A Review and a Critique of the National Math Panel Report - PART I

On March 13, 2008, the National Mathematics Advisory Panel presented its Final Report, Foundations for Success, to the President of the United States and the Secretary of Education.

As with any education report there is a both a technical and a political aspect. Each of us will view this type of document through the lens of our personal bias. As much as I'd like to believe I am fair and objective, I know that I bring my own perspective to this as one who passionately believes that one can expect both mastery of skill and conceptual understanding, the what, the how and the why. Instruction that combines skill practice with rich problem-solving and exploration. I know I am not the only one who holds these beliefs and, to me, they are self-evident.

Since the inception of this blog 15 months ago, I have not wavered from this view. You can see this in the many letters I wrote to the National Math Panel which have been published on this blog to my interview of Prof. Schmidt, to my interview with Prof. Steen, to my post in Jan 2007 describing the genesis and reason for this blog. Repeatedly I have called for more coherence and consistency in K-8 math curriculum and beyond. Repeatedly, I have asked those in a position to make a difference to listen to our dedicated professional educators who have always known the truth of what is needed for our children's math education. Perhaps some of this has not fallen on deaf ears...


You may want to view the short video of an interview with Dr. Larry Larry Faulkner,
the chair of the National Mathematics Advisory Panel. In it, he discusses the key findings of the panel, published in their recently released report.

In this post I will summarize and comment on this video interview. In later posts, I will comment on other aspects of the report.

Highlights of Dr. Faulkner's Comments
(much of this was taken verbatim from the video):

(1) We need to streamline mathematics education in the years leading up to algebra
(2) Streamlining is needed because it is well-known both inside the US and outside that we have too many topics in early grades that are covered too shallowly
(3) A principal recommendation is that we cover fewer topics and cover the most important ones more thoroughly.
(4) The panel focused on the preparation of American children for success in algebra. Algebra has a central role in mathematics curriculum, the first course leading into secondary mathematics.
(5) Success in algebra not only plays a central role because of its correlation to success in high school math courses, but also because of its correlation to college-attending and graduation rates and eligibility for the national technical workforce.
(6) Cognitive science informs us that children who believe that working harder can make one smarter, actually achieve more.
(7) To encourage their children, parents should make note of how common mathematics is in the world and how many jobs parents hold which make use of mathematical concepts.

MathNotations Reactions to Dr. Faulkner's statements:
(1) "...It is well-known outside the US and inside"! When did it become so well-known? Could it be when Prof Schmidt uttered WELL OVER A DECADE AGO his now-famous description of our math curriculum as "one inch deep and one mile wide"? You mean now his word will be heeded? Now that NCTM has recognized the need for a narrower focus as recently published in its Curriculum Focal Points? Now there will be new textbooks written that reflect this narrower focus? Is that what you mean Prof. Faulkner? Exactly who will be apologizing to the generations of students who have been exposed to such superficiality? Who will be apologizing to the dedicated educators who have been compelled to sacrifice mastery and deeper understanding for superficial coverage of a myriad of topics? Who exactly will be apologizing for not listening to Prof. Schmidt until now?

(2) We need to be informed by the research of cognitive scientists that a strong work ethic will lead to success, particularly in mathematics? That persistence and effort can lead to success in math!

This is the revelation I've been waiting two years to hear? Yes, I agree this is a wonderful message for all students to hear in all aspects of their learning. Yes, Prof. Faulkner, I believe we can also accumulate 400 million pieces of anecdotal evidence from our educators to support the truism that there is no substitute for hard work, particularly in mathematics...

Irving 'Kap' Kaplansky - He Strikes A Familiar Chord! The Mystery is Solved...

We had three winners this week who correctly identified Professor Kaplansky (1917-2006). As usual, I had many reasons for selecting this outstanding mathematician. I'm always partial to algebraists or number theorists and 'Kap' worked in both areas. I've also been told I bear some physical resemblance to Dr. Kaplansky! I was also influenced by the fact that music played an important role in his life.
A press release from the U. of Chicago provided the following excerpt:

Kaplansky loved working with young people, and he served as Ph.D. advisor to 55 graduate students, the most of any mathematics professor ever to have taught at the University of Chicago, said J. Peter May, Professor of Mathematics at the University.

He published close to 150 papers, the earliest appearing in 1939 and the last in 2003, an astonishing span of activity for a mathematician,” May said. “Kaplansky had a great sense of humor, or perhaps more accurately fun. He enjoyed life and lit up any room he was in. He liked quirky mathematical problems with a real life twist. For example, a 1943 paper gave an elegant solution of the problem of finding the number of ways that a given number of married couples may be seated at a round table, men alternating with women, so that no wife sits next to her own husband."

In 1984 he became Director of the Mathematical Sciences Research Institute in Berkeley, California, a post he held until 1992.

David Eisenbud, current Director of MSRI and a former student said, “I remember well his highly entertaining and beautifully polished lectures from my student days in Chicago. Whatever he taught, I signed up for the course, it was such a pleasure to listen to him.

“After stepping down as MSRI director, at 80, Kaplansky went back to full-time research mathematics, and returned to number theory, one of his first loves,” Eisenbud said.

As an avid musician and pianist, Kaplansky played in or directed many University musical and theater productions, including its annual productions of the works of Gilbert and Sullivan.

Music was a very important part of Prof. Kaplansky's life. Here is an excerpt from Ivar's Peterson's Math Trek:

A distinguished mathematician who has made major contributions to algebra and other fields, Kaplansky was born in Toronto, Ontario, several years after his parents had emigrated from Poland. In the beginning, his parents thought that he was going to become a concert pianist. By the time he was 5 years old, he was taking piano lessons. That lasted for about 11 years, until he finally realized that he was never going to be a pianist of distinction.

Nonetheless, Kaplansky loved playing the piano, and music has remained one of his hobbies. "I sometimes say that God intended me to be the perfect accompanist--the perfect rehearsal pianist might be a better way of saying it," he says. "I play loud, I play in time, but I don't play very well."

While in high school, Kaplansky started to play in dance bands. During his graduate studies at Harvard, he was a member of a small combo that performed in local night clubs. For a while, he hosted a regular radio program, where he played imitations of popular artists of the day and commented on their music. A little later, when Kaplansky became a math instructor at Harvard, one of his students was Tom Lehrer, later to become famous for his witty ditties about science and math.

He wrote A Song About Pi. Additional lyrics were added by one of his inspired students. If you ever get a chance to hear a performance by singer-songwriter Lucy Kaplansky (Irving Kaplansky's daughter), you might very well get a rendition of "The Song About Pi" as part of the program. A club headliner, recording artist, and former psychologist, Lucy Kaplansky has her own distinctive style but doesn't mind occasionally showcasing her father's old-fashioned tunemanship.

Now for our winners:

Eric Jablow

Irving Kaplansky...

Barton Yeary

Howdy -- the mystery mathematician is Irving Kaplansky. The reference to rings made me immediately think of him and a bit of googling located his obit from AMS Notices (http://www.ams.org/notices/200711/tx071101477p.pdf). The pictures matched up.

I don't have an anecdote about him. But one fact is this: he wrote a nice book for undergraduates, Set Theory and Metric Spaces.

Vlorbik

Irving Kaplanski.

i met him in '87 at herstein's festschrift:

http://www.math.niu.edu/~behr/Math/Yitz/

.

his daughter's semi-famous:

http://en.wikipedia.org/wiki/Lucy_Kaplansky

keep 'em coming!

I'll leave Prof. Kaplansky's picture up there for another week or so. Stay tuned for the next contest...

Classic AMC Contest Square Dissection Problem and more...

Recognize this diagram from a famous math contest problem which I first saw many years ago on an old AHSME contest (now known as AMC)? We'll start with this one and then modify it, creating variations on the basic theme. Finally, we will ask our readers/students to generalize the result algebraically.


In the diagram at the left, nothing is labeled, so we will describe it verbally and hope it will make sense.
We start with a square and dissect it by drawing 4 segments, each connecting a vertex to a midpoint of a side.

THE CLASSIC
Explain why the area of the shaded region is one-fifth of the area of the original square.

Notes/Comments:
(1) This is a wonderful exercise to develop spatial reasoning and to demonstrate a visual approach to a geometry problem when dimensions are not given. Of course, one could use an algebraic or numerical approach if one chooses.
(2) Students who 'see' the jigsaw puzzle approach of rearranging the pieces rarely consider what assumptions are being made. To make the problem even more meaningful, the instructor could ask why the shaded region is, in fact, a square.
(3) Simpler versions of this often appear on the SATs.

VARIATION #1

This time, both diagonals are drawn. The additional two segments join the midpoint of the bottom side to the midpoints of two other sides.

(a) The red shaded region (does it have to be a square?) is not one-fifth of the original square. What fractional part is it?
(b) The total shaded area is what part of the original square?

VARIATION #2

This time the smaller segments divide the sides into a 1:2 ratio. The figure is not drawn to scale. The 3 segments on the base are supposed to be equal!

(a) The blue shaded region (is it a square?) is now what fractional part of the original square?
(b) The total shaded area is now what part of the original square?



THE GENERALIZATION OF VARIATION 2
Use the diagram from Variation 2. Assume the original square has a side length of 1 unit. If the smaller segments divide the sides of the square into an x:(1-x) ratio, do parts (a) and (b) again, expressing your results in terms of x. What restrictions on x make sense here? Make sure your expressions agree with the results above.

A 'Simple' Traversal through a Number Grid -- Patterns, Functions, Algebra Investigation Part I

Here is an activity for Prealgebra and Algebra students. This introductory activity is not meant to be a conundrum for our crack problem-solvers out there, but the extensions below may prove more challenging.



Target Audience: Grades 6-9 (Prealgebra through Algebra 1)

Major Standards/Objectives:
(1) Representing numerical relationships and patterns algebraically
(2) Recognizing, interpreting and developing function notation
(3) Applying remainder concepts

A 2-column number matrix (grid) is shown above and assumed to continue indefinitely. We will be visiting (traversing) the numbers in the grid starting in the upper left corner with 1. Following the arrows we see that the tour proceeds right, then down, followed by left, then down and repeats.

First, some examples of the function notation we will be using to describe this traversal:
T(1) = 1 denotes that the 1st cell visited contains the number 1.
T(4) = 3 denotes that the 4th cell visited contains the number 3.
Similarly, T(6) = 6.

STUDENT/READER ACTIVITY/INVESTIGATION

(a) Determine T(1), T(5), T(9), T(13), T(17).
(b) 1, 5, 9, 13, 17, ... all leave a remainder of ___ when divided by 4. (Fill in the blank)
Therefore, these numbers can be represented algebraically as 4n + 1, n = 0,1,2,3,...
(c) Based on (a) and (b), it appears that T(4n+1) = _______, where n = 0,1,2,3...
(d) Determine T(2), T(6), T(10), T(14)
(e) 2,6,10,14,... all leave a remainder of ___ when divided by 4. Therefore, these numbers can be represented algebraically as ______, n = _________ (Fill in blanks)
(f) Based on (d) and (e), it appears that T( _____ ) = _____, n = __________.

Note: The instructor may choose to start n from zero or one throughout this activity. I will vary it depending on our needs. It is important for students to see how restrictions (domain of a variable) is critical for an accurate description and that more than one set of restrictions is possible (provided they are equivalent).

Since T(3) = 4 and T(4) = 3, we cannot say that T(n) = n for all n. The numbers 3 and 4 leave remainders of 3 and 0 respectively when divided by 4. We will need a different rule for these kinds of numbers. Let's collect some more data:

(g) By extending the table, determine T(7) and T(8); T(11) and T(12); T(15) and T(16)
(h) Without extending the table, make a conjecture about the values of T(35) and T(36).
(i) Numbers such as 4,8,12,16,... can be represented algebraically as ____, n= 1,2,3,...
(j) Numbers such as 3,7,11,15,... can be represented algebraically as ____, n = 1,2,3,...

Note: Again, the instructor may not like varying the restrictions here. Adjust as needed.

(h) Ok, so you're an expert now. Well, prove it:
T(100) = ______; T(153) = _____; T(999) = ______
Show or explain your method.

EXTENSIONS

Surely, a 3-column number grid or even a 5-column number grid can't be that much more difficult to solve using the same kind of traversal (move to the right until you come to the end, go down, move left until you come to the end, move down, lather, rinse, repeat...). ENJOY!

Ok, for our experts: Try an n x n grid!

DISCLAIMER: As with all of the investigations I publish, these are essentially original creations and therefore have not been proofread or edited by others. You are the 'others!'. You may not only find errors but alternate and perhaps superior ways to present these ideas.
Also, please adhere to the Guidelines for Attribution in the sidebar.

Odds and Evens - Week of 3-24-08

We've already had a couple of submissions for our Mystery Mathematician. I selected him for several reasons - see if you can intuit some of these!

• Take a look at the 29th edition of the Carnival of Mathematics over at Quomodocumque, an interesting 'non-blog' written by a professional mathematician who has a unique perspective on the role of blogging in math research. This edition of the Carnival is weighted in favor of research mathematics, not that there's anything wrong with that! Enjoy the quality and variety of excellent posts. I particularly like the name of the host's web site. Now whenever one of my adolescents utters the classic 'whatever', I can reply "quomodocumque!"
  

• One of our Mystery Math solvers shared a fascinating link to Dr. Hung-Hsi Wu's web site. Dr. Wu from the U. of California, Berkeley, has been a major voice in the counter-reform movement in math education. I particularly encourage readers to scroll down the page until you reach his 2006 paper on Professional Development: The Hard Work of Learning Mathematics. I may have more to say about this.
• Now that the National Math Panel has released its final report, entitled Foundations for Success, it behooves me to comment, considering all of my past correspondences with the Panel. There have already been reactions on several leading blogs including Edspresso and Joanne Jacobs.
  
• Still working on developing some rudimentary math casts using Mimio technology and SnapZ Pro X image capture software. I just don't have enough hours in the day!
• I'm also planning a review of Explore Learning, an excellent web-based set of interactive simulations for math and science. I'm in the process of using some of their Gizmos and I've received helpful background information directly from the company. I'm sure many of you are familiar with this product and will add your thoughts.
  
• I'm working on an in-depth investigation relevant to the Patterns, Functions, Algebra strand in most standards. It starts out as a simple winding traversal through a 3-column number matrix, then onto 5 columns and perhaps beyond. If you've ever counted on the fingers of one hand, then you know how easy this is: 1-2-3-4-5-6-7-8-9. Uh oh, I only have 9 fingers...
  

Mystery Math Icon Week of 3-24-08

Several compelling reasons for our choice of this luminary. In my opinion, he could run rings around anybody! Email me at dmarain at gee-mail dot com with your answer and an interesting anecdote. If this is your first submission, please include some background about yourself as well.

HOW MANY FASCINATING FACTS can you find about 97? about 153?

Note: There has been a revision in one of the properties of 17 below - I'm sure you already caught the error! Joshua caught another error involving the 4th powers - both have now been corrected.

For middle or high school students:

Number sleuths -- get into your detective groups. No calculators for the first 5 minutes.
You will have a total of 15 minutes to uncover as many interesting or fascinating facts as you can about the number 97.
One member of the group must record these and report back. Make two columns.
The first labeled: Discoveries found without the calculator.
The other column: Discoveries found with the calculator.

Ah but you're wondering what makes some fact interesting or fascinating. That's pretty subjective, right? Let's model one:

17

• Prime number
• Both of its digits are odd
• The product of its digits is prime
  
• Can be written as a sum of 2 consecutive integers: 17 = 9+8
• If its digits are reversed, the resulting number, 71, is also prime and the difference of 71 and 17 is 64 or 8²; oh, and 8 just happens to be 1+7!!
  Note: This is clearly incorrect - 3-point penalty!!
  
• Can be written as a sum of squares: 17 = 4² + 1²
• Can be written as a difference of squares: 17 = 9² - 8²
• Can be written as a sum of a perfect square and a perfect cube: 17 = 3² + 2³
• Can be written as a sum of 2 consecutive fourth powers: 17 = 1⁴ + 2⁴; in fact, 17 is the least integer which can be written as the sum of two distinct nonzero 4th powers
  
• You can get your driver's license in some states on your 17th birthday (this is clearly the only fact that's interesting to an adolescent - I had asked one of my students what's special about March 14 and he replied, "It's my birthday and it's also the Junior Prom.")
  
• 17 is the hypotenuse of a Pythagorean triple; 8-15-17 [high school level?]
  
• 17 is part of another Pythagorean triple: 17-144-145 [high school level?]
  
• Is there any end to this list?

Well, you get the idea.
You will have 15 minutes to complete your investigation of the number 97.

To rate each of your lists, we can assign a point value to each fact. For example, the first four facts listed above for the number 17 could each receive one point. Some of the other facts could receive 2, 3, 4 or even 5 points (max) for being more difficult to find or just more amazing. If any fact is incorrect, 3 points are deducted, so you'd better do independent fact-checking on your team! We will then determine the top 3 lists from this rating system and those teams will receive worldwide recognition by having their results appear on You Tube (just their lists, not names or faces for reasons of confidentiality of course). This will surely go viral in 17 seconds or less. Oh, so you'd rather receive 97 bonus points on your next test?

Oh, I forgot to mention. As a super extra credit bonus project/assignment, do the same this evening for the number 153. You will need to email me with your individual lists by 10 PM (yes, yes, we all know this is unfair to students who do not have access to email or whose parents may not give permission for this or for some other reason, so this is just an option!).

A, B are two points on a circle... Extending Student Thinking in Geometry

One of my stellar SAT students (who has been mentioned before) led me to develop the following extension of a challenging SAT-type question. Problems similar to this have appeared on previous SATs and math contests. All geometry students can benefit from both the spatial reasoning inherent in this problem as well as the algebraic analysis needed to demonstrate the results numerically.

Part of the issue of problems like this is that test constructors often attempt to develop questions similar to previous questions that had proved effective. An earlier version of the problem below read something like this:

A, B are 2 distinct points on a line. How many points on the line are twice as far from A as from B?

Occasionally, the ratio changes: three times as far from A as from B...

I've even seen other locus versions of this basic premise:

If l and m are parallel lines in a plane, how many lines are twice as far from l as from m?

In both cases, there are two solutions, one involving internal division and the other involving external division.

One can make this a bit more challenging by asking the more general version:

How many points are twice as far from one point as from the the other. (Similarly for the parallel lines problem).

By symmetry, there would now be 4 solutions. Nice questions that promote spatial sense and the ideas of internal and external ratios. We could stop there, but changing the line to a circle adds another dimension to the question:


The Challenge:
A and B are distinct points on a circle of diameter 2.
The length of arc AB is 1.
Note: For this question, all distances are arc lengths, not chord lengths.

(a) How many points on the circle are three times as far from A as from B?

(b) Now specify the locations of such points. Thus, if P is a point such that the length of arc PA is three times the length of arc PB, determine the position of point P in each case and the actual lengths of these arcs.


Comments:
Do you think most students or even the author of this question (I've altered the original problem slightly and extended it with part(b)), considered what the student with greater insight might have considered, namely, the issue of major vs. minor arcs? Do some students see the more profound subtleties of these questions -- the ambiguities that the author or others do not consider? Worth discussing the topological similarities and differences between a circle and a line?