kitchen table math, the sequel: Ideologically-motivated intellectual gatekeepers

Thursday, December 10, 2009

Ideologically-motivated intellectual gatekeepers

What Climategate and Discovery/Constructivist math have in common.

Narrow intellectual gatekeeping is omnipresent in academia. Want to know why the government wastes hundreds of millions of dollars on math and science programs that never seem to improve the test scores of American students?[3] Part of the reason for this is that today’s K-12 educators—unlike educators in other high-scoring countries of the world—refuse to acknowledge evidence that memorization plays an important role in mastering mathematics. Any proposed program that supports memorization is deemed to be against “creativity” by today’s intellectual gatekeepers in K-12 education, including those behind the Math and Science Partnerships. As one NSF program director told me: “We hear about success stories with practice and repetition-based programs like Kumon Mathematics. But I’ll be frank with you—you’ll never get anything like that funded. We don’t believe in it.” Instead the intellectual leadership in education encourages enormously expensive pimping programs that put America even further behind the international learning curve.

I like it. This will be widely distributed to our local educators.

Perhaps I should picket like this Climategate picketer (I have no idea who he is) at the entrance to NCAR (National Center for Atmospheric Research) in Boulder.

Imagine, that could be me!
Change title to "Discovery Math Protest Longmont, Colorado Day 1


Anonymous said...

The lure of the grants is the common corrupting influence behind both. As a PI said at a national MSP conference last year "there's big money" if you advocate for a certain type of math and science curriculum but not for building a new chemistry lab.

If you believe in something in spite of evidence to the contrary, it's not science.

Anonymous said...

Great find, Chris,

She gets straight to the point and names names.

Some of the comments are quite good, too. Of course, we here at KTM have known this stuff for a long time, but I'm always glad to see it spread.


PhysicistDave said...

Anonymous wrote:
>If you believe in something in spite of evidence to the contrary, it's not science.

I’d amend that to:
>If you believe in something *without* evidence, it's not science.

The problem with Climategate is not that the climate guys are crooks, are ideologically motivated, etc. No doubt some of them are indeed crooks, ideologues, etc., but that is true in all areas of human endeavor.

The idea of science is to rigorously test proposed theories/models with *new* evidence: whether the people proposing those new theories or models are saints or scoundrels does not matter. All that matters is whether or not the new evidence confirms the proposed theories.

If you got your new theory while you were high on LSD, but it is confirmed by (new) evidence, the theory is good. If you are the brightest scientist in the world, and new evidence disproves your theory, you still lose.

The climate guys have tuned their models to historical data (data we now know is flawed) and then announced that their models are “settled science.”

That’s not science at all: the next stage is to make firm, unambiguous, detailed predictions of future climate and wait and see if those predictions are confirmed. If they are not, they have to admit that their theories are in error, according to the rules of science.

They have tried to just skip that whole stage of attempting to refute their theory with later data.

And, when reality started catching up with them, they played the little political games that we now all know about.

(Incidentally, I do know that the "CRU Team" was involved in historical data reconstruction, not the modeling efforts, but it is all ultimately part of the same overall research project.)

I will leave the comparison to educational “research” to others here.


Paul B said...

I have a Wheaten Terrier; highly athletic and very smart. One year we had a skunk problem. As the family protector, he took it upon himself to get the skunk!

At first, he failed. Three times he was sprayed and I spent hours trying to rid him of the skunk scent each time. Then one day I went out on my deck and there he was, standing proud, his bobbed tail nearly touching the back of his head and the dead skunk hanging from his jaws. He smelled like he always does. No skunk spray!

I can only assume, of course, but my bet is that he used his repeated failures as lessons. He banked his experience and built upon it until finally he knew that to get a skunk you do it from the front.

If my wonderful dog had been granted his skunk learning in dog constructivism school, he would have been placed in a protective bubble that would have prevented skunk stench. He would have been trained with my other Wheaten (who is more like a beer barrel and not athletic) in group skunk hunting. They would have progressed in their bubbles at beer barrel speed with a coach on the side, admonishing them to think creatively and speculate about what it would feel like to bag a skunk.

I suspect that shielding my hunter from the hard work that is learning, in an artificially induced creative environment, would not have led to a kill.

If you want to extend your knowledge you have to do it on top of a foundation of internalized, solid, memorized, fact. You can't do it by standing on oatmeal, in a bubble that protects you from failure.

This is as true for people as it is for dogs.

SteveH said...

I studied the GCM models years ago and thought they were pretty crude (but fascinating). I also thought that they were very far away from having enough data for proper validation. Even now, I can't imagine how they can come close to showing any correlation to historical data without a lot of "tuning". The more accurately you discretize the geometry of the system (core, mantle, land, oceans, atmosphere, solar system), the more data you need to get the model off and ruuning, let alone having enough data for historical validation.

Considering the state of ecological correctness, I feel that I need to say that I'm making no comment here about public policy. I will say, however, that our small town now has a windmill committee which wants the town to build a windmill. Somebody wrote a letter to the editor saying that it would be more effective to focus on conservation. Darn. That's not as much fun as selling town tshirts with windmills on them. We can put the modern windmill right next to our windmill from the 1700's. What symbolism.

PhysicistDave said...

SteveH wrote:
>I studied the GCM models years ago and thought they were pretty crude (but fascinating)….
> The more accurately you discretize the geometry of the system (core, mantle, land, oceans, atmosphere, solar system), the more data you need to get the model off and ruuning, let alone having enough data for historical validation.

Oh, yeah, the science is really fascinating: I’ve always thought it would be really cool to work on this. I came close to going to a school where “cloud physics” was the major research field in the physics department (this was back before anyone had considered global warming). Had things turned out slightly differently for me, I might have been one of the climate guys in deep trouble right now, maybe even one of the senior people in deep trouble!

Steve also wrote:
>Considering the state of ecological correctness, I feel that I need to say that I'm making no comment here about public policy.

And, I similarly want to make clear that I am making no statement about whether the globe is or is not warming.

Unfortunately, by jumping to conclusions without following through on the whole scientific method, the climate research establishment “short-circuited” the research process, pretending that very preliminary results were “settled science,” when they were in fact at a fairly early stage of the research process.

They have thereby led to a situation where it is hard for anyone to know whether the globe is really warming, much less reach informed conclusions about public policy.

Rather sad, and I can honestly say, “There but for the grace of God…”

Of course, that does not let them off the hook, legally or morally: it was their job to do this right – professionally and honestly -- and they didn’t.


ChrisA said...

Steve, I was worried a little bit that bringing Climategate into the discussion would sidetrack the issue, but I felt it was worth the risk.

Back to the subject at hand. I did attend my 8th graders geometry (this unit is being team taught with the algebra class) class yesterday. The whole point of the fractions unit is that the teachers are absolutely convinced the students don't know how to apply fractions to the real world. I'm sure there's more then a grain of truth to that belief. The solution, it's the solution that sickens me. It's all about manipulatives. Triangles, squares, hexagons and representing fractions in this way.

Although not a direct analogy, this seems akin to teaching kids skip counting (and I'm sure you could do that with manipulatives somehow!) after they already know how to multiply.

The class has been working on a number of problems in groups for most of the week, with a write-up due today. Some of the problems are good problems, but solutions to manipulatives are tedious, and of course an essay must be written regarding the logic behind each step. I would much rather the teachers show them how to use algebra to solve these problems. That's what they learned algebra for isn't it?

It appears their worry is that algebra is so abstract that the students don't really understand what they are doing. In my mind, that's where the teaching comes in. I would much rather they start putting units into the calculations to help deal with the abstractions then dealing with manipulatives.

Not to mention their were "posters" of problems the kids had worked out. They were related to the Pythagorean theorem. Three were incorrect:

1. 10^2 = sqrt(100) (not quite!)
2. sqrt (225) = 5 (It sure didn't look like a 15, perhaps I was wrong)
3. I don't remember the 3rd at the moment, it will come to me.

'nuff said.

Robin said...


From the very beginning I too saw the analogies between the nonscientific attitudes and the refusal to look at evidence that was inconsistent with desired outcomes.

Is an early article that made me immediately think of inquiry math.

Especially note the last sentence in the article.

Maybe they are so similar because both involve political goals and trying to use science or "best evidence" assertions as cover to avoid any questions.

SteveH said...

I've seen the two main things that Barbara Oakley talks about over and over: lack of content knowledge and ethical issues.

Many teachers and administrators in K-8 simply don't know what I'm talking about. They parrot back the ed school line and claim that many parents just want what they had when they were growing up. I can't make them understand because they don't have the knowledge to understand.

But that's not all there is to it. There is an ethical component. It relates to the comment by the NSF program director; "We don't believe in it." I've never had anyone be so bold in their comments to me, but I have had comments like "Everyday Math is better for our mix of students." They know that rigorous and timely enforcement of the basics is needed, but they just don't do it. They know about the sort of arguments we raise at KTM, but they still fall back to their stale arguments and strawmen. They still try to claim the higher ground of critical thinking and understanding. This takes a certain amount of slippery ethical rationalization.

Paul B said...

I remember reading once about a concept called 'information cascade'. I may have the name wrong but the concept is that information exists in layers, each one built upon the others below it.

When a person jumps into this layer cake they use layers that extend out from their own personal knowledge domain. As they dive down through layers they often reach a place where their own knowledge weaknesses become overwhelmed by 'consensus'. That is to say, as their own expertise weakens they rely on the accumulated and 'accepted' knowledge of experts and their own journey through the murk ends.

The implication of this cascade is that if the low level knowledge becomes perceived as unassailable truth it will seldom be challenged, even by people who should know better. This can lead to all sorts of secondary science that is built upon falsehood.

The raw data underneath much of the AGW theories is thoroughly corrupt. The problem is that to know it, requires painstaking analysis that most people aren't equipped to make. So what happens is that most climate science is being done without turning over the lowest level rocks. The homogenized data produced by a handful of respected organizations is simply accepted as true and the keepers of this knowledge have gotten away with not revealing either the data or their methods of homogenization.

In education, the exact same thing is happening. I can't tell you how many times my quest for why we do certain things has been met with the inevitable, "Well, research shows that... blah, blah, blah."
When I hit this wall I ask for the citation but it never comes. The research is obscured in a fog of consensus. You're just expected to deny what your lyin' eyes are showing you.

If you are lucky enough to find some of this research gold and study it, you'll find that most of it isn't research at all. You won't find control groups. You won't find tabular findings of objective measurements. You won't find scientific methods described in a fashion that would let an experiment be repeated.

You won't find that lowest level of raw data exposed and assigned to experiments that demonstrate the truth of a hypothesis. What you'll find is a lot of opinion wrapped up in the semantics of research with a twist of survey information thrown in to make a chart or two.

We're just supposed to believe that drilling is bad and constructivism is good without a shred of proof to the assertion. The underlying data to support the hypothesis is either non-existent or too corrupt to be exposed to oxygen.

Paul B said...

Here's an interesting hypothesis to challenge the status quo.

In six years of teaching middle school math I have never had poor performance from the students who brought their lunch to school. I theorize that bag lunches create good students.

Do you think I could get funded for a project to study this phenomenon? Do you think I would be allowed to follow this trail wherever it leads?

It wouldn't take much money to prove or disprove my hypothesis. The interesting thing to think about is where you could drive this Trojan horse. What would happen if my correlation was stronger than the signal we eek out of constructivism vs DI?

Redkudu said...

There's a bit about that in "Fast Food Nation" - some schools turned around what they fed kids at lunch.

I've only ever worked in schools where the majority of kids had to eat at school. I've often thought there's an enormous correlation between what schools feed kids and their performance. After sampling the cafeteria food I resolved never to go back. The offerings are high-starch, high-sugar, and high everything that's poor for the body. (Bread, bread, bread, or breaded.)

Even the salad bar is all iceberg lettuce, "meat cubes", processed cheese, enormous croutons, and a little dried out broccoli (which, even if someone has a sudden fit of nutritional interest, is so old and dry it's unpalatable), topped with four varieties of creamy dressings.

Another problem is timing - many of the students must eat either before they catch the bus (6:30-7:30 a.m.) or at school (8-8:20 a.m.). Then they don't "eat" again until 12:30-1 depending on lunch scheduling. Even in the best-case scenario, that's 4 hours without any substantial protein. I know I'm starving by 12, and even eating a handful of almonds between periods helps stave off the craving. By the time kids get to officially "eat" again, they are ravenous, and eat too much of the carb-loaded food they are offered, making them sleepy and sluggish in the afternoons. (If they snack it's usually on chips and candy.)

I'm not sure it's such a Trojan horse, unless I misunderstood your meaning.

Paul B said...

I was thinking more about the connection between the bag lunch as a proxy for parental involvement/care but yours is a valid connection too.

My connection was more tongue in cheek with a bit of 'correlation is not causation' thrown in. Yours is probably better science, connecting proper nutrition to feeding the brain.

Both are reasonable avenues for valid research but neither will pass muster with the 'experts' who are the gatekeepers for such things.

Last week my kids were treated to pizza, peas, and tater tots. They lather up the tots with catchup, trade the pizza, and use the peas for target practice. They feed the kids, some of whom are 200 pounds or so, elementary school portions of junk while fretting over bulletin boards and the furniture arrangements.

My brother in law says that experts are has-beens that leak under pressure. He's more blunt than I like, but often on the money.

SteveH said...

"fog of consensus"

I like that. There is also: 50,000 Frenchmen can't be wrong.

At my son's school, the state mandates that kids get a balanced meal. Kids have to accept it. The balanced part ends up in the trash. Either that, or it gets slathered with creamy ranch dressing.

Catherine Johnson said...

oh my gosh!

I know her!

Remember -- way back when -- she wrote a couple of things for the old ktm!

OK, now I gotta try to round those up.

Catherine Johnson said...

well....I've been feeling overwhelmed by all this lately, possibly because I've begun the process of tackling reading instruction here in my town, God help me

The issue of memorization is fantastically destructive with 'balanced literacy' -- and directly tied to our fantastic levels of overspending on public education.

I've been planning to get a post up about this for 3 months now....

I will do it!

For the moment, the way it works is:

* you teach phonics 'on the side' via 'authentic literature,' i.e. story books that haven't been constructed to give children LOTS of repetition & practice in decoding the letters on the page

* as a result a good number of students never get to the point of simply REMEMBERING that the letter 'b' makes the sound b

* that opens the door to '3 cueing strategy' etc., etc. - guess and check reading. Instead of READING THE WORDS ON THE PAGE, the child uses "cues" to guess (or 'predict') what the word is

* THEN the "literacy specialists" ride in to the rescue: tenured teachers providing "intensive intervention" to kids who never memorized the sound-letter correspondence

What I am seeing is an effort across the country to hire an entire new tier of "specialists," "coaches," "facilitators" etc., whose job it usually is to respond to instructional failures caused by a complete rejection of the very concept of memory, let alone memorization

Public school ideologues scorn memory.

Catherine Johnson said...

That’s not science at all: the next stage is to make firm, unambiguous, detailed predictions of future climate and wait and see if those predictions are confirmed. If they are not, they have to admit that their theories are in error, according to the rules of science.

They have tried to just skip that whole stage of attempting to refute their theory with later data.

Great way of summing it up.

I, too, am not taking a position re: policy.

I am, however, taking a position re: science and the scientific method.

Catherine Johnson said...

btw, I'm seeing no references to 'creativity' at all around here -

I'm hearing "understanding" as the be-all and end-all

Our administrators constantly talk about students 'understanding' -- and, worse yet, they speak of student 'understanding.'

If a student understands, that's all that matters.

I remember encountering this back when C. was in 4th grade (sophomore in high school now).

I had no idea what was going on in the public schools, and I asked C's math teacher about a grade of 72 or something in that realm on a test.

She gave me a big smile and said, "Don't worry. He understands the concepts."

At the time, I thought, "Oh, good."

But 'he understands' didn't quite sit right with me.

I've been hearing "understands" the material ever since then about all the kids.

When you divorce understanding from knowing, as public schools do, you benefit twice:

* much easier to teach when it's not up to the teacher to provide deliberate practice, oversee deliberate practice, and assess mastery

* more money for the school when "specialists" have to be hired to "intervene" with students who are failing math

Speaking of more money, the middle school here, with around 450 students, had 5 math teachers until last year.

Now it has 6 because we need a new remedial class.

This coincides with the arrival of kids who've had Math Trailblazers all the way through.

Catherine Johnson said...

speaking of manipulatives, Singapore Math is coming to Westchester County

with manipulatives

and differentiation

Catherine Johnson said...

redkudu - there's a new book out on the history & politics of school lunches

from Princeton Press

Catherine Johnson said...

In education, the exact same thing is happening. I can't tell you how many times my quest for why we do certain things has been met with the inevitable, "Well, research shows that... blah, blah, blah."
When I hit this wall I ask for the citation but it never comes. The research is obscured in a fog of consensus. You're just expected to deny what your lyin' eyes are showing you.

I love it!

fog of consensus for sure

SteveH said...

"Public school ideologues scorn memory."

My son's schools have never liked the way he just absorbs knowledge. I have had numerous preemptive strikes because of it. Since it's such a visible attribute, they have to claim that other less visible things reflect real learning. They never bothered to test my son on those things, but they would claim that there is no linkage anyways.

SteveH said...

"He understands the concepts."

i.e. the general idea.

They are wrong about memorization, wrong about mastery, wrong about understanding, wrong about problem solving, wrong about real world problems, and wrong about critical thinking.

It's all just cover for lower expectations. They hope you won't ask for details. They just want you to go away.

Anonymous said...

My son "regurgitates" really well. Or so I've been told.

Teachers used to be happy when kids learned the material in front of them. If they got some critical thinking out of it, so much the better. But now it's a pejorative.


CassyT said...

PaulB & Redkudu-
Curious anecdote on the school lunch issues. Most IB students at my son's high school bring their own lunch.

Lunch period is 30 minutes (one way to solve the open campus lunch issue - no time to leave!)and the school has about 900 kids at each lunch period. Son & his friends learned early on that if they wanted to eat and get back to class on time, they better bring lunch from home.

Cranberry said...

The growth of "specialists" is troubling. In our system, the spots seem to go to the teachers with seniority. They get out of the classroom, and are paid more to be "Math Specialist" or "Literacy Specialist." I'd wager that it's easy to teach teachers than children.

I also think that, as a parent, it's a bad sign if the "literacy specialist" is "working" with your child. It means that there's a weakness in his or her performance which might lead you to ask for an assessment. (It means that you probably should ask for an assessment, if you can't remedy the problems at home.)

I don't know what will happen as the budgets tighten. Those specialists will probably choose to return to the classroom, bumping existing teachers out of their existing placements, wreaking havoc through the system.

The bag lunch question. Hmm. If the parents are organized enough to make lunches, then they're probably organized enough to pay attention to their kids. It's a correlation, in my opinion, rather than a causation.

Paul B said...

At first, I was humorously offering up the bag lunch correlation as a proxy to parental involvement. Now that I've had time to think about Redkudu's comment I'll throw out another correlation.

My school has a seriously high number of overweight kids and guess what? None of them bring their lunch. So there's another vector to add to parental involvement, better nutrition found in bag lunches improves learning.


Redkudu said...

>>I'd wager that it's easy to teach teachers than children. <<

I'm not sure they ARE teaching teachers. Does anyone have any evidence to share that these "specialists" ever participate in teachers' professional development? I've never seen it happen at any school I've been at.

It's one thing to teach struggling students (as a "specialist") in a smaller environment with specialized materials. It's another thing to teach teachers how to teach struggling kids in a regular classroom setting - read "differentiated" setting and all the problems that entails for the teacher.

At my last school, overburdened with 3,000 kids, they finally brought in a reading specialist who was a true reading specialist, not just an upwardly mobile senior teacher. She gave us practical advice and resources we could use, but was never asked to speak in professional development - as if reading wasn't a concern except in English classes.

Cassandra said...

"well....I've been feeling overwhelmed by all this lately, possibly because I've begun the process of tackling reading instruction here in my town, God help me" said Catherine.

Oh, please do keep us informed. It appears your school system uses Fountas & Pinnell's Guided Reading. Every parent should get a copy of the text...and weep.

lgm said...

The specialists here are growing as a result of adopting Response to Intervention, which the state is pushing. They won't be leaving -some of them will be awarded extra money for being dept chairs and team leaders. So far, the money seems to be coming out of general ed, and from savings by serving special ed. in district rather than transporting out of district.

One of the flaws in RtI is that it never address instructional quality or gaps ...all unsuccessful intervention is presumed to be because the child is low IQ, has an unidentified LD, or is not motivated. A classroom teacher is never held accountable for poor quality instruction or for skipping major grade-level concepts in the included whole class instruction.


Open house here involves a mention of statistics that students involved in activities are successful. many overweight students are involved in activities?

Bag lunches here are due to the scheduling and the poor selection. Many college-prep and AP students have to choose between lunch and the courses they need. The lunch selection is all processed garbage..not a thing is cooked on premises. So much for health class..we certainly don't practice what we teach.

VickyS said...

As a parent of high school students, I can tell you that most high school students we know who bring their lunches to school pack them themselves. Another data point.

Cranberry said...

"So far, the money seems to be coming out of general ed, and from savings by serving special ed."

Yes, we've seen a tendency to shift whatever they can from the "special ed" to the "general ed" column, for budget purposes. In these tight budget times, the general ed budget is being nibbled away.

It's counterproductive, though, as I think a lack of proper classroom instruction (due to tight gen ed budgets) will lead to more children being identified with learning disabilities. My kids aren't special ed, but I notice that it's very hard to correct something that's been learned "wrong" the first time. Cutting into the gen ed budget leads to larger class sizes, and fewer aides (or no aides) to help control chaos.

I don't believe that smaller is automatically better for all grades. I do believe, though, that above a certain number of bodies in the classroom, a teacher is practicing crowd control. The public school classrooms we have were not designed to house 35 students. (That's our state mandated maximum, and I'm very afraid that we'll reach that in this budget crisis.)

Add in the effects of mainstreaming, and it adds up to a day which is physically and mentally challenging--every day. At a certain point, I suspect the system breaks down, as the teachers start calling in sick much more often. Because they are sick! It is physically draining to try to teach in such a setting.

Cranberry said...

As a parent, I don't approve of forcing students to choose between lunch and academics or activities. Teenagers are not mature, and are still growing. If they skip a meal, they are more likely to eat junk at home, because they are hungry. And the girls who have eating disorders? Starving a brain isn't a recipe for success. If a child with an eating disorder can skip lunch, it's much easier for him/her to hide the disorder.

It's like the school (and athletic) schedules which assume that students don't need to sleep. Research is clearly showing that not getting enough sleep has direct health effects, including obesity and diabetes. Sleep helps the brain to learn.

I'm all for rigor, but some schools assume that students who want to take AP classes are robots, without human bodies. And yet, I remember my high school, which offered honors and AP classes throughout the day, and had a fine number of activities. Most of those activities met after school, and there was a late bus. I also think that no one would have tried to be on the football team and the Jazz band at the same time.

Some of the unhealthy practices come from unhealthy assumptions.

Robin said...

With respect to the comment on reading and RTI, I found the reading IES practice guide to be far less helpful than the math. It was as if the authors had certain team members who just couldn't quite bear to place the requisite emphasis on phonological awareness and explicitly teaching the phonetic code.

Anyone with an interest or practice with special ed or remediation should use the wrightslaw website.

It's a cornucopia of effective materials, explanations of rights and remedies, and also ideas on what are some dead ends and how to fight if necessary.

ChrisA said...

Wow... a lot of comments for my little post! Next up is team teaching fractions to Algebra and Geometry students. Manipulatives for addition and apparently division and multiplication of fractions.

And essays! Oh it's a wonderful thing. A lot like Katherine Beals posting...

ChemProf said...

""He understands the concepts."

i.e. the general idea."

The scary thing to me is that students hear this so often that they'll parrot it back to me -- oh, I understand the concepts, I just have trouble with the math.

Problem is, it is almost never true. If they don't understand the mathematical problems, they are confused about the concepts too. I always have both calculation-based and written answers on my tests (based on understanding definitions), and in fifteen years, I only had one student who did well on one section but not on the other.

SteveH said...

"Problem is, it is almost never true."

I find that when many educators talk about understanding, they really mean a high level, general understanding, like having some idea of how a computer works. I used to teach students exactly what happens when you press a key on the keyboard; from the interrupts to turning on the pixels. I guess that's not what they are talking about.

There are different levels of understanding that are appropriate for students in different grades. If you look at place value, you don't begin by talking about base systems. You can't even talk about an algebraic view. At best, curricula like Everyday Math spiral through the same basic concept every year. A friend complained to me once that she had 3 kids in different grades covering the exact same material on place value.

As best I can tell, this is all a very simplistic reaction to seeing kids try to solve problems using any sort of rote, mechanical process. This leads them to believe that the algorithm is the problem; that you have to solve things from some sort of general understanding level. No. The problem is that the kids were taught poorly to begin with. Don't blame the algorithm. Tools are tools. You either understand them or you don't. If you don't understand how a hammer works, you can wreak a lot of havoc. The problem is not the tool.

When EM has kids use the lattice method, most don't know how that works. What's the benefit. How about having kids design their own algorithm? Have them understand that you have to multiply each digit of the first number times each digit of the second. It doesn't matter how you do it as long as you get the results in the correct columns.

As I've said before, skills with little understanding can be fixed, but understanding with few skills is nowhere.