I often hook illustrative anecdotes into essays making a larger point. But this anecdote has so many applications that I’m just going to put it out there in its pure form.

A colleague who I’ll call Chuck is pushing the math department to set a department goal. Chuck is in the process of upgrading our algebra 1 classes, and his efforts were really improving outcomes for mid to high ability levels, although the failure rates were a tad terrifying. He has been worried for a while that the successful algebra kids would be let down by subsequent math teachers who would hold his kids to lower standards.

“If we set ourselves the goal of getting one kid from freshman algebra all the way through to **pass** AP Calculus, we’ll improve instruction for everyone.” (Note: while the usual school year doesn’t allow enough time, our “4×4 full-metal block” schedule makes it possible for a dedicated kid to take a double year of math if he chooses).

Chuck isn’t pushing this goal for the sake of that one kid, as he pointed out in a recent meeting. “If we are all thinking about the kid who might make it to calculus, we’ll all be focused on keeping standards high, on making sure that we are teaching the class that will prepare that kid–if he exists–to pass AP Calculus.”

I debated internally, then spoke up. “I think the best way to evaluate your proposal is by considering a second, incompatible objective. Instead of trying to prepare every kid who starts out behind as if he can get to calculus, we could try to improve the math outcomes for the maximum number of students.”

“What do you mean?”

“We could look at our historical math completion patterns for entering freshmen algebra students, and try to improve on those outcomes. Suppose that a quarter of our freshmen take algebra. Of those students, 10% make it to pre-calc or higher. 30% make it to trigonometry, 50% make it to algebra 2, and the other 10% make it to geometry or less. And we set ourselves the goal of reducing the percentages of students who get no further than geometry or even, ideally, algebra 2, while increasing the percentages of kids who make it into trigonometry and pre-calc by senior year.”

“That’s what will happen with my proposal, too.”

“No. You want us to set standards higher, to ensure that kids getting through each course are only those qualified enough to go to Calculus and pass the AP test. That’s a small group anyway, and while you’re more sanguine than I am about the efficacy of instruction on academic outcomes, I think you’ll agree that a large chunk of kids simply won’t be the right combination of interested and capable to go all the way through.”

“Yes, exactly. But we can teach our classes as if they are.”

“Which means we’ll lose a whole bunch of kids who might be convinced to try harder to pass advanced math classes that weren’t taught as if the only objective was to pass calculus. Thus those kids won’t try, and our overall failure rate will increase. This will lower math completion outcomes.”

Chuck waved this away. “I don’t think you understand what I’m saying. There’s nothing incompatible about increasing math completion and setting standards high enough to get kids from algebra to calculus. We can do both.”

I opened my mouth…and decided against further discussion. I’d made my point. Half the department probably agreed with me. So I decided not to argue. No, really. It was, like, a miracle.

Chuck asked us all to think about committing to this instruction model.

Later that day, I ran into Chuck in the copyroom, and lo, a second miracle took place.

“Hey,” he said. “I just realized you were right. We can’t have both. If we get the lowest ability kids motivated just to try, we have to have a C to offer them, and that lowers the standard for a C, which ripples on up. We can’t keep kids working for the highest quality of A if we lower the standards for failure.”

Both copiers were working. That’s three.

**************************************************************

I do not discuss my colleagues to trash them, and if this story in any way reflects negatively on Chuck it’s not intentional. Quite the contrary, in fact. Chuck took less than a day to grasp my point and realized his goal was impossible. We couldn’t enforce higher standards in advanced math without dooming far more kids to failure, which would never be tolerated.

Thus the two of us collapsed a typical reform cycle to six hours from the ten years our country normally takes to abandon a well-meant but impossible chimera.

Many of my readers will understand the larger point implicitly. For those wondering why I chose to tell this story now, I offer up Marc Tucker, whose two–part epic on American education’s purported failures illustrates everything that’s wrong with educational thinking today. I would have normally gone into greater detail enumerating the flaws in reasoning, facts, and ambition but that’s a lot of work and this is a damn good anecdote.

Some other work of mine that strikes me as related:

- Education: No Iron Triangle
- Why Higher Standards are Impossible (one of the first pieces I wrote for this blog, and it still mostly works.)
- On Graduation Rates and Standards

I think I’ve written about my suggested solution somewhere, but where…(rummages)….oh, yes. Here it is: Philip Dick, Preschool and Schrödinger’s Cat–the last few paragraphs.

*“Reality is that which, when you stop believing in it, doesn’t go away.” *

When everyone finally accepts reality, we can start crafting an educational policy that will actually improve on our current system, which does a much better job than most people understand.

But that’s a miracle for another day.

April 28th, 2015 at 1:21 pm

A good story. Many governments would not accept it. Most employers would. My plumber just called, with his apprentice. None of the other apprentices lasted. This one made several mistakes, even ones I could see. However, I think he has a chance of learning, because my plumber needs a helper, and is very patient.

April 28th, 2015 at 4:04 pm

I was completely lost on this comment until I realized that you think of me as the plumber? Which is why I was lost, because Chuck is the senior teacher. Also second career, interestingly. He’s the guy who is given all the curricular stuff. On the other hand, I’m the one selected to mentor teachers. Which is weird, because I’m not touchy feely and love curriculum. But I’m a good coach, and the admin wants more teachers who use my methods.

April 28th, 2015 at 1:50 pm

For the most able and motivated kids, there used to be honors-level classes that covered more and deeper material which, at my kids’ schools, prepared kids for AP calc BC, although there were some kids (who had not taken 8th grade algebra) who ended at honors pre-calc as seniors. Those schools still have honors; is this not possible at your school?

Schools didn’t always pretend that all kids could, or should, do the college-prep math sequence, but felt that it was better for some to MASTER k-8 arithmetic (without calculators) and some algebra and geometry concepts. Vo-tech math courses were tailored to specific fields; carpentry, sheet-metal, practical nursing etc.

April 28th, 2015 at 4:24 pm

But if we have honors classes, it will be increasingly obvious that they are not racially balanced. In fact, we still have plenty of tracking in rich school districts (whose parents know better than to sue) and in homogenous districts.

That’s what we can’t face.

April 28th, 2015 at 3:01 pm

I’ve agreed with this for a while. And, through my wordy explanations of the Finnish system (which you probably remember,) it has been used there for decades.

My sons’ own HS (last one graduates in 2 years) uses this same system quite effectively. There are not just 3, but 4 math tracks. Honors & AP ; Competitive College Prep; College Prep – (sounds better, I guess!) but the highest course is Algebra 2 or Pre-Calc for that Level III.

There is sort of a “Level IV,” that a carpenter I’ve known for years spoke about to me, but it is really not mentioned in the official booklet. It is created for the students who are entering Culinary/Construction & Manufacturing/Transportation/Communication/Food Service & Restaurant Management. These students identify these areas fairly early, and are not planning to go to college. However, if they change their minds, they will at least have Algebra 1 & Geometry completed, of course, certain “shop tracks” require Alg 2.

This system of 3 to 4 tracks has worked well for decades and our HS sends 90% to 4-year colleges. There is an extremely low drop-out rate. Granted, the student body is just 10% minority, with 5% of those considered affluent. We do have another 10% that are children of foreign-born parents representing many countries…but those parents are all college-educated. Real estate is high because of the school system.

The school is in a semi-rural area, and the social life revolves around school, sports, clubs, arts, nature. We have both Project Choice students and the ABC house in town.

Most students complain about boredom, but somehow they plug along well enough to get into these types of institutions after 4 years: Flag-ship State U (most); small private liberal arts colleges, Ivy Leagues, Engineering Institutes, large out-of-state public & OOS private universities, West Point, Air Force, Naval Academy, Coast Guard, Citadel, Army, Navy, Marines, community college, Culinary Institute…so, every institution. And, many of the “shop” kids apprentice right away with electricians, plumbers, carpenters, equine management people. And, there is so much value for kids to discover that they are really interested in cooking, farming, animal care, car & motorcycle mechanics, woodworking, etc. before they waste time and money in a college without ever knowing what they are doing there. I’ve met 30-year-olds that after 4 years of college, they actually have returned to their original interest which is farming, masonry, cooking, etc.

Kids are going to institutions in 42 states this year! So, having reasonable math tracks for all levels of ability has a lot to do with that 90% college-bound statistic year after year.

April 28th, 2015 at 3:47 pm

“I do not discuss my colleagues to trash them, and if this story in any way reflects negatively on Chuck it’s not intentional. Quite the contrary, in fact. Chuck took less than a day to grasp my point and realized his goal was impossible.”

Chuck comes off very well in your recounting of the story. The ability to come back in a day and say, “Hey, I’ve thought about it and changed my mind,” is very nice to see. Lots of people get stuck in “their” position and don’t like to change opinions because it means that they were wrong before.

Also, your backing off and giving Chuck time to think about it himself (which is what you did, whether you thought that was what you were doing …) was also well played. Sometimes people just need time to think about something a bit before they can change their mind.

I think everyone comes off looking pretty good … even the copy machines.

April 28th, 2015 at 4:01 pm

I absolutely think that my shutting up is related to Chuck’s willingness to reconsider. Unfortunately, while I understand this logically, my inability to shut up in the middle of an argument is ingrained. I’m much better at not starting arguments than I am walking away in the middle.

April 29th, 2015 at 12:00 am

Fellow math teacher here. I’m with Chuck on this one. It’s true that your method will be better for your school’s completion rates, but simply trying to move the grade distribution’s median a little to the right shortchanges the students whose progress in mathematics actually matters: the far right outliers.

The 10% of the population with the ambition and aptitude to finish a robust calculus course in high school represents almost everyone who will ever make any meaningful use of higher math. These students should be pushed to their full potential, even if it means leaving the bottom quartile (or two!) in the dust.

The student who scrapes by with a pity C in algebra II after failing it the first time may go on to lead a happy life as a decent and productive member of society–but they will almost certainly never use any mathematics beyond what is typically taught in about 5th grade. It makes no sense at all to retard the progress of the an elite math student (who may go on to actually use math in valuable work in science or engineering) so average students can get a few extra credits in a class that they fervently hate, don’t really understand, and will quickly forget.

I don’t blame you for prioritizing completion rates though–it’s impossible not to in the hyper-egalitarian cargo cult that is the modern public school system.

April 29th, 2015 at 4:12 am

Sigh. I deleted my snark.

Look, you don’t have a clue. For starters, it’s not about agreement or disagreement. And I didn’t say I prioritized anything. So wrap your head around that, and I’ll try not to be rude.

April 29th, 2015 at 5:04 am

A pity that you felt the need to censor yourself. The rude comment was almost certainly more interesting.

It sounds like you took my comment much more personally than it was intended though.

April 29th, 2015 at 5:28 am

No, not particularly interesting. Just dismissive. I didn’t take your comment personally at all. And I suppose it’s not surprising that the import of my response didn’t sink in, since you weren’t capable of reading the original post. So try reading again. Move your lips. Perhaps it will aid in comprehension. Or maybe find a 2 year old who can translate:

1) I did not state a preference.

2) Chuck and I were not disagreeing about goals.

3) I wasn’t prioritizing anything save, perhaps, helping Chuck realize the problem with his goal. The problem is not that I disagreed with his goal but that he didn’t understand why his goal wasn’t possible, given the outcome.

Edited to add: In honesty, given our current standards, I am more in favor of getting more students to pass. But my point to Chuck was not “your goal is wrong” but “your goal is impossible”.

April 29th, 2015 at 7:44 pm

“If we get the lowest ability kids motivated just to try, we have to have a C to offer them, and that lowers the standard for a C, which ripples on up.”

Isn’t it possible for the school district to just make a “D” a passing grade? Or for the teachers to broaden the ranges which correspond to Cs, B, and As and ask the school to mandate a B or B+ or A- minimum in order to go on to the next class in the sequence?

Couldn’t it even be possible to mandate a minimum grade in two math pre-reqs (e.g. “at most one C in the following two pre-req courses”, or “at least one A in the following two pre-req courses”) so that the lower ability students would still be motivated through a good deal of the sequence, but would ultimately be kept out of courses they likely couldn’t succeed in?

April 29th, 2015 at 8:02 pm

We give plenty of kids Ds. But we have to be able to motivate them, which means giving them a C if they work hard and get lots of D+s on tests. We’re mostly talking about kids who will get to trig or precalc by their senior year.

May 2nd, 2015 at 10:22 pm

Making c’s more accessible solved most classroom management issues. Copy these notes/examples/board work because it will be graded, to borrow from you, artificially gives some students skin in the game.

May 2nd, 2015 at 10:30 pm

That doesn’t work. All it does do is give compliant students a way to get a C. But then you’ve got to grade it, and if they don’t do it, you have to give them an F for that entry. Instead, I give the students help on the tests, let them get high Fs and Ds on the test, which participation get bumped to Ds and Cs.

May 2nd, 2015 at 11:10 pm

For me it made students who would otherwise not be – compliant! In 80 minutes, I’ll take universal compliance! I had students who checked out by Christmas last year whereas this year students are still plugging along. I have a lot to learn, no doubt, but I think our ends are similar.

May 3rd, 2015 at 1:30 am

I tell the students to show up in class and work hard, and they’ll pass. They do. No homework, no taking notes, just do the work in class.

May 3rd, 2015 at 1:26 pm

A little off topic but … If it is indeed true that most students won’t “use any mathematics beyond what is typically taught in about 5th grade” (Monic Binomial, April 29th, 2015 at 12:00 am), do you think it would be a good idea to make sure students understand that math and not require most of them to go any further, with smaller, more rigorous classes beyond that for the talented tenth? Is that true–or is something like it true?

If you have ideas that you care to share, what would your optimal math curriculum be?

May 3rd, 2015 at 1:35 pm

Not fifth grade, but through proportional math, which is 8th grade.

My optimal curriculum would be proportional thinking in increasingly complex situations, coupled with simple algebra.

May 4th, 2015 at 1:33 pm

That was the norm when I was in school. Only the college-prep kids took the HS math sequence. The kids on the general track and those on the secretarial track took a year of math in HS that was (IIRC) algebra/geometry lite. The kids doing the vocational track took math specific to their field. Everyone had literature, but the non-collge-prep was less complex and/or shorter and the writings were far less lit-based than the college prep, but were tailored to everyday use; business letters, advertisements, notices etc.

I’d like to see everyone get basic statistics (for understanding) and economics (Sowell’s Basic Economics would be a good guide). The advanced kids could do that in MS, and the others in HS.

May 7th, 2015 at 4:36 am

I’m going to add to my thoughts on Roger’s question.

When I say that kids should be taught proportional math in increasingly complex situations, I’m referring to the bottom tier. I think that we can teach the middle tier kids advanced math without necessarily making them calculus ready by senior year. We can make it rigorous and thought provoking without making it impossible. I think that’s a good mental challenge for kids, even if they never use the math.

But if we teach math assuming that all kids will be willing to devote 3 hours a day to struggling over problems, we’ll fail.

May 9th, 2015 at 12:44 pm

MONIC BINOMIAL ABOVE:

“The 10% of the population with the ambition and aptitude to finish a robust calculus course in high school represents almost everyone who will ever make any meaningful use of higher math. These students should be pushed to their full potential, even if it means leaving the bottom quartile (or two!) in the dust.”

ED REALIST:

“I think that we can teach the middle tier kids advanced math without necessarily making them calculus ready by senior year. We can make it rigorous and thought provoking without making it impossible. I think that’s a good mental challenge for kids, even if they never use the math.”

The payoff from pushing the talented tenth to their full potential is pretty obvious, since they’re the ones who might actually make explicit use of higher math later. The ones from that group who don’t later use math explicitly might still come away from their math education with some mental tools that enable them to understand the world better

Trying to teach math beyond basic arithmetic to the bottom quartile seems completely pointless to me. Having them memorize pages from a phone book would not be any more useless, and they’d probably find that less unpleasant than algebra class.

So what’s the societal payoff from teaching advanced math to the middle tier kids? Is there a long-term benefit from having them experience that “good mental challenge”? And of course most kids won’t be getting a high quality teacher like Ed Realist who might actually give them that good mental challenge. Algebra and beyond for most middle tier kids will just be memorization of meaningless algorithms. Of course there can be an individual payoff from learning meaningless algorithms in terms of getting credentials like a high school diploma or a college degree. Overall, though, as Charles Murray argued in his REAL EDUCATION book, our whole education system is not a good fit with the population we actually have, and I think that what we try to do with math education in high school is part of that poor-fit problem.

May 9th, 2015 at 2:25 pm

Is there a long-term benefit from having them experience that “good mental challenge”?Kids with, say, IQs of 95 to about 110? With an algebra II class that’s basically algebra I with a tougher spin? I think so. I’m willing to be convinced otherwise. I’m willing to believe that we might be able to come up with meaningful elective classes that did the same thing.

You bring up a good point about instruction, although I like to hope that more teachers do something other than “meaningless algorithsm”. But in this perfect world. I’m envisioning, we’re spending more time evaluating how to teach math to all ability levels.

I do not understand this obsession some people have with believing the bottom tier a waste. I wrote about my dad in an other essay, but his IQ is about 95, and was an outstanding airline mechanic for 40 years. He took trig through high school. He didn’t really “get” the advanced math, but it prepared him for the mental challenge of becoming a mechanic, where he also had to learn math.

The real intellectual difference between high and low cognitive people is their ability to abstract. It doesn’t mean it’s a waste to instruct them.

I agree that our educational system isn’t a good fit. I think we need to give most kids more time in proportional math and applications before they go on to algebra. I think we spend a useless amount of time giving most kids advanced literature and science and far too little time in an understanding of science in the world today and English composition. But some people assume that all kids are going through pre-calc or higher. In fact, a good chunk of kids are failing algebra and geometry two or three times–mostly because they aren’t doing homework. So I’m just suggesting we actually instruct these kids for four years in math, rather than have them repeat courses that they could probably pass–not excel at, but grasp the basics of.

May 15th, 2015 at 3:59 pm

ER: “Kids with, say, IQs of 95 to about 110? With an algebra II class that’s basically algebra I with a tougher spin? I think so. I’m willing to be convinced otherwise. I’m willing to believe that we might be able to come up with meaningful elective classes that did the same thing.”

I’m willing to be convinced that “an algebra II class that’s basically algebra I with a tougher spin” is a good thing for average kids, though I’d be hard pressed to say exactly why. A few of them will go on to careers in which a general facility with numbers is useful, and maybe your algebra ll class would give them a better facility with numbers, even if what they’re doing later is not algebra. But maybe not. And maybe the ones who don’t do explicit math later can just reason better because of your algebra ll, but maybe not. You’re a lot closer to the situation than I am and you’re a sensible person, so I take your opinion seriously.

The German education and training system seems to fit their population better than ours fits our population. I’m pretty sure they don’t share our insane notion that even the dumbest teenagers should be taking college-prep courses. I’m not sure how much math they try to teach to their average students. My guess is that the way they train their average students is more oriented toward stuff that those average students might use later. Maybe a student learning a trade learns math relevant to that trade. A baker probably does not need quite as much math as an electrician.

ER: “I do not understand this obsession some people have with believing the bottom tier a waste.”

I don’t think educating bottom tier students is a waste. I think we should teach them what they’re capable of learning (and willing to learn). For a lot of them, basic arithmetic is about all we can hope for with respect to math, and most of them will be pretty shaky even at that. isn’t that just reality?

ER: “In fact, a good chunk of kids are failing algebra and geometry two or three times–mostly because they aren’t doing homework. So I’m just suggesting we actually instruct these kids for four years in math, rather than have them repeat courses that they could probably pass–not excel at, but grasp the basics of.”

One problem seems to be that some students don’t care enough to put in much effort. I’m skeptical that anything you do with those students will have much payoff. But maybe for the ones who are average and willing to put in some effort, several years of general math (probably not following the traditional sequence of subjects, maybe with more probability and statistics) at a level they can grasp would be an improvement.

Art of Problem Solving is an outfit that is focused on providing math resources for smart kids.

http://www.artofproblemsolving.com/

There’s an essay there by one of the main guys about the “calculus trap”.

http://www.artofproblemsolving.com/articles/calculus-trap

The point is that the rush to get up to calculus is not a good path even for the smart students.

QUOTE:

For an avid student with great skill in mathematics, rushing through the standard curriculum is not the best answer. That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too. This is not to say that high school students should not learn calculus – they should. But more importantly, the gifted, interested student should be exposed to mathematics outside the core curriculum, because the standard curriculum is not designed for the top students. This is even, if not especially, true for the core calculus curriculum found at most high schools, community colleges, and universities.

Developing a broader understanding of mathematics and problem solving forms a foundation upon which knowledge of advanced mathematical and scientific concepts can be built. Curricular classes do not prepare students for the leap from the usual – one step and done – problems to multi-step, multi-discipline problems they will face later on. That transition is smoothed by exposing students to complex problems in simpler areas of study, such as basic number theory or geometry, rather than giving them their first taste of complicated arguments when they’re learning a more advanced subject like group theory or the calculus of complex variables. The primary difference is that the curricular education is designed to give students many tools to apply to straightforward specific problems. Rather than learning more and more tools, avid students are better off learning how to take tools they have and apply them to complex problems. Then later, when they learn the more advanced tools of curricular education, applying them to even more complicated problems will come more easily.

END QUOTE

I’d say that advice applies to the less smart students as well. Better to have a solid understanding of basic math concepts than algorithmic training in math that they don’t understand.

June 13th, 2015 at 2:46 am

Spare, I’ve only recently found this blog, and after reading a few entries, I find most of your comments to be pretty solid, but I do have to disagree here with the idea that “some students don’t care enough to put in much effort.”

I understand why you would say that, because it definitely seems that was sometimes, but if you do a deeper examination, and find the REASONS why they don’t seem to care, you see the danger in locking in on that assumption. For instance, at my middle school, most of our kids were one or more (often several more) lexile grade levels below their current grade. This impacted their learning in a host of ways. We implemented an online reading system that assesses lexile level, then levels topical articles for each student so that they are in the kid’s ZPD. Coupled with significant external reward systems for participation/achievement (free dress days, double lunches, candy, etc.) and significant direct instruction in the actual processes involved in close reading and referring to the text for evidence, we had kids who came into their final year of middle school at pre-literate reading levels finish with roughly a 5th grade reading level after just one year of the program, and kids who came into the last year reading at the 6th or 7th grade level finish with 8th, 9th, and 10th grade reading levels.

On average, kids gained around 3 years of lexile development in the one year. Here’s the tie-in: these kids started out without the ability or the interest to apply reading endurance to their academic reading tasks. For most of them, this inability/unwillingness came from years of being unable to be successful at the reading tasks that were put in front of them. Can you empathize? After several years of failing at something, my motivational level would be damned small, as well. However, after being given differentiated and direct instruction on the processes involved, and being externally motivated to attempt the tasks again, and given reading sources that were challenging but attainable for every individual student, kids found that for the first time in years, they actually COULD be successful, and it kick-started a fantastic change in their motivation level.

It made me reconsider all of the other areas that kids just seemed like they didn’t give a damn about doing the work. Was that truly the case, or were they just coming in SO LOW, with so many years of failure at academic tasks thrust at them, that they had given up on being successful? And if that was the case, how could I attack those skills that they hadn’t yet been given, and motivate them with grade level/content area tasks that they could be successful at, to help restart the not just confidence, but the belief in the feasibility of being successful at school, lost from being so far behind the curve of what they were being asked to do for so long…

June 13th, 2015 at 12:39 pm

Which brings up the practical question: how late can such interventions be successful in raising students’ reading level enough to substantially raise their level of motivation? Third grade? Fifth grade? Seventh grade? I’m guessing high school (ninth grade) is too late.

And there is another big problem with “motivation.” We’re talking motivation to do well at school and to learn the things taught in school. But much of what students are expected to learn (at least as of middle school) are things that they are not intrinsically interested in and that they see no practical use for–and they are often correct about that.

(Around middle school, young people change and schools change. Students become less docile. Many of the things that used to be interesting and cool to elementary kids are no longer interesting and cool to post-pubescent teens. At the same time, school becomes more “academic.” When it comes to motivation, that is a deadly divergence.)

It’s somewhat like asking an athletic kid who loves American football to play soccer instead. He may love moving around and playing a game, and he may feel that he has the capacity to be successful in athletics, but if he just doesn’t care for the game …

May 11th, 2015 at 3:49 am

Here, here!

May 13th, 2015 at 9:39 pm

I didn’t read this as trashing Chuck in any way, though no doubt some might. My current supervisor is one of those miracles: a person in a leadership position who doesn’t stop thinking about the content of contrary messages after she had publicly rejected them, and will sometimes come back a day or two later and say, “I’ve thought more about what you said…”

It’s a blessing.

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