Help Wanted, Dept.

NOTE: The article referred to in this entry was updated.  In keeping with this “moving the goal-post” type of journalism, I too have updated this post.

This article in USA Today purports to explain why the US lags other countries in math. Here’s one of the reasons their supposedly-researched article provides:

“One likely reason: U.S. high schools teach math completely differently than other countries. Classes here often focus on formulas and procedures rather than teaching students to think creatively about solving complex problems involving all sorts of mathematics, experts say. That makes it harder for students to compete globally, be it on an international exam or in colleges and careers that value sophisticated thinking and data science.”

Like most articles one sees in newspapers about math education, this one assumes that students are taught via rote memorization without any context of the conceptual underpinning.  The article argues that the problem with math education is that it fails to teach students “complex problem solving” skills in high school.  Actually the article fails to look at how students are being taught in the elementary grades (K-6).  As a reaction against traditional math–mischaracterized as rote memorization without understanding–students are subjected to a push of conceptual understanding. Students are often not allowed to learn and/or use standard methods or algorithms until they have learned the conceptual understanding in the mistaken but widespread belief that standard algorithms eclipse the holy grail of “understanding”.  Students who are ready to move on are held back until they can show what amounts to the “rote understanding” that teachers want to hear.  Memorization, in fact, would actually be far more easy and benefit many more students.

But the reporter of this article like most journos who don’t know what they don’t know about math education, offers the standard narrative of how to fix the problems that traditional math teaching has supposedly caused:

There is a growing chorus of math experts who recommend ways to bring America’s math curriculum into the 21st century to make it more reflective of what children in higher-performing countries learn. Some schools experiment with ways to make math more exciting, practical and inclusive. 

And there you have it: the standard troika of how to fix math education: 1) The magic bullet of how higher-performing countries do it: conceptual understanding. No mention of their reliance on the traditional techniques that are held to cause the low scores in math in the US, and that students there actually do rely on memorization.

2) Make math more exciting:  The article fails to mention how, but the usual reasons are to give problems that are more than just computation (and therefore useful not only in daily life but as a means to move on to solving more complex problems). Give them more mathematically oriented problems, as well as open-ended, multiple answer problems.  The belief is that there are core competencies (like problem solving) that can be learned independently of the type of problems one has to solve.  This is referred to as “habits of mind”. The belief is that a steady diet of challenging one-off type problems will develop a problem-solving “schema” that will allow students to transfer these skills to any type of problem they come across. This belief is carried out with the help of making no distinction between how novices learn differently than experts.

3) Make math more practical and inclusive: This is the other side of their mouth speaking. Students should be given more of the everyday problems held to be un-exciting and which turn children off of math. As far as inclusivity, problems should embody aspects of different cultures rather than the white western culture which has prevailed and oppressed free thought for centuries.

So to get more perspective on the problem and how to fix it, the reporter turns to Jo Boaler, who is regarded by many journos as the be-all end-all of math education.

Most American high schools teach algebra I in ninth grade, geometry in 10th grade and algebra II in 11th grade – something Boaler calls “the geometry sandwich.” Other countries teach three straight years of integrated math – I, II and III — in which concepts of algebra, geometry, probability, statistics and data science are taught together, allowing students to take deep dives into complex problems.

Let me talk about that for just a bit. You don’t have to have the geometry sandwich as she calls it. You could have students take Algebra 1 and 2 in sequence. The reason for not doing this is supposedly that geometry prepares them for the trig aspect of Algebra 2. This is nonsense. I had some time at the end of an Algebra 1 course so I taught the basics of trig–which by the way was at the end of the book, and it wasn’t uncommon for algebra 1 textbooks to do so. The prior knowledge the students needed to grasp it was already there via their understanding of similar triangles and right triangles.

As far as math being taught differently in high schools overseas, it is true that they use an integrated approach: i.e., a mixture of algebra, geometry and trigonometry throughout the year, with each year getting more advanced. Greg Ashman, who teaches in Australia and who writes a blog about education is embarking upon a PhD in education from John Sweller. Greg makes this comment about integrated math:
“In Australia, I teach integrated maths and I always have. However, I teach it directly and explicitly. Other schools may choose less effective approaches. It seems to me that the math reformers are trying to use integrated maths as a Trojan horse for discovery maths. That’s a hard tactic to counter because there is some logic to integrating maths – it should lead to more contact with the various concepts over time, more retrieval practice and better formed schemas.”
In the US, the integrated math programs are by and large discovery math programs as Greg alludes to in his note.  IMP is one of them as is Core Plus. The newest is MVP Math, an integrated math program heavy on inquiry-based, student-centered learning, with teachers serving as handmaidens/facilitators of an ineffective program. Blain Dillard, a parent who lives in North Carolina where MVP is being foisted upon an unsuspecting population was sued by MVP for speaking out against its ineffectiveness. (The lawsuit was dropped when Blain countersued).
Boaler is quoted throughout the article. At one point, she says:
“There’s a lot of research that shows when you teach math in a different way, kids do better, including on test scores,” said Jo Boaler, a mathematics professor at Stanford University who is behind a major push to remake America’s math curriculum.”
What research? She doesn’t say, nor did the reporter ask, apparently.  What different way of teaching math? She doesn’t say, and again it appears the reporter did not ask.  Boaler provides a hint in this indirect quote in the article:
“In higher-performing countries, statistics or data science – the computer-based analysis of data, often coupled with coding – is a larger part of the math curriculum, Boaler said. Most American classes focus on teaching rote procedures, she said.”
Which then leads the reporter to discuss a podcast of Freakonomics author Steve Levitt on math eduation:
“Levitt is engaged in the movement to upend traditional math instruction. He said high schools could consider whittling down the most useful elements of geometry and the second year of algebra into a one-year course. Then students would have more room in their schedules for more applicable math classes.”
A friend of mine who teaches high school algebra and majored in physics at Harvard has this to say about Levitt:
“Someone needs to engage Steven Levitt in active debate on that topic. Learning data science will do nobody any good if they lack the basic skills to apply and comprehend the underlying math, and conceptual overviews just don’t cut it. I’ve noticed a trend in software that has been dumbing down the features available to power users, with the idea that few people know how to use them in the first place, and if that widespread adoption of brain-dead approaches starts happening more in the realm of big data, then we are in for all kinds of complexity-related problems down the road.”
This article, like most that appear in the press, takes the word of math reformers as gospel and rarely questions their assumptions.  My friend characterizes the reform math stance as follows:
“They define traditional education as “that which does not work” and then co-opt any demonstrated success that was brought about from a traditional program as being tied to the “conceptual understanding” that said program fostered, as if nothing else mattered. They declare, by definition, that traditional doesn’t work, and then take credit for anything that has worked in the past, even as their present programs fail miserably.”
Well said.  And in fact if you have anything to say about the tropes in the USA Today article, or about math reformers in general, I need your help:  Please add your comments here.

Another Math Miracle, Dept.

This promo piece is about Nashua NH public schools adopting Eureka Math. In keeping with the tradition and style of such articles that pass as objective reporting, it contains the usual disparagement of algorithms, memorization, and of course tests that are not “formative”.

To wit and for example:

“It builds student confidence, year by year, by helping students achieve true understanding of math, not just algorithms,” said Fitzpatrick, adding students are focusing on applying math as opposed to memorizing math formulas. By implementing Eureka Math for kindergarten up to eighth grade, she said it will provide a continuous standard progression and help build conceptual understanding and abstract skills. It also encourages consistent math terminology and common assessments that are formative and summative, explained Fitzpatrick.

It is more than a little discouraging to see the premises of Kamii and Dominick (famous for their seminal piece that supposedly provides evidence that teaching standard algorithms is harmful) being taken so seriously.  The obsession with conceptual understanding continues, with little regard that the end product is usually students parroting back what the teachers want to hear — what I refer to as “rote understanding”.

Programs such as Eureka math consist of a steady diet of drilling confusing and ineffective strategies that supposedly ingrain the conceptual underpinning of the standard algorithms.  (For more on this, see this article.) In fact, the strategies being used to teach the conceptual understanding are not new at all, and were used in previous eras after mastery of the standard algorithms.  The standard algorithms were the main course, and the strategies, presented later were the side dishes, meant to then provide more perspective of the conceptual underpinning.  

Missing in all these discussions about the holy grail of “understanding” is that there are various levels of understanding, which are built upon in subsequent math courses over the years.  In freshman calculus classes, for example, the concepts of limits and continuity are presented in an intuitive approach, allowing students to progress to the powerful applications of derivatives and integrals. Later in upper level math courses, more formal and rigorous definitions of limits and continuity are provided–which would result in a lot of confusion for many first-time calculus students.

But the myth persists that the reason American students do poorly in math is because they lack “understanding”. A glance at how Singapore and other Asian countries provide math instruction would show that the approach used in teaching the standard algorithms is very similar to how math used to be taught in the US many years ago.

With the myth comes the programs, and with the programs come more help at home, tutors, and learning centers for those who can afford it.  And also with the myth comes a willing ignorance as indicated in this telling paragraph about a teacher’s comment on Eureka Math:

She acknowledged that with Eureka, fluency with math facts is not a daily practice, however teachers are finding other ways to introduce math facts into the middle school curriculum. There is also a 45-minute time crunch for Eureka Math, meaning there may not be as much time for remediation if concepts are not fully understood, said Porpiglia. There is currently a list of highly rated support resources that are being vetted to encourage visual models and a greater algebraic understanding for middle schoolers, she added.

So what the reformers mean by “understanding” are visual models–i.e., drawing pictures and going back to first principles each and every time you solve a problem. This supposedly provides the “greater algebraic understanding”– even if it takes remediation. And math facts?  Well, relax. Never mind that no one is bothering to express surprise that math facts need to be introduced as late as middle school.  It’s all for the greater good until the next bright shiny new thing is introduced.


How’s that been working out for you? Dept.

A recent article on what and how wonderful Common Core is supplies the ongoing narrative that just won’t quit no matter how many parents call bullshit.

A slump in math education through the early 21st century within the United States triggered the desire to improve how the country educated students in both math and language. That became known as the common core and was adopted by New York State in 2011. 

That slump has been going on for some time as has the desire to improve math education. Every generation disparages the previous as having taught math wrong, with the principle reason being that it is being taught as “rote memorization” without understanding.

While there may be certain aspects of the common core followed in private schools, there is more freedom to use different methods, like at Harley School in Rochester. “They might all be 8 or 9 years old,” said Margaret Tolhurst, third grade teacher at Harley, “but every single one of them needs something different. So, we try to provide activities that can provide to all those different needs.”  Tolhurst has been teaching at Harley for 28 years. 

They all need something different?  In fact they do given that the students are likely not getting a decent math education. What they need different is for math to be taught explicitly with worked examples without this obsession that students must “understand”.  To wit:

An understanding of math translates to liking math, according to Mathnasium co-owner Asiya Ali.

Well, what is meant by “understanding” math first of all? Actually, kids tend to like things that they can do successfully.

The math center helps all grade levels to become proficient in math by working closely with the student, their teacher, and family. “We’re in there because our main purpose is to help kids,” said Ali. She says a lot of her work does follow common core standards. “It’s more of a focus on conceptual understanding, so whereas traditional math is more like procedural.” 

And wait! How did Mathnasium suddenly step into this article? Aren’t they a learning center?  And don’t they teach procedures?

Wait, there’s more.  There’s also the standard refrain of “instructional shifts”.  Let’s listen.

 Some of the biggest changes may be a shift away from teacher centered at the front of the classroom to a more student-centered classroom.  According to students, this has been a different strategy than what parents have seen before.  

Well, no, the student-centered, inquiry-based fad pre-dates Common Core.  And for the record, Common Core doesn’t call for moving away from a teacher-centered classroom. In fact, the Common Core website’s FAQ’s explicitly state that the standards do not dictate how math is to be taught: “Teachers know best about what works in the classroom. That is why these standards establish what students need to learn, but do not dictate how teachers should teach. Instead, schools and teachers decide how best to help students reach the standards.”  

And no article about Common Core, or math education in general would be complete without a reference to real-life word problems:

Students see more word problems that connect the numbers to real life, something that is different thanks to common core standards. That puts some frustration into parents’ minds.  

And that frustration is also in students’ minds, considering that such problems tend to be dull, tedious, wordy, and frequently one-off types that do not generalize to anything useful. Based on my experience (yes, one person’s so those of you sold on reform type philosophies can ignore this) the simple straightforward type of problems that are held in disdain, students may find them difficult but ultimately enjoy being able to do them.

And to be able to do them requires some explicit instruction and worked examples. And practice. And since the article doesn’t  get into the “productive struggle” gambit, neither will I.  You’re welcome!

Mischaracterizations, Dept.

“Traditionalists” (as they like to style themselves) are incapable of grasping the fact that high school math exists, and that most high school math teachers aren’t constructivists.

The above quote was from a blog written by a math teacher, and was a post about an article that Katharine Beals and I wrote which was published in online Atlantic in 2015. It caused a stir among those who don’t like what “traditionalists”  have to say about teaching math.

In fact, we do know that most high school math teachers do not teach in the inquiry-based manner. What we also know is that in K-6, much of math has been dominated by the math reform ideology as embodied in many textbooks. Constance Kamii’s belief that teaching the standard algorithms to young children does them harm by eclipsing understanding has set the stage for how math has been and is being taught in the lower grades. What has happened in K-6 math over the past 30 years or so, is in part an increase in inquiry-based, student-centered learning, but in larger part an obsession with understanding.

I have written about this in several places, but one place to start is here (as well as my book!)

What we “traditionalists” do notice about high school math is that many entering freshman do not know basic computation rules, and are dependent on calculators. In an eighth grade algebra class which I teach (and which is equivalent to 9th grade high school algebra), I had a student who had great difficulty multiplying two-digit numbers. He used a convoluted method that took up much time.  Thirty years ago, most entering high school freshmen had the mastery of such elementary procedures.

The blogger whose quote I posted also states that Beals and I do not believe that “math zombies” exist. By “math zombies” the blogger is referring to students who can reproduce procedures but lack the “understanding” to apply the concepts underlying the procedures to new or novel problems. Yes, such students exist. They are on the novice scale of learning; there are levels of understanding that accumulate with experience. Judging novices in terms of the expectation of expert performance seems to be the goal of those who are in the “understanding uber alles” mode.

Also,–based on my observations and those of people I know who teach mathematics in college–it is interesting that the so-called “math zombies” are generally the ones who don’t need remedial math in college.

New boss = old boss, Dept.

According to the promo, Florida has jettisoned the Common Core standards and replaced them with a new set of standards called BEST. I haven’t done a thorough review, but I quickly looked through the math standards for mention of “the standard algorithm for multidigit addition and subtraction”.
The CC standards have that standard in Grade 4. Mention of multdigit addition and subtraction in the BEST standards appears in Grade 3 (not much better), but includes this rather slippery wording:
“Add and subtract multi-digit whole numbers including using a standard algorithm with procedural fluency” (Emphasis added)
It doesn’t refer to “the” standard algorithm but “a” standard algorithm. This choice of indefinite rather than definite article relies on the dubious reasoning that any strategy for adding two multidigit numbers that relies on place value can be considered a “standard algorithm”. So drawing pictures, using convoluted methods can all count. Which leaves open the typical reformer-based math interpretation that parents and others have protested against.
This same wording was proposed in one of the early drafts of the National Math Advisory Panel’s report on math education back in 2007. There was a big debate among the members regarding whether they should refer to “the” standard algorithm for addition/subtraction or “a” standard algorithm. “The” won out, but it was a bitter fight. Interestingly, Sybilla Beckmann, a math professor at U GA is of the view that many algorithms can be considered “standard”. She wrote a paper with Karen Fuson on this subject.   Bill McCallum, one of the lead authors of the Common Core math standards has the same view.
Stay tuned.

Awaiting word, Dept.


In Jo Boaler’s “Mathematical Mindsets”, the following paragraph appears in Chapter 2:

“In workshops with Carol Dweck I often hear her tell parents to communicate to their children that it is not impressive to get work correct, as that shows they were not learning. Carol suggests that if children come home saying they got all their questions right in class or on a test, parents should say: “Oh, I’m sorry; that means you were not given opportunities to learn anything.” This is a radical message , but we need to give students strong messages to override an idea they often get in school — that it is most important to get everything correct, and that correctness is a sign of intelligence. Both Carol and I try to reorient teachers so they value correct work less and mistakes more.”

I wrote the following to Carol Dweck:

“Based on articles I’ve read by you in which you clarify misinterpretations of your growth mindset theory, the above quote doesn’t sound like advice you would give to parents. Could you tell me if the above quote is accurate?”

Stay tuned for a response. But don’t hold your breath.