Who ya gonna believe and whatcha gonna do about it? Dept.

I saw this article about a school adopting enVisions Math for elementary grades. It was a typical “Everyone’s happy in Happy-Land” type of story complete with the usual accolades for how the program is “balanced”:

According to K-12 Mathematics Coordinator, Gregory George, “enVisionmath2.0 is what we call a balanced program. It emphasizes conceptual understanding, procedural fluency, and applications and problem solving with equal intensity. We believe this approach to math instruction provides a complete learning experience for students that honors understanding of concepts and the ability to solve math problems efficiently and accuracy. 

But then you have this story in the Baltimore Sun about the same program, saying this:

Some elementary school parents and students expressed concerns about the program to the school board earlier this year, however, saying its “abstract” nature made it difficult and time-consuming for parents to help their children with math homework, even with online support tools. Those who complained said the program had caused children who once enjoyed math to hate the subject.

So which story are you going to believe?

If this is the first time you’ve heard the words “conceptual understanding, procedural fluency, applications and problem solving” in one sentence then you’ll likely believe that this program does it all.  Those of us who’ve been around the block a few times recognize those words as meaning the program obsesses over “understanding” and provides inefficient, picture-drawing and/or convoluted approaches to addition, subtraction, multiplication and division in lieu of the standard algorithms, which are delayed until 4th, 5th and 6th grades.

This delay is justified by saying that’s what Common Core requires, even though it does not.  This belief and practice persists despite words to the contrary from Jason Zimba, one of the lead writers of the Common Core math standards who states in an article he wrote that “the Common Core requires the standard algorithm; additional algorithms aren’t named, and they aren’t required.”

But the real key word here is “equal intensity”.  In programs similar in look, feel and practice like enVision Math, students are made to drill (yes, drill) these inefficient strategies and in so doing they attain a “rote understanding” of the underlying concepts–an understanding that could  have been attained by teaching the standard algorithms.  For more about “equal intensity” see this.  But the prevailing group-think of educationists everywhere posits that teaching the standard algorithm “too early” eclipses the understanding with kids gravitating to the procedure. And they claim they have the evidence that this is so.

In real-life, kids tend to gravitate to the procedure no matter what.  I have seen this even with students in accelerated classes who are highly motivated and quite bright.  I teach for understanding like many teachers, despite statement made that traditional math teaching does not do this. Most students glom on to the procedure.  Procedure and understanding work in tandem; sometimes understanding comes first, sometimes it comes later.

The parents who complain that the approach used in enVision Math (and other comparable programs) don’t teach math as they were taught are castigated by those who are part of the pervasive edu-group-think.  Ironically, those doing the castigating are for the most part adults who have attained understanding after having been taught in the traditional manner. After seeing how enVision Math does it they exclaim “If only I had been taught math this way.”  They climbed the math ladder like the parents they put down for their mistaken beliefs. They then kicked it away when they reached the next level, and now insist on bad practices and eschew the practices used by traditional teachers.  Regarding the practices used by Kumon, Sylvan, Huntington and other tutoring/learning centers, they get very silent and say “Well, we just don’t know to what extent it’s the tutoring or the program used in the schools.”

Who ya gonna believe?



Principal Gladhand Verbatim, Dept.

I regularly read Principal Gladhand’s weekly missives on how great his school is. There’s something that bugs me about his missives, but there’s nothing distinct that I can point to. I think it’s the unspoken but ever present undertone of “Look at me and how great I am” .  I’m interested in your reactions and interpretations, so will be posting these regularly.  Here’s the latest one.

Many of our students participated in the 17-minute walkout on Wednesday, March 14, and many of our students didn’t. To me, this tells me students felt respected for whatever they chose to do that day and students were comfortable expressing themselves by walking out, or by remaining in their class.
For the students who did walk out, they were amazing. In the quad, we had set out markers and butcher paper with the title “17 things you can do to change the world” written at the top. Students filled the papers with wonderful ideas and expressions of positivity. Students in the quad were also encouraged to meet 17 other students that they didn’t know well. Students all over the quad were shaking hands, exchanging names, and lending a smile to new people. It was heartwarming and amazing. There were no student discipline issues during this time, no goofing around, and at the end of the 17 minutes, students returned to class as if it were a normal part of our day. They managed all of this beautifully, and we were all very proud of them.

What you learn in Ed School, Dept.

For those who are wondering what future math teachers learn in ed school, here is a concise summary:

Traditional mathematical teaching has never worked and has failed thousands of students.

The standard ed school catechism is that traditional math teaching is based on rote memorization with no understanding, and no connection between concepts.  Another is that the conceptual underpinning of math procedures are not explained. According to ed school teaching, procedures are presented as a “bag of tricks” (such as “keep, change, flip” for dividing fractions). The evidence presented is simply that many adults do not remember how to solve certain problems. This stands as proof that the traditional methods are not effective–if they were, they would “stay with us.”

That people do not maintain proficiency in math as they age says less about traditional or reform math than about the way in which a population’s knowledge and skill base is maintained over a lifetime. It is not evidence of failure of traditional math.  The results of not using math on a consistent basis can also be seen in a study conducted by OECD.  In the study, people from ages 16-65 in over twenty countries, including the U.S., were given the same exam consisting of math computations and word problems.  According to the study, “the percentage of U.S. adults between 55 and 65 years old who scored at the highest proficiency level (4/5) …was not significantly different than the international average for this age group. (Goodman, et al., 2013).”  These findings can be used in tandem with the first argument above since people in the U.S. in the 55 to 65 age group learned math via traditional math teaching—and the differences in proficiencies between the U.S. and other countries is not significant.

This argument ignores that in countries doing well on such international tests, students learn math  mainly via traditional means — and over the past two decades, increasing numbers of students in the U.S. have learned math using the reform-based methods. Reformers are quick to point out that Japan and perhaps other Asian countries actually use reform methods, ignoring the fact that many students are enrolled in “cram schools” (called Juku in Japan) which use the drilling techniques and memorization held in high disdain by reformers.

The argument also fails to consider that traditional math can also be taught poorly. There have always been good and bad teachers, as well as factors other than curriculum and pedagogy that influence the data.  In order for such arguments to work, one would have to evaluate how achievement/scores vary when factors such as teaching, socioeconomic levels and other variables are held constant and when pedagogy or curriculum changes. Studies have been conducted that examine how math is taught in specific areas of North America, as well as looking at the common traits of high-performing systems across the world.  They indicate that when both conventional and non-conventional (i.e., reform) math are taught by well-trained teachers, students learning under traditional mathematics instruction show much higher achievement than those learning under the reform math methodology. (Stokke, 2015; see here

Traditional math failed to adequately address the realities of educating a large, diverse, and rapidly changing population during decades of technological innovation and social upheaval.

This argument relies on the tracking argument, when many minority students (principally African Americans) were placed into lower level math classes in high school through courses such as business math. It goes something like this: “Most students did not go on in math beyond algebra, if that, and there were more than enough jobs that didn’t even require a high school diploma.  Few went to college.  Now most students must take advanced math, so opting out is not an option for them like it was for so many in the past.”

First, in light of the tracking of students which prevailed in the past, the traditional method could be said to have failed thousands of students because those students who were sorted into general and vocational tracks weren’t given the chance to take the higher level math classes in the first place — the instructional method had nothing to do with it.  Also, I don’t know that most students must take advanced math in order to enter the job market. And I don’t think that everyone needs to take Algebra 2 in order to be viable in the job market.

Secondly, while students only had to take two years of math to graduate, and algebra was not a requirement as it is now, many of today’s students entering high school are very weak with fractions, math facts and general problem solving techniques. Many are counting on their fingers to add and rely on calculators for the simplest of multiplication or division problems.  In the days of tracking and weaker graduation requirements, more students entering high school than now had mastery of math facts and procedures including fractions, decimals and percents.

Some blame the “changing demographics” on the decrease in proficiency, but this overlooks variables like poor curriculum and the reform-based approach to math which views memorization “workarounds” as deep understanding. Also frequently overlooked is the fact that students in low income families who make up the “changing demographic” cited in such arguments do not have access to tutoring or learning centers, while students in more affluent areas are not held hostage — dare I say “tracked”? — to poor curricula and dubious pedagogical practices.

Teachers use a combination of reform- and tradition-based methods so we are all saying the same things and there’s no point in making such distinctions.

I do not think that I am alone in drawing a distinction between reform and traditional modes of math teaching. While traditional math can be taught properly as well as badly, I believe that poor teaching is inherent in most if not all reform math programs. I base this on having seen good teachers required to follow programs that present content poorly, lack a coherent logical sequence and rely on questionable pedagogies.

I would like to see studies conducted to document how U.S. students who do well in math and science and pursue STEM majors and careers are learning math. The chances are fairly good that such investigations would show that in K-8, many students are getting support at home, from tutors, or from the many learning centers that are springing up all over the U.S. at rapid rates. Since tutors and learning centers (and parents) tend to use traditional methods for teaching math, I somehow doubt that the clientele are exceptions to some ill-defined rule.  In my view, as well as the view of many parents and teachers I’ve met, there are few exceptions to the educational damage reform math programs have caused, even when such programs are taught “well.”

In summary…

People may choose to use the information I’ve presented here — or persist in ignoring it. I don’t expect that I’ve changed anyone’s mind about anything, but I am always hopeful. that there are some exceptions.

Rationalizing memorization, Dept.

I thought this article would be in the category of “learning math through interpretive dance” but I’m happy to say, I was wrong. It’s about a math teacher who uses musical chants/rhymes to help students memorize (yes, memorize) particular formulas and procedures.

As she puts it:

“Memorizing basic formulas can make it easier for students to grasp larger, more abstract mathematical concepts because students’ minds aren’t mired in the minutiae, Jorgensen said.

For example, it’s easier to understand the square root of 36 if you already know the answer to 6 multiplied by 6.

Jorgensen’s method has yielded results. In her 8th-grade geometry class from last year, 23 of her 40 students had perfect scores on the Smarter Balanced exam, and in the 7th-grade algebra class, every student exceeded the standards, she said. Seven of those students had perfect scores.

I couldn’t believe I had actually read that and read it several times to make sure I wasn’t dreaming. Most teachers would be hung for saying something like that without mentioning the U word. (“Understanding”).

Oh, but wait.  Here comes an apologist to offer a strawman about understanding and memorization:

Mark Ellis, a math education professor at California State University, Fullerton, said he observed elementary teachers in Japan using songs and chants to successfully teach math to their students and he has used music to help low-performing middle-school students learn their multiplication tables. … But it’s not the end of the story, he said.

“Music itself cannot teach kids to understand mathematics,” he said. “Music can help students improve dramatically, but ultimately math is not about memorization. It’s about reasoning, seeing patterns, making conjectures. It’s about meaning.”

Memorizing formulas will only be effective in the long run if students understand the concepts underlying the formulas, he said.

Anything else, Mr. Ellis? Oh, yes, he does have something more to say:

Ideally, students should be able to come up with formulas on their own, with guidance from the teacher. In some cases, it’s not even necessary to memorize formulas because so many students have calculators on their phones, he said.


Nothing to see here, Dept.

From Bermuda comes this report about less than stellar results on the latest math exams in the school system there.  True to form comes this rubber-stamped mischaracterization of traditional math, serving as explanation of why students do poorly in math:

Mr Pitcher said: “This is not a new problem and will take dedication and hard work from everyone to make a real change.”

He said that student struggles with maths were not specific to Bermuda.

Mr Pitcher explained: “This has been a major issue for many countries, including the United States.”

He said that a number of factors had led to problems in the subject, including the way maths is taught, which he described as “counterproductive”.

Mr Pitcher explained: “We teach students several formulas they ‘need’ to memorise and introduce a litany of abstract symbols, then we strive to find application of what we have taught with meaningless word problems.

As noted in the article, and of particular importance is that Mr. Pitcher is the founder of tutoring services Planet Math.

The More Things Change, Dept.

The Education and Human Resources Dept (EHR) of the National Science Foundation (NSF) has been granting millions of dollars to beef up math education in the US. In the early 90’s they gave grants to entities to create math programs/textbooks that parents have been protesting for years:  Everyday Math; Investigations in Number, Data and Space, Connected Math Program,  Core-Plus Math (an integrated math program for high school) and other educational atrocities.

They continue pumping money into the education machine–universities and school districts–to ensure that the latest fads/trends in education (STEM) stay true to the educational party line which ultimately gets implemented in your children’s schools.

Their latest gift is $2.8 million to the University of Houston’s education school  The grant covers new courses in the ed school as part of a masters program whose title tells much of the story:  “Enhancing STEM Teacher Leadership Through Equity and Advocacy Development in Houston”

I just about stopped reading when I read the title but like a passer-by on a busy highway who says he will not look at the bloody accident on the shoulder, I looked.

Here are the highlights from the accident:

The courses (and of course coaching—what program would be complete without coaches that help teachers to stop teaching and start facilitating) include topics such as:

  • Culturally responsive teaching and addressing learning disparities in STEM education.
  • The roles of technology and inquiry-based instruction in STEM education
  • Engineering design

As long as we’re paused at the site of the bloody accident, let’s take a closer look. The first bullet doesn’t go into a lot of detail but I wondered if they were going to take the view that math is all about “white privilege” and should be taught differently.  Meaning, from what I’ve read of such arguments, that it be watered down or discarded entirely.

Technology is always a big one. One must use technology at all costs; if you aren’t using technology in education then you are not doing it right, apparently.  iPads, online textbooks, writing in text-based sentences rather than reports—the list goes on.  Inquiry-based instruction simply goes without saying. Since we put our students through the “read my mind and tell me what I’m trying to get you to say/discover” exercise, why not the same for our ed school students—fair is fair!

Engineering design sounds great but I’m willing to bet it has as much to do with engineering as the so-called “coding” programs schools are saying students must learn.  These coding programs are mostly pre-packaged, pre-coded software that allow you to draw pictures and engage in other amusing activities. I imagine the engineering design is similar to Project Lead the Way; not much math, not much engineering, but a lot of “maker-space” and “project-based learning” items that teach little about science, technology, engineering or mathematics.

Of course, I could be entirely wrong about all of this.




Clarifications, Throat Clearings, and Other Furthermores, Dept.

In my last “smart and thoughtful post” (the parlance that edu-pundits use when they refer to each other’s writings), I talked about “understanding vs procedure”. The quote at the end from a teacher in New Zealand seemed to ruffle the feathers of some who took to Twitter to state that they believed otherwise.

In all these discussions of “understanding”, those who believe it is not taught and that students are doing math without knowing math rarely if ever explain what they mean by understanding in terms of how it translates or transfers to problem variations or new areas in math.  For example, a student who has learned the invert and multiply rule for fractional division may not be able to explain why the rule works, but may have an understanding of what fractional division represents.  The student then uses the latter to solve problems requiring fractional division.

Anna Stokke, a math professor at University of Winnipeg has also addressed the issue of student understanding in math and echoes what the teacher in New Zealand said. She has  kindly given me permission to quote her:

When we teach, most of us generally do teach students why things are true but I sure don’t want my students going through the understanding piece every time they solve a problem. What a waste of time! The point, I think, to get across to students is that there is a reason why everything in math works the way it does and you could figure this out if you need to (because you WILL almost certainly forget).

With the consultants I’ve met, who always push this stuff and insist that kids aren’t fluent unless they can explain everything to you, it seems that they themselves just figured out that there are reasons behind procedures in math as adults. Then they’re angry that their teachers (supposedly) didn’t explain all these things to them. They’re certain that they would have liked math more and done better if only their teachers would have focused on understanding. So, their mission is to make sure that all kids are forced to explain their thinking at every step. Pure torture, really.

Funny thing is, that the understanding piece is a lot more difficult for students. They generally don’t like it and it’s something that really comes with much experience and mathematical maturity. It won’t make students like math more if we spend more time on understanding…it will just confuse and frustrate them more. In my experience, I’ve found that students like step-by-step procedures and algorithms more than anything else.

What people in the “understanding uber alles” crowd likely mean when they talk about understanding probably has to do with words. They would probably be happy with words that didn’t ensure that the kids could actually DO the problems: a “rote understanding”.

We’ve always been at war with Eurasia, Dept.

What with Robert Pondiscio’s welcome and well-written article extolling the benefits of Direct Instruction (Zig Engelmann’s method for instruction) and thereby praising direct instruction in general, there are indications that others may be following suit.  I just read a blog piece by a math teacher who has reached the eye-opening conclusion that conceptual understanding doesn’t always have to precede procedural fluency. In fact, procedures may not be the bogeyman that math reformers have been saying they are for the past hundred years or so.

And it isn’t as if math teachers have routinely refused to teach the conceptual understanding. It’s just that if you’ve spent any time at all in a classroom, you will have noticed that your students glom on to the procedures.  And unless the conceptual understanding piece was part and parcel of the procedure (as is the case with adding and subtracting with regrouping) few if any remember the underlying concepts.  This has led to math texts now “drilling understanding” by making students do the conceptual understanding piece as if it were the algorithm itself; i.e., 3 by 5 rectangles and shading the appropriate parts to represent 2/3 x 4/5 as a means to “understand” what fractional multiplication is.

The belief still persists that in order for students to understand, math must be made relevant.  It just can’t be that if students know the procedures and can do them, and use them to solve problems, they really do not care if the problems are relevant or not.  And so we have statements like this which appeared in a recent Education Week testimonial/polemic that passes as evidence-based, research-based, relevance-based, brain-based and any other kind of “base” you can think of:

Math lessons, on the other hand, have historically focused less on real-life connections. Like many students, I excelled in math by memorizing rules and tricks. In college, I trained to teach social studies, but became a math teacher by accident because I had earned enough math credits to qualify for a math teaching certification.

Never mind that the author of the article may have benefitted from what she calls “rules and tricks”.

In any event, it appears that there may be more teachers who had insisted we are at war with Eastasia now coming out from the woodwork to say that we’ve always been at war with Eurasia, though it comes out more like: “Hey, procedures aren’t that bad, and most kids don’t really get the understanding til later.”

I will leave you with the words of a math teacher I know from New Zealand who puts it this way:

A few years back I started explicitly telling my students “I don’t care if you understand it, provided you can do it” when they complained that they “didn’t understand”. I tell them that when their exam papers are marked there are no marks for “understanding”. I follow that up with saying that understanding will inevitably follow in time, provided that they could do the skills, but that it would not follow if they couldn’t do the skills.
Now that isn’t to say that I don’t teach the reasons for things — I teach invert and multiply explicitly, but I also explain why it works. What I don’t do is fret about whether they understood my explanation, and I don’t let them not do something because they “don’t understand”. I most certainly do not try to teach understanding of a procedure to a student who can do it accurately.
Some students find that truly liberating — they can get on with learning the Maths without any pressure to have to understand the whole picture first. Most just do what they always have done, which is do what the teacher asks them to do and not worry about understanding because they never have.  To the fury of many reformers, most kids really don’t want to understand very much.


Everyone’s Happy in Happy Land, Dept.

Another Happy Land story about how schools are seeing the light and not teaching math in the way it used to be taught.  It is a given in education and apparently also for reporters, not to ever challenge the premise that the way it was taught never worked.  This story is no different.

We start with the classic notion that math shouldn’t be rote memorization (as if that is what traditionally taught math is about) but about critical thinking.

At Fair Oaks Elementary in Brooklyn Park, teacher Michelle Kennedy pushes her math students to give her more than just the right answer.

Her tactic was on display during a recent lesson when she asked her class: What is three plus three?

“It’s a six!” a kindergartner blurted out.

“Why?” Kennedy asked, unsatisfied.

“I saw it in my head,” the timid student explained.

“How did you see it in your head?” Kennedy persisted.

“I see a three and a three,” the student answered.

Really, folks, kids really do get what addition and subtraction are about without having them explain it each and every time.  But memorization is a no-no unless students show that they “understand” what is going on.  And the pay off is evident; I see high school students counting on their fingers to get the answer to 7 + 8; they are definitely showing understanding of what addition is about by combining the numbers, rather than just pulling it from memory.

The new approach: less memorizing formulas and more focus on understanding math concepts and building up kids’ confidence to do math. School leaders say the changes are necessary to shift the emphasis from boosting test scores to better preparing students to excel in college and in the workforce.  “Our instruction needs to change to meet the needs of today’s workplace,” said Kim Pavlovich, director of secondary curriculum, instruction and assessment for the Anoka-Hennepin School District.

Most of the school districts mentioned in the article use some form of discovery-based programs, such as CPM. They also use other chestnuts such as Everyday Math, whose spiral approach–in which students partially learn a concept and then bounce to an unrelated topic the next day and eventually spiral back to the first concept which by now they have completely forgotten–has gone unquestioned by the powers that be.  The publisher simply tells the teachers to “Trust the spiral” and those words are repeated to parents.

There is one district using “Math in Focus” which is based on the programs used in Singapore. And though such series has been Americanized to include the aspects of discovery and explaining things that defy explanation, it is at least a step in a better direction.

But for the most part, people are happy in Happy Land with programs like CPM:

On a recent Thursday, eighth-graders at Roosevelt Middle School in Blaine tackled math problems together using the CPM program that focuses on teamwork. They sat in small circles in a classroom, decorated with motivational words such as “I’m going to train my brain to do math” and “Mistakes help me improve.” They measured the rebound ratio for a small ball in groups of four, carefully explaining their answers to each other. They demonstrated to the teacher who was walking around the classroom that they knew how to at least solve a decrease in a quantity using graphs and measuring sticks.

 Math teacher Carrie Paske peppered each group with such questions as “Tell me why you think that?” to gauge their critical thinking skills.

 In her class, students use objects like algebra tiles to help them visualize algebra. Homework also has become less of a burden because students take home no more than five problems.

Yep. Everybody’s happy all right!  They even have the Jo Boaler-inspired quotes to keep them going.  The question that’s never asked, however, is how many students who make it into AP calculus in high school and major in STEM fields have had help at home or from tutors or learning centers.  And how many students how have not had such help are still counting on their fingers and doing poorly in math in high school?