Allison on Math Nights

MSMI has done several parent math nights.

When we are asked to give a math talk by a school and it is well attended, it is because the parents are upset. If it is very well attended, it is because the parents are in an uproar about the math program.

Since we generally are going in to fix the math program, or to support a math change to it, our goal is first to name the problem. We explain the issue (nationally, not just locally, not just here, wherever we are, but nationally) is that US curricula are not preparing kids for algebra. We tell parents what they know intuivitely but can't name. We tell them what they've watched their older kids suffered through. Then we explain we need to change what we teach, when we teach it, and what the teachers know about the maththey teach to fix it. When we are done, generally, parents calm down and give us the benefit of the doubt.

Usually, the second math night (a followup) has 1/4 of the turnout the first one had.

If a math night has no attendees, it is because math is doing just fine--the parents are concerned about some other problem.

Parents don't have time to go to meetings if things are fine. They go to indicate their disapproval or their concern.

We also found if the *children* put on the math night, as one of the grade night programs, it is well received--so if we want parents to learn about the math program, learn the games to practice math facts, etc. then it needs to be a child-centered event. Parents come when kids put on a math carnival. They even enjoy it. It does not need to be fuzzy math--kids LOVE stumping their parents at mental math calculations and bar modeling.

We tell parents what they know intuivitely but can't name.

Knowing intuitively that something is amiss: this is the chronic problem parents face. You know something--your cognitive unconscious knows something, rather--but you can't name it.

I remember, when I first became politically engaged here, living in a state of chronic anxiety that a) I didn't know what I was talking about and b) I was about to be publicly called out on not knowing what I was talking about. I spent hours Googling and reading, and reading and Googling, to make sure everything I said and wrote in my district had already been said and written by someone who did know what they were talking about.

Kitchen Table Math was incredibly important to that effort. I wrote posts to put into words what my cognitive unconscious already knew (or suspected), and I said nothing, in district, without ktm commenters vetting it first.

Funny thing: at some point I stopped feeling anxious, and I stopped obsessively fact-checking myself.

I hadn't become an expert on math or math instruction or public schools in general, but somehow I knew enough to feel confident that anything I said -- even something I said off the top of my head -- would be in the realm. Which it generally was.

I also, and I hesitate putting this in print, developed a sense of how thin my adversaries' knowledge was. That's not a criticism. Administrators can't possibly know everything about every subject (that's the problem with central administrators choosing math curricula), and an administrator who went to ed school before constructivism was in full bloom may not actually know that much about the doctrine and its history, however committed s/he may be to "rolling out" one constructivist initiative after another.

In short, at some point it dawned on me that I could pretty much say whatever I wanted and get away with it. I could get sloppy and no one would know but me.

That came as a bit of a shock.

I see politicians and pundits differently now.

Politicians and pundits are churning out an awful lot of content.

How often do they actually know that what they're saying is true?

How much fact checking happens in politics?

I'm guessing not too much.

Lockhart's lament redux

"The essence of mathematics is recognizing interesting patterns in interesting abstractions of reality and finding properties of those patterns and abstractions."

Everything About The Way We Teach Math Is Wrong

This strikes me as completely wrong, not that I have much confidence in my intuitions concerning the essence of math. So take this as a confession, not an argument.

"Recognizing interesting patterns" -- even "recognizing interesting patterns in interesting abstractions of reality" -- strikes me as the toolkit approach to math.

I personally -- another confession, not an argument -- can't stand the toolkit approach to math. Blech!

I have no interest -- none! -- in endless iterations of function problems designed to determine how much profit the guitar teacher will make teaching x number of students while paying y rent on the studio for 20 weeks, or whether Jim should buy the monthly contract or the annual, as useful and important as those questions are in daily living.

Nor do I relish the thought of encountering yet another roller coaster depicted as yet another instance of geometry, or another textbook with a nautilus shell splashed across the cover. Enough with the nautili.

If you want to make me hate math, real-world math will do the job.

That reminds me.

I did not, as a child, solve two trains leaving the station problems, and I wish I had. I was utterly charmed by the two-trains problems when I encountered them as an adult, working my way through "Russian Math."^*

I don't remember whether we were asked to solve bathtub problems (I think we were), and I was charmed by those problems, too.

(Speaking of the real world, the bathtub problem is all you need to know to understand why fiscal stimulus doesn't work when the Federal Reserve targets inflation. One party is putting money into the system; the other party is taking it back out.)

To be fair, I'm not at all sure that "toolkit math" is what Lockhart actually means: "That’s what math is — wondering, playing, amusing yourself with your imagination."

Nevertheless, inside K-12, the toolkit approach is what look-for-patterns turns into.

Look for patterns may be a bit out of date.

Today, with Common Core, at least here in NY, we've moved beyond look-for-patterns to modeling, for pete's sake. In my district, the entire high school mathematics curriculum, the entire rationale for the entire mathematics curriculum, is modeling.

Lots and lots of function problems modeling stuff nobody cares about.

This is always the conundrum with constructivism and progressive education.

Progressive educators think the real world is fun and motivating.


^* Mathematics 6 by Enn Nurk and Aksel Telgmaa

most Berkeley students do not know

Brad De Long:

MOST BERKELEY STUDENTS DO NOT KNOW THAT 2^5=32 AND 2^10≈1000
How can they expect to survive in the modern world without knowing these things?
And why haven't they learned them?

To which I can only say: Brad DeLong has not been paying attention!

I've just started reading the comments. Calculators in grade school aren't faring well thus far.

From the first Comment:

The best students are great, but the quantitative reasoning skills of even the average student at a good university are worse than those of a typical waiter/waitress 40 years ago.
invhand

What Philadelphia 5th graders should know how to do

I recently came into possession of one the Philadelphia School District Parent Teacher Brochures, which breaks down the goals that each Philadelphia School District student should be able to meet by the end of each grade level.  Since I'm homeschooling my 5th grade daughter, I was particularly interested in the goals for fifth grade. And I was shocked, shocked, to find myself more baffled than enlightened after reading through these goals.

The language arts goals are all about process, purposes, and genres, with a developmentally inappropriate assignment thrown in in the form of a research project:

•Continue to build a reading, writing and speaking vocabulary
•Read to learn new information
•Read a wide range of stories, books and magazines for enjoyment
•Understand a problem or conflict in stories or books and talk or write about an appropriate solution
•Make connections between stories and texts that they have read and the world around them
•Tell and/ or write a summary that gives the main idea of what they read and the most important details or events
•Complete a research project including a written report
•Write stories with several paragraphs
•Write poems, plays, and reports

Not a word about specific reading skills (vocabulary level, sentence complexity, making deductions and bridging inferences within the context of the text) or writing skills (grammar and punctuation, sentence construction, paragraph construction).

As for the math goals, most are vague ("compute" and "find the relationships"), easy (locating numbers on a number line; comparing numbers; sorting shapes), and emphasize verbal explanations over mathematical performance. Four out of the 13 goals are about data and probability. Here the developmentally inappropriate goal (especially given what isn't covered here) involves algebra:

• Compute and find the relationships using whole numbers, fractions, and decimals
• Locate positive and negative numbers on a number line (integers)
• Explain to you what prime numbers, factors, multiples and compositie numbers mean
• Compare numbers (equal to, greater than, and less than)
• Collect, organize, display, and analyze data in a variety of ways
• Find mean (average), median (middle number), mode (most frequent) and range (difference between largest and smallest) of data
• Predict or determine all possible combinations and outcomes, such as, "How many outfits can be created with six shirts and eight pants?"
• Calculate the chance of a simple event happening
• Use a variety of methods to solve for unknown quantities in simple one-step algebra equations (solve for x)
• Sort polygons according to their properties and angles, such as triangles, rhombi, and parallelograms
• Define and compare perimeter (distance around) and area (amount covered inside) of shapes
• Understand properties of a circle
• Explain how they solved a math problem in their own words.

Not a word about which computation skills the child should develop, what sorts of numbers, fractions, and decimals the child should be able to do computations on (perhaps only the "friendly" fractions and decimals), and what level of computational fluency the child should have. Not a word about multiplication tables, long division, repeating decimals, ratios and percents, and multi-step word problems.

Turning to science, only one substantive topic is mentioned (solar energy) and goals pertaining to it remain vague ("build an understanding;" "recognize"). Most the goals pertain to process rather than achievement, many of them involving developmentally inappropriate activities that wrongly assume that children can function as little scientists:

• Develop skills that will emphasize the five senses while doing science
• Use prior knowledge when making observations
• Make predictions and hypotheses based on observations
• Design investigations with a control and one or two variables
• Gather, organize and display data independently
• Build an understanding of how solar energy is transferred
• Recognize that the sun is the main source of energy for people and they use it in various ways
• Design and conduct experiments with variables. Students should be able to explain cause and effect
• Study the relationship in an ecosystem that shows the relationship of an organism to its environment
• Conduct hands-on investigations to discover and understand their world
• Record observations in science notebooks

It would seem that, "goals" aside, the Philadelphia Schools are avoiding any commitment to help your 5th grader increase his or her vocabulary, reading level, sentence construction skills, or computational fluency with "unfriendly" numbers; or learn any scientific content other than a few vague propositions about solar energy.

(Cross-posted at Out In Left Field).

Mallard Fillmore lays it on the line

The cartoonist who does Mallard Fillmore either attended ed school at one point, or knows people who are in education:

Seattle School District Loses Lawsuit over Discovery Math Program

News report from Seattle Post Intelligencer is here.

From the article:

Those suing to stop Discovering Math in court were Martha McLaren, a retired Seattle high-school math teacher; Cliff Mass, a professor of atmospheric science at the University of Washington; and Da-Zanne Porter, mother of a Cleveland High School student.

Mass, a well-known local meteorologist, has said college students' math abilities have been decreasing over the past 10 years.

"I've been giving a math diagnostic exam in my 101 class, and the results are stunning; stunningly bad," Mass said in a story that appeared last month before the lawsuit was argued.

This is stunning news. Every school board should be made aware of this decision. That the ruling was based--in part--on the widening achievement gap attributable to the "discovery" series, raises the spectre of civil rights.

Congratulations to Cliff Mass, Martha McLaren, and Da-Zanne Porter.

The court decision goes a long way in giving parents the credibility they need to stand up to school boards across the country.

6th (actually 7th) grade holiday math project: Just Because I Care About You

A Just - Because - I - Care - About - You MATH PROJECT!

You have been given $2,000 to buy gifts for ten different people in your life. You must decide who you want to give a gift to, what you want to buy them, and why you want to buy them this particular item. You must find a picture of this item with the price. Every item you select has a discount. You must find the discount for each item, calculate how much you will save, and how much the item will finally cost you.

Each student must complete a booklet consisting of 13 pages
Page one is your title page. This must include your name, and title of this project.

Pages 2 - 11 will display:
* A picture of a gift
* The original price
* The discount
* The final price with calculated sales tax ***
* Your math work
* Who the gift is for and why you chose this item for this person

Page 12 will show the price you spent for each item, how much money you spent all together, and how much you have left.
On page 13 you will donate the remaining money to a charity of your choice and explain why you chose this charity.

DISCOUNTS
20% off all major appliances (refrigerator, washer)
...
50% off all jewelry and clothing

***Please remember, there is a 8% sales tax on everything but clothing.

...Your project will be judged on creativity, accuracy, and neatness.

Sample

[Picture of Lamp]

A lamp for my friend Nancy.
My close friend, Nancy, just got married. At the Craft Show last month, she admired a lamp which bears a resemblance to this one. She said it was the perfect lamp for her foyer. I could not pass it up.

[Various calculations]

[Final price]

The Revolt Against Lousy Math Instruction May Just Go Viral

This example was just posted at BoingBoing, a large group blog, at this post.

Do You Understand My First-Grade Child's Homework?

The blogger asks

My six-year-old told me she doesn't understand her homework. After studying it for 15 minutes, I *think* I understand what she's supposed to do, but I'd like a second opinion.

Is it from Everyday Math?

Go add to the BoingBoing comment fun, if you like.







(I have another question -- homework for six-year-olds? I'm ok with requesting reading at home, but that's it. Period. The end.)

Reactionary politics at Kitchentablemath?

In a recent comment on Out In Left Field, someone mentioned having "reactionary politics shoved in my face" here at kitchentablemath.com. As recent examples, the commenter in question cited this and this.

For my part, I've added the commenter's reaction to my growing list of associations between general political ideologies on the one hand, and, on the other hand, specific opinions/observations about the Powers that Be, status quo, and prevailing fashions, in grade school education.

My list also includes Barry Garelick's discussion of how Lynne Cheney's criticisms of Reform Math made democrats not want to touch it; rants at Rational Math Ed; the political branding of Mathematically Correct members and, of course, teacher's unions and the legacy of Progressive Education.

I'd love to write a longer piece about this political branding, because I think it, combined with what seems to me an unprecedented polarization in this country of political "debate," is one of the biggest obstacles to improving public education, I'd hoping at some point to write a longer piece about this.

So, if you have other examples of this, or thoughts about them, please share them here! Along with your thoughts, in particular, about reactionary politics at kitchentablemath.

If I had to guess, I'd guess that many of us here are politically moderate or eclectic, pragmatist, suspicious of big bureaucracies and big government (because of what these have done to education), and sympathetic to free markets (e.g., school choice). As for whether we tend to be hawks or doves, religious or agnostic, pro choice or anti-abortion, or for or against curbside recycling, I doubt that there's much here at ktm on which to base any firm conclusions.

The market steps in

As the parent of children in the public school system, I receive many solicitations for tutoring services. This pitch for a summer school session, in particular, caught my eye.

Math Facts Boot Camp
Pre-Algebra
Algebra I
Algebra II
Geometry
With the new emphasis on “real world” or “integrated” math, many educators agree that the skills that go into solving math problems, pure and simple, are being lost. Chyten’s math Facts Boot Camps are comprehensive and intensive courses in which the ability to solve equations is brought back to its rightful position, front and center in a student’s math mind.
Five 2-hour session

Although many of our government schools seem to be more focused on making sure students are “engaged” and “love learning” rather than on actually teaching vital skills and concepts, at least it’s reassuring to know that the private market has stepped in to help. Well, it can help those families that can afford to pay $550 [edited to correct price] for this “boot camp” class.

I notice that these boot camps are targeted to middle- and high-school students. For many, this is when the sh*t hits the fan, and parents may be ready to part with their hard earned money as they come to realize what vital skills their children have not yet learned (were not taught, perhaps?) in school.

http://www.chyten.com/ces/site/global/ads/summer_catalog_09.pdf

Bootstrapping their way to mediocrity

The National Science Foundation offered grants to school districts starting in 1995 (and whose funding ended in 2002) called Local Systemic Change (LSC). These grants provided professional development to science and math teachers. To be brief, the NSF funded PD programs were party-line instruction in "standards-based" teaching; i.e., how to teach the crap programs that NSF's Education and Human Resource Division funded (like Everyday Math, Investigations, IMP, CMP, Core Plus, etc).

There is a report that came out in 2006 that I just discovered. It evaluates the effectiveness of the LSC program. (It was written by Horizon, Inc., under a grant from NSF). I wish I had found it earlier. It can be found here.

This excerpt taken from page 43 of the report is quite telling:

"Other evaluators cited changes in teachers’ beliefs about who can learn science and mathematics. For example, prevailing attitudes among some teachers before LSC workshops included low expectations and the need for ability grouping. LSC professional development helped change these beliefs. Said these teachers about the impact of LSC professional development:

"Before IMP, I felt that there were mathematically unreachable students. I felt that students could not go on to more challenging ideas like algebra, statistics, probability, or trig without basic skills. Fortunately, with my IMP training, I have a different feeling about students. I strongly believe in access to mathematics for all. (Teacher, 6–12 mathematics LSC)"

The above quote from a teacher (in italics) is amazing. Before this teacher started using IMP, he/she felt that basic skills were necessary in order to proceed in mathematics. After IMP, which essentially avoids content whenever possible, he/she saw the light. Yes, wonderful things happen when you pretend that content doesn't matter, and that higher order thinking skills occur just by giving students "authentic" problems without the bother of all those and boring drills and instruction. They are able to reach for the stars. Unfortunately they do so by standing on a two legged stool. But NSF has done its duty and the people who wrote this report have confirmed what NSF always knew: Their reform math programs are an unparalleled success.

The only thing this report lacks is a chorus line kick and the ritual singing of Kumbayaa.

Everyday Math: Coming to a School Near You!

It appears that the marketing people for Everyday Math are earning their salaries. There's a new promo on the web for EM (I've reproduced it below) The main site is located here. There are also YouTube videos containing testimonials, etc. Of interest is the one here. It features people from Woodbridge NJ (other districts are there also). It's all sound bites. "Allows higher level dialogue in solving problems" etc.


In the EM promo (which I've included below) it looks like they've picked up on the criticisms of EM and are now using it in their advertising. To wit: "There's nothing fuzzy about it".

Excuse me? Nothing fuzzy aboout it? Well, maybe not as fuzzy as TERC, but still... Yes, a casual reading might reveal what look like good problems, but you wouldn't know from looking at the workbook that they don't teach the standard algorithms, that a particular page of problems may represent the last time such types of problems are seen that year, that there are far less computational problems in EM than in the highly disdained "traditional" textbooks, and the computational problems that are there do not cover a lot of 2-digit or 3-digit multiplication. Not to mention that calculators are allowed fairly often.

The sentence that really got to me was "Everyday Mathematics is better than traditional, textbook-centered programs that produced generations of students who hated math." While some of the traditional text books of the 50's and 60's had their bad points, I think this statement is over-generalized and extremely misleading. The textbook-centered programs of yore also produced generations of students who liked math, were good at it, and understood the underlying concepts. And ironically, many if not all of the students who hated math as a result of those "traditional textbook-centered programs" are probably more proficient in the basic skills than those who have received the EM treatment without benefit of Kumon, or outside help.

Of course, EM's solutioon to the "textbook-centered" approach is to do away with a textbook. Students only have workbooks. (Oh, and a reference manual. Which does have a good section on how to use a calculator). The EM promoters' disdain for such "scripted" approaches is pure hypocrisy, since EM does have a particular script. Students and parents can't see it, but it's contained in the teacher's manual and provides the outline of daily lessons. Not very good, mind you, but still, there is a plan there which parents and students do not get the benefit of seeing--except in the "family letters" (some in very poor Spanish as has been discussed here) that students bring home with them and which explain what they will be learning in a particular unit. Every unit is a hodge podge of topics, nested inside some main topic. There is no concentrated focus on any one topic that allows any kind of mastery learning. But the EM promoters have an answer for that one as well: I

"Content is taught in a repeated fashion, beginning with concrete experiences to which students can relate. Research shows that students learn best when new topics are presented at a brisk pace, with multiple exposures over time, and with frequent opportunities for review and practice. The sequence of instruction in the Everyday Mathematics curriculum has been carefully mapped out to optimize these conditions for learning and retaining knowledge."

They didn't bother to talk about what research it was that showed this, but there is another module on their "research base". Much of their research was conducted by William Carroll, who has been on the EM/U of Chicago payroll for some time. Seems to me the National Math Panel's final report seemed to address EM's approach head on when they said:

“A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided.”

Anyway, here's the promo. Read and weep.

How Everyday Mathematics Offers a Better Approach to Mathematics Mastery

There’s nothing fuzzy about it. Everyday Mathematics brings more clarity and rigor to math instruction, so students understand and appreciate the role of mathematics in daily life. Everyday Mathematics, a comprehensive Pre-K-6 mathematics curriculum, not only embraces traditional goals of math education, but also sets out to accomplish two ambitious goals for the 21st century:

• To substantially raise expectations regarding the amount and range of math that students
learn.

• To support teachers and students with the materials necessary to enable students to meet
these higher expectations.

To provide more rigorous, balanced instruction, Everyday Mathematics:

• Emphasizes conceptual understanding while building mastery of basic skills.

• Explores a broad mathematics spectrum, not just basic arithmetic.

• Is based on how students learn and what they’re interested in while preparing them for their
future mathematical needs.

Changing the Way We Teach Math

The accelerating demand for competence and problem-solving agility in mathematics requires
improved methods for teaching math in the classroom. Teachers are no longer preparing students for a lifetime of pencil-and-paper calculations, but for future careers that demand a true understanding of how mathematics works at much higher levels.

Everyday Mathematics is better than traditional, textbook-centered programs that produced generations of students who hated math. It is consistent with the ways students actually learn math – building understanding over time – first through informal exposure, then through more formal and directed instruction.

Content is taught in a repeated fashion, beginning with concrete experiences to which students can relate. Research shows that students learn best when new topics are presented at a brisk ace, with multiple exposures over time, and with frequent opportunities for review and practice. The sequence of instruction in the Everyday Mathematics curriculum has been carefully mapped out to optimize these conditions for learning and retaining knowledge.

Academic (Math) Slums

Economist Walter Williams weighs in on the academic atrocity also known as 'fuzzy math':

American education will never be improved until we address one of the problems seen as too delicate to discuss. That problem is the overall quality of people teaching our children. Students who have chosen education as their major have the lowest SAT scores of any other major. Students who have graduated with an education degree earn lower scores than any other major on graduate school admissions tests such as the GRE, MCAT or LSAT. Schools of education, either graduate or undergraduate, represent the academic slums of most any university. As such, they are home to the least able students and professors with the lowest academic respect. Were we serious about efforts to improve public education, one of the first things we would do is eliminate schools of education.

The inability to think critically makes educationists fall easy easy prey to harebrained schemes, and what's worse, they don't have the intelligence to recognize that the harebrained scheme isn't working. Just one of many examples is the use of fuzzy math teaching techniques found in "Rethinking Mathematics: Teaching Social Justice by the Numbers." Among its topics: "Sweatshop Accounting," "Chicanos Have Math in Their Blood," "Multicultural Math," and "Home Buying While Brown or Black." The latter contains discussions on racial profiling, the war in Iraq, corporate control of the media, and environmental racism.

If you have a fifth-grader, his textbook might be "Everyday Math." Among its study questions are: If math were a color, it would be —, (blank) because —. (blank). If it were a food, it would be —, (blank) because —. (blank). If it were weather, it would be —, (blank) because —. (blank). All of this is sheer nonsense, and what's worse is that the National Council of Teachers of Mathematics sponsors and supports much of this nonsense.

Mathematics, more than any other subject, is culturally neutral. The square root of 16 is 4 whether you're an Asian, European, or African, or even a Plutonian or Martian. While math and science literacy among white 15-year-olds is nothing to write home about, that among black 15-year-olds is nothing less than a disaster.

Few people appreciate the implications of poor math preparation. Mathematics, more than anything else, teaches one how to think logically. As such, it is an important intellectual tool. If one graduates from high school with little or no preparation in algebra, geometry and a bit of trigonometry, he is likely to find whole areas of academic study, as well as the highest paying jobs, hermetically sealed off from him for his entire life.

You can read the entire article here. For those interested, Williams has a number of syndicated columns over the past few years that are quite critical of the declining 'educational' departments in American universities.

The Educational Industrial Complex

Today I came across a great blog entry over at The Daily Kos by Nanoman, "a professor of electrical and computer engineering in Louisville, Kentucky and a parent who has had children attending K-12 public schools." He's taken the liberty of welcoming Education to "the old familiar Military, Tobacco, Pharmaceutical and Medical Insurance Industrial Complexes."

He prefaces a fabulous chart, a centerpiece of the blog entry, with this:

Please forgive the hyped intro, but the story that follows (which is summarized in the blue and yellow chart) is worth reading because it is being repeated in school districts across the country where choreographed ploys are used to bypass parent concerns about poor math texts.

Just take a look at Math Wars: The Educational Industrial Complex. The chart with Steve Leinwand at the center of all the hullabaloo is certainly worth the price of admission.

When you're done with that, you might want to check out "Is Our Children Learning" Math Texts too.

Patterns, probabilities and data analysis

There has been some discussion of this in comments, but I felt that it deserved more attention. The question has come up as to why many math texts, curricula , state standards etc emphasize patterns, probability and data analysis.

With respect to "patterns", someone got the idea in his/her head that algebra was about patterns. Maybe they said math was about patterns. It doesn't matter. In any case, it all reminds me of a twelve-year-old's version of Paradise. They seem to think that giving students exercises in which they are given a pattern like x=1, y=2; x=2,y=4; x=3, y=6. and asked to find the "next term" lays the groundwork for future understanding of functions. Well, it doesn't, really. It may lead students to believe that any pattern they can see must be it. So in the above pattern, the student comes up with "Oh, the rule is y = 2x, so the next term is 8." Well, what if I told you the next term is 14? It could be if y = 2x + (x-1)(x-2)(x-3). Actually, some people who advocate this approach have recognized this, and instruct teachers to watch for students who give "one" answer to the problem, illustrating "convergent thinking" and watch for students who give many answers to the problem, illustrating "divergent thinking". What this has to do with teaching about functions is another matter. But why teach about functions when life doesn't lend itself to functions, bringing us to our next topic: Data anlysis and probability.

Everyone knows that life is messy and that life's problems frequently don't lend themselves to "nice" solutions that are in algebra texts like Dolciani. Besides, students are bored learning the algebra that will be necessary to succeed in college science and engineering courses. So let's turn math into an empirical science. That way, expressing ideas in mathematically precise fashion need not ever be a concern.

The issue of why so much emphasis on data analysis and probability came up at the November meeting of the National Math Advisory Panel. Vern Williams questioned a member of the College Board why there was so much emphasis. Here is the exchange:

"WILLIAMS: Maybe one reason why students need more advanced courses to become successful in college is because so many things have been taken out of the basic courses because of the addition of topics like data analysis. I can't understand why data analysis would be a part of a geometry course. American students are extremely weak in geometry. In many cases, that is the only proof-based course, or at least it used to be a proof-based course, that students get. So, of all places, why would data analysis be included in geometry?

"DR. MANASTER: That's a very hard question to answer. We share your concern about mathematical reasoning and proofs being more apparent in geometry than in any other subject. This is a major concern of mine and has been for at least 10 years. I think that the only answer I can give here is that what we put into the geometry course was largely probability, for which there are geometric models. It still isn't traditional geometry by any means. But we tried to have a progression of treatments of data analysis throughout the curriculum."

In other words, "we found a way to make it fit, and whenever we can make data analysis and probability fit into any aspect of math, we make sure that is done." Which doesn't answer Vern Williams initial question but these guys apparently get paid big bucks to not answer the public questions, just like our teachers in "constructivist math" classrooms (pardon the expression for those of you who insist on maintaining that constructivism doesn't exist) do not answer students' questions.

Any questions? If so, please direct them to Tyrrell Flawn, Executive Director National Math Advisory Panel: Tyrrell.Flawn@ed.gov

why formalism is important

When you teach a course and you are responsible for creating the materials, assignments, and exams, you look at problems differently -- more analytically, and from various perspectives one wouldn't normally. One of the ways in which you analyze problems is in terms of difficulty or complexity. And there's more to it than most realize. (See here for a discussion of problem complexity from a "higher-order thinking" perspective.)

Let's look at this whole thing cognitively and take as our first couple of examples what most would consider to be relatively simple statistics problems:

1. Lessen Waist, Inc. produces low-fat cereals, which they sell in 12-ounce (weight) boxes. Because of settling and production scheduling, Lessen Waist cannot weigh every box of cereal, and 0.35 ounces (weight) is considered to be an acceptable variance from the advertized weight. Lessen Waist weighs a subset of boxes because the filling machines must be adjusted periodically. Use the sample weights below and the appropriate statistical tests to determine if the boxes of cereal are within the acceptable weight. If they are not, use the appropriate statistical tests to determine how much the filling machines need to be adjusted.


2. Jennie's Rugs has been aggressively marketing their products on the web with Google ads and popups over the last twelve months. Below are the ad costs for both types of ads and the sales revenue for the last twelve months, as well as the sales revenue (before Jennie's Rugs started advertising on the web) for the previous twelve months. First, use the appropriate statistical tests to determine if the web ads have had any statistically significant effect on the sales renenues. If so, use the appropriate statistical test to determine if the sales revenues for the second twelve months can be predicted from either of the web ad types. Report all relevant statistics, and if relevant, include the formula to predict the sales revenue from the web advertising.

The first problem students have -- because a problem is more complex than most realize -- is parsing the text of the problem. Far too many students experience some kind of frustration just reading the problem, and find it even more frustrating to try to get past the first reading (sorry to be cliché, but if I had a dollar for every time a student has come to office hours and expressed exasperation at being required to figure out how to figure out the "story problem," I'd have my own island in the Caribbean). And this problem is getting worse.

Let's return to formalism. Students -- again, judging from what those who come see me say and (don't) do -- shut down when math is involved (yes, even in a statistics class -- as if they expected it to be, well, I'm not sure what, though I've often wondered), and from what I've observed, much of the reason is because they see math as some kind of abstruse knowledge expressed in some kind of foreign language. Students ten years ago were much more likely to understand something when you wrote equations on the board than now, when more and more students give you the deer in the headlights.

I think that's partly because it is a foreign language due to lack of exposure, and a de-emphasis of formalism.

But once they get past the first reading of the problem, they have to do a number of things: Decide how best to solve the problem, extract any essential information, determine what additional calculations they may need to do, then set up the problem and solve it. So yes, if you're statistically literate, either of the problems seems almost childishly simple, but to an undergrad, both are actually pretty complex.

But the problems get even more complex. How about these two:

1. The Superbowl Company produces footballs. Superbowl must decide how many footballs to produce each month. The company has decided to use a 6-month planning horizon. The forecasted demands for the next 6 months are 10,000, 15,000, 30,000, 35,000, 25,000, and 10,000. Superbowl wants to meet these demands on time, knowing that it currently has 5,000 footballs in inventory and that it can use a given month’s production to help meet the demand for that month. (For simplicity, we assume that production occurs during the month, and demand occurs at the end of the month.) During each month there is enough production capacity to produce up to 30,000 footballs, and there is enough storage capacity to store up to 10,000 footballs at the end of the month, after demand has occurred. The forecasted production costs per football for the next 6 months are $12.50, $12.55, $12.70, $12.80, $12.85, and $12.95, respectively. The holding cost per football held in inventory at the end of any month is figured at 5% of the production cost for that month. (This cost includes the cost of storage and also the cost of money tied up in inventory.) The selling price for footballs is not considered relevant to the production decision because Superbowl will satisfy all customer demand exactly when it occurs—at whatever the selling price is. Determine the production schedule that minimizes the total production and holding costs.


2. General Ford (GF) Auto Corporation is developing a new model of compact car. This car is assumed to generate sales for the next 5 years. GF has gathered information about the following quantities through focus groups with the marketing and engineering departments.
   
   
   
   • Fixed cost of developing a car: This cost is assumed to $1.4 billion ($1,400,000,000). The fixed cost is incurred at the beginning of the year, before any sales are recorded.
   
   
   • Unit Gross Profit: GF assumes that in year 1, the gross profit will be $5000 per car. Every other year, GF assumes the unit gross profit will decrease by 4%.
   
   
   • Sales: The demand for the car is the uncertain quantity. In its first year, GF assumes sales – number of cars sold – will be triangularly distributed with parameters 100,000, 150,000, and 170,000. Every year after that, the company assumes that sales will decrease by some percentage, where this percentage is triangularly distributed with parameters 5%, 8%, and 10%. GF also assumes that the percentage decreases in successive years are independent of one another.
   
   
   • Depreciation: The company will depreciate its development cost on a straight-line basis over the lifetime of the car.
   
   
   • Taxes: The corporate tax is 40%.
   
   
   • Discount rate: GF figures its cost of capital at 15%

The general process is the same, of course, as the first two (decide how best to solve the problem, extract any essential information, determine what additional calculations they may need to do, then set up the problem and solve it), but these problems are even more complex because there is more information to extract, there are more calculations required that the students must perform (after they've figured out they have to perform them), the problems are more mathematically complex (the first requires linear programming and the second, a monte carlo simulation) and therefore the process to arrive at the solution is more complex, and unlike the first two, there are terms and concepts (sometimes with their own hidden calculations) students must know and understand: Planning horizon, inventory, demand, production and storage capacity, holding and production cost, production schedule, fixed cost (of developing a car, as opposed to fixed cost in general), unit gross profit, triangularly distributed (and parameters), depreciation (and straight-line basis), cost of capital, and NPV (net present value).

Then there is the covert information in the problem, such as "The fixed cost is incurred at the beginning of the year, before any sales are recorded," which actually is a hint on how to set up the problem, "The demand for the car is the uncertain quantity," which is another hint to tell students what the input variables for the simulation will be, "GF also assumes that the percentage decreases in successive years are independent of one another," another hint to tell students how to set up the problem. But students are poorly prepared in the problem-solving process (eek! another one of those hijacked phrases!) and many scan for numbers and ignore everything else.

See? I didn't even get to the mathematical knowledge required to know what additional calculations to set up, do them, or figure out how the information given fits together. But sure, they have to do that too. Solving a problem is much, much more than just coming up with the correct solution.

The only way to solve these problems without jumping off the roof of the nearest dormitory (actually yes, students do that -- I've had two students die during the semester, but neither committed suicide, I'm glad to say) is to approach the problem with the process that traditional math pedagogy has been teaching for several thousand years now. What kind of problem is it? What is the goal of the problem? What information is in the problem and what information is not in the problem? And so forth.

You can always tell the students in class who have been rigorously trained in formalism: They're the ones who immediately begin asking the questions and cutting it up into its components, and then solve it first, usually without much trouble at all. They read the problem and they know how to attack it. The students who have never mastered the thought processes behind solving problems are the ones that start then stop, start then stop, start then stop, and eventually give up, because they find it too frustrating just to try and get past reading the problem.

Trailblazers URLs

cross-posted from Irvington Parents Forum:

An A-Maze-ing Approach to Math by Barry Garelick
history of constructivist math

An Open Letter to United States Secretary of Education, Richard Riley
(anti-NSF funded curricula, signed by 3 Nobel prizewinning physicists & 200 colleagues)

CA Department of Education Content Review Panel review of Math Trailblazers

Math TRAILBLAZERS “backpack letters”

Math TRAILBLAZERS “backpack letters” and “backgrounds”

National Research Council - On Evaluating Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations (2004)
"These 19 curricular projects essentially have been experiments."

VIDEO: “Math Education: An Inconvenient Truth”
M.J. McDermott critiques TERC & Everyday Math; recommends parents "afterschool" using Singapore

I give up

Now these take-home worksheets are making me feel like an idiot. Here it is:

x² + 2y = 10

What do you notice about the expression?

Uh, and she's looking for ... what, exactly?

a sorry state of affairs

Here's a nice little letter to the editor concerning the sorry state of math instruction:

I'd like to address the MCAS test and math education together. If what I suggest is happening (inflated evaluations in our schools) is true, we do need to have some measuring instrument which will accurately assess the comprehension of material beyond what the schools are reporting. In 1970, 6 percent of college freshmen reported having an "A" average in high school. In 2005, this figure was over 22 percent. In 1970, most of my 11-year-old sixth graders could pass a test on fractions, decimals and percents. In 2004, many of my above-average freshmen struggled if I included those concepts on an algebra test. Recently, a college-age clerk at a food store asked if 99 cents a pound was "close enough" to "a little over a half a pound." A junior college student thought that there were 12 yards in a mile. Understanding of this degree, or should I say misunderstanding, is just not acceptable.

That does seem to comport with today's math reality, rather than the rosy rhetoric. Math is supposedly now being taught with understanding, yet what we see is that many students have little real understanding of math and little facility solving simple math problems.

The traditional curriculum was far from perfect--too few kids learned higher math skills. But. today's math instruction seems to have gone from mediocre to worse.

I, for one, blame NCTM who are largely responsible for this current state of affairs. They claim to know how to teach math to kids. In reality, they don't. Their bromides have been failures. Sure, you can go through their standards and twist the words to come up with some sound math principles, but, overall, the framework they've laid down has resulted in actual instruction that is worse than what we had before.

It's time to give them the boot.