A Textbook is Just a Resource, Dept.

This article explores how a school district in Decatur discards the traditional notions of “curriculum” and goes for the student-centered, small-groups, guided instruction approach.  Like most such articles and efforts that purport to do away with traditional approaches, the first thing to get attacked is the use of textbooks:

“Your curriculum is based on the learning standards and how you teach, the strategies you use,” Mahoney said. “A textbook is just a resource.”

This is one of those maxims that starts in ed school and persists beyond. It has been repeated often enough over the years that it has now taken on a life of its own and accepted by the education establishment as the absolute truth. And the general unquestioned consensus is that something that’s so organized, sequential, and linear, and contains problems that students have to practice is not something to plan a course around. (Extended discussions of this topic usually end with another maxim: “Research shows” that this is true.)

Which might explain why the textbook industry, faced with the maxim that textbooks are just resources, produce math books with sparse explanations, a sequence of topics that makes little sense, a dearth of word problems of any consequence–or explanations of how to solve them–and problems that do not scaffold or provide much in the way of practice and learning.  Key pieces of information that are not discussed in the body of the lesson are hidden in discovery type problems at the end of the problem set, ostensibly for the brighter students to conform to the current fad and practice of individualized and differentiated instruction.

Well, don’t tell anyone but in my eighth grade algebra class, I managed to secure multiple copies of Dolciani’s “Modern Algebra: Structure and Method” published in the 60’s that served many students well over the years despite claims about research on textbooks and traditionally taught math, and what such research shows. And that’s what I use instead of the textbook I’m supposed to use–part of the”Big Ideas” series of math textbooks by Larson and Boswell.  The students told me they like the Dolciani books much better and even like the problems.  One parent also told me this, having voiced the same concern about the “Big Ideas” textbook that I described above.

But the watchword in classrooms across the country continues to be “individualized” and “differentiated” instruction. This last term appears to be falling out of favor to be replaced by “guided instruction”:

“The idea behind guided math is small groups,” said Teri Cutler, math coordinator for the district. “Research is showing that teaching a whole group, 20 to 30 kids at the same time, is no longer as effective.”  Classrooms will have stations, just like they do in guided reading, and students will work independently or in pairs at those stations while the teacher meets with a small group of children who are at similar levels. Every 15 minutes or so, kids switch to a different station. Every student gets independent time on learning activities and time with the teacher for specific instruction.

“The good thing about guided math is, the teacher can group them by ability,” Cutler said. “Some are really struggling and can have that time with only those students at their level. Some are ready to move on to the next level, and can have the chance to excel and move on as well. It’s so much better for students that they can have that time in a small group with the teacher.”

Again, another idea that sounds great. But I’ve been in classrooms where the students work in stations. I was an assistant to teachers in such classrooms and tried to work with the students in each of the groups.  There is only so much time that one can spend with a student and frequently I was in the position of having to repeatedly explain to students what they were supposed to do, and provide explicit instruction for those students who weren’t able to discover what they were expected to discover. The idea that students move at their own pace and learn what needs to be learned in such a set-up is a seductive one.

But like most efforts, they are based on the assumption that traditional methods simply do not work–practice is “drill and kill” and mind-numbing, and textbooks are only resources.  What isn’t done–in practice and in articles such as these–is to find out what the successful students are doing that’s different than the strugglers.  Successful students often get help at home, or outside via tutors, using methods held in disdain by the education establishment.  And in fact, parents are exhorted to help their students at home, to learn things like math facts and other essentials for which no time is built in to the classroom. Classrooms are the province of the “fun stuff”: the discovery activities that students used to do at home.

Those who don’t have access to such help are at the mercy of ineffective practices that are touted as “the answer”.  And of course, if a student doesn’t succeed, there’s all sorts of excuses: The teacher isn’t teaching it right, or the student just isn’t trying, or the parents aren’t teaching their kids the math facts and other essentials at home.  That last one is a popular one; along with “there’s no cure for the ravages of poverty.”


Just Do What Japan Does, Dept.

In a recent article published in “The Hill” the author laments about the US performance in math and then comes up with what is now a bromide in math education circles: “Why not look at what other countries do to teach math–countries whose performance is math is in the top tier. Why not look at, say Japan, the article says and then cites Elizabeth Green’s article that was published in the NY Times Sunday Magazine in 2014: “Why Americans Stink at Math”. The big surprise of all this is that Japan’s techniques, according to Green, came from the U.S., and Japan emulates the reform math ideas promoted by organizations such as the National Council or Teachers of Mathematics, and which are subtly (and not-so-subtly) embedded in the Common Core Math Standards.

In Green’s view, reform math is how math should be taught because math is about “understanding”. In the world according to Green, if math is taught poorly it is either because 1) it is taught in the traditional manner or 2) it was reform math executed poorly. This leaves a nice out for both types of teaching. Pointing to reform math as a possible reason for the bad performance of students over the last two decades is responded to by saying “It wasn’t true reform math.” (See No True Scotsman fallacy )

Since, according to Green, reform math in the United States is done poorly, this accounts for the many examples of ridiculous problems and incomprehensible homework assignments attributed to Common Core, although such approaches were around long before Common Core. She agrees that such approaches are bad and not what Common Core is about. Green writes “With the Common Core, teachers are once more being asked to unlearn an old approach and learn an entirely new one, essentially on their own. Training is still weak and infrequent, and principals—who are no more skilled at math than their teachers—remain unprepared to offer support.” She holds in disdain the “I, we, you” model of classroom teaching (and which is one of the few techniques actually supported by research) and argues instead for “You, Y’all, We”/discovery learning. And to bring that off requires teacher training that is “meaningful”—i.e., allowing for properly implemented reform ideas and Common Core.

Green argues that reform math done right does work, pointing to Japan for evidence. She claims Japan is doing U.S.-style reform correctly. Japan’s method of teaching is admittedly different in how teachers collaborate to create “lesson plans” and anticipation of student misunderstandings and questions. But Green fails to acknowledge that Japanese instruction is strongly teacher-led and does not rely on discovery. A paper by Alan Siegel, (see http://www.cs.nyu.edu/faculty/… ) a math professor at New York University, provides valuable insight into myths about Japanese teaching and shows that it is based on whole class and direct instruction techniques—and much practice.

Wayne Bishop, a retired math professor, is familiar with what has been done in Japan and provides evidence that the approach that Green talks about (which she recommends should be done in the US) did not succeed and that they went back to the way math had been taught. His article about this can be found here. I suggest that anyone swayed by Green’s arguments, including the author of The Hill article read it.

One for the Old Schoolers, Dept.

In an article about a private Christian school that maintains teaching cursive and math facts, was this paragraph:

Learning cursive and strict memorization of fast math were dropped from Common Core educational standards around 2010 and 2014, respectively, but states and individual school districts still have the option of teaching them. Cursive writing was removed because of its declining use and the increase in technological communication. Focus on math memorization was discouraged because Common Core began placing emphasis on deep understanding of mathematics concepts, instead.

This is fairly accurate. Although CC doesn’t call for “strict memorization” of math facts, it does call for fluency with them, which of course is subject to interpretation.  Current implementation has students learning “strategies” for multiplication/addition facts, and a rather neurotic insistence that students “understand” what 2 + 3 is rather than just memorize it–as if such connection was never there and students just memorized blindly for years without knowing what addition and multiplication represented.

I would say that St. John’s School is in fact implementing Common Core. In fact, CC can be implemented in many different ways, including those that are deprecatingly referred to as “old school”.  Or maybe even “traditional”.

Don’t tell anyone. It’ll be our little secret.

Unpacking Learning Goals, Dept.

Education Week ran an article about what the author called “Lesson Imaging” that used such phrase. The basic thrust of the article (and phrases like “unpack the learning goals”) was based on the premises stated in this passage:

Inquiry Mathematics and Science. Leading mathematics and science educational organizations (e.g., NCTM and NSTA) have called for fundamental changes to instruction to include a more student-centered classroom. Also, most state standards for mathematics and science education include requiring students to create viable solutions to problems through inquiry and communicate their reasoning (CCSS-M, NGSS).

As I have indicated in articles I’ve written, the Common Core website contains statements that the standards do not dictate pedagogy.  For example here:

While the standards set grade-specific goals, they do not define how the standards should be taught or which materials should be used to support students.

And here:

Do the standards tell teachers what to teach?

Teachers know best about what works in the classroom. That is why these standards establish what students need to learn, but do not dictate how teachers should teach. Instead, schools and teachers decide how best to help students reach the standards.

Despite such advice, the prevailing interpretation is that Common Core does call for inquiry-based approaches as evidenced in the quoted passage from Education Week.

The Education Week article then notes that the publishing industry has responded to this “requirement”:

Textbook publishers have created materials that attempt to engage learners in more inquiry activities that foster the development of rich science and mathematics understanding. Along with a shift in educational goals, math and science teachers have been pressed to change from a traditional lecture, information-sharing classroom style to student-driven inquiry. Such a shift has been challenging for teachers because of the lack of resources to guide teachers in their pursuits.

Unstated in this article and in many others like it, is that the traditional practices of the past have not worked. And the articles that do state that, omit much in the way of evidence for such statement, aside from the standard “Many adults today cannot do basic math, so there’s your proof.”  Such proof seems to neglect that if someone hasn’t done something in a while, they tend to get rusty and forget certain procedures, like finding percent of increase or decreases, or the formula for finding the volume of a pyramid or cone, or surface area of a sphere. The so-called proof also neglects that just as many adults today do remember their math facts, know when to multiply or divide, and know basic procedures with decimals and fractions.  By contrast, many of today’s students–who have been taught using the student-centered, inquiry-based approaches championed by NCTM and other reform-based organizations over the past 28 years or so–cannot do basic math.

And this is why I found Tara Houle’s recent op-ed about the education problems in British Colombia so pertinent (and of which I wrote about a few days ago). Of note was this particular passage:

If we truly want to ensure our kids have a bright future, we must first build on the successes of the past, and bring that forward for them. That has not happened here. Successful methods of teaching mathematics have all been eradicated from British Columbia classrooms.

There is a rabid fervor promoting 21st-century learning, and insisting that inquiry-based learning take precedence over everything else. However, the fundamental principles of arithmetic are non-negotiable. Without mastering these crucial facts at the elementary level, any attempts at Math 10, pre-calculus or entry-level university mathematics will end in failure.

I wish the best for British Columbia’s education efforts as well as other provinces experiencing these same problems and hope they don’t continue to follow the pathway that the U.S. has been on.  “Rich problems” are fine and in their place fulfill an important purpose. But there is no clear-cut or convincing evidence that this gives any insight into a great way to teach core material in this subject.

Not much to say to this, Dept.

George Monbiot wrote an article in the Guardian about how our schools teach our students to be redundant. It is a wonder of writing in that he leaves out no cliche about how schools are broken (e.g,”why is collaboration on an exam considered cheating?”) as well as the usual mischaracterizations of traditional education (e.g., “factory model schools”, “teaching students to be robots” etc).

In the ensuing criticism of the article on Twitter (and there was much), there was also the usual “And he hasn’t taught a day in his life.” I once leveled that at someone spouting some similar nonsense and was rewarded with a rather hostile response back stating that the person did not have to be a teacher to be allowed to criticize.

And that’s true. Years before I became a teacher I wrote many articles criticizing how math was being taught. And there are many parents upset at how their children are being taught math–rightly so.

There are plenty of things wrong with what the author has to say about education in the article so that one does not have to rely on his not being a teacher in order to question his credibility. The reference to the Prussian model of education as the source of the factory model, for example. And there are others that many people have written about so I won’t waste any more of your time.

Oh, I guess it wouldn’t have hurt for him to be a teacher in the case of his question about “Why is collaboration on an exam considered cheating?” but I’ll let it pass.

Ralph Raimi on Proportionality, whatever that means…

Ralph Raimi, who recently passed away, was involved in his retirement years of fighting the battles commonly referred to as “the math wars”.  I am happy to say I knew him and corresponded with him frequently about the various battles in math education.

I’m also happy to say that his website is still accessible, and it contains much of his valuable writings on various subjects.  One that recently caught my eye is a letter he wrote to the Dean of Education at CUNY, Alfred Posamentier.  Dr. Raimi takes apart the focus of “understanding” proportions.  His letter synthesizes a lot of what’s wrong with reformist/progressivist math ed arguments that involve “understanding”, and comes to the conclusion that much of what is perceived as confusion is alleviated with the use of letters to represent values (via formulas) and eventually a proper teaching of algebra when the time is appropriate.

When I was a child I was able to do such problems easily enough —
sometimes — but at other times, e.g. the problem of two hoses filling a tank,
I was paralyzed by the boxes (chalked rectangles) into which my teacher put
the key numbers on the blackboard, and I could not remember whether to add
or multiply, or where to put 1/8, or maybe 8, if hose #1 took 8 hours, etc.
In other words, the attempt to teach me “proportional reasoning” collapsed
when the problem required the invention of a rate.  I did not overcome my
shyness about such problems (I had to draw pictures, rather than boxes with
numerals, and then couldn’t explain what I had done) until I learned some
algebra and was able to label every quantity in sight with a letter, writing
down all relations I could think of, that the data gave me, and then solving
for the quantity asked for.  What was proportional to what no longer needed
to concern me; the relations dictated by the problem led to airtight equations
about the meaning of which I no longer had to think. It was, to me, a

     Since learning about the current obsession with “proportional
reasoning” I have decided that the language of functions, input and output, is
the easiest way to understand such problems.  After all, Proportional is the
description of only one class of functions, and all science is the quest for
analogous relations.  What is there against the use of letters and equations
from the very beginning, when such real-life problems are first attacked?
First learn the number system itself, then observe that in the real world there
are many relations expressed by formulas, in which an input and an output
are related by a scale factor, or rate.  Write the relation and solve. 

The letter can be accessed here.


What are they afraid of? Dept.

My friend Tara Houle in BC has an excellent Op-Ed in the Vancouver Sun. Epitomizes much of what is bugging those pesky traditional math people like me:

Recently, my daughter’s dance studio did something revolutionary — they adopted an exam system for their pupils. Their reasoning for this was shocking: In order to ensure their student’s progress in a more meaningful manner, rigorous practice and assessment would be required, both for their instructors and their students. In short, students would be trained properly under the watchful eye of their knowledgeable instructors, and then be held accountable to demonstrate their understanding, and performance, by attending exam preparation sessions and a final exam. These students are in good hands.

Unfortunately, this same attention to detail isn’t happening in today’s math classrooms. Ample evidence illustrates there has been a significant decline in our student’s math performance over the past 15 years, and we also know that the percentage of our top math students has fallen dramatically. Tutoring rates have recently skyrocketed, as parents are now scrambling to ensure their kids learn the fundamentals properly — something that is lacking in today’s classrooms. This spike in enrolment correlates with an increased use of inquiry/problem based learning in our schools. Yet education leaders don’t want to acknowledge the tutoring phenomenon. They are silent when asked to investigate this issue, to determine how many kids are using tutors, and if so, why? Are our ministry officials interested, or are they afraid of what they might find?

Read the whole thing here.


The Best Writing on Mathematics, 2016

The 2016 anthology of writing on mathematics has been released by Princeton University Press.  I’m happy to say that the article that Katharine Beals and I wrote, which was published in The Atlantic (online) in November, 2015 made it to this selection.  The article (“Explaining Your Math: Unnecessary at Best, Encumbering at Worst“) received many comments–both hostile and supportive. In addition it propagated discussions on various math education blogs–again engendering hostile and supportive comments.

We argued that requiring students to explain their answer to math problems created more problems than it solved and did not offer the path to “understanding” that math reformers seem to think such practice achieves.  We stated that the math itself carries the explanation. It is one thing for a teacher to ask students questions about how they arrived at an answer, but quite another to require students–particularly those in lower grades–to do so in writing.

The introduction to the volume is online if you are interested. I was heartened to see that the editor (Marcea Pitici) was kind enough to also mention me in the same paragraph as e mentioned Jo Boaler and Carol Dweck.  This could be good, this could be bad.

“A great number of books on mathematics education are published every year; it is not feasible for me to mention all that literature. Here are a few recent titles that came to my attention: Confessions of a 21st Century Math Teacher and Math Education in the U.S. by our contributor Barry Garelick, What’s Math Got to Do with It? by Jo Boaler, Mathematical Mindsets by Jo Boaler and Carol Dweck, More Lessons Learned from Research edited by Edward Silver and Patricia Ann Kenney, Assessment to Enhance Teaching and Learning edited by Christine Suurtamm, How to Make Data Work by Jenny Grant Rankin, and the refreshingly iconoclastic Burn Math Class by Jason Wilkes.”