Articles I Never Finished Reading, Dept.


In this piece in the Huffington Post, the authors attempt to answer the question about why US performance in math is so dismal. They first state that “it is not class sizes, funding, or teacher qualifications” but rather:

It is the summer learning gap that brings down the educational outcomes for students from low-income families.  Many children from low-income backgrounds do not have the financial resources to engage in academic summer programs that help them retain and expand upon their learning from the previous school year. Meanwhile, their middle- and upper-income counterparts take part in summer schools, camps, and other enrichment programs. The result is a significant achievement gap between these two groups of students over the summer break; the students from families with resources to learn even when school is not in session gain one month in math and reading each summer, while lower-income students lose at least two months in math and reading every summer vacation. By the time both of these groups reach fifth grade, the gap is about three years, and by the time they reach high school, the gap has stretched to over five years.

Although they mention class size, funding and teacher qualifications as possible contenders, which they rule out quickly, they fail to mention curriculum, text materials, and ineffective pedagogical practices.  For almost three decades, math education in K-6 has fallen prey to reform math (aka progressive) ideologies that promote collaboration, group work, project/problem-based learning, student-centered and inquiry-based classrooms, as well as whatever edu-fad happens to be trending at the moment.

In the past, summer slide was not as noticeable, though I’ve heard reformers argue that that’s because in the past the first three months of every school year were devoted to re-teaching key topics from the previous year. The implication was that this does not go on now, although they fail to mention that such review built on what was learned the previous year so that students weren’t simply repeating learned material.

And since the article focuses on income disparity, do you think perhaps that outside help like tutors, and learning centers may play a role? Lower income families cannot afford these things but some higher income families tend to rely on it to ensure their children are learning what isn’t being taught with today’s rather dubious curricula.

Another reason cited by the authors is that in Asian countries, students spend on average 220 days in school compared to our 180 days.  Significant, sure, but also not mentioned is the quality and effectiveness of the curricula used in Asian countries.  They largely use traditional techniques despite assertions to the contrary from progressivists who insist that Asian countries are doing reform math right.

They then suggest that the US invest a “$400 voucher (or 3% additional spending per student per year for summer learning) to the 25 million students that come from households living in poverty.”

Yeah, throwing money at things always works. NSF threw $93 million in grant money in the early 90’s to develop such programs as “Investigations in Number, Data and Space”, “Everyday Math”, “Connected Math Project” and other poorly written and conceived textbooks that embodied the principles in NCTM’s 1989 math standards.  That worked out well, now didn’t it?



Edutropia, Dept

Edutopia (which if you don’t mind I’ll call Edutropia) ran a piece last year on “Visualizing 21st-Century Classroom Design”  that someone made me aware of.  The first line grabbed my attention:

“Problem-based learning, makerspaces, flipped learning, student blogging — these are becoming perceived staples of 21st-century learning.”

The phrase “Problem-based learning” of course is always a red flag for me, but followed so closely by “makerspaces” and “flipped learning”, I found myself puzzled as usual.

What do “makerspaces” have to do with teaching a particular subject like math? Unless of course they mean building a model of a bridge in order to solve a problem, (which I’ve seen done), or bending pipe cleaners in the shape of parabolas and glueing them to poster boards (which I’ve also seen done). All of this is supposedly engaging and fun for students not to mention allowing teachers to facilitate rather than teach, although I’m told it’s a lot more complicated than I make out. One teacher who is an adherent of this type of 21-century classroom wrote on his blog that there’s a lot of things going on that people don’t realize: facilitating, assessing, questioning–yes, everything but teaching.

And of course there’s the flipped learning aspect of things, in which students watch videos like Khan Academy in which information is presented in direct and explicit fashion in the manner held in disdain by those promoting 21-st century classrooms. (Not to mention the fact that some students tune out when watching videos, and if someone does not understand something in the video, there’s no way to ask for clarification. Playing back the same explanation repeatedly just repeats what the student found confusing in the first place. But not to worry: reformers have been quick to point out that things like Khan Academy are not really math–they are just how-to’s on procedures, and the real learning goes on in places like, well, like 21st-century classrooms where teachers facilitate, assess, question, and don’t teach.

“We need to be sure that we’re not catering to just one type of learner. Be mindful of your introverts, extroverts, collaborators, solo thinkers, writers, dreamers, and fidgeters — and design a flexible environment that can meet everyone’s needs.”  

Right. Good idea. Because assuming they’ll be able to get a job someday, they need to know that if they feel like writing and dreaming and fidgeting they should feel free to do so.  But let me ask one question. For the introverts and/or socially awkward student (for any number of reasons), are they exempt from group work, or doing posters or art projects, and allowed to obtain knowledge in conventional ways? Uh, what’s that? They can watch Khan Academy? And what about those students–uh, I mean learners–who prefer to sit at desks arranged in rows rather than in groups.  Is there a conventional or traditional zone in addition to the bean bag zone or the creation/inspiration zone? Or would that be anathema to learning?

“Explicitly teach and emphasize process over product, growth mindset, and metacognition. We cannot cultivate risk taking, failing, and perseverance — all essential characteristics of creativity — if we repeatedly demonstrate to students how all that really matters is neatly filling out our worksheets.”

Process over product–now there’s a phrase we don’t hear enough of these days.  I’m all for students showing their work and giving credit for writing an equation correctly even though they made numerical mistakes. But I do note the numerical mistakes. And I don’t give credit to “guess and check” approaches to solving a problem when an equation is called for. But I’m open to compromise. Although I have them do worksheets, I do ensure that they know there are correct answers to the questions and whether or not they got them right. I’d best end this now before some of you call Child Protective Services on me.





Comments on Elon Musk’s PBL-based advice

As frequently happens, the comments I received on my recent blurb on Elon Musk’s ideas for how to teach math, were better than the post itself. So here they are:

From Richard Phelps:

My high school physics class — in the early seventies — comprised project after project. Teacher called it “Harvard Physics.” Each class period was filled with setting up equipment and building contraptions, in small groups. Essentially, it was a “lab only” course. I learned less in that course than in any other over 4 years of high school, and I’m including gym in the comparison. Typically, an entire class period was devoted to delivering just one fact or concept — “authentically” — that could have been simply told to us in less than a minute or, with some discussion, in just a few minutes — un-authentically. Moreover, so much of our attention was focused on building the contraptions and getting them to work that, in most cases, the one factoid we were supposed to learn was lost in the morass of mostly irrelevant information.

From SteveH:

Traditional learning often includes year-end projects, but PBL’s approach is to use only that to hack their way to a vocational education. PBL is the definition of a non-liberal arts education. How does hacking create deep understanding and critical thinking?

I remember all of the projects my son had to do in K-6. One had him build a diorama of a national site with no preparation in art or graphic design. I distinctly remember taking the time to teach him how to plan (and limit) the project, perspective, graphic arts, and even how to label the project so that the text was centered. If the result was not crappy enough, then they knew the parents helped.

Why have teachers if all they’re going to be are potted plants on the side? We can save a lot of money that can be used to hire individual subject expert tutors starting in Kindergarten. Then again, why not do it right and teach the skills in class and leave it to after school clubs for opt-in PBL.

More of the same, Dept.

More of the same regarding math being taught wrong, but this time the words come from Elon Musk, the new patron saint of innovation and disruption.

Musk suggested learning be focused around solving a specific problem, such as building a satellite or taking apart an engine. Then students will encounter and master subjects such as math and physics on the path to solving their problem. Understanding how to use a wrench or screwdriver will have a clear purpose. “If you had a class on wrenches, ugh, why?” Musk said. “Trying to solve a problem is very powerful for establishing relevance, and getting kids excited about what they’re working on and having the knowledge stick.”

Translation: Project/Problem Based Learning is the way to do it. Students learning on a just-in-time basis will have the motivation to learn what’s needed in order to solve the problem. Or so the theory goes.

The rest of the article was about Elon Musk and how great he is.

Not this again, Dept.

An Education Week article titled “How Teacher Prep Programs Can Help Teachers Teach Math Conceptually”, starts out right away making the usual uncontested claim about how mathematics is taught:

“Future teachers are likely to teach as they were taught—which can be problematic, researchers wrote in a recent study, “because most teachers experienced school mathematics as a set of disconnected facts and skills, not a system of interrelated concepts.”  But even when prospective teachers are taught to teach math conceptually, a good content knowledge base is still important, the study found.”

In other words, it helps to be able to do mathematics in order to teach it.  The research study referenced in the article was one in which teachers at an ed school who were taught how to teach math “conceptually” were then observed in a classroom.  They found that of four conceptual techniques, the teachers tended to use two of them: “use of mathematical language to support students’ sense making and use of visual representations”.

Also, I wonder how true it is that the teachers were taught math as a set of disconnected facts and skills, totally unconnected with each other.  The research study focused on six first-year teachers, so I am assuming that these teachers were fairly young–say in their 20’s.  Being in their 20’s, these means that they received their elementary math education starting about 15 years ago, taking us to the 2000’s.  Of note, is that many elementary math programs in the 2000’s and late 1990’s had been influenced by the math reform ideologies starting with NCTM’s 1989 standards which gradually became incorporated in ed school programs and in textbooks–not to mention the burgeoning of textbooks written under grants from the National Science Foundation including Everyday Math, Investigations in Number, Data and Space, Connected Math Program, and others, so that maybe, just maybe, any shortcomings in the teachers’ mathematical proficiency and understanding could be attributed to the weaknesses associated with such programs. And even though such programs purported to “teach the concepts”, spiralling and beating about the bush rather than utltimately telling students what they need to know to solve the problems might be at issue here. Just guessing.

The article summarizes the research study:

“During the classroom observations of the six first-year teachers, the researchers found that the teachers were more equipped at teaching math conceptually when they had learned the topic in their preservice classes (which incorporated all four of those instructional practices). When they hadn’t been taught a topic in teacher prep, they focused on procedural talk rather than using academic language and conceptual meanings. They also weren’t sure of what appropriate visual representations to use to illustrate the concepts.”

I wonder what textbook these teachers using from which they were teaching the lessons. Based on what I’ve seen being used, explanations of what’s going on with math concepts seem to be in short supply.  In particular, the “Big Ideas” series of textbooks that I’ve had the misfortune of having to use, concepts are embedded within discovery-based activities, and some hidden within the problems at the end of the lesson.

Also of interest is that one of the conceptual strategies taught to these teachers was:

“Pressing students for explanations. Doing so allows students to further develop their understanding by working through obstacles and contradictions and reaching for connections across strategies. Teachers should establish classroom norms, researchers say, where a good explanation is a mathematical argument and not simply a description of the procedures, and errors are further opportunities to learn.”

I guess it would depend on what level of explanation we’re dealing with, wouldn’t it? There are different levels of understanding. Also, there is no simple path of understanding first and then skills; an idea which pervades a lot of modern math education pedagogy.  Adding to this, words can get in the way. A student may know how to do something but won’t know how to put it into words, except in those cases where he/she has been given the correct mathematical vocabulary. But is that parotting back the words the student thinks might make a teacher happy–i.e, “rote understanding”? I’ll go out on a limb here and say as I have many times: understanding is not tested by words, but by whether the student can do the problems.

It becomes harmful when students are expected to focus too much on the “process” and less on the subject matter of the problem solving.  Problem solving skill is highly contextual, so domain knowledge mastery is really the critical thing. If that means they know the procedures but are weak on the conceptual understanding, there are worse things that can happen–and are happening. 

Motivation for Learning: Is There a Point?

I am extremely grateful for this piece, and the well-articulated thoughts. Should be required reading for all teachers and in all ed schools. But until that happens, spread the word in the usual manner:

3-Star learning experiences

Paul A. Kirschner & Mirjam Neelen

Motivation 1

Motivation, engagement, commitment, drive, grit … Some people seem to be obsessed with these concepts. For them, they’re like magic wands that can solve almost all of the problems in education or learning in general and/or are primary objectives for education and learning (we’ve blogged about the topic before here). It’s a mystery to us why people are so fascinated, often to the point of obsession, with motivation and engagement because…

Motivation 2

Reason 1: Measurability

First, there’s a real problem with measurability. Motivation, engagement, and any similar affective state for which people experience positive or negative emotions or feelings, are almost never measured in a direct and objective manner[1]. In order to give it a shot anyway, researchers use various self-reporting methods, such as surveys, Likert scales, journals / diaries, log books, and/or semantic differential scales. Unfortunately, we’ve known for a…

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Let’s try this again, it didn’t work the last time, Dept.

This latest from Greenwich CT:

“A parent entering a second-grade math classroom in September will see students designing their own lessons. The children will be moving around, working at different tasks that they have chosen to explore the idea of, say, subtraction. They will be discussing math with each other, not just the teacher. The teacher will hop from student to student, making sure they are on the right track and answering questions, but only as a guide, not the autocrat telling the students how to learn. This is the vision Irene Parisi, assistant superintendent for curriculum, instruction and professional learning for Greenwich Public Schools and her team of teacher and administrators have for the widely anticipated “elementary math pilot.” “

It isn’t as if this hasn’t been tried before. And when it has been tried before with dubious results, the general fallback excuse is “Teachers aren’t doing it right” or other forms of “Teachers are traditionalists in reform clothing”.  When scores do rise, there have been (to my knowledge anyway) no studies adjusting for help at home via tutoring, or help from a learning center such as Sylvan, Huntington or Kumon.

But they’ve hired consultants to give workshops to make sure this goes off without a hitch.

“[The} workshops focused on how to teach students to reason their way through math problems instead of memorizing math algorithms and trying to apply them without understanding. “All I’m trying to do for kids is make sense of things,” Tang  [the consultant] told the math pilot teachers. “Solve through reasoning versus grinding: that is truly what nobody is teaching students to do.” “

OK, let’s examine this statement. Nobody is teaching students how to reason apparently. The popular mythology about the math teaching that apparently never worked and failed thousands of students is that students solved problems without knowing why the procedure worked and were unable to apply those procedures to other problems in new settings–thus becoming part of a growing body of “math zombies”, guilty of “doing” but not “knowing” math.

I have just finished a year of teaching my students how to solve problems using concepts and skills developed in class.  I taught the underlying concepts of how specific procedures and algorithms worked so it wasn’t as if they were devoid of the “understanding”.  But as often happens, the students were more interested in the algorithm or procedure than in understanding how or why it worked–so there weren’t many who remembered the derivations.  Nevertheless, their capability of working with the algorithm/procedures ultimately helped them solve problems.–even new ones in different settings.

What does it mean to “apply without understanding”?  If you know what fractional division represents, and that to find how many 2/3 oz servings of yogurt are in a 3/4 oz container one divides 3/4 by 2/3, does understanding the derivation of the invert and multiply rule help you to solve the problem? Or is it the understanding of what fractional division, and division in general, represents?  There are times when the underlying concepts are part and parcel to applying a procedure or strategy.  But just as importantly there are times when the concepts are not.

But such questions and issues remain in the province of the marginalized traditionalists, while people like Parisi hold sway with the oohs and ahhs of others who are subscribed to this particular groupthink. And they continue on, unabashed and unhindered:

“The teachers are also working on developing a “flexible curriculum,” including resources that will allow children to go deeper than the current math lessons if they desire, and designing lessons that will allow students to choose how they learn, Parisi said. They also must create new ways of assessing how students demonstrate what they learn — whether an oral presentation, illustration, video, written statement or another way.  ‘A lot of it is going to be the mindset, the student ownership, the student voice in all of this. That’s going to be significant because we’ve already seen the power of that,’ ” said Parisi. ‘So the question is, how do we do more of that.’ “

The question also is how do we get away from this magical thinking?

The Understanding Paradox

No Easy Answers

No one wants students who don’t understand the meaning of their subject; we don’t want our students to merely regurgitate facts devoid of context, or for them to know how to answer questions in an exam yet have no idea what these things mean outside of an exam hall. And yet, on the path to understanding it is unavoidable that our students will often have to learn things that perhaps they feel they don’t fully understand, and will have to memorise things devoid of context. This is what I call The Understanding Paradox.

My view is that teachers attempting to bypass the memorisation and rote learning part of teaching in order to ‘teach for understanding’ can have disastrous consequences for students.

I want to illustrate my point by discussing trigonometry. You might remember it from school, and if you do, you probably remember SOHCAHTOA, the mnemonic device used by pretty…

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