A Rant, An Update and Model Problems for You

And the seasons they go round and round
And the painted ponies go up and down
We're captive on the carousel of time
We can't return we can only look behind
From where we came
And go round and round and round
In the circle game...

Oh, how I love Joni Mitchell's lyrics made famous by the inimitable Buffy Sainte-marie. Oh, how The Circle Game lyrics above describe my feelings about the state of U.S. math education. I feel I've been on this carousel forever. But I do believe that all is not hopeless. I do see promise out there despite all the forces resisting the changes needed to improve our system of education.

Our math teachers already get it! They get that more emphasis should be placed on making math meaningful via applications to the real-world, stressing understanding of concepts and the logic behind procedures, reaching diverse learning styles using multiple representations and technology, preparing their students for the next high-stakes assessment, trying to ensure that no child is ... They've been hearing this in one form or another forever. BUT WHAT THEY NEED IS A CRYSTAL CLEAR DELINEATION OF ACTUAL CONTENT THAT MUST BE COVERED IN THAT GRADE OR THAT COURSE.

The vague, jargon-filled, overly general standards which have been foisted on our professional staff for the past 20 years is frustrating our teachers to the point of demoralization. THIS IS NOT ABOUT THE MATH WARS. THIS IS NOT AN IDEOLOGICAL DEBATE. JUST TELL OUR MATH TEACHERS WHAT MUST BE COVERED AND LET THEM DO THEIR JOB!

BY "WHAT MUST BE COVERED" I AM INCLUDING THE SKILLS, PROCEDURES AND ESSENTIAL CONCEPTS OF MATHEMATICS. NONE OF THIS CONSTRAINS TEACHER STYLE OR CREATIVITY. BUT WITHOUT THIS STRUCTURE THERE IS ONLY THE CHAOS THAT CURRENTLY EXISTS. AND IF YOU DON'T THINK THERE IS CHAOS OUT THERE, TALK TO THE PROFESSIONALS WHO HAVE TO DO THIS JOB EVERY DAY.


UPDATES...

Results of MathNotation's Third Online Math Contest

The Common Core State Standards Initiative

NCTM's latest response to the Core Standards Movement - the forthcoming Focus in High School Mathematics

Validation Committee selected for draft of Core Standards

The results of the latest round of ADP's Algebra 2 and Algebra 1 end of course exams

It will take several posts to cover all of this...


RESOURCES FOR YOU

MODEL PROBLEMS TO DEVELOP HIGHER-ORDER THINKING AND CONCEPTUAL UNDERSTANDING

Consider using the following as Warm-Ups to sharpen minds before the lesson and to provide frequent exposure to standardized test questions (SAT, ACT, State Assessments, etc.). I hope these problems serve as models for you to develop your own. I strongly urge you to include similar questions on tests/quizzes so that students will take these 5-minute classroom openers seriously.

I've provided answers and solutions/strategies for some of the questions below. The rest should emerge from the comments.

MODEL QUESTION #1:

For how many even integers, N, is N² less than than 100?

Answer: 9

Solution/Strategies:
Always circle keywords or phrases. Here the keywords/phrases include
"even integers"
N²
"less than".

This question is certainly tied to the topic of solving the quadratic inequality, N² "<" 100 either by taking square roots with absolute values or by factoring. Of course, we know from experience, when confronted with this type of question on a standardized test, even our top students will test values like N = 2, 4, 6, ... However, the test maker is determining if the student remembers that integers can be negative as well and, of course, ZERO is both even and an integer! Thus, the values of N are -8,-6,-4,-2,0,2,4,6, and 8.


MODEL QUESTION #2

If 99 is the mean of 100 consecutive even integers, what is the greatest of these 100 numbers?

ANSWER: 198
Solution/Strategies:
There are several key ideas and reasoning needed here:

(1) A sequence of consecutive even integers (or odd for that matter) is a special case of an arithmetic sequence.

(2) BIG IDEA: For an arithmetic sequence, the mean equals the median! Thus, the terms of the sequence will include 98 and 100. (Demonstrate this reasoning with a simpler list like 2,4,6,8 whose median is 5).

(3) The list of 100 even consecutive integers can be broken into two sequences each containing 50 terms. The larger of these starts with 100. Thus we are looking for the 50th consecutive even integer in a sequence whose first term is 100.

(4) The student who has learned the formula (and remembers it!) for the nth term of an arithmetic sequence may choose to use it: a(n) = a(1) + (n-1)d. Here, n = 50 (we're looking for the 50th term!), a(1) = 100, d = 2 and a(100) is the term we are looking for.
Thus, a(50) = 100 + (50-1)(2) = 198.

However, stronger students intuitively find the greatest term, in effect inventing the formula above for themselves via their number sense. Thus, if 100 is the first term, then there are 49 more terms, so add 49x2 to 100.



MODEL QUESTION #3: A SAMPLE OPEN-ENDED QUESTION FOR ALGEBRA II

If n is a positive integer, let A denote the difference between the square of the nth positive even integer and the square of the (n-1)st positive even integer. Similarly, let B denote the difference between the square of the nth positive odd integer and the square of the (n-1)st positive odd integer. Show that A-B is independent of n, i.e., show that A-B is a constant.


MODEL QUESTION #4: GEOMETRY

If two of the sides of a triangle have lengths 2 and 1000, how many integer values are possible for the length of the third side?


MODEL QUESTION #5: GEOMETRY

There are eight distinct points on a circle. Let M denote the number of distinct chords which can be drawn using these points as endpoints. Let N denote the number of distinct hexagons which can be drawn using these points as vertices. What is the ratio of M to N?

Answer: 1
Solution/Strategies: The student with a knowledge of combinations doesn't need to be creative here but a useful conceptual method is the following:
Each hexagon is determined by choosing 6 of the 8 points (and connecting them in a clockwise fashion for example). For each such selection of 6 points, there is a uniquely determined chord formed by the 2 remaining points. Similarly, for each chord formed by choosing 2 points, there is a uniquely determined hexagon. Thus the number of hexagons is in 1:1 ratio with the number of chords.

MODEL QUESTION #6: GEOMETRY AND THE ARITHMETIC OF PERCENTS

If we do not change the angle measures but increase the length of each side of a parallelogram by 60%, by what per cent is the area increased?

(A) 36% (B) 60% (C) 120% (D) 156% (E) 256%

43x27? Which Algorithm should be taught? A MathNotations survey will end the debate!

Update:
A poll has now been created in the sidebar! Please express your preference. (Thank you, Jonathan, for making this suggestion).

As if the educational fate of our children could be determined by a poll of our readers...
Hey, you never know.

What's even more incredible is that individual teachers, schools, districts, and states still feel that this is a 'local' decision. After over two decades of this debate, children and their parents still have little to say about these kinds of curricular decisions that will impact on the mathematical futures of another generation. After all, the experts know what is best, right? Well, there are many many experts who all feel sure they know the answer to this question. Trouble is, they do not all draw similar conclusions and spend most of their time defending their choices or their particular agenda or favorite set of materials.

Finally, MathNotations will settle this debate once and for all. The results of this survey (assuming a minimum of 3 responses) will be used to influence national policy for years to come!

Here are your options regarding your preference for how multidigit multiplication should be taught in Grades 3-5 :

(A) Teach only the traditional algorithm and expect mastery
(B) Teach the 'partial products' method to develop understanding of place value and the traditional algorithm; teach the traditional algorithm as a more efficient method and require it; expect mastery
(C) Teach the 'partial products' method to develop understanding of the traditional algorithm; teach the traditional algorithm as a more efficient method; give students a choice of methods; expect mastery of at least one method
(D) Model other methods (e.g., 'lattice method') and encourage students to invent their own method; do not require any particular method or mastery

Now, don't miss this opportunity to be part of an historic decision. Your vote does count...

Update on National Math Panel

As promised, I am posting, with the permission of the National Math Panel, the reply to my latest email to the Panel. Jennifer Graban sent me this last Friday and you can read it for yourself and decide if you believe that some of my recommendations and those of others are now being considered seriously. Below, I will also briefly discuss the conversation I had with Ms. Graban when I called her:

Dear Mr. Marain,

We apologize for the delay in responding to your e-mail of February 11 and acknowledge your ongoing concerns about the composition of the Panel. Understanding the experiences of teachers, particularly algebra teachers currently teaching in the classroom, is important to the Panel and its final report. I have shared your concern with the Panel Chair, Larry Faulkner, as well as other Panel members. We are currently considering additional opportunities that will involve teachers to obtain their vital input in the work of the Panel.

Please know that Panel is most concerned with and plans to carefully consider the perspectives of today's math teachers.

Once again, thank you for your interest in the National Math Panel.

Sincerely,

Jennifer



Jennifer Graban
National Math Panel Staff
U.S. Department of Education
400 Maryland Avenue, SW
Washington, DC 20202-1200


I decided to call Ms. Graban last Friday afternoon around 4 PM EST. She picked up the phone immediately and we spoke for about 10-15 minutes. To the best of my recollection here's how it went:
I began by saying that I felt it important to put a voice to all the emails and blogs and let her know that I am a real person and not the enemy, just someone who feels that not having at least one secondary teacher on this panel was a gross oversight. Continuing, I indicated that, although it would be better to appoint 2-3 dozen classroom math teachers immediately, the reality is that the panel has less than one year to complete its task and publish its final report. Therefore I suggested the importance of identifying a cross-section of a few current high school math teachers with broad experience in both urban and suburban/rural school settings. I also volunteered to provide whatever input I could to the Panel, but certainly this group of teachers should be allowed to serve in an advisory capacity, i.e., freely provide their ideas and be be able to review and make suggestions now and to the final draft of the report before its release. I expressed that this would significantly enhance the credibility of the report. She replied that this was already being considered and she seemed to concur with some of my statements. I also mentioned my call for a national math curriculum but she indicated that the panel has not yet endorsed this. I also asked her if she had noted that the textbook publishers, who were allowed to give extensive presentations to the Panel, had indicated they now had editions for EACH state or nearly so. I expressed how absurd I felt this was and she did not disagree! We left the conversation cordially.

National Math Curriculum UK

Just to see how England is handling a national curriculum (for all subjects), here's a link to the Mathematics section. You will need to navigate this site for awhile to get the feel for it. You will also want to download the 90+ page pdf math curriculum document. I've skimmed it and it is fascinating. Of course much of it is not new or surprising or that different from NCTM's many recommendations or reports, but it's the overall structure and expectations for the four key stages of learning that make it unique. Why can't we benefit from something like this so that we don't have to reinvent the wheel? I'm certain that members of the National Math Panel are familiar with this but I'd be interested in their views. Is it possible for our society? Better yet, perhaps we need to begin to realize the vision of Robert F. Kennedy:

There are those that look at things the way they are, and ask why? I dream of things that never were, and ask why not.

Things that never were? But they are already existing elsewhere!

Better yet, download
A Coherent Curriculum: The Case for Mathematics by William Schmidt, Richard Houang, and Leland Cogan is available online in American Educator, Summer 2002, pp. 1-17.

This was a call to action five years ago and we're still arguing about it today...