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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Education Endowment Foundation discovers small placebo effect (but not for maths)

Greg Ashman take an Extreme Close-Up shot of EEF’s study on dialogic teaching and concludes as I have that it’s one more strawman used to bash traditional teaching.

Filling the pail

The Education Endowment Foundation (EEF) have released a report into a three-year randomised controlled trial. Schools were allocated to a control or intervention group and the intervention teachers were provided with the a mentor, time off timetable for training, a number of books to read (I’m sure the authors were pleased about this) and, crucially, a video camera and microphone to record and then review their lessons. The control group received none of these things and carried on as usual.

The result showed that, on a standardised assessment, the intervention group slightly outperformed the control in English (mean of 13.76 vs 13.16, p=0.05, d=0.15) and science (mean of 26.67 vs 26.29, p=0.04, d=0.12) but not in maths where the result was not statistically significant. Note that the English and Science effect sizes were very small.

The most likely explanation for these results is an expectation effect such as the placebo…

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Articles I Never Finished Reading, Dept.

Quartz has yet another enlightening article about what we’re doing wrong in education:

Research, and common sense, show that kids learn more by being actively engaged in what they’re doing. When they engage in discussionteach others, and grapple with a math problem, they boost their ability to absorb and retain information.

The evidence is mounting against schools that fail to take this approach. Most are still built around making sure kids have the right answers to rote questions, rather than the tools to formulate meaningful questions that deepen their learning.

Putting aside the mischaracterization that teachers/schools teach by “rote” and that no understanding or dialogue takes place in schools that are considered to not use this “dialogic” approach as it is termed, let me ask this.  The inquiry-based, student-centered, and “dialogic” approaches have been upon us for the last 30 years. How has that been working out?  Oh, right. The teachers aren’t doing it correctly.

Let me ask some more questions. Did they control for tutoring, help at home? And another question. Do tutors, learning centers use the “dialogic”? And what are the results from tutoring/learning centers compared with classes using the “dialogic” approach. And can it be said that traditional teaching does employ so-called “dialogic” approach? And can it also be said that the “dialogic” approach sought after by EEF study mentioned in the article is another form of “rote understanding”?

Andrew Old who some of you know from his blog Scenes from the Battleground, also asks about the EEF study: “Looking at the results of that study earlier. Small effect sizes most of which weren’t statistically significant. Don’t get the hype at all.” Let me go out on a limb here: Can it be that the hype is to prove that the so-called dialogic approach is superior to the rote approach the trads are supposedly using?

Stop me if you’ve heard this before, Dept.

The Lowell Sun News (from Lowell, Mass) has an article about a new book on how to teach math. The book is called “Teach Math Like This, Not Like That” by Matthew Beyranevand and features examples from the local area of Lowell, where the author is from, with names of teachers who, according to the author are doing it right. Those who in the authors opinion aren’t doing it right, according to the article, are also offered as examples, but the names of those teachers are changed.  Hey, this author sounds like a real fun guy!

In case you’re wondering what’s the right way and what’s the wrong way, this paragraph gives a clue:

“Beyranevand’s mission is to move away from the practice of memorizing things such as equations and multiplication tables in favor of building conceptual understanding and increasing student engagement and interest in learning math. Rather than drive kids to hate math, he seeks to make it joyful and meaningful for them.  In the book, Beyranevand takes 40 different ideas, from communicating with students and parents to differentiating instruction, and provides examples of how each is done, traditionally and in more innovative, impactful ways.”

Sorry to be skeptical but I’ve heard this refrain more than a few times. (I’ve also heard the word “impactful” used; I’m guessing it means effective but I’m open to information anyone cares to impart.)

I’ve heard about “memorizing–bad; conceptual understanding good”, as if the two don’t work together and as if math is traditionally taught as rote. (And for those who maintain that memorizing IS bad and that it devalues learning the concepts, and that speed is not important, you might be interested in this column on Dan Willingham’s blog. )

I’ve also heard about lesson plans being about stating the objective and then summarizing what the students have learned at the end of the lesson.

I’ve heard about developing relationships with students as if teachers don’t ever show an interest in them as people.  I’ve heard that students have to be engaged which is accomplished by making things entertaining.

What I don’t hear is that students get engaged when they feel they are (and can be0 successful, and they feel successful when they are given the means by which to do so.  And so far, the means by which many are proclaiming is the way to do this, are so far off, they aren’t even wrong.  What I also don’t hear is how requiring students to solve problems in multiple ways can result in confusion and frustration–particularly when anchor methods (such as, say, oh I don’t know, standard algorithms or procedures) are given first so students can get their feet wet and feel they’re on the right track.  Not to say that this book has these faults, but having looked at the author’s website, I’m guessing I’m not too far off.

Of particular interest to me is that Rowman and Littlefield is the publisher of this book series (and yes, it’s a series of three books). They’ve published “Betrayed” by Laurie Rogers, about how the reform methods of math education are not working and what to do about it. They’ve also published Eric Kalenze’s book “Education is Upside Down” which takes a critical look at how the education establishment embraces fads and trends that are unproven and ineffective and their rejection of traditional methods of education.  I guess I shouldn’t be surprised that they’re publishing this new book about math education given what publishers are all about.

The beautiful simplicity of saying what you believe to be true

Filling the pail

Deans for Impact have released a blog post that is intended help learning scientists influence education. As with many publications by Deans for Impact, it seems to have been well received. However, I always react badly to pieces that position the authors as the adults in the room and my problem with this article runs deeper still. I am going to have to dissent.

I am, of course, sympathetic to the aim of convincing teachers that learning styles are a myth. However, I feel much the same about the Deans for Impact post as I do about this piece, the aims of which I certainly do not share. So what is my issue?

I don’t believe that proponents of evidence-based education should be in the business of consciously trying to spin concepts in order to manipulate their readers. At a basic level, it is wrong because it treats the intended…

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Education Shock: Actually Teaching Students Math is Effective

This 2014 story published in the Atlanta Journal Constitution reports the shocking news that teaching first-grade students math using the dreaded worksheets, and traditional modes of education was more effective than ” group work, peer tutoring or hands-on activities that use manipulations, calculators, movement and music.”

According to Maureen Downey in her article, “This is an important issue as I increasingly see schools – including those my children attend – tout group learning activities. In many classrooms now, you will see students working at tables together on math. A friend who teaches in a Title 1 school lamented that her students didn’t do as well in the math CRCT as the classroom next door where the teacher used worksheets all the time. My friend’s classroom was a beehive of fun activities around math, but the worksheet class continually outperformed hers. These new findings help us understand why that might have been.”

What I find interesting is the conclusion that the direct, traditional instruction benefits those students with math difficulties, implying that those students without math difficulties do just fine with student-centered approaches. The possibility that difficulties with math may be a result of the student-centered approaches is something that is not discussed even though the study by Paul Morgan and George Farkas of Pennsylvania State University indicates that “a higher percentage of [students with mathematics difficulties, or MD] in the first-grade classrooms were associated with greater use by teachers of manipulatives/calculators and movement/music to teach mathematics. Yet follow-up analysis for each of the MD and non-MD groups indicated that only teacher-directed instruction was significantly associated with the achievement of students with MD (covariate-adjusted effect sizes [ESs] = .05–.07).”

I recall in an article I wrote called “Being Outwitted by Stupidity” I suggested that the increase in students being diagnosed  with learning difficulties in math raises the question of whether the shift in instructional emphasis over the past several decades has increased the number of low achieving children. I also question whether the learning difficulties came about because of poor or ineffective instruction and whether such students would have swum with the rest of the pack in previous eras when traditional math teaching prevailed. I stated “I believe that what is offered as treatment for learning disabilities in mathematics is what we could have done—and need to be doing—in the first place.”

This article garnered about 80 comments, many of them hostile, including my all time favorite which named me a “conservative simpleton fraud”.

I continue to maintain that many of the difficulties we see students having in math may be attributed to insufficient and ineffective instruction. To put it as simply as I can, they may not be learning math because they aren’t being taught math.  But the Morgan/Farkas study is being interpreted in the usual manner: “Teacher-directed instruction is also linked to gains in children without a history of math trouble. But unlike their math-challenged counterparts, they can benefit from some types of student-centered instruction as well – such as working on problems with several solutions, peer tutoring, and activities involving real-life math.”

Not mentioned is whether and to what extent such students receive additional help in the form of parents at home, tutoring, or learning centers.  We might have to wait a while for that kind of study to surface.


Some comments from our readers

The previous post (Articles I wish I never finished reading, Dept) stimulated several insightful and useful comments which I am reproducing here. The first, from Greg Ashman, a PhD candidate in education who resides in Australia, and is outspoken on the issue of the ineffectiveness of constructivist teaching strategies, commented at the article itself.  He writes:

A few points to note:

  1. Finland is declining in its PISA maths performance and has been doing so since 2006. If we therefore wish to look to Finland as a model then we probably need to examine what it was doing prior to 2006. If we do so then we see it was pretty traditional:  Reference. 
  2. Teaching for conceptual understanding is actually quite a strong theme in American maths education and not something that has been ignored. A comparative study of the attitudes of mathematics teachers in different countries found that American teachers prioritised the idea of understanding-first more than teachers in East Asia. East Asian teachers still thought understanding was important but were relaxed about whether it came before or after a knowledge of procedures (East Asian countries do quite well in international tests): Reference
  3. When teachers prioritise understanding-first then they are often drawn to ‘constructivist’ pedagogies where students are asked to design their own strategies for solving problems rather than learn canonical ones. These pedagogies are ineffective:  Reference. 
  4. Canonical procedures have advantages over student invented procedures, particularly for more complex work:  Reference  
  5. Jon Star has done a great deal of work on the fact that what we often call ‘procedural knowledge’ actually contains a great deal of conceptual understanding. Procedural and conceptual knowledge are bidirectional and iterative:  Reference 

Far from a challenge to the prevailing orthodoxy, this opinion piece is a good expression of it. It is thus part of the problem.

And one of our readers, Chester Draws, has this to say about the belief (or poster child of the progressivists/math reformers) that the distributive rule leads students to a lack of understanding in solving certain algebra problems. Specifically, he is referring to this part of the previous post: “When they see an equation such as 3(x+5)=30, they will distribute rather than divide both sides by 3 to get the simpler equation of x + 5 = 10. The author claims ‘but a child who learned the distribution method might have great difficulty recognizing the alternate method – or even that both procedures are equally correct.’ ”

Chester comments:

But both procedures are not equally useful.

If you have a standard 14 year old and you give them 7(x + 5) = 30, then they are going to stuff it up if they don’t distribute first. x + 5 = 30/7 is not something you want them to be attempting. Whereas 7x + 35 = 30 produces no such problems, despite yielding the same fractional negative.

The divide first method is usually more difficult and much more prone to error. Why would you even want them to know about it? I have nothing but derision for teachers that show students methods that aren’t universally applicable so they have “choice”. They don’t need or want choice, they need and want to get the answers correct with a reliable method. The time lost teaching a trivially useful technique would be much better spent getting the ones that they do need properly organised in their heads.

I teach all my students that normally the first thing you do in any algebra solving problem is get rid of fractions and brackets. Then you can see what you have. They then have their minds freed of what to do first — remove brackets and fractions — and that leaves more brain power for the hard bits.

Finally, we have SteveH, addressing this same issue:

In a traditional approach to algebra, you learn that there is no one way to solve anything, even though pedagogues really push ideas of order of operations. Learning this is not an understanding issue. It’s a practice issue, where mastery of problem sets give you plenty of chances to solve problems in different ways. Practice for SAT also teaches you to look for tricks and short cuts, but that is neither necessary or sufficient. Practice, practice, practice on problem sets is the solution. That level of understanding is only driven by individual practice on problem variations, not transference of a few in-class group projects covering general ideas. Words are not understanding.

There are ways to talk and provide proper and more abstract algebraic understandings, but most of these rote pedagogues don’t have a clue. In the end, the only way to create proper understandings is via lots of individual practice on problem sets. Practice is not just about speed.