kitchen table math, the sequel: Gone Fishin'

Thursday, August 13, 2009

Gone Fishin'

Catherine, that is. She's on vacation for a week.

Will the rest of us will hold the fort while she's gone, or throw a rowdy party? :)


Beth said...

I've been reading this blog with great interest, and I would like to contribute my thoughts about math education.

Constructivist approach:

at best, helps kids understand the motivation for math; not just how, but why

at worst, results in confusion and a hollowed-out curriculum

Traditionalist approach:

at best, results in solid mastery of mathematical principles

at worst, dry and unmotivated; kids can crank through algorithms but don't understand what they're for or how they should be applied

I'd be happy to send my kids to a school that followed either of these philosophies really well. What I often see in practice, though, is just plain bad teaching that devolves to the worst-case result no matter what the original philosophy.

My daughter had a traditional-type teacher who went so fast and put so much pressure on the class that she wound up seriously depressed and anxious, and not learning much. Now she goes to a different school that follows a more constructivist approach, and actually I think she has benefited from having more of the "why" questions addressed. On the other hand, I worry about whether she's getting a good foundation in the formal aspects of math. I tried a little Singapore math with her and it's clear that their curriculum is much more sophisticated than what she's done in either school.

SteveH said...

"I'd be happy to send my kids to a school that followed either of these philosophies really well."

These approaches are not equal, but opposite. They are also not mutually exclusive. You have to look at the details.

You can always improve understanding if you slow down the coverage or spiral through the material many times - at least you should be able to do that. As for understanding, there are many levels, from the concrete picture and manipulable level to the abstract, identity and proof level. Most constructive approaches fail to advance beyond conceptual understandings of pictures and manipulables. They can't make the transition to algebra. While things might seem nice in K-6, trouble appears in 7th grade when students have to really understand what the identities mean and how to manipulate algebraic expressions.

In fact, most schools make an abrupt change in 7th grade to using more traditional math textbooks, with teacher lectures and larger homework sets. My son's school finally (!) got rid of CMP and started using the Glencoe Pre-Algebra and Algebra textbooks for 7th and 8th grades. This was required to properly prepare kids for geometry in 9th grade and the calculus track.

Many schools start tracking kids in math in 7th and 8th grades based on a test they give in 6th grade. The better students use more rigorous and traditional textbooks and the rest (unfortunately in many cases) get funneled into math tracks to nowhere.

We've talked about things like the Socratic Method and Harkness tables as high expectation discovery or constructive approaches to education, but the goal should always be to get from point A to point B in terms of content knowledge and mastery of skills. Too often, constructive approaches see the process as the goal itself. It might be fun or interesting (to the teacher), but the students never get to point B.

Algebra in 8th grade is a very reasonable goal for most students without having to resort to speed, pressure, or lack of understanding. For those students who need more time, the alternative should not be a math track to nowhere.

Allison said...

Beth, I want to emphasize what Steve said about how these methods are not mutually exclusive, at least for the positive elements of constructivism.

If by "constructivist", you mean: some student-based-instructor-led-inquiry, some spiral learning where you deepen your curricula on the next iteration, or some hands-on-problem solving, then the truth is, good constructivism REQUIRES mastery.

It simply doesn't work without it. The only way to get anything out of the inquiry method or deeper problem solving is to have mastered the underlying mechanics previously.

If you don't, then constructivism is nothing more than math appreciation--not math. A "math appreciation" class doesn't allow you to do a single calculation, but it might make you mistakenly think you know some math, because you see Fibonacci numbers in flowers.

Likewise, any really good traditional math class DOES do problem solving and at more advanced levels does do interesting problem solving, because doing real math--proving mathematical theorems are true--is one of the most constructive things a person can do! You simply start from first principles, and try to construct an argument out of thin air.

So when you say "I'd love to see either philosophy followed well but I don't", that's really because you CAN'T do either philosophy really well without the other! That's why "all you see" are bad approaches--because it's not possible to do math teaching well without both pieces.

Now, at worse, the constructivsts don't even believe in teacher-guided inquiry, because at worst, they don't really believe in helping students reach the truth. And the main problem is that their ideology so trumps the teaching of math that they refuse to allow their students to gain mastery. Their students aren't allowed to ever get enough practice to gain anything at all out of the constructivist garbage put in front of them.

Allison said...

Now, to your daughter's experience:

I caution you that you may have been misled that your daughter may have benefited from having more "why" questions asked.

That's because it feels good to hear the answer to a why, and feel "oh, that makes sense".

It's like eating a cookie. Tastes great right then and there. Feels good. It's rewarding.

But like that cookie, there's no substance to build on. The equivalent to eating a real meal means would mean learning how to DO a problem from beginning to end. The feeling "oh, that makes sense" might temporarily increase confidence, but it undermines the later work, when you try to actually DO something and find you HAVE NO IDEA HOW to do it.

There are lots of times that students say "I understand the material, but I can't do the problems." They are mistaken. They DO NOT understand the material at all, but they feel that they do, because that "made sense" feeling was there when that why stuff got asked.

In truth, there is nothing in the Singapore Math curriculum that is sophisticated. It's actually one of the clearest, simplest curricula out there. It just builds on itself over and over and actually gets somewhere. That your daughter can't do it based on what she's been getting at school is a sign that what's at school is a disaster.

Beth said...

Allison, I'm happy to say that my daughter can do the Singapore math, at least the little we've looked at so far. When I say it's more sophisticated, I just mean it contained ideas she hadn't seen before. But she caught on to the new ideas quickly with a little help from me.

This whole discussion has made me determined to work through more of the Singapore curriculum with her.

I'd like to clarify that part of my daughter's experience was that the traditional approach can be really terrible in the hands of a bad teacher. She had a teacher whose method was pretty much, "Memorize this algorithm. Go home and do 35 practice exercises. Now memorize this algorithm.", etc. The material was going by way too fast for her, and she really couldn't see any overall purpose to it. My daughter needs to see at least a glimpse of the big picture first before she can make sense of the details, and she wasn't getting that at all. Again, a really good teacher might have been able to make this work for her, but we didn't have a chance to find out.

I agree with your point that the goal is to use both constructive and traditional techniques to work toward mastery.

Allison said...

I also need to see the big picture before I really get something.

But constructivism per se doesn't give the big picture. It can actively deny it to the students by forcing them into "student led insights" and guess what? If the student can't guess their way to the right insight, then they still won't see the big picture.

That's because it's hard to figure math on one's own. There's a reason it took 10,000 years from the development of stable human agrarian culture to Euclid and our first comprehensive knowledge of geometry, and why it was still almost 2000 more years until someone invents calculus. People who need to see the big picture are a lot better off learning from their elders than trying to discover it on their own.

But this comment thread again shows the disadvantage parents are in when they use the words the ed school world uses. What is meant by "constructivism" is a moving target, and inquiry-led interactive whatevers aren't any more clear.

There is nothing untraditional about saying "here's the big picture". It's quite traditional. It just requires someone who actually KNOWS it in the first place, can articulate it clearly, and a textbook to back it up. (and there you have strike three for most math classrooms today.)

Constructivism doesn't claim it's going to give the big picture at all. Unless it's coupled with an explicit idea like modeling instruction, where all of the handouts, lecture notes, labs, examples, and assignments are driving exactly toward an explicitly stated big picture, then your daughter is just going to have to get lucky, hoping to wander down some paths in a maze, and figure out the big picture for yourself.

You're right--there are terrible teachers out there using traditional methods too. That's the state of the system, regardless of curricula: there are precious few good math teachers out there in elementary or middle school math today. But there is hope!

Prof. Wu at UC Berkeley is now teaching a series of courses in math education, which is now at minor there. He's writing a textbook too, and then there will at least be a way to help math teachers become capable of seeing the big picture themselves.

SteveH said...

I find it interesting that at my son's school everything changes in 7th grade when they start to track in math. Now that they have gotten rid of CMP and are using real math textbooks, gone are the simple notions about discovery and understanding. Schools might still talk about discovery or different learning styles, but the reality of high school looms.

Prof. Wu has a great paper called "From arithmetic to algebra" that shows the non-linear jump in knowledge and skills that many schools require to go from the fuzzy math ideas and expectations of K-5 to the more rigorous expectations of algebra in 8th or 9th grade. Schools usually make this steep or discontinuous jump by giving a placement test in 6th grade. Prof. Wu says:

"Implicit curricular message - If students cannot negotiate the
steep climb to algebra in grade 8, that is their problem!"

As I've said before, if you wait long enough, you can always blame it on the kids, parents, or society. Even the kids will blame themselves. They will say that they hate math or they are "just not good in math".

When kids get to high school and struggle with algebra, schools look to remediation (it's the student's fault) rather than go back and fix the lower grades. In our town, there seems like a curriculum wall between high school and our K-8 schools. Seventh and eighth grades are a non-linear transition that will make or break the future educational success of the students. As Prof. Wu says, the implicit school message is that if some kids can do it, then it's not our problem. At my son's middle school, they talk about having the students take on more responsibility for their own learning. This comes after years of fuzzy, developmentally-appropriate talk and low expectations.

Schools might still talk about understanding and discovery, but underneath it all, the math classes (on the top track) are completely different. Reality pushes down from high school, but K-6 schools are not getting the message. That's why I say that this is not about understanding or discovery, but about low expectations. Only those kids who are very bright or have a lot of support at home can make the transition.

SteveH said...

"I also need to see the big picture before I really get something."

There are different types of big pictures, just as there are different levels of understanding. Most kids understand the idea and need for multiplication - the big picture, but what they might want is a little better understanding of the particular algorithm they are using.

That's why many reform math curricula tackle this by teaching the partial products method. It's easier to see what's happening. This is good, but that's where they drop the ball. They don't require mastery of the procedure because they think that's what calculators are for, but then they don't use the power of the calculator to do more complex problems. The calculator is used as an avoidance tool rather than a magnifier tool.

Also, they don't use the partial products method to motivate the understanding of the traditional multiplication algorithm with carrying. Everyday Math jumps off to teach the "lattice method" with little or no justification or understanding of the big picture.

What is a big picture for multiplication that is appropriate for K-6 kids? It's not an algebraic one. It's not a base (octal, binary) one. You could try to get the kids to understand that for multiplication, you have to multiply every digit of the first number with every digit of the second number. Then it's a matter of getting the numbers in the right column and adding them together. You could then ask the kids to design their own algorithm for multiplication. What do they get instead; the lattice method with no justification.

As Allison said in another thread, it would be nice to do what they proclaim to do, but do it properly. That would be a way to show the details of what is going on (or not). What I see, however, is that not only are they bad at achieving mastery, they are bad at real understanding. Mastery with little understanding is workable; understanding with little mastery is math appreciation - nowhere.

palisadesk said...

In our town, there seems like a curriculum wall between high school and our K-8 schools.

That is certainly true in our whole district (several hundred schools).
The rationale is hard to figure out, because the same curriculum pooh-bahs are in charge of both elementary and secondary math programming. You would think they would be addressing the problem.

Unlike your school, we have no tracking at all in K-8 math, and not usually in 9th grade either. However, it is in 9th grade that the excreta hits the air circulation device. The math chairperson from the local secondary school came to one of our professional activity days and described what can only be called a crisis situration -- kids (who have passed every year and are not Sp. Ed) entering with math skills so low they don't register on any of the tests the high school gives. "We can sort students into groups for instructional purposes at grade levels like sixth, seventh, eighth, even fourth-fifth -- usually those are Sped kids. But we have no way to calibrate students at levels below fourth grade, and those are students we are seeing in ever greater numbers." He went on to describe how even with intensive remediation those kids could not possibly get even one math credit for secondary school graduation. They need to cover 4-6 years work in math before they can get credit at a 9th grade level, let alone beyond (I think 3 math credits are required). They are already dropouts, even if they stay in school and keep the seat warm. They will not earn a diploma, merely a certificate of attendance.

Even our "A" students do poorly on the math test at the end of 9th grade (which does sort students into levels to separate the college-bound from the workplace or junior-college-bound, and the ones aiming for science or math fields from those planning on liberal arts or service careers). He said of the students in the advanced math classes in 9th gerade, only 40% passed the exam -- which covers middle school pre-algebra skills.
Only 1% of the ones in the regular or remedial math classes pass it. They have a high failure rate in the 9th grade math classes because they do not track them and students can enroll in the advanced class regardless of previous achievement, but of course the unoprepared will flunk. He said parents often had no awareness that their kid was so poorly prepared, and insisted on the advanced classes -- at least in the beginning.

This chap said they are usually able to steer students into a more appropriate class by the second semester of 9th grade, but some kids keep trying to pass a too-difficult class for several semesters; they get tutoring, but are way too far behind to catch up..

I could see that this man saw the issue as a crisis, but most of my colleagues did not appear to see it the same way. They see themselves as teaching the required curriculum well and demanding high levels of performance from the students (which they do -- but the curriculum itself is seriously flawed, hence the poor results in the high school). Indeed, I don't see what the K-8 teachers could do much differently, given the imperatives they have to follow. Without changes in curriculum the changes in student results in secondary math will not be forthcoming.

The disconnect between grades 8 and 9 is like some kind of void separating parallel universes.

Anonymous said...

At least you have a wall where for some kids there is a high school math track that is designed to get them ready for college.

In Georgia with its new state wide mandate to offer only integrated math in high school, the wall is disappearing so that only math appreciation courses will be available in most districts. They can in fact remake high school math to support the lack of foundational work that came before. When the kids inevitably struggle, it is simply credited to the "high" standards adopted instead of the lack of explicit instruction, example filled textbooks, or procedural skills.

Be glad if your district still offers solid high school math classes even if they have to fill a lot of K-8 holes to do well.

SteveH said...

"In Georgia with its new state wide mandate to offer only integrated math in high school..."

Only?!? Do they have a list of authorized math programs somewhere online? Is there one that is supposed to support AP calculus?

Allison said...

Anon in GA,

Do you have some links to their integrated curriculum, or news reports on it? Anything? I'd like to put it up on the main page so more folks will see it.

Anonymous said...

The best place to find the integrated curriculum for high school is at the site maintained by the State Department of Education.

Georgia talks a lot about its Performance Standards. The high school math standards can be found at . As you will see there are no more Algebra or Geometry courses. Everything is integrated and everyone is expected to learn the same math concepts. The only question is how quickly.

The real curriculum and the heart of what is being pushed into high school math classrooms statewide is found in the Instructional Frameworks. It is here you will find the recommended activities to illustrate the math concepts mentioned in the standards.

For example in the Math 2 Curriculum Map to teach Quadratic Functions, you have the Henley Chocolates, Protein Bar Toss, and Paula's Peaches Learning Tasks described.

See .

These activities are "designed to allow students to build their own algebraic understanding through exploration" .

Georgia's important even to those living elsewhere because it is being held up at national education conferences as a model to emulate.

SteveH said...

I went to the Georgia state DOE site and saw their frameworks and some of the discussions of quadratic functions. But I also picked a high school at random and saw traditional-looking AP classes and IB programs. It seems to me that if a student doesn't get on the top track, then they get condemned to the slow, integrated track of minimal expectations. You're stuck in the peloton with nobody willing to help you bridge the gap back to the leaders.

Anonymous said...

This is the 2nd year of the rollout into high school so some 10th graders and all 11th and 12th graders are on the old track.

Everyone in 9th grade and younger is now in the new integrated math track.

Does the instruction described on the DOE site look likely to get anyone ready for Calculus, AP Chemistry, Physics, or the SAT? It's hard to see much of a realistic future for these advanced courses.

Anonymous said...

This thread just reiterates an earlier thread that public school is daycare while parents work and afterschooling is now required.

Allison said...


For that level of crisis, can curricula alone account for it, in your opinion? Or at least, assuming a majority of teachers are well meaning but not driven to do stealth DI-like instruction, certeris paribus, can the curricula alone account for it?

What is the curricula your k-8uses?

Is there any chance a better implemention of the same curricula would achieve better results, or does that question not really mean anything?

Are the teachers required to give nothing outside of it? how restricted are their lessons?

How are the students in middle school graded?

palisadesk said...

I'll try posting this in two parts -- Blogger is giving me grief, saying I have too many characters (even though I was 1000 characters under the limit.)

For that level of crisis, can curricula alone account for it, in your opinion?

Of course curriculum cannot be the only factor but it is a major one. Other factors impacting academic results include inadequate instructional time (200 minutes/week),student mobility, absenteeism, absence of math remediation via resource room or math support teacher, district-wide poor math preparation/knowledge on the part of middle school teachers.

What is the curricula your k-uses?

It is standards-based in the usual way. The "standards" for each grade are fairly woolly and emphasize "concepts." Students are not expected to learn math facts or standard algorithms to automaticity.

We do not have a text-based math curriculum (as in Catherine's district, "we write our own curricula"), because the district frowns on the use of textbooks. Most teachers cobble together what they can using bits and pieces of texts they have scrounged, photocopies of pages from resources they buy at teachers' stores, and so on. Many of us, myself included, subscribe to sites like, or where you can download and print (sometimes customize) resources in a variety of subjects.

The district requires teachers to teach three "strands" of math every reporting period, which usually means about three weeks per strand (each strand incorporates multiple topics).

This makes teaching to mastery very difficult because limited reteaching or review can be done: at the end of the allotted time teachers must move on to the next strand.

palisadesk said...

I think the latter -- the curriculum is so loosey-goosey that the phrase "better implementation" doesn't really apply.

Are the teachers required to give nothing outside of it? how restricted are their lessons?

If teachers are being monitored, they are reprimanded for going outside the specs of the curriculum documents. We have "math coaches" who come around and require teachers to do constructivist-type activities that are supposed to enhance student engagement and develop "higher order thinking" but which minimize actual math skills.

At the school level, if the administration backs what you are doing you can get away with subtly flouting the district feel-good mandates; however, if the admin wants the latest fads implemented, you are [bleep] outta luck. Some teachers have a lot of leeway because the admin leaves them alone -- usually because they have no discipline problems and stay below the radar (don't complain, show up for supervision duty, don't ask for money). While this permits individuals to be effective, it does not allow for any large-scale effective teaching initiatives. People have to be covert about what works.

How are the students in middle school graded?

Grading is supposed to be done on a rubric basis, with standard achievement levels from 1-4. 3 is "meeting expectations" and 4 is "exceeding expectations" but does not mean "above grade level" (students who are achieving well are not supposed to be "advanced" i.e. given work from higher grades, but rather "enriched" -- given additional topics, say, topology or set theory.

The high-SES areas in the district do better, but there is a booming private-sector market to provide the tutoring and afterschooling required. One of my co-workers moonlights in a well-to-do neighborhood as a math tutor (he has no special skills in math, but is smart and probably a good tutor) for about $75/hr, a little above the going rate for a private, non-specialist tutor. A local DI tutoring outfit charges closer to $100/hour, while Sylvan and other franchises are in the range of $60/hr for group instruction.

To turn things around would require: a better curriculum, with more clearly designed sequences and achievement targets, a tiered system of assessment and instruction, with some flexibility in allocation of instructional time, better teacher training and assignment overall, and more access to basic resources (not just manipulatives, but quality textbooks, computerized support materials, possibly media resources).

I think all the research I have seen points to time on task, quality of instruction/engagement, and effective use of massed and distributed practice as key components. My district engages more in "math appreciation" (love that term, will be using it) than math instruction. Without a basic change in orientation, I don't see how results will improve much for students who depend on in-school instruction to prepare them for math in secondary school and beyond.

Catherine Johnson said...

Will the rest of us will hold the fort while she's gone, or throw a rowdy party? :)

I'm laughing -- I had an impulse, which I squashed immediately, to put up a post saying, "I'm on vacation - be good!"

I may have to come back & delete this Comment later on when I've had a chance to consider just how mortified I am by that confession.