This interview with James Wilson, a retired math teacher, is worth listening to. He gives good advice to parents who ask how they should go about making sure their kids are learning the math that isn’t being taught in schools. I’ve worked with James for many years, and he and I wrote comments on the initial and final drafts of the Common Core standards. He gives very good advice.
There are certain narratives in education that are repeated so many times, that they become viewed as the absolute truth. The idea of “learning styles” comes to mind. Although articles and papers have been written debunking the idea of learning styles, the idea persists in ed schools and in other fora. I hear it at my school when, for example, a teacher will say that some student is a “visual learner” and thus finds listening to the teacher on a video to be disconcerting. This is finished up with “Too much ‘teacher talk’ and not enough guidance.”
This last I can almost buy, because I believe students do need guidance, particularly in the form of direct and explicit instruction. How you do this without talking, however, remains a mystery.
One of the many other myths is the one about “productive struggle”. I saw this in an article about the difficulties of distance learning.
“Sometimes when you think, ‘oh my kid’s not getting it, they are having a hard time, they aren’t getting to the answer’, you might even know the answer, but they can’t make that connection and that’s okay because that’s what we call productive struggle,” Musick said. “We want them to struggle because when they are struggling, that’s when they are learning the most.”
A friend of mine wrote me recently about his “struggles” to learn about web programming. He writes:
Good teaching makes learning easier, not harder with more struggle. Students will struggle plenty with regular textbook problem sets even if you do everything possible to make learning easy. This email comes after spending a good long time online trying to (and still trying to) understand what Node.js is and how it’s installed and used to simulate client-server programming. I have a master’s degree in computer engineering and taught computer science for years. I still don’t get it. Creating struggle is just an excuse for really bad teaching.
That’s the part you don’t read about in newspaper articles. If you do, someone will point out that that’s merely an opinion of what the person thinks is true. But the person provides no cites to studies, no evidence to support what is being said.
Right. Say that enough times and people believe that as well.
From an article about the Russian Math School:
“Meanwhile, Hilary Kreisberg, director of the Center for Math Achievement at Lesley University and a former fifth-grade teacher turned math coach, says her experiences with RSM students have led her to question the claim that Russian Math focuses more on developing a deep understanding of math instead of memorization. In fact, she has seen the opposite. “From what I’ve seen, they come in well above their grade-level standards in terms of memorization, but not in terms of content understanding,” she says. “Many of them very quickly get to an answer or can compute in a fast way, but they can’t necessarily explain to me what they’re doing or why they’re doing it.” And explanations, she says, are a critical component of mathematics. “In public school teaching, we are very strictly taught that the goal is not to accelerate,” Kreisberg says. “The goal is not to extend their thinking into another grade level, but to go deeper with the current grade-level standards because there’s always more you could learn about a topic.” “
There is no discussion in the article, or from the experts proclaiming math zombie-ism, of levels of understanding and how that figures into the novice/expert spectrum. The general excuse given for programs like RSM that are successful is “Well the kids are good at memorizing procedures but there is no deeper understanding.” The unanswered question is how deep is the understanding of the students who are supposedly benefiting from the “deeper understanding” approach the math reformers seem to think works so well? More to the point, how many of those in the “deeper understanding” category have to take remedial math in college compared to those from the RSM program?
And why is it that the US is lagging behind other nations (who used traditional techniques or those used by RSM) with all the “deeper understanding” the US students supposedly have? In articles I’ve read about reform methods, a common answer is “We’re just not doing reform math correctly.” (See in particular this article, which received criticism for its inaccuracies.)
These are questions that need further delving. In my opinion, what passes for “deeper understanding” in the educationists’ realm, frequently amounts to “rote understanding.”
Again, an op-ed showing that people know something’s wrong while being told by school administrators and departments of education that how it was done in the past (when it worked) was wrong, and now we have a better way (which doesn’t work). The current term for such subterfuge is called “gaslighting”. Another term is “willful ignorance”.
It is prevalent in education and the reason why many people advocating for change, sooner or later, give up trying to change the trend of ineffective faddish practices.
I heard a fellow teacher say recently that she uses “stations” in her class (i.e., a project-based activity in which students rotate among various tables doing various tasks, in groups) because middle schoolers’ attention span does not exceed 13 minutes. While short attention spans cannot be denied, there are other ways of breaking up a lesson, such as asking questions, having students solve problems, or a host of other activities, such as has been done effectively.
Productive struggle
Inquiry-based learning
Project-based learning
Problem-based learning
Growth mindset
Grit
Student-centered learning
Three-before-me (teacher doesn’t answer a question until student has asked 3 other students the question)
Depth of Knowledge
Bloom’s Taxonomy
The USA Today story I talked about the other day is quite long and addresses many facets of the problem. Unfortunately, it is horribly misinformed, and even more unfortunately, many people reading it will believe it.
There’s one paragraph that was disturbing to me about the results of a test called PISA:
“The approach has led other countries to success. Teens in the Netherlands post some of the strongest math scores in the world on the PISA assessment. That’s largely because the exam prioritizes the application of mathematical concepts to real-life situations, and the Dutch teach math rooted in reality and relevant to society. Some longtime Dutch math experts were involved in the design of PISA, which began in 2000 and is given every three years to a sample of 15-year-old students in developed countries and economies.”
The method that the Dutch use to teach math is known as RME: Realistic Mathematics Education, and originated in the Hans Freudenthal Institute. Despite valid criticism of it, like most things in education, bad ideas get touted as the silver bullet and RME is no exception. So I refer you to a post by Greg Ashman in which he analyzes PISA results, showing that inquiry-based learning (which is a large part of the RME method) is not what its touted to be by so called experts, nor USA Today.
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.
“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.”
“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.”
“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.”
“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.”
“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.”
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.
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!
“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.
traditional math 