kitchen table math, the sequel

(Here's his blog. And here's an early review!)

I told Phil, as soon as I laid eyes on the book, that it has the look and feel of an authored work, nothing like the many-authored splendor of a typical commercial textbook.

Plus (and I have to say this) the cover and pages are creamy and smooth, a throwback to the physical beauty of books pre-crash (although of late I've had the sense that decent paper may be making a comeback).

The book is so compelling I may have to buy a second copy to work through myself so the other one can stay on the coffee table.

From the Introduction:

For 26 years, I have been a high school physics teacher. I work in an excellent, well-regarded high school and I have been fortunate to have many talented students who soak up all the physics I can teach them, and more. But every year, I also teach students who struggle to master the topic, despite their great efforts and mine. And I know from discussions with colleagues, both within my school and arose the country, that we are not the only ones struggling. There is something getting in our way. Maybe this will seem obvious to anyone who has struggled in physics, but here's what I think: I think it's the math.

Physics applies math. It's all about finding relationships and solving the puzzles that the laws of physics present. For the most part, this work is done in the language of mathematics, and more specifically, the language of algebra. So to be comfortable learning physics, a student has to be fluent in that language. Algebra cannot just be a memorized set of procedures for finding 'x'. It has to be a symbolic way of representing ideas. But for many students, that level of fluency is not attained unjust one year of algebra--which is all that many students have had when they start studying physics. It's no wonder that some struggle.

It is not only physic students who struggle. For even more than 26 years, I have been teaching students how to prepare for the math portion of the SAT. What I have seen over the years is that most students are not fluent enough in algebra to successfully apply algebra on the SAT. One goal of my SAT course is to teach alternative, non-algebraic approaches to SAT problems. It is also a major theme of my math SAT book, The New Math SAT Game Plan. And I will tell you something you may find surprising (or even distressing): on the SAT, these non-algebraic methods work very nicely. They won't get you to an 800, but they will take you pretty far. And even my top scorers report that they like to mix in the non-algebraic methods along with the standard approaches (which, as top students, they also know how to use).

The non-algebraic methods, however, won't get you very far in physics. In fact, a student who does not really learn the language of algebra is going to struggle in all later math and science classes: physics, statistics, computer science and beyond. That STEM door is swinging closed because one year of algebra class did not lead to sufficient fluency. So why spend only one year? Why not start earlier?

I am not saying every 7th grader should be in a high-school version of Algebra I. But I am saying that every middle school student should, over the course of the middle school years, start learning about and thinking about the ideas of algebra (even some ideas that won't reappear until Algebra II or Pre-calculus). These are ideas that take some time to ponder.

From Algebra II and The Declining Significance of Coursetaking:

...taking and successfully completing an Algebra II course, which once certified high school students’ mastery of advanced topics in algebra and solid preparation for college-level mathematics, no longer means what it once did. The credentialing integrity of Algebra II has weakened.

The declining significance of successfully completing Algebra II highlights a dilemma. Pushing students to take more advanced coursework has been a mainstay of American school reform for several decades. That prescription has worked in boosting enrollments. In 1986, less than half of all 17 year-olds (44%) had completed Algebra II, and for Black and Hispanic students, the rate was less than a third. Completing Algebra II is now commonplace. In 2012, about three-fourths of students completed Algebra II, and the race/ethnicity gaps associated with taking the course have narrowed significantly. (All NAEP data below are from the NAEP data explorer.)

Getting more students to take higher level math courses may be a hollow victory. It has not coincided with students learning more math.

(Cross-posted at Out In Left Field)

In an article in the most recent issue of American Scientist entitled "The Music of Math Games," Keith Devlin (head of the Human-Sciences and Technologies Advanced Research Institute at Stanford University and NPR's "math guy") says that learning math should be like learning to play the piano. In doing so, he recalls (but does not credit) Paul Lockhart's Lament ("A piano student's lament: how music lessons cheat us out of our second most fascinating and imaginative art form"), which I blogged about here.

Though Devlin is no literary virtuoso, not all of what he writes here is mushy metaphor. He begins with a discussion of educational software, and here his points are clear and consistent with my own experience. Most "math games" and "math education" software programs I've seen don't make mathematics an organic part of the games or activities. Instead, math problems--mostly arithmetic problems of the "mere calculation" variety--are shoe-horned into non-mathematical situations. Here they serve simply as tasks you must complete before moving through the current non-mathematical activity or on to the next non-mathematical activity.

As Devlin writes:

To build an engaging game that also supports good mathematics learning requires... understanding, at a deep level, what mathematics is, how and why people learn and do mathematics, how to get and keep them engaged in their learning, and how to represent the mathematics on the platform on which the game will be played.

The same is true of language learning. Most linguistic software taps only superficial aspects of language, and, as I know from personal experience, it takes great effort to build a program that does more than that.

Where I begin to part ways with Mr. Devlin is in his discussion of traditional math and what he thinks is an excessive emphasis on symbols:

Many people have come to believe mathematics is the memorization of, and mastery at using, various formulas and symbolic procedures to solve encapsulated and essentially artificial problems. Such people typically have that impression of math because they have never been shown anything else...

...

By and large, the public identifies doing math with writing symbols, often obscure symbols. Why do they make that automatic identification? A large part of the explanation is that much of the time they spent in the school mathematics classroom was devoted to the development of correct symbolic manipulation skills, and symbol-filled books are the standard way to store and distribute mathematical knowledge. So we have gotten used to the fact that mathematics is presented to us by way of symbolic expressions.

This approach to math, Devlin suggests, is at odds with the resolutions of a "blue-ribbon panel of experts" serving on the National Research Council’s Mathematics Learning Study Committee ("Adding it Up: Helping Children Learn Mathematics," National Academies Press, 2001). In Devlin's words: these resolutions hold that math proficiency consists of:

the aggregate of mathematical knowledge, skills, developed abilities, habits of mind and attitudes that are essential ingredients for life in the 21st century. They break this aggregate down to what they describe as “five tightly interwoven” threads. The first is conceptual understanding, the comprehension of mathematical concepts, operations and relations. The second is procedural fluency, defined as skill in carrying out arithmetical procedures accurately, efficiently, flexibly and appropriately. Third is strategic competence, or the ability to formulate, represent and solve mathematical problems arising in real-world situations. Fourth is adaptive reasoning—the capacity for logical thought, reflection, explanation and justification. Finally there’s productive disposition, a habitual inclination to see mathematics as sensible, useful and worthwhile, combined with a confidence in one’s own ability to master the material.

Ah, "21st century skills," "habits of mind," "conceptual understanding," "real-world situations," "explanation," "disposition"...--all this makes me wonder about the ratio of mathematicians to math eduation "experts" on this blue-ribbon panel. (It should be noted that Devlin himself is not, strictly speaking, a mathematician; he holds a Ph.D. in logic from the University of Bristol, and, while affiliated with Stanford, is not a member of the Stanford math department.)

Standing in the way of these lofty goals is what Devlin calls the "symbol barrier":

For the entire history of organized mathematics instruction, where we had no alternative to using static, symbolic expressions on flat surfaces to store and distribute mathematical knowledge, that barrier has prevented millions of people from becoming proficient in a cognitive skill set of evident major importance in today’s world, on a par with the ability to read and write.

To the rescue comes... Devlin's math education software program:

With video games, we can circumvent the barrier. Because video games are dynamic, interactive and controlled by the user yet designed by the developer, they are the perfect medium for representing everyday mathematics, allowing direct access to the mathematics (bypassing the symbols) in the same direct way that a piano provides direct access to the music.

Devlin's notion that a well-designed math video game can help students meet the National Academy's goals for math education rests on two assumptions. One is that students can achieve a sufficient level of mastery in mathematics without symbols. The other is that playing such video games is to math what playing the piano is to music.

To address the first claim, Devlin elaborates the analogy to music:

Just how essential are those symbols? After all, until the invention of various kinds of recording devices, symbolic musical notation was the only way to store and distribute music, yet no one ever confuses music with a musical score.
...
Just as music is created and enjoyed within the mind, so too is mathematics created and carried out (and by many of us enjoyed) in the mind. At its heart, mathematics is a mental activity—a way of thinking—one that over several millennia of human history has proved to be highly beneficial to life and society.

But there's an important difference between math and music--and a reason why no one confuses music with a musical score. Music has a privileged place in subjective experience. Along with sensations like color, taste, and smell, it produces in us a characteristic, irreduceable, qualitative impression--an instance of what philosophers call "qualia." Just as there's no way to capture the subjective impression of "redness" with a graph of its electromagnetic frequency, or of "chocolate" with a 3-D model of its molecular structure, so, too, with the subjective feeling of a tonic-dominant-submediant-mediant-subdominant-tonic-subdominant-dominant chord progression. Embedded in what makes music what it is to us is the qualia of its chords and melodies.

Like most other, more abstract concepts ("heliocentric," "temporary"), mathematic concepts don't generally evoke this qualia sensation. What makes math beautiful are things like eloquence, patterns, and power. Unlike a Bach fugue translated homomorphically into, say, a collage of shapes, mathematical concepts can be be translated into different representational systems without losing their essence and beauty.

Devlin argues that while we might write down symbols in the course of doing real-life math, it is primarily a "thinking process," and that "at its heart, mathematics is a mental activity—a way of thinking." I agree. Indeed, math is much more appropriately compared with thoughts than with music. But this makes math symbols the mathematical equivalent of linguistic symbols. While thoughts, like math, can be expressed in a number of different symbol systems, you need some sort of symbol system in order to represent your own thoughts and to understand the thoughts of others.

This is especially true of abstract thoughts--and of abstract math. As Devlin himself admits, "the advanced mathematics used by scientists and engineers is intrinsically symbolic. "What isn't intrinsically symbolic, Devlin claims, is "everyday mathematics":

The kind of math important to ordinary people in their lives... is not, and it can be done in your head. Roughly speaking, everyday mathematics comprises counting, arithmetic, proportional reasoning, numerical estimation, elementary geometry and trigonometry, elementary algebra, basic probability and statistics, logical thinking, algorithm use, problem formation (modeling), problem solving, and sound calculator use. (Yes, even elementary algebra belongs in that list. The symbols are not essential.)

OK, but what does this mean for education? Are we going to decide before the end of middle school which students are going to become scientists, engineers, and mathematicians, and only help those students scale the "symbol barrier"? For a barrier it certainly is, as Devlin himself notes: "people can become highly skilled at doing mental math and yet be hopeless at its symbolic representations."

But Devlin is too busy appreciating the (well-studied) math skills of Brazilian street vendors, who do complex arithmetic calculations in their heads with 98% accuracy, and supposedly without the help of symbols (even mental ones?), to realize the educational implications of the fact that "when faced with what are (from a mathematical perspective) the very same problems, but presented in the traditional symbols, their performance drops to a mere 35 to 40 percent accuracy." No, not everyone is going to become an engineer. But not all non-engineers are going to become Brazilian street vendors.

It's ironic how deeply Devlin appreciates the difficulty that "ordinary people" have with the symbol barrier without appreciating what this says about their educational needs:

It simply is not the case that ordinary people cannot do everyday math. Rather, they cannot do symbolic everyday math. In fact, for most people, it’s not accurate to say that the problems they are presented in paper-and-pencil format are “the same as” the ones they solve fluently in a real life setting. When you read the transcripts of the ways they solve the problems in the two settings, you realize that they are doing completely different things. Only someone who has mastery of symbolic mathematics can recognize the problems encountered in the two contexts as being “the same.”

Instead of seeing this as a reason for exposing children to mathematical symbols early and often, Devlin sees this as reason to create computer games that somehow teach math non-symbolically.

He calls this "adaptive technology," a term that should raise red flags. In a recent blog post, I wrote about how assistive technology often becomes yet another excuse not to teach basic skills. Kids with dyslexia struggle mightily with the symbol system of written language; should they instead learn everything through text-to-speech and speech-to-text devices, and never learn how to read and write?

Devlin makes a few other strained comparisons to the piano:

The piano metaphor can be pursued further. There’s a widespread belief that you first have to master the basic skills to progress in mathematics. That’s total nonsense. It’s like saying you have to master musical notation and the performance of musical scales before you can start to try to play an instrument—a surefire way to put someone off music if ever there was one.

No it's not; it's like saying you have to master simple scales and exercises before you move on to Rachmaninoff.

The one difference between music and math is that whereas a single piano can be used to play almost any tune, a video game designed to play, say, addition of fractions, probably won’t be able to play multiplication of fractions. This means that the task facing the game designer is not to design one instrument but an entire orchestra.

Can one create a video game that functions "as an instrument on which a person can 'play' mathematics?"

Can this be done? Yes. I know this fact to be true because I spent almost five years working with talented and experienced game developers on a stealth project at a large video game company, trying to build such an orchestra.

What does Devlin's software do? The last two paragraphs of this article function as an extended but not very informative infomercial. Here's the most informative excerpt:

Available in early March, Wuzzit Trouble is a game where players must free the Wuzzits from the traps they’ve inadvertently wandered into inside a castle. Players must use puzzle-solving skills to gather keys that open the gearlike combination locks on the cages, while avoiding hazards.

Puzzle solving? As I argue in my last post on math games, existing games already offer some version of this, and it isn't math. This, indeed, is one of the other problems with so-called math education software.

Devlin suggests his software is different:

Unlike the majority of other casual games, it is built on top of sound mathematical principles, which means that anyone who plays it will be learning and practicing good mathematical thinking—much like a person playing a musical instrument for pleasure will at the same time learn about music.

Wuzzit Trouble might look and play like a simple arithmetic game, and indeed that is the point. But looks can be deceiving. The puzzles carry star ratings, and I have yet to achieve the maximum number of stars on some of the puzzles! (I never mastered Rachmaninov on the piano either.) The game is not designed to teach. The intention is to provide an “instrument” that, in addition to being fun to play, not only provides implicit learning but may also be used as a basis for formal learning in a scholastic setting.

If you say so. But I wonder how much it will cost schools (and society) to find out whether this latest incarnation of "math education" software helps prepare students to become mathematicians, scientists, engineers--or Brazilian street vendors.

A couple of weeks ago, James Milgram, an emeritus Professor of Mathematics at Stanford University, updated me on some recent developments in the controversy over Jo Boaler's "Railside Study." It was only after I reviewed the various critiques, accusations, and rebuttals that I remembered what an enormously consequential case of educational malpractice is afoot here--one that deserves much wider attention than it's gotten so far.

Professor Milgram is known in the education world for his comprehensive critique of a study done by Jo Boaler, an education professor at Stanford, and Megan Staples, then an education professor at Purdue. Boaler and Staples' paper, preprinted in 2005 and published in 2008, is entitled Transforming Students’ Lives through an Equitable Mathematics Approach: The Case of Railside School. Focusing on three California schools, it compares cohorts of students who used either a traditional algebra curriculum, or the Reform Math algebra curriculum The College Preparatory Mathematics (CPM). According to Boaler and Staple's paper, the Reform Math cohort achieved substantially greater mathematical success than the traditional math cohorts.

In early 2005 a high ranking official from the U.S. Department of Education asked Professor Milgram to evaluate Boaler and Staples' study. The reason for her request? She was concerned that, if Boaler and Staples' conclusions were correct, the U.S. department of education would be obliged, in Milgram's words, "to begin to reconsider much if not all of what they were doing in mathematics education." This would entail an even stronger push by the U.S. educational establishment to implement the Constructivist Reform Math curricula throughout K12 education.

Milgram's evaluation of Boaler and Staples' study resulted in a paper, co-authored with mathematician Wayne Bishop and statistician Paul Clopton, entitled A close examination of Jo Boaler's Railside Report. The paper was accepted for publication in peer-reviewed journal Education Next, but statements made to Milgram by some of his math education colleagues caused him to become concerned that the paper's publication would, in Milgram's words, make it "impossible for me to work with the community of math educators in this country"--involved as he then was in a number of other math education-related projects. Milgram instead posted the paper to his Stanford website.

This past October a bullet-point response to Milgram's paper, entitled "When Academic Disagreement Becomes Harassment and Persecution," appeared on Boaler's Stanford website. A month ago, Milgram posted his response and alerted me to it. I have his permission to share parts of it here.

Entitled Private Data - The Real Story: A Huge Problem with Education Research, this second paper reviews Milgram et al's earlier critiques and adds several compelling updates. Together, the two papers make a series of highly significant points, all of them backed up with transparent references to data of the sort that Boaler and Staple's own paper completely lacks.

Indeed, among Milgram et al's points is precisely this lack of transparency. Boaler and Staples refuse to divulge their data, in particular data regarding which schools they studied, claiming that agreements with the schools and FERPA (Family Educational Rights and Privacy Act) rules disallow this. But FERPA only involves protecting the school records of individual students; not those of whole schools. More importantly, refusals to divulge such data violate the federal Freedom of Information Act. Boaler's refusal also violates the policies of Stanford University, specifically its stated "commitment to openness in research" and its prohibitions of secrecy, "including limitations on publishability of results."

Second, Milgram et al's examination of the actual data, once they were able to track it down via California's education records, shows that it was distorted in multiple ways.

1. Boaler and Staple's chosen cohorts aren't comparable:

It appears, from state data, that the cohort at Railside [the pseudonym of the Reform Math school] was comprised of students in the top half of the class in mathematics. For Greendale, it appears that the students were grouped between the 35th and 70th percentiles, and that the students at Hilltop were grouped between the 40th and 80th percentiles. [Excerpted from Milgram; boldface mine]

2. Boaler and Staple's testing instruments are flawed:

Our analysis shows that they contain numerous mathematical errors, even more serious imprecisions, and also that the two most important post-tests were at least 3 years below their expected grade levels.  [Excerpted from Milgram; boldface mine]

3. The data comparing test scores on California's standardized tests (STAR) comes from a comparison of test scores from students not involved in Boaler and Staple's study:

The students in the cohorts Boaler was studying should have been in 11th grade, not ninth in 2003! So [this] is not data for the population studied in [Boaler and Staple's paper]. This 2003 ninth grade algebra data is the only time where the Railside students clearly outperformed the students at the other two schools during this period. There is a possibility that they picked the unique data that might strengthen their assertions, rather than make use of the data relevant to their treatment groups.   [Excerpted from Milgram; boldface mine]

4. The most relevant actual data yields the opposite conclusion about the Reform Math cohort's mathematical success relative that of the traditional math cohorts:

o The most telling data we find is that the mathematics remediation rate for the cohort of Railside students that Boaler was following who entered the California State University system was 61%
o This was much higher than the state average of 37%
o Greendale's remediation rate was 35% o and Hilltop's was 29%.

5. School officials at "Railside" report that the results of the reform math curriculum are even worse than Milgram et al had originally indicated:

A high official in the district where Railside is located called and updated me on the situation there in May, 2010. One of that person's remarks is especially relevant. It was stated that as bad as [Milgram et al's original paper] indicated the situation was at Railside, the school district's internal data actually showed it was even worse. Consequently, they had to step in and change the math curriculum at Railside to a more traditional approach.

Changing the curriculum seems to have had some effect. This year (2012) there was a very large (27 point) increase in Railside's API score and an even larger (28 point) increase for socioeconomically disadvantaged students, where the target had been 7 points in each case.

6. Boaler’s responses to Milgram et al provide no substantiated refutations of any of their key points

In response to comments on an article on Boaler's critique of Milgram, Boaler states:

"I see in some of the comments people criticizing me for not addressing the detailed criticisms from Milgram/Bishop. I am more than happy to this. [...] I will write my detailed response today and post it to my site."

However, as Milgram notes in his December paper:

As I write this, nearly two months have passed since Boaler's rebuttal was promised, but it has not appeared. Nor is it likely to. The basic reason is that there is every reason to believe [Milgram et al's paper] is not only accurate but, in fact, understates the situation at "Railside" from 2000 - 2005.

In a nutshell: under the mantle of purported FERPA protection, we have hidden and distorted data supporting a continued revolution in K12 math education--a revolution that actual data show to be resulting, among other things, in substantially increased mathematics remediation rates among college students. Ever lower mathematical preparedness; ever greater college debt. Just what our country needs.

Nor is Boaler's Reform Math-supporting "research" unique in its lack of transparency, in its lack of independent verification, and in its unwarranted impact on K12 math practices. As Milgram notes,

This seems to be a very common occurrence within education circles.

For example, the results of a number of papers with enormous effects on curriculum and teaching, such as [Diane Briars and Lauren Resnick's paper "Standards, assessments -- and what else? The essential elements of Standards-based school improvement"] and [J. Riordan and P. Noyce's paper, "The impact of two standards-based mathematics curricula on student achievement in Massachusetts"] have never been independently verified.

Yet, [Briars and Resnick's paper] was the only independent research that demonstrated significant positive results for the Everyday Math program for a number of years. During this period district curriculum developers relied on [Briars and Resnick's paper] to justify choosing the program, and, today, EM is used by almost 20% of our students. Likewise [Riordan and Noyce's paper] was the only research accepted by [the U.S. Department of Education's] What Works Clearinghouse in their initial reports that showed positive effects for the elementary school program ``Investigations in Number, Data, and Space,'' which today is used by almost 10% of our students.

As Milgram notes:

Between one quarter and 30% of our elementary school students is a huge data set. Consequently, if these programs were capable of significantly improving our K-12 student outcomes, we would surely have seen evidence by now.

And to pretend that such evidence exists when it doesn't is nothing short of educational malpractice.

When I read this article, it made my blood boil! Amazing that this junk makes it into print! (Since it's Monday, you may want to put reading this one on hold...) Is Algebra Necessary? NYTimes Sunday Review, Opinion Pages I agree with rknop that "the core of his argument is the ultimate in anti-intellectualism"

from California, JC writes:

UNDER THE HEADING OF THEY DO WHAT THEY DO

N will be entering the 9th grade this fall and will hopefully be enrolling in the local public high school (or maybe not). Currently in the 8th grade, N is taking his final today in Honors High School Geometry. N attained this level of competency by working year round on math ever since the 1st grade. He has not had to skip a year of instruction as the local public students have.

The local district, in its infinite wisdom, will not accept N’s transcripts showing straight A’s and Standardized test scores in the 99th percentile for mathematics as proof of his ability. In their defense, N does not have state administered test scores on the CST Algebra 1 or Geometry tests – those tests aren’t made available to home schools that operate independent of a school district. The local high school is requiring N to take a 50 question, calculation heavy, 2 hour, Geometry course challenge exam. The test will cover material from a classroom he’s never attended and a textbook he has never used. The test won’t be a neutral exam such as the standardized exams.

According to “district policy” my son must pass the challenge exam with a 90% grade or better in order to continue on in the honors track. We were not to be supplied with the answers to the study guide that was provided. We were also informed that the study guide was missing ¾ of the materials that could be included on the exam. I wouldn’t be able to correct the study guide questions, as N has passed me by mathematically speaking. Thankfully Barry G came to my rescue and worked the problem sets, and provided fantastic comments and notations which helped me to provide my son with a proper review where he was weak.

The district, county and state all refused to let N take the CST tests this spring because he wasn’t enrolled with the public schools. In May I gave N the CST retired questions for Geometry (posted on the CDE website) and he only missed 2 out of 64 questions. We also gave him the on line Algebra 1 test, and uncovered an area of weakness, which was remedied with one evening’s chalk and talk thanks to purple math - he then only missed 3 questions on the whole Algebra 1 exam.

Needless to say, I’m a bit worked up because our high school won’t accept N’s test scores and grades as proof of his abilities. It seems terribly unfair when you consider that the district students only need a B- grade on their report cards to advance to the next honors level class.

If you investigate further - our high school’s CST scores for the 9th grade single honors Geometry class indicate 75% of those honor students fail to score in the advanced category. That’s not a stellar record -- especially knowing how low the proficiency bar is set by the California Department of Education. N is trying to gain entry to the 9th grade Honors Algebra 2 course. The Algebra 2 double honors track has better CST scores with only 22% of the honors students failing to score at the advanced level. The improved performance of the upper track is likely a product of after schooling.

The administration is slamming the gate shut on students who transfer in from schools other than the local district. UPDATE: It isn’t “all” out of district transfer students who must take this exam as I had previously been informed. No, it’s just the students who come from “non traditional” high schools that must pass a challenge exam. So apparently, private home school students must out perform the majority of the local high school honors math students in order to gain admission to the honors math track. Perhaps, hypothetically speaking, it’s just the home school children of veteran math warriors who are expected to perform at this level.

This all seems so horribly unfair. N is an excellent student and has worked incredibly hard, and the standardized tests all place him at the very top. I wouldn’t be so bent out of shape if the local high school CST scores indicated that the honors classes were full of elite math stallions, but that just isn’t the case.

N finished the high school’s study guide exam and took 40 minutes longer than the 2 hour time limit they are setting on the exam. Hopefully the multiple choice exam he will be given next Tuesday will be easier to complete within the time limit otherwise, N will be getting the gate slammed shut in his face.

UPDATE: I’ve just learned the ruling we will get as a result of this test is not the final end – I can always appeal the decision! This is great! I can spend my summer vacation fighting with the school district -- and my son isn’t even in their classrooms yet.

THERE IS HOPE:

On a happier note, I do have some good news to report. N took the California High School Proficiency Exam last Saturday. The CHSPE allows students to exit high school early, and continue their education at the community college level. Though the exam is designed for students 16 years and older, N (only 14) was able to take the exam with the permission of his “non traditional” school (heh). We will get the results in mid July.

Previously parents have posted on KTM that skipping the high school math program and moving a student up to the college level for math isn’t perhaps the best move. I do remember Wayne Bishop having once written about saving a student from the public high school system and getting him placed into a college math program early. I would like to know what those of you who frequent KTM think about taking this approach.

With the CHSPE proficiency certificate in hand, N will be able to enroll in community college courses without permission from the high school and receive dual credit for his courses. Some local home school parents even send their middle school students to the community college. N will still be able to attend the HS and play sanctioned sports as long as he attends 4 classes. A few of the ambitious local students have managed to graduate high school having also completed an Associate Arts degree.

The CHSPE does NOT allow a student to stop attending school. Students must (by law) attend classes until they reach their 18th birthday. Additionally, it’s possible the HS will refuse to allow N to receive a HS diploma and be in the graduation ceremonies if we go this route (though there may be a way to work this out). The CHSPE certificate is supposed to suffice as an equivalent document in lieu of a high school diploma and is accepted by the state of California and, I believe, the armed services though I am not sure about that.

I would appreciate hearing what KTMers think about this opportunity and look forward to reading your comments.

A friend of mine is a special ed teacher at a highly selective science-oriented magnet school. She has observed a number of incoming freshman who are unable to handle beginning (9th grade) algebra. I'm guessing that some (most? all?) of them have no inherent math disability, but have merely been poorly instructed (most come from elementary schools that use Everyday Math/Investigations and from middle schools that use Connected Math).

Anyway, when she asked me what I knew about math remediation programs that might help prepare these kids for algebra, I realized I had absolutely no ideas and should turn to ktm for help. Any suggestions?

Why does accelerating algebra for everyone not help? Moving on the same curve with a relabelled x axis does not change the curve.

In summary, our data show a clear decline in the level of Michigan State University mathematics courses taken by Core-Plus graduates. The existence of that decline is statistically significant at any reasonable level. The decline in course level is accompanied by a decline in average grades for all but the very top students, as well as a decline in the percentages of those who eventually passed a technical calculus course. These trends occur in data that include students from a variety of communities. The data from individual high schools show that the timing of these declines corresponds precisely to the implementation of the Core-Plus program.
A Study of Core-Plus Students Attending Michigan State UniversityRichard O. Hill and Thomas H. Parker Thomas Parker

I'm tossing old papers, or trying to, and in my rummaging came across this Times article from 2005.

I was tickled to see that one of the problem types added to the revised test was the absolute value inequality word problem, a category I had never seen or imagined until I encountered one in Dr. Chung's SAT Math:

For pumpkin carving, Mr. Sephera will not use pumpkins that weigh less than 2 pounds or more than 10 pounds. If x represents the weight of a pumpkin, in pounds, he will not use, which of the following inequalities represents all possible values of x?
a. | x - 2 | > 10
b. | x - 4 | > 6
c. | x - 5 | > 5
d. | x - 6 | > 4
e. | x - 10 | > 4

Pop Quiz; New and (Maybe) Improved
Published: November 7, 2004

Talk about inflexible knowledge. Somehow I had concluded that absolute value inequality calculations were just that: calculations. Arithmetic. I was stunned to discover that you could have an absolute value inequality word problem.

Wonders never cease.

Elizabeth King's explanation of these problems is excellent.

 

I just came across this old post by Ken DeRosa!

Apropos in this summer of SAT math prep.

Maybe PWN will tell us what level of difficulty this problem would be rated on the SAT. I'm thinking 3 or possibly 4, and it would be a 4 only because a lot of students haven't taken algebra 2.

from Glencoe Algebra 1 Skills Practice (pdf file):

the difference of 17 and 5 times a number

How do you write this?

17 - 5n

I don't think I've seen a number problem worded that way.

at JD2718

This year's math course has been a disaster. Beyond disaster, actually. Where math is concerned, this year has been epically^* bad.

Some kind of emergency repair has to happen this summer - but what?

Any suggestions?

I could conceivably sign C. up for an algebra 2/precalculus course at the local community college - but are the teachers there going to be any better? At this point, we desperately need an actual math teacher: a person who can teach math to a student who doesn't teach math to himself.

We could do ALEKS, but ALEKS is super-slow and overwhelmingly procedural; the 1 1/2 courses I took on ALEKS taught me one disembodied procedure after another. At this point disembodied procedures might be better than nothing -- but then again to the extent C. learned anything this year he learned disembodied procedures.

I could insist that the two of us work through Foerster (I own the teacher's edition).

We could work our way through Saxon.

I could advertise for a math teacher or check out the various tutoring companies....

We also have to do serious SAT prep (though that's not going to be onerous & time-consuming).

I'm thinking this wasn't the summer to sign up for the precision teaching institute at Morningside.

^* My neighbor's son, who is a terrific writer, is constantly inventing words he says ought to exist, and having watched him do this a few times, I think he's right. Epic is an excellent word, and its non-existent cousin epically is the word I need today.