Monday, June 29, 2009

NJDOE Math Task Force Dies without DOE Action

The NJDOE convened a math task force which met for 4 full days and to date, DOE, has ignored the task force and has not released one iota of information from the task force.  Thus, I wrote the following letter to DOE:

Now that the Math Standards Task Force has completed its work, a number of us want to address the issue of what will happen next.  For the Department of Education to develop a set of math standards that will provide teachers, schools, and districts with the guidance that they need to enable our students to achieve high expectations in mathematics, it is critical that the next steps, following the efforts of the Department’s Task Force, build on the good work already done and involve people who are best able to craft the standards New Jersey deserves.

Concurrent with the Department’s Task Force, the New Jersey Mathematics and Science Education Coalition convened a working group of experienced math educators to take the two early versions of the standards, the December version prepared by the Department of Education’s writing team and the February version prepared by the Department of Education’s staff, and develop a set of recommendations that combines the best of them.  That document came to be valued and used by a number of members of the Task Force as it deliberated; it should be integral to any subsequent development of the math standards.

There are many people who can examine and critique individual indicators in the standards, including many of the members of the Task Force.   But only those with a more intimate knowledge of the curriculum, such as math supervisors and curriculum specialists, are able to see the standards as a whole, so that an individual indicator is not only analyzed in itself, but also in regard to how it fits in the K-12 flow within its standard and how it fits it with indicators in other standards at its and nearby grade levels.  Those are the people who should be called upon to provide advice on the content of the standards.

Writing standards requires a team whose members:

ª have a deep understanding of mathematics and of the PreK-12 mathematics curriculum

ª understand the curricular consequences of what might appear to be a modest change of language

ª can write concisely and clearly

ª understand assessment and have experience with a wide range of types of questions including multiple-choice and extended response

ª have knowledge of a wide variety of curricula in order to ensure that appropriate, high-quality instructional materials are available

ª have knowledge of the psychology of learning mathematics so that content is developmentally appropriate.

Even those who have experience with the content cannot necessarily translate their recommendations into the language of standards.  The people in New Jersey who have the greatest experience and expertise in doing that are Janet Caldwell, Warren Crown, Bob Riehs, Joe Rosenstein, and Bill Smith (Crown and Smith are now retired and may therefore not be available).  In addition to and because of their qualifications, they have the trust of their colleagues across the state. They should be key players in the further development of math standards which are both credible to the field and of the highest quality.

Friday, June 26, 2009

Big News at Rowan

With the stroke of a pen, Gov. Jon S. Corzine Thursday changed the landscape of higher education and health care in South Jersey.  Through an executive reorganization plan submitted by the governor to the state Legislature on Thursday, Corzine requested the creation of a four-year allopathic medical school on Broadway in Camden that would combine the resources of Cooper University Hospital and Rowan University. The result would be the fourth medical school in New Jersey, and the only one not to be affiliated with the University of Medicine and Dentistry of New Jersey.

Saturday, June 20, 2009

The Structure of Educational Revolutions

In his widely influential book, The Structure of Scientific Revolutions [7], Thomas Kuhn argued that, while science usually proceeds by taking small steps based on the accepted paradigm at the time  ("normal science"), occasionally it is recognized that the current paradigm is incapable of solving important problems and what is needed is a new paradigm (a "paradigm shift") One example is the Copernican revolution in the 16-17th centuries where the paradigm of a geocentric universe was gradually replaced by that of a heliocentric universe.  A second example is the chemical revolution caused by Lavoisier's discovery of oxygen that replaced the phlogiston theory that posited that all flammable substances contain a material ("phlogiston") that enables them to burn.  A third example is the Einsteinian revolution that made Newtonian mechanics a limiting case of relativistic mechanics for small velocities.

What does this have to do with mathematics education.  Well, read on...

Calculators and the Elementary School Mathematics Curriculum
Arithmetic has been at the heart of the elementary school mathematics curriculum for many centuries.   Since counting has been important for as long as there has been human civilization and since, except for geometry, there was - and still is - just about no other mathematics accessible to children of elementary school age, arithmetic has been and still is a crucial, indeed a dominant part of the elementary school mathematics curriculum.  And so, I believe, it should remain.  The relevant issue here is:  How should arithmetic be taught to students?

There is (almost?) no one who questions that an understanding of what addition, subtraction, multiplication and division are and an ability to discern when each of these operations is the appropriate one to use should be in every child's mathematical arsenal at some point in his/her elementary school career.  Neither is there any substantial disagreement among mathematicians or mathematics educators or anyone else that students need to achieve virtually instant recall of the addition and multiplication tables as early as possible in their educational careers.

For at least the past two centuries the heart of the elementary school arithmetic curriculum has been instruction in the pencil-and-paper algorithms (hereafter PPA) of addition, subtraction, multiplication and division that all readers of this article surely recall from their own schooldays.  Until the advent of the hand-held calculator these algorithms were of substantial practical value, being essentially the only way for almost everyone to add columns of figures, subtract one sizable number from another, multiply two multi-digit numbers or perform long division .  But now the practical value of these algorithms has all but disappeared.  The question is:  Are these algorithms still the best way to impart the understanding of arithmetic necessary to prepare students for the further study of mathematics?

The Math Wars

Mathematics education, particularly in the United States , has been a subject of controversy for some 40 years now since the time of the New Math in the 1960s.  Indeed, the New Math has been called the first of three "revolutions" in mathematics education [2], the other two being the back-to-basics reaction to the New Math and the changes wrought by the NCTM Standards (hereafter the Standards) [10].  The first two of these, however, whatever you may think of either, were not revolutions in the Kuhnian sense and the last, whatever the long term impact of the Standards, proposed only a modernization of school mathematics education rather than anything revolutionary.

One thing the Standards did do, however, was to spark the so-called Math Wars which for nearly 20 years now have pitted mainly research mathematicians but also some parents, business groups and teachers and others (Traditional Math Warriors - TMWs) against mainly mathematics educators but with significant support from some parents, teachers and others (Reform Math Warriors - RMWs) [5, 12].  At times these wars have led to acrimonious exchanges between the two sides; at other times the exchanges have been more genteel.  There have even been recent attempts at truces [3] and fudges [9].  But an end to the Math Wars is not in sight nor, I believe, should it be because the essential issues are too important and the essential positions of the two sides are so far from each other that what is needed is victory for one side, not a pale compromise that, in the long run, would not be good for anyone.

Calculator technology has been the catalyst for almost all the disputation in the Math Wars.  Since this technology almost wholly diminishes the practical value of PPA, it is essential to answer the question above about whether these algorithms are the best way, even an effective way to prepare students for further study of mathematics.  In one sense, the answer to this question is clearly, yes, since there is no alternative to these algorithms that has ever been tried in schools anywhere.

Some places, notably California, have, after some experiments with calculators in elementary school mathematics, returned to a "back-to-basics" curriculum in which calculators are effectively banned in the classroom through the sixth grade.  In many other places curricula are in use in which calculators are used in conjunction with traditional instruction in PPA.  There are lots of variations in these curricula.  Judging only by test results and anecdotal evidence, some of these curricula seem just as or more effective than traditional PPA curricula while others do not seem to perform very well.  In some partly-calculator curricula, the traditional algorithms for addition, subtraction and multiplication are taught but long division is no longer taught.  This is in line with a 25-year-old recommendation from Great Britain [4] that is, however, anathema to TMWs.

The More Distant Future

It is only reasonable to conclude this paper by asking the question:  If the computer revolution presages a paradigm shift in elementary school mathematics education, will there also be paradigm shifts in other aspects of mathematics education?

It is true today that essentially all the manipulations of mathematics taught not just to elementary school students but also to secondary school students and to university undergraduates can now be performed by hand-held devices and, in virtually all cases, more rapidly and accurately than (almost) any human can perform them.  What value then to teaching any of the old, almost entirely pencil-and-paper manipulations taught almost universally in secondary schools and universities?  The answer to this question is not so simple as I believe it to be in the case of elementary school arithmetic.  As with arithmetic, the practical valueof learning to do these manipulations is rapidly decreasing.  But it cannot be said, at least it cannot be said yet, that learning these manipulations is not the best way to understand the underlying mathematics and, therefore, the best way to proceed to the study of further mathematics.  Nevertheless, I would be surprised if more and more of the current secondary and university mathematics curricula were not replaced in coming decades by new curricula in which calculators and computers are fully integrated into them (mental algebra?  mental calculus?).

Thursday, June 18, 2009

Wolfram|Alpha Search Engine

Sum Help: New Search Engine for Mathletes

Wolfram Alpha Site Automates Arithmetic Drudgery for Students, but Teachers Worry It Does Homework, Too

I bet the anti-reformists want this "banned" from schools too.

Wednesday, June 17, 2009

STEM Careers

A previous post referred to an article by Gerald Bracey in Phi Delta Kappan (March 2008) entitled "On the Shortage of Scientists and Engineers".  The full article can be found here.

Education Week: Subject-Matter Groups Want Voice in Standards

Education Week: Subject-Matter Groups Want Voice in Standards

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Tuesday, June 16, 2009

Abolish Paper & Pencil Arithmetic

Let's Abolish Pencil-and-Paper Arithmetic by Anthony Ralston (Imperial College, London).  The paper appeared in Volume 18, Number 2, pp. 173-194 (1999) of the Journal of Computers in Mathematics and Science Teaching.  (The anti-reformists will love this article!)  The article proposes that paper-and-pencil arithmetic no longer be taught in elementary school and that it be replaced by a curriculum which emphasizes mental arithmetic much more than at present and in which calculators are used for instructional purposes in all grades including kindergarten. The article analyzes and refutes the arguments made by "back-to-basics" proponents against the use of calculators and for traditional instruction in the algorithms of pencil-and-paper arithmetic. The value of mental arithmetic in achieving all the aims - and more - of the traditional curriculum is argued. Also considered is the outline of an elementary school mathematics curriculum without pencil-and-paper arithmetic. As well, the impact of such a curriculum on secondary school and college mathematics is discussed. Finally, the barriers to achieving what the article advocates are assessed.  To read the full article, click here.

Monday, June 15, 2009

On the shortage of scientists and engineers

The anti-reformists have continued their rhetoric with respect to the failures of the US in math and science.  
Gerald Bracey in Phi Delta Kappan (March 2008) does a wonderful job of responding to them. 
His article, On the shortage of scientists and engineers, touches on many aspects.
He response to the U.S. TIMSS data is below:
While the U.S. is middling in ranks in math and science assessments, the differences among countries in terms of scores are often quite small. The most extreme occurrence of this phenomenon was in the eighth-grade science assessment for the 1995 TIMSS (Third International Mathematics and Science Study). American students got 58% of the items correct, compared to 56% for the international average. This ranked them 19th among the 49 participating nations. Middling performance, no? Had they gotten a mere 5% more correct answers, they would have ranked fifth, and had they gotten 5% fewer correct, they'd have slumped to 30th.
He also points out two other major fallacies of the anti-reformists:

1. The U.S. suffers serious shortfalls or shortages of scientists and engineers, and this bodes ill for both creativity and international competitiveness.
RESPONSE: There is no shortage. Several RAND Corporation studies found surpluses. There might be shortages in some new fields or fields growing explosively, but not overall.
2. The number of newly educated scientists and engineers is insufficient to fulfill employer needs. Thus the need to hire from   overseas.
RESPONSE: There are substantially more scientists and engineers graduating from U.S. universities than can find attractive career openings in S & E fields. Indeed, the S & E opportunities seem unattractive to many holders of S & E degrees. "Into the Eye of the Storm" (no doubt a pot shot at the National Academies), a paper by Lindsay Lowell of Georgetown University and Harold Salzman of the Urban Institute, found roughly three S & E graduates for every new S & E job (not counting openings created by retirements). They also found that two years after graduation from S & E programs, 20% of the grads with bachelor's degrees were in school but not in S & E programs and 45% were in the work force but not in S & E jobs. The attrition rate for that time period for those with master's degrees was about 38%. One can only imagine how critics would howl if education lost 65% of its work force in just two years!

Nor are fewer students following S & E paths in universities. From 1977 to 2002, the number of citizens and permanent residents earning bachelor's degrees in S & E grew from about 300,000 to about 400,000, those earning master's degrees increased from about 60,000 to about 70,000, and those earning doctorates held steady at about 20,000.

Other studies have concluded that the decline in the pool of citizens and permanent residents with S & E credentials may reflect a weakening demand, a comparative decline in S & E wages, and market signals to students about low relative wages in S & E. I'm not sure exactly what the "market signals" are, but real wages for S & E workers have declined over a 20-year period. And students can see older scientists spending more time writing grant applications, getting fewer of them funded, and having a tougher time getting tenure. They can see the post-doc headed for what science writer Dan Greenberg calls the newest title: post-doc emeritus. And students, not surprisingly, head to greener fields.

Finally, there is some evidence that the nature of the engineering profession has become less appealing. Lowell and Salzman observe that projects today are often larger team efforts that require more coordination and management. In their interviews, engineers often commented that the field was not as challenging as it once was because it contains less "real" engineering.

Sunday, June 14, 2009


A hilarious parable on NCLB written by William Tomhave is available here:  A Track and Field Parable

Saturday, June 13, 2009

New York City Shows Gains in Math

New York City’s public school students showed large gains on state math tests this year, particularly in the middle school grades, and black and Hispanic students continued to edge closer to their white counterparts, the city and state education departments announced on Monday.
To see the full article, click here:  New York City Shows Gains in Math

Thursday, June 11, 2009

The Opportunity Equation

The Opportunity Equation

Transforming Mathematics and Science Education for Citizenship and the Global Economy 

Report of the Carnegie Corporation  of New York–Institute for Advanced  Study Commission on Mathematics and Science Education 

The United States must mobilize for excellence in mathematics and science education so that all students— not just a select few, or those fortunate enough to attend certain schools—achieve much higher levels of math and science learning. Over the coming decades, today’s young people will depend on the skills and knowledge developed from learning math and science to analyze problems, imagine solutions, and bring productive new ideas into being. The nation’s capacity to innovate for economic growth and the ability of American workers to thrive in the global economy depend on a broad foundation of math and science learning, as do our hopes for preserving a vibrant democracy and the promise of social mobility for young people that lie at the heart of the American dream. 

Guiding Principles for Mathematics Curriculum and Assessment

NCTM releases Guiding Principles for Mathematics Curriculum and Assessment

A national curriculum for school mathematics is a topic of growing interest at state, national, and policy levels. The development of a common national curriculum and assessment in mathematics should be driven by the following basic principles for designing an excellent curriculum to avoid the risk of producing a negotiated list of standards that is merely an intersection of those that are currently addressed in each of the 50 states. Therefore, NCTM recommends the following guiding principles for the potential development of any set of common curricular expectations and assessments across the nation.

The National Council of Teachers of Mathematics (NCTM) ushered in the standards era with the 1989 publication of Curriculum and Evaluation Standards for School Mathematics, which was updated in 2000 as Principles and Standards for School Mathematics. Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (2006) addressed the related issues of curricular focus and coherence, which are at the heart of the growing call for common standards. The forthcoming Focus in High School Mathematics: Reasoning and Sense Making (2009) will address mathematics education in high school. The following guiding principles are adapted from these NCTM publications.

A curriculum is more than a collection of activities: It must be coherent, focused on important mathematics, and well articulated across the grades.

Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. Learning mathematics with understanding is essential.

If a voluntary national mathematics curriculum is developed, the topics studied in that curriculum must be taught and learned in an equitable manner in a setting that ensures that problem solving, reasoning, connections, communication, and conceptual understanding are all developed simultaneously along with procedural fluency.

Wednesday, June 10, 2009

Connecticut District Tosses Algebra Textbooks and Goes Online

WESTPORT, Conn. — Math students in this high-performing school district used to rush through their Algebra I textbooks only to spend the first few months of Algebra II relearning everything they forgot or failed to grasp the first time.

So the district’s frustrated math teachers decided to rewrite the algebra curriculum, limiting it to about half of the 90 concepts typically covered in a high school course in hopes of developing a deeper understanding of key topics. Last year, they began replacing 1,000-plus-page math textbooks with their own custom-designed online curriculum; the lessons are typically written in Westport and then sent to a program in India, called HeyMath!, to jazz up the algorithms and problem sets with animation and sounds. “In America, we run through chapters like a speeding train,” said John Dodig, the principal of the 1,728-student Staples High School here. “Schools in Singapore and India spend more time on each topic, and their kids do better. We’re boiling down math to the essentials.”

NCTM and National Standards

NCTM on Standards: Don't Forget About Us!

As state leaders and school chiefs from 46 states press ahead with plans for common standards, an organization that has held major sway on how math is taught in this country is asking for a seat at the table.

The National Council of Teachers of Mathematics, which has 100,000 members, this week released Guiding Principles for Mathematics Curriculum and Assessments.” It's basically a statement describing the organization’s extensive work on standards, and more importantly, its beliefs about the core content and ideas that should guide math teaching. NCTM released voluntary national standards in 1989 and a revised version of them in 2000. For two decades, those documents have shaped how math is taught in classrooms from coast to coast, from elementary through high school.

Sunday, June 7, 2009

NJ Join National Standards Effort

NJ joins effort to draft US math regs

"I think it would be foolhardy not to be at the table and not to be part of the discussion," said state Commissioner of Education Lucille Davy. "Our children in New Jersey are entering a world where they are not competing against just each other, they are entering a global economy."  There is no difference in the math that fourth-graders should learn in New Jersey or anywhere else in the country, Davy said.

Wednesday, June 3, 2009

More on the national standards movement

My opinion on the voluntary national standards effort led by Achieve, with support by the College Board and ACT, is that it will do nothing more than push Achieve’s agenda and rubber stamp Achieve’s Common Core document from 2008.  In the table above, from Achieve, NJ 2008 math standards (these are the current NJ standards, not the Feb 2009 ridiculous draft) received very high grades from Achieve with respect to Achieve's Common Core. The scale is from 0 to 3, with 3 representing the closest alignment on each strand.  Note that NJ receives a perfect alignment in the Number Sense strand and the Data, Prob strand.  Again this alignment is based on the 2008 NJ mathematics standards NOT the Feb 2009 revisions made by NJDOE senior personnel without input from the field.

Monday, June 1, 2009

Common Standards

New Jersey Joins 49 States and Territories in Common Core State Standards Initiative

Governor Jon S. Corzine and Education Commissioner Lucille E. Davy today joined the Common Core State Standards Initiative, a state-led process to develop common English-language arts and mathematics standards. The Common Core State Standards Initiative will be jointly led by the National Governors Assoc. Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO).