kitchen table math, the sequel: 4/11/10 - 4/18/10

Friday, April 16, 2010

Alternative Learning Theories to Constructivism

Nothing is going to dislodge constructivism from its pedestal until a competing theory understands what's so appealing about constructivism.

Constructivism is a coherent way to understand ourselves and how we learn. Our intuitive experience of mastery is that we gain it through a highly inefficient process. Whether it's learning that letting go of something makes it fall, or that not all animals have teeth, our primary way of learning about the world is not from books or even from a master, but from our ambling around. first, by bumping into ideas/new concepts, where we don't even know what we don't know. Then, as we dimly start to figure out what the elements are to this new notion, we explore. Maybe we vary the conditions a bit. Maybe we play with things in our hands, build a model. If we finally get somewhere on this without giving up, we start to build hypotheses that can be tested and hopefully test them. If we hit upon a relatively correct one, we go with it, applying it to everything around (until we hit a problem again, and then we start over...)

Watching children learn to walk or to speak reinforces this notion of constructivism: kids just learn these things, quite inefficiently, but no teaching (outside of very rarefied conditions) would have helped. Kids do it at their own pace, with great variation, yet they nearly all arrive at the same place.

Constructivism is at its core, natural.

Ed school theories can articulate constructivism using a variety of words, but in the end, all of them posit a cycle, often with 4 stages.

One version is:
awareness --> exploration --> inquiry --> utilization -->awareness

Another is:
engagement -->exploration -->explanation --> elaboration -->engagement

These theories are coherent. But being a coherent theory doesn't make it right, and even if it is, it doesn't make it *the best way to help children* reach mastery. One reason it is viewed as best is because it is natural to our experience: why fight nature? The history of humans gaining knowledge of the world is clearly constructivist from fire and the wheel to geometry to public sanitation. Another reason is the theory can be taught in a lecture, from a textbook, and then by observation. It's easy to express. It fits our intuitive experience, at least in part. And what other theories compete with it?

Constructivism will never leave ed school until some other theory replaces it.

Lemov's taxonomy might work as a competing theory, if it were a theory. But it's more a proto theory of the interaction of student-teacher learning, and could still all be fit into the constructivist learning theory. Here's one example from Lemov's book:
There's a consistent progression to the lessons of the champion teachers who informed this book. It's best described as "I/We/You." This name refers to a lesson in which responsibility for knowing and being able to do is gradually released from teacher to student. It means beginning with "I" by delivering key information or modeling the process you want your students to learn as directly as possible, then walking your students through examples or applications. In the "We" step, you first ask for help from students at key moments and then gradually allow them to complete examples with less and less assistance on more and more of the task. Finally, in the "You" step, you provide students the opportunity to practice doing the work on their own, giving them multiple opportunities to practice

Many people might say that the above is explicit instruction/direct instruction. His "model directly as possible, walk them through examples" is to the well informed a vastly different model than constructivism.

But to the not-fully-versed-in-ed-school, it may not seem different at all. A constructivist would say it fits his model, too: first you introduce a carefully tailored environment for the awareness phase. The tailoring delivers key information. A student and teacher go through the exploration phase by walking through examples or applications, using them as focal points and areas for jumping off. Then, in the inquiry phase, you ask for help from students at key moments, gradually allowing them to complete a problem with less and less assistance, as you turn the inquiry from teacher-inquiry to student inquiry. And then the utilization phase is the practice opportunity phase, where you allow them to practice and reinforce as they interact with the world.

Cog Sci ala Willingham doesn't compete either, mostly because it's not coherent, and the individual elements can be shoehorned into constructivism. He calls for understanding the need for distributed practice, but certainly proponents of the cycle can argue that each time around the loop, you're getting another round of practice, each at a deeper level. He calls for recognizing the difference between how children learn and how adults do, but I think this supports the constructivists more: children, especially young children, have even more bumping-into-things to do. Constructivism supports giving kids more free range to do this bumping. Many would intuitively see that and think only adults can learn from books. He calls for getting the reward system tuned just-so, but that can fit if you build the environment correctly, and if the teacher does nothing but reward.

Engelmann and Carnine's work won't, because they created a theory of instruction, not learning. It doesn't concentrate on explaining how *we* get to mastery; it says how we teach *them* to get to mastery: isolate the necessary concept with enough positive and negative examples to disambiguate all possible misconceptions from this core concept. It is a method for *efficiency*, and rests on the notion that in order to help the low performing student to catch up, you will need efficiency. But in isn't focused on learning. Engelmann and Carnine argue the *number of examples* needed for a low performing child is exponentially more than for a high performer--they aren't at core arguing the low performer learns differently.

They violate constructivism because it isn't *natural* to push for efficiency. It doesn't speak to how we go around making sense of our own world. And in our post-baby boomer world, we're all about ourselves and how we experience the world.

Are there competing theories to this? I'll address one in the coming days.

Teacher Training No Boon for Student Math Scores

Sigh. The more it changes, the more it stays the same.

Published Online: April 16, 2010
Teacher Training No Boon for Student Math Scores
By Debra Viadero

First-year findings from a federal study of 77 middle schools suggest that even intensive, state-of-the-art efforts to boost teachers’ skills on the job may not lead to significant gains in student achievement right away.

The "Middle School Mathematics Professional Development Impact Study," which was released April 6, is the second major experimental study by the U.S. Department of Education’s Institute of Education Sciences to find that a high-quality professional-development program failed to translate into any dramatic improvements in student learning. A two-year study of efforts to improve teachers’ instructional skills in early reading reached a similar conclusion in 2008.

“What accounts for this somewhat consistent pattern of results? We don’t really know,” said Michael S. Garet, a vice president at the American Institutes for Research. His Washington-based organization conducted both studies with the MDRC research group of New York City. “I think what we’re learning,” Mr. Garet added, “is that it’s challenging to make a big enough difference in teacher knowledge and instructional practice to have an impact on student learning.”

The results are already providing some intellectual ammunition for finding better ways to select and retain effective teachers—and shedding those who are ineffective—as the best way to improve instructional quality in schools.

The new study shows that “you can’t change teacher effectiveness very well with the tools that we have, and that you can’t change ineffective teachers into effective ones,” said Eric A. Hanushek, a senior fellow at the Hoover Institution, based at Stanford University. He is also the president of the IES advisory board, which heard a presentation on the new study’s findings last week.

But other scholars said it is too soon to issue a verdict on the effectiveness of professional development.

“We know teacher change takes time,” said Hilda Borko, an education professor who is also at Stanford. “The general belief is that it takes a while for teachers to take ownership of change and really incorporate change into their instruction.”


Findings in Detail

By the end of the school year, the researchers found, teachers who participated knew slightly more about rational numbers, overall, than their control-group counterparts, but the effect was not statistically significant. Those slight improvements were most notable on a test of pedagogical content knowledge, where teachers stood a 55 percent chance of getting a question right, compared with a 50 percent chance for their counterparts in the control group. (In comparison, the trainers’ chance of answering a question correctly on that test was nearly 93 percent.)

On the plus side, the training did lead to changes in teaching practice. Compared with the control group, the teachers in the experimental group were more likely to try to draw out students’ thinking by asking students whether they agreed with a classmate’s response, or inviting them to share their mathematical strategies.

The changes in practice were not dramatic enough, however, to translate into student-learning gains on the computer-adaptive tests that students took in the spring. The students, most of whom came from schools where more than half of students qualified for federally subsidized school meals, continued to score, on average, at the 19th percentile on the tests, which were developed for the experiment by the Northwest Evaluation Association of Lake Oswego, Ore.

Mr. Garet said the researchers are now analyzing the results from a second year of similarly intensive teacher training in the same schools, which they hope to publish this year.

If those results show learning gains from the training, they might suggest the need for even more sustained professional development.

“You might need to have pretty intensive professional development all the time, every year, and then slowly get schools into a culture such that the expectation is that you always keep working on your knowledge,” said Sybilla Beckmann, a math professor at the University of Georgia in Athens and an adviser to the study.

The Case With Reading

The results from the 2008 reading study were based on two years of data, but just one year of professional development. That study also differed from the latest one in that it provided in-class coaching, randomly, to only about half the group in an effort to see whether that approach yielded any added benefits for teachers or their students. But that did not turn out to be the case.

The reading study also found that, while the skills of the 2nd graders in the study did not improve, there were some measurable gains in teachers’ knowledge.

For the middle school math study, Mr. Garet said, the researchers tested various possible explanations for why the training had failed to affect student achievement. They ruled out the hypothesis, for instance, that the tests were too hard or too easy for the teachers or their students.

“As you move away from the study itself,” Mr. Garet added, “one hypothesis worth testing would be to see what would happen if the professional development aligned with what teachers were evaluated on.”

Ms. Borko, who is leading her own study of teacher professional development, said it also would have been instructive if the federal studies had measured the quality of the instruction that teachers were getting.

Teacher Training No Boon for Student Math Scores

Thursday, April 15, 2010

international math test for future teachers

America’s future math teachers, on average, earned a C on a new test comparing their skills with their counterparts in 15 other countries, significantly outscoring college students in the Philippines and Chile but placing far below those in educationally advanced nations like Singapore and Taiwan.

The researchers who led the math study in this country, to be released in Washington on Thursday, judged the results acceptable if not encouraging for America’s future elementary teachers. But they called them disturbing for American students heading to careers in middle schools, who were outscored by students in Germany, Poland, the Russian Federation, Singapore, Switzerland and Taiwan.

On average, 80 percent to 100 percent of the future middle school teachers from the highest-achieving countries took advanced courses like linear algebra and calculus, while only 50 percent to 60 percent of their counterparts in the United States took those courses, the study said.

U.S. Falls Short in Measure of Future Math Teachers
by Sam Dillon
Here's the report: The Preparation Gap: Teacher Education for Middle School Mathematics in Six Countries (pdf file)

NCTM response
“There are so many people who bash our teachers’ math knowledge that to be honest these results are better than what a lot of people might expect,” said Hank Kepner, professor of mathematics education at the University of Wisconsin, Milwaukee, who is president of the National Council of Teachers of Mathematics. “We show up pretty well here, right in the middle of the pack.”
NCTM motto: Good enough is good enough.

I could do this problem, which is a good thing, given that I've completed ALEKS geometry.

On the other hand, I had to think about it -- and I don't remember having learned that the angle bisector also bisects the opposite side in a parallelogram.

Erin Johnson on public school management

in a comment on Flypaper, Erin Johnson wrote:

Given the wave after wave of ill-conceived curricular reforms that have been imposed upon teachers (Whole language, Constructivist math, etc.) how would eliminating tenure affect teachers that are “being held accountable for student performance” and yet are being told by their administration that they must use poor curricula. Talk about being caught between a rock and a hard place.

If we wanted principals to “stop acting like middle managers and start acting as true CEOs of their school” then we should be holding principals accountable for performance not the teachers. Or how about holding district superintendents or curricula specialists accountable for the misguided materials that they insist that teachers are to use in the classroom? In business, the leaders are always more responsible than their direct reports.

there is more

Wednesday, April 14, 2010

I'm going to msmi

I've finally got my reservations made!


I hope some of you can come -- I'd love to put faces to names after all this time.

charter school for Westchester?

Laura asked:
Catherine, I'm wondering if you have any thoughts about how realistic it might be to start a small charter school in Westchester.

Something like a mixed-age, low-tech, low-frills charter school with large class sizes, and a curriculum that students can work through as quickly as they are ready for, or as slowly and intensively as they need to.
I would love to see a mixed-age, low-tech, low-frills, warm/strict charter school here in Westchester County.

Hogwarts on Hudson.

A happy prep school.


from Teach Like a Champion: 49 Techniques that Put Students on the Path to College:
We're socialized to believe that warmth and strictness are opposites: if you're more of one, it means being less of the other. I don't know where this false conception comes from, but if you choose to believe in it, it will undercut your teaching. The fact is that the degree to which you are warm has no bearing on the degree to which you are strict, and vice versa. Just as you can be neither warm nor strict (you may teach the children of parents who are this way and see for yourself the cost), you can also be warm and strict. In fact, as this Warm/Strict technique shows, you must be both: caring, funny, warm, concerned, and nurturing -- and also strict, by the book, relentless, and sometimes inflexible. It's not, "I care about you, but you still must serve the consequence for being late," but, "Because I care about you, you must serve the consequence for being late."

...Not only should you seek to be both: you should often seek to be both at exactly the same time. When you are clear, consistent, firm, and unrelenting and at the same time positive, enthusiastic, caring, and thoughtful, you start to send the message to students that having high expectations is part of caring for and respecting someone. This is a very powerful message.

page 213
Doug Lemov is describing the approach used by authoritative parents, whose children fare better academically and socially than those of permissive, authoritarian, and neglectful parents.

A good teacher is like a good parent.

Lynn G - more on public school expulsions

I just got a call last night from another parent looking for some school that will take her son. He's been kicked out of the regular public school for discipline issues. He comes from a great family (mom was big in the PTO a few years back). The public school couldn't care less -- kick him out and tutor is fine by them. I told her to try the Big Picture school, which also has great success with kids tossed out of public school.

There isn't a KIPP in the area to recommend, or I would have. My anecdotal experience is that the Charters and Magnets are where the suburbs ditch their trouble makers.
That's my experience too, here in Westchester. Except we don't have any charters and we've lost the one magnet school we had. (If there are other magnets here in Westchester, let me know.)

cranberry on public school choice

cranberry wrote:
Which of the town's employees or elected officials have a long term interest in keeping town finances in check? In controlling school spending? In our town, the superintendent sets the budget. The school committee presents the budget to the town. They can give the superintendent guidance as to what they wish to see preserved, but they aren't making decisions about hiring and firing staff. They can fire a superintendent who doesn't follow their desires, but that's a very blunt instrument. There's no guarantee that the next superintendent will choose to anger all the other pressure groups in the school system for the good of the students and town budget.

The average superintendent's time in office lies between 2.4 and 6 years. So, a newly hired superintendent arrives in a district, and discovers how this district functions. He fires or transfers the people he can't work with, and transfers or hires people he can work with. They propose changes to the system. For the changes to work, the teachers must agree with them and implement them. By the time the changes are being implemented, he may very well be interviewing for his next job. The structure of the system works against long-term planning.

Parents support larger school budgets because one avoids the need to decide between competing interest groups within the system. By filling all buckets, a little of the overflow might improve things for the average students, the kids who do not have defenders within the system, other than the teachers. The typical kids are often shunted into as many study halls as possible, because that's the only way the budget will balance.

Allowing funding to follow the student would give the parents of typical kids more negotiating power with schools. "If I like your program, your budget expands by $6,000. By the way, I have 8 friends who like the thought of foreign language classes in elementary school. Hmm, study halls instead? Well, I could drive 25 minutes in the opposite direction. That principal offers Latin and Saxon math."


from Flypaper:
"Many of us who support school choice do so because of our hope that competition will force recalcitrant districts and unions to reform."

I left a comment.

Two comments, actually.

Creative Students In The Classroom -- What Teachers Really Think

Jonah Leher discusses a study on elementary education
Everybody wants a creative child - in theory. The reality of creativity, however, is a little more complicated, as creative thoughts tend to emerge when we're distracted, daydreaming, disinhibited and not following the rules. In other words, the most imaginative kids are often the trouble-makers.

Eric Barker recently referred me to this interesting study, which looked at how elementary school teachers perceived creativity in their students. While the teachers said they wanted creative kids in their classroom, they actually didn't. In fact, when they were asked to rate their students on a variety of personality measures - the list included everything from "individualistic" to "risk-seeking" to "accepting of authority" - the traits mostly closely aligned with creative thinking were also closely associated with their "least favorite" students. As the researchers note, "Judgments for the favorite student were negatively correlated with creativity; judgments for the least favorite student were positively correlated with creativity."

Link to study:

You should really go read Leher's post and the comments there, plus the study.

Early Introduction to High School

My son is heading to high school next year and it has already started. A pink note came home that told us that because the town council wants to keep the budget increase down to zero (instead of almost 2%), the school will be forced to cut all freshman sports, all afterschool groups and clubs, and all extra music offerings (plus more). The town council says that the school can cut whatever it wants, and the school (and school committee) offers up what they think will rile up the parents the most. In response to the complaints, the town council tells parents to complain to the school committee. No wonder many people think that more money = better. Is this common in most places? Is there some other model that allows a better analysis of cost versus benefit?

Tuesday, April 13, 2010

military schools

Terrific comments about sending kids to or attending schools on military bases -- hadn't read them until this moment.

Having read them, I now take back what I wrote about military schools doing a better job of classroom discipline - !

Joanne Jacobs on KIPP study

The study compared students who won the lottery to get into KIPP Lynn with students who applied but lost the lottery. To the extent that families who apply to KIPP are more motivated, there was no difference between the two groups.

The study also found that KIPP lottery winners were no more likely to leave KIPP schools than lottery losers were to leave their Lynn Public Schools placement.

Students applying to KIPP were more likely to be black or Hispanic and less likely to be Asian than other Lynn students. They were lower performers than the public school average.

Lynn G on whether KIPP kicks kids out

Public schools kick kids out. It is standard policy. They have to provide tutoring and/or pay for alternative placements, but they are quite happy to do that and bill the taxpayers for it. As the public schools struggle to show improvement in test scores, I expect to see more low performing kids shuffled off to alternative programs where their scores won't count against the sending school district.

I've looked at the expulsion and graduation rates at inner city schools, and KIPP does a better job of keeping kids in and getting them to graduation than the public schools they came from. KIPP has a lower expulsion rate than say, the Hartford public schools, where kids are expelled or encouraged to drop out at much higher rates than the charters.

My $32K per pupil public school has some species of "zero tolerance" policy, which means the administration kicks kids out and pays for home tutoring. e.g.: If a student is caught smoking pot in the bathroom, he's gone. Period. I assume most parents support this policy, but that's beside the point. The point is: if a student in my district steps across the line, he's out.

I suspect my district has a higher rate of student expulsions than KIPP:
With children who had more serious behavioral problems, who would tell a teacher to f*** off without feeling any remorse, the Porch [in-class detention] had little power....Levin was resolved to expel students only in the most extreme circumstances, which, it turned out, happened only once or twice a year, far less than in many regular public schools that forced students to transfer to special programs for discipline problems. Levin and Feinberg considered each student they could not teach a failure on their part. They kept looking for ways to get the number of dismissals down to zero.

Work Hard. Be Nice. by Jay Mathews
page 171
KIPP doesn't do what it does by cherry picking students or kicking students out or having parents sign contracts.

It does what it does through excellent teaching and through the creation of an "authoritative" school culture that keeps kids in line the same way authoritative parents keep kids in line. Lemov calls this culture "warm/strict," and I recognize his account: he is describing the school culture of Hogwarts, my son's Jesuit high school.

Voices from the Classroom: Mrs. Bluebird and the Calculator Rant

Mrs. Bluebird teaches middle school science in the southern US. I believe she has both general ed students and some in special ed. I really enjoy her blog, as she is funny, practical, and honest about the challenges. I've condensed this post ( .... indicates edited text) and I've also added emphasis

So what do you do, as a teacher, in middle school if your students are lacking essential skills that they should have mastered in previous grades? Here's her take.

My Calculator Rant, or My Kids Can't Do Math Without One

Next week we take our Very Big Deal Government Mandated Tests. Oh yipee....

We're pretty confident they have a fairly good grasp (as far as seventh graders with the puberty brain freeze can grasp) of most of the basic science concepts.

However, the math is going to kill us.

Part of our new standards this year include Newton's Laws of Motion and simple machines, and fun little things like acceleration, velocity, work, force and all that wonderful little physical science stuff which I find really cool. However, there's a lot of math and calculations involved, such as figuring out that work equals force times distance, and power is work divided by time, and momentum is mass times velocity. ...

We're talking simple math here - multiplying and dividing. That's it. However, we began to notice on our quizzes, our tests, and our Benchmarks that our kids can't do math without a calculator. They can punch in numbers and solve math problems until the cows come home, but ask them to do math with a pencil and paper (and their brain) and they go into shut down mode. Heck, they're not even sure how to set up a math problem without a calculator. They would read a question, say, Power = work/time, and they'd write it out and then MULTIPLY IT. Not just a handful of kids, but huge numbers of kids. Mrs. Eagle, Mrs. Hummingbird and I were shocked...and promptly ran to our math teachers.

"Do you mean to tell me," I asked Mr. Math, "that without a calculator, these kids can't do math?"

"Pretty much," he said. "Welcome to my world. They don't know their multiplication tables by heart, and they depend on a calculator for everything. They may have learned their multiplication tables in fourth grade, but then they stuck a calculator in their hands and they promptly forgot everything. And we're encouraged to have them use calculators."

Oh good gracious. They don't even remember that a line between two numbers means to divide.....

So, we have the kids learn their multiplication tables, and then give them a calculator. How stupid is that?

About as stupid as the State Department of Education's Decree that No Calculators Will be Allowed on Any Test Except for Math. Period.

We tried, when this first became apparent to us earlier in the year, to see if we could get the Special Ed kids that have "use of calculator" written into their IEP's permission to use calculators. Not only was the answer NO, it was a Big Fat NO.

I have kids who, quite honestly, cannot tell you how many times 3 goes into 24, who need calculators as a life skill because 2 times 6 is a challenge. These kids will be forced, along with all my other kids, to do math problems on the Very Big Deal Government Mandated Science Test, without a calculator. Even though they use calculators every freaking day in math class. And this year, about 20% of that test will be math. (I do have a few good special ed parents who are annoyed at this and asked me what to do - I suggested that as parents the state may listen to them a bit more than they listen to us teachers. Perhaps if they complained loud and long, we'd see a change.)

When we did our datachat for our last Benchmark, the kids did really well. Except for the standards that were math-based. They, bluntly, sucked. Badly. Why? They lack the basic math skills to do basic problems. And it's not just my kids, but apparently it's an issue across the entire district. And I'd guess, across the state, and most likely the country.

So, when Mrs. Eagle and I went and judged the science fair at the local elementary school a few months ago, and we found out that they'd spent some grant money buying calculators for their 2nd and 3rd graders, we pretty much told them to send them back and get a refund. They were appalled when we told them the issues that we were having with the lack of basic math skills. Again, if you don't use a skill, like doing math with a pencil, paper and your brain, you aren't going to be good at it.

Which is why our team remediation class has been doing multiplication practice, just like they did in fourth grade, several times a week (and grading those is frightening, they're so awful.) Hopefully, this practice will help a few of them.

However, I'm still incensed, that my kids are going to be, in a way, penalized because they don't have the ability to do math without a calculator. And at the same time, we stick a calculator in their hands and encourage them to use it. It makes no sense to me that they can use one for the math part of the test, but not the science part which also has math.

The politics of testing just irritates the bloody hell out of me.

So there you have it. What are teachers in the higher grades to do, if the students in their prior education have been handicapped in this way?

Monday, April 12, 2010

the "traditional" math sequence

Well first there is algebra in 9th grade. But then take the entire sequence geometry --> algebra I --> algebra II --> precalc --> calc I --> calc II etc. (Maybe if you are advanced enough you get to do Calc III or linear algebra in the last term of high school or maybe the opening semester of college.) Why necessarily in that order? Is it necessary to have fully mastered all the theorems of geometry before moving on to algebra I?

Furthermore, there is a perspective lost in the entire sequence, because generally, a lot of fields from number theory to graph theory are not even considered until college, except as shallow "enrichment" activities. Are kids capable of doing hardcore number theory in eighth grade? Well take the International Math Olympiad (for young students) or your average state math competition. You can get straight As on your calculus exams and totally get creamed at these competitions by a bunch of ninth graders from a school with a more dedicated math team.

Formal linear algebra is mind-expanding, but it occurs to me it doesn't require college-level understanding to master its basic concepts (like transformations). Arguably, you /really/ understand what differentiation and integration is all about when you learn to map from one polynomial space to another. The most important thing about solving math is abstraction, and a rather powerful abstraction is the ability to rotate entire shapes by a simple entity called a matrix; ever had that annoying geometry problem where you had to tediously map all the points of some crazy polygon after it was rotated an arbitrary angle? Well now, what a perfect time to bring up a concept that would systematically map every point to its new point after the rotation...

Here is the place to use a calculator. Take a polygon with 20 points (or some inconvenient number). Students would then have to design a matrix that carried out said transformation. Perhaps an arbitrary transformation whose formula wouldn't generally be memorised (like with standard rotations). The calculator is the "reward": simply input all the points, the appropriate transformation and then report the results of the magic. Here, the calculator ensures that students actually understand how to use the abstraction being learnt (rather than say manually applying an algorithm to each point). On the other hand, it is not being used as a crutch for bad understanding.


There has been much said about "holistic" educations. Many ostensibly "holistic" curricula are not holistic curricula at all. One of my disenchanted teachers put it -- you can't just toss a bunch of department heads together into biweekly meetings and call the result "holistic". (Now mind you, this teacher was from a top Singaporean school and he had been "released" from duty by being too vocal about certain teaching practices.) There is so much reteaching of math concepts from applied subject to applied subject. And oh, all the forgetting! Imagine what could be done if all the inefficiency was removed...

See, I don't actually disagree with the actual idea of integrated math. Integrated math should be a vehicle for exploring important life concepts (like, ooh, equilibrium! optimisation under constraints! recursion! etc.) taught tightly with useful, complex, mind-expanding applications. You could pack a lot of learning into a year. It shouldn't be about using matrices as some sort of organisational table for keeping scores of random soccer matches.

special ed students at KIPP

Charter schools affiliated with the Knowledge is Power Program (KIPP) are emblematic of the No Excuses approach to public education. These schools feature a long school day, an extended school year, selective teacher hiring, strict behavior norms and a focus on traditional reading and math skills. We use applicant lotteries to evaluate the impact of KIPP Academy Lynn, a KIPP charter school that is mostly Hispanic and has a high concentration of limited English proficiency (LEP) and special-need students, groups that charter critics have argued are typically under-served. The results show overall gains of 0.35 standard deviations in math and 0.12 standard deviations in reading for each year spent at KIPP Lynn. LEP students, special education students, and those with low baseline scores benefit more from time spent at KIPP than do other students, with reading gains coming almost entirely from the LEP group.

Joshua D. Angrist
Susan M. Dynarski
Thomas J. Kane
Parag A. Pathak
Christopher R. Walters
Working Paper 15740

algebra in 8th grade

Take the guideline that most college-bound students should proceed through the K-12 math curriculum--a hierarchy of classes stepping from simple arithmetic to calculus--at a pace that lands them in algebra in ninth grade. This is too easy for most children. In a regular middle school two of my children attended, all of the students took algebra in or before eighth grade and over 80 percent mastered it, putting them one or more years ahead in math.

Entire countries of students accomplish this routinely. The Third International Mathematics and Science Study, conducted in 1996, found that the material taught in U.S. eighth-grade math classes was taught in the seventh grade in many other developed countries and even earlier in Japan and Germany. As a result, U.S. eighth graders performed significantly poorer on a standardized math test than eighth graders in twenty other countries, and far poorer than Japanese students, who scored highest. Overall, U.S. elementary and middle school math education lags a full year behind that in dozens of countries and one and a half years behind Japan and Germany.

Math Coach by Wayne Wickelgren, p. 4


Teach Like a Champion: 49 Techniques that Put Students on the Path to College on the subject of inquiry:
There's a consistent progression to the lessons of the champion teachers who informed this book. It's best described as "I/We/You." (As far as I know, Doug McCurry, founder of Amistad Academy Charter School, coined this phrase. Others use the terms direct instruction, guided practice, and independent practi to describe what McCurry means.) This name refers to a lesson in which responsibility for knowing and being able to do is gradually released from teacher to student. It means beginning with "I" by delivering key information or modeling the process you want your students to learn as directly as possible, then walking your students through examples or applications. In the "We" step, you first ask for help from students at key moments and then gradually allow them to complete examples with less and less assistance on more and more of the task. Finally, in the "You" step, you provide students the opportunity to practice doing the work on their own, giving them multiple opportunities to practice


The recipe may sound obvious to some, but it doesn't happen this way in many classrooms. Often students are released to independent work before they are ready to do so effectively. They are asked to solve a problem before they know how to do it on their own. They're asked to infer the best solution by "inquiry" when they have little hope of doing so in an effective and efficient way. In many cases, they independently and industriously practice doing a task the wrong way. They reflect on "big questions" before they know enough to do so productively.

Teach Like a Champion: 49 Techniques that Put Students on the Path to College
by Doug Lemov
p. 71 - 72

These are the teachers whose practices he is describing:
[H]e decided to seek out the best teachers he could find — as defined partly by their students’ test scores — and learn from them. A self-described data geek, he went about this task methodically, collecting test-score results and demographic information from states around the country. He plotted each school’s poverty level on one axis and its performance on state tests on the other. Each chart had a few outliers blinking in the upper-right-hand corner — schools that managed to squeeze high performance out of the poorest students. He broke those schools’ scores down by grade level and subject. If a school scored especially high on, say, sixth-grade English, he would track down the people who taught sixth graders English.

He called a wedding videographer he knew through a friend and asked him if he’d like to tag along on some school visits. Their first trip to North Star Academy, a charter school in Newark, turned into a five-year project to record teachers across the country.

Building a Better Teacher
by Elizabeth Green
New York Times | March 2, 2010

Lemov's taxonomy
Teach Like a Champion is out
more words you don't see in Teach Like a Champion

Paved with Good Intentions

UPDATE: I guess I wasn't clear. Here's the gist of the post:

In order to meet the "algebra 1 in 8th grade" state requirement, the St. Paul school district has made algebra 1 a two year course that begins in 8th and ends in 9th.

Since the NMAP final report suggested that states move as much as possible to a math curriculum that taught Algebra 1 in 8th grade (thereby matching many international countries' standards), many states have decreed Algebra 1 will be taught in 8th grade. Like many states, Minnesota has mandated such.

The new law states that by the 2010-2011 school year, all public schools will offer "algebra 1" in 8th grade. That course will no longer be available for credit in high school--if a student needs it again, it will not count toward grad requirements.

The MN standards for this 8th grade course, however, are not a full year of algebra 1. They instead cover the algebra of the line: linear functions, linear equalities. At least one of the authors of the standards told me himself that he felt a full fledged algebra 1 course was not appropriate in 8th grade. The standards for an "authentic algebra" course are actually defined by the end of an algebra sequence--including algebra 2.

So what of the gap between?

The rest of an "authentic algebra" course as defined by NMAP
must go somewhere right?

Well, here's St. Paul's solution:

Grade Level: 9 High School

Subject Area: Mathematics

Course Number: M403011

Course Title: Intermediate Algebra (Reg 9)

Course Length: 1 Year

Prerequisite: Algebra 1

Course Description:
This course covers the second half of a traditional Algebra 1 course. It extends from polynomials, through quadratic and exponential functions.
Topics covered include:

1. Exponents and Polynomials
2. Factoring Polynomials
3. Quadratic Functions and Equations
4. Data Analysis and Probability
5. Exponential and Radical Functions
6. Rational Functions and Equations

To recap:

the first half of algebra 1 is now taught for an entire year in 8th grade. The second half of algebra 1 is now taught for an entire year in 9th grade. Why, we've sure accelerated our math program up to that international standard, haven't we!

To the parents, though, this is a fraud. Their children are taking 8th grade algebra 1, they think. But they do not know that algebra 1/2 is now being taught for a year, and algebra 1/2-1 is another year. Their kids are still not going to reach AP calc.

And if you're interested in how you get to algebra in 8th grade:
Unfortunately, MN didn't change the standards leading up to 8th grade. Those changes to curricula in grades 4-7 are required by 2014. No changes in teacher certification have been mandated at all.