“Math of Least Resistance”

by Robert Pondiscio
March 26th, 2010

At Teacher in a Strange Land, Nancy Flanagan tells a fascinating story of a Michigan community college that received a big grant to build a wind turbine and create a green energy careers program.  Education! 21st century careers! In Michigan, no less.   Just where we need to be headed, right?  There was just one problem. 

“They received over 500 applications. The minimal prerequisite for application was successfully completing Algebra II–but the 16 candidates selected for the prestigious pilot program all had top-flight math coursework credentials, including success in national AP math exams. When they got into the actual hands-on learning in the program, however, the selected students had great difficulty in applying the mathematics content they had aced the previous year.”

In short, these looked-good-on-paper students couldn’t integrate and utilize what they knew. Flanagan sees in this story a problem with what grades and test scores tell us about student preparedness–and a cautionary tale for ed reformers.  The disconnect between the students’ credentials and what they could actually do with what they supposedly learned ”can’t be traced back to lack of market-based schooling options, teachers who didn’t get merit pay, or the fact that none of their teachers came from Teach for America,” she notes.  “Their troubles are directly tied to curriculum and instruction–the way they learned to ‘be successful’ in math.

“Education wonks and armchair pundits hate this kind of traditionalist thinking. Curriculum and instruction are dull and unsexy, and push wide-scale policy-lever solutions to the periphery of the discourse. This is why we hear lots of pontificating about recruiting and rewarding quality teachers–including those break-the-mold whiz kids who cut their teeth in the country’s toughest classrooms–but almost nothing about consistent, quality teaching.

Flanagan puckishly headlines her story “The Math of Least Resistance,” which is clearly a comment on ed reform’s tendency to take on the “easy” challenges of structures and incentives while ignoring the much harder task of getting curriculum and instruction right.  Take the lessons from Flanagan’s fascinating anecdote that you will.  But when students that our schools deem successful–those who would almost certainly meet any current definition of “college and career ready”–struggle in situations in which they should have been well-prepared, something’s clearly not right.


  1. Maybe what this really shows is that curriculum is relatively useless, because people forget just about anything that they learn after a year’s time anyway.

    Comment by Stuart Buck — March 26, 2010 @ 1:46 pm

  2. Very interesting story. I look forward to Nancy’s followup post, where she will look at why the students were unable to apply their knowledge. I have one idea, which may or may not apply.

    It is not only a question of spending time on “applied math.” It is a matter of being so familiar with a math concept or problem that one can tilt it in the mind.

    Paradoxically, some of that requires abstract thought as much as practical application. It is like being able to recognize not only a musical chord but its inversion, its role in a melody and progression, its relation to the key of the piece. Much of this comes from the practice of toying with things in our minds, becoming familiar with their shapes. And that in turn requires spending time with problems and concepts, not just “succeeding” at them and moving on.

    When I was little we made long trips to Maine on weekends. On those long trips, my father would give me difficult math problems to solve in my head. They went beyond arithmetic to algebra, probability, number theory. I learned to think and think about them until something came clear. There was no rush; I could think about them for hours if I wanted.

    One of my concerns about the “School of One” and other skill-progression models is that it assumes you have “mastered” a skill when you can answer the questions correctly. This is not necessarily so. You have mastered it when you can hold it in your mind and can turn it backwards and forwards, inside out and back, around and around.

    This is just one thought. It may or may not be different from what Nancy will say–I am eager to find out!

    Comment by Diana Senechal — March 26, 2010 @ 1:58 pm

  3. This is an odd story. Why would anyone expect a mathematics course sufficient to do complex engineering? In addition to those necessary Algebra II courses (or AP), to be successful at installing a wind turbine would require rigorous Physics and Engineering, including fluid mechanics, electrical generation and systems controls, as well as some good hands-on experience with mechanical tools. This is not the fault of the mathematics course at all, but of an ill-conceived notion of what is required to be successful with this complicated project.

    Rather this is a great illustration of the difficulties with “project based learning” which assumes that with the proper motivation, students will be able to solve any problem, no matter how ill-prepared. Not so. The proper in-depth academic background knowledge is also needed to be effective at solving real-life problems.

    Comment by Erin Johnson — March 26, 2010 @ 6:31 pm

  4. Why aren’t kids given the opportunity to demonstrate mastery in high school,, middle and elementary of what they are supposedly being taught?

    Comment by tim-10-ber — March 26, 2010 @ 6:52 pm

  5. It seems to me that this story doesn’t tell us much about math education one way or the other. The most likely explanation for the disappointment in the students’ performance, it seems to me, would be a mismatch between expectations and reality. As Erin aptly points out high school math, even with AP calculus I would think, hardly prepares one to do or understand the math and science of wind technology. I would expect that when those who run the program learn just what their students can and cannot do, and learn how to maximize what they have to work with, the problem will be seen as simply a normal frustration of starting a new program. The program is for technicians, not engineers or scientists, is it not?

    I like Diana’s perspective on this. But, Diana, how about a name for this idea that you’re talking about? I call it “levels of fluency”, and have argued that it is a very important idea in pedagogy at all levels. To get a question correct on a test might indicate a very low level of fluency in that topic, or a very high level of fluency, or something anywhere in between. Good teachers are aware of this and teach accordingly, but, as I have complained again and again, are not necessarily any good at all in analyzing it or verbalizing that analysis. We need to be aware of levels of fluency in anything our students learn. A topic learned to a low level of fluency may get a student past a test, but form a very poor foundation for future learning.

    As a teacher of college freshman math I often wonder why I am teaching concepts that students had, and presumably learned, in eighth, ninth, or tenth grade. One possibility is that we have drifted into sort of a bad habit. We teach a lot of math, but we leave each topic when students have achieved only a low level of fluency. So we teach it again, and again. Then the students come to college and learn it again, but again to a level of fluency so low that six months after the course is over they have forgotten everything, once again.

    How can we get out of this, if indeed this is what is happening? We can’t. We can’t slow down and teach more thoroughly and insist that students achieve a higher level of fluency, because then our courses would cover less material than similar courses everywhere else. We do have to keep up with the Joneses, so to speak.

    And so I very much agree with Nancy also. Curriculum and instruction are very important.

    Comment by Brian Rude — March 27, 2010 @ 1:10 am

  6. Brian,

    I find the term “levels of fluency” fitting, as it suggests an analogy with language. When you learn a language, you may be able to say a sentence just as you were taught it, or you may understand the many possible variations on the sentence and the differences between them.

    I like this observation: “To get a question correct on a test might indicate a very low level of fluency in that topic, or a very high level of fluency, or something anywhere in between.” This point should get more attention: students with the same scale score may differ widely in fluency and understanding.

    Why can’t schools slow down and teach fewer topics in greater depth, provided they do it right? Such a school may fall behind the others temporarily but will gain over the longer term. At some point schools have to stand up for good instruction.

    Comment by Diana Senechal — March 27, 2010 @ 8:07 pm

  7. Diana, Why do you think that schools need to slow down?

    Certainly, Singapore and the other top Asian countries do not slow their math instruction down, but rather speed it up so that students are learning Algebra starting in 6th grade. They are able to do so because they teach less irrelevant topics (e.g. probability and stats in the lower grades) and focus substantial time upon number operations and problem solving (topics that really do matter in understanding Algebra).

    Getting to fluency is of course a great goal, but is it really necessary to reach that for each and every class? I didn’t really understand arithmetic until I studied Algebra. I really didn’t understand Algebra until I studied Calculus. And I really didn’t understand Calculus until I had ro use numerical analysis for solving complex equations. The point being, sometimes the real understanding of topic takes a long time (longer than a semester) and sometimes it is the applications of that work/idea that flesh out the concepts to enable mastery of the material.

    If we slow do too much, there is of course the danger that we would grind to a halt and never get to those complex ideas/projects that allow for fluency and mastery of a subject.

    Comment by Erin Johnson — March 27, 2010 @ 10:01 pm

  8. Erin–of course. I simply meant that we shouldn’t hop from topic to topic in the style of “Everyday Math.” Of course certain concepts and topics become clearer in retrospect.

    Comment by Diana Senechal — March 27, 2010 @ 10:22 pm

  9. Diana, Agreed. That insane jumping of topics by Everyday Math is almost as bizarre as their distain for the mastery of the fundamental concepts and operations of arithmetic. How could anyone in their right mind think that that would be beneficial for children?

    Comment by Erin Johnson — March 28, 2010 @ 12:49 am

  10. For a project like that, I would never pick someone merely on the basis of math skills. I would pick someone who had done real things before, the kids who had built hybrid cars at home, renovated the kitchen, geez, even just making a treehouse would be more relevant!

    But, in all likelihood the types of things I think are important were also considered. This sounds to me like a large design project. Engineering design education is difficult to do well, it’s difficult to figure out exactly how to do it well and there are relatively few educators who are consistently good at it. It’s also true that the best students are not necessarily the best at design. Still, I’d suspect the engineering design guidance given to the students first, before blaming their math education for any shortcomings.

    Comment by kcab — March 28, 2010 @ 8:25 pm

  11. “I really didn’t understand Algebra until I studied Calculus”… etc. (Erin)

    I didn’t really *understand* a lot of things about mathematics (as opposed to the ability to accurately replicate processes) until I taught 7th grade math using Connected Math Project materials. Of course, the point wasn’t the eventual fluency level of the teacher (me)–but the growth of mathematical understanding in my students.

    First–a thank-you to Robert for highlighting my blog.

    Second–I really like Diana’s idea of knowing content so well that you can expand, contract, oppose, amend, replicate, deconstruct and reconstruct it. You own it, and can create with it. That could involve fluency levels, as Brian suggests–which, at the upper end might look a little like Mihaly Csíkszentmihályi’s flow theory.

    But–it’s also possible to apply (for lack of a better word) new, imperfectly understood content knowledge to tasks and problems at lower levels, even if the results aren’t spectacular–and learn a lot from the experience. My middle school students composed music frequently– using pretty sketchy ideas of theory, derivative melodies, mistake-prone rhythmic notation.

    They knew just enough to write their own pieces; the learning came from struggling to figure out how to notate a melody, analyze harmony, put X beats in a measure–plus my corrections on their mistakes. With every composition, they got better. But I didn’t wait until they had mastered all the relevant knowledge before turning them loose to take a stab at creating.

    I realize we’re drifting into the great Skills vs. Knowledge arena here (a.k.a. False Dichotomy Stadium) but my essential point is that, however defined and intertwined, curriculum and instruction are what matter most in improving student learning. It’s very difficult to evaluate curriculum and instruction dispassionately, and almost impossible to create consensus around “best practice” in them.

    So–we argue that sifting out the 16 students with the highest math scores was pointless, since wind turbine technology is all about…physics. Or get diverted–once again–into rants about particular math curricula or “project based learning” (which I define very differently than Erin).

    Side note: My own children used Everyday Math curriculum in grades 3 through 6–and CMP in grades 7 & 8–I didn’t notice any, umm, insanity. I do agree with Diana (and just about every scholar who studies standards and curriculum) that we would be vastly better off studying fewer mathematical topics, in considerably greater depth, as other high-achieving nations do.

    Comment by Nancy Flanagan — March 28, 2010 @ 9:06 pm

  12. Nancy, Feel free to define what you mean by “project-based-learning”. As far as I can tell, the term has been used in two connotations: 1) using projects to motivate learning and 2) using projects to integrate academic knowledge and concepts with solvable problems. My opinion: the former is usually a waste of time as students are already intrinsically motivated (or not) while the latter is essential for quality learning. How would you characterize that term?

    Also, it is a shame that you have not had the opportunity to teach using a quality mathematics program such as Singapore math. While CMP (or EDM) purports to get to fundamental mathematics concepts, unfortunately the concepts that they promote are largely mistaken and contrary to those needed to solve problems as complex as the wind turbine problem (or any other mathematics, engineering, science or technology problem). Complex engineering installations are rather unforgiving in their operation and real mathematics is needed to ensure proper installation and operation.

    Considering that to solve our very complex energy/technology problems (such as your wind turbine example) that will be facing us over the coming years, we will require extraordinary numbers of talented, bright and *educated* engineers, it is troublesome that the actual number of engineers who have graduated from US colleges has greatly decreased in acutal number (not just percent) in the last 25 years!

    While we may never know if this decrease was due to “reform math” or not, certainly those initiatives have *not* helped to increase the number of students interested in math intensive careers. And even more troublesome, those “reform math” programs have hurt the mathematical prospects of minority students by increasing the math achievement gap. I can’t see how you (or any reasonable person) would support math programs that increase the achievement gap.

    You aren’t the first teacher to look at CMP vs. conventional math and see that learning concepts are better than just learning procedures. Who wouldn’t want their students to understand what they are doing and not just follow the rules/procedures? But the best countries in the world do both: understand concepts while ensuring computational mastery. Both CMP and EDM are completely inadequate when compared to the conceptual understanding and mathematical proficiency underlying much of the Asian mathematics programs (e.g. Singapore math). And they are completely inadequate in preparing students for work in the STEM fields.

    Comment by Erin Johnson — March 29, 2010 @ 1:37 am

  13. I really believe that if students were able to use more virtual education and online resources, and if they were brought into the collaboration on how to make education work, these students in K12 could find this kind of social project emphasis on their own.

    Good article today in Washington Post about the online ed world: http://www.washingtonpost.com/wp-dyn/content/article/2010/03/26/AR2010032602224.html?sub=AR&sid=ST2010032603631

    Points out that some stakeholders in “traditional” education are keeping things old-fashioned.

    Comment by Douglas — March 29, 2010 @ 7:01 am

  14. Nancy, the wind turbine project I would think would require instead fluency with Physics and familiarity with Engineering rather than just fluency in Algebra. Of course, you need a solid math basis, but by itself basic math is not a sufficient requirement for a project like this.

    Have the candidates been screened for AP physics exam grades? As you probably know, Physics is the ugly duckling of high school subjects. In my state MA (the state that everybody lauds these days as exemplary for education performance) it has been relegated as an optional discipline in the high school sequence, taught in the 12th grade if at all.

    In view of the sorry state of Physics in our schools, I am not surprised of stories like this one in Texas, Kalamazoo.

    “Dr. Schlack used the story to illustrate the point that good grades and high scores on standardized and national tests were no guarantee that students could integrate and utilize what they knew–sometimes, a credential is just a credential, a test score merely a number.”

    I’d be interested to look at Dr. Schlack’s presentation, if it is available in written form. I wonder if the presentation drew the right conclusion from this debacle after all.

    In your response to Erin, you write that “…I didn’t really *understand* a lot of things about mathematics (as opposed to the ability to accurately replicate processes) until I taught 7th grade math using Connected Math Project materials.” This is very concerning to me – I would caution against expressing a judgment about CMP and Everyday Math simply from a 7th grade view point of math.

    CMP and E.M. are both terrible curricula, that have been repeatedly reviewed by professional research mathematicians and have been found to be full of flaws. Have you seen the open letter to US Secretary of Education, Richard Riley, signed by over 200 people, mostly professional mathematicians but including other noteworthy people in education and scientific fields – among them Nobel laureates and Fields medalists?


    What good supporting continuous attention to curriculum and instruction – if this translates in support for known problem-curricula like CMP and Everyday Math.

    Comment by andrei radulescu-banu — April 1, 2010 @ 11:09 am

  15. @Erin and Andrei: I can’t speak to specific details of the wind turbine project, or to what core content knowledge might be involved/essential, beyond what was shared by Dr. Schlack in her brief remarks. What she described however–screening students by their standardized math scores (keeping in mind that HS science is not tested in many states), and their AP scores–is precisely how we characterize “student achievement” in this country. The way her institution screened candidates for an applied STEM program–quickly, and based on readily available numeric data–is how we assess student learning (and make Grand Proclamations about instruction and curriculum, BTW).

    Over at “Teacher in a Strange Land” and on my Facebook page, there are dozens of comments about this story. Most of them are from teachers, and all but one strongly support the idea that a constructivist curriculum (like CMP) makes a genuine difference in the way students learn math–using the second kind of project learning, the kind that Erin keeps saying is rare. Most of the discussion there is practice-based, with teachers sharing what they’ve observed about student learning. I am curious about why teachers–foot soldiers in this discussion–see this issue from a different perspective than the supposed “experts.” This would also not be the first time THAT has happened–see the makeup of the Common Core Standards committee, for example.

    BTW, I was not making grand proclamations myself, re: CMP. I was merely observing that–as a person who taught a relatively low-level math class using two very different curricula–I found one stilted and artificial and the other application-based and engaging for students. There probably are better curriculum models–I certainly wouldn’t have the expertise to argue that one. But it’s worth considering that we measure the value of a particular curriculum using standardized tests where item selection biases results. If the assessments and curriculum aren’t aligned–then results will not tell us which curriculum is “best.” Only which curriculum aligns most closely with test format and content.

    Finally, I think the reason there are fewer engineers being trained is because so many engineering jobs are now being outsourced. My son began his degree at a prestigious private engineering college in Michigan, long considered the Ivy League of automotive engineering. The career placement and internship information he received in his first year was pretty dire. He dropped out of Private U and shifted to a state university, where he’s studying Media Design and Technology and has two internships lined up. Why study something when the entry-level positions are being filled by $14K employees in India?

    Comment by Nancy Flanagan — April 2, 2010 @ 2:38 pm

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