Silver Lining, Dept.

In this recent article, we learn about a math ed professor from Southern Illinois University who is going to Tapei, Taiwan to bring them American methods of teaching math.

“I’m looking forward to sharing my perspective on mathematics knowledge and education with an international audience,” Lin said. “I will be telling them about American teaching techniques and learning more about their methods. I want to know more about their recent teacher training and compare it to ours.”

Lin noted that in recent years, American mathematics education has shifted toward a focus on teaching for understanding, assuring that students comprehend what they are doing rather than just learning to apply formulas and procedures. The teaching approach used in Taiwan is a more traditional format that relies heavily on written computation.

To readers who are unaware of the battles over math education in this country, this sounds just dandy.  There is the usual misconception packaged as an implicit assumption, that in the US we have never taught for understanding, and that students are proceeding by rote (i.e., “math zombies” as some clever bandwagon math teachers have dubbed it).

To readers who are more aware of what has been happening, the obsession over “understanding” has been going on for more than a century.  But during the last 3 decades, such obsession has grown legs, due in large part to the National Council of Teachers of Mathematics’ (NCTM’s) standards written in 1989 and rewritten in 2000. Students in lower grades must now demonstrate understanding of the mathematical concepts in simple arithmetic operations and are taught the standard algorithm later.

I’m seeing the results of such “deep learning”, “deep understanding”, “deep dives into math” and other 21st Century nonsensical edu-jargon in the 7th grade math class that I teach. Students are reluctant to do double digit multiplication, or to divide. Many of those who do double digit multiplication use an inefficient “partial products” method, and are relatively unaware of the standard algorithm method for doing so.  Math facts for many are not mastered, and I get many requests to use a calculator.

Through a series of lectures, workshops, seminars and other activities, Lin will work with teachers-in-training, teachers and administrators regarding theories and practices used successfully in mathematics education instruction in the United States. He will also assist in developing appropriate tools for testing mathematical skills and knowledge and help analyze the results. In addition, Lin will assist colleagues in preparing students for Taiwan’s new compulsory national tests for graduating teachers and will make recommendations for course revisions or other changes to assist students.

Last time I looked, Asian countries were doing just fine on international math tests despite the highly disregarded and despised “traditional methods”.  I guess the silver lining is that if American methods are adopted hook, line, and sinker in Asia, we’ll come out smelling like a rose!

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How to Write a Pro-Common Core, 21st Century-Skills-Based Polemic

You’ve probably seen this before.  Someone, probably in their twenties, thinking they are holding their own among a group of highly educated people, prattling on about the uselessness of most college courses and disciplines and the value of a PhD.  The group listens politely and after the outburst continues their conversation as if the young orator hadn’t said a word. The young person thinks that they are pretending they didn’t hear the polemic because they didn’t want to hear the truth.

The type of foolishness that one engages in during one’s twenties can be forgiven because of youth, inexperience, and the envy that comes from realizing but not accepting one’s place in society. Fortunately most such people grow up.

But the same thing is occurring with alarming frequency in the field of education in general and math education in particular.  This phenomenon may be attributed in part to the ease in which one can air one’s views on internet-based platforms, such as blogs and social media.  But such views are also published in so-called peer-reviewed journals, in which the peers have known each other and have been taking in each other’s laundry for years.  One reads their polemics in places such as Phi Delta Kappan, the journal of the American Education Research Association (AERA) and various publications of the National Council of Teachers of Mathematics (NCTM).

While people who have done legitimate educational research in the field of cognitive science have been critical of what has been written, the people who should be ignored are the ones holding court.

I recently read a piece published by Achieve.org which unequivocally and uncritically supports the Common Core standards.  One such piece caught my attention and since the writing of poorly informed and unscientific polemics seems to be the new standard, I thought I would provide a guideline on how to construct such papers, using this particular atrocity.

State that STEM is more than just “technical”. That is, STEM workers include support staff, like lab techs, technical writers, people who don’t necessary know math or science in other words.  Using such logic, one can say that the medical profession also includes medical secretaries and custodians, which would give me some relevance in the medical field when I worked as a janitor at the University of Michigan medical school during the summer.

Recent claims that the market for STEM workers is saturated are based upon a narrow definition of STEM.  When I advocate for STEM thinking and STEM skills, I have the 4 C’s in mind: collaboration, communication, creativity, and critical thinking.  While I realize some STEM jobs are in higher demand than others and that some sectors are, indeed, saturated, I don’t think most 21st Century employers prefer employees that can’t work together, can’t communicate well, and can’t figure their way out of a paper bag. I suspect market analysis for STEM jobs does not include all the support roles such as technical writers (need physics) or quality assurance (operating coordinate measurement machines).  But, as Hacker points out, those skills are not developed by performing tedious math processes, especially those largely performed through a memorized sequence.

 

Point out that Common Core fixes these problems by leaving out what this author and others of her ilk thinks are tedious and useless (but from which she and others benefitted in their careers). Instead, CC focuses on (wait for it) “deeper learning” like exponential functions, which it pushes down into Algebra 1 when students need considerably more experience with basic algebraic procedure.  Having taught exponential functions from a CC-aligned algebra textbook, I decided to leave such lessons until the last part of the year to get to it if I have time.

 The problem could be all but fixed if teachers were using materials accurately aligned to the Common Core State Standards (CCSS).  Love them or hate them, CCSS weeds out most of the minimally-extensible, boring, tedious procedures and leave room for explorations and developing numeracy.  But the extraneous procedures remain in unaligned texts and are encoded into curricula, leaving teachers with little choice other than to teach them.

As such, I agree with Hacker who wrote “The Math Myth” that Algebra 2 is not only unnecessary for graduating high school but that it is unnecessary altogether given the arcane, and tedious things it teaches that are of questionable use.

 I believe most teachers are doing the best they can with sometimes impossible situations.  Most explain the procedures before showing students what to do. However, students quickly figure out they can pass tests in the short-term by zoning out during the explanation if they focus on the steps.  Hacker points to many examples of boring, tedious procedures that are in traditional textbooks.  He argues that it is wrong to require all students to learn those procedures, and I agree with some of his examples. 

Ignore that education is about giving students opportunities, not slamming doors shut. Make arguments about the requirement for Algebra 2 to graduate high school being onerous and is preventing students from graduating and ignore that some of the topics that used to be taught in algebra 1 are left out of CC.  Also do not say that the CC treatment of algebra 2 similarly waters down what is presented; rather state that it focuses on few topics in more depth.  Omit mentioning that such action jeopardizes those who are truly committed to majoring in STEM and that Jason Zimba once remarked that CC does not present a path to AP calculus, nor a path to more selective universities (not to mention to a STEM career).

However, it is difficult to imagine why one would want a sixteen-year-old to make academic decisions that could set him or her back a year or two at college. After all, it is not uncommon for college students to change their majors multiple times.  I believe the better idea is to limit topics in high school math to those in the CCSS and connect those topics to thought processes we all use in real life.  Some, like Hacker, argue for two tracks:  one for calculus and one for statistics.  The CCSS balances the two with respect to content, keeping students’ options for both.

Follow these steps and you too can be an education hero–and maybe even be invited to give talks at NCTM! Who know what the future has in store for you.  Goodness knows we know what’s in store for our students.

 

We’re Saying the Same Thing, Dept.

One well-known gambit that is used by those seeking math reforms in debates with those who take a traditional view of math education is to show how much both camps have in common. That is, traditionalists don’t rely solely on whole-class and direct instruction; they incorporate discovery and have students work in groups from time-to-time. And progressives don’t rely solely on discovery approaches and collaboration. The reformer will use this in arguments and say “I think we’re both saying the same things.”

Actually, speaking as a traditionalist, we’re not.  Yes, some people have a lot of success using student-centered, inquiry-based techniques, and others have success using traditional approaches.  But there is the question of “balance”: how much of each is being used.  There’s also the issue of misrepresenting how traditional math is and has been taught: i.e., traditional math teaching relies on “rote memorization” without context or understanding, or connecting concepts to prior ones.

Another gambit used in the argument is one that is featured in a guest blog at Rick Hess’ “Straight Up” blog.  In it, the guest blogger, Alex Baron, states:

[G]iven the multifarious nature of students, teachers, and school contexts, it seems clear that no single prescription would work for all, or even most, students. However, policymakers proceed with “research-based” inputs as if they would work for everyone, even though this contravenes our foundational sense that no two students are the same.

There are variations in effectiveness depending on the student. But that doesn’t negate research that demonstrates positive effects on large populations of students, such as Sweller’s “worked example effect”, scaffolding and “guided discovery”.  Despite such evidence, there will always be those who claim “Ah, but there are exceptions.”

Yes, there are.  One rather inconvenient exception is that students with learning disabilities seem to do better with explicit instruction than with discovery-oriented and other reform type approaches. And in fact, such exception raises the inconvenient question of whether and to what extent the reform-type approaches may be causing the increases in math learning disabilities over the years. And if that is the case, why not rely on the remedy in the first place? (See in particular this article, an updated version of which appears in this collection of articles on math education.)  In addition, there are many exceptions to the approach that “understanding” must always come before procedure.  With respect to the latter, many people have interpreted Common Core standards as requiring “instructional shifts” that place understanding first. (See this article.)

I am reminded of something said by Vern Williams, a middle school math teacher who served on the President’s National Math Advisory Panel in 2006-2008:

I have always stated that if a reform minded teacher produces competent, intellectually passionate students, they will absolutely escape any criticism on my part. But the opposite seems never to occur. Regardless of stellar results, the traditional teacher will always be criticized for being a self centered sage on the stage, controlling student learning and running a draconian classroom. Their students may be the happiest most accomplished students of all time but the teacher will never be good and pure until they cross over to the reform side.

I guess there is no end to the gambits of “we’re all saying the same thing” and “no one size fits all”.

Articles I Never Should Have Started Reading, Dept.

Ed Source has published the latest in a seemingly never-ending series of articles on how best to teach math.

Nearly two decades ago, international math and science tests revealed mathematics instruction in the United States as an inch deep and a mile wide. Since then, we have grappled with how to get depth over breadth in classrooms. 

This is confirmed every time I work with teachers or parents, most of whom remember the procedural, answer-based mathematics that they were taught, and the results of that approach. I often hear phrases like: I was really good at math, and then I just didn’t get it anymore; I was never good at math; I was dumb.”

And of course in the world of edu-groupthink, the only reason for this is because the students were taught procedures and nothing else. No other reasons will do. On the other hand, students whose “answer-based mathematics” served them well, are regarded as exceptions; they would have done well in any learning environment because they liked math and were interested in it.  The idea that instruction that resulted in success in problem solving served to motivate students to go further is definitely not in the group-think dogma or lexicon.

The parents of students who major in STEM fields understand that well-organized mathematical solutions are their own explanations. Many of the math reformer crowd including the author of these folks seem to regard translating “of” to “multiply” as rote, or mechanical decoding.  I, and many others like me, regard it as precisely the kind of “understanding” that is appropriate. The student who goes straight to a mathematical encoding of the problem is the one who likely has the best functional understanding.

The thinking amongst math reformers is that one indication of “understanding” is if a student can solve a problem in multiple ways. Thus, the reformers then insist on having students come up with more than one way to solve a problem. In doing so, they are confusing cause and effect. That is, forcing students to think of multiple ways does not in and of itself cause understanding. They are saying in effect that “If we can just get them to do things that LOOK like what we imagine a mathematician does … then they will be real mathematicians.”

The “answer-based” classroom is now the latest perjorative description along with Phil Daro’s view that math has been taught as “answer getting” with no regard for process or underlying concepts.

Instead, math classrooms become discussion groups.  I’ve been told by more than one edu-expert that the content standards of the Common Core math standards are there to serve the eight Standards of Mathematical Practice. Thus, critiquing each others’ work and developing the “habits of mind” outside of the math courses in which instruction would naturally develop such habits is thought to make students look like they’re thinking like mathematicians.

A friend of mine has a son who is majoring in math at MIT. The father had to work with him every night in the lower grades (K-6) to ensure he was mastering the math procedural skills that were not being taught in the son’s classes.  When the father was in school, he made it to AP Calculus in high school without any parents’ help. He has remarked that this is not possible today–despite the student’s interest in math. Students don’t just learn it anyway. They need to know how to (dare I say it?) “get answers”. And to use procedures.

Our PreK-12 math curriculum is taught using principles of “growth mindset,” a concept developed by Carol Dweck, a professor of psychology at Stanford University. Taught with this framework, students learn mathematical reasoning; embrace mistakes as learning opportunities; and work together to build the flexibility and resiliency required for success in math. The goal is to help students stay motivated in the face of challenging work. We’re working to reframe the question, “What does it mean to be good at math?”

Presenting students with open-ended problems with many possible “right answers” is neither necessary nor sufficient to be “good at math”. Getting them to make mistakes by tripping them up with “divergent thinking” type questions is also not necessary in order to obtain the brain growing effect that Jo Boaler has popularized in her writings.

Just teach the students what they need to know, even if it means they are “getting answers”.

Where it all began, Dept.

I began getting involved with math education during a six month stint, working in Senator Ron Wyden’s office (D-OR) from April to November 2002. Shortly before I started, there was a panel discussion hosted by American Enterprise Institute in Washington DC on the state of math education in the US.

The panel included two members of NCTM (Gail Burrill and Lee Stiff, both former Presidents of NCTM), David Klein, a math professor at Cal State U at Northridge, Tom Loveless of Brookings Institution, and Michael McKeown a medical research at Brown University who cofounded a website called Mathematically Correct (to inform parents what was going on in math education).

Lynn Cheney (Dick Cheney’s wife) moderated the discussion. It is interesting to listen to the opinions expressed. Not much has changed in terms of the arguments, except that at that time, NCTM’s standards ruled the roost, and now Common Core standards do. Common Core’s standards have a lot of ties to NCTM’s particularly in the area of the 8 Standards of Mathematical Practice, which used to be called “Process Standards” in NCTM’s standards.

David Klein and Tom Loveless were two people I learned much from during my stint on the Hill, as well as some of the people who were in the audience and who spoke during the Q&A at the end.

For your info and reading pleasure, my experience during that time culminated in a widely read article that was published in Education Next. 

Count the Tropes, Dept.

Counting the tropes in this article is a bit like doing those puzzles where you have to make as many words as you can out of some word.  This article was like such a puzzle but with a word that has so many letters in it that you can come up with thousands before things start getting tough.

As a result, I don’t even know where to start. It’s about a new school started by Elon Musk of Tesla called “Ad Astra” that addresses what in his mind schools ought to be:

[It]seems to be based around Musk’s belief that schools should “teach to the problem, not to the tools.” ‘Let’s say you’re trying to teach people how engines work. A traditional approach would be to give you courses on screwdrivers and wrenches. A much better way would be, here is an engine, now how are we going to take it apart? Well, you need a screwdriver. And then a very important thing happens, the relevance of the tool becomes apparent.’

Ignoring for the moment his rather banal observation, presented as if it is new and innovative is a gross mischaracterization of what traditional education is about, let’s focus on the many more edu-tropes the article contains:

Education today really isn’t that much different from what it was a hundred years ago. It’s still classrooms crammed full of students all learning the same thing at the same pace from overworked, underpaid and under-appreciated teachers who spend thirty years teaching more or less the same thing.

Of course, some of the “under-appreciated” teachers have no problem teaching these same things at the same pace, and holding students accountable for mastering the material in the time alloted.

The world that the next generation will grow up in will be radically different from anything we have seen in the past. A world filled with artificial intelligence, genetic engineering, automation, virtual reality, personalized medicine, self-driving cars, and people on Mars. A world where people might not even have jobs and where society itself may be arranged in fundamentally different ways. How are parents, and society for that matter, supposed to know how to prepare them to succeed in a world that we cannot predict?

The same problem about the future has existed for many years, and students still need to know basic facts and procedures–but that hasn’t stopped the above ever-popular trope from flourishing.

The role of school should no longer be to fill heads with information, rather it should be a place that inspires students to be curious about the world they live in. Kids are born explorers, when they are young all they want to do is push boundaries and explore the limits of what they can do. Let’s not suffocate that curiosity by making them spend their childhoods preparing for one test after another while adhering to rigid school policies that stifle creativity and independent thought.

Wait a minute; is this Elon Musk talking or Sir Ken? The schools kill creativity trope is  taking on a life of its own with everyone taking credit for it, apparently.

All active learning should be task driven. No more lessons where you jot down notes off a blackboard, rather students are assigned tasks to complete and given all the tools they might need to figure out how to solve the problem. (3d printers, virtual learning environments, interactive displays, a connection to labs and research facilities all around the world, etc.)

The “just-in-time” learning model. Throw a kid who can’t swim in the deep end of the pool and shout instructions from the side on how to swim. “Now’s a great time to learn the breast stroke.”  How has that been working out for everyone? Or more precisely, how much business has been generated for Kumon, Sylvan and other companies of similar ilk?

Teachers become facilitators of learning. Rather than lecturing everyone, they go from student to student or group to group helping them figure out how to learn what they need to know. Teachers no longer need a deep understanding of the given topic but they should know how to learn about it. Students eventually should also be supplied with their own virtual learning assistant to answer any question they may have and help them stay on task.

Yes, this is an old chestnut of a trope.  And how liberating that teachers no longer need to know anything about the topic they teach.  More “just in time” learning. It never gets old.

In addition, education should give people an understanding that the world is not divided up into discreet subjects.

Yes, God forbid we should study one subject at a time so we can eventually apply it to other disciplines. Just meld it into one big coloring book activity for teachers to facilitate. And of course, it is understood that mathematics is just white privilege but I’m stepping into other territory so I guess it’s time to stop.

Believe I’ll Pass, Thanks.

“Linked In” is not only famous for reminding me to congratulate people I barely know on work anniversaries at firms doing nebulous things, but also for exciting job offers.  I recently received a message offering me an opportunity to be “paid well in the future”.

Honored that they thought of me, but I believe I’ll pass.  Here’s the letter:

This is an invitation. You have been carefully chosen to be part of a special project. My name is Shane Mesa and I am the President of a cloud-based K-12 school for kids and adults. There is a lot of work to be done but I believe it all starts with curriculum.

I am looking for brilliant teachers who are to be paid well in the future. For now, I need your help in developing a simple curriculum in your field of mastery. There will be some required classes in this program but we will be focusing on the student’s strengths rather than pulling them in all academic directions, setting them up for failure.

My mission (like many others) is to create a better future for our children and our country. Eventually we will be building schools around the world but we are starting small. There is monetary incentive for you to join this cause. I only ask for your faith in this project. They say the sky is the limit but the truth is; they aren’t looking high enough!

If this spikes your curiosity, please write, call or email me for more info. Thank you for reading!

Count the Tropes, Dept.

This article about a “math festival” held at an elementary school contains all the usual tropes about what math education is supposed to be about. I almost stopped reading here, but like being stuck in a traffic jam because of an accident, I found myself staring at the gory site at the side of the road.

A dozen parents gathered around veteran math educator Leanna Baker, moments before students show up for what is billed as a math “festival” for students at Allendale Elementary School in Oakland. “Do your best not to give them an answer,” Baker told the dozen parent volunteers about how best to help the transitional kindergarten to fifth grade students participating in math activities arranged for that day. “We want them to be problem solvers.”

The tropes of “to problem solve” and “problem solvers” have emerged as the latest shiny new thing over the past several decades.  As if math was never about teaching students how to solve problems. In the past, there was instruction given with worked examples on how to solve problems, but now it’s all about transfer of prior knowledge to new situations.  If a kid can’t do that, then it is generally assumed that (a) the teacher is teaching it wrong or (b) the parents aren’t doing enough at home. (Translation of latter statement: Parents aren’t teaching their kids what isn’t being taught at school).  Of course blaming it on the student is an option too but thanks to Jo Boaler and others of her ilk, saying a child is “not a math person” is a no-no.  (Not that that stops anyone; it’s just said in other ways.)

Then of course there’s the “I wish I learned it this way when I was in school” trope:

It began with a math night at single school, which later expanded to eight elementary schools. Zaragoza said she started math nights because “families didn’t understand what (teachers) were doing and that was causing a disconnect” between schools and families when it came to math instruction. “Once parents understand what we’re doing and what is happening you’ll hear them say ‘why didn’t someone teach me this way ?’ or ‘why didn’t I know this before?’”

Decades of experience with math provides adults a different view of the subject than kids have who are going through it for the first time.  The adults who benefitted from the methods now held in disdain are being subtly programmed to reject such methods as injurious and inadequate–a view reinforced by Alan Schoenfeld, who has been vocal for many years against the traditional mode of math education:

“We lose so many kids in elementary school because they get convinced math isn’t for them,” said UC Berkeley professor Alan Schoenfeld, a leading expert in early math education. “I’ve seen some really engaged kindergartners get to fourth grade and just get turned off from math because it’s boring or it’s taught in a way that makes them think it’s boring.”

And of course, no story like this would be complete without the “rich task” trope:

“One of the goals (of math nights) is to show early learning teachers and parents just how easy it is to provide really rich mathematical experiences,” she said.

For more about “rich tasks” see this. Then hire a tutor for your child.

“Rich Tasks” defined

Facilitator of Professional Development course answers my question of “What is a rich task?”

“It’s a problem that has multiple entry points and has various levels of cognitive demands.  Every student can be successful on at least part of it.”

My translation:  A one-off, not generalizable, ill-posed, open-ended problem which can be answered in many ways.  Thought to gauge understanding, as opposed to traditional “only one right answer” problems that are viewed as helping only a few students–students who are viewed as “getting it” no matter what.

My Innermost Circle of Hell

Whatonomy


I hate a staffroom with prickles in it. I hate the freedom to choose any cup in the staffroom cupboard that is in reality an elaborate but secret system of cup choice only hinted at by the odd, fleeting grimace and veiled umbrage at my choice of cup. I hate – but hate – the sinking feeling (pants already long around ankles and minutes before the bell chimes for my next lesson) as my hand reaches deeper into the toilet roll holder and my fingertips frantically scrape around the cardboard roll – around and around and around, as if one, two, three turns of the screw will begin to set in train a miraculous process of the regeneration of the now nonexistent toilet paper.

Oh, but I reserve a special, most sadistic circle of Hell for the poorly-made education resource. I’m not shy to admit that I use textbooks in…

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