Not Much to See Here, Dept.

When I read the headline “A Passion for Making Math Make Sense to Kids” I thought it was another of those “students need to understand not memorize” type of articles, with homages to growth-mindsets, memorization is bad, and traditional-math-never-worked-because-it’s-taught-as-rote memorization-and-a-bunch-of-tricks-and-look-at-all-the-adults-who-say-they-don’t-like-math.
But no. It was about Susri Anuradha, a woman with an engineering degree who opened up a Mathnasium franchise. Mathnasium is one of many of the learning centers to which parents send their kids to learn the math that isn’t being taught in schools, in the manner held in disdain by math reformers of all stripes.
“In 2015, she opened her first “Mathnasium”, a math learning center for children ages 5 to 18, in Acton, MA. Earlier this month, she opened opened her second franchise location in Burlington, MA. Anuradha holds a bachelor’s degree in engineering and a M.S. in Information Management Systems from Harvard University. After a rewarding career in Information Technology, Anuradha followed her passion of “Making Math Make Sense to the kids” and opened Mathnasium of Acton. Her decision to open and expand Mathnasium is driven by her desire to help children build confidence and excel in the subject she enjoys most – math.”
Of note was this quote from Ms Anuradha: “I love it when I meet my students in social gatherings and they come and hug me. I love to see kids getting empowered by the gift of education.”
I have to say I love it too. For the record I teach math in a traditional manner. Not to brag too much, but I just received a card from the mother of one of my students during teacher appreciation week. The card said “Thanks for bringing math back.”

Everyone’s Happy in Happyland, Dept.

I did a long-term sub assignment in San Luis Coastal school district in California a few years ago. I wrote about it in “Confessions of a 21st Century Math Teacher”.

I taught during the year in which California was in transition to the Common Core. We were told quite often that “next year would be different”. No more teacher in front of the class saying to open books to such and such page and do the following exercises. Teachers would facilitate learning, students would learn to “problem solve” and to “think” and “understand”. This assumed that the status quo was rote memorization and teaching without understanding or conceptual context.

The Superintendent of San Luis Coastal who was in charge then and still in charge today has a personal philosophy that aligns with the above bromides.  He wrote about his personal philosophy at length here.  An excerpt follows:

I believe students in the 21st century are different. They are digital natives and live in a world where “any knowledge” can be found immediately on Google. Therefore, why regurgitate knowledge (like an “academic rationalist”) when it is far more reasonable to expect a student to apply this knowledge and to make new meaning from this knowledge. (This is my “cognitive processor” or “social reconstructionist” coming out.) Relevance is critical among this generation of students in order to motivate them to move beyond what I see as low-level thinking.”

His constructivist viewpoints are bolstered by the school district’s hiring policies which use the Danielson Framework for evaluating potential new teachers. The webpage for this framework states right at the beginning that “The Framework for Teaching is a research-based set of components of instruction, aligned to the INTASC standards, and grounded in a constructivist view of learning and teaching.” What then follows is a description of 22 components (and 76 smaller elements) of what they consider teaching.

This framework, coupled with the Superintendent’s philosophy lays the groundwork for hiring and firing. If you are  an advocate of student-centered, inquiry-based, project/problem-based learning, c’mon in.  Traditional type teachers need not apply.

If you wish to teach skills, they better be learning, critical thinking and problem-solving skills.  Top-down, open-ended, ill-posed problems with many possible answers are preferable to the stuff that this particular cadre of educationists hate; i.e., distance/rate, work, mixture, and number problems.  No relevance to what kids really care about.

To my knowledge there has been little to no parent, teacher, or student backlash in this school district. So it appears that everyone is happy in happy-land. That said, I refuse to teach there. Not that strong a statement considering what their response would likely be.

A Comment Worth Reading

SteveH, a frequent commenter on issues relating to math education, left a comment on the piece below on “Hidden Figures” that is worth reading. So I’ve reproduced it here:

Traditional education pushes and values incremental mastery of skills along with understanding. That still happens in high school AP Calculus tracks, but not in K-6. Facts are “mere” and skills are “rote.” Add to that the use of social promotion and full inclusion where curicula like Everyday Math “trust the spiral” and assume that kids will learn when they are ready. Meanwhile, STEM parents and those who know better hide the tracking at home and ensure mastery of basic skills so that their kids are ready for a proper algebra I class in 8th grade. CCSS has now officially made K-6 a NO-STEM zone (PARCC actually states this) and educators claim that students can catch up by taking summer classes or doubling up in math in high school. Right. I had to work with my math brain son in K-6, but didn’t have to do a thing for his traditional AP Calculus track high school classes. Not a thing.

Is math a natural learning process in K-6? Does reform math provide a better “understanding” base for faster improvement later on? There is absolutely no proof of that. In fact, after 20+ years of reform “understanding” math, quite the opposite is shown – that if one fails to get on the advanced (algebra in 8th grade) math track, then any sort of STEM career is all over. I got to high school calculus in the old traditional K-6 days with absolutely no help from my parents. I had algebra I in 8th grade followed by geometry, algebra II, trig, and calculus. What’s different now? K-6. The women in the movie would have a much more difficult time of it now.

CCSS officially increases the academic gap. Parents who make up the difference at home and with tutors hide this systemic K-6 failure and those educational pedagogues never, never ask us parents what we had to do at home even though it would be a very simple task. (All of my son’s STEM friends had help outside of school.) They just claim that their process works, point to our kids as examples, and then blame the other kids or claim that they just need more hands-on real world engagement. They do not understand the importance of pushing and nightly individual success on homework problem sets. That’s the fundamental problem I see with the students I tutor. They don’t value homework. When you get to college, it’s ALL about the P-sets. My son stays up all night to finish them if he has to. This is likewise true for programming classes. It’s ALL about doing everything you can to finish your individual (not group) program with no errors. THAT is where true understanding is achieved. Back when I taught college math and CS, it was NEVER about engagement or any sort of group or class work. It was about the hard, individual work put into P-sets and programs. Success on homework and tests REALLY helps engagement, not in-class group work that does nothing for grades.

In this age where we can’t have any sane, fact-based discussion on health care choices, let alone understand even what insurance means, I have no hope for change in education, especially when some claim that it’s a liberal/conservative issue. Some of us are actually unaffiliated and quite willing and able to separate issues from political party ownership. I push educational choice, but that apparently means that I believe in all sorts of other baggage. Some people alter reality to fit their simplistic view of the world. You can’t argue with these people. We can only appeal to parents who want to understand what’s going on.

I Must Be Missing Something, Dept.


The movie “Hidden Figures”, about three African American women who were instrumental in the space program in the 60’s, has garnered lots of “STEM is not just for white guys” types of promotions.  (See this, or this or this.  There are many more.)  Also there are many blog posts by various progressivist/educationist types, praising the movie.

It seems strange to me then that in all this “conversation” about the themes of the film, there is nary a mention that the three women rose to their prominence based on the traditional math education they received. This is the type of math teaching, prominent in the era the women were from, that is so derided and despised by math reformers and given blame for “failing thousands of students”.

Is there an unstated progressivist narrative going on? How does it go again? The women were interested/gifted/talented in math and science to begin with and thus were destined to succeed in it no matter how it was taught? Have I got the right words? Or am I missing something?

I ask this because of this excerpt from an article in The Atlantic about the film:

“Math, in that sense, is in Hidden Figures a tool of meritocracy. It is a symbol of the power of education (chalk being handed from one person to another is a recurring motif in the film), but it is also, more broadly, a metaphor for a world that could be so much better if we would just let everyone, equally, have a say in its improvement. Math’s equations double, in Hidden Figures, as a hope for equality.”

I agree. Interesting that the reform math that passes as education and brings people flocking to NCTM and other conferences to adulate various math reformer as if they were rock stars actually penalizes the very people the reformers/progressivists think they are helping.

Stop me if I’ve said this before: The inequity arises from those who can afford to do so paying for the appropriate education offered at learning centers and the like.  And those who cannot afford it being deprived of what they need.

OK. I’ll stop.

Gee, That’s the First Time I’ve Heard That, Dept.

At a math blog, I came across the following paragraph:

“Word problems in math textbooks often give much of the information needed to solve them. There is no mystery. Students walk away from a math course with the only skill acquired being the ability to decode the textbook. They are just swapping numbers and plugging in different information. As a result, the so-called problems are no longer problems. They are routine and predictable. The problems are too scaffolded and the students realize that it’s an exercise in futility. An insult to their intelligence. While practice is indeed a fundamental part of math, when problems are variations of the same one, the motivation to complete them is lost. They don’t see the point of it all.”

This is the standard complaint levied against the typical word problems one sees in math textbooks. Well sorry to disagree with this grand master, but I’ve been using Dolciani’s algebra book for my 8th grade algebra class and believe me, the students are not finding the problems predictable. Dolciani, like other good math book authors, does vary the problems so that one is not working the same problem over and over, which is the standard complaint–i.e., it’s just plug and chug.

But this blogger doesn’t even like such variations and scaffolding. Sorry, but that’s how you understand and master the basic skills in math–by getting students to see the structure of problems and how to solve them through the initial worked examples, and then stretch their capability and extend the problem solving principle to situations that are just a bit different, and more difficult.

No, what this blooger likes are what are called “Fermi problems”

Look to Enrico Fermi. The Italian physicist had a gift for making accurate estimates of seemingly unsolvable questions. Given little information, he was able to provide educated guesses that came very close to the actual answer. His most famous question, “how many piano tuners are in Chicago?” seems to make no sense, but through a series of questions, estimations and assumptions, he arrived at a reasonable answer. Legend says that Fermi calculated the power of an atomic explosion by looking at the distance his handkerchief travelled when he dropped it as the shockwave passed. He determined it within a factor of 2. For a discipline that is always looking for realistic applications, math class would do well to use Fermi problems. It doesn’t get more real-life than that!

While there’s nothing wrong with such problems per se, they should not be used as a starting point or replacement for learning math nor as the fundamental definition of what math is used for. Fermi problems are percent/scale-up problems. How many golf courses are there in the US? How many molecules are in a mountain? These are classic IQ and job interview types of questions. The fallacy is that one cannot memorize (!) a lot of facts to make these estimates easier. You can practice these problems to get better at them. They can convince others that you are a genius. A big fact to memorize is how many people are in the US. A second one is how many people are in your state. Then estimate the number of golf courses (or whatever) in your state and scale it up. Often, when someone asks you one of these questions, they are happy if you can come up with some reasonable process for estimation. 

A steady diet of these things does not teach students general and transferable problem solving skills that they will need in other math courses. The belief seems to be “Give them top down type problems that force them to learn things on a ‘just in time’ basis, as if there is a problem solving schema that will emerge, given enough time and enough off the wall problems. Solving Fermi problems depends on memorizing simple facts and using simple math. It is not what math is all about.

Talk to the parents of the students who are on the track to AP calculus and on to STEM majors. They solved lots of the traditional word problems people like this blogger hold in disdain. Dolciani’s algebra books didn’t skimp on problems. Every chapter had word problems tailored to the particular math skill that was the focus of that chapter. If the chapter was on algebraic fractions, then the rate/distance problems and mixture problems given in that chapter relied on knowledge of algebraic fractions to set them up and solve.

My students are finding the problems challenging. It took some time before we were at the point where a certain type of problem was familiar, and for me to then up the ante. But unfortunately, the beliefs espoused by this blogger are very typical and people who teach the traditional problems are viewed as doing their students a disservice.

A math professor I know has this to say about teaching students how to solve problems using things such as Fermi problems:

I WANT my students to look at the sophisticated work I give them and say “Huh, this is no problem I just do such and such and so and so, and this will get me to the answer”. It is the students who have to struggle and fret over straightforward stuff that I worry about. Why do they insist on making easy things hard and putting roadblocks in students’ way?”

But what does he know? He’s just a mathematician who happens to think like one.

Shut the Hell Up, Dept.

This article focuses on the changes a school district is making in its classrooms to encourage collaboration. The unstated and unproven assumption is that working in groups is a 21st century thing, that it prepares students for getting along in the work-force, and is superior to a traditional type arrangement.

The sub-headline reads “Only ‘matter of time’ before similar spaces pop up across Iowa, expert says”.

At least they didn’t call them “maker spaces” but that, too, is probably a matter of time.

“Laura Wood, a 21st Century Learning specialist for the area education agencies that oversee much of southern Iowa, has helped advocate for those spaces.”

I find it interesting that one can now have a job title of “21st Century Learning specialist”. I might try calling myself that too and see if people find me more credible.

The school district’s vision of this brave new world in which students can move desks around at will and “collaborate” is summed up in these statements:

“While students can collaborate in traditional classrooms, 21st century spaces treat group work as the default, said Cindy Green, Cardinal’s director of curriculum and instruction. “Our focus and our push is to get kids collaborating,” she said, “because when they leave and go out into the workforce, they’re going to need to be cooperating with their peers.” “

Well, you may not know this but I worked in the real world for 40 years, up to and including 2010 when the modern day workplace was in existence. Collaboration consisted mostly of having some meetings in the beginning of a project to hash out ideas. But once things got going, we worked pretty much on an individual basis. People brought various levels of expertise to the table. Engineers worked on engineering aspects; lawyers worked on legal aspects and so on. In school, students are still novices and do not have levels of expertise to offer other students. It’s pretty much the blind leading the blind, unless there is some guidance being provided by “teacher as facilitator”.

In this brave new world of collaborative, student-centered, inquiry-based learning, instruction from the teacher is generally kept to a minimum. Direct instruction is considered inferior; getting instruction from a fellow student, however, is considered a good thing. Why direct instruction from a teacher is considered inferior to direct instruction from a student is a mystery to me. I did have someone invested in this theory explain it to me. “Students have more faith in something they think they came up with than something the teacher tells them.”

Not mentioned is to what degree such “collaborative instruction” is coming from expertise gained from Kumon, Sylvan and other sources of outside tutoring. Might be good if reporters asked such questions, but I don’t see that happening anytime soon.

Articles I Never Should Have Started Reading, Dept.

Once in a while I visit Edutopia’s website to find out what the current educationist trends are. If you’re looking for bad ideas, Edutopia will never let you down. To wit and for example, this article: Brain-Based Strategies to Reduce Test Stress in which “A neurologist shares ideas for beating stress before and during test time.”

The article has no shortage of platitudes and trendy edu-thoughts, including this gem of a paragraph. It starts off well enough:

“When the goal of learning is only test preparation, students will not be prepared to apply their learning to novel questions or problems.”

Yes, very true. But is anyone saying that the goal of learning is “only” test preparation? The author continues, undaunted with what is REALLY needed:

“But engaging students in authentic performance tasks and project-based learning helps deepen their understanding on both the factual and conceptual levels. In addition, when students experience their learning as personally meaningful, their intrinsic motivation strengthens long-term, durable memory networks. These are far more accessible for test retrieval (and longer term access) than rote memory.”

In other words only if students engage in “authentic” activities and PBL, will learning really take hold. Otherwise nothing is “personally meaningful” and is therefore relegated to the catch-all category reformers love to use: “Rote memory”.

Apparently, the author has written books on education. Well at least the article didn’t mention “maker spaces”.

If You Repeat Something Long Enough, Dept.

This article in The Windsor Times (which serves Windsor, California) explores the changes in math education going on in that school district. There is a discussion of how the “integrated math” option of Common Core is superior to the “old ways” of doing things:

“Part of the issue is the substantive changes to the curriculum itself, which everyone agrees is better in the long run and significantly more rigorous that previous versions but is also significantly different. Gone are the old paradigms of Algebra I, Geometry, Algebra II/Trigonometry, followed by AP Calculus for the elite few. In its place now exists Integrated Math 1, which incorporates early concepts of both Algebra and Geometry, with slightly more emphasis on Algebra, Integrated Math 2, which introduces more advanced concepts of Algebra and Geometry, with slightly more focus on Geometry, and Integrated Math 3, which takes both sets to the next level and along with some Trigonometry.”

Excuse me, did you say “everyone agrees” the new curriculum is better? Everyone, you say? Sorry. Go on.

” “The belief is that students who have grown up in the integrated model will be much better equipped and have a much more extensive math education that those without. However, students at all levels are struggling with the staggered roll out from the old model to the new.

” “We are no longer asking them just to compute,” Director of Educational Services Lisa Saxon said. “They have to think about numbers in a different way, explain, defend and justify their answers, and undertake a completely different level of rigor.”

Excuse me again. I teach algebra and believe me, I don’t just ask students to “compute”. Where are you getting that? As for defending and justifying their answers, and this completely different level of rigor: I’m aware some people think there are such things as “math zombies” who operate from a “rote” level of executing procedures. I would suggest that there is also what I call a “rote” level of understanding in which students mouth the explanations the teachers want to hear. Is that what you meant?

Shut the Hell Up, Dept.

I was going to file this under “Articles I Never Finished Reading, Dept” but I felt that “Shut the Hell Up” was more apt, considering this quote:

Often what’s lacking for U.S. students, and Massachusetts students as well, is a conceptual understanding,” Chester said. “They may learn the mechanics, but without that conceptual understanding it’s not as clear to those students how that math gets applied.” Mitchell agrees. It’s the skills to apply math that are lacking. She says students need to be taught how to think with math and not just memorize.”

Given that this has been the complaint for the past 28 years or so, don’t you think the excuse that “we’re just not teaching reform math right” sounds a little lame?

On the issue of “understanding” vs “procedures”, a math teacher I communicate with in New Zealand has this to say:

“I tell them that when their exam papers are marked there are no marks for “understanding”. I follow that up with saying that understanding will inevitably follow in time, provided that they could do the skills, but that it would not follow if they couldn’t do the skills.

“Now that isn’t to say that I don’t teach the reasons for things — I teach invert and multiply explicitly, but I also explain why it works. What I don’t do is fret about whether they understood my explanation, and I don’t let them not do something because they “don’t understand”. I most certainly do not try to teach understanding of a procedure to a student who can do it accurately.

“Some students find that truly liberating — they can get on with learning the Maths without any pressure to have to understand the whole picture first. Most just do what they always have done, which is do what the teacher asks them to do and not worry about understanding. Most kids really don’t want to understand very much. Every now and then I have a student who refuses to learn a new skill until they “understand” it — and that causes problems, largely because they learn so unnecessarily slowly as a result, which I find difficult as a teacher.”

I recognize that this quote will cause cognitive dissonance. That’s OK.  Productive struggle is good for you.