SteveH on problem solving transference and “understanding”

SteveH, a frequent commenter at this blog, has made some cogent observations about the transferability of problem solving skills, and about “understanding” in general.  Like me and others I know, he feels that far too much importance is placed on understanding in math in K-6 than is necessary. He also posits that mastery of procedural skills in the early grades and even high school, doth not a “math zombie” maketh.  I’ve taken two of his recent comments and placed them here for general interest.

I. Problem solving and transference

I want to say more about problem solving and transference. Polya is worthless, so what problem solving skills are transferable? You can draw pictures, label variables, and write equations to look for m=n. Everybody does that, but still, it’s difficult. You can study governing equations and their variations. This is classic homework p-set work. However, does D=RT problems and variations easily transfer to work problems? Yes and no. Both are amount = rate * time sorts of things, but there are lots of odd variations. Look at AMC test problems. You could know a lot about logarithms, but the testers are really good at finding odd problem variations. I look at some of their questions and feel really stupid. Success on the AMC is not really about general transference, but very specific and detailed preparation of problem classes for a timed test. You don’t study general problem solving skills. You study in detail every past test question you can get your hands on. Only that reduces transference distance.

How about general problem solving transference? First, it’s not a timed test issue. The question is whether you can do the problem eventually. I’ve seen cases where the smart kids in a class take longer to figure out the trick or angle. Knowing and working with lots of governing equation variations is probably the best foundation. How about this problem?

You are sitting in a row boat in a big tank of water that has a scale showing the level of the water in the tank. You take out the anchor and drop it into the water. Does the level of water in the tank go up, down, or stay the same?

What general problem solving skills help with this if you don’t know the governing equation? Do you have to do a JIT (thanks Barry) discovery of Archimedes? There really is no such thing as general magical transference and problem solving outside of m=n, and that’s not a big help even with a LOT of p-set practice. If you figured out this problem quickly, does that mean that you will do the same for any other problem? I doubt it. Even Feynman used to study up on trick problems so that he could impress people when he pretended to figure them out off the top of his head. Colleagues called him a faker.

Feynman’s true understanding brilliance was based on not only on knowing many governing equations in physics, but making sure he had a real physical sense and connection of those equations to reality. However, even that ability has limited trasference. You have to develop that sense for every different governing equation. Feynman struggled a lot. I still struggle a lot with problem solving. Epiphanies of understanding come to me, but only after I struggle and work on problems for a long time. Then again, there is a huge gap of hard work and discovery between insight and any sort of final solution.


II. Understanding and challenging assignments

This is from my son’s old Glencoe Algebra 1 textbook on factoring differences of squares – Exercises. (page 451)

Section 1 – Basic skill problems like this – Factor or prime?
256g^4 – 1

Section 2 – Factor and solve
9y^2 = 64

Section 3-6 various word problem applications

Section 7 – Open Ended
“Create a binomial that is the difference of two squares. Then factor your binomial”

Section 8 – Challenge
Section 9 – Find the error
Section 10 – Reasoning – find the “flaw”
a = b
a^2 = ab
a^2 – b^2 = ab – b^2
(a-b)(a+b) = b(a-b)
a+b = b
a+a = a (remember that a=b)
2a = a
2 = 1

Section 11 – Writing in Math

Section 12 – Standardized Test Practice

Section 13 – Spiral REview questions

Section 14 – Reading Math
Learn to use a two-column proof for algebraic manipulation

So what’s the problem here? Is it just that teachers only assign the problems in the first two sections? Clearly, those are the most important sections and they are not dumb, rote, busy work just for speed. You have to understand the basic concepts and see the variations. However, there are many layers of understanding and nobody can fully understand the implications with just a few problem variations. That’s why it’s common for many to not really “understand” algebra until sometime during Algebra 2.

You can get good grades in math, but still feel like you are struggling. It’s not a “zombie” issue. It’s normal. You can get poor grades and not understand, but that’s another issue. If you get good grades and really don’t understand, then probably your grades are OK, but dropping. Some honors (proper math) classes say that you can only enter if your grade is 80+ or some such thing. Gaps and weaknesses in understanding will eventually cause you to fall off the cliff. Math is tough in high school and college. However, it’s NOT that in K-6, but too many kids struggle. It’s NOT an understanding issue.


And just what are you after exactly, Dept.

Sorry to keep harping on the Education Trust “study” that finds middle school assignments lacking in the “conceptual understanding” department.  In the Education Week article on the “study” (and no mention whatsoever in the study of what mathematicians, engineers and/or scientists were consulted in writing it) a commenter agreed wholeheartedly with it and said:

And I COMPLETELY agree that my experience in MS (and HS) classrooms focuses way too much on procedural over conceptual understanding. 

Which caused me to wonder: Just what conceptual understanding do people think is missing from middle school math?  It isn’t that students are just given problems to solve without explaining what the concepts are.  Percentages are explained, as are decimals, as are fractions, and why one uses common denominators, and even the why and how of multiplication and division.  Anyone who teachers middle schoolers or even high school students, knows that students gravitate to the “how” rather than the why.

SteveH who comments here frequently notes the following about procedural vs conceptual understanding:

Kids LOVE being good at facts and skills. They are easier to ensure and test, and that success drives engagement and much deeper learning and understanding. Skills come before understanding and engagement, not the other way around. The best musicians are the ones with the most individual private lesson skill instruction. They didn’t get those skills top-down by playing in band or orchestra only. The process is difficult and many don’t like it and drop out. This is true for any real life competitive learning – it is not natural.

Such disputes are usually settled in the same manner as the one about whether inquiry or direct instruction is better, and someone says “You need a ‘balanced’ approach” without defining what that balance is.  In the argument about understanding vs procedure, the usual bromide is “they work in tandem”.  This tells us absolutely nothing.

Sometimes the conceptual understanding is part and parcel to the procedure like place value and carrying and borrowing (two terms for which I make no apology for using).  Other times it is not.

Having understanding is only part of the process. If I may talk about calculus for a moment. In upper level math courses in college, one learns the concepts behind why calculus works–including the delta-epsilon definitions of limits and continuity.  A student may be able to recite the definition of continuity and tell you what needs to be done to show continuity at a specific point  in a function.

(I.e., a function f from R to R is continuous at a point p ∈ R if given ε > 0 there exists δ > 0 such that if |p – x| < δ then |f (p) – f (x)| < ε.)  But having a student prove that the function y = x^2 is continuous at the point x=2 involves a procedural knowledge of how to go about doing that.  Just knowing the theorem is not enough.)

So I’d like to know.  Do these people who claim they focus too much on procedure think that the majority of middle school students are just operating blindly as “math zombies” as some bloggers like to call it, without any knowledge of what it is they’re actually doing?  Really?


More just in, Dept.

Looks like Education Dive isn’t the only one to write about how middle school math assignments lack “high cognitive demand”; Education Week is reporting on the “study” as well.

They give an example from the study of two assignments. Assignment A is considered purely procedural, while Assignment B is considered to require more cognitive demand:

A: Factor completely, and state for each stem what type of factoring you are using.   

x4 + x3 – 6x2

B: Create expressions that can be factored according to the following criteria. Explain the process you used to create your expression.

A quadratic trinomial with a leading coefficient of 1 that can first be factored using greatest common factoring. The greatest common factor should be 2x.

I find the wording of Assignment B a bit confusing but that’s besides the point. It appears that the people who did this “study” (and yes I will keep using quotes around that word, however offensive that may be to some) are not happy with a focus on procedural type problems. We are not given a complete view of the homework problem sets, so we don’t know if the problems scaffolded in difficulty. The “study” also did not examine the textbooks/assignments in K-6, which from what I have seen take an “understanding first, standard algorithmic procedures later, and only when understanding is attained” attitude. (See here for a clarification of what I mean).

Since I teach in a middle school, I see directly the casualty cases from what passes as mathematical education in K-6.  Middle school teachers in general realize they have to prepare students for high school math.  Given that burden, and having to teach students who are still counting on their fingers to add and subtract, don’t know their times tables and are flummoxed by the simplest of problems, it doesn’t take a brain surgeon to figure out what’s going on.  And what is going on is that middle school teachers are having to focus on the basics rather than the “critical thinking, depth of knowledge problems” held so dear by those who believe Common Core’s content standards are only there to support the platitudes known as the Standards of Mathematical Practice (SMPs).

I’ll also make a distinction here.  We are trying to teach students to solve problems, not “problem-solve”.  The latter is a term generally used to describe the process of solving one-off problems with little or no instruction in how to even approach them.

As for students taking algebra in 8th grade, I mentioned I use a 1962 textbook by Dolciani.  Here are two problems taken from the book. The first is about factoring:

“In the following problem, the given binomial is a factor of the trinomial over the set of polynomials with integral coefficients.  Determine “c” .    2𝑥−3; 10𝑥2−3𝑥+𝑐  “

And this is a word problem students are expected to solve:

“The plowed area of a field is a rectangle 80 feet by 120 feet. The owner plans to plow an extra strip of uniform width on each of the four sides of the field, in order to double the plowed area.  How many feet should he add to each dimension of the field?”

My students are not concerned with the “relevance” of the problem or whether it meets “real world” criteria. They want to solve such problems and draw on solid, explicit instruction and mastery of procedures in order to do so.

This just in, Dept.

From “Education Dive” (as in “deep dive”, “deep understanding” and other ridiculous jargon which unfortunately permeates the edu-world), a summary of a shocking new study:

Less than 10% of math assignments in the middle grades require “high levels of cognitive demand,” and only about a third of tasks expect students to show their thinking when providing their answers, according to a new analysis of more than 1,800 assignments, released today by The Education Trust.

Oh dear! Say it ain’t so.  What kind of high level cognitive demand do you want from a homework assignment?  What’s wrong with practicing procedures or solving word problems that escalate in difficulty–even if they aren’t the “open ended” variety? (Open ended, as in “The area of a rectangle is 24? What are the dimensions of the rectangle?”  Things like that, which supposedly get at “depth of knowledge” rather than the dreaded procedural “plug and chug”, which supposedly never scaffolds to higher difficulty problems.)

And what do they mean by showing their thinking?  A written paragraph? Showing work is not enough, I guess.  Can’t the teachers assess students’ reasoning by asking questions in class, like “Why did you subtract those two numbers? How did you come up with that approach?”  No, students now have “some ‘splainin’ to do!” Assignments that are “answer-focused” to use the jargon of the study, do not allow students to communicate their thinking.

And from the report itself:

Unfortunately, our analysis revealed that although roughly three-fourths of all assignments at least partially aligned to the grade- or course-appropriate math content, they also tended to:

  • Have low cognitive demand
  • Over-emphasize procedural skills and fluency
  • Provide little opportunity for students to communicate their mathematical thinking

And this tendency was often worse in higher poverty schools.

Which concludes with:

This analysis of middle-grades math assignments show that
schools and districts across the country are falling short when it
comes to providing their students with high-quality math tasks
that meet the demands of college- and career-ready standards.
The high percentage of aligned assignments demonstrates
that teachers are adjusting from the “mile-wide” philosophy
of previous standards movements and embracing the focused
prioritization of content that the math standards provide. These
high rates of alignment should be celebrated and strengthened.
However, alignment on its own is not enough to meet the high
bar set by rigorous college- and career-ready math standards.

And another “conclusion”:


Wow, it has all the right words doesn’t it?  And how does data show that we need to engage students with “opportunities for choice and relevance in their math content”? It might show that there is not much opportunity for such choice, but does it show that we need to provide such opportunities?  There are teachers (not just me) who will tell you that if students know enough to be able to tackle the problems given, they won’t care if it’s relevant or not.  OK, don’t believe me.

Look, I use a 1962 Dolciani algebra textbook to teach my algebra class. The word problems are plenty challenging for my students, though I’m fairly certain that the authors of said study would find such problems lacking in “real world relevancy” (as if my students care) and low cognitive demand.  Yes, I hear you saying “But they’re not from poverty and they would do well anywhere.”  Really? Got proof of that?

For my 7th grade class, I use JUMP Math, which uses micro-scaffolded approaches, but doesn’t skimp on the conceptual understanding behind the procedures either.  It has been given bad reviews by those who hole math reform ideologies in high regard as being “too procedural”.

Which brings me to one final question. Did the study in question look at how the students are doing on standardized tests?  And, oh yes, what types of approaches are used at Learning Centers, by tutors and by parents at home.  What is it that successful students are doing? Do they explain their work? Spend time on open-ended problems? Are do the stuff that’s held in disdain?  Any data on that anyone?



Mischaracterizations, Dept.

I, and others like me, talk about how traditional math teaching has been mischaracterized. I’ve written a few articles about this, so in case you haven’t seen them, you might want to take a peek:

Traditional Math: The Exception or the Rule

The Myth about Traditional Math Education and

Poster Children of Math Education

I also mention this in a talk I gave recently, which you can see here.

And if you are still alive at the end of all of that, and crave more, then buy my book: Math Education in the U.S.: Still Crazy After All These Years 


Principal Gladhand, Dept.

Another missive from the principal, this time about the dangers of “vaping”.
I am hearing more and more about students vaping. We’re hearing about students who are vaping and posting it on Instagram and Snapchat, we’re hearing about students being concerned about her friends, and had a few instances at school where we had students using vaping paraphernalia on our campus. 
I’ll spare the readers his explanation of what vaping is and cut to the dramatic dialogue he has with a student:
And here’s where parents need to be aware. Our children, who are LARGELY anti-cigarettes see vaping as being different. I pointed this out to a student just the other day. The student had admitted to “hitting the jewel” (a term for vaping). 
I told the student, “Wow, they’re really working you.” 
“Who?” the student asked.
 “The tobacco and cigarette industries. They’re working you.” 
The student looked confused, so I continued. “You don’t even see it, but you don’t talk about smoking, you don’t talk about vaping, you talk about ‘Hitting the Jewel’.” “So?” The student said. “So, you are so duped by an industry, you won’t even call it what it is. ‘hitting the jewel’…those three words don’t even make sense together. And that’s just what marketers want. They don’t want you to be thinking about what you’re doing, because you know putting things in to your lungs other than air is likely unhealthy. They’ve got you calling it something else. Something that doesn’t even make sense. They’re training you to smoke.”
This was the first time in the conversation where I think I had the student’s full attention. No middle school student EVER wants to feel they are being made to do something. They think they are smarter than that. And they can be. But they need our help. Now that you know, talk with your student about this. Ask what they know about vaping. Check their bags if you think they might be trying it. Check their social media if they have it and ask their opinions. Let them know they are being aggressively marketed to. Help them make good choices, keep those lines of communication open, and not be used by others.
Once again, I am inspired by Principal Gladhand’s intervention into the lives of students and parents.  He has uplifted me from the doldrums I was in because I feel my messages about education are not getting through. But following in Gladhand’s footsteps, I too will continue to warn parents against the marketing forces.  But not the marketing of the tobacco companies; rather the marketing undertaken by the educationists, driven by their progressivist agenda. I will continue to let parents know that in the age of the internet, students still need to acquire knowledge before they can learn how to think with that knowledge, let alone think critically.
I’ll continue my message that knowledge and content are not “low level” and Google is not a substitute for its acquisition.
I’ll continue letting people know how parents, schools and teachers have been “duped” by the educationist marketing arm that calls ineffective teaching approaches by different names like “inquiry-based” , “student-centered”.
I’ll tell them how the marketers disparage what has worked in the past, saying those traditional approaches to teaching have failed thousands of students (despite data to the contrary).
I’ll tell them the why’s of this marketing strategy: crafted to convince parents and teachers that such approaches result in “deeper learning” and “deeper understanding”, and all without even defining what those terms mean!
And if you need definitions of those terms, here they are: deeper learning and deeper understanding mean that the parents will be doing a lot of the research, report writing and teaching at home.
It also means you should buy stock in Kumon, Sylvan, Huntington, and other learning centers.

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.