The 21st Century Workplace Model of Education

There’s a new “study” out (which will soon be pointed to by various reformers and edu-pundits as “research shows” in their various diatribes) that purports to instruct teachers how to make their classrooms happier. In fact it is called “How to Create Higher Performing, Happier Classrooms in Seven Moves: A Playbook for Teachers”

The playbook/study explores why companies such as Zappos, Geico, and Google
are ranked among the best places to work in terms of happiness and success. It extrapolates what the Googles of the world are doing to what teachers should be doing in classrooms.  This is all done to “improve their students’ happiness and performance, not to mention their graduates’ readiness to work in America’s top organizations someday.”

I’ll save you the time of reading this “study” about how classrooms should look more like Google and just give you the bare bones version of it.  One hallmark of one of the many sub-classes of ed reformers is that they believe that the traditional model of education worked only for the industrial age to prepare students for mind-numbing work in factories and to be cooperative unquestioning workers. This is sometimes called the “factory model” of education.  Even though the traditional mode of teaching predates the industrial revolution, this particular narrative has gained momentum and proves the adage that if you repeat nonsense long enough, it is taken as truth.  Those who dispute it are often categorized as naysayers (or in modern parlance, “trolls”).

The study states at the outset that:

“[T]here’s a growing sense that classrooms aren’t cutting it for all students, not just those in fragile circumstances. Too many graduates can’t find work, and employers can’t find the right people to hire. Teachers can see that their schools are not consistently producing the types of graduates that today’s workplaces can readily employ. That’s why in this past decade teachers have voiced increasing concern about the need to move beyond basic reading, writing, and math and help students develop high-order skills like critical thinking, creativity, collaboration, and communication—the so-called “21st-century skills” that a knowledge economy demands.”

Thus, in the modern incarnation of how schools should be run, the factory model of education has been replaced by the 21st century workplace model. As stated in the study, “Many students will one day look for jobs in workplaces that embrace these management principles. Classrooms would do well to prepare students by resembling future workplaces more intentionally.” And what better protege of a 21st century workplace is there than Google and companies like unto it?

What this paper does is then lay out how schools should be run, modeled on companies like Google.  You’ll find the old familiar refrains of “working in teams”, “collaboration”, “creativity” and of course “critical thinking”.  Teachers shouldn’t micromanage, but rather facilitate students to find out what they need to know on their own, using online resources. In other words, teachers shouldn’t teach, but should manage their classrooms the way good CEOs manage their companies.  And here is their prescription for what teachers should be doing in that vein:  (Comments in italics are mine and are my rather biased opinions. Ignore them if they bother you.)

1. Teach mindsets. Develop the mindsets of agency, creativity, growth mindset, and passion for learning. (And what better way than to use things like Google docs, Google Classroom, anything Google, anything internet, anything online, and nothing like a textbook or direct instruction?)

2. Release control. Provide content and resources that students are free to access without your direct instruction. This control gives them ownership, develops their agency, and frees up your time.  (Frees up your time to check your email, and search for more fulfilling work–like a tutor at Mathnasium, or Sylvan, Kumon, or Huntington.)

3. Encourage teaming. Foster peer-to-peer learning and dynamic, team-based collaboration.  (Because everyone knows that students learn better from each other than from a teacher at the front of the room. Particularly if your peers have benefitted from after school hours at Mathnasium, Sylvan, Kumon, or Huntington.  Hey, isn’t your tutor at Huntington someone who used to work at this Google-school? (See no. 2 above).  )

4. Give feedback. Create a culture of feedback so that students receive personal, frequent, and actionable feedback in the moment, in small groups, and in one-on-ones.  (I am assuming that such feedback does not include things like “You need to learn your times tables” and “Please do problems 1-15 odd on page 360 to get some practice with this procedure”.  Actually that last one couldn’t possibly exist because there would be no textbooks in this 21st century classroom.)

5. Build relationships of trust. Show interest and concern in students as individuals and trust in their ability to drive their own learning, given the right structures are in place. (And what might those “right structures” be? Actual instruction perhaps?)

6. Help students hold themselves accountable. Give them tools to set goals, track their progress,and follow through.  (Of course this would not include things like homework assignments, or tests would they?)

7. Hold yourself accountable. Use reflection time, peers, student surveys, and self-assessments to make sure that you are on track personally.  (See item 2 regarding working on your resume during reflection time.  Oh, and who puts together the questions for the student surveys of teacher performance?  Just asking.)

Education models have been toyed with since John Dewey.  With this report, it appears there’s no end to it.  It will be interesting to see if the Kumons, Sylvans, Huntingtons, and Mathnasiums follow suit.

Thinking like a mathematician…or someone like one

This article makes the point that the emphasis on getting students to “understand” by using alternatives to standard algorithms is a subterfuge. The purpose, the article contends, is to make students look smarter than they are.  They reason as follows:

The problem is this: “number bonds” is a counterfeit of the way kids who are genuinely good at math act by the time they get into elementary. While the other kids are counting on their fingers, kids who’ve been playing with numbers in their heads since they were two or three have figured out all the relationships and will take numbers apart to make it easier to solve. Not something stupid like seven plus seven, of course. More something like 115 + 115.

Having figured out that number-gifted children will do this as 100+100=200, 15+15=30, so 115+115= 230. This is quite nifty for a first-grader, but the left thinks it can skip all the work getting there. If they just teach perfectly normal, average children to think in terms of taking numbers apart, voila! Everyone will be a math genius!  

I don’t know that it’s “the left” who thinks this way–I have run across plenty of apolitical math reformers who seem to be on this wavelength– but I take the article’s point.   The thinking amongst these reformers is that one indication of “understanding” is whether a student can solve a problem in multiple ways. Reformers then insist on having students come up with more than one way to solve a problem. In doing so, they are confusing cause and effect. Forcing students to think of multiple ways or using “number bonds” does not in and of itself cause understanding.  They are in effect saying  “If we can just get them to do things that LOOK like what we imagine a mathematician does … then they will be real mathematicians.”

Robert Craigen, a math professor at University of Manitoba sees this type of thinking as wrong-headed: “Mathematicians don’t think that the ephemeral truths of higher reasoning have any validity when disconnected from the basic, mechanical foundation on which they are built.”

Maybe if we call the traditional approach something that sounds more appealing, we can get on with teaching math properly.  Something like “alternative math”.  What that alternative is (and to what it is an alternative) will be our little secret.



The Squeakiest Wheel, Dept.

According to this recent news article, a group of students at Sobrato High School in Morgan Hill, California, signed a petition objecting to the way math is taught at that school.  A number of complaints were raised.  One was that “the math department does not tailor its teaching needs to every learning style of its students.”

Another complaint was that students were forced to work in groups.

According to one student, “Common Core is based on groups, so the teachers aren’t as involved as they would be in another curriculum. Sometimes, we do not even get to ask questions at all from the teacher, we have to work in groups and ask others who are just as confused as we are. So if your members don’t care, then you’re out of luck.  Teachers won’t even give you credit for completing classwork if some of your group members didn’t do their work exactly as instructed, and they won’t help your group if no one knows how to solve a problem.”

A third complaint was  that “teachers move on from one lesson to another before students can even grasp the concepts; they don’t allow students to retake tests that often have subject matter that was not previously covered in class.”

The valedictorian of the graduating class commented that he tutored other students  “going over the basics that the teachers should’ve gone over.” He said he supports the common core curriculum, “but the basics must be taught before you delve into the deep thinking that common core requires.”

While I am not sympathetic to the theory of “learning styles”, I think this complaint will likely get the most attention. Despite the lack of evidence for the learning style theory, it has garnered credibility, starting in ed school and continuing beyond. It is to education as “blood letting” was to medicine, and there’s no sign of the belief abating despite papers having been written that challenge the notion.

Also interesting is the belief that Common Core requires that classes be conducted in groups. It is not surprising that this would come up, given that Common Core has been interpreted in ways that align with math reform ideologies and ineffective practices, despite the statement on the Common Core website that the standards do not dictate any particular pedagogy.

The comment about mastery of basics is also pertinent, but it is not clear whether the valedictorian was speaking about basics that should have been mastered in K-8, or whether he meant foundational aspects of algebra.  The article doesn’t mention what textbooks are used, but it does allude to an integrated approach being used at the school. This means that instead of individual subjects like algebra 1 and 2, geometry, pre-calculus and so forth, there is a blending of all of these topics at various levels within each year. Although integrated math has been used overseas successfully, it has had problems of implementation in the U.S.  Two such programs (that received grants from the NSF in the early 90’s for their development) are Core Plus, and Interactive Mathematics Program (IMP), both of which have received substantial criticism for their poor coverage and implementation. (See for example this article that describes the effects on student performance in college attributed to Core Plus in high school.)

I suspect that despite the credibility the petition might gain within the school’s administration by its mention of learning styles, that will not be enough to compensate for the complaints about curriculum, use of groups, and grading policies.  My prediction is that if the administration does anything, it will look focus on what they think is the squeakiest wheel and will integrate “learning styles” into its practices.  They will hold on to everything else, however, dismissing the students’ other complaints by relying on this old chestnut: “Since when do students know more than teachers”.

Such complaint is part of an arsenal of arguments school administrations use, which also include “What do parents know?”,”What do mathematicians know?” and “If you’re not a teacher you have no right to criticize.” Such arguments are as cherished as the concept of learning styles.  And like learning styles, they show no sign of abating any time soon.

Drilling “rote understanding”

In working with a group of fifth graders in need of math remediation at my school, I had them do the exercises in their book.  It involved multiplication of fractions, and it used the area model of a square as the means to illustrate what multiplication of fractions represents, and why one multiplies numerators and denominators.  A problem like 3/4 x 2/3 then is demonstrated by dividing a square into three columns, and shading two of them, thus representing the 2/3.  Then the square is divided into four rows, with three of them shaded–the 3/4.  Thus 3/4 of the 2/3 have common shading, and the interesection of the shading of the 2/3 and 3/4 portion yields 6 little boxes shaded out of a total of 12 little boxes which is 6/12 or 1/2 of the whole square.  The students see what 3/4 of 2/3 means in this model in terms of area and the reasoning behind multiplying numerators and denominators.

This was the explanation used in my old arithmetic book from the 60’s (and in other textbooks from that time in an era denigrated as being about “rote memorization” without understanding”.)

fraction of fraction

Source: “Arithmetic We Need” by Brownell, Buswell, Sauble; 1955.

It is also the method used in Singapore’s math textbooks. But in the current slew of textbooks claiming alignment with the Common Core, after the initial presentation of the diagram to show what fraction multiplication is, and why and how it works, students are then required to draw these type of diagrams for a set of fraction multiplication problems.  The thinking behind having students draw the pictures is supposedly to “drill” the understanding of what is happening with fraction multiplication, before they are then allowed to do it by the algorithmic method.

Where is this interpretation coming from? One possible source are the “shifts” in math instruction that are discussed on the website for Common Core. One of the shifts called for is “rigor” which the website translates as: “Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity”.  Further discussion at the website mentions that students should attain fluency with core functions such as multiplication (and by extension, multiplication of fractions). There is nothing wrong with that. But they are also combining fluency with understanding: “Students must be able to access concepts from a number of perspectives in order to see math as more than a set of mnemonics or discrete procedures.”

I had a conversation recently with one of the key writers and designers of the EngageNY/Eureka Math program that started in New York state and is now being used in many school districts across the US. I noted that on the EngageNY website, the “key shifts” in math instruction described on the CC website, were broken out from the original three (Focus, Coherence and Rigor) , to six.  The last one, called “dual intensity” was, according to my contact at EngageNY, an interpretation of “rigor” and states:

“Dual Intensity: Students are practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. The amount of time and energy spent practicing and understanding learning environments is driven by the specific mathematical concept and therefore, varies throughout the given school year. “

He told me that the original definition of rigorous at the Common Core website was a stroke of genius that made it hard for anti-intellectuals to speak of “rigorous” in loosey-goosey ways. And,  in fact he was able to justify the emphasis on fluency in the EngageNY/Eureka math curriculum.  So while his intentions were good (use the definition of “rigor” to increase the emphasis on procedural fluency) it appears to me that he was co-opted to make sure that “understanding” took precedence.  In our discussion, I pointed to EngageNY’s insistence on students drawing diagrams to show place value in adding and subtracting numbers that required regrouping (aka “carrying” and “borrowing”–words that are now anathema in this new age of math understanding). I asked if students were barred from using the standard algorithm until they acquired “mastery” of the pictorial procedure. His answer was evasive, along the lines of “Of course we want students to use numbers and not be dependent on diagrams, but it’s important that they understand how the algorithms work.”  He eventually stated that Eureka “doesn’t do standard algorithms until students know the prerequisites needed to do them”.

Thus, despite Common Core’s proclamations that the standards do not prescribe pedagogical approaches, it would appear that in their definition of “rigor” they have left room for interpretations that understanding must come before procedure.

Understanding and procedure work in tandem—you need both. Understanding sometimes comes first, sometimes later. As evidenced by EngageNY/Eureka Math, and other programs making inroads in school districts across the US, the Common Core way is “understanding first, procedure later” which aligns with the reformers’ view of math education and their mischaracterization of traditional approaches being a set of “meaningless drills”.  So instead, we now have a nod to both camps. In the reformers’ view, students are made to use procedures that supposedly impart understanding. Ironically, as much as the reformers disparage “drill and kill” they have no qualms about “drilling understanding”.  And while it may work to give the adults who design such programs a mental visualization, they’ve had the advantage of many years of math experience (and brain growth) that students in 5th, 6th and even 7th and 8th grades do not have.

The major problem with this approach is that not all students take away the understanding that the method is supposed to provide. Some get it, some don’t. Robert Craigen, a math professor at University of Manitoba who is active in the issue of K-12 math education has described this process as “the arithmetic equivalent of forcing a reader to keep his finger on the page sounding out every word, with no progression of reading skill.”

The scary part about all of this is how easy it is to get swept in to the recommended methods.  I was working with the fifth graders and insisting that they draw the diagram to go along with each problem, when midway through the period I realized that I was forcing them to do something that I felt was ineffective.  The next day, I announced to them that instead of them having to do the rectangle diagrams, they could just do the fraction multiplication itself.  I couldn’t help but picture reformers shaking their heads in dismay, believing that I was leading the students down the path of ignorance, destined to be “math zombies”.  But in making my decision and announcement (which was met by cheers from the fifth graders), I believed that had I continued, I would just be giving them little more than a “rote understanding” for procedures they would not be able to perform for problems they would not be able to solve.

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.