Saturday, March 22, 2008

A Blueprint for the Foundations of Algebra
could be the first step in SPS math repair

Dear Seattle School Board Members, 3-22-2008

The recently released National Math Panel Report should be used in giving you appropriate guidance in the coming high school math adoption. The panel’s recommendations reveal that the primary materials from the last two math adoptions are in serious misalignment with the panel’s views. I urge you to consider not just a high school adoption at this time but how the current math direction of the SPS can be corrected. I hope you will find the following information useful.

The recommendations included in “The Final Report of the National Mathematics Advisory Panel” and the “Report of the Task Group on Conceptual Knowledge and Skills” released on March 13, 2008 should be carefully considered in any Seattle Schools Math Adoptions, especially the items listed here as they are the most pertinent to standards:

A.. Focused, Coherent Progression of Learning
....i...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. (Get someone to work on D44.00 and D45.00, this neglect is malpractice)
....ii...Avoid approaches that revisit topics superficially, year after year without closure. (This means Everyday Math should be avoided.)

B.. Authentic Algebra
....iii...As called out in NMAP Table 1: “The Major Topics of School Algebra”
....iv...While the Panel finds no basis in research for a preference of integrated versus single-subject approach, an analysis of high school mathematics standards suggests that high school students enrolled in mathematics courses using an integrated approach to mathematics may find it more difficult to take advanced mathematics course work (e.g., calculus or pre-calculus) in their senior year than high school students who are able to enroll in an Algebra II course in their sophomore or junior year.
....v...Universal availability of a basic math education for more students than at present preparing them to enroll in Authentic Algebra by Grade 8. Major Topics of School Algebra should be the focus for Algebra I and Algebra II standards in state curriculum frameworks.
...vii...Algebra II by Grade 11.

C.. K-8 Critical Foundations
viii. As called out in Table 2: K-8 Critical Foundations
ix. Yearly progress to match or beat by the grade level indicated in the critical foundations.
x. The Critical Foundations are not meant to comprise a complete mathematics curriculum leading to algebra; however, they deserve primary attention and ample time in any mathematics curriculum

D.. Use of Calculators
....xi. The Panel cautions that calculators impede the development of automaticity and adversely affect fluency in computation. (What can be the possible benefit to calculators prior to grade 6 vs. risks?)
xii. Calculators should not be used on test items designed to assess computational facility.

E.. Fluency with Whole Numbers
....xiii. Computational proficiency with whole number operations is dependent on sufficient and appropriate practice to develop automatic recall of addition and related subtraction facts, and of multiplication and related division facts.
xiv. Fluency with the standard algorithms for addition, subtraction, multiplication, and division.

F.. Fluency with Fractions
....xv. A major goal for K–8 mathematics education should be proficiency with fractions (including decimals, percents, and negative fractions), for such proficiency is foundational for algebra and, at the present time, seems to be severely underdeveloped.
....xvi. Ensure acquisition of conceptual and procedural knowledge of fractions (including decimals and percents) and of proportional reasoning. This should include:
1. Representing and ordering fractions on a number line
2. Judging equivalence and relative magnitudes of fractions with unlike numerators and denominators
3. Solving problems involving ratios and proportion.
4. Knowing that sums, differences, products, and quotients of fractions are fractions and carry out these operations confidently and efficiently.
5. Understand why and how (finite) decimal numbers are fractions and know the meaning of percentages

G.. Particular aspects of geometry and measurement, especially for middle grades
....xvii. Knowledge of similar triangles
....xviii. The slope of a straight line and of linear functions
....xix. The properties of two- and three-dimensional shapes and the use of formulas to determine perimeter, area, volume, and surface area.
....xx. Finding unknown lengths, angles, and areas (geometry and not trigonometry).

H.. Less focus should be placed on the patterns, data and probability, especially in the early grades.
....xxi. In the Major Topics of School Algebra set forth in the report, patterns are not a topic of major importance; therefore, patterns should be de-emphasized.
....xxii. In asking for redesign of the NAEP, the Panel noted the importance of fractions for the conceptual understanding of probability. The Panel questioned the appropriateness of items related to probability within NAEP at Grade 4.
....xxiii. The Panel recommends that the strand on Data Analysis and Probability in the NAEP at Grade 4 emphasize well-organized representations of data pictorially and numerically and be re-titled as “Data Display”.
Sincerely, Danaher M. Dempsey, Jr.

Table 1: The Major Topics of School Algebra
.... (EM and CMP2 are not preparing enough students adequately for these topics.

Symbols and Expressions
• Polynomial expressions
• Rational expressions
• Arithmetic and finite geometric series

Linear Equations
• Real numbers as points on the number line
• Linear equations and their graphs
• Solving problems with linear equations
• Linear inequalities and their graphs
• Graphing and solving systems of simultaneous linear equations

Quadratic Equations
• Factors and factoring of quadratic polynomials with integer coefficients
• Completing the square in quadratic expressions
• Quadratic formula and factoring of general quadratic polynomials
• Using the quadratic formula to solve equations

• Linear functions
• Quadratic functions—word problems involving quadratic functions
• Graphs of quadratic functions and completing the square
• Polynomial functions (including graphs of basic functions)
• Simple nonlinear functions (e.g., square and cube root functions; absolute value; rational functions; step functions)
• Rational exponents, radical expressions, and exponential functions
• Logarithmic functions
• Trigonometric functions
• Fitting simple mathematical models to data

Algebra of Polynomials
• Roots and factorization of polynomials
• Complex numbers and operations
• Fundamental theorem of algebra
• Binomial coefficients (and Pascal’s Triangle)
• Mathematical induction and the binomial theorem

Combinatorics and Finite Probability
• Combinations and permutations, as applications of the binomial theorem and Pascal’s Triangle

I urge you to begin Math Repair mode with the adoption of the Algebra Readiness Textbook and Courseware from the Mind Research Institute. At least 40% of entering 9th graders in Seattle High Schools would benefit from these materials, as these children are too unskilled to undertake the Authentic Algebra recommended by the National Math Panel. These materials could also be effectively used for many children in SPS middle schools. These materials in addition to preparing students for Authentic Algebra will close the Math Achievement Gap that has continually widened in the SPS over the last decade.

I urge you to look at the textbook “A Blueprint for the Foundations of Algebra” and the accompanying software. This can be the foundation of building a strong math foundation for SPS students. It can be the anchor point for real Authentic Algebra and a successful High School math program.

Recommendation: The Benchmarks for the Critical Foundations in Table 2 should be used to guide classroom curricula, mathematics instruction, and state assessments. They should be interpreted flexibly, to allow for the needs of students and teachers.

Table 2: Benchmarks for the Critical Foundations

Fluency With Whole Numbers
1) By the end of Grade 3, students should be proficient with the addition and subtraction of whole numbers.
2) By the end of Grade 5, students should be proficient with multiplication and division of whole numbers.

Fluency With Fractions
1) By the end of Grade 4, students should be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals.
2) By the end of Grade 5, students should be proficient with comparing fractions and decimals and common percents, and with the addition and subtraction of fractions and decimals.
3) By the end of Grade 6, students should be proficient with multiplication and division of fractions and decimals.
4) By the end of Grade 6, students should be proficient with all operations involving positive and negative integers.
5) By the end of Grade 7, students should be proficient with all operations involving positive and negative fractions.
6) By the end of Grade 7, students should be able to solve problems involving percent, ratio, and rate and extend this work to proportionality.

Geometry and Measurement

1) By the end of Grade 5, students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., trapezoids).
2) By the end of Grade 6, students should be able to analyze the properties of two-dimensional shapes and solve problems involving perimeter and area, and analyze the properties of three-dimensional shapes and solve problems involving surface area and volume.
3) By the end of Grade 7, students should be familiar with the relationship between similar triangles and the concept of the slope of a line.
Source: National Mathematics Advisory Panel, 2008.
(please let the above be reflected in D44.00 and D45.00 by next Fall)

A Need for Coherence
There seem to be two major differences between the curricula in top-performing countries and those in the U.S.—in the number of mathematical concepts or topics presented at each grade level and in the expectations for learning.
U.S. curricula typically include many topics at each grade level, with each receiving relatively limited development, while top-performing countries present fewer topics at each grade level but in greater depth. In addition, U.S. curricula generally review and extend at successive grade levels many (if not most) topics already presented at earlier grade levels, while the top-performing countries are more likely to expect closure after exposure, development, and refinement of a particular topic. These critical differences distinguish a spiral curriculum (common in many subjects in U.S. curricula) from one built on developing proficiency—a curriculum that expects proficiency in the topics that are presented before more complex or difficult topics are introduced. (Everyday Math is a spiral curriculum not built on developing proficiency.)

The Singapore standards (Singapore Ministry of Education, 2006) provide an established example of curriculum standards designed to develop proficiency in a relatively small number of important mathematics topics, as validated by a recent analysis (Ginsburg et al., 2005). The desirability of emphasizing fewer important mathematics topics in greater depth has also been recognized by some U.S. educators. (but not those making math curriculum decisions in Seattle )

In 2005, the Fordham Foundation report on state mathematics standards (Klein et al., 2005) ranked state mathematics curriculum standards based on mathematics content, clarity, and reasoning, as well as negative qualities, assigning different weights to each criterion for the overall assessment. The standards of California, Indiana, Massachusetts, Alabama, New Mexico, and Georgia achieved the highest ranking. The curricular profiles of the standards of these six states do, on the whole, provide an emphasis on fewer important topics per year than most states; but compared with the “A+ countries” (Singapore, Japan, Korea, Hong Kong, Flemish Belgium, and the Czech Republic), they all spend a great deal of time in the primary grades on topics other than arithmetic.

A more recent development in the national discussion is the publication of Focal Points (National Council of Teachers of Mathematics, 2006), which offers curricular direction to teachers and administrators by suggesting areas of emphasis for the concepts, skills, and procedures that connect important mathematics topics from grade to grade, and form the foundation for more advanced mathematics, beginning with Algebra. The message of Focal Points is also one of curriculum coherence with an emphasis on fewer important topics per year. Focal Points does not represent a set of standards but calls for a curriculum which reduces the number of important topics per year. In effect, Focal Points asks for greater emphasis on key topics, particularly with whole numbers and fractions and particular aspects of geometry and measurement. Yet Focal Points still implies more time on non-number topics, especially in the primary grades, than is the case in the A+ countries but less than the intended mathematics curriculum as represented in the frameworks of the six states.
The Panel also notes that a state’s (or a country’s) mathematics standards, however highly their quality may be judged, cannot ensure high student achievement. For example, the six leading states in the Fordham study exhibit a wide range of student achievement on the 2007 NAEP mathematics tests for Grades 4 and 8. The quality of a state’s assessments and the extent to which its standards drive sound school curricula, as well as appropriate programs for teacher preparation and professional development, are intervening variables that strongly influence achievement. They may well override the quality of the standards. (It appears that the Everyday Math alignment with failed state standards was the primary motivation for its adoption -- Now what??? -- Mind Institute software could be a start to repairing the Reform Math damage done to students in grades K-8)
MIND Research Institute’s Algebra Readiness is a new, one year curriculum that rebuilds a solid math foundation for struggling middle and high school students who would otherwise enroll in, and fail, Algebra I. The innovative learning sequence and distinctive visual diagrams of its textbook fully explain and interconnect all the essential math concepts and skills from grades 2 through 7 which form the foundation for success in Algebra I. The text is fully integrated with MIND’s unique Spatial Temporal courseware, which engages students who struggle in conventional math programs.

Please click the above link to find out what will bring math success for ELL and most students who are unprepared for Algebra in SPS schools.


dan dempsey said...

This is a program designed as an intervention program for secondary level students lacking the foundation for success in Algebra I.

Think of the market for this product in the USA. Must be at least 50% of entering 9th graders most places.

Think of what a reasonable market share of that looks like.

Here is a program that actually will work. Everyday Math does not work very well in a great many places and has about a 30% market share in Elementary Schools.

That means this Blueprint could have a market share in excess of thirty percent of the math needy ninth graders who are about 50% or more of the current 9th grade population.

Of course given the quality of decision making in regard to math curricula there may not be a reflection between the efficacy of materials and whether they are purchased. I offer Seattle's purchases in Math recently as proof.

Anonymous said...

Blueprint was already been hijacked by the reformer camp - "Blueprint for success" That might have some bad memories for some people.

Also, something not new, but should be mentioned are the software folks of which there are four major companies, including Jostens. I believe these are the companies who helped create this market which I will call 'transition math'

Paul Alan has a software company, that owns the rights to a smart program called Boxer. I think the market share for this program is about a million students.

At the other end of the spectrum, you have Integrated Learning Environments (ILEs) - Roy Pea. These are Pea-brain programs (repetition and skill building) evolved from Novanet and Plato. This is what is going to make education cost 10x more for kids.

Cognitive Tutor (Diane Briars, Carnegie Learning)
SimCalc(Jim Kaput, Dartmouth)
TERC (Ricardo Nemerovsky)
SRI (Jeremy Roechelle, Syracuse)

dan dempsey said...

The beauty of the program that I mentioned is that the Book is written by Dr Matthew Peterson an electrical engineer with a PhD in Neuro-Science.

He has been developing software (for a decade) that actually works as in producing statistically significant improvement.

These three volumes of 111 lessons are coherent focused and on the mark to prepare students for authentic algebra.

Anonymous said...

I have to disagree, a program will never replace a talented teacher. Learning requires intelligent conversation. The difficulty with all group work, is getting students to converse in a way that will have some meaning for the classroom. The teacher cannot be an effective monitor. The best dialogue occurs between a teacher and what I call gatekeepers. These are students who can act as leaders for the classroom - a good teacher makes effective use of the available talent to keep the entire classroom engaged.

Computer Assisted Instruction (CAI) assumes students are motivated independent of their surroundings, this is not backed by research or observation.

dan dempsey said...

Before you disagree. You need to look at the materials. The book with a teacher is the primary tool. The software activities are for reenforcement of the Material that needs to be learned.

This is a great package for the ELL students. A student can become a real math stud without great reading ability in this program.

It has been so long since I've seen a book with precise mathematical language, these materials were a joy to look at. They are filled with numbers and ideas instead of excess verbage.

I've never been this impressed with a first look at a text before. This is totally on the mark for the population targeted.


Here is a sample:
Chapter 4 - Rates and ratios

5 lessons
L 15 Converting Units
L 16 Finding rates from Situations
L 17 Generalizing Rates
L 18 Expanding the meaning of Multiplication
L 19 Multiplication with rates

L 15 9 pages
L 16 8 pages
L 17 9 pages
L 18 7 pages
L 19 7 pages
Chapter Review and Summary 1 page

each lesson is broken down:

by for example Lesson 17
Vocab: input-output machine, ratio


California Content Standards

Algebra Readiness Topics

Remeber from Before

Get you Brain in Gear


Then come several pages of diagrams and examples of input output machines with checks for understanding

then a 30 problem set.

four challenge problems

one multiple choice practice

two math journal questions

three find the error questions

Two self assessment looking back questions.

There are no color pictures of penguins jumping off cliffs.

Looking at a lot of math books many appear to be designed to keep ADD kids off task.

Anonymous said...

My assumption would have been that a curriculum would be developed and approved by the state. And that would have been incorrect, because Bergerson would have been in charge and I can't trust anything I read from OSPI.

I'm not sure what to expect anymore when I read about a 'new' curriculum. Singapore is not new and it has a track record - as a teacher I have to trust in one thing only - every year, the kids I receive are just the same as the kids everywhere else. The only difference logically could be the curriculum.

The UW psycho-ologists keep trying to pin problem on students, teachers, and parents. Where is the logic behind that? Why have curriculum at all?

Anonymous said...

It's important to remember that the instructional materials developed by the MIND Research Institute were built to meet the specifications of the rigorous Algebra Readiness standards as described in the California Standards Framework (The California standards, of course, were written by mathematicians Milgram and Wu). This isn't just a 'new' curriculum that a publisher thought up.

Anonymous said...


Nothing unusual about this press release:

The program was developed and authored by MIND Research Co-Founder, Dr. Matthew Peterson, the neuroscientist who created MIND's research-based ST Math(TM) line of software. In developing the program, Dr. Peterson studied and analyzed the mathematics teaching methods and textbooks of countries that are highly successful in teaching math, especially Singapore.

He also consulted with nationally renowned mathematicians, including W. Stephen Wilson, professor of mathematics at Johns Hopkins University, as well as highly experienced middle school math

Does this sound like an educator?

Wilson - "My field of interest and research is homotopy theory, generally from an algebraic viewpoint. I tend to specialize in the use and development of complex (co)bordism, Brown-Peterson (co)homology and Morava K-theory.

In my spare time I have taken up an interest in K-12 mathematics education and this led to a full time position for 8 months of 2006 when I served as Senior Advisor for Mathematics in the Office of Elementary and Secondary Education, United States Department of Education. I own lots of suits and ties now."

What is so special about being aligned to the California Framework?

Where is the independent research?

Milgram and Wu might have been on the panel that developed California's standards, but so what? The claim is their standards are rigorous. That is not true.

Name dropping is something this reform movement is becoming famous for. Software developers are epidemic. Every writer could make the claims made by Mind Research Institute, including Uri Treisman.

This is an intervention program, not mainstream. What this state needs is a uniform curriculum from grades 1 - 12 and it should prepare students for careers in science and engineering.

What is going to stop this reform movement dead in its tracks will be the lack of qualified teachers. Computers are not going to be able to keep up with the demand for good teachers. For one thing, they have to work all the time.

Schools will be operating in crisis mode for many years to come - you can see it happenning right now.

Anonymous said...

Lots of suits and ties and how many hours spent in the classroom testing problems with students.

People have to get out of the lecture halls and into the classroom. The curriculum writers are looking at the textbook, not the process of learning. Teaching is more than standards. Its not only about the content.

A textbook is a design process, it goes through many revisions before you get to the final product. And it must be comprehensible for even the lowest academically achieving students.

Comprehensible is not used in the the same sense as comprehension or mastery. In a linguistic context, it means more like accessible or decodable.

Jie said...

Regarding calculators, my personal experience with it is that when you get used to using it you'll habitually use it even for the simplest computation.

You might be interested in the Young Entrepreneur Society from the An interesting documentary about successful entrepreneurs.

dan dempsey said...

What we eventually need are focused coherent grade level appropriate real mathematics materials.

Unfortunately due to a decade plus of neglect what we really need right now are materials to help the children transition from the nonsense largely masquerading as essential math knowledge to actual learning of mathematics.

This is the point at which "A Blueprint... from MIND" enters the conversation.

This is the absolute best book to prepare the large percentage of our students who know very little math because they have been taught very little math to take Authentic Algebra.

If you find something else let me know.


Anonymous said...

So Dan, we're agreeing there are two issues - Foundations is a support program. Singapore is designed as a core curriculum, for instance.

The concern here, is if you introduce a support program at this late stage, will the opposition claim it is a victory.

There are reasons, structurally, why this will probably fail as a support program. For instance, it is one program of many, why not cognitive tutor (children dislke this program). Support teachers are seldom math certified and not trained to use thsi particular textbook. This is a support program that is designed for one year of instruction. This hardly offsets the ten years or so of bad instruction.

By the time, kids get into a support program it is too late. It would probably look like a PAS class.

As you know, PAS classes would take a few years off any teacher's life. Passing an algebra class does not exactly prepare one for college, but I guess we have to be satisfied with leadership's baby steps - nothing I would like to face a school board with.

Its sort of like the chicken or the egg. Without a good core curriculum, how do you know what is needed in a support program. The current curriculum 1-12 fails to accomplish anything right. The majority of students could only take two years of math anyway. They haven't learned enough to take intermediate algebra or pre-calculas. That's far worse than when we attended school.

I grew up in a small community and know from my friends that my class (200 students) now has former students working at Microsoft, JPL, HP, Boeing, etc. On the SATs, we were pretty average, but we took higher math and science. Today and for the past 10 years this has not been the case. Students that take support math are far less likely to enroll in other math or take science.

So while Foundations might be a good program, the idea that you can augment it to a bad program and have a significant effect on kids is unsupportable.

Anonymous said...

Support programs are designed to support curriculum, not replace it. Foundations should not be the culmination of ten years of bad instruction.

Why should students have to fritter away ten years of school, just to finally get a year of basic algebra? Most of my failing students are getting married or having kids before they're 18.

The leadership in this state is moronic. They are so out of touch with the world.

dan dempsey said...

My point is simply that at this point in time we have a large number of students, who because of educators disregard for producing improved results know very little math.

I would prefer that these students had been provided a decent curriculum and learned appropriate material, but they were not and they did not.

Mind's Algebra Readiness program is the absolute finest thing I have seen for this mathematically abandoned group.

It is so far superior to any remedial program I have seen, there is no way to compare it to anything. The software makes sense and is designed as an essential complement to the book.

It would be wonderful if remedial programs were hardly needed because schools were doing their job correctly. That certainly has not been the case in math for the last decade most places.

I am a great fan of Singapore math but right now we need to find a way to help the kids that were essentially unserved by the reform math propagandists.

Anonymous said...

Its probably too late for this group. They're out in the community working and having families. This is sweet bitterness. They'll have something to say in the polls, and I'm sure it won't be good for elitists. I'm not bitter, but I want blood. Democrats will by no means be moderate when it comes to punishing criminals, even so-called educators.

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