A Reply to John Derbyshire

The following is a comment on a posting by John Derbyshire entitled The Dream Palace of Educational Theorists.. Derbyshire seems to relish being a provocateur, assailing the bastions of American public education (not to mention almost every other institution along the way!). He is also an exceptional writer and has written highly regarded popularizations of serious mathematics. I'm a believer in listening to all sides, no matter how extreme and then making reasoned judgments of my own.

John, Since I referred to myself as an iconoclast before finding your fascinating blog, there may be some common ground here. I'll leave it for you to discover the rest if you choose to visit my blog... The particular post in the link above addresses a critical piece of the 'big picture' of the expression of intelligence that has been systematically (intentional use of that term) overlooked. Here are a couple of excerpts I'd like you to consider: This is an expansion of comments I made in Richard Colvin's excellent Early Stories blog... The Carnegie Report, Starting Points, alluded to by Richard Colvin stressed the importance of nurturance and stimulation of the infant in the first 6 months to 1 year - a lot of prewiring and hardwiring of neural pathways is taking place in that first year and the simple verbal and non-verbal interaction between mommy or daddy with baby appears to be more critical than any Baby Einstein tape! Further, the first year is surely the most critical for developing a sense of stability, security and trusting one's environment - just feeling 'safe!' Picture an infant removed from the primary caregiver and institutionalized or placed with different caregivers during that first year and how terrifying that can be to that child. This is a commonplace occurrence for the thousands of babies born crack-cocaine addicted each year whose mothers are unable to care for their children. The bonding process with the mother or grandmother or one primary caregiver is so vital for the later psychosocial development of the child. John, I absolutely accept the importance of genetics, but not its preeminence over all other factors. If generations of a societal group have been systematically deprived of the physical and emotional necessities for proper brain development, it is poor science (not to mention discriminatory) for anyone to conclude anything about group inferiority. Paul Tough made it clear that these children need significant advantages to compensate for their initial start in life. This goes way beyond HeadStart programs, selfless dedicated teachers and unique programs. Neither you nor anyone else in my limited readings is willing to tackle the pernicious effects of poverty and deprivation during the first 3 years of life. Trying to reverse the damage that takes place in this critical period may be insurmountable, but we don't really know, do we? It hasn't been done yet. Other nations provide far more during the prenatal and postnatal period than we do. Again, I do not discount the significance of heritable factors. However, how would you have turned out if your mother was a crack addict and you were born drug addicted and suffered violent withdrawal symptoms for the first 12 months of life. How would your genetics have overcome this? Having swaddled a dozen or more of these infants in the past 25 years provides a limited but telling empirical base for my own theories. BTW, I think I'm also enjoying the advantages of Senior Tourette's to which you referred in an interview! It's delightful to not give a **** about what anyone thinks of me!

A Reply to Michael Paul Goldenberg re Proof

Apologies to you, Mike...
I was really replying to Anita’s post, however, I was indirectly referring to your comments. Back in the 90’s I bought into the NCTM argument that as long as we ask students to explain their reasoning and continued to ask those ‘Why does it have to be a rectangle?’ types of questions, this would be just fine for conveying the meaning behind those theorems and being able to apply the principles of Euclidean geometry. I watched as more and more geometry teachers moved away from formalism and, even though something bothered me in my gut, I said, ok, fine. But then I noticed some teachers who didn’t compromise and some of these were young teachers who felt the necessity to include some formal proofs, keeping them short (under 8 steps). When I asked why they were doing this even though 2-column proofs were optional, they simply replied, “Are you telling me, Dave, I’m not allowed to!” Of course they were and they argued that it was hypocritical of me to be calling for higher standards while not advocating we continue to include this type of rigor. They felt that, despite the fact that there were opportunities for proof in other subjects, geometry lent itself to developing logical arguments and 2-column proofs provided a structure that paragraph proofs could not. I attempted to refute that by saying some students will find it easier to write a paragraph proof than the structured kind, but they replied that these students were few and far between, since the teaching of writing a coherent paragraph is enough of a challenge for the English teachers!

Since I was trained to be rigorous in math (pronounced, ‘rigormathis’ I believe) it has always been difficult for me to abandon this upbringing, however I did compromise back then. Now, I believe I should have held firmly to my principles. Mathematics is one of the most beautiful structures ever devised, a product of human curiosity and invention but held together in place by the mortar of deductive logic (ok, that metaphor was way overblown, but it’s still XMAS and I felt inspired!). Mathematicians search for patterns and attempt to make beautiful generalizations (inductive reasoning) but they then must prove these conjectures most often using the power of logic and deductive reasoning. Perhaps there is a compromise here but again I will reiterate that famous quote: THERE IS NO ROYAL ROAD TO MATHEMATICS! Oh, and yes, I’ve taught proof in number theory and loved every minute of it, particularly when achieving that aha! reaction. But the students were more mature and they were not ready to venture on their own with these proofs for awhile, although they could begin to imitate Proof by Mathematical Induction (which actually uses deductive methods!). I think you would argue that younger students could explain their reasoning fairly well if given the right models and opportunities, but that isn’t the same as what I’m talking about.

Reply to Euclidean geometry/Proof Question on Math Talk

The following is a reply to a question posed on the excellent math discussion group Math Talk on Yahoo groups.

Anita,
Here’s my view. Pls don’t be offended by its didactic nature, but I need to use formal logic to defend the need to teach formal logic! I know you and Mike already know all of this but I need to state the obvious anyway.
Mathematical reasoning and the structure of mathematics depends to a large degree on DEDUCTIVE REASONING. As mathematicians, we either accept a statement as true (definition or postulate) or we PROVE it is true, most often by deduction. The ‘statement-reason’ proof format was designed, IMO, to help students develop a structured approach to logic and deductive reasoning. The key to this structure is what it means for a conditional ‘if-then’ statement to be true, since every theorem can be stated as a conditional or biconditional. This is profound and requires development ONE STEP AT A TIME.

Consider the conditional
If P, then Q. Every application of this conditional MUST formally have a structure similar to:

1. P is TRUE. Given
2. Therefore Q is TRUE. Reason: If P is TRUE, then Q is (also) TRUE.

This is a variation on the 3-part law of logic entitled ‘modus ponens’ or the Law of Detachment which used to be taught in a geometry course. No, Mike, teaching syllogisms didn’t cause brain damage! Complex multi-step proofs are just compound applications of this basic argument form. This can also be taught using symbolic logic and I’ve seen those curricula too, replete with truth tables! Ok, I’m not proposing revisiting the 60’s!

For example, suppose we want to prove that, in triangle CAB, Angle A is congr to Angle B and we’re given that side CA is congr to side CB. The proof then has the basic structure outlined above (the spacing may be off here):

1. Side CA congr to side CB. Reason: Given (assertion)
2. Therefore, Angle A congr to Angle B. Reason: If two sides of a triangle are congruent, then the angles opposite these sides are congruent.

A ten-step proof is of course nothing more than a chain of structures like this.

The fact that so many students struggle with PROOFS (therefore we should avoid teaching them!) and can’t seem to supply much more than the Given and the Conclusion is that most humans cannot build a FIVE-step structure before developing MANY MANY two, three and four statement arguments! Call this the Piagetian or the van Hiele Model or just plain commonsense! There are no shortcuts to this development. Either we teach this structure or we don’t. I propose that it is still important for all students to be able to make a valid argument using sound reasoning. The 2-column method is not the only way to do this but it wasn’t arrived at whimsically or to abuse young minds despite contrary views! Certainly, any college prep curriculum should continue to expect this, but I would argue that simple proofs are accessible to most students. I completely agree with you, Anita, that the message has gone out to geometry teachers that they can deemphasize formalism. I strongly disagree with this message. As far as the argument that proof is not inherent to geometry courses, of course that is true. But Euclidean geometry was built on deductive reasoning and I just don’t buy that most algebra (or anywhere else for that matter) teachers will infuse this kind of reasoning into the curriculum!

For more comments on this and other related matters of math curriculum, I invite you to read my blog at
http://www.MathNotations.blogspot.com. I will probably post this entire reply there. I hope this isn’t poor protocol, Mike. Feel free to delete it if it is.
Dave Marain

An Angry Reponse to rightwingprof et al on edspresso

The following was posted as a comment on edspresso's 'having a laugh'link:

Why the mean-spirited ad hominem arguments? There's enough vitriol to go around to poison any meaningful debate but I'll take the high road. The real issues are not political. While all the 'experts' and amateurs bluster to hear themselves heard and gain their 15 minutes of fame, the true victims of this rhetoric are our most precious national commodity.

Let's decide on a set of skills and concepts to which ALL children must be exposed and get on with it. Those who have been on the front lines as long as I have know what works for children and what they need. They also know everyone else out there are phonies looking to make a name for themselves and cash in on the MathWars, whose only real casualties are another generation of children. Children are being taught to look for the ''easy way', for the 'short-cuts to learning'.

If you folks just don't get it, then get the **** out of the way. There will be a revolt and the charlatans and hypocrites ON BOTH SIDES who have perpetrated this mess we're in will be trampled upon and left in the dust. HOW DARE THESE PHONIES TAMPER WITH MY CHILDREN'S EDUCATION!

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

Response to Joanne Jacobs and John Dewey...

Imagine the following activity, Joanne. You guess the grade where this was actually demonstrated. Also, how would you characterize this lesson? Traditional? Reform? Constructivist? Discovery-based? Active learning? Ah, labels!

Children working in pairs draw, with a ruler: a triangle, quadrilateral, pentagon and hexagon.
The instructor tells each group to take their scissors and cut the three corners of the triangle and rearrange the pieces to form a straight angle. The teacher is demonstrating this on the overhead with color transparency film or with an opaque projector. This activity is repeated for the other figures, except the teacher doesn't tell them how many straight angles can be formed. She tells them to make as many as possible with the pieces, but pieces cannot be used more than once. The children are told to record their findings in a data table, which is also demonstrated by the teacher.
# of sides # of vertices (corners) # of straight angles formed
[Note: This will not appear spaced correctly]
3 3 1
4 4 ??
5 5 ??
6 6 ??

The teacher is circulating, assessing, guiding, asking many questions... She asks the students to predict how many straight angles could be formed from cutting the vertices of a decagon. She doesn't tell them what a decagon is -- someone in the class will make an educated guess and she'll guide them by relating the prefix deca- to common words.
She will then ask each group to formulate a rule in words for the relationship between the numbers of sides and the number of straight angles. After 10-15 minutes, groups volunteer to discuss and display their findings. The teacher is asking many questions of varied levels of taxonomy.

Later she has the children combine pairs of straight angles to form 'circles' and begins to formulate the numerical version of this important rule.

So, what grade level? Could it be introduced as early as 4th or 5th? Should all children have had similar experiences BEFORE taking high school geometry? Is this lesson far too time-consuming just to get at a simple algebraic formula 180(n - 2)? Personally, I have seen lessons like these at the middle school level, but, in other countries, younger children (4th grade) have these kinds of experiences. My Korean students told me so! If you would like to read further views like this, pls visit my blog (http://www.MathNotations.blogspot.com). I will repeat this comment there and expand on it...

Ok, I'm expanding! Just to invite any reader to offer useful lesson plans like this and make them accessible to many. Yes, there are many web sites for lesson plans, study guides, submitting lessons (and getting paid for them), etc. For math, we need to have common standards for each grade for ALL our students and to support these goals with effective lessons that develop conceptual thinking, communication and get kids excited about learning. Discovery-based lessons like the one described above are NOT the goal five days a week. That's obvious. But why we do we see fewer and fewer of these kinds of lessons in higher grades. Is it because most educators and administrators would view this as appropriate only for younger learners? Do they continually echo the refrain, "In the upper grades, there's just SOOOO much more material to cover -- theres' no time for 'fun and games'... " Now what did the TIMSS study reveal and suggest about math lessons in our country?? Could he have been suggesting that 'LESS IS MORE'!?! Will anyone out there ever get it? National Math Panel, ARE YOU LISTENING?? By the way, does everyone remember that I am not a reformist, not a traditionalist, just an iconoclast, who expects to be ignored in my own land... I detest labels! HAPPY HOLIDAYS - PEACE IN OUR TIME...

Joanne, I enjoy reading your excellent edublog as much as anyone's. You cut through the b******* and make sense. I hope you will direct your many readers here as well.
Dave Marain

Thoughts on Stay-at-Home Moms vs Preschool

This is an expansion of comments I made in Richard Colvin's excellent Early Stories blog...
The Carnegie Report, Starting Points, alluded to by Richard Colvin stressed the importance of nurturance and stimulation of the infant in the first 6 months to 1 year - a lot of prewiring and hardwiring of neural pathways is taking place in that first year and the simple verbal and non-verbal interaction between mommy or daddy with baby appears to be more critical than any Baby Einstein tape!

Further, the first year is surely the most critical for developing a sense of stability, security and trusting one's environment - just feeling 'safe!' Picture an infant removed from the primary caregiver and institutionalized or placed with different caregivers during that first year and how terrifying that can be to that child. This is a commonplace occurrence for the thousands of babies born crack-cocaine addicted each year whose mothers are unable to care for their children. The bonding process with the mother or grandmother or one primary caregiver is so vital for the later psychosocial development of the child.

However, my feeling is that the issue for 2-4 year olds is somewhat different. If there is an appropriate blend of social and intellectual activity occurring in a nurturing preschool environment, this may be highly effective and difficult to simulate at home. The Canadian study seems to have arrived at a fairly obvious conclusion since most stay-at-home moms by necessity are multitasking - not that there's anything wrong with that! I think the argument is similar to the homeschooling debate for school-age children. To simulate social interactions at home, one would need a daily playgroup of similar-age children. The academic stimulation would require a mom or dad who has the appropriate background which usually requires training and early childhood certification. Some could do this, but for underprivileged children whose moms or grandmoms are just trying to survive economically, this would be very difficult.... All of this again underscores the critical importance of providing the necessary funding for the early years. This means that our society has to come to believe that this is just as important or even more important than K-12 funding. Are we there yet? Don't think so! Those who are more privileged will usually have the funds to invest in the preschool years and therefore may not see the urgent need for tax money to be directed towards urban areas and those children who need it most. It's all about social conscience, but it's also about recognizing the disastrous consequences of ignoring these children.

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