constructivism doesn't work, part 1: little scientists

New study out from the University of Virginia re: science education, which David Klein once told me is in even worse shape than math education. Gauging by the first sentence in Tai & Sadler's report, David is right:

Inquiry-based instructional practises are a mainstay of the National Science Education Standards (National Research Council, 1996) and Benchmarks of Science Literacy (AAAS, 1993) in the USA.

Same Science for All? Interactive association of structure in learning activities and academic attainment background on college science performance in the USA
Robert H. Tai; Philip M. Sadler
International Journal of Science Education
Vol. 31, No. 5, 15 March 2009, pp. 675–696

Here's a nice summary of where things stand, drawn from O'Neill & Polman:

In recent years, a number of curriculum reform projects have championed the notion of having students do science in ways that move beyond hands-on work with authentic materials and methods, or developing a conceptual grasp of current theories. These reformers have argued that students should come to an understanding of science through doing the discipline and taking a high degree of agency over investigations from start to finish. This stance has occasionally been mocked by its critics as an attempt to create ‘‘little scientists’’—a mission, it is implied, that is either romantic or without purpose. Here, we make the strong case for a practice-based scientific literacy, arguing through three related empirical studies that taking the notion of ‘‘little scientists’’ seriously might be more productive in achieving current standards for scientific literacy than continuing to refine ideas and techniques based on the coverage of conceptual content.

Why Educate ‘‘Little Scientists?’’ Examining the Potential of Practice-Based Scientific Literacy
D. Kevin O’Neill, Joseph L. Polman
JOURNAL OF RESEARCH IN SCIENCE TEACHING
VOL. 41, NO. 3, PP. 234–266 (2004)

I have not read O'Neill & Polman's study as yet.^* However, a mere glance at the final section turns up the phrase "student-designed research projects," accompanied by a vote for Deborah Meir, "the principal who has led school reforms in New York and Boston [and recommended] that educators foster 'the capacity to hazard an opinion on matters of science that may pertain to political and moral priorities, and a healthy and knowing skepticism toward the misuse of scientific authority'^** (Meier, 1995)."

Rubbish.

"Student-designed research projects" and "the capacity to hazard an opinion on matters of science that may pertain to political and moral priorities" have nothing to do with each other.

In fact, I would go so far as to say that the typical student-designed research project is likely to render a teen-aged student less able to hazard an opinion on a matter of science that may pertain to political and moral priorities than a solid, book-based, content-rich science course would do, while at the same time causing him to consider himself more able. Not knowing what he or she doesn't know: that's your little scientist.

In any event, Robert Tai and Philip Sadler's analysis of survey data from more than 8000 high school students produced the following conclusion, which will come as a surprise only to ed-school trained educators:

Self-led, self-structured inquiry may be the best method to train scientists at the college level and beyond, but it's not the ideal way for all high school students to prepare for college science.

This is the kind of thing parents and taxpayers do not need a peer-reviewed study to figure out, mostly because parents and taxpayers have a clue.

Data show that "autonomy doesn't seem to hurt students who are strong in math and may, in fact, have a positive influence on their attitude toward science" Tai said. However, "Students with a weak math background who engaged in self-structured learning practices in high school may do as much as a full letter grade poorer in college science," he said.

[snip]

According to Tai, many secondary science classes are turning to a self-structured method of learning with the notion that students will discover science on their own. "Advocates should be sobered by this study's findings," Tai said.

Sobered, hell.

Advocates should be overcome by guilt and remorse; advocates should get down on their hands and knees and beg forgiveness of parents and taxpayers for the countless thousands of young people lost to scientific and science-related careers because they arrived at college having spent 13 years pretending to be little scientists instead of acquiring the content knowledge they needed to study science in college.

But I don't see that happening.

^* If you'd like me to send you the study, email me: cijohn @ verizon.net
^** I guess pure research is out.

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.

Bellevue, WA teachers being instructed on how to deal with parents

This is a video of Phil Daro and Uri Treisman of the Dana Center, during a visit to Bellevue, WA. They are instructing teachers on how to deal with parents opposed to reform math. In a nutshell: "Lie".

The synchronization between sound and image is off, so lip movement will not match the words. Doesn't matter; even if it did, the effect would still be nauseating.

Phil Daro does this kind of thing routinely, and he is part of a group called the MARS team. MARS stands for Mathematics Assessment Resource Service, which operates in part from a grant from NSF. ( Grant No ESI0137861). The MARS team has a website, and it contains all kinds of useful information including "tools". One such tool is setting up a "peace treaty". According to the web site, the peace treaty

"...is a disarmament tool – a brief statement of common beliefs that the non-specialist public can understand. It is not a compromise, but a response to the legitimate concerns of the public, especially parents. It can be adopted as a policy of the district, circulated to the press, or by any other means of getting it to the public as a representation of what the district math leadership believes about the issues spelled out in the Treaty. To build such a consensus, it is important also to bring in mathematicians from the local academic community who are interested in mathematics education in schools but have not adopted extreme positions. The letter and meeting agenda are designed for this purpose, complementing the text of the Peace Treaty."

Note that the key qualifications for mathematicians from the local academic community are that they not adopt "extreme positions"; loosely translated this means they should not be against fuzzy math.

The peace treaty is "to help change agents, if and when the 'Math Wars' impact their system, to take the heat out of the exchanges by seeking common ground and civilized discussion of areas of disagreement."

I think that's what Daro is doing here in this video, though he is talking only to teachers, and he seems to definitely be quite defensive against those pesky parents.

The last blurb on the web-page on the "peace treaty" provides the following advice:

"Do not worry about getting people to sign the treaty. You sign it and make it public. Then challenge anyone who attacks you on the grounds they are attacking what you stand for in the treaty. Dare critics to sign on or publicly expose their disagreement with it. Do not add divisive language, even if it warms your heart. Do not use this tool as a public education tool. It is a public engagement tool. It is already understood by the public. They want to know if YOU understand it. This is not an expression of your philosophy. It is a direct response to issues framed by the public. Don’t change the subject."

Prince William County , Virginia votes to keep Investigations

Sad news from PWC, VA. The school board voted to keep Investigations, though that's not how they phrased it. They voted for a "blended approach". Which means Investigations. An excellent summary of the night's events can be found here.

The parents have not given up, however, and school board elections are not far away. There are many disgruntled parents there.

Pearson's Claims of Success with Investigations are Put to Rest

The parents against "Investigations in Number, Data and Space" in Prince William County, Virginia, have been busy lately. Some of them put together a well written report refuting the "success stories" about Investigations touted by Pearson Publishing. The report has been posted on the Teach Math Right web site, and can be downloaded here.

Towards the end of the report in the section called "Success Stories" there are emails and records of phone conversations with people in the school districts that were touted by Pearson as having success with Investigations. They were fairly candid in their emails--perhaps they didn't know just how public this report was going to be. One of my favorites is this one:


Fairfield City School District, OH - Elementary Curriculum Coordinator:

"I did ask teachers to keep their thoughts to themselves and not express their dislike of the program to the parents. And that actually worked. I told them they could have the parent call me or they could say whatever they wanted about the program when we had our PDs [professional development]. I asked them to do their venting any way but [not] with the parents and the children. Another thing in our favor was that the school that did the pilot was one of our lower achieving schools, but their test scores went up significantly."
[Email of 2/9/09]

Pay no attention to the man behind the curtain!

Forbes on the Uselessness of Math

Forbes has an interesting guest editorial on its online page. Mr. Joseph Tartakovsky has decided to write an op-ed on the uselessness of studying math. I suppose that Forbes decided to run this in the interest of equal time, given that they've linked to an essay by Diane Ravitch. Tartakovsky is a law student who has no use whatsoever for math, stating "Why teach math in the age of the calculator? The device is available everywhere, from cellphones to fashionable watches."

Like any good lawyer, he backs up his thesis with facts:

"Once a visitor to the Indian prodigy Srinivasa Ramanujan (1887-1920) noted that his cab number, 1729, seemed "rather a dull one." "No," replied Ramanujan, "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways." He did that in his head. So what? Give me two minutes and my calculator watch, and I'll do the same without exerting any little gray cells. "

Look Tartakovsky, I hate to be the one to rain on your parade, what with publication in Forbes and all, but I think that even with two calculators and a laptop you wouldn't be able to prove Ramanujan's thesis.

He goes on and completes his proof of the uselessness of math by showing that both Benjamin Franklin and Winston Churchill were bad at math but went on to illustrious careers in spite of it.

Other than the fact that Tartakovsky has no use for math, I don't know what point he is trying to make. I only hope that as our recession worsens, he doesn't write an essay scolding big business for hiring engineers from China and India when so many people in the U.S. need jobs.

Mathematician weighs in on Investigations

In a December 19, 2008 letter to the editor of the Frederick News Post, Steve Wilson, a math professor from Johns Hopkins University, issues a warning "not to the newspaper or to the board or the teachers, but to the parents. If your child goes to a school that uses TERC Investigations, you should understand that it means your child's school has abdicated its responsibility to teach your child mathematics. By doing so, the responsibility now rests with the parents. Good luck."

Such a letter has long been overdue. Hats off to Dr. Wilson.

Update: Independent of his letter to Frederick News Post, Dr. Wilson has been invited to be on a panel with other mathematicians and scientists to talk at Sidwell Friends School in mid January. The topic is on the mathematical skills, background and preparation necessary for success in mathematics and related mathematics fields in college. For those who don't know, Sidwell uses Investigations as well as Everyday Math. Most everyone knows, I think, that the Obama kids will attend Sidwell. And if you haven't seen it, I wrote about this here. And yes, Dr. Wilson intends to talk about Investigations during the panel discussion. Wish I could go but I think it's open only to parents of students who attend Sidwell.

W. Post article about Montgomery County's acclerated math program

The Washington Post Magazine in the Sunday edition has an article by Emily Messner about the accelerated math program in Montgomery County, Maryland.

In the article she profiles Eric Walstein, a revered high school math teacher who complains that many students coming into the accelerated math courses in high school do not have mastery of the basic skills. And these are the accelerated students!

Maryland has long had a "pretend" algebra exam, which produces results showing that many Montgomery County students are proficient in algebra in 8th grade. Yeah, if your exam concentrates on non-algebra type problems, you'll get all kinds of good results. Why not call it a calculus exam and really brag?

Fans of KTM may remember that Montgomery County piloted Singapore Math in 4 schools in 1999 - 2000 and then said if they wanted to continue with it, to pay for it on their own. Although the County's own study showed the pilot was successful, adopting it County-wide would have raised questions about what was going on before, plus it had the potential of not eliminating the achievement gap. Though the program would have floated and raised all boats, the achievement gap would have still existed. The usual course of action is to dumb things down and eliminate the achievement gap that way. If you can't raise the water, lower the bridge. Some schools in Montgomery County are using Everyday Math. I know Woodfield Elementary uses EM; they were one of the schools piloting Singapore Math.

The Post article talks about students who took

above-grade-level math and getting good grades, yet did not seem to have a firm grasp of the material. The curriculum is being "narrowed and shallowed," Walstein said. "The philosophy is that they squeeze you out the top like a tube of toothpaste. That's what Montgomery County math is."

This thesis has become Walstein's obsession: In its drive to be the best, please affluent parents and close the achievement gap on standardized tests, the county is accelerating too many students in math, at the expense of the curriculum -- and the students. The average accelerated math student "thinks he's fine. His parents think he's fine. The school system says he's fine. But he's not fine!" Walstein declares on one occasion. On another, Walstein is even less diplomatic. " 'We have the best courses and there's no achievement gap and everything is wonderful,' " he says, parroting the message he believes county administrators are trying to project.

"The problem is, they're lying!"

Another interesting quote from the article:

"You would have a hard time finding one math teacher in this county who supports the scope and sequence of the way math is taught,"says Billie Bradshaw, the math and science magnet program coordinator at Poolesville High School [in Montgomery County, Maryland].

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.

Having it both ways; Everyday Math Responds to NMP report

I received a copy of a response that the good folks at Everyday Math prepared in response to the National Math Panel report. Cheryl Van Tilburg who lives in Singapore was kind enough to provide it to me. She got it from the curriculum head at the Singapore American School which her children attend. Yes, they use Everyday Math at that school. Right. "Water, water everywhere but not a drop to drink."

One of the authors was Jim Flanders who has been with EM for many years. It is too long to post the whole thing, so here are excerpts, plus some comments written by a math professor who is heavily involved in the issue of K-12 math education.

The basic message that EM puts forth is we do everything the NMP wants but we disagree with it all.

"For many reasons an Algebra course is a "gateway to later achievement", but for reasons we detail later the EM authors do not believe it is the gateway as the panel recommends. There is evidence, in fact, that the gateway is much more flexible than the panel maintains. For example, in the mid- 1990s the U.S. Military Academy changed its first (gateway) mathematics course for all freshmen from Calculus (a primary reason Algebra is so important) to a Modeling course based primarily on discrete mathematics and embedded in computing technology. In short, exactly what are the gateway or critical topics of a 21st-century mathematics education is a matter of considerable debate."

The mathematician's response: "They neglect to mention that as soon as the Military Academy students are done with their Modeling course they have to all take 2 or 3 semesters of Calculus, for which you need that algebra.

"The authors believe that a curriculum focused solely on the panels "Critical Foundations of Algebra" (i.e., arithmetic with whole numbers and fractions) would be a step backward and would not prepare students for success in tomorrow's world. Further, many of the parents of todays children had very unhappy experiences with the panels limited definition of mathematics. Most of them want their children to have the richer mathematical experience that EM has to offer."

Mathematician's response: "Here we have K-12 educators redefining mathematics, partially based on the logic that some parents had"very unhappy experiences" when they were kids learning math. "

"... the authors believe that the paper-and-pencil skills championed by the panel are simply the ones that students learned in the mid 1900s and are insufficient preparation for careers and daily life in the 21st century."."

Mathematician's response: "The necessary math hasn't changed or been redefined. If they were getting all kids to learn the basics, then this branching out might make sense, but such is not the case."

"EM also requires that students explore several computational algorithms. Knowing a variety of algorithms can (1) help with a variety of computational tasks, including estimation, in which a standard algorithm might be inefficient; and (2) help students better understand the concepts behind standard algorithms. Yet EM also encourages and supports teacher in being sensitive to individual differences. Some students may need to focus on one algorithm over all others and suggestions for how to identify such students are in the Teachers LessonGuides."

Mathematician's response:

"Note that like in TERC Investigations, there is no emphasis on learning efficient algorithms. Worse,note that it is what the students themselves "need tofocus on" that determines what a student uses. This student chosen algorithm might work well in an EM class in elementary school, but it ismy understanding that it is very difficult to get students to change once they are comfortable withan algorithm, and such an algorithm may not even be remotely comfortable when the student getsto college. This is a real, and unbelievable, disservice to students."

"They [fractions] are also represented in fraction-manipulating calculators, which are primarily tools for allowing students to do many more calculations with fractions than can be done on pencil-and-paper.

Mathematician's response: "Maybe so, but they won't get the "conceptual understanding of fractions" the NMP wants. "

Perhaps Andy Isaacs and Jim Flanders would care to offer their thoughts?

NEA weighs in on Who Wants to be an Engineer?

The NEA has always been quite helpful in providing parents with guidance on how to teach their children what isn't being taught in school, so I was delighted to see that they are offering online brochures on a number of topics.

Parent guides on a number of topics, including helping your child with math are available here.

I was particularly impressed with the guide on what children need to know to become engineers.

Here are some excerpts:

"What do kids do in a technology class?
They think about and solve problems like:
• Cleaning a polluted lake or river
• Creating an invention to solve a household problem
• Designing and building a habitat for a unique
situation
Second-graders might design and make a home for
their favorite bug. They would draw a plan (complete
with measurements) and use boxes and other
materials to build the home. They would have to think
creatively about how to keep the bug in the house,
how to provide water and food, and how to make
sure the home was the right size for their pet."

Silly me; I thought for sure they would need to learn something about bugs in their science classes. But, after all, we’re talking about future engineers here. Mere facts are of no use in the modern day classroom.

"Fifth-graders might design and make their own paper
airplanes. They would test them to see which ones
flew the furthest or the highest and then revise the
design to see if they could make a better paper
airplane. They would use mathematics, learn aviation
science, and practice reading and writing skills
throughout the design process."

Just curious; what type of mathematics would they be using other than measuring the distances? Also, any clue about how “aviation science” would be used? I don’t think I have to ask about reading and writing skills used throughout this activity. Writing: How I Feel About Today’s Assignment. Reading: Various essays about paper airplanes in a rascist world.

"Eleventh-graders might investigate the idea of
growing plants in a hydroponic system (without soil).
They would design, build, and test the system. They
would study the effect of this type of growing on the
environment and figure out whether this system was
more cost effective than growing plants in soil. They
would become engineers!"

Yes, it’s axiomatic that when students design, build and test hydroponic systems, they grow up to become engineers. No one really knows why, but there is a flurry of “action research” taking place in our classrooms to find out.

Wait. There’s more!

"When those juniors in high school study
hydroponics, they think creatively about ending
hunger and about how to grow food in places where
the soil is not ready for planting."

RIGHT. Of course, how to get the water to deserts is part of another class. Creative thinking is the order of the day here.

What content should I expect my child
to be learning?
What students should know and be able to do
is identified in standards developed by the
National Science Foundation (NSF) and NASA–
Standards for Technological Literacy: Content for the
Study of Technology. Standards for K–12 were formally
reviewed by the National Academy of Engineering,
the National Research Council, and the technology
teaching community.
The standards address content for K–12. Content is
integrated into thematic units at the elementary
levels, while course titles at the middle and high
school levels may include:
• Exploring Technology
• Innovation and Engineering Design, Technological
Systems
• Engineering Design Fundamentals
• Inventions/Innovations
The standards also address medical, agricultural and
related bio-technologies, energy and power,
information and communication, transportation,
manufacturing, and construction topics."

Yep. No need for math, biology, chemistry or physics. Just get them up to speed on technology and engineering design. It’s gonna be a beautiful world!

Linda Seebach on the whale problem

The whale problem again:

A whale swims 40 miles in 1 1/4 hour. How far does he swim in 1 hour?

The math mom I mentioned solved this problem mentally in a couple of seconds, then explained that "1 1/4" is 5 parts.

Linda writes:

What is apparent to someone with good number sense is that 1 and 1/4 is 5/4, that is five of something called "one fourth." There's no work to show, though you could explain that if five somethings equal 50, then one something is 10 and so four somethings are 40.

Since the numbers are easy, you can do it in your head; no need for visual models or bar models or algorithms. But then, I was an algebra person, not a geometry person.

It works just as well when the whale swims 37 miles in 16/13 hours except you probably can't compute (13)(37/16) without writing it down or asking the calculator. Knowing what you need to compute, however, is exactly the same.

I have to say....it's pretty sad that I still can't put it this way (or, rather, don't think to put it this way).

Of course Linda is right; 1 1/4 is 5/4. That's what a person with a good number sense "sees" or, more accurately, grasps.

Remembering back to when I first started reteaching myself K-12 math, seeing that 1 1/4 is 5 one-fourths was an obstacle. I had good fraction knowledge for an American, I think; I could create a correct word problem for 1 3/4 ÷ 1/2, the famous challenge given to 22 math elementary school math teachers in Liping Ma's book. I had no math phobias, SAT math scores were good, took statistics in college.... in short, I am a math literate person, certainly for the U.S., and was before I started studying K-12 math.


An aside: hmmm... It's interesting that I keep using the word "see." I'm going to assume that means I personally do need to "see" this as opposed to "grasp" it.

Interestingly, as I've moved into algebra 2, I don't feel this way about many of the topics I've encountered there (practically all of which are new to me). Will have to give this some thought. At some point, the abstractions of math came to seem natural or "real" to me. At least, I think they did.


Back on topic: In spite of the fact that my own understanding of fractions was perfectly appropriate to my needs, I had trouble with the idea that 1 1/4 is five one-fourths. The "one-fourth" is the unit; the 5 is multiplier and 1/4 is the multiplicand. I remember my neighbor explaining it to me one night in the context of another question. (Wish I could remember it now. I may be able to find it. She was trying to explain something about fractions, I think, by pointing out what Linda has just pointed out --- and I stumbled over the explanation.)

I think bar models are a way of teaching the kind of number sense Linda is describing. Which is why C. is going to carry on doing the bar models in 3rd grade Primary Mathematics.

Sybilla Beckmann's textbook for students in education school teachings bar models, as does Parker and Baldridge's text. (I've found Parker and Baldridge much easier to study and understand than Beckmann's book, but that may not be a fair comparison since I began at the beginning of P&B, and only dipped into Beckmann.)


procedural "versus" conceptual knowledge

Given my experience of (relearning) fractions, I now feel strongly that schools must teach the addition, subtraction, multiplication, and division of fractions to mastery in all kids, bar none.

I think, too, that schools should give kids word problems to solve using all four operations -- and a good number of these word problems should be "real world" word problems, by which I mean the actual real world, as opposed to the constructivist real world, which seems to consist mostly of roller coasters and hay balers (see below).

Non-GATE kids should be given fraction word problems about things like measuring a room for tile or cutting fabric from a bolt to sew a dress (I don't care if the kid actually sews or not. Sewing is something he/she might do one day, especially if he/she has kids and sends them to a public school where the curriculum is project-based.)

This approach, which I think is what I probably had, may not give non math-brain kids a good conceptual understanding of fractions. It didn't give me a good conceptual understanding.

But it did give me all the foundation I needed to use fractions in adult life, and to pick up the study of math later on, when I wanted to.

I now believe that non-GATE kids (not sure about the GATE kids) should be taught bar models, too. No matter what curriculum or approach a teacher/school is using, consistent teaching of bar models should be included.

Cassy T has mentioned that 3rd grade is the big year for Singapore Math; that's the year bar models are introduced. (pls correct me if I'm wrong)

I would ideally have U.S. kids work through either the 3rd grade Challenging Word Problems book, or work all the bar model problems in Primary Mathematics 3.



the hay baler problem

A post from Barry a couple of years back:

Here's a problem that appears in IMP for 9th grade It is known as the "Haybaler Problem"

“You have five bales of hay. For some reason, instead of being weighed individually, they were weighed in all possible combinations of two: bales 1 and 2, bales 1 and 3, bales 1 and 4, bales 1 and 5, bales 2 and 3, bales 2 and 4 and so on. The weights of each of these combinations were written down and arranged in numerical order, without keeping track of which weight matched which pair of bales. The weights in kilograms were 80, 82, 83, 84, 85, 86, 87, 88, 90 and 91. Find out how much each bale weighs. In particular, you should determine if there is more than one possible set of weights, and explain how you know.”

David Klein, a mathematics professor at California State University at Northridge comments on the problem. “The process of solving this problem made me resentful of the stupidity and pointlessness of it. There is nothing ‘real world’ about it. It is completely inappropriate for kids who likely have not been taught how to solve simultaneous linear equations, or exposed at most to two equations in two unknowns. If I had been given such problems at that age, I think that I would have hated math.”

Consistent with much of the philosophy of “real life math”, the goal of the exercise is to explore strategies and to be able to write about it. This is made apparent by the “student guide” that accompanies the problem. It is essentially a scoring sheet, containing categories, with points awarded for each, such as “Restate the problem in your own words” (4 points); describe all the methods you tried before reaching your solution(s) (4 points); describe the process that lead to your solution(s) (4 points); describe all assistance provided and how it helped you (2 points); state the solution (2 points); describe why your solution(s) is correct, include all supporting data (6 points). Out of a total of 50 points, only 2 are given for the solution. In fact more points are given for describing why the solution is correct.

The Skipper gets off Gilligan's Island

The Skipper in the title is none other than Skip Fennell, President of NCTM who testified before a subcommittee of the House Science and Technology Committee. This is a guy you'd definitely want to be stranded on an island with, because he would surely know how to get rescued. Here's an excerpt from his testimony:

"I would like to address an element of a child’s education that is often overlooked by policy experts and elected officials. As members of the National Science Board and other prominent education leaders have noted, a child’s first—and perhaps most—influential teacher is a parent. Any call to action—small or large—must recognize the crucial role that parents play in encouraging children and exposing them to knowledge and ideas about any topic or subject, including mathematics. Without parental support and involvement, it will be very difficult to convince young people of the urgency and importance of STEM literacy in this country."

His politburo is quite good. The party-line answer of "the influential role played by parents" is designed to make everyone feel good. But what it really means is, if you want your child to learn math, you better do your part at home. Which means more than providing a support for learning. It means that the principle role that parents play is to make up for the lack of instruction provided by schools forced to use inferior math programs, and taught by teachers with inadequate math knowledge.

Which brings me to the Focal Points. The focal points are like an ink blot test. You can see whatever you want to see in them. If you want to adopt TERC in your school district, just say it conforms to the Focal Points. Why not? That's what Prince William County in VA did last year when they adopted Investigations. Well, actually they said it after Scott Foresman, the publisher of Investigations said it. Check it out: on the County's website is a "correlation document" prepared by Scott Foresman (publisher of Investigation) showing how Investigations meets the Focal Points. It's located here. This is for the new edition of Investigations.

You may also be interested in the School Board's document that they prepared to rebut the protests from one lone parent (an engineer) in the County against Investigations. Go here and scroll down to where it says: Clarifying Misconceptions about Investigations in Number, Data, and Space and click on it. It's described as: "This position paper was written by members of the PWCS Office of Mathematics in response to some criticisms and concerns about Investigations."

Here's only one of many gems: "The crisis in mathematics education in the United States is at least twenty-five years old. Programs like Investigations did not create the problem, but were developed after 1989 to address the problem." Reminds me of that old joke:

Mother: Stop pulling the dog's tail.
Child: I'm not pulling, the dog is pulling.
(Cue laugh track)

As they say in blog land: "Read the whole thing". Then weep.

Real world vs mathematics for math's sake

Arguments about how math should be taught frequently include the issue of "real world" math problems. I.e., students need relevance, otherwise they'll tune out. This attitude excludes a whole host of problems that one might find in a geometry book, let's say. So a problem that is real world and makes use of the Pythagorean Theorem is OK, but a proof of the Pythagorean Theorem is not. Well, no, people might object to my extension. So let's use another example. "For any point within an equilateral triangle, the sum of the perpendicular distances of the point to each respective side of the triangle is constant, and equal to the altitude of the equilateral triangle."

This wonderful theorem probably wouldn't qualify as "real world" and even though understanding the theorem and how it is proven opens up higher order thinking skills and mathematical reasoning, a math reformist would not teach it in its pure form. They would have to find some application, however contrived, to provide relevance.

Math for math's sake is out. All math must be relevant. In fact, when the Fairfax County Public School Board (in Virginia) was adopting math textbooks back in 2001, they argued about the criterion for "real world" applications. During the ensuing debates, two school board members found the following criterion too narrow: "Materials and concepts are related to real world situations". They argued for, and lost, the following (excerpted from the minutes of a School Board meeting found here):

"Mrs. Brickner moved, and Mrs. Thompson seconded, to amend the main motion to add the words, “mathematics and” to the eighth bullet under criteria #2, so that it would read, “Materials and concepts are related to mathematics and real-world situations.”

"Mrs. Brickner said that a mathematics textbook could not relate everything to a real-world situation; that there should be a balance between the presentation of math concepts and their relationship to the real world to help students understand the need for those concepts; that applications of mathematics should come from both within mathematics and from problems arising from daily life; that a strict application of the criteria, as originally written, would cause an evaluator to find a textbook less effective than they otherwise might on the basis that the text did not wholly focus on the real world; and that the objective was clearly to teach math concepts and skills.

"The motion to amend the main motion to add the words, “mathematics and” to the eighth bullet under Criterion #2, so that it would read, “Materials and concepts are related to mathematics and real-world situations” failed 4-7, with Mr. Braunlich, Mrs. Brickner, Mr. Reese, and Mrs. Thompson voting “aye”; with Mrs. Belter, Mrs. Castro, Mr. Frye, Mr. Gibson, Mrs. Heastie, Mrs. Kory, and Mrs. Strauss voting “nay”; and with Mrs. Wilson absent. "

So, the upshot is that math textbooks must base all examples and applications on real world situations; mathematics for mathematics sake does not count. That would make calculus textbooks rather challenging to write!

2001 NCTM editorial on constructivism

I don't know why I Googled on "Constructivism; math". I should know better at my age. But it did take me here to an editorial written by Lee Stiff, president of NCTM from 2001-2002. In it, he talks about a familiar theme: Constructivism doesn't exist. Gee, where have I heard that before? Well, last time I heard it, Jay Mathews of the Washington Post was talking about it.

Lee Stiff says all the right buzzwords:

"Constructivist math is a term coined by critics of Standards-based mathematics who promote confusion about the relationships among content, pedagogy, and how students learn mathematics. It is how they label classes where they see students engaged and talking with one another, where teachers allow students to question and think about the mathematics and mathematical relationships. Critics see these behaviors and infer that the basics and other important mathematics are not being taught."

[Reform-minded teachers] "promote making connections to other ideas within mathematics and other disciplines. They ask students to furnish proof or explanations for their work. They use different representations of mathematical ideas to foster students' greater understanding. These teachers ask students to explain the mathematics. Their students are expected to solve problems, apply mathematics to real-world situations, and expand on what they already know. "

So, the Hay Baler problem in IMP expands on what students already know? Is that true?

So, asking a student the solution to 5 divided by 1/2 in the midst of a problem set of whole number division problems prior to students receiving instructions on fractional division (as Everyday Math does) expands on what students already know?

He prattles on about how working in groups allows "students help one another create richer meanings for new mathematical content." And "that students should be encouraged to create their own strategies for solving problem situations."

First of all, I don't know what a "problem situation" is. And I probably don't want to know. But I do know that when I see essays talking about how math should be taught so that students "make connections" between "concepts" and "real world" situations, I head for the Dolciani, Saxon and Singapore texts and even my old Arithmetic We Need texts from my grade school days. Somehow these "traditional" texts make connections.

I really don't think this 2001 essay is outdated, even with the Focal Points in place.

out of the woodwork

Some anonymous knucklehead, claiming to be an NCTM insider, just left a comment to this post on D-Ed Reckoning discussing the new NCTM focal points. Either he or she screwed up the terminology and/or players or I am very confused:

Re: the phrase "quick recall"--as someone involved in reviewing the CFPs, I should let you in on the fact the the phrase was a compromise to those traditionalists who couldn't deal with the phrase "automaticity of facts" or "automatic recall" or even "fast recall"--etc. [Ed: Huh?] "Quick recall" is the phrase used by those who support what appear to be the ideas expressed in your blog.... You cannot believe how many versions of this phrase were discussed and debated before NCTM made this compromise with those who are part of the Mathematically Correct camp....

I guess I would be critical too if I opposed the efforts of Skip Fennell and NCTM. I would be kicking myself for not coming up with and publishing the Focal Points myself.

For those who criticize the CFPs, and only comment on statements printed in various biased articles (some of which are taken out of context), I encourage you to read the 41 page document. [Ed: which I linked to and did, in fact, read] It's a free download. Most of the reporters, radio-show hosts, and journalists that I've spoken to in my state have not actually read the entire document, but focused more on the reform-traditional controversies that have been part of the Math Wars over "how to teach mathematics"--and not "what to teach."

I encourage you to spend some real time with the document (http://www.nctm.org/focalpoints), then develop a real opinion....

constructivism -- just a hoax?

In an email from a noted education writer for a major metropolitan newspaper, this writer said the following with regard to constructivism:

Life is too short for me to try to distinguish for readers the difference between true constructivism and guided discovery. It really is not an interesting issue, because readers only care if their kids are learning, and I dont think I have ever seen anything in a public school that could qualify as true constructivism, and there is almost nobody out there with any influence pushing for t. c. It doesnt exist except in a very few private schools, if there.

This is from a writer who no matter what arguments you proffer in opposition, will respond, "oh, but what I meant was..." From what I've seen, the theory of constructivism has manifested itself in textbooks such as TERC's Investigations, or Connected Math Program, or Everyday Math, or IMP, or Core-Plus. One can make arguments that EM is not really constructivist. Well go ahead and make them. It sure isn't guided discovery. And it isn't mastery learning either. So you tell me what it is?

Adherents of constructivism will of say "Well of course we don't use constructivist techniques every day; it would be impractical." But they use them often enough. Giving students problems for which they do not have enough information or skills to solve them is but one example.

But enough of me talking. How would you respond to this darling of the edu-journalism community in 50 words or less?