## A few days ago I posted on MathNotations a comment I had left on Edspresso, a reply to Barry Garelick's essay, Living in a Post-National Math Panel World.

## My post was entitled, Balancing the Equation.

*From edspresso:*

*Barry Garelick is an analyst for the U.S. Environmental Protection Agency in Washington, D.C. He is a national advisor to NYC HOLD, an education advocacy organization that addresses mathematics education in schools throughout the United States.*

## I just replied to Mr. Garelick's reply to my original comment. Unfortunately, I received the following automatic response from edspresso:

## Your comment submission failed for the following reasons: In an effort to curb malicious comment posting by abusive users, I've enabled a feature that requires a weblog commenter to wait a short amount of time before being able to post again. Please try to post your comment again in a short while. Thanks for your patience.

I assume I will be able to re-post that comment in another day or so, but I thought MathNotations' readers may want to see a preview.

Here is my original comment to Mr. Garelick's post and his recent reply:

"There are also teachers who maintain a truly balanced approach and who, while rejecting the discovery-oriented and textbook-less programs being foisted on schools across the country, are admonished by their administrators to do as they are told."

Although now retired, I was one of these educators for the past several decades.I believe the Panel paid lip service to these educators. Mr. Garelick, just what benefit does this report have for this group of math teachers? There are many dedicated professionals who have always balanced the need for 'correct answers' with conceptual understanding. Educators who always knew that there must be mastery of essentials before one can move on in mathematics. Educators who continue to find creative ways to satisfy their administration and their personal integrity...

The problem is that it is just not easy to blend skill practice, mastery and rich problem-solving experiences and explorations when one has to essentially create one's own materials. Particularly when the rewards for going 'above and beyond' are purely intrisic in the teaching profession. Experienced math teachers know that computational proficiency is absolutely essential but, when confronted with problems that are not formulaic and require recognition of essential concepts and making connections, many of our students flounder. Yes, it is really hard to do the right thing, isn't it?

In your opinion how will textbook publishers respond to the Panel's report? IMO, skills-based texts that neglect exploration and more challenging problem-solving would be just as damaging to this next generation as many of the reform texts have been to the current generation. Perhaps such 'skills' texts will not be the response to the Panel's report from textbook publishers. Perhaps...

But that's ok, the most dedicated of our profession will compensate for whatever materials they are handed. They'll continue to write their own and do what's right, just as they always have.

Dave Marain

MathNotations

Posted by: Dave Marain | April 9, 2008 08:17 PM

Mr. Garelick's Reply:

Mr. Marain,

Thank you very much for your comment. I've seen your writing on MathNotations and am glad you wrote.

I am aware that there are people who hold that "problems which are not formulaic" are not well-addressed by teaching students the components of math and algebra delineated in the NMP's report. Such problems are the so-called "messy" problems that have a range of answers or are open-ended, and so forth. Problems such as the "work" problems and others in math textbooks are held in disdain and thought not to lead to problem solving skills. Proper presentation of the solution of say, work problems, however, opens the door to "rate problems" in general, and which generalize to the solution of a great many problems in engineering and science. In fact, many of the standard so-called "formulaic" problems in algebra and other math classes are widely generalizable and have their purpose as I can attest as one who majored in math and work in a field that requires knowledge of scientific and engineering principles.

Providing students the opportunity to solve non-formulaic problems does not in and of itself prepare them to solve problems. Analytic and procedural skills and knowledge of form, which generalize do in fact provide such preparation. I tend to think the term "balanced approach" is one that is not well defined. I used the term "true balanced approach" in my essay, meaning an opportunity for student-centered instruction (such as discovery) that makes use of prior knowledge, rather than the melange of "just in time" skills, procedures and concepts that some teachers, textbook writers and policy makers seem to think students will discover because they need them to solve a problem.

It is my hope that those teachers who use textbooks that are written topics presented logically, sequentially, with expectation of mastery, and which builds upon concepts, will not be punished for doing so. Vern Williams who I quote in the essay is one of those teachers. He gives students very tough "out of the box" problems that are not in the textbook necessarily, but he makes sure they have the requisite skills and information (which he imparts via instruction) before giving them such problems.

Posted by: Barry Garelick | April 14, 2008 11:06 AM

Here is my latest reply of 4-18-08 which was temporarily rejected:

Dear Mr. Garelick,

Your quote:

"I am aware that there are people who hold that "problems which are not formulaic" are not well-addressed by teaching students the components of math and algebra delineated in the NMP's report."

My quote:

"The problem is that it is just not easy to blend skill practice, mastery and rich problem-solving experiences and explorations when one has to essentially create one's own materials..."

I never suggested that direct instruction of powerful models such as rate problems is not desirable, nor do I hold these traditional rate problems in disdain. In fact, in my classroom, I always used the same RATE X TIME = DISTANCE chart that I learned in high school, several decades ago. The emphasis on my blog is on rich open-ended investigations that one normally does not see in most textbooks. Problems that teachers have to search for in most cases. Most of my regular readers know that success with these problems require a strong base of skill and knowledge of traditional algorithms. I've stated this repeatedly.

Consider what happened when the AP Calculus Exam changed dramatically a few years ago. The tried-and-true methods still were needed and still were assessed. But the test changed dramatically from an emphasis on technique to an emphasis on application and deeper understanding of fundamental concepts. The decline in scores was predictable and occurred until textbooks and instruction were revised to address this shift in focus. The most capable students were able to 'generalize' from formulaic problems - they always will. However, the scores on the AB Exam initially reflected that many others could not. It has been my experience that students perform well on non-routine problems when they have that strong base of skill AND experience with many nonroutine problems that require more than a superficial understanding of content. When presented with such problems, students need TIME TO EXPLORE. Direct instruction initially will usually fail in this case. After giving a reasonable amount of time to grapple with the problem and discuss it, the solution is explained clearly and thoroughly, with alternate methods presented if time permits. One cannot rush this process, that's why it is called a process...

That's my definition of 'balance.' There's nothing fuzzy about it. I've never suggested that students be given challenging problems and left entirely to their own devices to invent something out of nothing. I have suggested they need to know their trig values, their trig identities, their differentiation and integration formulas well. After they have demonstrated this KNOWLEDGE, they can be challenged with problems they have never seen before. Exploration does not mean one is "a blind man, searching in the dark for a black cat." (Luv that Escalante quote!). Exploration means that one is PREPARED to explore and then given the OPPORTUNITY to EXPLORE!

Me. Williams and I share many common beliefs about teaching. But he doesn't represent even a majority of math teachers out there. This is why I repeatedly emailed the Panel asking for more frontline teacher representation on the NMP (and, of course, I was politely dismissed). There was not a single high school math teacher represented, not to mention an underrepresentation of research mathematicians.

I appreciate your thoughtful reply, but you attempted to pigeonhole my beliefs. No one yet has been able to do that and I'm afraid you did not succeed either! I welcome an opportunity to discuss this further with you, perhaps in another venue. How about a debate on my blog? We'll probably discover we have far more in common than one might think.

Oh well, nothing has changed. Does anyone out there really get my balanced perspective, other than my faithful readers of course!

## 31 comments:

IMO, skills-based texts that neglect exploration and more challenging problem-solving would be just as damaging to this next generation as many of the reform texts have been to the current generation. Perhaps such 'skills' texts will not be the response to the Panel's report from textbook publishers. Perhaps...But that's ok, the most dedicated of our profession will compensate for whatever materials they are handed. They'll continue to write their own and do what's right, just as they always have.Oh, man. What a great quote. I feel a post coming on.

Thanks, Dave.

My apologies if I pigeon-holed you. I think the report will serve teachers like you and Vern Williams well. My point is there are interpretations of "balance" that I don't agree with, and from what you have written, you probably would not either.

I too would have liked more research mathematicians and teachers on the panel, but all in all, it was a fairly good group.

No apology needed, Barry, but it is appreciated.

I've been thinking about the best way of conveying my approach to you and others. What makes it difficult is that one really needs to be in the classroom to observe this in action. If I could get clearance from parents, I would video a lesson demonstrating some of my techniques -- that is a possibility.

However, here is one way of looking at it --

Your argument (and I'm paraphrasing) is that if students receive a strong grounding in well-established methods and skills, they should be able to apply this to other types of problems. Here is what I believe from working with ALL ability levels of students for over 35 years, including the most mathematically talented. To borrow from mathematical logic, the conditions you have stated are NECESSARY BUT NOT SUFFICIENT, except for the very few who can take the leap by themselves and make the connections. From my personal experience, most students learn how to do nonroutine problem by doing many nonroutine problems! They are then developing a set of tools from which they can draw upon and apply and generalize.

There is another extreMely important piece here. most children do not develop Liping Ma's "profound understanding of fundamental mathematics" without being given the opportunity to explore, make sense of the mathematics for themselves and ultimately internalize the concepts. Again, knowledge of facts and traditional algorithms are NECESSARY for this to occur, but are NOT SUFFICIENT. Helping children conceptualize by creating an environment of research and experimentation in the classroom may be the greatest challenge for the mathematics teacher. Asking effective questions, leading students to a deeper understanding and reaching that "Aha!" moment (different moments for different children) is immensely difficult but the most rewarding aspect of teaching. Direct instruction must therefore be complemented with teaching conceptually. From what methods book do preservice teachers learn this? What student text can you name develops both sides of the brain as I'm describing?

I would love to continue this discussion. Let me know...

"Does anyone out there really get my balanced perspective, other than my faithful readers of course!"

Not unless you provide a lot more details. Problem solving from a foundation of mastered skills is what I expect, but you don't get something for nothing, and this isn't the problem in schools.

The problem in schools is that they want to approach math from the top-down. They assume that mastery of basic skills will evolve out of that. One indicator of this is the appearance of two equation and two unknown problems that kids have to solve using guess and check.

Everyday Math is used by millions and is based on the idea that mastery is not necessary at any particular point in time. You seem to be talking about some sort of ideal, but most of us are hoping for common sense and more than a little bit of rigor.

Besides, nobody in their right mind can be against "balance". I've been in parent-teacher meetings when the talk turns to some vague idea of balance. Everyone nods in agreement and then the school continues to decide on all of the details.

A top-down approach leaves many kids unable to achieve the "NECESSARY" condition, even though their talk is full of "problem solving" and "understanding". Many, like myself, are struggling with getting the process flipped around to a bottom up one. A bottom-up approach that focuses on mastery might be weak on the "SUFFICIENT" condition, but at least it gets to "NECESSARY". But you have to be careful when you talk about "problem solving" because the term includes a lot of baggage.

If you do build problem solving on top of mastered skills, then how much time do you take away from advancing to new material to spend on problem solving, and what type of problem solving are you talking about. You will have to write it all down in detail before I can make any comment. I might like some of the problems, but I might not like the slow pace. I think the panel properly adressed a lot of content and mastery issues, but they left a lot open for optimization. Have at it.

"Asking effective questions, leading students to a deeper understanding and reaching that "Aha!" moment ..."

All good teachers try to lead kids down the primrose path, so to speak. It's nice to achieve the light bulb moment, but it's not a necessary condition and you can waste a lot of time trying to achieve it, especially if the basic mastery "NECESSARY" condition hasn't been met yet.

It also comes in various forms. I've had quite special moments when I was doing homework or was directly taught. I distinctly remember in high school calculus class when I was directly taught that integration gave you the area under the curve! I didn't have to discover it myself. I remember it after 40 years. I also remember learning (with homework) about the wonders of line integrals.

The devil is in the details.

Steve H--

I've read your insightful and passionate comments many times both here (in your exchanges with mathmom) and over at KTM and elsewhere. I don't believe that we are really at cross purposes. I certainly do not advocate a 'top-down' approach to learning. I've always believed as you and and many others believe, that mastery of any skill comes from clear instruction and repeated practice of that specific skill.

However, for as long as I've been in education, there has been an implict assumption that most children are not able to think conceptually and abstract bigger ideas until much later on in school, until they have a rock-solid foundation in essential knowledge and skill. It's as if there is a belief that, at some magic age, the brain develops the requisite number of neural pathways for conceptual abstract thinking to take place, and a switch can then be turned on. I do not share this belief.

Here is what I have OBSERVED, Steve:

If a child is given only low-level cognitive tasks, they will develop cognitively at a slower rate than when the child is given a variety of cognitive activities of different levels of difficulty. I had the pleasure to be a Staff Developer in K-5 classrooms in the early 90's. I can still recall the stimulation provided for these young children that incorporated both traditional learning and higher-order thinking. Children were expected to learn their basics but were also given time to work with pattern blocks and pentominoes.

In a 3rd grade class, I recall a teacher writing a long column addition problem on the board consisting of a string of NINE 15's. She asked the children to find at least 4 ways of doing the addition. At first, many of the children looked puzzled, since they had already practiced 2-digit addition problems and knew the 'method.' I recall that, after a few minutes, a child came up to the board and wrote a column of nine 10's which she 'knew' added up to 90, since they had practiced counting by 10's. Then she wrote a column of nine 5's which she said added up to 45, since they had been practicing their fives times tables recently. She then added the results together, 90+45, the traditional way. Now you and Barry might argue that this young lady was only able to do this because she had a strong foundation of skill and knowledge and I would say this was absolutely NECESSARY! But what if the class as a whole had not been given the opportunity to tackle this question and had not been exposed to this particular approach. Do you really believe this is a waste of time or that textbooks generally include these kinds of questions in the body of homework exercises? The question asked by the teacher created cognitive dissonance or disequilibrium (sorry for the fancy jargon but it makes sense to me here) that caused students to leave their comfort zone of safe methods. The teacher's questions and subsequent behaviors lead to deeper thought on her students' part. While expecting her students to know their basic facts, she then went beyond by challenging them to think about numbers and operations in other ways, before providing direct instruction for computing 15x9 = 135. Which she did...

The problem is that there simply isn't a single text that does both well. Everyday Math and similar curricula do not provide sufficient mastery practice to go along with their higher-order activities. Although the developers talked about the importance of automaticity of facts, there was an assumption that playing daily games like Multiplication Baseball, could replace the drill and practice needed to store the knowledge in long-term memory. We know that for most children this simply will not happen. However, Steve, our dedicated math teachers want to challenge even our youngest children with questions, puzzles and activities that do more than just develop automaticity. Where do they get these materials? What most of our dedicated and experienced teachers do is to use worksheets for skills and other resources for 'thinking'. This is probably why I should stop writing this blog and develop at least a unit for 3rd, 4th or 5th grade that incorporates the best of both approaches. Of course, I expect others are already doing it or perhaps these materials already exist. I doubt it though, since many keep searching for extra challenges for their children and land on my blog...

Listen, I know you will dissect my arguments and we'll end up going in circles just like in the past. But the dialogue is healthy as long as we don't take it personally. Progress will ultimately evolve from these debates.

Well, my long post got lost in the ozone. I suppose it doesn't matter.

You remind me of those who talk about constructivism in an idealized sense, when in reality, it's used to hide all sorts of bad curricula and teaching. We will continue to go in circles until you can be more specific about your ideas. I'll take Singapore Math with its "Challenging Word Problems" supplements. Show me something better. This can't be just a unit. Anyone can write a great unit. You have to have a curriculum that makes sure that the "NECESSARY" job gets done.

I don't know that you are really speaking to very many on this. I think, essentially, the minority that has classroom experience and has a somewhat strong grasp of elementary mathematics is it.

And in the end, we are the ones who will be stuck teaching what the balance of forces amongst the two sides dictates.

During 1999-2001 I was able to drive a worse than typical constructivist program out of my (large) district.

We'll see what the back to basics folks throw at us.

Back to content, I can't imagine cognitively low level stuff for the first N years, followed by enriched content after. It only makes sense to weave them together.

So we need a curriculum that is arranged in a logical way, a developmental way. And we need the number of topics pared down, way down. We need to have room, to have time, to explore. And we need to enrich as we go, not only after some (unclear) point.

Steve, your discussion of, if I am reading correctly, "mastery first," seems to miss this. It reads like a reaction to the constructivists that is now aimed at all who don't agree with you. And it just won't do to lump Dave into the same group.

Jonathan

I appreciate that, Jonathan. I do expect mastery, but not 'mastery first' to the exclusion of higher-order thinking introduced at an early age. I might not have made that clear. The mastery first approach is just as 'top-down' as what Barry and Steve find so distasteful in reform math.

The NMP Report and the central theme of NCTM's Curriculum Focal Points is to reduce topics and sharpen our focus. This is very encouraging provided it leaves more time for critical thinking and exploration. I really like the 'enrich as we go' phrase, Jonathan.

"Steve, your discussion of, if I am reading correctly, "mastery first," seems to miss this. It reads like a reaction to the constructivists that is now aimed at all who don't agree with you."

I expect mastery. I don't care how you do it but you better get there. The problem right now is that schools don't get there. They hide behind a lot of fancy talk of understanding and problem solving.

Mastery first doesn't mean no enrichment. It means that you don't use a time-wasting constructivist process to achieve mastery. You practice skills and then you apply them, not the other way around.

"The mastery first approach is just as 'top-down' as what Barry and Steve find so distasteful in reform math."

Oh please. Show me the details. Do a comparison with Singapore Math. Do you think it's just a mastery first curriculum? My spin meter is turning very fast.

This is what you said above about 'top-down'.

"I certainly do not advocate a 'top-down' approach to learning. I've always believed as you and and many others believe, that mastery of any skill comes from clear instruction and repeated practice of that specific skill."

You redefine 'top-down' to spin what I'm saying.

"I really like the 'enrich as we go' phrase, Jonathan."

Let's see the details. Let's see how you get the job done. You seem to be carving out a position that you can't clearly define. It sounds like a turf issue. You're not one side or the other. You're better. Let's see the details.

Actually, Steve, I've demonstrated dozens of conceptually-based lessons on this blog right down to some of the questioning techniques an instructor could employ. I can do that because I was a professional educator who developed these ideas over many years and implemented them in my classes for all ability levels.

I think the burden now is on you to demonstrate a lesson that develops conceptual understanding. It's easier to provide direct instruction of procedures and the teaching of basic facts. I want you to show us how you would give the opportunity for children in grades 3-6 to make sense of numbers and operations. Include the questions, the activity or whatever you feel would be needed to enable children to develop some big idea of mathematics, say, multiplication by powers of 10.

By the way, you might want to take a look at videos of 8th grade Japanese math lessons from the TIMSS Video files, wherein children devote an entire lesson to tackling ONE challenging nonroutine math problem, with strategic guidance from the instructor. These children have developed skills for sure, but those developing the curriculum there know that tackling difficult problems requires time, discussion, some frustration and a highly trained educator who knows intervention and questioning techniques.

"I want you to show us how you would give the opportunity for children in grades 3-6 to make sense of numbers and operations."

As I said before, look at Singapore Math and their Challenging Word Problem supplements. The onus is on you to show how your ideas compare with this well-known standard. I think it does a great job of teaching mastery and problem solving. You obviously think you have a better way.

But you also have to show how you are going to get the NECESSARY mastery part done. You can't just focus on enrichment examples and say that mastery is no problem. You need to show the complete process and number of classes for a unit; mastery and problem solving. Compare them with specific Singapore Math grades and units so everyone can judge for themselves. If you can top Singapore Math, then many people will beat a path to your door.

let's see if i can offend everyone.

if i understand what's going on here

at all, it's something like this.

steveh wants textbooks

& curriculum design to be

so well designed that

teachers could just

*follow the script* and win.

he hasn't got any problem with

teachers being made to use

methods dictated by administrators

(and textbook publishers etc. --

curriculum designers, let's say),

just wants the administrators

(etc.) somehow to wake up and adopt

*his* (or singapore's) methods.

dave, on the other hand,

pretty clearly seems to feel

that math teachers actually

might know more about teaching

than administrators or publishers.

and wants *less* pushing around

from the boss so we can do the job.

so they're at cross purposes.

and should change the subject,

or, better still, *clarify*

what the heck they're talking about

(which, on my view, is the struggle

for power ...).

"we"?

If only teachers were given the freedom, then everything would be fine? Dream on. If you think this means I support administrators or textbook designers (as a class) you would be wrong again. Is that clear enough?

Why do you bother using "anonymous"?

As for Singapore Math, it's low on the "follow the script" scale and may not be good for typical K-6 teachers. Saxon would be better for that, or maybe enVision Math. Everyday Math is very scripted, but it doesn't get the job done. However, given the state of teacher preparation, having no script sounds very scary.

Most K-6 curricula are script-like because many of these teachers don't have a clue. In our state, teachers have to have certification in a subject starting in seventh grade. They normally use traditional textbooks and are given much more leeway in how they enrich the material. This can be good or it can be bad. Our school used to use CMP. You could call that scripted enrichment with low expectations. I'm definitely more in favor of scripting in the early grades because of the lack of teachers who know math. In the later grades, I would give more leeway to the teachers, but that doesn't mean that they should get to design their own curriculum. If you've got a lousy administration, go teach somewhere else.

As for clarity, that's the purpose of using Singapore Math as a basis of comparison. I haven't figured out yet what is so unique about Dave's approach.

As for power (money), it should be given to the parents.

"we" teachers.

i sure don't mean you, steveh.

i bother "using anonymous"

because it's less trouble

than signing in (or appending

my chop). why do you bother

signing in?

so. you're ticked about math ed.

who among us is not?

should that exempt you from

even *trying* to figure out

what somebody's getting at

before you just jump up

on your hobbyhorse and start

screeching incoherent abuse?

i notice that you don't even bother

to deny that your message is that

teachers need to be pushed around

*better* (not less). thanks for that.

by the way. anybody but an infant

ought to have figured out that

power is never "given" to anyone.

only taken from (the cold dead hands

of) the recently powerful.

for some reason, you've chosen

a pretty powerless group -- teachers --

to try and seize some from.

ho-hum.

seems to me like what little power

*i've* got in this sorry state

of affairs -- the power to talk

face-to-face with students and

influence 'em to the best

of my ability -- i've got only

because hardly anybody *wants* it.

you'd gladly see me lose it,

i suppose. but i'll bet *you*

don't want it ...

signing off. sorry to've

flamewarred all over your blog, dave.

the entity known far and wide

to various internet backwaters as

vlorbik

"i sure don't mean you, steveh."

I see you hit the flameout button.

I taught algebra for years (and other math courses) to students who never mastered the basics. I am also the father of an 11 year-old who has suffered through years of MathLand and Everyday Math. He has great math scores and the school doesn't (want to) know that I've been supplementing for years with Singapore Math.

The problems facing a teacher don't define the problems of education. You have a very self-centered and self-righteous view of the problem.

As far as the issue with Dave, I suspect that we would agree on many things, but he is making a distinction I can't see. I don't see how it's different from a common approach that introduces a new topic, practices standard solution methods, and then discusses how these techniques fit into a larger view of math.

This last part is often missing because many (especially K-6) teachers don't have the ability, they don't have the time, or they have to fix problems left over from other years. Math then becomes a trade-off. Do you use your time just to enforce mastery of the basics or do you try to add enrichment and hope that mastery will somehow happen? A lot depends on exactly which skills are required and what level of mastery is enough.

Vlorbik--

I forgive you for ranting. I do it too - passion is sometimes hard to restrain!

Steve, I believe you and I have a great deal in common and the distinctions are subtle. Singapore Math is excellent and it continues to evolve (new editions, revisions). There appears to be a greater attempt to reduce some of the basics and bring in more critical thinking and enrichment, consistent with some of the most recent recommendations from NMP and NCTM. Very encouraging...

I know the investigations I am writing are just another resource. I know they are effective because, when I was in the classroom, I used these frequently with classes of various ability levels. Further, wonderful educators such as Mathmom and Jonathan have been kind enough to use them in the classroom and have posted students' reactions and how the lessons played out. I've received many helpful suggestions from many others and words of encouragement, even from parents whose children are currently using Singapore Math!

All I've ever wanted is a balance between teaching procedures and developing conceptual understanding. The latter is very difficult to do, but I've never stopped trying. Many students have come back to tell me that, because of the kinds of questions I asked and the activities I used in class, they were able to grapple with the higher challenges at the undergraduate level. There are many other educators who do the same as I've done. I respect them and hopefully they respect me...

I see the issues as very basic at this point. I've dealt with the frustrations of not being able to teach what I want because some or many of the students needed remediating. This hurts the better students, but I had to make a choice. A focus on mastery always won out, but procedural understanding does not have to be devoid of understansing. I'm sure I see more connections between the two than you do, and the necessary part has to get done.

There are other issues too, like why are kids entering sixth grade not knowing their times table? If a teacher has to spend all of his/her time on mastery, then where does that leave the better prepared kids. This happened to my son in fifth grade Everyday Math. The teacher (good for her) even pushed an after-school session for the kids needing extra help. In spite of that, she told me that she had no time for extras. By the end of the year, she only covered 65% of the material. This was no urban school! This was not good for my son, but it was better for those kids who lacked basic mastery.

One parent took it in his own hands and held after school math enrichment classes. But what many of the kids needed at this point was a faster pace rather than enrichment of material they easily understood.

Why did these kids get to fifth grade not quickly knowing 7 + 8 = 15? It's not just the curriculum and it's not just the administration. In my son's current school, there are top seniority teachers who should have retired long ago. The school can't get rid of them. Parents talk about "lost" years. There are lots of problems. Teachers see a one-instant snapshot of the problem, but parents see the longitudinal or historical problems.

Steve, you and I

bothsee how procedural and conceptual understanding can and should be linked. Children do not make connections by accident, however. The key is the kinds of questions the instructor asks at each stage. The 'what do you do next' level of question is necessary, but, again, not sufficient, to deepen understanding. I typically asked questions that made students challenge their beliefs and even their intuition. The art of asking questions is not easily learned or mastered. Most of us develop this skill over the course of a career. This is why I sometimes model the kinds of questions one could ask (note my non-scripting phrasing there).You and I share many of the same frustrations as Mr. Williams has, the lone teacher from the NMP. However, I have never lost my deep abiding respect for the members of my profession. I know how hard their job is and I celebrate them every day. Anything I can do to help them, I will. I don't dictate style or methods. I only suggest a direction they could take.

Your frustration is shared by me both as a parent and an educator. I see the more capable being held back and I see how many move on from grade to grade without learning the fundamentals of mathematics. BUT for years I also observed the glaring lack of conceptual understanding of mathematics in our students even they had mastered the basics. Yes, Steve, one can teach BOTH procedural knowledge and concept at the same time, but it's darn hard. Ask any teacher!

Steve, I started this blog as an outlet for years of frustration and I try to remain positive and uplifting whenever possible. I have always shown you respect, even when the dialog became heated. I hope this is mutual.

Steve,

I just realized I've overlooked a possible important distinction between our views of mathematics. Years ago I began to feel that there are important similarities between the science and math classrooms, i.e., I felt strongly that there was an experimental nature to mathematical learning. I felt there was a strong parallel to be made to the way science instruction balanced delivery of knowledge with experimentation in a lab environment. This is a major part of my belief system and goes beyond involving kids in a math lesson. Steve, think of how Mr. Wizard delivered his 'lessons': He created an environment of discovery, provided the background information the child needed, but then let the child work through the experiment herself. He guided the child toward understanding with beautifully thought-out questions until the child could 'see' the idea for herself!

I have a feeling that this view of learning mathematics may separate us. Does it? This is probably the underlying reason for my explorations. It's not just those Extra for Experts pages that often went overlooked in textbooks.

I'd love to hear from others on this as well. BTW, I'm certainly not suggesting that such lessons can occur ALL of the time, but there must be a place for them IMO.

I think you are glossing over huge problems with mastery in schools to focus on your ideas of understanding. I see schools that don't ensure that the basics get done, schools that don't deal with bad teachers, and schools that talk all about "understanding" even though kids don't know the times table. This is not just a subtle distinction and this is not just the administration. It's the teachers too. I remember one teacher telling us parents how good it was for little Johnnie and Suzie to write an explanation about why 2 + 2 = 4. This is a school that doesn't require mastery of adds and subtracts to 20 until the first part of third grade. Part of this has to do with full inclusion. Many K-6 teachers see education more in social terms than academic terms. That's a big issue in itself. Low expectations.

That's the biggest problem I see in constructivist-type math curricula. The fancy talk of understanding just hides low expectations. When I was in college, the change from slide rulers to calculators made life more difficult. We were expected to do much more complicated problems. In K-6, calculators are used to avoid hard work, not magnify it. With calculators, K-6 classes could tackle much more complex problems. They don't. That's a big sign.

Schools and curricula now make it almost impossible to talk about what understanding really means. But talk of understanding doesn't mean much when so many kids flunk trivial tests. You seem to be up on some lofty cloud while parents are just struggling to get their kids the basics at school.

However, let's assume that a school is not screwing things up too badly with basic mastery. Then there is the trade-off between speed of coverage and enrichment. You can get a lot of understanding while mastery is being achieved. In fact, proper mastery should ensure a certain level of abstract understanding. If you speed up the pace, the understanding level should not go down.

Then there is enrichment, which might involve more wide-open (and time consuming) topics. My position is that this should not be a required part of the curriculum. If a school wanted to add an extra math lab class (just like in science), I would not disagree, but that is no guarantee that the time is well spent, and it has to be above and beyond the regular class time. Our high school has an algebra course with an extra lab class, but that's quite different. It's for fixing K-8 problems.

This is all so vague because nobody defines proper mathematical understanding. However, I look at it from a mastery standpoint. Show me the problems a student can do (including word problems), and I will tell you what level of understanding he/she has. Proof of mastery is tied directly with understanding. Many seem to view mastery as just speed and not understanding. I consider mastery more equivalent to experience, which gives it a much richer connotation.

I was involved as a coach for my son's First Lego League robotics team last year and his Science Olympiad team this year. Both are interesting examples of content, mastery, and enrichment. I have very mixed feelings about the time/benefit payoff for each of these programs.

The fine line is how much leading, prodding, or pushing do parents and coaches have to do. I'm in the camp that thinks that the adults have to provide a lot of help to make the learning and process useful. (This is for middle school, not high school.) I like the structure of the Science Olympiad much more than the robotics. With the robotics, only a few get to do the real work. Their technical report requirments are way too open-ended. For the Science Olympiad, however, my son had to study anatomy; specifically the circulatory system and the brain and senses. It was much more specific and a better learning experience.

Continuing on with the lab idea, I remember my chemistry lab in high school. We did chemistry-like projects for each lab. One was developing black and white film. That was fun, but I didn't really learn any fundamental concepts. We just followed the instructions.

Chemistry always seemed to me like some vague, amorphous topic. There were basics like the periodic table, but I felt that I had very little tangible knowledge and skills that I could hang on to. All I ever remember from the labs is that you had to measure very carefully. That's it.

I saw that with the robotics program. Just do it and you will learn. The coaches are supposed to give guidance, but it was up to the students to figure out how to program the robot. The kids who can figure it out naturally gravitate to that role, and the others, who might like it if given more instruction, are left out. It all looks really good from the outside, but what real learning is happening on the inside?

In a lab environment, everyone would get to do the same task, although most labs I have been in are done in teams and some kids do the work and others just follow along. If Mr. Wizard leads the kids along to discovery, it still doesn't mean that they are discovering anything worthwhile. The process doesn't guarantee success.

In the robotics program, they set it up like real life, with specific tasks and deadlines. Everybody specializes and has their own job and responsibilities. This was not reasonable for the 4th and 5th graders we were working with. I suggested to the school that the First Lego League should be just for the older kids who have some basic skills. The younger kids could start in an after-school robotics club where all kids had a chance to learn basic skills without specific deadlines or goals.

Actually, our team did quite well and everyone thought it was great, but still, some of the kids learned very little about robots or programming. Just because things look good from the outside doesn't mean that everyone is learning what they should.

I liked the Science Olympiad more. There were 15+ topics and project areas that the students could select. You could have one or two kids per group and the task was all theirs. The downside is that much of it is self-learning, and the material is way beyond anything they see in class. It would be better to see the material as an extension of what they see in class. There is also the issue of learning bits of science here and there and nobody to put it into a larger picture.

Some teams at the Science Olympiad seem to run like well-oiled machines. One middle school won the state championships for the fourth year in a row. One parent-helper for our team said that in North Carolina, her old school assigns one teacher or parent to take charge of each task. This could be good or it could be bad.

Our team pretty much let the kids take responsibility for their own tasks, including the responsibility of making sure they followed the task instructions. This isn't necessarily good either. One student was very unhappy on the day of the competition because he didn't read the instructions carefully. Most kids aren't automatically self-motivated learners and need lots of guidance.

I had to monitor my son's progress on his tasks. Sometimes the deadline of the task got in the way of learning. I would want to stop and give him a lot of background knowledge or skills, but there wasn't any time. For the anatomy task, I helped him with reducing the material he had to study into a reasonable amount. The rest was up to him.

After all of that, I would say that I don't see an "experimental nature" need for math. "Need" is the key word. I see the experimental part only as enrichment over and above the regular classroom learning. Experimental is not necessary for understanding. Is this semantics? Perhaps, but I see schools and curricula hide behind this sort of vague talk and never get the basics done. It's as if they think that all you need to do is work on some special thinking skill.

I have a better sense that this is not what you mean, but I'm not sure how your ideas would work in a real curriculum. An experimental approach takes time, and if it's a part of the daily classroom, then I would be concerned that the basics would suffer.

Steve,

Take a look at MathWorld's article on Experimental Mathematics: http://mathworld.wolfram.com/Experimental

Mathematics.html

On a more elementary level, these are the kinds of investigations I write. I certainly do not imagine these would occur more often than once per unit but I see benefit accruing from this without sacrificing mastery of basics. These investigations are more extensive than anything I've seen in Singapore Math. Traditionally, math teachers would see these as something extra to look at if time permitted (which rarely occurred).

As we move toward reducing topics and emphasizing the core, there should now be more time for such 'experimentation'. This is also a natural place for technology of all kinds (including the traditional straightedge and compass constructions).

I've been writing my own own investigations for over 30 years and I've learned a lot. I did not write them exclusively for honors or high ability students. I always included extensions for those youngsters.

Steve, this is not pie-in-the-sky. While I have, like you, bemoaned the erosion of skills over the past 2 decades, I did not abandon my belief that exploratory mathematics deepens understanding as much as if not more than problem-solving. When Reform Calculus became popular in the 90's, I realized that I did not have to adjust very much. I moved into the Demana, Waits, Kennedy text without terror. I still required technical facility from my students, but I managed to cover all of the topics and infuse explorations as well. Can this be done in lower grades? I believe the answer is yes WITHOUT sacrificing skills. In fact, these activities require application of these skills.

"Can this be done in lower grades? I believe the answer is yes WITHOUT sacrificing skills."

I don't mean to be snarky, but there are no skills to sacrifice in the lower grades. They just aren't there to begin with. I've already talked about full-inclusion and how K-6 teachers emphasize the social aspects of education over the academic ones. Many lower grade teachers have a much more vague idea of education than my wife and I.

You're talking as if this is not a huge problem. Many parents might really like your ideas, but it's hard to consider them when the system is so broken. Look at the standardized test scores. What other knowledge or skills makes it OK for kids to really screw up these basic tests?

Most parents like me are just trying to deal with issues like a school not covering 40 percent of what is in the workbooks. I wish the problem was as simple as evaluating the benefits of "experimental" math. I see that as a minor issue for me and most parents.

I was able to get my 6th grade son to skip 6th grade Everyday Math (it's horrible) and move up to 7th grade Pre-Algebra using the new Glencoe text. This text isn't bad, but my son just told me today that the class (the top-level math class) won't finish the text and will continue with it next year! Forget experimental mathematics. Let's talk about the basics. They can't even get through the regular material. At this rate, they might only get through one-half of the algebra text next year. What happens when these students try to jump to geometry in 9th grade?

I'd love to be in a position where experimental mathematics was a realistic topic of discussion. It just isn't, and I don't want it confusing people about the huge problem with mastery of the basics. Experimental math won't speed up or fix this problem.

Ignore SteveH.

I admire his passion, but he's fighting a war of ideas that ended at least a decade ago.

At this point, he's a distraction--a lapel pin.

Following the advice of the other jd, I see 2 questions:

1) How much demand is there 'out there' or could there be for an "enriched traditional" curriculum?

2) How safe is it to assume that the whole world knows what the base traditional curriculum looks like? Is it enough to fill in with enriched activities/problems/chunks (sort of what Dave has been doing)? Or do we also need to describe more basic Stuff, what Steve wants, the day to day end?

"...he's fighting a war of ideas that ended at least a decade ago."

This is completely unsupported.

Did the problem of basic school and teacher competency end a decade ago? Are there no problems now to get in the way of enrichment and higher order thinking, however they are done? Ideas are meaningless if they are not based in reality. Results, not process, is what the last decade has been all about.

Teachers look at workbooks and textbooks and know that there is no way in hell that they can get through all of the necessary material. And all you want to talk about are fresh new ideas as if these problems just don't exist.

"How much demand is there 'out there' or could there be for an "enriched traditional" curriculum?"

I'm not sure what people mean when they talk about a traditional curriculumum. It's mostly used as a strawman to justify almost anything and everything. "Parents just want what they had when they were growing up." Actually, I didn't like many things about my traditional math education, but many of the the modern math curricula I see are going in the wrong direction; low expectations. This is not just about two equal, but opposing views of the same goal. They are two different goals.

"Demand" is an interesting word though. There is a lot of latent parent demand, but few or no choices. Some parents want a fast-paced curriculum that focuses on the basics and problem solving. Others (obviously not me) might like an un-schooling type of approach. Some might like slow-paced, full-inclusion approaches, and some might like grouping by ability. As anyone who has read my postings know, I'm all for parental choice.

That's the big difference here. I say that parents should be given options. Schools say no. A school could set up a faster math track that provided more expectations of mastery and more problem solving or enrichment (however it's done). They won't do that. What do we get? Our school only requires mastery of adds and subtracts to 20 by the beginning of third grade. What does enrichment mean at this level when some kids are ready to go so much faster?

Schools try to cover up for low expectations and a slow pace by talking about enrichment and critical thinking. Meanwhile, their standardized test scores on trivial tests stink.

These ARE old ideas, but few are dealing with them directly.

"I'm not sure what people mean when they talk about a traditional curriculumum."

Then take a very deep breath, and ask.

If you are annoying an anti-constructivist here (and you are, believe me, no one likes getting trolled, least of all me), how will you ever get anyone to listen to you?

Deep breath. Take a break. Try again. If you can respond calmly after a few days break, (and only if), we can answer some questions and discuss.

Jonathan

I've asked lots of questions and have gotten very few answers, or even opinions. At least Dave attempts to make progress.

I know what a traditional math curriculum is. I wasn't asking for a definition. That should have been obvious. But you can't just talk about it as a baseline when so many spin it to their own advantage.

I'll assume that your sarcastic reply was based on your inability to deal with the issues and trade=offs of content, competence, assumptions, goals, expectations, and results that I raised.

You brought up the idea of "demand" and I expanded on it, but you ignored it completely. You just picked on silly semantics to try to get me to go away. Your insincere interest in discussing these topics is quite clear.

After days of successfully avoiding getting drawn in, I finally broke. But I wrote a whole megilla, so I posted it at my own blog instead of ramming it into the comments here...

Your insincere interest in discussing these topics is quite clear.As is yours, SteveH. You're fanning the flames, nothing more.

Get a life. And get real.

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