Articles I didn’t finish reading, Dept.

This article in Business North was titled “Teachers Learn to Make Math Fun, Engaging”, which alone would have caused me not to read any further, but I was curious as to how many tropes I would run into. The article doesn’t disappoint.

It starts off with an opening as classic (and nausea inducing) as Denny’s bacon and egg eye-opener:

“When will I ever use this in my real life?” is a plaintive question students often pose to their math teachers. For over 20 area elementary and middle school teachers, the “Engage, Learn, and Connect Math Topics” summer workshop at The College of St. Scholastica (CSS) provided teachers some solid ways to respond to that perennial question.

I usually find that students ask this when they are frustrated and/or having difficulty with a particular type of procedure or problem. When they are capable of doing the procedure, they tend to be just as engaged as they would with any activity. Not to mention that the prevalence of this question is helped along by TV sitcoms that may feature such a situation. The question is met with the predictable laugh track as the camera zooms in to a close up of teacher’s frustrated expression.  

“Engagement is key. There is no one size fits all approach to teaching. Workshops like ours help to bridge some gaps and help teachers dispel some misconceptions of what it means to study math.”

Of course, what is ignored is what I pointed out above; that proper instruction that allows students to be able to do problems and stretch beyond the initial worked example breeds success.  And success in turn breeds motivation. But the pervasive group think is that engagement comes first, and then success and motivation follow.  

And part and parcel to engagement, is this gem of a trope without which no article would be complete:

“It is important for our society to prepare kids for the jobs of the future. There are new jobs that don’t even exist yet that will come as a result of the new ideas in the math and science fields. We want to engage students so they know they can succeed in those fields and to prepare them to be ready for the new opportunities yet to come.”

Oops, I guess I did finish the article, didn’t I?



Oh, Really, Dept.

There is a very long article in Quanta about various schools in the New York City area and how teachers go about teaching science and math.  The context of the article is the usual “The US has not done well in science and math”, and the not so subtle sub-text of “That’s because we’re teaching it wrong”. Per the article, “Wrong” means teaching by rote, by cookbook, by telling students formulas instead of having them derive it, by having them do experiments with predictable results. In short, the traditional model of science and math treats students as novices. The collective wisdom of the teachers featured in this article, however, treats students as if they are experts.

The emphasis is project-based/inquiry-based structures in which students are given an assignment or problem with some prior knowledge, and expected to make up what they don’t know on a just-in-time basis.  Guidance is kept to a minimum, with the student expected to discover what they are supposed to learn–or, in the case of science experiments gone awry, to discover what was the cause, and correct it.

The article profiles several teachers who have used these various methods. If a teacher can have success using such techniques, that’s great. But in reading the article, I do have to wonder about such methods and their basic assumptions and mischaracterization of the conventional/traditional techniques which they hold as “not working”.

For example:

Too often, said Vice Principal Elizabeth Leebens, students “get the history of science rather than getting an opportunity to do it for themselves.” Still, Leebens was surprised when, after a group of science club students asked Comer for help with their ice-cream-making experiment, which had already failed seven sticky times, Comer told them, “Go back and check your process.” Off they went to the bathroom to dump the latest batch and start over.

“I call it productive struggle,” Comer said. “That’s where the growth happens.”

In meetings, Leebens said, Comer has “challenged me to stop doing the cognitive work for kids: ‘Let them do it themselves. They can do it; they can do it.’” For Comer, she added, it’s about “life lessons and also high expectations for kids in letting them see what they can do before the adult decides what they can do.”

The “struggle is good” philosophy prevails at these schools and among these teachers. I am guessing that because such approach has come under criticism in the past, that it is now called “productive struggle”.  In any event, there are good ways and poor ways to make students struggle.  One hears about “scaffolding” problems so that students stretch beyond initial worked examples or experiments but have enough prior knowledge to make the leap.  But there are those who consider “scaffolding” as handing it to the student, giving the student too much information so they don’t get the “deep understanding” nor learn how to “problem solve” (to use the edu-trendy parlance that has gained popularity over the past few years).

The article talks about a math teacher and how she came to her particular philosophy of productive struggle.

Midha has always loved math — it came easily. She suffered some doubts about her abilities in 10th grade, thanks to an algebra teacher who did the standard “chalk and talk” at the blackboard. But during freshman year at Wesleyan University, she took calculus with a professor who considered mathematics to be an art. “Everything he said was so profound,” she recalled. “I was like, ‘Oh, my god. This is the coolest thing ever.’” And that was that. She became a math major — and also a music major; she plays classical guitar.

With this quote, the article then lays the groundwork  that the standard “chalk and talk” doesn’t work (and making the general assumption that all teachers using traditional methods never ask questions of students, but just lectures in a boring and uninspiring manner).  Rather, teachers are required to be inspiring and profound and math is to be taught as an art. Or something like that.  I don’t know much else about her calculus professor or what he did in class, but I suspect that he relied just a little bit on “chalk and talk”. 

The article goes on to describe a lesson the Midha gives her students in which they have to determine the heights of buildings outside their classroom using an iPhone app that served as an inclinometer, and using what they know about trigonometry.  She calls this a “struggle problem”:

Using two iPhone apps, they measured the angle of elevation from point A to the top of the building and the distance the walker traveled from point A to point B, at the base of the building. These two measurements gave them crucial pieces of information about a right triangle, and from there — using what they’d learned so far in class about trigonometry — they are now charged with the task of calculating their building’s height. But first there are ponderous stares, frowns, diagrams drawn and redrawn amid plumes of eraser dust, and a collective buzz of puzzlement:

“I dunno, man. I really don’t know.”

“Help! Help!”

But they won’t get much help from their teacher.

“I’m the teacher who stopped giving them the answer,” said the 30-something Midha. “In every unit that we do, I warn them: ‘I’m going to give you the tools that you need, but I’m never going to tell you how to do something. You have to figure out how to do it, you have to figure out the answer, and you have to prove to me why you think that answer is what it is.’” She also offers reassurance through an oft-repeated mantra: “The only way that you can fail is if you give up. If you continue to persevere, if you continue to try, if you continue to work through this, you will get this. But if you give up, you will fail.”

But after a few days of this, with students not getting anywhere she decides to give them some hints:

“Remember: Where was the angle of elevation measured from? The eye. So when you are drawing and calculating, remember that. Your job is to calculate the total height of the building. … Remember: There is a reason we measured the eye height. There is a reason we measured the eye height.” Repetition, and more repetition, is key for penetrating the adolescent brain.

Midha also provides a bonus hint of sorts, pointing out to her students that they measured the eye height in inches and the building distance in miles, and that the worksheet asks for the height of the building in feet. With that, she leaves her students to their own devices —“Good luck!”

She circles the room, surveying the progress, asking simply: “Does it make any sense? … Why doesn’t it make any sense?” Despite her hints, the relevance of the eye height is proving elusive, and the inch-feet-miles conversions are confusing. She reiterates her tips one-on-one with the groups, and then lets them loose again, declaring: “I’m going to walk away now … ”

Which raises the question of whether it would do any harm to just tell students what they need to know when after some grappling with the problem they are clearly in a receptive mode to receive and process such information.  No, I guess not.  Best to just let them struggle.

I will stick with the more controlled variety of struggle, and provide the scaffolding needed and when it appears fruitful, tell them what they are now ready to hear.  And in so doing, it appears that many will brand me “old school”. I recall a professor I had in ed school saying to a student that such direct instruction may be good in the short term, with students scoring higher on tests than those using inquiry-based techniques. “But in the long term, those using inquiry have a deeper understanding.”

This appears to be the premise that guides the teachers described in the article, as well as an ever-growing body of teachers indoctrinated into this type of teaching.  

I’ll risk being called old school–and other names.






Count the Tropes, Dept.

Yet another in an unending series of articles about math education and what parents should (and should not) be doing to help.  This particular one is from Chicago Parent and contains the usual tropes/mischaracterization about how math used to be taught and why the new ways are so much better.  We start with the usual ever-popular one:

STOP TEACHING THE TRICKS: A large amount of research has gone into the progression of teaching students mathematical concepts. The shift has moved away from teaching students to blindly follow rules and toward making sure they understand the larger mathematical ideas and reasoning behind the processes.

I have written extensively about how math was taught in the past, citing and providing examples from textbooks from various eras. It might be interesting to look at some of the books used in previous eras that have been described as teaching students to “blindly follow rules”.  Many, if not most, of the math books from the 30’s through part of the 60’s were written by the math reformers of those times.  It makes the most sense to start with the series I had in elementary school: Arithmetic We Need.  The reason is because not only is it from the 50’s, but also one of the authors was William A. Brownell, considered a leader of the math reform movement from the 30’s through the early 60’s. Today’s reformers also hold Brownell in high regard, including the prolific education critic Alfie Kohn, who talks about him in his book The Schools Our Children Deserve .

In arguing why traditional math is ineffective, Kohn states “students may memorize the fact that 0.4 = 4/10, or successfully follow a recipe to solve for x, but the traditional approach leaves them clueless about the significance of what they’re doing.  Without any feel for the bigger picture, they tend to plug in numbers mechanically as they follow the technique they’ve learned.”  He then turns to Brownell to bolster his argument that students under traditional math were not successful in quantitative thinking: “[For that] one needs a fund of meanings, not a myriad of ‘automatic responses’. . . . Drill does not develop meanings.  Repetition does not lead to understandings.”

Brownell, however, requires students to do the practice and exercises held in disdain by those who believe traditionally taught math did not work.  The series contained many exercises and drills including mental math exercises. Such drills might appear to run counter to Brownell’s arguments for math being more than computation and “meaningless drills,” but their inclusion ensured that mastery of math facts and basic procedures was not lost. Also, the books contained many word problems that demonstrated how the various math concepts and procedures are used to solve a variety of problem types. Other books from previous eras were also similarly written—most authors were the math reformers of their day—and provide many counter-examples to the mischaracterization that traditional math consisted only of disconnected ideas, rote memorization, and no understanding.

But let’s move on. It gets worse:

…By teaching the trick before a child has this foundation, you may be inadvertently doing more harm than good. Students become reliant on tricks and fail to master the conceptual understanding needed to use the tricks appropriately.  Remember, kids will be more successful in the future as a problem-solver than as a memorizer.

What this article and many look-alikes caricature as “tricks” are actually mathematically sound algorithms. The idea that teaching standard algorithms “too early” eclipses the underlying conceptual understanding most likely stems from Constance Kamii’s infamous study “The Harmful Effect of Teaching Algorithms to Young Children” which was published by the National Council of Teachers of Mathematics (NCTM) in an annual review.  It has become the rallying cry that has garnered more believers than the idea that the substance called “laetrile”, extracted from apricot pits, is a cure for cancer.

With respect to the math books of earlier eras, they started with teaching of the standard algorithm first.  Alternatives to the standards using drawings or other techniques were given afterwards to provide further information on how and why the algorithm worked.  This is opposite of how reformers are advising it be done now.  What happens, typically, is the first way a child is taught to do something becomes their anchor, with everything else being supplemental.  By teaching the supplements first, there is a mix-up of main course versus side dish, with many students unable to tell the difference.  The popular theory is that students now have a choice and can pick the method that works for them.  I have tutored students showing profound confusion, asking me what method they should use for particular problems, feeling that various problems demand different versions of the same algorithm.

Note also the ever-popular warning against memorization: Memorizers are not problem solvers apparently. Well, sure, there may have been teachers who taught math poorly and had students memorize day in and day out with no conceptual context. To listen to the people who write these articles, it seems that the nation was plagued with such teaching, as if poor teaching was/is an inherent quality of traditionally taught math.  I would argue otherwise and go so far as to say that many of the “understanding-based”, student-centered, collaborative techniques that dominate many of today’s classrooms are inherently ineffective and damaging.

Memorization is the seat of knowledge. Eventually students just have to know certain facts and procedures and do them automatically. The idea that memorizing eclipses the understanding of what, say, multiplication is presumes that students are taught the times tables with no connection to what multiplication is, and what types of problems are solved using it.

Other than that, the article is pretty good, I suppose.


Nothing New To See Here, Dept.

The thrust of a recent article at “The 74” argues that parents have an erroneous view of how well their children are doing in school:

“Parents rely more on their child’s report card (86 percent) than on annual state test scores (55 percent) to understand whether their child is on grade level. Two-thirds believe report cards provide a more accurate picture of achievement than standardized tests.  “The amount of weight that parents place on report cards, they truly see it as gospel,” said Elizabeth Rorick, deputy executive director of communications and government affairs at the National PTA. “We need to push our parents to look at the overall performance. It’s not just what’s happening in the classroom. It’s important they review their annual state test results as well.”  “

The bottom line for the article is that parents need to be more involved in their children’s school and in understanding what it is that their children are learning or not learning.

To this end, the Washington based non-profit, Learning Heroes, which commissioned the survey had sage advice to give:

Learning Heroes partnered with the National PTA and Scholastic to create a back-to-school action list of ways families can engage and communicate with schools. It gives advice for analyzing state testing data, preparing for parent-teacher conferences, and practicing skills at home.

All well and good I suppose except for the fact that the article appears to omit any culpability on the schools’ part over why the report cards don’t match up with the state test results, or why students need remediation when they get to college. Nor for that matter do they address whether and to what extent the curriculum used for math, as well as the pedagogies employed play a role.

The article resorts to the usual advice of help your kids by practicing skills at home. That parents need to make up for what students are not  being taught in school is not in the province of this particular article, I guess. Math facts, apparently, are to be taught at home, and not practiced in school. Of course, students can get such practice by enrolling in outside learning centers such as Kumon or Sylvan, but why should “The 74” be concerned with that rather inconvenient equity and social class issue?