Everyone’s Happy in Happy-Land, Dept.

Chicago Tribune carries this story about a controversial approach to education being tried at various high schools in Illinois :

They’re abandoning most aspects of traditional classroom instruction and reshaping the way kids learn.

The approach, called competency-based learning, puts the onus on students to study and master skills at their own pace, making their own choices along the way and turning to peers and online searches for answers before they lean on teachers for help. Students may show proficiency on a topic not simply through traditional testing but by using projects, presentations or even activities outside school.

Competency-based; another “base” in the long line of bases; “inquiry-based”, “research-based”, “evidence-based”, “brain-based”. The list goes on.

The pilots are in various stages of planning and implementation, but all focus on instruction changes in high schools, likely a tough sell for parents who learned the traditional way: Teachers lecturing to a group; quizzes, tests and homework; letter grades and GPAs.

Right; everyone knows that doesn’t work. The educationists have been saying it for years, and the press eats it up and apparently many people believe it.

In contrast, students in competency-based programs take the time they need to master skills and make their choices in their academic journey.

Right, this sounds like a winner, especially if they’re trying to get into college and they need certain courses completed and mastered. But “take your time”, we’ll just make believe there’s no time element anymore and everyone’s happy in happy-land.

At Ridgewood High School, math teacher Tristan Kumor started off geometry class on a recent day by asking his students, “How do you want to learn today?”

His 9th and 10th graders sat in groups, with one student using Popsicle sticks to build a bridge. The lesson related to triangles. Kumor traversed the room guiding students individually on their progress, which is part of the way competency-based education plays out.

Yeah, right, that’s the ticket. But I think this has been tried before. PBL or something like that. Or is there some twist to this approach that I’m just not getting?

Computer testing and other work show how kids are progressing, and teachers provide individual feedback to students, acting as facilitators or coaches who monitor student growth and ensure kids are self-directed enough to assist and even teach their peers. The idea is that if a student can teach a peer, it’s clear they know the material.

At Proviso East, signs in classrooms direct students to do several things before going to the teacher: They need to reread the question, check their notes, ask other students for help and search Google for information. If their question is not resolved after all that, they may go to the teacher.

Yes, it all sounds happy and fruitful. Teachers as facilitators or coaches. Students teaching other students.  A “three-before-me” approach; God forbid they should ask the teacher anything. We don’t want teachers teaching  handing it to the student, now do we?

There’s no such thing as an F in the competency-based learning world, because failing is considered an attempt at learning that helps lead to mastery.

Right, the Jo Boaler approach which holds that mistakes grow your brain. And if mistakes are that powerful, then failing is even better!

Depending on the school, students might not receive letter grades on report cards. Some Illinois schools already use a numerical approach, such as 1 to 4, rather than letter grades to show academic progress.

And let me guess; no one gets a 4. At least that’s how it’s played out in other schools that have tried similar things.

Educators say the grades and transcripts have been a source of concern, with high schools reluctant to change because that data is used in college admissions.

Well yeah, there’s that, but let’s not let that stop such a promising program.

And even the strongest supporters of competency-based learning acknowledge the challenges ahead as educators, parents and kids process how the program will work.

Right, but let’s just keep doing it until the next shiny new thing comes along and this approach will be tossed on the dust bin with the usual lamentations: “Well we tried competency based learning but that didn’t work because –list of reasons follows — but this approach fixes those things.”  Just as long as it’s not traditional ed. Because God knows, that certainly hasn’t proved out, now has it?

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The Rote Memorization/Standard Algorithm/Lack Understanding Narrative

This particular rant/polemic is by someone who rests her authority on being good at math, and most likely because she learned it in the way she now says is ineffective. Like many of the defenses of Common Core math, hers rests on a denigration of traditionally taught math.

I’ve written much on Common Core math and I maintain that it contains watchwords (or dog whistles per Tom Loveless of Brookings) of reform math that cause it to be interpreted along those lines. That is, emphasis on alternatives to the standard algorithms prior to teaching the standard algorithms in the belief that teaching the standard algorithms first eclipses the “understanding” of the algorithm. And the “understanding” of the algorithm is believed to be essential for students to solve problems.

I’ve also maintained that Common Core can be taught using traditional methods, and the standard algorithms can be taught first, with alternative approaches later (as it used to be taught).  Jason Zimba has agreed with me on this in articles.

To wit and for example:

The criticism that I referred to earlier comes from math educator Barry Garelick, who has written a series of blog posts that aims to sketch a picture of good, traditional pedagogy consistent with the Common Core. The concrete proposals in his series are a welcome addition to the conversation math educators are having about implementing the standards. Reading these posts led me to consider the following question:

     If the only computation algorithm we teach is the standard algorithm, then      can we still say we are following the standards?

Provided the standards as a whole are being met, I would say that the answer to this question is yes. 

With that, let me turn to the blog piece I had linked to originally. The argument starts out with the usual:

The traditional way involves rote memorization and algorithms performed on paper. They require little to no understanding of why the algorithm works. It simply works.

This statement is so far off, it’s not even wrong. In fact, older textbooks do explain why the algorithms work as they do, and later, after students master the algorithm, alternatives to the standard algorithm are introduced. (See for example, this article.) In many cases, students discover these methods by themselves. And it isn’t as if teachers do not teach the understanding; most do. But as many teachers will tell you, many students do not glom on to the reasons, and instead rely on the procedure.  Understanding is a process that works in tandem with procedural fluency. Many students will understand in time.  And some will never fully understand. There are varying levels of understanding. But being able to solve problems using a procedure or algorithm does not necessarily make the student a “math zombie”.

She goes on:

We forget. All the time. Especially when we don’t understand why the algorithm works or if it has been a long time since we last used it. I can guarantee if I took a survey at Starbucks right now, and asked people how to perform long division, convert a mixed number to a fraction, recite the quadratic formula, factor a binomial, and complete the squaremost would fail. Not because math is too hard or people are bad at it, but because memorization and algorithms are not the best ways to retain information. We remember through contextunderstanding and application.

We also remember through continued application. For example, if I haven’t worked with percent calculations for a while, I have to brush up on it. Same with finding derivatives of certain functions. The survey of people at Starbucks might be different if the majority of customers were practicing engineers. That people forget how how to do something if they haven’t worked with it for years is not evidence that the traditional method of math teaching is ineffective. And like many authors of similar rants and polemics, she also does not provide evidence that her methods are superior to those that she feels do not work.

She then goes into a demonstration of how mental math provides a superior and quicker way of adding two three digit numbers than the standard algorithm and states:

Number sense allows us to have an arsenal of ways to problem solve, including but not limited to the traditional algorithm.

Quite true. And many of us got to that point by discovering these shortcuts ourselves after having been taught the standard algorithm, not to mention that the shortcuts she mentions (and gives a demonstration of) were also included in the old textbooks in the era of traditional math that she and others find so destructive.

She leaves us with:

The Common Core math standards are an attempt to expose your child to this flexible way of thinking. It may not be perfect, but it is in the right direction.”

Yes, and the standards can also be met quite effectively perhaps more so, using traditional methods. The “understanding uber alles” approach to math is likely to result more in “rote understanding” and an inability to solve many simple problems. Interestingly, students in the 80’s and 90’s entering algebra classes in high school knew how to do basic computations with fractions, decimals and percents, whereas now many teachers will attest that this is not the case.  Traditional math is the usual culprit, but such finger pointing fails to acknowledge that the reform methods have been around for almost three decades.

Oh right. It’s because reform math hasn’t been taught correctly. I keep forgetting that.

 

 

 

 

 

 

Shut the Hell Up, Dept.

In Education Week “Teacher”, a recent article reproduces the responses of four high school math teachers to the question: “What are some best practices for teaching high school mathematics?”

I got as far as the first teacher who starts off by telling us:

“Focus on the processes and connections between different processes rather than just finding the answer.”

Those who believe the big lie told at school board meetings and repeated ad nauseum by math reformers (in publications like Education Week and others of similar ilk) that “traditionally taught math failed thousands of students” likely nodded their heads at this with a big “Yup, that’s what’s wrong with how math is taught.” What goes unnoticed and unacknowledged is that a good textbook does everything claimed to be his, but in a more efficient way and one that focuses on individual success, not feel-good student reactions in class.

To be more specific, in elaborating on this idea about processes and connections, he starts off with: “Teach to big ideas rather than 180 disconnected lessons.”

I don’t know about you, but I haven’t heard of or witnessed any high school teacher who teaches 180 disconnected lessons. And though math books from previous eras like the 50’s and 60’s are held in disdain by today’s math reformers and others who claim the high ground in “best practices in math teaching”, the lessons contained therein were well sequenced, well connected and well scaffolded. Such textbooks taught to big ideas; in fact, my old algebra book was called “Algebra: Big Ideas and Basic Skills”.

But then he gets to the heart of what’s really bugging him:

 The answer is important, but the processes that lead to answers are far more important to learn. The connections between processes and between different representations of that process are where the core mathematical ideas you want your children to learn reside. This means that while teaching processes, you should use these processes to teach students mathematical principles they can apply to solve problems, not how to solve problem types.

The connections he thinks are not happening have always been a part of traditional math. We learned big ideas from unit material. Given a governing equation, like distance equals rate times time, we solved straightforward problems with worked examples to get us started.  Then, in working the homework problems, there were many problem variations. We learned how to rearrange the equation and solve for the unknown, to draw diagrams, label unknowns, and start writing down known equations to find an equal number of equations as unknowns. There was never one type of problem for any new unit of material.

There was no special process or set of ideas that guaranteed transference other than practice. From the unit material homework variations, we saw patterns of solutions that could be applied to other types of problems. The variations took us well past the initial worked example and required us to stretch our prior knowledge and apply it to new situations within the context of the original problem type. What if you had five people working on a job at different rates? What if they started at different times? If they could only work for 5 hours, how much might be left over for you to do? (For more on the topic of non-routine problems within the context of problem types, see this article. )

The teacher next exhorts us to:

Use instructional routines to support all of your students in having access to the mathematics. Instructional routines shift the cognitive load for students as they focus less on what their role is and what they are supposed to next, since these tasks are delegated to long-term memory, and are therefore more able to focus on learning mathematics.  

He provides a link to his blog to provide an example of instructional routines.  There, we see the typical “growing tile” type of problem presented as an instructional routine.  He states that

[t]he high level goals of Contemplate then Calculate are to support students in surfacing and naming mathematical structure, more broadly in pausing to think about the mathematics present in a task before attempting a solution strategy, and even more broadly in learning from other students’ different strategies for solving the same problem. A critical aspect of this and other instructional routines is that they embody a routine in which one makes teaching decisions, rather than scripting out all of the work a teacher is to do with her students.
 
I find that “growing tile” type problems promote inductive reasoning as if that is all that is needed in math and deductive reasoning takes a back seat if given any seat at all. That aside, he evidently believes that the effect of such problems is to get students to pause before leaping into solving something. In turn, this “thinking skill” is viewed as an inherent part of problem solving–i.e., a “habit of mind” like those of the eight Standards for Mathematical Practice contained in the Common Core Math Standards–that will allow students to solve problems independent of type and without the practice one needs to achieve any kind of flexible thinking.
One only needs to “learn how to learn”; all the rest can be Googled, I guess.
(Special thanks to SteveH for his thoughts on this topic which are incorporated into the above.)
  

Articles I Unfortunately Finished Reading, Dept.

The opening to this article about students in Wake County, North Carolina, was almost enough to get me to stop reading, but apparently I felt the need to punish myself, so I pressed on.

Wake County students taking high-school-level math courses are now finding that getting the right answer isn’t always as important as the process they use to solve the problem.

Yes, as a math teacher, I often give partial credit on test questions where a student has made a numerical error but has set the problem up correctly.  But I do like right answers, and from what I’ve seen, the definition of “process” varies from teacher to teacher.  Drawing pictures amounting to a time consuming and inefficient process is sometimes considered to be just fine. In a high school level math course, in my opinion it is not just fine.

Wake began rolling out this school year new classroom materials from the Mathematics Vision Project, a nonprofit education group that provides resources designed to align with the Common Core learning standards. Wake school leaders and teachers say the new materials have led to major changes in how math classes are run to shift from lecturing to having students work in groups to learn concepts through problem solving.

“We have to teach failure as a part of learning,” said Brian Kingsley, Wake’s assistant superintendent for academics. “If we’re going to be college and career ready, the answer isn’t always the most important thing.

Well, to be college and career ready, let me just say that the right answer is a lot more important than what Mr. Kingsley and others of his ilk seem to think.  Learning how to work with algebraic expressions may seem like senseless manipulations of meaningless symbols to those subscribed to the group-think this article extols. But to the rest of us, it’s rather important to learn how to factor, and work with algebraic fractions.  Students will undoubtedly make mistakes, so if “failure” is a goal of these people, the traditional method they hold in such disdain provides opportunities to make mistakes.  Making mistakes is not the province of the “new way of teaching math”, I can assure you.

Math I students got workbooks this school year, but a lot of what they experienced was unexpected. Instead of students copying what their teachers write on the board, they’re working in groups to solve problems such as how long it would take to drain water from a pool. Teachers are guiding the students instead of lecturing them on subjects like quadratic functions.

“Instead of me saying, ‘Here is a linear equation: It’s y=mx+b,’ it’s much more, ‘Let’s get to this equation,’ ” Herndon said. “It’s very much a different approach. Rather than me giving them all the answers, they’re having to work towards them themselves.”

The opinion of those who believe traditional math has never worked is that teachers give the students all the answers, and students do not do any work for themselves.  There is general belief among the prevailing educational forces that initial worked examples with plenty of practice is nothing more than “rote memorization” of procedures and that the process produces “math zombies” who cannot think for themselves.

I have observed the teaching methods described in this article.  A problem about how long to drain water from a pool is rather straightforward, but I’ve seen problems like this take the better part of a class period when with proper direct instruction and worked example, it should take about 10 to 15 minutes.  I’ve seen a class in which students worked on slope, graphing and the connection to linear equations for five to six weeks. There is no reason for taking that long and I wonder what evidence there is that such students have “deeper understanding” (a term that remains undefined, but generally means a “rote understanding” of explanations that students learn to give the teacher in order to satisfy them) than students taught in the traditional manner.

Do the people in the article have any evidence that their methods are producing better results than the methods they hold not to work?  I mean other than their opinions and seeing what they want to see.

 

 

 

 

 

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!

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 knows 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.