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”.

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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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Interviews I never finished reading, Dept.

This one is from a blog called “Cool Cat Teacher” that ostensibly presents hip, new, ideas about education as a refutation of the stuffy old traditional methods that people have claimed “never worked”, or if they did, “only for a few people.”

This particular blurb is an interview with a history teacher called Keith “Hip” Hughes, who has some ideas about “flipping the classroom”.

The basic idea is to try to reduce the amount of time you’re talking to kids from the front of the room. I think sometimes we have this illusion that the kids are learning in the space between our mouths and their ears, and I’m not sure how much that might be happening. So “flipping” is saying, “Let’s have the kids get the content somewhere other than the classroom.” Many times, that could be a reader, but many times it could be a really great video that explains a concept that you would normally be explaining in front of the room. You’re probably still going to have to review it. But the idea is to free up time in your class so kids that are working through skill-based activities that might in the past have been done as homework. Now (those) can be doing that in class with your facilitation.  The next step is designing projects and really having kids doing authentic inquiry-based awesome stuff in your classroom, using the content.

First of all, I’m a traditional based teacher and if I facilitate, I do so using direct instruction methods followed with questions as has been done in the past.  For centuries actually.  Effectively, I might also add.

Second of all, I make sure to leave enough time in my class for students to get started on the “skill-based” activities known as homework so they can ask questions about it in class, rather than going home and not knowing how to do it.

Third, videos only go so far. Yes, you can rewind and replay as much as you want, but if you didn’t get something the first time, hearing the same explanation again without clarification isn’t going to help too much.  And if they’re that sold on doing learning at home rather than at school, have them read the textbook as preparation for the next lesson as is done in many university classes. It teaches them how to read a textbook rather than rely on videos and come to class with questions for the teacher.  Oops, I forgot. Textbooks are bad. Never mind!

Biting the Hand that Fed You, Dept.

Keith Devlin, outspoken spokesperson (known as “the math guy” on NPR) on mathematics and who apparently has never backed down from a stated position, has this to say in a column written originally in 2011 and revised in 2014:

“As a nation, we should stop the current suicidal cry to turn back the clock to a form of math teaching that did no one any good and which those of us who became professional mathematicians first had to unlearn, and focus on making good, practical, sensible use of centuries old teaching tools (such as word problems) to produce a generation well equipped for life in the Twenty-First Century.”

He is not the only one to make such statement. What is more alarming than the statement itself is how often such statement is taken as truth and rarely if ever challenged.  The older forms of math teaching did no one any good? Really? Including him? Oh, he’s got that covered.  People like him (a professional mathematician) had to unlearn what he learned in his classes.  The word problems that he learned from did him no good whatsoever and he had to unlearn the problem solving techniques?

I get tired of the canard that “it never did anyone any good” particularly as uttered by those who, despite earnest claims with no evidence to support them, benefited from what they feel harmed many.  The ongoing mischaracterization of traditional math as “mindless computing” needs to be challenged.  Take a look at the students who go on into STEM majors and careers and the many problems they solved at home. Yes, they took on challenging problems, but also the very problems he says do not promote thinking and are mechanical and non-thinking in nature.  The “worked example” effect has its merits despite the claims of professional mathematicians who climbed the ladder and then kicked it down after they reached their levels of success, claiming that the ladder never did anyone any good.

Well, yes, but, Dept.

From Madison, Wisconsin comes this report about yet another parent math night, this time to introduce parents to the new math curriculum Bridges in Mathematics.  The math night consisted of math games that were actually used as part of the new curriculum so that the kids could guide their parents through them

Emerson teachers were on hand to speak with parents about the lessons and provide guidance as needed. Maria Cavicchio, a kindergarten teacher at Emerson, said students build on concepts taught in games like Spill Five Beans throughout the year. Although Hannah used the game to learn about number recognition, number writing and counting, the game also teaches the fundamentals of addition, subtraction and probability. “They do these sorts of games and they learn (the concepts,)” she said. “It’s not rote memorization, students really like (the activities) and learning is more meaningful.”

Yes, God forbid anyone memorize anything; that would be rote and would eclipse any “understanding”. Because math is about learning to problem solve (as opposed to learning how to solve problems–sorry for the digression but I still fail to understand how problem solve became an infinitive verb with a life of its own) and of course “explain your answer.”  Everyone knows if you can’t explain your answer, you lack understanding. I mean everyone:

“(Bridges) focuses on developing the students’ understanding of math concepts,” Davis said. “It is not about how students can memorize certain skills, but really around their ability to problem solve and look at math in more complex ways…and explain their reasoning to their teachers and peers.”

Never mind that understanding is many layered and true understanding can ONLY be built on mastery of skills. As my friend and frequent commenter here, SteveH, has said:

“If students can do problems without words, that is meaningless and rote? It’s all rubbish. There are too many skills to learn and understanding first is clearly not necessary or sufficient. However, true understanding is not possible without mastery of skills. Conceptual understanding (what they are talking about) is at best only the motivating baby steps of proper understanding.”