Proposed math standards unteachable

Sacramento Bee
Viewpoints: Proposed math standards unteachable


Algebra I is taught in eighth grade in high-performing foreign countries, and this is also recommended by America's 2008 National Math Panel. California has made immense progress in this direction in the past decade, and we now lead the nation in the percentage of algebra-takers in eighth grade. Regrettably, all these gains are in danger of being reversed because of these ill-advised standards recommendations.

Bill Evers is a research fellow at Stanford University's Hoover Institution and member of the institution's Koret Task Force on K-12 Education. He was formerly U.S. Assistant Secretary of Education. Ze'ev Wurman is an executive at a Silicon Valley high-technology company. He was formerly a senior adviser in the U.S. Department of Education.

National Standards (CCSSI)

The high school math standards seem to me to leave a whole lot to be desired - far below our current expectations for college preparedness.

If you have the time, compare these standards to the Major Topics of School Algebra in the National Mathematics Advisory Panel's Report, then take a look at current ACT math content.

Wurman and Stotsky: New Standards will set schools back
[They must be talking about me!]
High school math teachers will look in vain for course standards in Algebra II, pre-calculus, or trigonometry. The drafters deem algebra, which the prestigious National Math Advisory Panel identified as the key to higher math study, as an outdated organizing principle.


You can find CCSSI standards HERE

Comments are due to CCSSI by April 2, but you might also consider posting them publicly somewhere else.

More information available HERE

United States for World Class Math

Just in: a website for people like us who want a seat at the table when it comes to national math standards:

United States Coalition for World Class Math

Check out their Design Principles for K-12 Mathematics Standards:

1. All students should be expected to master foundational concepts and skills – especially in arithmetic – that are prerequisite to an authentic Algebra I course in a logical progression from grade to grade in the elementary and middle school years. The Final Report of the National Mathematics Advisory Panel (NMAP) should be the guiding document describing appropriate mathematical content.

2. The K-7 standards should be designed to prepare as many students as possible for an authentic Algebra I course in Grade 8. K-7 standards should be based on the "Critical Foundations of Algebra" described on pages 17-19 of the NMAP’s Final Report. Standards for authentic Algebra I and Algebra II courses should be based on "The Major Topics of School Algebra" described on pages 15-16 of the NMAP’s Final Report.

3. Standards-based alternatives could be written for less prepared students and alternate paths after algebra and geometry for high school students, depending on student achievement, interests, and career goals. For example:

a. The standards document could outline the possibility of a two-year course spanning Grade 7 and Grade 8 based on Grade 7 standards for students who, at the end of Grade 6, are judged to need more time to master foundational concepts and skills for Algebra I.

b. The standards document could outline a two-year course spanning Grade 8 and Grade 9 based on authentic Algebra I standards for students completing Grade 7 who are judged to need two full years to master Algebra I standards.

4. As emphasized by the National Mathematics Advisory Panel, "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." Placement of the standards should reflect the grade level at which mastery is expected, and standards should not be repeated from year to year.

a. The sequence of the standards should be logical and hierarchical, following the structure of mathematics itself and should be modeled after the strong standards in California, Indiana, and Massachusetts.

b. "Benchmarks for the Critical Foundations" (pages 19-20 in the National Mathematics Advisory Panel’s Final Report) and recommendations from the National Council of Teachers of Mathematics’ Curriculum Focal Points should be used for grade level placement.

c. Concepts and skills, once mastered, should be used in subsequent years with a minimum of review.

5. In order to focus on building solid foundations for the more advanced mathematics – including algebra – that occurs in Grades 8-12, extraneous topics including aspects of geometry such as tessellations, nets, statistical approaches to geometric properties, much of data analysis, probability and statistics, and non-algebraic concepts such as pattern recognition should not be present in the K-7 standards.

6. In Grades K-7, the distribution of content by strand should be stated explicitly as percentages at each grade level and should change as students move up through the grades.

a. Early grades should concentrate on the arithmetic of whole numbers and measurement, with a limited amount of geometry and graphing. Certain aspects of algebra, as well as preparation for algebra, should be present from the earliest grades, as is the case with the California and Massachusetts standards.

b. Students should be expected to acquire automatic recall of basic number facts at least to 10 x 10 and 10 + 10.

c. Students should be expected to understand and use the standard algorithms of whole-number arithmetic in the early elementary grades (i.e., addition, subtraction, multiplication, and long division).

d. Students should be expected to understand and use the standard definitions for operations with fractions in conjunction with the standard algorithms of whole number arithmetic to compute sums, differences, products and quotients of fractions, including fractions expressed as decimals and percents.

e. The algebra strand gains emphasis in the middle grades, focusing on the content specified by the National Mathematics Advisory Panel.

7. The organization of the standards should change at Grade 8.

a. In grades K-7, standards should include multiple strands of mathematics, with their relative weight appropriately adjusted through the grades.

b. For algebra and beyond, standards should be given for a single-subject course sequence (Algebra I, Geometry, Algebra II, Pre-calculus, etc.) and their components re-ordered for alternative integrated mathematics courses. The standards for the Geometry course should require students to do proofs and to understand postulates, theorems and corollaries.

8. Mathematical problems should have mathematical answers.

a. In general, students should learn techniques for problem solving that can be applied to many contexts. Problems should be contextualized in the "real world" only when the context is sensible and relevant and contributes to an understanding of the mathematics in the problem.

b. Standards documents should include example problems. The level of difficulty of these problems should reflect mathematical complexity rather than non-mathematical issues.

9. K-12 math standards should meet the criteria specified by the American Federation of Teachers. They should be:

a. Clear and specific enough to provide the basis for a common core curriculum.
b. Rooted in the content of mathematics.
c. Clear and explicit about the content and the complexity students are to learn.
d. Measurable and objective.
e. Comparable in rigor to the standards of A+ countries, with grade-level specificity.

10. Standards documents should appropriately emphasize the attainment of procedural fluency. Students must be competent in performing all K-7 tasks without using a calculator.

11. Standards documents should only address mathematical content; language pertaining to pedagogy should be excluded.

12. As emphasized by the National Mathematics Advisory Panel, mathematicians should be included in greater numbers, along with mathematics educators, mathematics education researchers, curriculum specialists, classroom teachers, and the general public, in the standard-setting process and in the review and design of mathematical test items for state, NAEP, and commercial tests.


CO Coalition for World Class Math
CT Coalition for World Class Math
NJ Coalition for World Class Math
PA coalition for World Class Math
United States Coalition for World Class Math
Parents' Group Wants to Shape Math Standards
Common Core Standards: Who Made the List?

The "Global Achievement Gap" muddle

An editorial by Sandra Stotsky, member of the National Mathematics Advisory Panel, reveals misrepresentation in the author's "study" of MIT graduates. He supposedly found that only a few MIT students mentioned anything "more than arithmetic, statistics and probability" as useful to their work.

It's a great read!

Benchmarks per the Nat'l Math Advisory Panel

In response to a question in one of the comments regarding the spirited discussion at Eduwonk's site regarding Everyday Math, here is the link to what students should know by what grade. Table 2 (page 20) of the National Mathematics Advisory Panel's final report, contains the panel's recommended benchmarks for the critical foundations of math skills/concepts that students should master in order to be prepared for algebra in the 8th grade



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 percent, 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.


UPDATE: On page xvi of the NMP's report is the following:


"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"

Congressman Miller comes up with a plan to subvert historical thinking through interdisciplinary teaching

Barry attended the House Committee on Education and Labor hearing on the National Mathematics Advisory Committee report on May 21, 2008.

Not much to cheer about, it seems. Math coaches, technology, interdisciplinary studies.... and Project Lead the Way:

Another crowd pleasing testimony came from Dr. Wanda Talley Staggers, Dean of Manufacturing and Engineering, Anderson School District Five, Anderson, South Carolina. She is involved in a program called "Project Lead the Way™". Pardon my cynicism but it's hard for me to not notice the trademark symbol after the title. This program provides pre-engineering type classes in middle and high school that can only be taught by teachers who have been certified by this trademarked program trains such teachers, and testified that "After six years of training high school teachers during the Project Lead The Way™ Summer Training Institutes across the nation, I have heard overwhelmingly from teachers that they come away from the experience with a rejuvenated interest in teaching."

Maybe so, but the key is whether the students in Project-Lead-the Way™ courses will learn any math or simply be entertained.The program is based on the premise that students don't take a lot of math because it's too removed from the real world.The solution? Hands-on project-based courses that involve the students.In order to take such classes, however, they must be enrolled in regular math classes—thus a carrot and stick approach. The program has not been field-tested or validated, and is too young to have data showing whether this strategy is likely to work.Plus problem-based learning, tends to be rather discovery-heavy despite teachers' best and stated intentions that they use a "balanced" approach—the type of balance where someone's finger is on the scale.

Chairman Miller (D-CA) saw the value of this type of program and made a leap from math connecting to applications of math, to math connecting with everything.He rhapsodized about a school he saw in Oregon that became a math academy through an interdisciplinary approach.All courses were interrelated—students had to understand the mathematics of history: distances between cities, depth of the oceans.

I ran that one by Ed, a person who is, by profession, an actual historian as opposed to a Congressman. Paraphrasing: Ed says that measuring the distance between cities in a history course defeats the purpose of history because throughout most of history distance is relative to technology. For example, the distance from Marseilles to Alexandria is 1500 miles; the distance between Marseilles and Paris is 400 miles. But in the ancient world in the Middle Ages, Marseilles was for all intents and purposes much closer to Alexandria because of the ease of shipping in the Mediterranean and also because, culturally, until fairly recently the Mediterranean ports had more in common with each other than Marseilles did with Paris. Marseilles wasn't integrated into the French kingdom until the late Middle Ages. There were no decent roads from the Mediterranean parts of France to Paris. It was an incredibly arduous trip. But it was relatively easy to get from Marseilles to Egypt. The Mediterranean Sea was protected from storms and bad weather, and it was fairly rare for a ship to be lost at sea in the Mediterranean.

Or take the words, "Dr. Livingstone, I presume."

It took 6 months for these words to travel from the center of Africa to the nearest telegraph station. From there, it was only a matter of hours until these words had traveled to all parts of the world linked by telegraph.

To have students in a history course "calculate" the physical distance between cities is to willfully have students overlook the cultural and technological factors that make distance meaningful. It's not just a waste of time, it undermines and subverts historical thinking; It is the opposite of historical thinking. It is harmful.

Mathematically Sane's review of the NMP Report

For fans of Mathematically Sane, here is their review of the National Math Panel report. My editorial comments are in green. They are not meant to intrude but to "enhance" your reading experience.



Friday, March 21, 2008
Really Says it ALL!
Presidential Math Panel Vows to Increase Learning Disabilities
Tuesday, March 18, 2008 5:17 AM
Gary Stager

In the last year of his term, the President of the United States and theDepartment of Education are now trying to do for math what they did for reading. The notable achievements of Reading First include massive fraud, profiteering, junk science, federal control over classroom practice, fear and hysteria. [Ed: They also found unprecedented success in school districts that had been written off by many.] While the National Reading Panel was stacked with ideologues sharing the same educational philosophy, the National Math Panel co-opted the National Council of Teachers of Mathematics (NCTM) by appointing the organization’s President to serve on the committee.

The National Council of Teachers of Mathematics, never known for its radicalism, swung hard towards “the basics” last year in its Curriculum Focal Points and now finds itself in the uncomfortable position of having to disagree with NCTM’s President and the President of the United States. “Skip” Fennell did neither his members nor millions of American schoolkids any favors by participating in this unnecessary process. These federal education expeditions seek to narrow both the range of content and pedagogy permissible in public schools. The private and religious schools the GOP wants to support with taxpayer-funded vouchers are immune from these intrusions. The one-size-fits-all prescriptions for what ails public education are justified by claiming that schemes are research-based. [Ed; "Research-based". Where have I heard that before? Aren't Everyday Math and Investigations "research-based"?]

The rigid definition of “scientific evidence” enforced by Department of Education may be fine in testing remedies for restless leg syndrome, but is ill-suited for the complexities of education. [Ed: Such reasoning may explain why teachers believe that teaching algorithms in the lower grades is harmful to such students]. But hey, these are the folks who have mangled the English language to imply that theory is merely an unproven guess.

There is a lot wrong with the recent math report, but making Algebra the holy grail of K-8 mathematics is wrong-headed and goes unquestioned. Stressing the importance of fractions as critical prerequisites forAlgebra adds insult to injury. In a world-class display of side-splitting math teacher humor, panel member Frances “Skip” Fennell told the New York Times , “Just as“plastics” was the catchword in the 1967 movie The Graduate, the catchword for math teachers today should be ‘fractions.’“ What Fennell doesn’t realize is that the person who said, “Plastics,” in The Graduate was emblematic of everything wrong with society. “Plastics,” was a metaphor for a shallow, superficial, inauthentic culture focused onthe wrong values. The National Math Advisory Panel’s greater focus on fractions represents a “plastic” version of mathematics that will do more harm than good.

It’s easy to see how someone might think that several years worth of fraction study prepares a child for Algebra. Fractions have numerators over denominators, separated by a horizontal line. Many algebraic equations have something over something else, also separated by a line.That’s all you need to know. Right? [ Um, uh...]Not only is the progression from arithmetic manipulation of fractions to Algebra tenuous, but neither of the assumptions underlying the value of teaching fractions or Algebra are ever questioned. [Uhh, well, they did discuss that at the very first meeting. Were you not there, or did you not read the transcript?] The President’s Math Panel, like most of the math education community maintains a Kabbalah-like belief in an antiquated scope and sequence. [Ed: A scope and sequence which managed to be successful for a great many people despite claims to the contrary in Mathematically Sane and other places] Such curricular superstition fuels a multigenerational feud in which educators fight over who has the best trick for forcing kids to learn something useless, irrelevant or unpleasant.

Despite the remarkable statement in the 1989 National Council of Teachers of Mathematics Standards, “Fifty percent of all mathematics has beeninvented since World War Two,” the NCTM has been in full retreat ever since. [Ed: Not everyone remains outwitted by stupidity.] Although much of this “new” mathematics is playful, practical, beautiful or capable of being visualized via the computer, little new content has made its way into the curriculum. [Ed. Well at last we agree. There is little to no content at all in NSF-funded programs like Investigations, Math Trailblazers, CMP, and others.] Against this backdrop of unimaginative heuristics and a leadership vacuum, math class has become increasingly torturous for too many students.

Children who struggle to manipulate fractions do so because the skills are taught absent a meaningful context in a culture where fractions are rarely ever used. [Ed: Rational numbers are fast becoming passe as well.] Fraction fans might argue that fractions are important in following a recipe, but little cooking is done during fraction instruction. Even if kids did get to learn fractions by cooking, they might add, subtract or even multiply fractions, but one hardly ever divides fractions. The fact that there are four arithmetic functions doesn’t justify drilling kids for several grade levels. [Ed: Hats off to you, Mr. Stagers. Just when I think you can't outdo yourself, you do!] I wonder how many members of the Presidential panel can coherently explain how division of fractions works beyond repeating the trick – multiply the first fraction by the reciprocal of the second fraction? [Ed: Can't give you an exact number but I believe Wilfried Schmid, Hung-Hsi Wu, Vern Williams, Tom Loveless, Sandra Stotsky, and Liping Ma can do so. Maybe Deborah Ball. Not sure about Skip Fennel.]

The Report of the National Mathematics Advisory Panel does not dispute that teachers spend lots of time teaching fractions. The report merely urges that teachers do even more of the same while hoping for a different result. A definition of insanity comes to mind. It would be bad enough if wasted time was the only consequence of the fanatical fraction focus, but too many students get the idea that they can’t do math. This damages their inclination towards learning other forms of mathematics. Given the importance of mathematics and the widespread mathphobia sweeping the land, students can ill afford to a diminution in their self-image as capable mathematicians. Educators should not be complicit in creating learning disabilities regardless of what the President or his friends say. [Ed: I guess this means you won't be voting for George W. Bush again any time soon!]

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?

you can't cram math (or anything else)

Numerous parents here have spent years lobbying our high-performing, generously-funded district ($22,000 per pupil spending) to move to the international standard for math education. That being: algebra in the 8th grade.

We have approximately 30% of our 8th graders taking algebra. The figure at KIPP, in the Bronx, is 80%. ($10,000 per pupil spending, roughly)

They're not going to do it. They're so not going to do it they're not even going to say 'no.' They're just not going to do it.

While we're on the subject of well-funded school districts saying 'no,' I should add that the middle school is also not going to allow more students to take Earth Science in the 8th grade. Only forty-eight students, of 150 or so, currently take Earth Science, compared to 100% of students in Pelham. However, in the view of the school that is 48 students too many. As the chair of the science department told us, "If it were up to me, I wouldn't offer accelerated courses to any students in the middle school, but this community demands it."

The district argues, in meetings with parents, that learning depends upon maturity. Not all students are mature enough to learn Earth Science in the 8th grade. Or algebra.

Of course, maturity has nothing to do with ability to learn, as the National Math Advisory Panel reports. However, maturity has everything to do with a student being able to monitor his learning instead of depending on his teacher to perform this function. So, yes. It's easier to teach Earth Science to a high school sophomore than to a student in the 8th grade.

So why do parents continue to lobby school districts across the land to teach serious courses to younger kids?

What is the big deal, after all, about taking algebra in the 8th grade?

What's the difference when you take algebra so long as you get around to it sometime before college?


Brain Rules

It turns out there is a very good answer to that question.

It takes years to consolidate a memory. Not minutes, hours, or days but years. What you learn in first grade is not completely formed until your sophomore year in high school.

Rule # 6: Remember to repeat
John Medina

Bingo.

Here we have one of those facts of life many of us have picked up over the years but can neither verbalize in conversation with school officials nor defend as true, primarily because we don't realize we know it.

We don't know what we know. ^*

On the other hand, when we hear someone else verbalize it, we recognize it as true of our own experience. At least, I did, when I read this statement by James Milgram:

First of all, I claim that taking -- even asking to take it out of the curriculum -- shows a profound ignorance of the subject of mathematics. The point is, in mathematics, many, many skills develop over an extended period of time and are not really fully exploited until perhaps 10, 12, or even 15 years after they've been introduced. Some skills begin to develop in the first or second grade and they do not come to fruition or see their major applications until maybe the second year of college. This happens a lot in mathematics and long division is one of the key examples.

I'm going to guess that this is another reason why Singapore students are so far ahead of American students. Singapore students are doing simple algebra in the 5th grade. They're doing simple algebra in the 5th grade, and they're not dipping in and out of simple algebra, either; they're not being "exposed" to "algebraic thinking."

They're learning what they're learning to mastery.

At age 10.


^* not to be confused with known knowns, known unknowns, and unknown unknowns.

For those who wanted specifics from NMAP

This document has it for algebra.

http://www.ed.gov/about/bdscomm/list/mathpanel/report/conceptual-knowledge.doc

The exec summary says this:

What is usually called Algebra I would, in most cases, cover the topics in Symbols and Expressions, and Linear Equations, and at least the first two topics in Quadratic Equations. The typical Algebra II course would cover the other topics, although the last topic in Functions (Fitting Simple Mathematical Models to Data), the last two topics in Algebra of Polynomials (Binomial Coefficients and the Binomial Theorem), and Combinatorics and Finite Probability are sometimes left out and then included in a pre-calculus course. It should be stressed that this list of topics reflects professional judgment as well as a review of other sources.

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

Functions
• 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



The doc itself is much more specific, as you'd guess. The tables and figures are quite insightful.

APPENDIX A: Comparison of the Major Algebra Topics in Five Sets of Algebra I and Algebra II Textbooks with the List of Major Topics of School Algebra
APPENDIX B: Errors in Algebra Textbooks

List of Figures
Figure 1: Percent of Students At or Above Proficient in Mathematics Achievement on the Main NAEP Test: 1990, 2003, and 2007
Figure 2: States with Standards for Algebra I and II Courses
Figure 3: Topics in a 1913 High School Algebra Textbook
Figure 4: Singapore 2007 Algebra Standards for Grades 7-10
Figure 5: Algebra Objectives for NAEP’s Grade 12 Mathematics Assessment
Figure 6: Topics to Be Assessed in the ADP Algebra II End-of-Course Test
Figure 7: Mathematics Topics Intended from Grade 1 to Grade 8 by a Majority of TIMSS 1995 Top-Achieving Countries
Figure 8: Mathematics Topics Intended from Grade 1 to Grade 8 in the 2006 NCTM Focal Points Compared with the Topics Intended by a Majority of TIMSS 1995 Top-Achieving Countries

List of Tables
Table 1: Frequency Counts for Broad Topics in 22 States’ Standards for Algebra I and II Courses
Table 2: Algebra Elements Covered by State Algebra or Integrated Mathematics Frameworks, by State and Two-Thirds Composite
Table 3: Comparison of the Major Topics of School Algebra with Singapore’s Secondary Curriculum
Table 4: Comparison of the Major Topics of School Algebra with the 2005 NAEP Grade 12 Algebra Topics
Table 5: Comparison of the Major Topics of School Algebra with ADP’s High School Algebra Benchmarks, Core Topics in its Algebra II Test, and the Topics in the Optional Modules for its Algebra II Test
Table 6: K-8 Grade Level Expectations in the Six Highest-Rated State Curriculum Frameworks in Mathematics Compared with the Topics Intended by a Majority of TIMSS 1995 Top-Achieving Countries

Barry Garelick on the post-NMP world

David Geary, a cognitive developmental psychologist at University of Missouri, said that the reason a panel such as NMP was formed was because of the failure of schools of education to do what the country wants: Train teachers using research-based techniques, rather than running a playground for untested methods. Schools of education should be held accountable for their work, he said.

Vern Williams noted the current state of affairs in math education in which correct answers have been deemed over-rated and algebra has been redefined to include statistics and pattern recognition. He expressed his hopes that as a result of the NMP report teachers will feel it is once again crucial to consider content - and correct answers.

During a break in the meeting, however, an event occurred which to my mind simultaneously underscored and transcended the importance of NMP's report. Williams' 8th grade algebra class which had assembled at the back of the gym gathered, in rock fan fashion, around Hung-Hsi Wu - a panelist and math professor from Berkeley - to get his autograph and take pictures.

"I guess this shows that kids can get excited about math without sitting in groups doing projects and using math textbooks that look like video games," Williams said.

I hope for the best in this post-NMP world.

Living in a Post National Math Panel World

That's beautiful.

Number Sense—Right Now!

Number Sense—Right Now!
NCTM News Bulletin, March 2008

"Is 4 × 12 closer to 40 or 50? How many paper clips can you hold in your hand? If the restaurant bill is $119.23, how much should you leave for a tip? How long will it take to make the 50-mile drive to Washington? If a 10-year-old is 5 feet tall, how tall will the child be at age 20?"

OK. I've generally ignored it when people talk about "number sense", but here it is from the president of NCTM. He says: "Number sense is important and needed—right now." What is it, exactly? Is it estimation? No, it seems to be more than that. Does Mr. Fennell define it? No. He just gives the examples above. Let's look at each one.

"Is 4 × 12 closer to 40 or 50"

Do it exactly in your head. It's part of the times table.

"How many paper clips can you hold in your hand?"

How accurately do you have to make this estimate to show number sense? He doesn't say, but I don't think he is talking about plus or minus 25% accuracy. I don't think I could guess that closely.

"If the restaurant bill is $119.23, how much should you leave for a tip?"

Does he think that those who have mastered the traditional method of multiplication are stuck doing this calculation right-to-left on paper? This is a straight estimation problem, and my traditionally-taught wife takes pride in calculating these things down to the penny in her head.

Is number sense more or less than estimation? It seems to be both more and less. Number sense means more than just estimation, but it doesn't require you to provide accurate estimations.

"How long will it take to make the 50-mile drive to Washington?"

About an hour? What are the assumptions? Is number sense equal to estimation with common sense added in? Apparently the common sense level is not very high. How about the number sense to determine how long it will take to drive 400 miles on the highway, accounting for stops for gas, eating, and traffic? What if I gave you the exact times for stops and the lower speed in the traffic? Mastery of the basics leads to number sense, not the other way around.

He seems to be making the case that there is no linkage between mastery of the basics and number sense. But then he really isn't talking about mastery of estimation. Schools could hand out "Arithmetricks" and practice, practice, practice. No, he seems to be talking about some sort educational number sense osmosis. Low expectations.

"If a 10-year-old is 5 feet tall, how tall will the child be at age 20?"

He goes on to say:

"Students who have a good sense of number are able to provide a reasonable response to the examples above, including the driving example. And they know that there is no proportion-driven response for the final example."

Sure there is. If a 10 year old child is 5 feet tall, then proportion tells you that there is some other effect going on when they get to 20. If you were talking about some unusual species of tree, then proportion or number sense is not going to help. You need content knowledge, and we all know what schools think of content. I think I'll coin a new term: Content Sense. That's what we need-right now! Just like with fractions on the real number line, kids need to be able to place major historic events on a timeline.

After all of this, I still don't know what he means by number sense or what performance level is required. Whatever it is, it seems pretty low. Mr Fennell can't define it, but he wants it fixed "right now".

There is another example:

"A sense of number emerges that is built on the foundations discussed above, which
yield responses such as, “I knew 3/4 was more than 3/5 because the pieces were bigger in fourths.” This is what all math teachers want. Such “aha!” classroom moments remind us about the importance of understanding."

"the pieces were bigger in fourths"?

So number sense is something other than math; something other than mathematically knowing why 3/4 is greater than 3/5. And "understanding", according to him, is something other than mathematical understanding. What if the student said that 3/4 = .75 and 3/5 = .6? Is that number sense? Does that show understanding or is that just rote knowledge?

Math is all about tools and methods that you can rely on to give you correct results in spite of the fact that what you might be doing defies common sense. That's the power of math. As I've said before, let the math provide you with the understanding. If you're worried about estimation, teach it directly. Number sense, whatever it is, will take care of itself.

Equity in Mathematics Education (January 2008)

It's amazing what you find on the NCTM site.

"Equity in Mathematics Education (January 2008)"



NCTM Position

"Excellence in mathematics education rests on equity—high expectations, respect, understanding, and strong support for all students. Policies, practices, attitudes, and beliefs related to mathematics teaching and learning must be assessed continually to ensure that all students have equal access to the resources with the greatest potential to promote learning. A culture of equity maximizes the learning potential of all students."

[This sounds vaguely nice enough, but they don't leave it at that.]

"A culture of equity depends on the joint efforts of all participants in the community of students, educators, families, and policymakers:"

"All members of the community respect one another and value each member’s contribution.

[Except for the contributions of parents and mathematicians and anyone else they disagree with. We're not allowed to contribute. All we get are open houses and the opportunity to be "informed". Even the national math panel only gets to define "a first step". NCTM gets the rest.]

"The school community acknowledges and embraces all experiences, beliefs, and ways of knowing mathematics."

["Ways of knowing mathematics"? Did the national math panel define this? Did they define various ways to know algebra? They did the opposite. They defined what algebra is.]

"All necessary resources for optimal learning and personal growth of students and teachers are allocated."

[This is the more money escape clause.]

"High expectations, culturally relevant practices, attitudes that are free of bias, and unprejudiced beliefs expand and maximize the potential for learning."

[As long as they are in charge of defining what all of this means.]

All students have access to and engage in challenging, rigorous, and meaningful mathematical experiences."

["Meaningful mathematical experiences"? How about having access to rigorous curricula, quality teaching, and no excuses? How about making sure that kids actually learn math, not experience it?]

"Such practices empower all students to build a relationship with mathematics that is positive and grounded in their own cultural roots and history."

[OK, I reject the zero because it wasn't grounded in my own cultural roots and history. I want Roman Numeral Math.]

"Different solutions, interpretations, and approaches that are mathematically sound must be celebrated and integrated into class deliberations about problems."

["Must be?" As long as they are not the traditional algorithms.]

What Algebra? When?

I came across this message from Mr. Fennell when I went to the NCTM site to see if I could find any information about the math panel. I only found their press release about the panel report, but there was a big emphasis about data on the home page: "Focus on Data/Probability". Keep that in mind when youy read this message.


"President's Messages: Francis (Skip) Fennell"

"What Algebra? When?"

NCTM News Bulletin, January/February 2008"

"Currently, about 40 percent of eighth-grade students in this country are enrolled in first-year algebra or an even higher-level math course (for example, geometry or second-year algebra)."

[Really? What kind of algebra? Surely not the math panel type.]

"As Chambers (1994) notes, algebra for all is the right goal—we just need to make sure that we’re all targeting the right algebra in our teaching. This algebra would focus on topics like expressions, linear and quadratic equations, functions, polynomials, and other major topics of algebra. (Note that these ideas will be discussed in the National Math Advisory Panel’s report on algebra topics.)"

[OK, what is the percentage now, and how many of these kids get help outside of school. How difficult is that kind of research?]

"At a time when maintaining our nation’s competitive edge means encouraging more students to consider math- or science-related majors and careers, should we address the challenge by moving more students into higher levels of mathematics earlier? Well, I am not so sure."

[If students aren't ready, then it can't be the fault of the school or curriculum? It's also not a matter of when you get to algebra, but how well you are prepared for algebra. Many kids can't even handle algebra when they get to college! So, the problem is that many kids can't get to algebra at ANY time, but he sees the problem as a "when" problem. The rule is (apparently) that if you want to deflect criticism, redefine the problem.]

"Yes, we have more students taking higher-level courses in mathematics, and yes, the path to a good job often begins with algebra. But is mandating algebra for all seventh- or eighth-grade students a good idea? Teachers of algebra frequently tell me that far too many of their students are not ready for algebra, regardless of how it is defined (first- or second-year algebra, integrated mathematics curriculum, etc.)."

[Well, if they are not ready for algebra, then figure out why. Will they magically be prepared by ninth or tenth grade? No, our high school has to have lots of remediation to fix K-8 school problems. This is not a "when" problem. "When" allows them to avoid fixing problems.]

".. most teachers indicate that their students don’t know as much about fractions as they would like. By fractions, I mean fractions, decimals, percents, and a variety of experiences with ratio and proportion."

[Duh? And this is NOT a curriculum and/or a teaching problem?]

"Of course, we must not overlook the importance of integrating the essential building blocks of algebra in pre-K–8 curricula, especially during the middle grades. Work with patterns is probably overemphasized in some quarters as the defining component of algebra with younger learners, but early experiences with equations, inequalities, the number line, and properties of arithmetic (such as the distributive property) are foundations for algebra. Silver (1997) notes that integrating algebraic ideas into the curriculum in a manner that helps students make the transition from arithmetic to algebra also prepares them for what occurs later in algebra."

[So why does NCTM support curricula that don't meet these goals?]

"So is early access to algebra a good idea? Sure—for some—probably for many. More importantly, however, all students who are working to secure this valuable "passport" should begin their study of algebra with all the prerequisites for success, regardless of when the opportunity comes their way."

[They don't see that they have anything to do with this lack of preparation? They should have a big section called Focus on Algebra, not Focus on Data/Probability. Low expectations and blame shifting permeate this message from the president of NCTM. They lack any ability for critical self-analysis.]

turning lead into gold

Well, at least the Wright Group/McGraw Hill, publisher of Everyday Math, is attempting to clean up their own mess. On the same day the National Math Advisory Panel was busy making their report available to the public, Wright Group was introducing Pinpoint Math.

CHICAGO, March 13 /PRNewswire/ -- Wright Group/McGraw-Hill (News) has published a new math intervention curriculum, Pinpoint Math. The supplemental program, with both online and print components, was designed for students in Grades 1-7 who are one to two grade levels behind in mathematics.

Pinpoint Math can be used successfully with any basal mathematics program. It incorporates the three essential elements necessary for improvement of mathematics performance among struggling students:

-- Diagnostic Assessment: Identify areas of weakness for individual students.
-- Targeted Instruction: Provides content in an individual Student Action Plan that meets the needs of the student with both print and animated tutorials.
-- Progress Monitoring: For ongoing assessment of students' advancement on individual topics in both formal and informal formats.

Of course, it will come at a pretty penny to districts who buy the supplemental program. The Wright Group has a nice customer base to tap into considering the 175,000 classrooms they've wriggled their way into. That's a whole lot of potential sales.

Conveniently, Pinpoint Math "can be used successfully with any basal mathematics program." That way, schools can help those struggling students "who are one to two grade levels behind in mathematics."

So first you sell schools a math curriculum that results in a significant population of struggling students, and then you sell them a scaffolding tool to remediate the problem you created in the first place. Nice job Wright Group/McGraw-Hill. Way to keep the stockholders happy.

That's just wrong on so many levels, I don't even know where to begin.

I think I'm coming down with something...

The inability to understand and compute fractions, decimals, and proportions has important real-life implications, and has been linked to poor health outcomes, among other harmful effects.

Report of the Task Group on Learning Processes
Draft 3/6/2008

You can lead a horse to water ...

A presidential panel said yesterday that America's math education system was "broken," and it called on schools to ensure children from preschool to middle school master key skills.

...

F. Joseph Merlino, project director for the Math Science Partnership of Greater Philadelphia, which runs a research program involving 125 schools in 46 school districts, said that while he agreed with the finding that "you can't teach so many topics that you aren't able to get into depth," he disagreed with the report's focus on improving algebra instruction as central to better math education for all students.

He said he favored tailoring math instruction to the learning styles of students more than the report does. (emphasis added)

Philadelphia Inquirer, Panel: Math education is 'broken' The presidential panel called for ways to improve teaching and fight "math anxiety."

Bear in mind that in 2005, only 15.8% of black 11th graders in Philadelphia performed at the proficient level or above on the state math test. This placed them 2.61 standard deviations below the mean pass rate of 52.8% in Pennsylvania. This places these students below the first percentile.

See here.

I guess they haven't found the right learning style for these students yet.

Natl Math Panel: increase algebra in 8th grade

All school districts should ensure that all prepared students have access to an authentic algebra course—and should prepare more students than at present to enroll in such a course by Grade 8. The word authentic is used here as a descriptor of a course that addresses algebra consistently with the Major Topics of School Algebra (Table 1, page 16). Students must be prepared with the mathematical prerequisites for this course according to the Critical Foundations of Algebra (page 17) and the Benchmarks for the Critical Foundations (Table 2, page 20).

FINAL REPORT
p. xviii

Table 1: Major Topics of School Algebra

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
  

Functions

• 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

Recommendation: Proficiency with whole numbers, fractions, and particular aspects of geometry and measurement should be understood as the Critical Foundations of Algebra. Emphasis on these essential concepts and skills must be provided at the elementary and middle grade levels.

Recommendation: The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected.

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 threedimensional 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:
FINAL REPORT, 2008.

In a fraction fix

Today in their final report, the National Math Advisory Panel said:

Difficulty with the learning of fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It also has been linked to difficulties in adulthood, such as failure to understand medication regimens. Algebra I teachers who were surveyed for the Panel as part of a large, nationally representative sample rated students as having very poor preparation in “rational numbers and operations involving fractions and decimals” (see Panel-commissioned National Survey of Algebra Teachers, National Mathematics Advisory Panel, 2008). The Final Report of the National Mathematics Advisory Panel, p. 30

Today in the blogosphere, a student teacher said:

Today I taught a lesson about fractions from Everyday Mathematics. Fractions were not something I was good at in school, so I was nervous about teaching it. It went okay, but some of the students gave answers that were not correct, and I was having trouble explaining why they were correct. Luckily my master teacher was in the classroom so she was able to help me give an explanation on why that students answer was not correct.

Houston, we have a problem.

multiple choice vs. constructed response

The Panel first examined whether constructed-response formats measure different aspects of mathematics competency in comparison with the multiplechoice format. Many educators believe that constructed-response items (e.g., short answers) are superior to multiple-choice items in measuring mathematical competencies and that they represent a more authentic measure of mathematical skill. The Panel examined the literature on the psychometric properties of constructed-response items as compared to multiple-choice items. The evidence in the scientific literature does not support the assumption that a constructed response format, particularly the short-answer type, measures different aspects of mathematics competency in comparison with the multiple-choice format.

National Mathematics Advisory Panel FINAL REPORT
p 60

This comes at a timely moment as I'd been planning to put up a post asking for opinions on the "explain your answer" items on last year's NY state 8th grade test.

e.g.:

Zach earns $160 per week at a local market. He makes a payment of $12 per week for a new bike. He spends $75 each week on food and entertainment. Zach deposits the rest of his money in a savings account. Zach estimates that he deposits about 25% of the $160 into his savings account each week.

Is Zach's answer correct?

On the lines below, explain how you determined your answer.

How much of his weekly earnings would Zach need to deposit in order to save 40%?

source:
NYSED 2007 tests
Mathematics Test Book 3
Grade 8
March 12-16, 2007

To me, this looks like an opportunity for kids who don't know how to solve the problem to gain a point or two because they "understand" the concept. (This item is worth 3 points, the highest possible on any items on the test.)

By the same token, it is also an opportunity for kids who do know how to solve the problem to lose a point or two because their verbal explanation was incoherent.

The extended response item as equalizer.