ODDS and EVENS Week of 4-20-09

• Don't miss the latest Math Teachers at Play #5. Denise somehow is pulling this together and producing a delectable and high-quality carnival of K-12 Math every two weeks and this one is no exception. I particularly enjoyed reading her intriguing quotes. One of my favorites came from Paul Halmos who quoted a cynic on student learning. Here's an excerpt...
  "...since, he said, students on the average remember only about 40% of what you tell them, the thing to do is to cram into each course 250% of what you hope will stick"

• I've already received several requests from teachers for the registration form for MathNotation's 2nd FREE Online Contest. One request has made me re-think my rule about one team per school. So here is the latest revision:
  Schools may field up to TWO teams.
  
  A couple of other points about the contest...
  (1) Even if you're not sure you will be able to field a team, don't hesitate to email me (dmarain at gmail dot com) to request a registration form - no obligation here!
  (2) I want to reiterate that this contest is designed for the student who has completed Algebra 2 and knows some trigonometry.
  (3) Please include the phrase 2nd MathNotations Contest in the subject line of your email to me. Just copy and paste that to make it easier for me to organize all of my emails and not miss any.

• Walter Isaacson, former managing editor of TIME magazine has written a definitive piece on the National Standards movement and the need for improved quality of standards in this week's edition of TIME. It's entitled How to Raise The Standard In America's Schools and I strongly urge all my readers to read the online version at TIME.com.

Click Read more for further comments on the TIME article...


This one article justifies purchasing the print version of the magazine in my opinion. Mr. Isaacson documents the history of the Standards movement but more importantly makes two compelling points which my readers will surely recognize from the dozens of essays I've written on the subject:
(1) The existing standards are not working (most are of poor quality and there is considerable inconsistency of expectations from state to state).
(2) There is, not surprisingly, a significant gap between the proficiency levels being reported by many states (based on their own assessments) and the NAEP results which set a higher and more uniform standard.

How do I feel about my passionate beliefs and advocacy for uniform standards beginning to come to fruition two decades after I ranted on every forum for these changes? Relief, not self-satisfaction. It's long overdue and we better move quickly now. Right, Mr. Duncan?
...Read more

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...

A Message to Edspresso (John Dewey)

The following is a comment made on edspresso's blog and expanded on here:

Happy Holidays Mr. Dewey!
I invite you and your faithful to read my expanded comments in my 'newish' blog MathNotations.
At any rate, I've just become aware of your excellent writings and I commend you for your thoughtful insights. Ok, enough of the brownnosing... Here's the skinny. I believe you're on the right track with your thoughts about constructivism. My comments regarding this appeared in Joanne Jacob's blog so I won't go into detail here. As children move up the ladder, they should need less of the hands-on manipulatives and tactile experiences like the one described in my post. This is why I suggested that cutting off the corners from the vertices of a polygon is a worthwhile activity for FOURTH graders, but I would not spend that kind of time for middle schoolers. Once they've experienced the hands-on approach, they can quickly revisit this for a triangle, then move onto a more abstract pattern-based approach in the middle grades, e.g., dividing a polygon of n sides into n-2 triangles (they can formulate this for themselves within 5 minutes), thereby developing the standard formula. By the time they reach a traditional geometry course in hs, they've had the spatial experience from 4th grade, the pattern-based approach in middle school and therefore they can quickly review this and focus on APPLYING the formula to regular polygons and more sophisticated algebraic exercises. But this discussion has important implications:
0. None of my remarks makes any sense unless 4th, 8th and 10th graders in downtown Chicago are exposed to the same learnings as those in the affluent suburbs of Chicago off Lake Michigan, if you get my point. THERE MUST BE ONE NATIONAL MATH CURRICULUM and it cannot represent one side or the other in the Math Wars. Your approach is a good one, Mr. Dewey, because it is a blend, but you might need a bit more field experience before deducing general principles. I'm not being patronizing or condescending here, so pls don't take it that way. Radical solutions from either camp can be dismissed but how we combine the best of traditional and reformed approaches is not so obvious.
1. More hands-on in lower grades (assign any label you want!) with teachers who are committed to this and properly trained
2. Gradual development of abstract formulations of patterns with algebraic representations starting much earlier in Middle School than is the norm in the USA and ONE of the reasons why we lag behind other nations.
3. More challenging applied problems for the hs students instead of merely rehashing formulas and doing the standard problems
FINALLY, SOME APHORISMS (mock me if you will!):
4. Despite 'cutting the corners' for polygons, THERE ARE NO SHORTCUTS FOR DEVELOPING A PROFOUND UNDERSTANDING OF FUNDAMENTAL MATHEMATICS! Here's what I tell my students and they don't think the reference to their grades is amusing:
The only shortcut in math is from 'A' to 'F'! [ok, you can groan loudly now!]
HAPPY HOLIDAYS AGAIN AND BEST WISHES. WHEN YOU'RE LOOKING FOR A POSITION, LET ME KNOW!!
Dave M

Statement to National Math Panel

The following is excerpted from a statement sent to the National Math Panel on 9-1-06:


Comments to National Math Panel
Cambridge MA Sessions 9-13-06, 9-14-06

From: Dave Marain, Supervisor of Mathematics

Although I would prefer to be present and read this statement personally, the risk of not getting on after making a trip of several hours and the limitation to 5 minutes makes it somewhat prohibitive.

As I’ve noted previously, the limited opportunity for concerned educators and others to express their sentiments balanced against 30-45 minute presentations for textbook publishers and ‘established’ organizations does not send an encouraging message to those who feel the outcomes from this panel are predetermined. I, for one, am more optimistic than that, but the proof will be in the black and white recommendations.

From discussing this with the secondary math teachers in my department, with many concerned parents and with students over some time now, there is broad consensus on the following points:

* Problem: For some time now, we have observed and endured students’ deficiencies in arithmetic and their impact on the ability to handle algebraic processes, comprehend the rules of algebra and retain algebraic skills and concepts. Our educators see glaring deficiencies in students’ understanding of fractions and ratios. This is not acceptable.

Recommendation: Students should master the facts of arithmetic and develop proficiency with fractions, decimals and percents WITHOUT the use of the calculator. Using the calculator to promote conceptual understanding and solve real-world problems with ‘messy’ decimals or irrationals is however strongly recommended, We believe this should COMPLEMENT the mastery of arithmetic skills – no more than that.

* Problem: Algebra for All in 8th grade? The problem is that Algebra in New Jersey is not exactly the same as Algebra in New York, Algebra in California, and Algebra here in Cambridge! Were it not so deleterious to our children’s development of mathematics, it would be almost ludicrous to consider that textbook publishers are developing state-customized textbooks for Algebra 1 and other math courses. Although the differences are minor, they are nevertheless a reflection of a serious disconnect between what ALL of our students need and the need for publishers to meet the needs of individual education departments of 50 states. Why does it appear so obvious to our educators that it is insanity to have 50 different sets of state math assessments times several grade levels, yet it seems perfectly natural to governors and state commissioners of education. Testing companies are reaping the benefits of this, but are our children? Ironically, testing companies appear to be having difficulty keeping up with the demand for quality assessments. What’s wrong with this picture?

Recommendations:
* First we have to make sure that we have ‘Arithmetic for All’ in K-7! Our consensus here is that the concepts and skills from arithmetic and prealgebra must be far more standardized than they are now. The ONLY way to insure equity for all is to standardize the curriculum and set the bar higher than it is now.
* Instead of developing massive texts containing beautiful pictures and wonderful applications to every vocation and applications that address every states’ requirements, we strongly believe that the time has come to step back and demand that essential content be given the highest priority. We feel that each of you on this panel needs to ask yourself the following question: How have our esteemed national math and science groups responded to Bill Schmidt’s concerns over a decade ago about a curriculum that is too broad and too shallow? Translating a text from Singapore is not the answer. We need to take the best knowledge we currently have about how children develop mathematical understanding and balance that with the skills needed for those ideas to take root and have meaning. How many of you on this panel have observed numerous math lessons in this country that reflect students’ ‘profound understanding of fundamental mathematics’. We can all cite a few instances for sure, particularly if you are personally working on such a project. We’re talking however about more than a small handful of classrooms. Why are Japanese students in an 8th grade classroom spending an entire class period tackling sophisticated problems that require analysis, conceptual understanding and skill? In fact, we believe that this kind of activity is precisely what enables a child to develop that profound understanding. Isn’t it all about the kinds of questions we ask and the questions we generate and encourage from our students? Isn’t it all about setting the bar higher? We believe it is. Problem-solving however should only be part of the picture. One cannot solve a problem without the proper tools. We do not believe that the majority of our children are currently provided with those tools. This must change!

Thank you for the opportunity to share our concerns. We fervently hope that the Panel will respond to these concerns and make the bold recommendations needed for our children to survive and compete in the 21st century. We await your response…

A Reply to Susan's Response

I won't quote Susan Ohanian without her permission but my comments below were in response to her comments to me yesterday:

Susan,
Yes, there other camps as there are several political parties but there are the two major ones. From my communications with those who are opposed to national standards as I believe you are, their major fear/concern is that the standards will be shaped by individuals who do not understand the needs of children, individuals who will bring us back to the Dark Ages. I certainly believe that such conservative individuals are out there and they’re not all from CA and TX! However, I have been calling for both National Math Standards and a balance between procedural and conceptual understanding for the past 20 years. I’d like you to read my statements to the National Math Panel at some point.

I knew that in the absence of clearly defined objectives with abundant examples, the NCTM Standards would be misunderstood or, worse, subverted. This is exactly what has happened. Way before NCLB, states like NJ where I reside decided to have Core Curriculum Content Standards. As one of its architects, I fought to keep the verbiage simple, concise and crystal clear so that curriculum leaders and classroom teachers would not have to guess what was meant by an objective or expectation. BUT the standards we and other states produced were just as ambiguous as NCTM’s. Teachers did not and still do not know exactly what is to be covered at each grade level or course. It was left to the authors and textbook publishers to interpret and I believe they have made serious errors. Forget the testing frenzy for the moment. For 50 states to have 50 sets of noncongruent standards (of course they overlap), and textbook companies trying to include some or all of this to make everyone happy is insane. This is what happens when we don’t have a coherent national vision for mathematics. NCLB had nothing to do with this nightmare since this started in the early 90’s in many states. Testing evolved naturally as states needed to measure how the standards were delivered. Little thought was given to the deleterious effects of testing kids to death, particularly those students who already have 2 strikes against them and/or are emotionally fragile. But some form of assessment has always been necessary and always will. Teachers need to know what has been learned and how to help those who are struggling. Districts, states and federal governments need to have measurable results to determine policies and funding, although we know how political and dangerous this can be.
Yes, I knew you were not opposed to goals! But I used that term euphemistically for National Standards (and I do suspect that is anathema to you). If the groups developing these standards are ‘balanced’ and I do keep very close tabs on the Panel and the American Diploma Project group, then there is hope. I see some good people on these committees who share my beliefs in a balanced view of curriculum. The outcomes are not yet predetermined despite the politics (no I’m not naive!).
Thanks you for your quick response.
Dave Marain

An Open Letter to Susan Ohanian

A comment I sent to Susan Ohanian:

Susan,
I plan on expounding on the following comments in greater detail in my soon to be created blog and I invite you to visit it and comment often when it's ready...
I've only read some of your thoughts and feelings regarding standards, testing, NCLB, etc. I admire your passion and courage to take on the establishment and challenge the current 'de-constructive' movement as you might perceive it. I agree wholeheartedly that there is now a testing mania in our country that can potentially do more harm than good. Note that if I weren't a centrist I would have phrased that differently! [I bet you’re reacting negatively to my self-characterization since you may see only 2 camps here in this 'holy war’.] You're probably trying to read between my lines and predict whether I am your friend or foe. I hope we can be friends and enjoy our similarities and differences. I know we will disagree on my next few comments but I need to say them to you with the same emotion and passion you exhibit. Ok, here goes...
First, I do not see it as inconsistent that one can have reasonable clear goals for students and still nurture and allow children to develop in their own unique fashion. Since math is my specialty, I will use ladders with its rungs as my metaphor for the acquisition of mathematical skills and concepts. There are different math ladders to climb for the different parts of the whole of mathematics and these ladders are in fact dependent upon each other, but each ladder must be climbed rung by rung. You can;t get to the 5th rung from the bottom if the 2nd, 3rd, and 4th rungs are missing, This is the nature of mathematical knowledge as I see it. Now each child can make it to the next rung in a myriad of ways but she still needs to get there if she wants to continue climbing and eventually move on to more sophisticated math ladders that are even steeper and with more rungs. A child can climb math ladders at her own pace, stopping along the way or even needing to return to lower rungs or starting all over again to regain her strength.
This is how I view the need for math standards for bands of grade levels and ultimately for specific math courses at the high school. From my perspective, the mathematical exposure of a student sitting in classroom X in district Y should be approximately the same if that same student moved to classroom Z in district W. To me, this is a no-brainer because it's about content. No one tells me HOW I must teach my Advanced Placement students in Calculus. The College Board does insist however that my students must be exposed to the same core content -- the main ideas and principles of Calculus -- as every other student of AP Calculus. Never once in over 3 decades have I ever felt constrained by this or by the test. My creativity is not restricted, nor do I expect all my students to solve problems the same way. The dialogue in the class is fruitful and thought-provoking. I don't race through the content to finish ahead of everyone else because I understand the nature of how children learn. I believe with conviction that less is truly more when it comes to helping students develop conceptual understanding, but they still need to know their trig identities and the unit circle 'cold'!

In summary, for math at least, the journeys may be different but there are certain destinations one must have in K-8. After that, there can be many different destinations!

Again, good luck with your mission...
Sincerely,
Dave Marain