focus hard”.

Let’s put aside the fact that the last sentence is an impenetrable stylistic disaster.

How about this: how about you take your “focus hard” condescension and save it for your students? How about you take the time to actually READ what I wrote, which was only that 114 hours of actual math instruction would likely have had better results than the training that they were doing? I said absolutely nothing about teachers

needingmore training. My point was merely that more math would be better than more training in how to teach math.Coursework, letters, your publication record, and where you went to school are going to be MUCH, MUCH more importantMichael, go look up the word “assessment”. Those aren’t assessments. I use words precisely.

You also seem to take my opinion of you much more seriously than I take your opinion of me. Every so often you write these long screeds where, I gather, the upshot is you’re mad at me for not being respectful enough of your opinion. Whatever. I don’t read them. I skimmed until I saw the above sentence, then stopped. You have way too high opinion of yourself, dude. All I know of you is that you post here often and you’re incredibly pompous and usually full of yourself. Whine away.

And when I say mastered, I mean mastered. Rigorous standardized testing, not just having taken the classes in college. After mastery has been established, a year of practical pedagogy is all that is really needed followed by an intensive on-the-job apprenticeship under a master teacher.Eh. I’m very good at standardized tests, but I’m not a mathematician, and I would never argue that I’ve mastered mathematics. It’s not necessary, and anyone who thinks it is isn’t being realistic. Not only isn’t it necessary, it’s not necessarily aligned with teaching skills.

In other words, math teachers need to know how to explain math and they must know how to successfully solve math problems. A math degree is usually sufficient for both. My only point is that it’s not necessary. But none of this is about knowing the mysteries and the rigors and the beauty blah blah blah of math.

Cal,

You misread what I wrote… I did NOT say a major or even a minor should be required. But yes, MASTERY of courses a level or two above the level one would teach. And I didn’t mean mastery in the “PhD level, deep philosophical meaning”… but mastery of the content that will be taught (and mastery of content a few levels above). Are you saying that a high school math teacher shouldn’t have mastered first year calculus and some linear algebra???

Jab–no, I didn’t misread, and I don’t altogether disagree with your point. I particularly like your emphasis on tests, rather than coursework. But recall (as you read below) that I have passed tests on both linear algebra and calculus.

Are you saying that a high school math teacher shouldn’t have mastered first year calculus and some linear algebra???I am rather famously weak at both linear algebra and calculus. I have, nonetheless, tutored several students from failing to As in both subjects. I would not teach calculus without some intensive study to improve at it. However, I am entirely capable of teaching up through pre-calculus right now, with or without linear algebra. I’ve even taught the matrix successfully to Algebra II students, and no one hates the matrix more than me (except, maybe, Hugo Weaving. Joke.)

Am I an outlier? I doubt it. One of the strongest algebra teachers at my school is only authorized to teach algebra (has no credential). She works wonders with struggling kids.

The current requirements are, in my view, entirely appropriate. Teachers must pass a rigorous test in math–not master, just pass. The cut scores are reasonable.

This, on the other hand:

An elementary school teacher who has absolutely mastered algebra/trigonometry/pre-calc/geometry would be sufficient… a middle school math teacher who has mastered at least one semester of calculus, and a high school teacher should have completed and mastered calc I, calc II, multivariable calculus, and linear algebra.strikes me as absurdly high for elementary and high school, and too mild for middle school–after all, middle school math is almost entirely overlapped with high school math at this point.

Incidentally, I don’t often offer up my own situation at all, and I’m certainly not offering it here as proof–merely an example. I am by no means a mathematician, but I am extremely good at applied high school math (if by “applied” you mean solving math problems) and very good at explaining math, even in areas where my own expertise is not solid. When I am assessed, my subject matter expertise is usually mentioned as one of my strongest points.

I’m not the norm, but I’m not fringe. I’m just using my own circumstance as an example doesn’t make sense to demand that math teachers have mastery in anything other than the math they teach–which is high school math, not college math.

I wonder if the reason that teaching teachers how to present algebraic concepts more clearly didn’t improve achievement is that MOST OF THE FAILING STUDENTS AREN’T ready for Algebra!

If you haven’t mastered concepts like 6/6 is the same as one whole, and 3/6 is the same as one half, or that multiplying by a half is the same as dividing by two….. well, all the algebra-teacher-training in the world isn’t going to help.

The problem is not that the algebra teachers can’t teach algebra to algebra students. it’s that they can’t teach Algebra to students who still struggle with 4th grade math.

Maybe they should try offering “rational numbers training” to the 3rd, 4th and 5th grade teachers (some of who are not comfortable with fractions, areas and volumes) and see if THAT improves scores down the line.

A novice teacher who majored in Greek and Latin can excel at teaching Algebra to students who have elementary math down.

A veteran teacher with an advanced math degree cannot excel teaching Algebra to kids who still can’t handle 1/3+1/4 = 7/12

Maybe we should go back to trying to get all kids through Algebra by SENIOR year instead of 8th grade. And, to avoid the nasty ‘tracking’ problems, we could just give all students a math placement test at the beginning of each year, and allow switching between courses at several points during the year. (My college allowed switching between Calculus tracks…. they went from the 120’s (Calculus with extra class time and mandatory tutoring,purely practical, no theory) up to the 160s. (Spivak, glorious Spivak!) Kids moved up and down…though, honestly, mostly down….)

Based on my experience some of the algebra teachers truly cannot teach…even nationally certified ones…the certification is a joke…

But…that aside…I agree with Deirdre…teachers teaching K-8 math need to have mastered math through algebra II or maybe pre-calc. Yes, even those that are teaching K-4. The problem starts early and is out of control by middle school if the kids don’t have a solid foundation in order to be ready for true pre-algebra in 7th grade and algebra in 8th (9th at the latest).

If we have to pay more for solid math and science teachers then it is high time we do it…

Cal,

In what world isn’t a letter of recommendation an assessment?

In what world aren’t your grades assessments?

And even where you went to school or which courses you took — although themselves not assessments, but merely facts — can be used as an “assessment mechanism”, that is, as a mechanism for assessing the worthiness of an application.

Perhaps you’re meaning to say that you were using “assessment” as a term of art of some kind, in the sense that the SAT, the GRE, and your final exam in Algebra are “assessments” properly so-called. But then why tack on the word “mechanism”?

And if you wrote “assessment mechanism”, why tell me to go look up “assessment”, which is defined in every dictionary I looked at as “evaluation or appraisal” (along with various technical but irrelevant definitions relating to taxes and evaluative intelligence reports).

Why not tell me to go look up “assessment mechanism”?

It’s all too precise for me to understand, I suppose. I wish I could be that precise, but I guess I spend to much time

actually readingthe things on which I comment.teachers teaching K-8 math need to have mastered math through algebra II or maybe pre-calc.First, that’s just silly. Second, even if it is silly, teachers who teach k-4 currently have to demonstrate competency through geometry, even though they won’t teach that far. 7th and beyond, in California, have the same quals as high school teachers (new ones, anyway). So we already have most of what you demand as necessary, even if it’s silly.

The problem starts early and is out of control by middle school if the kids don’t have a solid foundation in order to be ready for true pre-algebra in 7th grade and algebra in 8th (9th at the latest).You really don’t have a clue what stats say, do you? You’ve just got it in your head that the problem “starts early and gets out of control”. You probably heard someone say it on TV.

Most kids are ready for pre-algebra in 7th grade and algebra in 8th or 9th, and they are made that way by the teachers we have now. The ones who aren’t ready aren’t unready because of their teachers, for the most part, and won’t be made more ready by your utterly absurd requirements.

For what it’s worth, in my non-random sample of two ordinary classes at an ordinary suburban high school in the northeast: Many ninth graders can’t multiply by 10 without a calculator. A large percentage of the class is flumoxed when a quiz comes back with a grade like 18/25. “What’s my grade?” they ask, meaning “what is that out of 100.” A similar number don’t know how to solve a problem like, “Johny ran for 1500 meters at a rate of 5 meters per second. How much time did he spend running?”

There’s knowing the subject and there’s teaching the subject. Two separate skills. Knowing makes teaching easier, presuming the teacher can teach.

Since “teachers’ general math knowledge wasn’t affected by the training [and it is] correlated to significantly higher student achievement” then the content of the training was insufficient. Too often teacher training is focused on pedagogy rather than subject content – something most ed schools simply refuse to acknowledge… Without substantive content, so-called “training” is a waste of everyone’s time and money.

“Every so often you write these long screeds where, I gather, the upshot is you’re mad at me for not being respectful enough of your opinion. ”

It is not the lack of respecting opinion, it is the lack of respecting others as people. Attack the argument, which you often do, without attacking the person. A counter-argument does not require an attached “idiot” or “moron.”

Is this how you treat your students? When they disagree with you do you simply belittle them?

Just goes to show that before you even begin to think about pedagogy, content must be thoroughly mastered.

From a high school math teacher to all 7th grade math teachers: Stop with the calculators. Stop it!! Students need to be able to add, subtract, multiply and divide fractions without calculators. The students I am getting have learned to do fractions with calculators, but they don’t UNDERSTAND fractions. I need them to understand fractions to understand the math that I teach. I shouldn’t have to be teaching fractions to 11th graders (those in Algebra II and Pre-Cal).

I have always had an issue with the poor quality of math instruction when I was teaching. Most people, teachers included, fail to realize that there needs to be an understanding of numbers, patterns and relationships when it comes to math. We can’t just teach standard methods and hope the kids learn to mimic us. What has to be taught is confidence, strategies and ‘thinking like an artist’. By that I mean math needs to be creative. Kids needs to explore and find the way that works for them.

The calculator comment is just a symptom of the change in math instruction. We no longer teach or expect fluency or mastery of basic underlying mathematical skills. This has huge repercussions for later grades that take these skills and explore them in depth. As mentioned, without mastery of basic of fractions, large sections of an algebra class will make no sense to students.

I believe part of this is a technological shift, where people no longer expect students to master basic skills since they’ll always have a calculator. Why know facts when you can just look stuff up? The second part is that a lot of the popular math curriculum are incredibly weak. Textbooks use the “spiral” concept, revisiting an idea multiple times. However, this leads people to never expect mastery, because the students will see the concept again. Instead, students should be expected to master a different application or view of the concept each time they revisit it, not just repeat the same idea over and over.

I used to think that was true, but the reality is that failure to understand math facts is not the cause. It’s just another symptom. The complete inability to do multistep equations or graph a linear equation has nothing to do with math facts.

Too often teacher training is focused on pedagogy rather than subject content – something most ed schools simply refuse to acknowledge…This is nonsense. Ed schools aren’t responsible for content knowledge requirements. The state is. Teachers have to have demonstrated content knowledge to be accepted into most ed school programs, at least in California.

I would suggest that you have a narrow view of “math facts”. How to graph an ordered pair, for example, is a mathematical fact/skill that needs to be mastered (factual knowledge is x,y, axis, etc.. Skill is applying these facts to put the dot in the right place). These types of fact/skills are minimized under current pedagogy, and should be emphasized. We need to return to an equilibrium between concepts and facts/skills, instead of going 100% concept like a lot of people do.

Knowledge of mathematical operations is incredibly useful for multistep equations, and a failure to have mastered fractions will make factoring or complex simplifications impossible to learn.

I would suggest that you have a narrow view of “math facts”. How to graph an ordered pair, for example, is a mathematical fact/skill that needs to be masteredI would suggest that anyone who talks about calculators should accept that the response will involve calculators. Your entire second paragraph, the one I was disputing, involved calculators. I was responding to that narrow point.

However, more generally, the same thing holds for ordered pairs. Teachers teach students how to graph–some of them just don’t understand it.

a failure to have mastered fractions will make factoring or complex simplifications impossible to learn.Actually, it doesn’t. Many of my students still don’t know fractions, but they are able to factor quadratic equations–without a calculator. That doesn’t mean they will be successful in algebra II, but they were able to factor and graph parabolas. And of course, this isn’t because I didn’t teach them fractions for the tenth time, but they just didn’t understand it.

Perhaps you should actually read my second paragraph, and in fact my entire post before responding in a curt manner. My second paragraph was mostly not about calculators, but about curriculum. In fact, I started the post saying that the calculator issue was a symptom of other changes, and was describing some of those changes.

Actually,

A huge problem occurs when the student ‘apparently’ learns the skill, but when it comes time to actually apply it, they cannot do it. I consider basic math skills as the following:

Addition, Subtraction, Multiplication, Division, Percentages/Fractions, whole and real numbers, and operator precedence. (the first five items should be mastered before the student leaves elementary school, along with operator precedence).

A student who does not understand fractions is going to have one h*ll of a time succeeding in most higher math (including trig, pre-calc, statistics, and so forth).

Now they are talking about something called ‘dyscalculia’ which approximately 7-8 percent of the population suffers from (in 2003, a court in Italy allowed a girl to be promoted to the next grade, without passing math in that country).

I’d also argue with the idea that the students who don’t ‘get’ fractions are cognitively unable to get them.

When I was teaching, I had classes of cognitively average kids who had made it to 11th grade and Algebra 2 and claimed that they were completely unable to understand fractions.

I think a lot of the actual problem is that they didn’t have the proper mental habits–they weren’t used to applying themselves, persisting with a difficult problem, and practicing until they understood a concept.

Many of the kids had just accepted that ‘math was too hard’ and that it wasn’t worth trying. A lot of the worst ‘fraction-disabled’ were from homes that didn’t value education, that felt that it was OK ‘as long as you get a D,’ and that assumed any failure to learn was the teacher’s fault– no kid should be expected to do homework or come in for extra help–they had sports practice!

Cal seems to argue that the kids who don’t get it are just too cognitively impaired to get it— I think in many cases it’s actually a SOCIAL impairment masquerading as a cognitive one– these kids can’t learn because they’ve been so poorly socialized they can’t even accept that there are certain steps the STUDENT must take to master material.

these kids can’t learn because they’ve been so poorly socialized they can’t even accept that there are certain steps the STUDENT must take to master material.That doesn’t sound like a cognitive problem to you? You’re just so used to claiming environment you don’t see a cognitive problem when you see it.

Yes, I think many more students are capable, with lots of work, of becoming competent in algebra I and algebra II. However, it will take them much more work than we usually expect these courses to take. We teach algebra in 8th grade with a certain expectation of the cognitive load it will take. Only 20-30% of the kids have the ability to handle it.

If you are capable of learning algebra with five times as much work as we expect it to take, five times as much work as the kids getting proficient and advanced, the difference between you and the other kids isn’t social impairment, but cognitive ability.

That said yes, at the margins, there are kids who have the cognitive ability and simply choose not to apply themselves at all. They are not more than 10-20% of the total, though.

Ummmm, this might sound dumb, but why does a school offer 8th grade algebra to students when only 20 to 30% can actually handle it?

In my day, the 20-30% would have been offered algebra, and the remaining students would have been prepped to get ready to ‘possibly’ take algebra upon reaching high school (the nasty concept known as grouping by ability).

I’m sorry, but the concept of trying to get students to succeed in higher math (algebra and above) is moronic without a grasp of the fundamentals of basic math (i.e. – those students who understand it will succeed at algebra, those who don’t will likely fail at algebra).

Reminds me of the deli clerk who couldn’t convert 2/3 of a pound to a decimal value on a digital scale (never mind that she told me she barely passed college algebra)…30 years ago, a deli clerk or butcher would have had no problems figuring that out in their heads (and that’s elementary school math, fractions that is).

California requires (?) algebra in 8th grade. The kids are tested for it.

http://articles.latimes.com/2008/jul/10/local/me-algebra10

8th grade algebra may only work for 20%-30% of US kids, but I think that 8th grade (or 7th grade) algebra is fairly normal in Europe and Asia.

Bill:

Long story short, eduwonks did some research that found students who had algebra in middle school had higher graduation rates from high school and college. So, ignoring the whole correllation/causality problem; it was decreed that Algebra for All would lead to 4 year degrees for all. We’re on the verge of having a universal Pre-Calc requirement, you should see the rate at which those kids graduate! Of course you still won’t get your 0.666666 pounds of deli meat and forget about 20% off our already marked down prices (no Johnny, 20% off a 20% markdown is NOT 40% off). But there you are.

The ed world seems to be perpetually incapable of differentiating between correlation and causation. At the time of the study Sean mentioned, 8th-grade algebra was offered only at honors level, for those kids ready for it. Surprise! They did better in several ways. Ditto for Latin; those kids did better but only the academic elite took Latin. The MS next to the one my kids attended actually added Latin; the kids did no better. Both the algebra and the Latin (and debate, foreign languages, etc) merely served as a proxy variable for identification of the top students.

Now it’s precalc, chem and/or physics; the state where I now live has just made chem and physics a HS grad requirement. Idiots. It’s a very rural state, with many very small schools (many less than 100 in HS). The “big” schools in the “cities” will offer two classes (real and “lite”), but the small schools can’t do that so everyone will get the watered-down, physics-appreciation version: tough bananas for the kids that can and should be taking the real thing. How is this progress?