Good grief! Dept.

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.


Having it Both Ways, Dept.

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.

New boss, old boss, Dept.

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.

Nothing really that new under the sun, Dept.

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.

Still relevant after all these years, Dept.

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.

The Vultures Descend, Dept.

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!

Out on Good Behavior, Dept.

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!

Happy talk, Dept.

Those who read Education Week are probably familiar with the breathless reporting that Catherine Gewertz did when Common Core was being adopted by state after state. Her latest breathless report about math education is about how “talking about math” helps students learn it. Or something to that effect.

“Research suggests that when students talk more about their math thinking, they are more motivated to learn and they learn more. Talking about math thinking can also serve as a stealth form of assessment, giving teachers insight into what students have mastered and where they still need help.”

First question: What research?

Second question: Have you ever worked with middle schoolers? Articulation of what they did is not their strong suit.

Oh, you have an answer for what I just brought up? OK, let’s hear it.

Learning to say things like, “When Robert uses this strategy, it makes me think of …” or, “This makes sense to me because … ” can help students learn how to “get mathematical ideas out into the classroom space” and build respectfully on one another’s thinking, Berry said.

I hate to burst anyone’s bubble, but I teach middle school math. My method is to “model” articulating the structure of the problem. “Are both cars traveling the same amount of time? What can we say about the distance each car travels?” In other words getting the mathematics that matters to solving the problem “into the classroom space”. Or whatever.

“The good news, according to experts, is that math discourse is a technique that works as well virtually as it does on paper or in face-to-face classrooms. And now, when students and teachers risk feeling disconnected and adrift, there’s even more reason to consider using “math talk” techniques to help students feel engaged and see themselves—and their classmates—as valued mathematical thinkers.”

Here’s some more news, though I doubt that believers in “math talk” will find the news good. Explicit instruction in procedures and problem solving techniques, with worked examples provide students with what they need to solve problems. Expanding from a worked example to solve similar problems demands much critical thought, and does exactly what these folks pretend that “math talks” accomplish.

The Solution to Inequity, Dept.

From Education Week, there’s this:

“A study released this week in the journal Educational Researcher found teachers cover significantly less algebra material in those classes at predominately black schools than their peers in schools that are mostly white or have no racial majority.”

The solution to this in some school districts, such as San Francisco, is to eliminate algebra in 8th grade entirely. That way no one benefits, and both black and white students are disadvantaged equally.

In other districts such as San Luis Coastal in California, students must score high enough on a rather poorly constructed test. The test is developed by Silicon Valley Math Initiative (SVMI) and the questions are typical of those thought to require “deep understanding”, but which are largely formative, one-off type problems which are treated as summative.

I wrote about how you no longer have to be economically disadvantaged or a minority to be given the short end of the stick. The comments on this story have long since disappeared but they included one from an African American teacher who claimed I was (paraphrasing here) a pandering white savior racist.

There’s no shortage of names to be called in this era of equity for all.