Can Khan Move the Bell Curve to the Right? asks June Kronholz on *Education Next*. She visits “affluent, tech-savvy” Los Altos, which is using Khan Academy math for all fifth and sixth graders and seventh graders with average or below-average proficiency.

While teacher Rich Julia met individually with fifth graders to refine their learning goals, everyone else logged on to the Khan web site to work on the math concept they were learning.

Some watched short video lectures embedded in the module; others worked their way through sets of practice problems. I noticed that one youngster had completed 23 modules five weeks into the school year, one had finished 30, and another was working on his 45th.

As youngsters completed one lesson, an online “knowledge map” helped them plot their next step: finish the module on adding decimals, for example, and the map suggests moving next to place values, or to rounding whole numbers, or to any of four other options.

Julian, meanwhile, tracked everyone’s progress on a computer dashboard that offers him mounds of data and alerts him when someone needs his attention. He showed me, for example, the data for a child who had been working that day on multiplying decimals. The child had watched the Khan video before answering the 1st practice problem correctly, needed a “hint” from the program on the 3rd question, got the 7th wrong after struggling with it for 350 seconds—the problem was 69.0 x 0.524—and got the 18th correct in under a minute.

. . . The classroom buzzed with activity, and amazingly, all the buzz was about math.

At Oakland’s Envision Academy of Arts and Technology, an inner-city charter, Ruth Negash uses Khan in her ninth-grade algebra class. While some students are learning geometry, “other students struggled with addition and subtraction, and one quarter don’t know their multiplication tables.”

Khan Academy developers want students to learn basic skills, then move forward, but Gia Truong, superintendent of Envision Schools, worries that students are too far behind. “If you do that, you might never get to the algebra standards” that California students must pass in order to graduate.

If you don’t know how to multiply, you’re not going to learn algebra.

Wait, is Truong saying that if students learn basic skills before moving on to algebra, then they’ll never get to algebra because there’s no time? How does she think anyone can learn algebra without basic skills?

You can learn something CALLED algebra on your transcript. You can then go on to flunk the college placement test spectacularly. But you’ve got your algebra credit.

Sometimes I wish that grading would be taken out of the teachers hands entirely, there’s too much of an agency problem. At the least, there could be an “end of course” grade for the standardized state EoC right next to your school district grade. Then Timmy’s parents can ponder why Mr. Jones gave Timmy an “A” in Algebra, but he scored in the 5% on the state EoC exam.

There’s also the PSAT, SAT, and ACT. Why did Timmy do so poorly on those exams if he earned A’s and B’s in his math courses? Duh – grade inflation.

I’m not familiar with the SSAT, beyond awareness that private schools (many/most?) use it as part of their application/placement process, but can’t help wondering what percentage of public-school 8th-graders would be deemed “HS-ready” based on their SSAT scores. At least, it might give kids and parents a much-needed wake-up call about their actual performance level (will they be college/work ready?), while there’s time to do something about it. Also, even my small-town ES gave the ITBS several times, so kids and parents did get that feedback – and that was decades before grade inflation was an issue.

Because, just like at the lower levels, they can take and pass an algebra course without the pesky reality of needing to actually learn algebra. The have an algebra, geometry, and algebra II credit on their transcript. That’s all that really matters.

This is what math education is for many students. They take and pass grade level math classes without actually learning and retaining the material. Thus, the need for remedial course at the college level. This is way my youngest son, who struggles with math, can never attend a public school. They cannot be trusted to actually teach the material they pretend like their teaching.

This is why requiring algebra for all is such a bad idea. Kids who enter HS not knowing basic math facts, let alone anything above that, are exceedingly unlikely to be able to learn real algebra by the end of their senior year. So, now we have algebra and “algebra.” I’d like to get the weak-to-horrible ES-MS math curriculum and instruction fixed, so ALL kids must master basic arithmetic. THEN, think about algebra for all.

I’m old enough to remember the original reason the ed world is pushing this lunacy. About 20 years ago, a study discovered that kids who took algebra in 8th grade did significantly better on a number of future measurements; what a shock! It never occurred to the ed world that, at that time, 8th-grade algebra was honors-only, for those kids who had mastered all the prerequisite knowledge and skills; of course those kids did better in HS, SATs etc., because they were the kids at the top of the academic pile. Instead, the ed world announced the magic bullet; 8th-grade algebra for all.

Of course, they fell for the same correlation/causation problem with Latin, debate, foreign languages, APs etc. Now they’ve discovered that the kids who take algebra II, chem and physics do better on SATs, college etc. The new magic bullet; require all three for everyone – SD has just done that for the current freshman class.

Okay, I get the general point of the disdain for the guy’s comment, but you’re all just wrong.

You can indeed learn algebra even if you have weak multiplication skills. And I know many students who can multiply and do all sorts of basic arithmetic in their heads yet cannot manage algebra. I have also come to the conclusion that fractions are largely unnecessary in learning algebra. The kids will run into trouble during rational expressions and freak out whenever they see a fraction, but there’s nothing about fractions that hinders the ability to learn how to graph and solve linear and quadratic equations.

It doesn’t mean you’ll be an expert in algebra or geometry (which is even easier to work without multiplication) and it doesn’t mean you’ll be an engineer. But lack of multiplication and fractions just means you’ll use your fingers and skip certain problems. It doesn’t mean you can’t learn some degree of algebra and geometry.

There were kids, back in the dark ages, who just barely passed algebra and geometry. Not all of them were like you worthy people, mastering everything you did simply by looking at it, and now sneering at the little people.

Finally, the woman is expressing a real concern. They have to pass algebra. Full stop. The state says so.

The issue is, naturally, that anyone with weak math skills is never going to go far in math. So why not spend the time teaching them the basic skills *rather* than algebra? Teach them using interesting problems that will help them think mathematically, algebra or no? And the answer is because the state won’t let us, because the racial imbalances in the populations of kids who don’t want to learn algebra are intolerable.

You can know all that and stop pretending that everyone in the past who took algebra was a master at it, or that you must have conquered basic math to understand algebra.

Oh, and Khan Academy is overrated hooha.

The issue is, naturally, that anyone with weak math skills is never going to go far in math. So why not spend the time teaching them the basic skills *rather* than algebra? Teach them using interesting problems that will help them think mathematically, algebra or no? And the answer is because the state won’t let us, because the racial imbalances in the populations of kids who don’t want to learn algebra are intolerable.I assume the reason it’s intolerable is the Robert Moses criticism, that algebra is the gateway to lots of science, technology, engineering, math, etc. courses and jobs. So to the extent that “people of color” don’t take algebra, they’re frozen out of those jobs. But students who take algebra and don’t understand it very well, aren’t they going to find the same thing happening? Haven’t there been a number of stories here about people who have to take remedial math in college? Not many of them make it to a degree.

Cal, I understand that schools and teachers are teaching it because the state requires it, thus my contempt for college for all. I also understand the racial component to the requirement.

And, while some folks may sorta/kinda get it and benefit from their exposure, the question is about lost opportunity really. What are they not doing because they’re sota/kinda getting a half-assed version of algebra, geometry, chemistry and physics? Would they benefit from more practical and useful math and science? Are we sacrificing this practicality at the altar of our need for feeling that we offer equal out-come. Yes, we are.

I’m sure that doing the half-assed version is better than doing nothing, but there should be better, richer options.

Sarah, I can’t find a single thing in your post to disagree with.

Here’s my only point: you can struggle with fractions and basic math while still having some skill at abstract thinking, and you can be awesome at basic math and never be able to handle abstract thinking. I see both constantly.

We teach “half-assed” versions of algebra in part because many kids are simply incapable of learning or caring about it–not necessarily because their basic skills are weak.

Chicken-egg.

Are they incapable of learning and caring about it because they’ve received poor elementary level instruction and feel inadequate and frustrated?

This leads us into another level of Dante’s hell – how do we get them to care about it?

” Are they incapable of learning and caring about it because they’ve received poor elementary level instruction and feel inadequate and frustrated?”

No. There’s no evidence to support that, and plenty to argue against it. But it’s something people like to pretend is true, so they can pretend that we can actually teach everyone algebra.

Cal, Evidence like Project Follow Through isn’t enough? DI doesn’t do a much better job of teaching low achievers, at least in the elementary years?

So the quality of the instruction doesn’t matter – it’s IQ and culture only?

I wonder how much of the difficulty is age. Ten year olds aren’t very good at abstraction. Twenty year olds are much better.

Years ago, people tried to toilet train kids when they were a year or a year and a half old. It took a lot of time and tears. But several decades ago, the advice changed. Parents were told that their kids’ muscles and nerves weren’t sufficiently developed until age two or so, and little Jason shouldn’t be forced to use a toilet until he was ready. Like magic, the tears disappeared and toilet training became a fairly straightforward–and short–process.

Perhaps 12/13/14 year old math should be about getting a feel for number and quantity, so that when abstraction comes, there is a certain comfortableness and familiarity.

Roger,

A totally reasonable suggestion, therefore it would be totally ignored by the “reform” crowd and the politicians.

Kids who enter HS not knowing basic math facts, let alone anything above that, are exceedingly unlikely to be able to learn real algebra by the end of their senior year.Something similar can be said for kids who enter HS not being able to read or write at a high school level.

We’re all tempted to be dictators, to make people to do what we know is right. In the United States, that usually takes the form of, “The federal government should require …” So here is my descent into dictatorship: the federal government should require all eighth graders to demonstrate that they can read, write, and do math at an eighth level. At that point, they will get a federal certificate of proficiency. No high school that enrolls anyone without a certificate can get any federal money–period.

I’m with you, Roger, but let’s do away with the false category of “8th grade”. Let’s devise a basic skills test that can be taken at any age. A 4th grader or 12th grader could take it. Those who pass it then have access to a college to vo-tech high school track based upon their score.

Study guides exist for the military’s aptitude test (ASVAB, I think it’s called) and I assume they have practice tests. One of these might be a useful resource for kids who aren’t very interested in college and haven’t a clear idea what other options might be a good fit for them. A lot of the military specialties have civilian equivalents AND, I’ve seen at least five recent articles about the need for skilled workers, both in local/regional and national papers.

Also, shouldn’t guidance counselors be able to offer non-college options and/or aptitude tests to interested students? They certainly used to do so.

When you walk in to the guidance office at my high school, everything on the walls is about college. Many students who aren’t interested in college aren’t real interested in anything else specific, so the counselors tend to push them to apply to college, in the hope that they will find some interest there and that they will increase their options. Alas, many our graduates go to college for a while and then drop out, having spent time and money and not having much to show for it.

Stacy — like this!

Not all of them were like you worthy people, mastering everything you did simply by looking at it, and now sneering at the little people.Mastery came from doing many problems and being taught how to do them.

You seem to think you have a point. If you do, it is unrelated to the sentence you quoted.

The bottom line in mathematics education is that results matter. There are those teachers who don’t have to face students down the road when they are excluded from fields of study because of weak math skills. Those teachers are the ones who can so easily blow off basic foundational skills that students need in order to declare power over a discipline for their personal and professional benefits. They are the ones who think math can be learned episodically, rather than as a linear and solid progression of knowledge and skills.

I propose that the folks who are so cavalier with their answers try teaching angry and frustrated students in a community college with its 70% to 90% of students in remedial math, simply because they can’t do basic math problems in their heads. The education community ought to be held professionally negligent for its part in helping create a uniquely American cultural phenonmenon that thinks it’s cute for people to joke that they “can’t do math.”

The parents and students in Asia and India are not joking like this. They are, in fact, expected to overtake America economically in two decades, and it’s largely because they know math is the language of science, which is the basis for a society’s development and growth. Who will be laughing then and making flippant remarks about mathematics education’s major role in a society’s success?

Yes! It wasn’t unusual to lose 10-20% of the students in our labs because the first 3 weeks were spent using the metric system, doing conversions, and calculating dosages (pre-nursing course). They weren’t dumb, but didn’t understand multiplication/division by 10, exponents, decimals, or solving for x.

They assured me that there was an app for that when they became nurses. I haven’t taught the class since I saw one of those medical shows on TV where they were calculating dosages in their heads based on how much of an IV bag had gone into the patient, ie the final dose not written on the side of the bag.

Cal says:

“I have also come to the conclusion that fractions are largely unnecessary in learning algebra. The kids will run into trouble during rational expressions and freak out whenever they see a fraction, but there’s nothing about fractions that hinders the ability to learn how to graph and solve linear and quadratic equations.”

Who’s algebra are you talking about?

“It doesn’t mean you’ll be an expert in algebra or geometry (which is even easier to work without multiplication) and it doesn’t mean you’ll be an engineer. But lack of multiplication and fractions just means you’ll use your fingers and skip certain problems. It doesn’t mean you can’t learn some degree of algebra and geometry.”

How will this “some degree of algebra and geometry” be of any use to this person?

“There were kids, back in the dark ages, who just barely passed algebra and geometry. Not all of them were like you worthy people, mastering everything you did simply by looking at it, and now sneering at the little people.”

Ah, so this is your real problem. Perhaps the sneering is directed at pathetically bad K-6 math curricula that cause many of these problems in the first place. Maybe the sneering is directed at those who think they are helping urban kids by setting low expectations.

“So why not spend the time teaching them the basic skills *rather* than algebra? Teach them using interesting problems that will help them think mathematically, algebra or no? ”

Why not try fixing the underlying problem first before you give up? Do the schools separate the willing and able from those who are not? Do the lower grades take any responsibility for ensuring any of the basics. A solid course in algebra is a proper requirement for all kids. Note, I could have said the CCSS pseudo-algebra II, but I didn’t. Are all urban kids condemned by their peer group and by poverty?

“You can know all that and stop pretending that everyone in the past who took algebra was a master at it, or that you must have conquered basic math to understand algebra.”

The first is a strawman and with the second, you redefine algebra.

“Who’s algebra are you talking about?”

The algebra in the state tests, in which fraction knowledge is useful, but not so essential that anyone not knowing fractions can’t still get a “basic” or even (unlikely) proficient/advanced score. Perhaps you’ve heard of the state tests?

“How will this “some degree of algebra and geometry” be of any use to this person?”

Who said anything about it being of use? Pay attention.

“Ah, so this is your real problem.”

Wrong. The issue to me is not idiots yammering on about the wrong thing, but the goals.

“Why not try fixing the underlying problem first before you give up? Do the schools separate the willing and able from those who are not?”

The “underlying problem” is that we are required to teach kids, regardless of whether they are willing and able. This renders separation a total non-issue to anyone capable of thinking. Like I said. Pay attention.

” Do the lower grades take any responsibility for ensuring any of the basics. ”

Yes. They do. Some kids can’t learn. Others learn and forget. Do NOT say “ah, if you’re properly taught you won’t forget” because all that means is you don’t understand what it’s like to be of low cognition. All current evidence to the contrary.

“A solid course in algebra is a proper requirement for all kids. ”

No, it’s not. We went for generations without requiring it for all kids.

” Are all urban kids condemned by their peer group and by poverty?”

No. All low ability kids, regardless of their race or location (since you seem to think only urban kids are poor), are incapable of learning abstract math to the degree that our standards require.

“I have also come to the conclusion that fractions are largely unnecessary in learning algebra.”

“Who’s algebra are you talking about?”

“The algebra in the state tests, …”

“It doesn’t mean you can’t learn some degree of algebra and geometry.”

“Who said anything about it being of use? Pay attention.”

This sequence of comments makes absolutely no sense.

“Wrong. The issue to me is not idiots yammering on about the wrong thing, but the goals.”

Which goals are those?

“The “underlying problem” is that we are required to teach kids, regardless of whether they are willing and able. This renders separation a total non-issue to anyone capable of thinking. Like I said. Pay attention.”

You’re only concerned about your own problem.

“Yes. They do. Some kids can’t learn. Others learn and forget. Do NOT say “ah, if you’re properly taught you won’t forget” because all that means is you don’t understand what it’s like to be of low cognition. All current evidence to the contrary.”

Which kids can’t learn, the ones who don’t? Ninth graders who are still trying to learn the times table? If you can prove they have low cognition, then why are schools trying to do something that can’t happen? What test are you going to use to determine that?

“A solid course in algebra is a proper requirement for all kids. ”

“No, it’s not. We went for generations without requiring it for all kids.”

We’ve gone for generations setting low expectations for many kids. You’re way out there on this one.

“No. All low ability kids, regardless of their race or location (since you seem to think only urban kids are poor), are incapable of learning abstract math to the degree that our standards require.”

How can you pick out the low ability kids? If whatever teachers do doesn’t work, then they are low ability or that they just don’t care? It couldn’t be the curriculum or teachers?

Go ahead and put the blame on kids, but give their parents the choice to just say no.

“There are those teachers who don’t have to face students down the road when they are excluded from fields of study because of weak math skills. ”

This is a big problem. They want to be nice and pump kids along, but when they hit the big filter later on, there is little anyone can do about it. That’s the major problem with Everyday Math’s “trust the spiral”. It’s a pedagogical Get-Out-of-Jail-Free” card.

“The education community ought to be held professionally negligent for its part in helping create a uniquely American cultural phenonmenon that thinks it’s cute for people to joke that they “can’t do math.””

At a recent open house for AP classes at our non-urban high school, some students on a panel joked that “ha, ha, ha”, they would never take a math or science AP class. Many in the audience went “ha, ha, ha” too.

“At Oakland’s Envision Academy of Arts and Technology, an inner-city charter, Ruth Negash uses Khan in her ninth-grade algebra class. While some students are learning geometry, “other students struggled with addition and subtraction, and one quarter don’t know their multiplication tables.””

I can’t imagine how Khan is of any importance in this discussion. The question is why are so many ninth graders still trying to learn things covered in 3rd grade? Seventh time’s the charm?

“The question is why are so many ninth graders still trying to learn things covered in 3rd grade? ”

Because they aren’t very bright, and it will take much longer for them to learn it. It’s only “covered in the third grade” for those who have the intellectual ability to grasp it.

Good grief. You’re pulling our leg, right? Having taught special ed, gifted ed, and regular ed students, and having been a K-12 principal for all of those groups during a 30-year period with 22 of those years in high-risk school populations, I can tell you that success is based far more on the curriculum/ text material used and teacher competency than it is about a student’s “brightness.”

When the president of the National Council of Teachers of Mathematics said in a radio interview in 1996 that American girls and minorities couldn’t learn like “white males” because girls and minorities (except Asians) don’t “think” in a linear way, that should have been a red flag that U.S. math education was in trouble. The 1989 NCTM curriculum standards that were designed to produce equity in the math classroom rather than excellence, were sexist and racist. And, worst of all, their reform methods and ideology haven’t worked for anybody. Yet, those same people are being allowed to make decisions that impact millions of school children everyday.

“I can tell you that success is based far more on the curriculum/ text material used and teacher competency than it is about a student’s “brightness.””

No, it’s really not. And the fact that you’re a principal and taught blah blah blah has nothing to do with it.

I’m not saying that every single student can’t improve at math. I am saying that the vast majority of students who haven’t mastered basic math by 8th grade have failed to do so because of their low cognitive ability.

We seem determined to ignore cognitive ability and pretend that kids aren’t learning because of bad teachers. Some of the teachers are bad, no doubt. But they aren’t impacting outcomes as much as cognitive ability.

Cal, you’re not considering the accumulated affect of not really terrible but kind of poor instruction.

Year after year of passing math with a C, understanding 65% of the material snowballs into an overwhelmed student.

If we were significant more careful with assessment and placement every year, by the time you got the kids at the algebra level they’d be better prepared.

Your rudeness is one thing; your closed mind is another. If a teacher is so willing to make sarcastic and cutting remarks to adults–and in print–It’sawful to think what he/she will do to students. I never had to look far to see why my gang kids believed and acted as they did. Most of the time, they were mirroring the adults in their lives.

Stacy,

By the time they got to algebra might be five years later. And yes, they might be better prepared. And they might still be incapable of grasping algebra (See other comments). Or they might not.

But in any case–as you seem to miss–you are arguing that kids, based on ability and demonstrated progress, should be kept from entering algebra until they are ready. We are in complete agreement.

Your only unsupported allegation is that bad teaching or bad curriculum is preventing them from making progress. There’s no evidence that this is so. But so long as we agree that kids who can’t demonstrate readiness shouldn’t take algebra, who cares why?

Niki,

Try not to be silly. Anyone who thinks they know anything about anyone else based on Internet interactions is very much a fool. Like you, for instance. Your smug and condescending willingness to attack me–twice–might make someone think you were an awful principal.

Newsflash, toots: You’ve done nothing but attack me in your two posts.

Ms. Hayes-

I’m not a huge fan of Cal’s general online demeanor, but this sentence of yours is jaw-droppingly silly:

Do you really think that teachers are

nastierwith students than with adults? Of course you don’t. Teachers modulate their behavior around their students to a ridiculous degree. You know this. Think of the parallels: many teachers drink, smoke, swear, or shoot guns. They might watch R-rated movies, or evenhave sex.But they don’t do those things around their students. (Usually.)

So don’t pretend like we can take Cal’s online misdemeanor* as evidence for her classroom persona. It’s a cheap, rhetorical tactic, and it’s really dishonest to boot.

* Since it’s the internet and I’m sure to have people saying I’m illiterate, let me point out that this is me having fun with language.

Cal says:

“Your only unsupported allegation is that bad teaching or bad curriculum is preventing them from making progress. There’s no evidence that this is so. But so long as we agree that kids who can’t demonstrate readiness shouldn’t take algebra, who cares why?”

There is plenty of evidence for this. You just don’t want to believe it. I talked about specific issues in other posts.

I care that many kids, who have great potential (whether via IQ or hard work), have career doors closed because of bad math curricula. You just don’t care … “who cares why?”

“The question is why are so many ninth graders still trying to learn things covered in 3rd grade? ”

“Because they aren’t very bright, and it will take much longer for them to learn it. It’s only “covered in the third grade” for those who have the intellectual ability to grasp it.”

You should frame this and put it on your wall. It makes a great case for school choice.

You should frame this and put it on your wall. It makes a great case for school choice.Oh, and let’s not forget that Cal’s clearly in favor the sort of “cherry-picking” that Cal’s previously complained charter schools engage in.

Or maybe Cal’s in favor of both positions; public education should only be available to kids of sufficient intelligence to take advantage of the institution while being available, gloriously egalitarian institution that public education is, to all.

Then there’s Khan Academy which our “always certain” correspondent has airily dismissed as “hooha”.

The effectively monolithic public education system is falling apart and with it the uniform response to technological innovations. While, in previous decades, school districts could milk technical innovations for budget bumps without ever showing any return on the investment due to both the isolation from comparison of the school district coupled with the essentially political nature of the school district, that era appears to be coming to an end. The reason is, of course, parental choice.

Given a choice parents will always opt for schools that, first, keep their kids safe and second, do a decent job of educating their kids.

Since the school district is by its nature indifferent to parental concerns more then a few urban schools don’t do a good job at either. Consequently charters only have to be a little better to attract parental patronage.

But with the puncturing of the school district comes a puncturing of indifference to education as Khan Academy, even if it falls by the wayside as newer, even better alternatives emerge, demonstrates.

So, Cal’s two-syllable dismissal is more an indication of a lack of worthwhile, or even adult, response then of any shortcoming on the part of Khan Academy.

Given a choice parents will always opt for schools that, first, keep their kids safe and second, do a decent job of educating their kids.It’s interesting that you put “safe” first. I suspect that they also factor in a lot of other non-educational factors. Is the school close by? Do my kid’s friends go there? Would he feel uncomfortable at a different school?

When it comes to education, it is very difficult for parents to tell how good a job the school is doing. If your kid is passing and isn’t complaining too much and occasionally tells you about some interesting thing they did, you may well be satisfied.

I’m not sure whether school choice would have much effect (especially since many parents already exercise school choice by where they choose to live).

“The education community ought to be held professionally negligent for its part…”

In fifth grade (Everyday Math), my son’s teacher had to slow down to make sure that very bright kids mastered their basic math skills. Some were still adding 7+8 on their fingers. Did she march down the hall and collaborate very loudly with the lower grade teachers about why they didn’t do their jobs? No. Did she inform them that some of the kids (like my son, who got proper math at home) would not get the math they expected. No. At least she didn’t trust the spiral. She was not able to cover 35% of the material by the end of the year, but sent home a note claiming victory over critical thinking and problem solving. Yeah, right.

Why would anyone believe that this doesn’t happen in urban schools? You can’t separate this variable from the poverty one? Baloney.

The other problem with pumping kids along is that by the time they get to 7th grade, it so easy to blame them, poverty, parents, peers, and everything else. The kids will even blame themselves. “I’m just not a math brain.”

It was probably 10 years ago that my sister-in-law, who had been teaching in the same public school for 25 years, moved from 3rd grade to 5th. When the first kids who hadn’t had her in 3rd arrived in 5th, she noticed that few of the kids from her old classroom knew their times tables; the exact opposite of her usual experience. The other 5th-grade teacher had the same problem, so they met with the young grad who had taken over my SIL’s former classroom. She informed them that drill and kill was unnecessary and undesirable and that the kids would learn when they were ready. They told her that it was her responsibility to make sure they learned the times tables in 3rd AND the principal backed them up on the issue! A small ray of light…

Well Steve, the problem seems to be your kid’s school is using Everyday Math. It is one of the most God-awful curriculums out there and nothing is taught to mastery.

Funny how EVERY school that adopts it immediately begins to root out any other Math materials and insist the script be followed completely. I suspect payoffs and fear of teacher input.

But hey it makes the publisher boatloads of money.

“Well Steve, the problem seems to be your kid’s school is using Everyday Math. It is one of the most God-awful curriculums out there and nothing is taught to mastery.”

Worse yet was MathLand that they used before. I was at an open house when my son was in first grade where they talked about this curriculum. Parents got to sit in little chairs and listen to a teacher, in her best first grade voice, tell us about the wonders of having kids explain why 2+2=4, but they didn’t try to ensure adds and subtracts to 20 until the middle of third grade. If you look for MathLand now on the web, all you see are the remaining bad reviews.

When we started looking at private K-6 schools, many of them used Everyday Math, but they were really defensive about it. I still remember the look on the administrator’s face when we brought it up. She quickly started talking about supplementation. I tell other parents that they have to ensure that learning gets done the first time through each loop. Their spiral is not scaffolding. It’s repeated partial learning.

Mike,

You realize that Everyday Math comes from the University of Chicago’s “Center for Elementary Mathematics Education”, right? McGraw-Hill runs the printing presses, but the text comes out of academia.

Cal, you’re not considering the accumulated affect of not really terrible but kind of poor instruction.Year after year of passing math with a C, understanding 65% of the material snowballs into an overwhelmed student.

If we were significant more careful with assessment and placement every year, by the time you got the kids at the algebra level they’d be better prepared.No, I’m not. In fact, you’re doing nothing but confirming my own point.

You’re saying that we need to treat kids of lower cognitive ability differently, that we need to assess them and teach them more differently. And if we did that, they’d learn more thoroughly and–in many cases–actually be ready for algebra.

That’s entirely possible although we really don’t have any idea of people with IQs less than, say, 100 are capable of learning algebra. But I’m certainly on board with the idea that if we taught low cognitive ability kids more slowly and thoroughly, allowing them extra years to grasp concepts that higher ability kids figure out quickly, then they would eventually internalize it and algebra would then become a possibility. Sure. But it’s going to take several years longer, and for most of these kids algebra, or maybe geometry, is the end of the line.

But that confirms my response to Steve’s query as to why ninth graders are unsure of material that they “should” have learned in third grade–namely, that they didn’t learn it because they aren’t terribly bright–and yes, I could have been more diplomatic but Steve’s a pill. But most of the rest of the time I said it as I usually do, which is that the kids are simply being submitted to curriculum they haven’t the ability to master in the allotted time and at the allotted age. They needed more “assessment”, more “placement”, and more time to learn the material.

That’s the end result of placement, after all–put the students in a place where they can learn at a speed that works for their ability.

Unreasonable expectations and goals for low ability kids–not bad teachers, not bad curriculum, but absurd and harmful expectations–are what lead to ninth graders “not knowing what they should have learned in 3rd grade.”

I agree with you that unreasonable expectations for lower performers is a problem. I differ in that I think bad teaching and curriculum contributes to the problem, and there is evidence to suggest I’m correct.

I know next to nothing about Khan. I haven’t used it. I continue to simply teach my kid with a textbook, a whiteboard, and some workbooks. But, it seems to me, that this type of program/structure maybe the beginnings of what we need. It’s better at assessment, placement, and allows for better individualization.

Look, a big part of Cal’s arguement is that people vary in ability, take it as an axiom and ignore WHY it arises; for now. In a typical classroom of 30 kids, what will be the range in cognitive aptitude (I’ll shy away from saying IQ)? A factor of 2? 4? even 8?

The system today is set up so that Timmie and Janey spend about 120 hours warming a chair to earn a credit. If Janey “gets it” in 30 hours, well too bad; she can read under the table or something. If Timmie needs 180 hours, well too bad. His teacher will probably give him a pity grade to move him on. Maybe he’s “mastered” 75%. There’s plenty of aufbau in math. If you do anything other than the simplest problems, you’re pulling lots of stuff together and doing multiple steps.

For the sake of simplicity, think about a problem with a 3 step solution. Timmie has been passed along with 75% mastery and has NEVER forgotten anything nor had his fine edge worn off (amazing!). For Timmie to get the right answer, he must execute each step correctly so his chance of getting the right answer is something like (.75)^3 or 42.2%.

I’ve seen “grading rubrics” and partial credit used to give students most or even full credit when the answer they obtain bears no resemblance to the correct solution; they think this is OK, they think they’re doing well; not so. Sure it’s grade inflation as others have pointed out; BUT – don’t place all the blame on the teachers, parents and kids and administrators have their hands in this problem in a big way.

Maybe everybody could master real algebra and geometry, it’d be nice. But expecting everybody to make it in the time alloted by the seat hour paradigm is absurd. Khan may not be the answer, but it’s closer than mass instruction aimed at some hypothetical middle ground. Develop a list of skills, have the kids actually demonstrate mastery of the skills, give credit for a group of them and move them on once they’be completed a theme (algebra, geometry, etc). If you need 2 years to get your multiplication squared away, then so be it. Perhaps we compensate in high school by giving you extra math time and pulling an elective. If it’s important enough, let’s say that and cut back on underwater basketweaving. It’s hard, it’s way different than the way we think of things today. It’d will certainly cost money.

“Steve’s a pill.”

“I am saying that the vast majority of students who haven’t mastered basic math by 8th grade have failed to do so because of their low cognitive ability.”

“Unreasonable expectations and goals for low ability kids–not bad teachers, not bad curriculum, but absurd and harmful expectations–are what lead to ninth graders “not knowing what they should have learned in 3rd grade.””

“absurd and harmful expectations”

I suggest that others go online, look at actual questions on their state tests, look at the raw percent correct scores, and judge for themselves. We parents who teach and reteach at home set much higher expectations. If my son didn’t get my help at home, Cal would assume that he had low cognitive ability.

From the article, we have:

“While some students are learning geometry, “other students struggled with addition and subtraction, and one quarter don’t know their multiplication tables.””

“multiplication tables”

So, which is it? Do they have low cognitive ability, or is it that they were not separated so that they can learn in their zone of proximal development? Do you have a category for those who disrupt class? Is this separation done with full inclusion or with separate classrooms with different curriculua? What about advanced students in K-6? Do you separate them too? What cognitive test do you use for this separation process?

“You’re saying that we need to treat kids of lower cognitive ability differently, that we need to assess them and teach them more differently. And if we did that, they’d learn more thoroughly and–in many cases–actually be ready for algebra.”

Now you’re saying that not learning has to do with pedagogy and curricula, not cognitive ability. You can’t have it both ways.

After you separate the kids and remove that variable, does curriculum still not matter? I’ve shown that Everyday Math’s trust the spiral doesn’t work at any level, even if you separate the kids. Pedagogy and curricula and low expectations can cause kids to look like they have cognitive difficulties. Even worse, students will believe it.

“Look, a big part of Cal’s arguement is that people vary in ability, take it as an axiom and ignore WHY it arises; for now.”

But you’re going to make decisions on expectations and educational opportunities for individual students based on those criteria. How is that done? What criteria do you use? How do you tell the difference between poor teaching, curriculum, low expectations and cognitive ability?

“Maybe everybody could master real algebra and geometry, it’d be nice. But expecting everybody to make it in the time alloted by the seat hour paradigm is absurd.”

“Absurd” is only your judgment, and you’re in the minority. Even the Common Core Standards set higher expectations. I’ve argued against these standards, but for other reasons.

Traditionally, schools start separating kids in 7th grade, especially for math. Full inclusion in the lower grades, however, creates a bigger need for separation there. This is typically handled using differentiated instruction. It doesn’t work and can make kids seem like they have lower cognitive abilities. It widens the range of abilities and the gap keeps growing. You won’t get any argument from me about the need for separation in the early grades. You will, however, get an argument from me if you set low expecations or discount the effects of teachers, curriculum, and pedagogy.

And it’s clear to me that your expectations are way too low.

Steve: I was trying to argue that we need to address the student where he is AT. Not pretending that they are widgets that all soak up instruction at the same pace. I’m saying the seat hour paradigm is absurd, I guess we can argue it, but I don’t think it’s a good model. I’d much rather see mastery of concepts and skills as a basis for promotion.

I’m sorry if I said anything that implied that my expectations are low. I’m a firm believer that most all children can attain proficiency in at least a decent pre-calc. Given good instruction, good teachers and the motivation to reach for it. We don’t have those things today much of the time and the wasted potential is worrisome.

I’ll agree with getting to a decent pre-calc, but I would allow for alternative paths after algebra depending on what they want to do after high school. (That’s my opinion.) I would, keeping as many doors open as long as possible, require students to get though a proper algebra course by 9th grade. The problem with the CCSS standards is that they still allow the lower grades to pass the burden and the skills gap on to the later grades so that kids struggle to get though algebra by the end of high school rather than by 9th grade.

” I’d much rather see mastery of concepts and skills as a basis for promotion.”

This is the source of so many problems. “Basis for promotion”. One shouldn’t care about IQ or cognitive ability to make decisions about individuals. Someone who scores low on some sort of IQ test, could just have the work ethic to go very far in math, but they have to be offered the path and the expectations. There is very little that is natural about the process. You don’t just pass kids along without trying really hard to see what they can do. If you do pass these kids along without ensuring mastery of skills, you end up with kids in 9th grade who don’t know their times table. That takes away from the other kids who are prepared.

However, the big pedagogy in lower grades is to devalue mastery of skills as if there is some other sort of reasoning or thinking technique that will make up the difference. They pass kids along with minimal to no skills. At an Everyday Math parent/teacher conference we had once, everyone agreed on balance of skills and understanding, but the school was unwilling to define even the basics of what skills meant on a grade-by-grade basis. By definition, Everyday Math says that it is OK to pass kids along to the next grade. They will see the same material during the next loop of the spiral. Kids will learn when they are ready. It doesn’t work.

In the old traditional days, math was kind of like sink or swim, but they focused on basic skills. You had to demonstrate those skills or else you had to stay back a year or go to summer school. There might have been problems with the teaching, but flunking kids kept the problems from going on until there was nothing you could do about it. Now with curricula like Everyday Math, kids and parents don’t know there is a problem until the end of 6th grade when their final Math Boxes fail and the kids get put on the low math track to nowhere. A STEM career is all over by 7th grade.

“Decent pre-calc” is a shorthand. Frankly, most kids would be better served by a good “Statistics around You” class”. I’d like to see us support multiple paths and outcomes. You know how hard it can be to get a good plumber? Much easier to get a Flaubert scholar. I doubt it will happen. Shop is expensive. Capital expense for tools, relatively smaller class sizes. I can’t imagine 50 kids in woodworking being safe. And the liability; DON’T even go there.

Someone who scores low on some sort of IQ test, could just have the work ethic to go very far in math

Exactly, which is why I’d like it tied to what you can DO. If we could imagine 100 skills you need to be competent at “Algebra I”, you pass the drills for 95 of them, you get to go to “Geometry”. Then it doesn’t matter if you’re high cognitive skill and lazy or low and hard working. You’ve done it. Ideally the paradigm of skills based promotion articulates all the way forward and backward. Sure; Timmie took 3rd grade off; but he came back and did double time in 4th grade and he’s in the groove.

A STEM career is all over by 7th grade…

Only if you worry about skills. There are plenty of terrible HS math programs out there; you can cover the deficiency until 13th grade; maybe longer.

If you do pass these kids along without ensuring mastery of skills, you end up with kids in 9th grade who don’t know their times table. That takes away from the other kids who are prepared.

Yes, BUT: There are social and developmental considerations that have to be addressed. What do you do with the “intentional non-learner”? Schools today don’t seem to do well with that problem. Many districts have such lax social promotion policies that the promotion decision is effectively taken out of the teachers hands.

“Frankly, most kids would be better served by a good “Statistics around You” class”. ”

The key is at what point do you allow kids to start closing doors? I disagree with the CCSS pseudo-algebra II by the end of high school. What if a student wants to go into electronics at a vocational school? It’s easy enough to define other paths (not necessarily statistics) that are tied directly to what students want to do in the future. Those paths should deifne what high schools offer after algebra.

What I see, however, are K-6 math curricula that set low expectations and allow doors to be closed. They don’t ensure mastery of the basics. These skills are very difficult to fix once students get to high school. Back when I taught college math and CS, K-6 math skills were still haunting brignt students. It wasn’t a cognitive or developmental issue.

A STEM career is all over by 7th grade…

“Only if you worry about skills.”

Skills are king. There is linkage between mastery and understanding. Trace the AP calc track back to algebra. Look at the homework assignments. Look at the SAT I & II tests, look at Accuplacer, and look at the math requirements of vocational schools.

“Yes, BUT: There are social and developmental considerations that have to be addressed. What do you do with the “intentional non-learner”?”

You don’t just pass them on, thereby making their problem worse and making it worse for those who are prepared. It’s not a lot of fun, but what do we have now? We have what Niki talked about; schools pumping kids (and their problems) along for other teachers and students to deal with.

Schools today don’t seem to do well with that problem. Many districts have such lax social promotion policies that the promotion decision is effectively taken out of the teachers hands.Oh, stop. How many “special reports” do you have to see castigating public schools for their high rate of expulsion for black and Hispanic students? How about their failure rates for both?

They

can’tdo well with that problem. And so long as charter schools aren’t held to the same standard, it goes nowhere. Charter schools boot their non-performers back to public schools because they can. Public schools take them because they have to and bend over backwards not to expel them because they will be criticized for doing so. And not just by reporters.If we ever had a policy that said “misbehave in school and you’re out” the cops would complain, and right after that the taxpayers would.

It’s the racial imbalance issue.

Meanwhile, people like you scratch their heads and say Jeez, for some reason, schools don’t do well at handling kids who don’t want to be in school.

“And so long as charter schools aren’t held to the same standard, it goes nowhere. Charter schools boot their non-performers back to public schools because they can. Public schools take them because they have to and bend over backwards not to expel them because they will be criticized for doing so. And not just by reporters.”

It may not “go” anywhere for public schools, but it surely goes somewhere for those students who get to leave. High schools separate kids by ability or willingness to learn, and there is nothing stopping the lower grades from doing the same thing. If lower schools are really prevented from doing so, then forcing charter schools to accept the same conditions has nothing going for it. If regular public schools are not allowed to make these decisions, then their days are numbered.

“Absurd” is only your judgment, and you’re in the minority.No, it’s not only his judgment and even if it were true he was “in the minority”, what does that have to do with anything? Reformers have no data to back up their claims. Neither do progressives.

What is well-established is the impact that cognitive ability has on academic outcomes. This is not a matter of opinion, and it is irrelevant whether or not the “minority” agree with me.

Now you’re saying that not learning has to do with pedagogy and curricula, not cognitive ability. You can’t have it both ways.The sentence of mine that you quoted had the words “low cognitive ability” in it. Did you notice?

If my son didn’t get my help at home, Cal would assume that he had low cognitive ability.No, I wouldn’t. (I might think he had low cognitive ability simply because he’s your son. Yes, that’s just me thinking you’re a smug twit. No, I’m not serious.)

But you’re wrong. I would not make that assumption about your son or any student. I am not saying that all kids who do poorly in math are doing so because they have low cognitive ability.

I was going to repeat myself, but figured why bother? Go back and read with something approaching comprehension without boring the world with your tedious screeds and maybe you’ll grasp it. (Hint: Sean does).

. I’m a firm believer that most all children can attain proficiency in at least a decent pre-calc.How much time have you spent teaching students with low cognitive ability? You get uncomfortable using IQ, and I am by

nomeans in IQ absolutist, but it operates as a useful metric, so ask yourself, honestly, this question:Can anyone–adult or child–with an IQ lower than 100 master pre-calc?

If you asked that and answered it honestly, then ask yourself if anyone with an IQ under 90 can master pre-calc.

As I said, I’m not an IQ absolutist, but it’s worth thinking about those questions before you say that you think most kids can learn through pre-calc. And ask yourself honestly if by “most kids” you mean “most kids who are like me”.

I agree that we should meet the student where he is at. And yes, there’s no question that this has an impact on academic outcomes. It’s also true that most people–progressives and reformers alike–think that students can “catch up”, rather than that the student has been learning at the pace that works for him–based usually on his cognitive ability.

Oops–the middle part of my answer got italicized.

From “No, I wouldn’t.” to “Hint.Sean does” is my answer to Steve.

Then I quote Sean about proficiency in pre-calc and then answer him.

” Reformers have no data to back up their claims. Neither do progressives.”

Then you are all for letting parents decide?

“What is well-established is the impact that cognitive ability has on academic outcomes. This is not a matter of opinion, and it is irrelevant whether or not the “minority” agree with me.”

Are you going to calibrate that for me? How? Who gets to make those decisions for individual students?

“I am not saying that all kids who do poorly in math are doing so because they have low cognitive ability.”

“All” is a weasel word.

“The question is why are so many ninth graders still trying to learn things covered in 3rd grade? ”

“Because they aren’t very bright, and it will take much longer for them to learn it. It’s only “covered in the third grade” for those who have the intellectual ability to grasp it.”

That seems pretty clear.

“Can anyone–adult or child–with an IQ lower than 100 master pre-calc? ”

It’s clear where you stand and I’m guessing that you won’t let parents make that decision about their own kids. Are you going to have all kids take an IQ test? What policy and individual student opportunities are going to rest on that infomation?

Cal: If teaching, curriculum and motivation converge; I’ll stick with my statement that MOST people have to potential to pass pre-calc. Within spitting distance of 1 standard deviation from the norm. (So orthodox IQ believers would put that at around 85) Overly optimistic you might argue, but that’s where I’d put it.

Can anyone–adult or child–with an IQ lower than 100 master pre-calc?

Not to nit pick, but I’m sure at least ONE must exist below nearly any arbitrary cutoff we might choose. Finding them and executing on it might be challenging. Master pre-calc? I don’t know, I said a decent pre-calc. We’d have to resolve our differences in committee as they say in Congress.

The experience question is relevant. I had three years in the general ed classroom. Two in Boston Public. One in northern CO. Specifically working with low skill kids, no; but I had them and again, the goal was to meet them and motivate them. They all made it past their high stakes test (MCAS), none on more than two tries. There’ve also been a couple years tutoring. I wish I was suited to teaching younger kids; because by they time kids get to high school, it’s darn hard to change their trajectory; especially in math and science.

. (So orthodox IQ believers would put that at around 85) Overly optimistic you might argue, but that’s where I’d put it.I find that unlikely, but let’s accept that number as, indeed, overly optimistic.

The mean IQ of African Americans is 85. The mean IQ of Hispanics is around 90.

So your overly optimistic number, the number that even you think is unlikely, means that half of all African American students, close to 40% of all Hispanic students, and around 16% of whites would not be expected, with your critieria, to be able to do “decently” at pre-calc.

Does what’s left count as “most” students?

(I agree, by the way, that there will be students with low IQs that can do very well in advanced math. One of the things I work on is figuring out to make abstract math concrete, because kids of lower cognitive ability do best when they can see it in concrete terms.)

I wish I was suited to teaching younger kids; because by they time kids get to high school, it’s darn hard to change their trajectory; especially in math and science.See, this thinking is exactly what I’m fighting against. There’s no evidence for your assumption that the “damage is done” in elementary school. We have no evidence, for example, that KIPP students who got to grade level in fifth grade went on to thrive in algebra. And there’s no reason why KIPP couldn’t provide the data instead of the much softer data of “college completion”. Are they actually better, or not?

A high or medium cognitive ability kid who had ten years of crappy teachers–assuming this happens, which is unlikely–is not likely to be behind in the first place. If said kid hadn’t been exposed to a challenging curriculum–more likely–then that kid would generally be able to deal with the work. And before you dismiss these assertions, remember that all evidence shows that intelligence (measured however) trumps income in academic and life outcomes.

So your whole assumption is oh, no, these kids were hurt by years and years of bad teachers or bad curriculum, and if I’d only gotten to them earlier! And there’s just no evidence that it’s true. First, accept as a real, genuine possibility that these kids didn’t have bad teachers and they were taught all the things you imagine they weren’t. Second, whether they were or weren’t, their ability to grasp what you’re trying to teach them is determined primarily by their own abilities, not whether or not they had bad teachers.

I’m making this sound too cut and dried. I know that we’d be better off if kids were given reasonable learning goals and of course, keeping them interested in school and working hard is important. That’s why I think we should change the goals.

But all of this is predicated on accepting that not everyone can learn advanced concepts, and most of high school expectations are beyond a significant portion of our school population. And that’s using your optimistic number.

“A high or medium cognitive ability kid who had ten years of crappy teachers–assuming this happens, which is unlikely–is not likely to be behind in the first place.”

“…I’m not an IQ absolutist…”

Yes you are.

See, this thinking is exactly what I’m fighting against. There’s no evidence for your assumption that the “damage is done” in elementary school. We have no evidence, for example, that KIPP students who got to grade level in fifth grade went on to thrive in algebra…

Consider the obverse: Is there not evidence that kids who DIDN’t get to grade level in 5th grade (etc.) failed to thrive in algebra? Sure, just because you get multiplication doesn’t mean you’ll grasp polar coordinates. At some point, you’re likely to hit a wall. Although I don’t have a peer reviewed paper showing it; being behind in 5th and 6th sure seems bode ill for Algebra I completion. Do you believe that with younger kids you’d have more chance to impact the trajectory? To build the habits and concetps that will be of benefit later on?

Although I don’t have a peer reviewed paper showing it; being behind in 5th and 6th sure seems bode ill for Algebra I completion.Sure. But I’m pretty sure you aren’t arguing that all students can be ahead, right? Or even at grade level? There will always be kids below grade level. So isn’t it also true that there will also be kids who aren’t able to do algebra?

More important is the indisputable fact that many kids who are at grade level in fifth and sixth grade, who were taught all the things you want to teach, fail to do well in algebra. That suggests that algebra not only requires a good grounding in basic skills, but an additional cognitive load that not everyone is capable of, regardless of their grounding in basic skills.

Do you believe that with younger kids you’d have more chance to impact the trajectory? To build the habits and concetps that will be of benefit later on?No, not really. You’re also supposing that the kids you taught had bad teachers, and you’d probably find out that’s not true a lot of the time.

Look. Teachers don’t teach groups. They teach individuals. As a teacher, I want to help all kids reach their potential, and potential is not some absolute standard in algebra or geometry or whatever. As a teacher, I know that my interest and support and the lessons I choose and the explanations I make will have an impact on the margin. I certainly know that my homework policies will have a difference in academic outcomes, although not much to ability levels.

But when you look at groups, you have to accept that all the little things you’re focusing on are just that. They’re little. They’re drowned out by the larger factors.

There’s tons of evidence supporting cognitive ability’s link to academic outcomes. It’s well established that teachers make a difference within the classroom, but it’s dwarfed compared to parents, SES, and–yes–cognitive ability.

And there’s very little evidence for anything except DI in reading that curriculum has much to do with outcomes.

None of that means we shouldn’t teach. None of that means we shouldn’t do everything we can to give kids a sense of purpose and possibility.

My point–and I think we agree on this–is that the worst thing to do to kids is constantly give them work they aren’t able to do and let them think that school has no value to them.

BTW:

before anyone start syammering about me insulting people, I’d like to point out that Niki, Steve, and Allan spent quite a bit of time insinuating various horrible things about me that were quite insulting. In contrast, I said Steve was a pill. So keep it in perspective, if you’re going to complain about insults.

“I’m not an IQ absolutist”

No, I’m not.

More importantly–and this is to the board–you are letting your own preconceptions about low cognitive ability color your response.

I am not dismissing people of low cognitive ability. It’s neither a sin nor an insult to have an IQ below 90. It doesn’t mean you can’t be a productive, happy, human being, nor does it mean that you can’t be a healer, a helper, or a host of other jobs .

If you feel I’m insulting students by saying that they have low cognitive ability, that’s your bag. It ain’t mine. I want to teach all kids. I don’t just want to work with the “smart” ones, which is what many of you seem to think is the goal. You just want to define everyone as “smart” because you don’t like the alternative.

“A high or medium cognitive ability kid who had ten years of crappy teachers–assuming this happens, which is unlikely–is not likely to be behind in the first place.”

You’re an absolutist in that you deny other major sources of problems in teaching, pedagogy, and curriculum. Everything is dominated by IQ. Poor K-6 math curricula are a huge problem, but you apparently can’t see it. Now you are thowing up strawmen to cloud the issue.

Starting a new comment thread because I’m not sure where to reply–

I have to agree with Cal on this one– While you need basic math facts to solve algebra problems quickly and well, algebra is not actually part of the same skill set as arithmatic. (Or, as one of my mathematician friends used to say: “What I like about higher math is that I never have to deal with arithmetic harder than “2n+1″ “)

A bright second grader who hasn’t learned her times tables yet can still grasp the principles of algebra, if she’s fairly mathy. There are ninth graders who’ve always been good at arithmetic and who had A’s all through elementary school who just can’t make the jump. Somehow, their brains just won’t grasp the idea that ‘x’ can stand for any number, or that you can manipulate the expressions without knowing what x IS. It’s an abstraction issue.

On the other hand, I can’t explain where the gap is. (Which is why I sucked as a math teacher for the non-gifted.) There seems to be a gulf between those who can handle the fact that it doesn’t MATTER whether you’re dealing with 2x+7=12 or 2(cows)+7 = 12 or 2@#%&@#+7=12, and those who get hung up on the fact that x and (cows) aren’t THE SAME, so how can you say they’re both 2.5???????

On the other hand, if you’re a bright kid who never learned fractions, getting the RIGHT answer in Algebra and subsequent classes will be tough, even if you understand the underlying math. Of course, a daily timed test is enough to bring most bright kids up to speed on the arithmetic pretty quickly, except for the kids who have math-dyslexia….

You probably didn’t suck at all; you were just realistic. The “good” teachers out there allow students to “demonstrate” their knowledge of algebra by writing essays and using words like “evaluate” and thus proving that they’re thinking at the highest level of Blooms.

“There are ninth graders who’ve always been good at arithmetic and who had A’s all through elementary school who just can’t make the jump.”

What percent have this difficulty? What percent of that amount do not have their problems resolved with just a little bit of effort? Difficulty with new concepts goes with math at all levels. Practice and mastery make them go away. Teachers who have taught algebra for any length of time have seen these issues over and over and know how to deal with them.

As for arithmetic, plugging numbers into equations also goes with the territory. The goal is not just some sort of symbolic mastery of math. A direct and singular connection between arithmetic and algebra is not in question. The problem is that some popular K-6 math curricula do not ensure mastery of much of anything. Bright kids do poorly when they get to algebra. The problem cannot be blamed only on IQ.

“There seems to be a gulf between those who can handle the fact that it doesn’t MATTER whether you’re dealing with 2x+7=12 or 2(cows)+7 = 12 or 2@#%&@#+7=12, and those who get hung up on the fact that x and (cows) aren’t THE SAME, so how can you say they’re both 2.5???????”

Actually, Deirdre, you hit the problem square in the target. This problem, and a lack of fraction skills, are the two main reasons that people don’t understand Algebra I & II or any Math above that level.

I have to agree with Cal on this oneHaveto? Bite your tongue, woman. Youchooseto, having seen all the errors in your previous ways. Welcome to the path of righteousness. Don’t stray.Cal says: ‘Have to? Bite your tongue, woman…’

I take it you’re one of those types of guys who would disregard advice from someone because they have mud on their boots, with you sitting in your ivory tower in a suit and tie. I also take it you’re one of those types of guys who treats waiters & waitresses horribly when you go out to eat.

Students run into many conceptual problems. A few may not slow them down much, but if they start to add up, or if they are not resolved, then the students are dead. These are not concepts that defy the intellect of certain students. They just might require a different approach. The problem with many K-6 math curricula and teaching pedagogies is that they think these problems will resolve themselves automatically if they keep going over the material. They do not. The curricula do not take into account the immediate learning needs of individuals. When I used to directly teach courses like algebra, I would look into the students’ eyes and get immediate feedback. I would see individual homework sets several times a week. I could respond immediately to any conceptual issue that came up. The variable name issue is a common one, and it’s not difficult to fix. I remember having an issue with that when I was in 8th grade. That is something the Khan videos can’t do. Videos can cover for poor teaching or the lack of help at home, but they should not be a replacement, as in the flipped classroom model. That doesn’t mean that I think they aren’t useful, but they can’t provide the immediate feedback that a teacher can give in a direct teaching environment with a homogeneous mix of students. I just viewed some videos yesterday on PHP programming that were fine until I had questions they didn’t answer.

With full inclusion, teachers are pulled in too many different directions. They might help a few individuals as the guide-on-the-side, but they will miss so many other kids with issues, especially with quiet students. They are lost behind the dominant extroverts in the active learning groups. With full inclusion, all kids are allowed to be on their own page. Individual gaps in understanding build up in all sorts of ways until it gets to the point where it’s difficult to untangle the mess. Just ask any tutor. You have to go back to the beginning. In many cases, however, tutors can make amazing progress in a very short amount of time if they can untangle the mess. This is not a cognitive issue, it’s an issue of pedagogy.

There is also a problem with moving from simple concepts to more abstract ones, like the move from simple fractions to the complexities of rational expressions. One might understand that a*b= b*a, but get confused because they can’t “see” the factors in a rational expression. One might understand that a^(-n) = i/(a^n), but can’t see what that means for a rational expression. One might know that X^1 = X, but can’t see that X = X^1, or why seeing that would be useful.

Conceptual problems are a fundamental part of all learning. Don’t blame the student.

So how DO experienced teachers get kids over the hurdle with “the name of the variable DOESN’T MATTER!” When I taught, I tried using a variety of letters, symbols, and silly pictures, and demonstrating that the solution was the same regardless. I tried frequent reminders. I tried “well, just let b=x, replace all the b’s with x’s, and then solve for x” There were some kids for whom nothing worked — they could not get their head around the fact that the variable in an equation stood for something that HAD a value, and we just didn’t know what that value WAS yet, so we were calling it the mathematical equivelant of “Hey You, over there!”

So, what DOES get these kids over the hump, other than time? I did see kids who couldn’t get variables in 8th grade, 8th grade, or 10th grade who suddenly got them in 11th grade—I could say it was my excellent teaching that someohow worked for them but not the eight graders who couldn’t get it….

OR I could say the problem is pushing kids into algebra before they’re ready— and that ‘ready’ varies not just based on elementary math curriculum (these kids had all gone to the same elementary school and had the same teachers) but on something else… and anecdotally, at least, the higher IQ kids were ready a lot earlier (most could have handled algebra in 7th, though the school made them wait until 8th) than the lower IQ kids, who took an extra 3-4 years before they got there….

Since IQ is, definitionally, your ability compared to your age mates, is it really such a huge leap to say “If a kid with an IQ of 100 is ready for this concept in 9th grade, a higher IQ kid will be ready earlier, and a lower IQ kid will be ready later?”

And how does it help either kid to declare that Algebra will occur for everyone at the same time?

I used a box that had a name on the box. I told the students that the box contained some fixed number, but I could give the box any name. Changing the name didn’t change the number in the box. If this didn’t work for some, I ended up working with them one-on-one. Acutally, this was one of the simpler concepts to deal with.

A more difficult one was what to do with minus signs as expressions got more complicated. When you have a minus sign between two expressions and the minus sign precedes a rational expression, what does it mean? What can you do with it?

If students didn’t “get” something, I saw that as my failing. I saw that as a challenge to come up with some new approach. I NEVER recalled thinking that the students did not have the cognitive ability or that they needed to let time fix the problem. This is perhaps not the case in K-6 with full inclusion, but it’s really dangerous to to start blaming the student.

This is now tangled with curricular ideas of trusting the spiral and that kids will learn when they are ready. That philosophy really denies the possibility that there are problems with the curriculum, pedagogy, or the teacher. All of the variables are mixed up. Curricula like Everyday Math just pump kids along and when they get to the tracking divide in 7th grade, it oh so easy to blame the students.

The solution is to not use full inclusion. One K-6 charter school in our area uses something called a full inclusion environment. For the main academic courses, students are separated by ability level. For everything else they are mixed up. The ability grouping doesn’t care about cognitive ability, only effort. If teachers find that certain techniques don’t work, they are not apt to start blaming the students. The other issue is trusting the spiral and the idea that learning is some sort of natural process. It isn’t. You have to push and set high expectations. There are extremes to the push equation, but for many subjects, learning is not natural.

You have to separate the variables or else it’s too easy to blame the student.

Steve, it’s not a “failing” on anyone’s part. Or blame. You just want someone to blame.

I take it you’re one of those types of guys who would disregard advice from someone because they have mud on their boots, with you sitting in your ivory tower in a suit and tie. I also take it you’re one of those types of guys who treats waiters & waitresses horribly when you go out to eat.Given how spectacularly wrong you are in every respect, you probably shouldn’t be let out of the house unaccompanied by a responsible adult. See previous posts about the idiocy of anyone who thinks it possible to judge personality from Internet interactions. Add to that I’m not the ivory tower person, but the person with muddy boots. Oh, yeah, and I’m not a guy. See how moronic it is to think you can know anything from a blog comment?

“Steve, it’s not a “failing” on anyone’s part. Or blame. You just want someone to blame.”

I blamed myself before I blamed the student. I blame curricula that set low expecations and assume that students will live up to their abilities. You blame students, but you couch it in such a way that it seems like you are really just trying to help them. You can separate the variables, but you’ve already decided that you know the answer. Why don’t you have students tested for IQ early on. Then, when they don’t do well in 9th grade math, you can tell the kids who meet your IQ/algebra calibration that they are just lazy.

To the extent that we consider something is a problem, we’re always looking for a cause, which I suppose means we’re always looking for someone or something to blame.

If a parent tries to toilet train a one-year-old and is failing, we can blame the kid, who is quite obviously not getting it. We can blame the parent, who is making the kid–and herself–crazy. But we can’t “solve” the problem unless we know what’s possible and when. Actual research solved that problem. Once a kid’s muscles and nerves have developed enough, it’s fairly easy to do. Most kids will be ready about age 2, but if it doesn’t work then, hold off for a while and then try again.

Is algebra like toilet training–eventually 98% of kids can do it but you have to wait for the right time? Does it instead require a lot of preparation? If so, what? How do you determine when is the right time for a given kid? Even with the best pedagogy and timing, will you get close to 98 per cent? 80 per cent? 50 per cent?

The thing that frustrates the hell out of me in this business is that we really have no idea what is possible, so we generalize from our own non-random samples and our political preconceptions.

If you ‘blame’ the teachers for the fact that some kids aren’t ready for Algebra in 8th or 9th grade (more are ready in 9th than 8th…) do you also ‘blame’ them for the kids who are ready in 5th or 6th grade? Because in my experience those kids were more likely to have experienced educational NEGLECT for a few years before someone finally threw an algebra book at them and said ‘fine. sit in the corner and work on this.” Does that mean that “Hey kid, instead of Math, play Oregon trail on the computer for two hours every day!” is a valid form of teaching to prepare kids for algebra?

If only the other students had had the same ‘instruction!’ They could have ALL done independent study Algebra as 6th graders!

Regarding the “arithmetic and fractions aren’t necessary for algebra” crowd…

how else do you explain the all-too-frequent student answers that inform me that x=180 in 3x=60? There’s a complete lack of both number and procedural sense there. By not achieving mastery of basic skills, students become nothing more than button-mashing monkeys with calculators who cannot take a simple second to judge the calculator’s response.

That’s because the brass, usually educated in “language arts”, think kids are learning “technology” as well as math when they mash buttons on the calculator. Don’t you know that by using calculators from an early age these kids will grow up to be creative geniuses who will go on to work for Texas Instruments? Was professional development wasted on you?

Supersub,

I think that the algebra they are talking about is not the simple single-variable type, which really can be considered a subset of arithmetic rather than algebra proper.

Still, I take your point and it’s a good one. It’s hard to understand how function graphing and quadratics and so forth work if you can’t solve for

x.Supersub, I don’t think you understand the point.

I am saying that there isn’t a particularly good link between knowing your math facts and being able to deal with the abstractions of algebra.

How is that you confuse “knowing math facts” with “knowing the difference between multiplication and division”?

What boggles my mind is that you are describing

exactlythe sort of learning issue that hasnothingto do with calculators. How is it you don’t see that?Honestly, it looks a lot like the kids who say 3x=60, so the answer is 180! aren’t getting the point of algebra at all. They’re overapplying the commutative property or something so that 3x=60 is the same as 3×60= and going from there.

Is this a conceptual issue? or a “I don’t care and I’ll just write down any number” issue?

Either way, it doesn’t look like either a teaching issue or a curriculum issue.

Also, I’m not saying that a huge number of kids can’t learn algebra. I’m saying a sizeable portion can’t learn it by 8th grade. Remember, 8th grade algebra used to be the accelerated option for kids who were ready for ‘high school math’ early. Now we force it on everyone, and overall math achievement hasn’t increased. Why? Because the old way was more realistic.

Also, 8th grade algebra for all HURTS the kids who could handle acceleration, because it dumbs down what used to be an honors class. When the teacher needs to spend 3 weeks getting across the concept of variables or equation balancing, that’s 3 weeks that the accellerated kids are NOT getting to work at their level…

For someone to be able to make use of the abstractions in algebra, one must be able to manipulate variables and equations. Yes, abstraction is a separate skill/capability, but to make use of it one needs a solid foundation in the basic skills.

Steering a car and pushing the gas pedal are separate skills, but you need both to drive a car.

Also, 8th grade algebra for all HURTS the kids who could handle acceleration, because it dumbs down what used to be an honors class.Oh, don’t you know what’s happening to them? They’re now taking algebra in 7th grade. It’s quite normal to see ads for middle school geometry teachers.

Which is insane. You now see sophomores taking Calculus.

Over 25,000 California 8th graders took the Geometry CST. Acceleration happens (and it happens through Khan as well). It’s also the built-in problem with attempting to design a system that eradicates all distinction between rich and poor, but that’s another issue.

The real problem with 8th grade Algebra–in California at least–is that the standards are not aligned to support it. 7th grade math standards are NOT pre-algebra: 1) the “algebra strand” is of equal proportion to all other strands (no ramping up) and 2) the “number sense” strand is the one most represented on the CST, causing the previous year’s score to lack predictive quality.

Kids can do Algebra in 8th grade. Many more of them would land there prepared if the states like California designed an instructional framework vis-a-vis state standards that considered the end goal in mind.