Community & Editorial
In defense of the new math
Want to make mathematics exciting? Then play's the thing
Since Real Change recently presented the views of UW Atmospheric Scientist Cliff Mass on mathematics education [“Storm Front,” May 20-26], I would like to take the occasion to present some contrasting views held by a number of mathematicians in the UW Mathematics, Applied Mathematics and Statistics departments, as well as among education researchers in the UW College of Education.
For a start, the interview references Seattle Public Schools’ recent selection of a new standard math text on the basis of a recommendation made by a committee of teachers, parents and community members who had put in many hours of study and discussion.
The one selected has plenty of solid mathematical content in a format that is quite different from textbooks written a generation ago. The most telling argument presented in its favor was that teachers from the full range of teaching philosophies felt they could teach from those texts, whereas the alternative supported only traditional teaching. Teachers need and deserve every kind of support we can give them; the more we avoid asking them to teach from a textbook they are actively at odds with, the better off we’ll be. No textbook can make all teachers happy, but it sounds as if this one comes close for many.
Closer to home for me and a number of colleagues in the Mathematical Sciences and Education Dept. is the statement by Mass that “The basic tenet of [Constructivist Math]—and it’s completely unproven—was that the only way kids can learn a math principle is by discovering it themselves.” This description of our “basic tenet” is far enough from the truth to be barely recognizable. And while his statement that it is “completely unproven” may well apply to what he claims we believe. Several decades of highly respected research support the actual beliefs that many of us share.
Mass is not alone in misinterpreting the constructivism that underlies a lot of what we do. It does indeed have to do with viewing knowledge as something personally constructed rather than externally imposed, but how that construction can be induced varies widely with the knowledge and the constructor. I did some heavy-duty and very effective constructing of complex variables while making sense of the notes I had taken in a warp-speed lecture. My granddaughter is constructing the concept of orientation as she works with her wooden puzzle pieces.
The corresponding view of the job of a teacher is that it is to set up a situation where the student is motivated to take his or her existing knowledge and use it to develop specific new knowledge—which is a lot tougher proposition than working an example and telling students to imitate it with minor variations 15 times over.
There are few things more intellectually exciting than actively engaging with an idea. Babies do that full-time (except when sleeping!)—and look at the amount they learn in their first few years. And there are few things deadlier than being told to park your brain outside the door and do what the teacher tells you to do. This is what became the model of mathematics teaching in the U.S. over many years, which in turn is a major cause of the self-perpetuating societal rejection of mathematics.
How do you fix it? Not instantly, and not in one prescribed and flawless way. But fix it we must. That’s what many of us have been trying to do since the ‘90s, and where we will continue to invest our time and energy and talent.
It is good to see mathematics education being taken seriously and discussed. My hope is that the discussion will extend to
CommentsDr. Warfield states that: "The most telling argument presented in its favor was that teachers from the full range of teaching philosophies felt they could teach from those texts, whereas the alternative supported only traditional teaching." This is incorrect on two fronts; 1. The committee was selected largely on the basis of their enthusiasm for reform math. I have both the 5 questions asked and the scoring rubric. You may find it here: http://mathunderground.blogspot.com/2009/04/how-to-stack-and-adoption-committee.html 2. The alternative to which she refers would be Prentice Hall 2011. It will support explicit instruction and inquiry far better that the Discovery Series. Here are results from Shoreline Teachers: Shoreline's process included an all high school teacher review of three programs. This is how the teacher preferences turned out: 69% PH 2011 28% Holt 3% Discovering ----------------------- Seattle has a large and growing achievement Gap in mathematics for which the UW is at least partially responsible. This gap and the failure of the Seattle School District to act responsibly toward educationally disadvantaged learners is precisely why three individuals have appealed this decision in Superior court. --------------------------- Perhaps Dr. Warfield would care to respond to two items. 1. The NMAP call for explicit instruction of students struggling with math. http://mathunderground.blogspot.com/2009/06/nmap-pg-xxiii-explicit-instruction.html 2. The massive failure of an expensive UW led inquiry based program in math at Cleveland High School from 2006-2009. ---------------------------- To improve a system requires the intelligent application of relevant statistics. It appears that all Dr. Warfield has to offer is recommending allegiance to a failing philosophy. I would urge others to intelligently apply the relevant data, even though UW appears incapable of doing so. --------------------------- Passing rate for english language learners at grade 10 WASL (2008) Seattle District average 19.5% Cleveland 4.8% Garfield 0.0% Both Cleveland and Garfield had two years of UW direct guidance in mathematics (part of a $7.1 million dollar N.S.F. grant). West Seattle was also part of the professional development piece but rejected being part of the program. E.L.L. pass rate at WSHS was 19.5% ----------------------------- The UW pushed the Everyday Math adoption in may 2007. In its first year of implementation. Achievement Gaps for Blacks, Hispanics, Asians, American Indians, and E.L.L. students all increase that is 5 out of 5. There was adequate data available in May 2007 to reject this EDM choice but as usual the needed intelligent application of relevant data failed to occur. This is a recurrent theme in the SPS. Dan Dempsey State Board of Education math Advisory Panelist Correction the above WSHS E.L.L. pass rate was 19.0% WSHS was a project participant but declined to have a school based project. Here is the link to the legal action in Superior Court: http://mathunderground.blogspot.com/2009/06/in-court-discovering-math-appealled.html It is amazing that the relevant data examined by NMAP and the largest study in Education History "Project Follow Through" outline an effective methodology and practice for educationally disadvantaged learners ( one that produces strong positive results) but the UW prefers the exact opposite and the UW recommendations bring failure.. Look here: http://mathunderground.blogspot.com/2009/06/seattle-data.html Go figure. I figure that pushing this defective inquiry/discovery/exploratory math plan brings the UW N.S.F. grants and that is good for the UW and Dr. Warfield but a disaster for Seattle kids. It is time to reject Dr.Warfield's expensive failing direction and follow the rest of the high achieving nations with more explicit instruction than we currently use. Several decades of highly respected research support the actual beliefs that many of us share. ......Really! respected by whom?..... the same people who depend on the money stream from other bogus educational research? Educational research is not taken seriously by most of those outside of the educational research community..... The NMAP researchers could find little if any valid educational research despite examining more than 16000 educational research submissions on mathematics. The bottom line is you can point to bogus research all you want.... the real life data doesn't support the claims. We see this in Seattle and we see it in every honest assessment of student mathematical progress. And by the way Virginia.... I taught out of the Discovery series for 7 years....Everyone I know called it Algebra and Geometry Light.... It does not work well for students who really need to learn mathematics. It left students and teachers looking for the real "meat" of the course. I would predict that the vast majority of students taking Discovery Algebra I and II will be in remedial math courses when they enter college....... the students don't like the Discovery methodology any better than the teachers. Do some teachers make it work? Sure.... I don't need any math text at all to teach high school mathematics.... The Discovery books are a stumbling block to both students and teachers.... The pedagogy it pushes is both inefficient and boring to the students who will ever pursue careers that have mathematics as a foundation. If the reform education you push is so good then you should push to allow students, teachers, and parents to have a choice between reform and traditional..... of course we know reformers will never advocate choice because that would be the end of reform. The only way that reform has spread is by the point of a gun (figuratively speaking). Virginia Warfield is one of the leading supporters of reform/constructivist math. The textbooks she has pushed...from Everyday Math to the Discovery Series...and the IMP series in between...have failed an entire generation of students, including my own. I have heard the complaints of hundreds of parents about these texts and a review of these books reveal that key material is missing. She tries to suggest that there is broad support for her views in the math and statistics department....this is plainly false...I have talked to many of her colleagues and know better. She and her math education colleague (Jim King) have not done any real research in years...they are education specialists who careers are centered around pushing reform math. It is tragic that work of Dr. Warfield and allies in the School of Education are undermining the futures of thousands, if not tens of thousands, of students. Dr. Warfield should provide readers with more detailed information about the textbook, Discovering Algebra (DA), and the manner in which it was developed in partnership with Key Curriculum Press. The Discovering series were all written by different authors chosen from different regions of the US. So none of the textbooks lend themselves to spiraling or building upon knowledge learned from its predecessor. One of the difficulties with DA is that it was developed in isolation at Interlochen Arts Academy. This is perhaps the most prestigious liberal arts private school in the US, but not exactly a model for what works in public education. Traverse City piloted the math program (about 2 hours from Interlochen) but failed to adopt DA because of the controversy the textbook generated from within the community. A more appropriate description of DA is that it looks very similiar to Core Plus. And indeed, the calculator activities were a rollup from an NSF-funded calculator project (1991) authored by Waits and DeMana. W. and D. wrote this curriculum for Texas Instruments to popularize the use of graphing calculators in classrooms. At this point, I will interject, that the TI-interface is beyond the grasp of most high school age children, let alone teachers. And teachers who honestly attempt following the program will find themselves tethered to a television for most of the 50-minute period focused on teaching calculators and not algebra. Most math departments, stuck with Core plus gave 4-week mini-courses on basic algebraic problem-solving methods starting the end of January, so students stand some chance of passing the WASL. Given the state's low test scores, I'm mystified that cities don't leap for a traditional program. You may recall that Core Plus was adopted in 1997 by Bellevue teachers. In return, the district was selected as the regional training center. This year, BSD adopted both DA and Holt - this is a dual track system. The lower track will use DA and it will not be possible to cross tracks once the textbooks are in the classrooms starting 2010. Constructivism is not a discipline or a pedagogy as it is being described by Warfield and some members of the MAA. That is very misleading. In fact, some notorious districts use the NCTM's constructivist lesson plan to silence or put the kabosh on their biggest critics, the teachers. WEA members take note - this might happen to other departments someday, You should be doing everything you can to protect math and science teachers from coward administrators. One basic objection to the reform math is its zen-model of a line. It is deeply flawed, because it is statistical and uses calculators. With this math, one can form virtually any correlation to anything. Students learn to draw lines using recursion and that does not fit with the modern geometric definition of a line. This is one reason that the textbooks fail to meet Washington's D-rated math standards. You can't easily draw a parabola using iterations. Why teach a philosophical Greek absurdity to hs students? So what is driving this math reform and why has it always been so toxic to its critics? Orthodox reformists? Like the elementary textbooks, which teach multiple non-standard methods of basic operations, the hs textbooks do not solve for the intersections of lines using standard algebraic methods. Imagine hs honors seniors solving ALL systems of linear equations, using subsitution. Traditional textbook series provide at least ten different standard methods for algebraic problem solving. Discovering algebra is only constructivist in a general sense that could easily apply to any math classroom. DA is foremost a poorly written textbook that has little or no value in practical mathematics. Vygotsky understood ethics and he applied his model in a practical administrative sense to overcome the myriad of obstacles the Soviets faced after the revolution - which included many diverse cultures. Warfield and her supporters are making smoke out of their straw ideas. There isn't any balance in this debate and so long as NSF-supported researchers continue publishing menusha, instead of serious research, the US will never a sound math program. I suggest readers follow the trail of money - much of it goes to Michigan, Wisconsin, and Colorado. Interested parties might decide to form their own conclusions by looking inside the first Algebra text in the Discovery Series found at http://www.keypress.com/x5265.xml The publisher makes available Chapter 3 on linear equations. The flavor of the text becomes immediately apparent with the authors' emphasis on recursion and technology, and should be judged as appropriate in light of the new Washington State Mathematics Standards, http://www.k12.wa.us/Curriculuminstruct/mathematics/pubdocs/Alg1Geomalg2Final.pdf. It is noteworthy that a review of this chapter contributed to Harel and Wilson's joint judgment that the text is "mathematically unsound" (http://www.k12.wa.us/Curriculuminstruct/mathematics/pubdocs/Alg1Geomalg2Final.pdf). My bailiwick is the community college mathematics curriculum. As a member of the SBE Math Advisory Panel, my perspectives were shaped by an interest in ensuring that the new K-12 mathematics standards would prepare students to be college ready in mathematics as defined in the College Readiness Standards (http://www.transitionmathproject.org/standards/index.asp). I am thankful that Dr. Warfield responded to Dr. Mass' original Real Change article, which I read with interest several weeks ago and felt made several sweeping and somewhat inaccurate characterizations of constructivist teaching practices. At the same time, a review of the aforementioned text chapter, with its emphasis on recursion and reliance on technology, makes me suspicious of claims that the text caters to different teaching styles, and makes me concerned that this curriculum is out of alignment with the new math standards and college mathematics curriculum. Of course, this all presumes that textbooks matter. As the Seattle School Board's decision is already made, and as other districts seem to be considering the Discovering series, the pragmatist in me believes it wise to continue support of cross-sector partnerships between college mathematics faculty and high school mathematics teachers (e.g., http://www.transitionmathproject.org), and to support data collection initiatives focused on the high school to college transition. First of all, textbooks do matter in other countries, but apparently not here in the US. They are so important in terms of equalizing education in these other countries that all students use the same textbooks and take the same exit tests - they have to matriculate before moving forward. That is not true in the US, where we promote students. Why? That's an interesting question, but I believe its because traditionally decisions about graduating and curriculum selection were left up to local school boards. The program has since changed substantially with standards. Always, districts have had multiple academic tracks for students. Bellevue has at least two main paths for students. One for failures or probables, and the other for successes or possibles (Washington's reform model). State adoption committees are forced to choose from hundreds of different textbooks. The selection process is not only time consuming, but favors textbook publishers. The Texas adoption committee last year selected all but one textbook for adoption - 3rd grade Everyday Math and they were sued by McGraw-Hill for censorship. Examine the members of the committee who mostly created the New Washington Math Standards and you will find many of the same persons benefiting from the NSF research grants that stipulate what textbooks should be adopted in order for districts to participate in staff development opportunities. I witnessed first-hand, professors from the college of education using the Boeing Achieve Standard (an as-yet unvalidated standard) as the template for Washington's new math standard. Intermediate Algebra under the new standard is missing many of the topics from traditional algebra - such as trigonometric and logarithms. Members of the committee (professors from education colleges) admitted as much to concerned parents and teachers who were present. By voting for lower standards, the committee members are creating a larger market of lower quality textbooks. Without a doubt, the lower quality of the curriculum is resulting in the resegregation of school districts, schools, and classrooms, and an increase in math failure rates. This is making it harder for students to finish high school . Being myself a pragmatist, if SPS must insist on using the Discovering series, then they ought to look into creating more successful alternatives for their failing students. If SPS has the money to fill in the gaping holes in their curriculum, then go for it. Students will have to spend more time at school learning math and shouldn't be recommended for physics or chemistry until after they've completed intermediate algebra with a B or better. Good luck. You can roll your lies into some truth, but after two decades, seeing your results, so far, makes me a non-believer.
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