A recent article announced that the National Science Foundation (NSF) funded a grant for West Virginia University College of Education and Human Services The grant is to help educate math teachers on a new way of teaching math to teachers. For those of you new to all this, NSF spent billions of dollars in grant money in the early 90’s to fund (in my opinion and the opinion of many others) ineffective and damaging math programs including Investigations in Number, Data and Space; Everyday Math; Connected Math Program; Core Plus; and Interactive Math Program.

Of particular interest to me was this sentence: “The hope was for math teachers to find ways to teach students how to problem-solve.”

It used to be that students solved problems.  But now in today’s era of math reform, they “problem-solve”. Popular use of this rather irritating verb form harkens back to NCTM’s 1989 standards which downplayed the importance of procedural skills, and replaced those with students achieving “deeper understanding” and being able to problem-solve.

The core belief behind the current math-reformers’ use of the term “problem-solving” is that it is a core competency that can be taught independent of the domain in which a problem appears. Little to no importance is given to mastery of procedural skills, instruction on how to solve particular types of problems, nor sufficient practice solving such problems.

The typical problems of the past (distance/rate, mixture, number, coin) are being replaced with what reformers believe are problems that students are interested in wanting to solve. These are typically one-off problems that don’t generalize and for which little to no prior problem solving procedure has been taught. 

One math reform approach has been to present students with a steady diet of “challenging problems” that neither connect with the students’ lessons and instruction nor develop any identifiable or transferable skills. The following problem from Hjalmarson and Diefes-Dux (2008) is one example: How many boxes would be needed to pack and ship one million books collected in a school-based book drive? In this problem the size of the books is unknown and varied, and the size of the boxes is not stated.

While some teachers consider the open-ended nature of the problem to be deep, rich, and unique, students will generally lack the skills required to
solve such a problem, skills such as knowledge of proper experimental approaches, systematic and random errors, organizational skills, and validation and verification.  The belief is that just as students develop problem solving habits for routine problems, a similar “habit of mind” or problem-solving schema occurs for solving non-routine problems.

Based on my experiences as both student and teacher, as well as the experiences of veteran math teachers, I submit that a substantial education in mathematics should steer a middle course between the proliferation of routine problems and reliance upon unique, complex projects. Students should learn to apply basic principles in a much wider variety of situations than typically presented in texts. Such problems, however, should not be as
complex or as time consuming as the example above. A math problem is not necessarily useful just because it requires outside-of-the-box insight and/or inspiration and will generally not result in a problem-solving “habit of mind” or schema.

Problem solving techniques taught independent of the domain in which they occur include such things as “work backwards”, and “find a simpler but similar problem”. But without experience, practice and mastery of domain-specific problems, asking a student to find a simpler but similar problem is as useful as telling a novice bike rider to “be careful” when taking a ride on their own.

Sweller et al. (2010) state that problem solving cannot be taught independently of basic tools and basic thinking. Over time, students build up a repertoire of problem-solving techniques. Ultimately, the difference between someone who is good and someone who is bad at solving nonroutine problems is not that the good problem solver has
learned to solve novel, previously unseen problems. It is more the case that, as students increase their expertise, more nonroutine problems appear
to them as routine.

Looks like the idea of problem solving as a core competency will be taught to a bunch of lucky teachers in West Virginia thanks again to the misguided largess of the National Science Foundation.

References

Margret A. Hjalmarson and Heidi Diefes-Dux (2008), Teacher as designer: A framework for teacher analysis of mathematical model-eliciting
activities, Interdisciplinary Journal of Problem based Learning, Vol. 2, Iss. 1, Article 5. Available at http://dx.doi.org/10.7771/1541–
5015.1051.

John Sweller, R. Clark, and P. Kirschner (2010), Teaching general problem-solving skills is not a substitute for, or a viable addition to, teaching mathematics, Notices of the American Mathematical Society, Vol. 57, No. 10, November.

Because of school closings due to Covid 19, there has been a flurry of articles about distance learning, and the difficulties that parents face when having to explain “Common Core” math. The articles take the opportunity to show that parents are just not “with it” and that the new way is actually better because it confers “deeper understanding” rather than rote memorization.

This article is typical as is the following quote from it:

“Amberlee Honsaker remembers learning only one way to add or subtract in elementary school. It was the standard algorithm: stack numbers vertically, add the digits in columns, and carry the ones where necessary. For her daughter, Raegan, math instruction extends far beyond that. In first grade, Raegan is using number bonds, making place-value charts, drawing out 10s and ones  — illustrating multiple methods for solving simple addition problems.”

Actually, in my elementary school as well as for many others, there were alternate methods taught. But they were taught after mastery of the standard algorithm. The alternate methods in addition to being taught were also often discovered by students themselves as an outgrowth of the mastery of the standard algorithm.  A problem like 76 + 85 could be solved by adding 70+80 to get 150, and then 6 +5 to get 11. Adding 150 and 11, the final sum of 161 is obtained. 

Number bonds were called “fact families” and place value charts were abundant as a glance through textbooks from the 60’s, 50’s, 40’s, and further back easily show. (See this article for examples)

But now, alternate methods are taught first in the belief that it imparts a “deeper understanding” of what is going on with standard algorithms and procedures which are taught later. Teaching the standard algorithm first is thought to obscure the understanding and is viewed as a “rote” procedure. As a result, what is mischaracterized as “rote memorization” has been replaced with “deeper understanding” as math reformers term it. I think a more accurate term is “rote understanding”.

 The so-called “deeper understanding” is measured by having students show more than one way to add or multiply numbers, and to explain in writing why it works.

From the article:

“Over the past 40 years, education research has emphasized that teaching math should start with building students’ understanding of math concepts, instead of starting with formal algorithms, according to Michele Carney, an associate professor of mathematics education at Boise State University.”

The article does not do us the favor of providing us references to the research but I’ve seen some of it. Most of it is based on “action” research done in classrooms with questionable controls, and authored by the same people who have been taking in each others’ laundry for years. (e.g. Fenema, Carpenter, Hiebert, etc)

Common Core codified much if not most of the reform math ideology that has been at work for more than three decades.  Reform ideology got its first big boost with NCTM’s math standards in 1989 which was predicated on the notion that traditional math teaching sacrificed conceptual understanding on the altar of procedural fluency. It put an emphasis on “understanding” and viewed procedures as nothing more than “rote memorization”.

The other catch-phrase of the math reformers is “problem-solving”; so much so, that it has become a verb. It used to be that students solved problems.  Now they “problem-solve”. Again, this harkens back to NCTM’s 1989 standards which downplayed the importance of procedural skills, and replaced those with students being able to “explain” their answers. “Math talk” has emerged as an indicator for whether students “understand”.  If a student cannot explain how they solved a problem, they are held to lack understanding. Also, if a student cannot solve a problem in more than one way, that too is held to show a lack of understanding. 

The typical problems of the past (distance/rate, mixture, number, coin) are being replaced with what reformers believe are problems that students are interested in wanting to solve. These are typically one-off problems that don’t generalize and for which little to no prior problem solving procedure has been taught. 

The “problem solve” mentality has made its way into ed schools where I heard the philosophy espoused. That is, there is a difference between problem solving and exercises. “Exercises” are what students do when applying algorithms or problem solving procedures they know. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation. Moreover, ed school catechism states that students’ difficulty in solving problems in new contexts is evidence that the use of “mere exercises” or “procedures” is ineffective and they are overused in classrooms.

It is more likely that students’ difficulty in solving new problems is because they do not have the requisite knowledge and/or mastery of skills—not because they were given explicit instruction and homework exercises.

Those who make such a differentiation and champion “true” problem solving espouse a belief in having students construct their own knowledge by forcing them to make connections with skills and concepts that they may not have mastered. But, with skills and concepts still at a novice level, students are not likely to be able to apply them to new and unknown situations. Nevertheless, the belief prevails that having students work on such problems fosters a discovery process which the purveyors of this theory view as “authentic work” and the key to “real learning.”  One ed school professor I knew summed up this philosophy with the following questions: “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure? How do we teach students to apply mathematical thinking in creative ways to solve complex, novel problems? What happens when we get off the ‘script’?” 

In fact, as Rittle-Johnson, et al. (2015) have shown, procedural fluency does not exclude conceptual knowledge—it can ultimately lead to conceptual understanding. Also, “Aha” experiences and discoveries can and do occur when students are given explicit instructions, worked examples, and scaffolded problems.

While some educators argue that procedures and standard algorithms are “rote”, they fail to see that exercising procedures to solve problems requires reasoning with such procedures — which in itself is a form of understanding. This form of understanding is particularly significant for students with LD, and definitely more useful than requiring explanations that students do not understand for procedures they cannot perform.

Reference:

Rittle-Johnson, Bethany; Michael Schneider, Jon Star “Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics.” Educ. Psychol Review; DOI 10.1007/s10648-015-9302-x

Conrad Wolfram is a brilliant mathematician. He has written a book which argues that math education should not focus on how to compute various things, but on the thinking behind the computation. This article describes in breathless wonder Wolfram’s equally breathless idea to change how math is taught in order to keep up with the real world.

Wolfram makes the case that computation thinking is required in all fields and in everyday living—and that no one does calculations by hand.  We’re living in what Wolfram calls a “computational knowledge economy” where the education question is, “How to prepare young people for a hybrid human-machine world?”  In this new age, it’s not what you know, “it’s what you can compute from knowledge,” argues Wolfram. 

It is a brave new world that Wolfram envisions, getting away from what he views as rote memorization and to the actual solving of real-world problems.

And perhaps for Wolfram, he had a “deep understanding” of mathematical processes at an early age, though I find it hard to believe that he never had to learn the basics somewhere along the line to get to his present state of development.

A key red-flag in this article is this:

Wolfram joins leading math educator Jo Boaler and economist Steven Levitt as leading voices advocating for change.  “Put data and its analysis at the center of high school mathematics.” That’s the conclusion of a paper by Boaler and Levitt. They recommend that “every high school student should graduate with an understanding of data, spreadsheets, and the difference between correlation and causality.

Boaler and Levitt argue that we need to get away from the traditional sequence of algebra-geometry-precalc-calculus, and focus more on data and statistics.

The problem with brilliant people like Wolfram is that they often fool themselves with their own brilliance and convince themselves that they know more than they do about subjects in which they have no expertise. Such a person is called ultracrepidarian which is defined as “noting or pertaining to a person who criticizes, judges, or gives advice outside the area of his or her expertise”.

Like many math geniuses, Wolfram appears to have forgotten his own consolidation phase. He makes it sound as if mastery of mathematical concepts is a lot simpler if we strip out the computation aspect of it.  But a person who may be extremely talented at doing computations, may not move through unfamiliar material with the same ease.

For the multitude of people who lament that they were never good at math, the pie-in-the-sky revelations of people like Wolfram, Boaler and Levitt have appeal. Their arguments are seductive and draw people in to an “if only I had been taught math this way” narrative. The Wolframs, Boalers and Levitts are welcomed to an edu-establishment that continues to extol ineffective practices to an ever-growing audience that unquestionably embraces them.

From the Algebra I section of the Mathematics Framework for California Public Schools: Kindergarten Through Grade Twelve:

“Modeling problems have an element of being genuine problems, in the sense that students care about answering the question under consideration. In modeling, mathematics is used as a tool to answer questions that students really want answered. Students examine a problem and formulate a mathematical model (an equation, table, graph, etc.), compute an answer or rewrite their expression to reveal new information, interpret and validate the results, and report out. This is a new approach for many teachers and may be challenging to implement, but the effort should show students that mathematics is relevant to their lives. From a pedagogical perspective, modeling gives a concrete basis from which to abstract the mathematics and often serves to motivate students to become independent learners.”

(I can’t be sure, but the above passage sounds as if it were written by Phil Daro.)

I’ve seen this “make math relevant” and “problems that students really want answered” line of reasoning before from those who supposedly know what’s best for students. Out of the other sides of their mouths, they lament that math is not just about computation and push for problems that explore the relationship between perimeter and area of polygons and other concepts. Using the same logic about making math relevant one could then argue that students may not find such topics relevant to their lives. But people in the edu-establishment often have things both ways.

Extending this Phil Daro-ish logic that students only like to solve problems they really want answered, one would conclude that students do crossword puzzles and sudokus, because they really care about having them answered.  Also breakout video games, Tetris and D&D.

In my experience and the experience of teachers who actually know what math is about and how to teach it, students care about problems if they’re able to solve them. Otherwise they write them off as irrelevant–sour grapes.

The problems that so-called math ed experts believe are so fascinating to students are generally one-off open-ended type problems which often involve gadgetry and ultimately number crunching. The fact that they don’t generalize to anything useful mathematically matters little to the people who write these frameworks.

Ontario’s math program for K-12 has come under fire the past few years. So much so that the current Premier of the province (Doug Ford) ran on a platform that included a “back to basics” math program.

The new math program was unveiled last week. A glance at its features showed that aside from the requirement that students know their multiplication facts, it appears to be the same mix of rhetoric for achieving “deeper understanding” of math.

A recent article talks about how a key aspect of the new standards is the Social and Emotional Learning (SEL) component.

Educators say the key innovation in the new curriculum involves teaching “social-emotional learning skills” throughout math. According to Ministry of Education documents, this means helping students to “develop confidence, cope with challenges and think critically.” For example, students will learn how to “use strategies to be resourceful in working through challenging problems,” says the parents’ guide to the curriculum.  … Teaching those skills is a far cry from drilling times tables into students’ heads. 

Interesting that the parent’s guide to the curriculum downplays the memorization of times tables, which was probably the biggest change in the new math curriculum from the older one. Actually, providing students with the necessary instruction to achieve success is what ultimately leads to confidence, motivation, engagement and–yes–critical thinking. Much of the thinking behind SEL, however, places the cart before the horse. The strategies talked about in SEL frequently include such things as telling students to say “I can’t do this…yet” and other motivational cliches. These so-called strategies are thought to give students a “growth mindset”.

The components of SEL are spelled out in the new standards. Specficially, they are:

1. identify and manage emotions
2. recognize sources of stress and cope with challenges
3. maintain positive motivation and perseverance
4. build relationships and communicate effectively
5.  develop self-awareness and sense of identity
6. think critically and creatively

The standards state that these components will come about through implementation of the standards as they apply “mathematical processes”. What does that mean? Well, here are the mathematical processes the standards cover:

• problem solving
• reasoning and proving
• reflecting
• connecting
• communicating
• representing
• selecting tools and strategies

Taking just the first item in the bulleted list: “problem solving”. The reform-minded thinking is that if a student learns how to “problem solve” (the current lingo for what used to be called “solving problems; apparently the term “problem solve” confers more meaning and implies that there is “deeper understanding” rather than just “finding an answer” ) they will automatically be attending to the six components of SEL

Nice and neat, tied in a bow, and ready to use. The only thing missing, it seems, is the instruction for how to solve problems. For that matter the tools that allow one to reason and prove, or even to reflect also seem to be missing from the standards. The new standards leave out learning things like, say, the standard algorithms for adding/subtracting multidigit numbers, or multiplying and dividing. Instead, it talks about students learning “algorithms” for same–not the “standard algorithms”. This may seem like a nit-pick but it is not. “Algorithms” in the lexicon of the math reformer can be any particular procedure that produces an answer. This usually includes methods that are typically taught after mastery of the standard algorithms.

For example, adding 75 + 56. Rather than teach students to stack the numbers and to carry the excess to the tens place (or regroup, using a more reform-minded term) they teach students to first add 70 + 50 and then 5 + 6. Then add the two sub-totals of 120 and 11 to get 131. This is nothing new, and I’ve seen it taught in a 5th grade arithmetic book from the 1930’s (an era said to be when math was taught by “rote memorization” with no understanding). The method makes sense once mastery of the standard algorithm is accomplished. But teaching the strategy first rather than the standard algorithm is thought to provide the “deeper understanding” that the standard algorithm is believed to obscure.

The new standards supposedly provide students with the skill of making “connections among mathematical concepts, procedures, and representations, and relate mathematical ideas to other contexts (e.g., other curriculum areas, daily life, sports)”. Traditional or “back to basics” approaches are, according to Mary Reid, (assistant professor of math education at the Ontario Institute of Studies in Education). “just following procedure without really understanding why you’re doing it.” This “understanding uber alles” approach prevails in the math reformers’ view of how mathematics should be taught. It fails to recognize that procedures and understanding work in tandem, and also confers the mistaken belief that understanding must always come before allowing students to use more efficient procedures. In the case of the new standards, it looks doubtful that efficient procedures (i.e, standard algorithms) will be taught at all.

As far as the holy grail of “connections” is concerned, Robert Craigen, a math professor at University of Manitoba who has been involved in improving K-12 math education says this: “It’s amusing when they speak about “connections” as if this were something different from “isolated facts”.  Actually it is the facts that provide connections.  Everything else is only the educational analog of a conspiracy theory.”

We’ll see how this latest conspiracy theory plays out in Ontario.

There is a continuing chorus of complaints about how math is taught from those who seek to reform math education. The chief complaint is the lack of transfer of knowledge. That is, students cannot seem to take their prior knowledge and apply it to problems that rely on the same knowledge but are in new or novel settings.

The reformers then talk about how we need to build students’ “depth of knowledge” to get to the holy grail of “deeper understanding”.

I’ve taken a sample of a PowerPoint which is similar to many others that have been making the rounds over the years. In it, the problem is presented as follows:

Students appear to demonstrate “deep, authentic command of mathematical concepts” when given commonly used problems.
However with more challenging problems, the same students seem
to no longer demonstrate that command.

First, we must have a clear understanding about why these problems are different from one another.
 Next, we need to practice using these problems so that we understand how students may react to them.
 Last, we need a source that can provide us with a variety of free problems.

The underlying message is that we haven’t been doing these things and students are getting “superficial knowledge”. The supposed proof are charts of problems that correspond to varying depths of knowledge. What’s misleading about these charts is that we look at them with many years of experience under our belts thinking “Yeah, kids should be able to do these.”

What seems to be neglected in all this, is the distinction between novice and expert and that problems seen for the first time (i.e., “new” problems) are naturally going to be harder to solve. What is needed to get students over the hump are 1) worked examples, and 2) scaffolded problems that increase in variety and difficulty.

A look back at math books from previous eras shows that in fact, we have been doing these things. Below is an excerpt from a fifth grade arithmetic textbook from 1937 called “Modern School Arithmetic” by the reformers of that era: John Clark, Arthur Otis and Caroline Hatton.

Interesting to note that the traditional modes of education seem to address the very concerns that the current slew of reformers claim has been missing. Something to keep in mind the next time you attend an NCTM conference or their equivalents.

The latest chapter entitled “Instructional shifts, Formative Assessments, and Taking Matters into My Own Hands” is now up at Truth in American Education.

The “Still relevant” part of the title refers to a book I wrote called “Letters from John Dewey/Letters from Huck Finn”. The first part of the book is a collection of columns I wrote for a blog called Edspresso that described my experience in a math methods course I was taking in ed school at night, when I was on my way to becoming credentialed.

I was looking through one of the old posts and found this one particularly beguiling:

In the afterglow of celebration and in between semesters I am getting ready for my next class: Human Development and Learning.  I am a bit concerned about one aspect of the course as described in the syllabus:

“The course examines the processes and theories that provide a basis for understanding the learning process.  Particular attention is given to constructivist theories and practices of learning, the role of symbolic competence as a mediator of learning, understanding, and knowing, and the facilitation of critical thinking and problem solving.”

OK, it may be another long haul, but I am happy to say that my stint in ed school so far has taught me superior vomiting suppression skills. The issue of constructivism is a perplexing one.  For example, Jay Mathews, the Washington Post reporter who writes the “Class Struggle” column, addressed this in his book of the same name.  Calling John Dewey a “squishy brained dreamer,” he states, “I have yet to observe a teacher who is not putting considerable emphasis on specific information and skills…If you know of a study that shows that Dewey’s principles are actually practiced in any serious way in many American classrooms, I would like to see it, because it conflicts with what I have found.”

I find this post of interest because nothing much has changed. Ed schools still teach that constructivism is still “the way” to go in classrooms. And Jay Mathews continues to believe that such practices don’t exist anywhere and that the “math wars” are just two groups of “smart people” calling each other names. (As he once told me in an email).

So I am taking this opportunity to shamelessly promote this book because it is as timely and relevant as ever. The second part of the book is a collection of letters written under the name “Huck Finn” which were serialized on the “Out in Left Field” blog. They chronicle my experiences as a student teacher, and then as a sub, when I went out into the real world of teaching.

I am hoping that this book will become required reading in ed schools, but it hasn’t happened yet. So until that occurs, please order your copy today.

Just received a message at my school email address from someone claiming to be the Head of Deeper Learning at some outfit called Thrively.

She stated: “Over the past few months, the ability of independent schools to deliver a personalized, strengths-based, academically rigorous and deeply engaging remote education has become critical to their mission.”

Uh, “strengths-based”? That’s one I’ll have to add to the ever-growing list of things that are “based”.

She went on:

“In preparation for the unknown parameters of Fall, many schools are considering 3-tiered plans: remote learning, traditional school, and a hybrid of remote/physical models while addressing:

• Social-emotional health of our students and families
• Student engagement and personalized, differentiated instruction
• Making learning deeper and more meaningful for our students”

Then she gets to the point:

“Please connect me with your principal so that we can explore how Thrively can support your work. Here is my calendar (link was provided), if you want to invite your principal and schedule some time with me.”

It gives me great faith to know that a deadly virus is no match for our market-driven economy!

This is Chapter 15 in a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder” by Barry Garelick, a second-career math teacher in California. He has written articles on math education that have appeared in The Atlantic, Education Next, Education News and AMS Notices. He is also the author of three books on math education. Says Mr. Garelick: “At its completion, this series will be published in book form by John Catt Educational, Ltd. If it is made into a movie I will be played by either Jeff Bridges or Harrison Ford. The part of Ellen will be played by Jamie Lee Curtis; Diane will be played by Helen Mirren.”

As usual, your efforts at disseminating information about this series will be greatly appreciated!