"We don't want to privilege the word."
That was the justification that the director of a college writing program gave to a friend of mine for why he should allow his students the option to draw a picture rather than write an essay. Even back then, over two decades ago, our education experts were being careful not to privilege symbols.
But it has only recently occurred to me how this concern is playing out in today's Reform Math. Along with group work, group discovery, multiple solutions, and, of course, explaining answers to easy problems, there's doing math visually. This explains:
1. The large percentage of elementary school math problems today that involve classifying shapes.
2. The large percentage of K12 math problems that now involve reading charts and graphs.
3. The replacement of geometry proofs by visual "demonstrations," complete with spatial "translations," "reflections," and "dilations."
4. The growing reliance on graphical representations to solve algebra problems.
We see 4 at play, for example, in Brett Gilland's comment about the PARCC problem on my last Problems of the Week post. My inclination was to answer the question algebraically rather than appealing to geometric intuition. But, superficially, the problem really begs to be solved graphically, given how it's spelled. Consider the specific letters used to represent variables in the equation g(x) = mx + b. Because "m" is conventionally used to represent slop and "b," to represent the y intercept, many students will instantly recognize this as an easily graph-able line. The algebraic solution, via the Quadratic Formula, would prefer a different spelling: the standard spelling of polynomials (ax2 + bx + c), has the letter b representing the co-efficient of x rather than the constant.
While the specific spelling details of symbolic representations are superficial, symbolic representations themselves are anything but. They are what allow you to abstract away from two-dimensional space towards that which is increasingly difficult to represent graphically. Functions in 3-d are already tough; what about 4-d spacetime, or n-dimension space more generally? And consider other branches of mathematics like number theory and logic. Again, visual diagrams can represent the simple stuff (e.g., Venn diagrams for simple logical relations like if, only if, iff; or the number lines for simple numerical relations). But they do not take you very far at all. That's why, if you take a look at a math article, or even a physics article, you generally find many, many more lines of symbolic expressions than you find shapes, graphs, and diagrams.
But today's math teaching experts think otherwise. Their assumption seems to be that shapes, graphs, and diagrams are what make math meaningful, and that everything else is mere symbols. Thus, symbolic manipulation must be as mindless as "mere calculation."
The upshot is that what mathematicians and physicists use as tools for powerful abstractions are viewed by others as being as superficial as the decision to use the letter "m" to represent a slope and the letter "b" to represent an intercept.
Wednesday, November 25, 2015
"We don't want to privilege the word."
Monday, November 23, 2015
While catching up with this past week's New York Times Sunday Review, I was thinking of something Paul Bruno wrote on my last post:
If only fans of reform math were this concerned with rigorous controls and falsification exercises when considering their preferred education research.If only. Especially when it comes to practices that potentially waste large amounts of student time. (People seem in general to worry a lot more about policies that waste money than they do about policies that waste time--particularly other people's time, and most especially other people's children's time).
Accordingly, many of today's instructional practices would seem to do just that. One potential time-waster that's not getting nearly the scrutiny it should, besides making students explain their answers to easy math problems, is so-called "social and emotional learning." According to an article by Julie Scelfo in last week's Week in Review article, entitled Teaching Peace in Elementary School:
In many communities, elementary teachers, guidance counselors and administrators are embracing what is known as social and emotional learning, or S.E.L., a process through which people become more aware of their feelings and learn to relate more peacefully to others.
Feeling left out? Angry at your mom? Embarrassed to speak out loud during class? Proponents of S.E.L. say these feelings aren’t insignificant issues to be ignored in favor of the three R’s. Unless emotions are properly dealt with, they believe, children won’t be able to reach their full academic potential.
S.E.L., sometimes called character education, embraces not just the golden rule but the idea that everyone experiences a range of positive and negative feelings. It also gives children tools to slow down and think when facing conflicts, and teaches them to foster empathy and show kindness, introducing the concept of shared responsibility for a group’s well-being.This trend has been going strong at least since the 1990s, albeit under a variety of different acronyms (from PATHs to RULER). The general justification, in part, is stress among students, which, while invariably unprecedented, has ever-changing causes. The stressors, this time around, are:
not only the inherent difficulty of growing up, but also an increasingly fraught testing environment, a lower tolerance for physical acting out and the pervasive threat of violence. (President Obama last year characterized school shootings as “becoming the norm.”) Poverty and income inequality, too, create onerous emotional conditions for many children.Of course, poverty and income gaps predate public schools. School shootings are an entirely different matter, and I would not presume to know the general psychological effect their incidence, and publicity, has had on today's school children. As for the other two factors, I'll get back to those later.
First let's look ask what scientific basis there is for any of this. As Marc Brackett, director of the Yale Center for Emotional Intelligence, explains:
The neural pathways in the brain that deal with stress are the same ones that are used for learning,Right: stress is a form of learning. But how does this get us to school-based S.E.L.s? The best I can come up with is this: neural pathways go with stress and stress goes with learning and learning goes with classrooms, so learning to de-stress goes with classrooms. But this kind of "jerk is a tug and a tug is a boat and a boat floats on water and water is nature and nature is beautiful" argument can take us pretty much anywhere we want.
Author Julie Scelfo, however, also cites studies, in particular a meta-analysis of many studies:
Studies have found that promoting emotional and social skills correlates with improved outcomes in students’ lives. A 2011 analysis of 213 S.E.L. programs involving 270,034 kindergarten through high school students published in the journal Child Development found that the participants demonstrated significantly improved social and emotional skills, attitudes and behavior compared with a control group, as well as an 11-point gain in academic achievement percentiles.However, as the meta-study itself notes in its conclusion:
only 16% of the studies collected information on academic achievement at post, and more follow-up investigations are needed to confirm the durability of program impact.Scelfo cites a second study in which:
researchers from Penn State and Duke looked at 753 adults who had been evaluated for social competency nearly 20 years earlier while in kindergarten: Scores for sharing, cooperating and helping other children nearly always predicted whether a person graduated from high school on time, earned a college degree, had full-time employment, lived in public housing, received public assistance or had been arrested or held in juvenile detention.
Moreover, positive relationships, emotional competency and resilience have also been widely identified as helping to prevent mental illness.But that's relevant only if S.E.L. programs really do raise social competency scores long term. After all, it's not exactly headline news that social competence predicts success in the classroom, success on the job, and success in staying out of trouble. In some cases, regardless of social competency, S.E.L. programs might have the opposite effect, stressing kids out more by forcing them to air their emotions in class and engage in role-playing activities with arbitrarily chosen peers.
Here's the one specific S.E.L. moment described in the article:
At P.S. 130 in Brooklyn, where most students qualify for free lunch, a class of third graders recently sat in a circle and brainstormed, for the second day in a row, about steps they could take to prevent an aggressive boy in another class from causing problems during lunch and recess: A 9-year-old girl said she “felt scared” when the boy chased and grabbed her; Leo, an 8-year-old with neon orange sneakers, described, with agitation, how the boy sat down, uninvited, at his table and caused so much commotion that it drew sanctions from a cafeteria aide.
“How does he really bother you?” a girl in a pink sweatshirt asked, seeking clarification, as she’d been taught.
“Because,” Leo responded, his voice swelling with indignation, “it took 10 minutes from recess!”It's easy to talk about a kid behind his back; whether and how the class that actually has the offending boy in it has gone about handling him is left unclear. As for taking time away from recess, one has to wonder what these S.E.L.s activities are taking time away from.
Scelfo does acknowledge concerns about the time taken from academics. She cites Robert Pondiscio in particular, whom she characterizes as "a senior fellow at the Thomas B. Fordham Institute, a right-leaning education policy group in Washington." (I know Robert Pondiscio and wouldn't call him "right-leaning;" since no one in the article is identified as left-leaning, it's hard to guess Scelfo's reference point.) But, no matter: to Pondiscio and his kind Scelfo has this to say:
Skeptics of using school time to tend to emotions might consider visiting P.S. 130, where the hallway outside a third-grade classroom is decorated with drawings made by students showing their aspirations for the current school year.
One child hopes “to make new friends.” Another wants to “be nice and help.”
And as for Leo, who is frustrated about losing 10 minutes of recess?
Underneath a watercolor self-portrait, in which his body is painted orange, he wrote: “My hope for myself this year is to get better at math.” If S.E.L. strategies work, he will be better equipped to reach that goal.And here Scelfo ends her piece, apparently convinced she's addressed Pondiscio's objections. Which were:
It’s easy to recognize the importance of S.E.L. skills. It’s much harder to identify and implement curricular interventions that have a measurable effect on them. Thus ‘what works’ tends to be defined as ‘what I like’ or ‘what I believe works.Let's return, now, to the two other factors that Scelfo cites as stressing kids out these days: "an increasingly fraught testing environment" and "a lower tolerance for physical acting out." Instead of potentially wasting kids time with S.E.Ls classes, let's give them back the time they're wasting on standardized tests, and let's give them back their time for physically acting out, which used to be called recess. And, in particular, let's stop suspending everyone's recess whenever some kids get physical.
For a final alternative to S.E.L.-based stress reduction, let's return to Scelfo's opening paragraph:
For years, there has been a steady stream of headlines about the soaring mental health needs of college students and their struggles with anxiety and lack of resilience. Now, a growing number of educators are trying to bolster emotional competency not on college campuses, but where they believe it will have the greatest impact: in elementary schools.When it comes to college-level anxiety, one source that remains under-appreciated is the increasingly poor academic preparation kids get in high school. And here we come full circle. For one big reason why kids are increasingly ill-prepared academically is because they are wasting their time in the service of the latest education fads, from explaining their answers to easy math problems to working through emotional issues in S.E.L. sessions.
Saturday, November 21, 2015
The American approach is to build conceptual understanding through time-consuming student-centered discovery of multiple solutions and explanations of relatively simple problems. An internationally more successful approach is to build conceptual understanding through teacher-directed instruction and individualized practice in challenging math problems.I got a little flack for my sweeping statement about an “American approach” so I followed up with:
I should clarify what I mean by “American approach”: the approach inspired by national movements like the Common Core and the NCTM standards.The various objections fell into several categories:
1. The pedagogy I’m calling “American” is rare throughout the U.S.: most classrooms still follow a traditional model.
But even if most students are still sitting in rows with the teacher in front, more and more are using Reform Math textbooks like Everyday Math and Investigations, which solicit multiple solutions and verbal explanations for relatively simple math problems. Even if teachers matter more than textbooks, textbooks can place a ceiling on how challenging the material is. That's why traditional texts that date back to the 1960s and earlier are so much better than today's textbooks: they don't place such a low a ceiling on mathematical challenge. Instead they provide math-expanding opportunities for those who can handle them.
2. International comparisons based on test scores are unfair because Europe is “white” and Chinese students cheat. (Yes, one commenter actually said this, repeatedly).
But being white doesn’t make you good at math; China is only one of several Asian countries I discuss; and the many Chinese (and other Asian) nationals who disproportionately populate the top PhD programs and math-intensive careers here in the U.S. probably didn’t get where they are by cheating on math tests.
3. International comparisons based on performance on the PISA test are unfair because other countries track out their lowest performing students prior to age 15-16, the age range of students taking the PISA.
I’d be curious to see statistics on how large this effect is; I’ve looked around a bit and found nothing. Presumably our scores, too, are affected by dropouts and no-shows.
4. International comparisons based on the relative mathematical difficulty of high school exit exams are unfair because these don’t tell us how most students actually did on the various problems.
I’d argue that the predominance on some of these exams of much more challenging problems than American high school students ever see on any standardized test or graduation test tells us something about what kinds of mathematical opportunities students from other countries are getting that their American counterparts may not be.
5. In addition to international comparisons being unfair, a comparison within a province of one country of student performance before and after a student-centered discovery-oriented curriculum was introduced is also unfair. Why? Because it ignores what was going on concurrently in the rest of the country at large.
Then what kind of comparison is fair?
6. The Finnish exam and the Chinese Gao Kao are no more difficult than our Common Core-inspired exams.
My impression is that people who believe this haven’t looked closely at the mathematical demands of these tests, and/or believe that applying math to real-world situations and “proving” things using graphs (common in America's Reform Math and Common Core-inspired exams) to be of a mathematical challenge equal to or greater than the “mere” manipulation of abstract symbols. People with this impression should take a look at the research produced by professional mathematicians and check out the ratio of graphs and “real-world” situations to sequences of abstract symbols.
7. Students at an elite private high school do really well with a discovery-based curriculum.
If I were forced to enact a student-group-centered, discovery-based curriculum somewhere, I’d do it at a highly selective high school whose students were admitted, in part, based on their aptitude for (and therefore their solid foundational knowledge in) math. Such students stand the greatest chance of learning additional math independently, and from one another, and without too much loss in efficiency compared to what’s possible in more teacher-directed, individualized-problem-solving classrooms.
Thursday, November 19, 2015
1. A problem from the Common Core-inspired PARCC exam, which a commenter on Dan Meyer's blog thought I would particularly like, and felt was comparable to the sample problems on the high school exam in Finland that I blogged about earlier:
2. The Finnish problems I blogged about earlier:
1 c. Simplify the expression (a2-b2)/(a - b) + (a2-b2)/(a + b) with a not equal to b or –b.
5. A circle is tangent to the line 3x-4y = 0 at the point (8, 6), and it touches the positive x-axis. Define the circle center and radius.
6. Let a1…an be real numbers. What value of the variable x make the sum (x + a1)2 + (x + a2)2 + ….+ (x + an)n as small as possible?
9. The plane 9 + x + 2y + 3z = 6 intersects the positive coordinate axis at the points A, B and C.
a) Determine the volume of a tetrahedron whose vertices are at the origin O and the points A, B and C.
b) Determine the area of the triangle ABC.
13. Let us consider positive integers n and k for which n + (n +1) + (n + 2) +… + (n + k) = 1007
a) Prove that these numbers n and k satisfy the equation (k +1)(2n + k) = 2014.
III. Extra Credit
1. Consider what is involved in solving the PARCC problem (recognizing that it's a quadratic; seeing what the a, b, and c coefficients of the standard form of the quadratic equation correspond to here, and seeing that 4ac must be positive). Compare these recognitions, apparent with minimal symbol manipulations, with what's involved in solving the Finnish problems.
2. Some problems can appear mathematically abstract without involving much math. Discuss.
3. About Finnish problem 13, the commenter on Dan Meyer's blog writes:
Problem 13 is a variation on the proof for the sum of an arithmetic sequence. Algebra 1 in the CCSS, btw. We don’t require a proof, but if we were going to require one, that would be one of the easiest to pick. And reform mathematics programs will typically explore that proof (geometrically, and then with algebraic symbols to formalise). Traditional American textbooks just give the formula as something to memorize because, get this, accurate procedures without understanding are considered sufficient.Discuss.
Tuesday, November 17, 2015
In my last post, I categorized the critical comments on Barry and my Atlantic article into 7 categories. A similar, though generally more sophisticated set of critical comments appeared on Dan Meyer’s blog.
Again, there were those who took issue with our specific examples of expected explanations. They agreed that requiring such explanations isn't reasonable, that teachers should be flexible, and that explanations could be oral and informal. But they also argued that explanations in general are a good idea. And they are. But explanations are most effective and efficient when solicited in a teacher-centered discussion, or when used to help a student understand why he or she got a particular answer wrong.
There were those (including Dan) who conflated explaining answers and thought processes verbally and diagrammatically with doing math proofs (the latter is something we think there should actually be more of).
And there were those who conflated explaining answers and thought processes verbally and diagrammatically with showing work. Others seemed to think that the kind of work displays Barry and I were endorsing was work consisting only of mathematical symbols. But there are plenty of words that go into work-showing (and proofs), including reasons (“given"; “side-angle-side"; “without loss of generality”) and units (“miles per hour”; "liters of water loss per minute"; “pounds of salt per pounds of total mixture”).
There were those who cited student testimonials about how producing explanations enhanced learning. Many students would beg to differ, though not everyone wants to listen to them.
In the "communication skills necessary for math-related professions" category, there were those who specifically discussed how mathematicians themselves
use words in describing their discoveries all the time – and have for a long time. That’s why some doctorates in mathematics require a foreign language so that the candidate can read the mathematicians’ writings in the original language.The question, however, is whether requiring the kinds of verbal and diagrammatic explanations we critique in our article will help prepare future mathematicians to communicate with other mathematicians. The mathematicians I’ve talked to are skeptical. A related question: are mathematicians who learned math in pre-answer-explaining times deficient in their communication skills?
True, teachers, including mathematicians who teach other mathematicians what they have discovered, should be able to explain the math in question verbally and diagrammatically, as needed. But that doesn’t justify requiring K12 students to provide such explanations in their assignments. Teacher prep programs exist for a reason. And are teachers who learned K12 math in pre-answer-explaining times worse at teaching math concepts than their contemporary counterparts?
In addition, there were those (including Dan) who argued that a student with correct but unexplained answers, even my hypothetical student who “progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus”—still might not understand the underlying math. My response is twofold. One: this ultimately depends on what we mean by “understanding,” and matters only if the student’s understanding is insufficient preparation for the next level of math. Two: the problem would be remedied by assigning more conceptually challenging math problems--of the sort that simply can’t be solved if you lack the requisite depth of understanding. Such problems do exist: indeed, if they didn't, none of this would matter.
Perhaps the most-discussed argument was that of the supposed meta-cognitive benefits of explaining your answer (and of students listening to each other explain their answers), and, relatedly, of the purported need for teachers to understand what is going on in students' heads at a level of depth for which mere answers and work-showing are insufficient. One commenter simply writes “Metacognition! I can’t imagine anyone not seeing the value of that, but you never know."
In support of meta-cognition, some cited students who can crunch numbers or apply formulas but lack conceptual understanding. I’d say these students simply need practice with problems that require more conceptual understanding than mere “number crunching” and “formula applying” assignments do. In general, assigning more conceptually challenging math problems is a much better, and more efficient, way to help students develop conceptual understanding (and, as one commenter put it, to “ferret out” those “math zombies”) than bogging them down with verbal and diagrammatic explanations to problems that aren’t conceptually challenging.
One person cited kids who do problems incorrectly, but, when explaining them, realize their mistakes. That’s a benefit that can be achieved simply by soliciting explanations specifically for wrongly-answered problems for which an explanation is likely to lead to this sort of realization.
In short, my answer is more teacher-led discussion of underlying concepts, with teachers calling on students, as appropriate, to develop concepts and explanations (for which Japanese classrooms as described in this discussion are a good model); and more individualized practice with conceptually challenging math problems.
A comment I made to this this effect on Dan's blog led to a whole new thread of comments, which I will digest in my next post.
Sunday, November 15, 2015
There are now over 450 comments on Barry Garelick and my article in the online Atlantic, Explaining Your Math: Unnecessary at Best, Encumbering at Worst.
Apparently, this is a rather controversial topic. Critical comments appear to boil down to 7 categories:
--What the Common Core actually says: Some people stated that there’s nothing in the Common Core itself that requires students to explain answers and thought processes verbally and diagrammatically. (We noted that there are parts of the Common Core that nonetheless can be, and are being, interpreted in this way).
--Our specific examples: Some claimed that examples we used to illustrate work showing aren’t representative of what’s generally going on, and that good teachers would be more flexible and reasonable about what constitutes adequate explanations. (That would be nice, if so.)
--Who is actually affected: Some claimed that our objections apply only to very small subsets of kids. (We pointed out that these practices are problematic for all students, and in particular for second language learners and children with language impairments).
--The virtues of showing your work: Some people conflated showing work (which we agree is reasonable wherever there’s work to show) with explaining answers and thought processes verbally and diagrammatically.
--The virtues of doing math proofs: Some people conflated doing math proofs (which we agree there should be more of in high school math) with explaining answers and thought processes verbally and diagrammatically. (We pointed out that there is actually relatively little emphasis on mathematic proofs in both the various Common Core-inspired curricula and tests).
--Communication skills necessary for math-related professions: Some people believe that having students provide the sorts of verbal and diagrammatic explanations we critique in our article will help prepare future engineers and scientists for the communicative demands of their jobs. (The question then is whether engineers and scientists who learned math in pre-answer-explaining times are deficient in their communication skills compared with their more contemporary counterparts.)
--Counter-exemplary anecdotes: Some people described how well explaining answers and diagramming thought processes work for their students or kids.
--Faith: Some people are sure that meta-cognitive processes are the best way to develop conceptual understanding. (We would say that a better way is to emulate the countries that outcompete us in math, giving kids more direct instruction and individualized practice in conceptually challenging math problems).
Friday, November 13, 2015
…in an article Barry Garelick and I just published in the online Atlantic.