A traditional approach to a traditional word problem in algebra

I saw this problem a few days ago and it got me thinking how I would present it in my algebra class: 
“An airplane can fly 550 miles with the wind in the same amount of time it takes to fly 425 miles against the wind. Find the speed of the wind, if the airplane is flying at a constant speed of 195 miles per hour.”
First I would ask my students what are we equating? Is it a distance=distance problem or time=time?  You could make it into distance = distance, but the time=time is simpler to deal with.  Time is expressed as distance over rate.  We know the distances, and if we let w = the wind speed, then we can express the rates as 195+w and 195-w.  
We get this equation:    550/(195+w) = 425/(195-w), which we can simplify into 
550(195-w) = 425(195+w).
Big numbers, so before they can ask whether they can use calculators, the first thing I ask the students is whether we can whittle down 550 and 425 by dividing both sides by a common factor? Dividing by 5 two times in a row (or 25 once) yields:  22/(195+w) = 17(195-w).
They still want to use their calculators.  I forbid their use for this problem. 
“Let’s just write it out without doing any multiplying,” I tell them.
17(195) + 17w = 22(195) – 22w
Let’s combine like terms:  39w = 22(195) – 17(195)
Is there a common factor on the right hand side?  Yes; so now the equation looks like 39w = (5)(195) and w = 195(5)/39
Do you think 195 can be divided by 39? Let’s try.  In fact it can; it’s 5.  Now we have w = 5 x 5 = 25 mph.  The only big number calculation we did was dividing 195 by 39.
This type of approach helps develop number sense, which some claim traditionally taught math doesn’t do a good job developing. I present it here as an example of what I consider a rich problem.  It’s not the problem that is rich so much as the way it can be used to address different strategies that may be employed within the same problem.

A Tale Told by an Idiot, Dept.

Yet another article on Singapore Math, which in the end signifies nothing. In fact, this one could probably be used as a template for so-called education reporters when they write about math education.

It starts out well enough with a very concise history of the program that was developed in Singapore:

What is referred to as Singapore Math in other countries is, for Singapore, simply math. The program was developed under the supervision of the Singaporean Minister of Education and introduced as the Primary Mathematics Series in 1982. For close to 20 years, this program remained the only series used in Singaporean classrooms.

In fact, the “Primary Math” series is simple and effective and has been used by many homeschoolers in the US for many years.  Some mathematicians (Richard Askey from U of Wisconsin among them) revamped the series for use in U.S. classrooms so that it used U.S. currency and English system measures (in addition to the metric system). It was called Primary Math, U.S. edition.

Over the years, the Primary Math series in Singapore was replaced by “new and improved” versions that in the opinion of many users of the original series, was watered down and prone to some of the U.S. reform ideas, such as increased use of calculators. Nevertheless, enough of the main tenets of the original Primary math series have remained in the subsequent editions so that the textbooks still stand out.

The article leads with this precis of the framework of Singapore Math:

The framework of Singapore Math is developed around the idea that learning to problem-solve and develop mathematical thinking are the key factors in being successful in math.

Ignoring for the moment the trendy use of “problem-solve” as a verb (instead of saying “learning to solve problems”), the above intro is not bad unto itself. But, not to disappoint, things go predictably downhill with the analysis of “pros and cons” of Singapore Math.

Actually the listing of the “pros” of Singapore Math are not bad; hard to disagree with these, for example:

  • Textbooks and workbooks are simple to read with concise graphics.
  • Textbooks are sequential, building on previously learned concepts and skills, which offers the opportunity for learning acceleration without the need for supplemental work.

But then we have the obligatory “not like traditional math” narrative:

  • Asks for students to build meaning to learn concepts and skills, as opposed to rote memorization of rules and formulas.

Yes, it does require students to learn concepts and skills and provides a contextual background for how the rules work. But so did many textbooks of the past which I’ve talked about extensively.  And it isn’t like teachers don’t teach the conceptual underpinnings.  Caricaturizing traditional math approaches as “rote” never gets old, though, as the writer of the article shows.

But moving on to the “cons” of Singapore Math, things get really interesting.  We start with “Requires extensive and ongoing teacher training, which is neither financially or practically feasible in a number of school districts.”

Yes, it does require some teacher training, but if you’re going to complain about financial burdens and practical feasibility, why not take a look at the Professional Development (PD) seminars teachers are forced to attend to learn the “Common Core way” and other topics.  For example, I was asked to attend six off-site workshops to “collaborate” with other math teachers in the county in order to learn innovative approaches to achieve the Common Core’s “Standards of Mathematical Practices (SMPs)”.  Aside from the impracticality of missing six days of instruction time, there was also the financial burden of hiring a sub for me for six days.  Then there was also the PD itself which was all trendy talk about how math has been about “answer getting” when the process is key and other balderdash that passes as educational wisdom while padding someone’s resume.

Then there’s this:

  • Less of a focus on applied mathematics than traditional U.S. math textbooks. For instance, the Everyday Mathematics program emphasizes data analysis using real-life, multiple step math problems, while Singapore Math’s approach is more ideological.

Less focus on applied math than traditional textbooks? Really?  Primary math is all about applied math, but the term “applied math” has a different meaning when talking about math education these days. It used to be we could talk about students learning to solve problems.  But “solving problems” in today’s edu-lexicon means the standard one-answer, non-open-ended type of problems (like “Johnny had 5 apples and gave some away. He has three apples left. How many did he give away?”) which is viewed as “dull, boring and deemed to not imbue students with a problem solving “schema”.  Furthermore, the problems are viewed as contrived, and not something they would use in “real life” (even though they are very applicable to real life).

Actually, Singapore Math provides challenging multi-step problems which enable students to generalize problem-solving procedures to solve a variety of different problems.  But the author would rather compare Singapore Math to Everyday Mathematics whose “discovery” and “spiral” approach provides little to no background in how to solve much of anything.  See this article to get an inkling or talk to any parent whose child has had to endure Everyday Math.) Apparently, Everyday Math is now incorporating “data analysis” to go along with the latest shiny new trend to teach statistical concepts in K-6.

What the author means by Singapore’s approach being more “ideological” is anyone’s guess, but it presents a damned-if-you-do, and damned-if-you-don’t mentality. On the one hand, presenting problems that involve computation is held in disdain because it doesn’t present the real beauty of math. But then there are complaints that math is presented as too abstract (a complaint generally leveled at algebra courses) and the reason students don’t like it is because it isn’t relevant.  So I guess data analysis is the new middle ground–everybody is happy learning about frequency histograms, box and whisker plots, and mean absolute deviations in the sixth grade.

Finally there is this:

Doesn’t work well for a nomadic student population. Many students move in and out of school districts, which isn’t a big problem when the math programs are similar. However, since Singapore Math is so sequential and doesn’t re-teach concepts or skills, using the program may set these students up for failure, whether they’re moving into or out of a district using it.

Apparently, the author of the article and many others believe the premise upon which Common Core was sold: i..e., that the U.S. student population is largely nomadic with huge movement into and out of states and school districts on a constant basis.  This belief persists despite evidence to the contrary obtained by a glance at U.S. Census data.  And even if it were the case that the U.S. is nomadic, that shouldn’t be a problem since there are now Common Core aligned Singapore Math books which teach to the same standards as all the other states.  Of particular interest, however, is the statement that “Singapore Math is so sequential and doesn’t re-teach concepts or skills.”  Actually, it does–it just takes up where it left off in the previous grade, like most textbooks do.  Maybe it doesn’t spend as much time with review as others, but one can always build in reviews as needed.

But finding problems where none exist is the bread and butter of many edu-writers these days.  And unfortunately, ignoring the real problems that accompany many of the student-centered, inquiry-based approaches that are increasingly popular in K-6 is another.



Skin Deep Understanding, Dept.

Ed Week, the self-professed “newspaper of record in education” does it again, with an article on San Francisco’s decision to do away with algebra in the 8th grade.  The article does everything but post banners and host a confetti parade for such decision, but is also careful to provide “nuance” and the semblance of balanced reporting.

The article sets the stage and tone of San Francisco’s decision and what it meant:

That means no “honors” classes. No gifted track. No weighted GPAs until later in high school. No 8th grade Algebra 1. In terms of curriculum, this is about as controversial as it gets. And that’s not just because of its math implications, but because of the parental pushback such a plan is guaranteed to generate. In effect, by de-tracking math classes, San Francisco has done away with one of the key avenues that the well-connected use to give their children an academic advantage.

That last sentence should win an award.  It’s all about equity, and algebra in 8th grade was inequitable because it benefitted the well-connected.  These same “well-connected” families shell out money for tutoring and learning centers to make up for what their children are not being taught in the earlier grades. I guess it’s “shame on them” time, and the Ed Week definition of equity is therefore to lower the tide for all.  And of course no mention of the non-well-connected families who might have benefitted from algebra in 8th grade.  That might be too “nuanced” an argument.

No, much better to rely on the tried and true messages:

Federal data show that white and Asian students disproportionately take Algebra 1—long seen as a critical gateway to advanced math—before high school, while African-American and Latino students are overrepresented among those taking it for the first time in grade 9. Many of them take it as late as their junior or senior year.

And then, of course, to accompany it with a chart showing –goodness–that the repeat rates for algebra 1 for all ethnic groups declined dramatically since the enactment of the ban on algebra1 in 8th grade.

Very impressive.  It brings to mind a question though.  How is algebra 1 taught in high school now?  And I almost hate to phrase it in this un-nuanced way, but is it “watered/dumbed down?”

I bring this up because of how they present the “before” picture, showing two contrasting  (i.e., “nuanced”) views (they even use the word “nuance” here) thus doing a good job of appearing to be balanced:

Research paints a far more nuanced picture than either side in such debates typically acknowledge. For the average student, researchers say, early exposure to a challenging class like algebra probably does pay off.

But in a 2015 study, the University of North Carolina’s Thurston Domina and colleagues tracked how California’s uneven 8th grade algebra-for-all rollout played out across districts. In a surprise finding, they discovered that higher enrollments in early algebra were linked to a decline in students’ scores on a state math test.

“I think the failure of 8th grade algebra was one of just not preparing teachers and school leaders to understand the policy and to implement it well,” said Domina, an associate professor of education policy and sociology. “The whole idea was to have heterogeneous, rigorous classes, and schools didn’t have the capacity to pull that off.”

The article then talks about “instruction vs textbook” and how important the method of instruction is in all this.

As for implementation, San Francisco administrators have shaped day-to-day teaching and curriculum to support the district’s focus on equity. Heavily based on work by Jo Boaler, a Stanford University professor of math education, the curriculum emphasizes having groups of students work through a series of ambitious math tasks.

Traditional math teaching, the thinking goes, tends to reinforce rather than break down inequities.

“If you have a procedural textbook, not only is there nothing to collaborate about, the ‘smart kid’ in the group is always the one who gets the computation right,” said Lizzy Hull Barnes, the mathematics supervisor for the San Francisco district. But when students wrestle over problems together, they can use different methods, compare approaches, and figure out why some work and others don’t, making all of them active participants in the learning, she said.

Jo Boaler’s methods seem to be accepted by Ed Week and other education reporters as beyond reproach. She claims her methods are backed by cognitive science research, making claims such as “making mistakes makes your brain grow”.  Though in all honesty, since she was challenged by scientific-minded people on such statement, there is now a more “nuanced” version of said statement. Nevertheless, “getting the right answer” is now symptomatic of “procedural” instruction and making mistakes is the watchword for ensuring “deep understanding” in this brave new world of math education.

Collaboration is key to 21st Century learning. Without it, you’re simply doing procedural stuff and “answer getting” which is stated with references to dubious research that show that such approaches have failed thousands of students.  “Procedural textbooks” are assumed to provide no pathways to understanding; it’s all rote memorization, unconnected ideas and bags of tricks.  No one in these types of articles ever asks parents of students who make it into STEM fields what these students do to gain mastery of the content. It’s simply assumed that such students are gifted, or highly intelligent, and are destined to understand no matter how the material is presented.

Well, the article does hint at this.  It admits that there are some students who have taken algebra elsewhere–and that this poses a problem in student-centered, inquiry-based, collaborative classrooms.

Most teachers praise the social-justice impetus behind the math plan. But they also say that heterogeneous classes pose unique problems.

Students bring vast achievement differences to class, a situation that’s not helped by ambitious parents who, now, shell out thousands of dollars for students to take non-district algebra classes over the summer in the hopes of getting their children into geometry early.

“We have kids who have seen some of the math before. Their knowledge may not be deep, it may be procedural, but they come in thinking, ‘I know this already.’ You have to authentically challenge them, too,” said Daniel Yamamoto, an algebra teacher and the math-department chairman at Burton High. “And there are other kids who say [in response], ‘I have nothing I can add to this discussion.’ “

And there you have it: Procedural knowledge is never “deep”.  It goes without saying, I guess, which would explain why our reporter didn’t question such statement. It would be nice to get interviews with some of the teachers who disagree with the statements above, rather than just relying on a statement that “most teachers praise” the this and that of banning algebra in 8th grade and about the false dichotomy between procedures and “deep” understanding. Or course, not too many of those teachers want to be interviewed for fear of appearing on record–and possibly losing their jobs.

Interesting that Phil Daro, one of the authors of the Common Core math standards and the person who holds “answer getting” in disdain, is hedging his bets on the algebra in 8th grade question.  On the one hand, the article shows he was instrumental in San Francisco’s decision:

“Tracking is an evil. But fear of tracking is a problem, because you do have to talk about differences in students’ backgrounds,” said Phil Daro, a common-core-math writer who helped San Francisco design the new course sequence.

But then it goes on to say:

“[H]e continues to worry that San Francisco leaders’ decision four years ago not to offer a limited amount of Algebra 1 in 8th grade might someday backfire. “I thought politically it was a mistake,” he said. “It may still turn out to be one.”

Oh, and here’s some balanced reporting, way at the end:

District leaders, for their part, are focused on more immediate concerns. Asked what challenges remain, Barnes points to the progress of black students as an area in which the city needs to double down. Those students have gained in math and science credits, alongside their peers, but those gains aren’t yet showing up on state test scores or in enrollments in AP Calculus.

Ah ha!  Interesting.  So maybe it isn’t working after all! But wait; there’s this:

Her colleague Angela Torres, a math-content specialist, cites the difficulty in ensuring that all teachers feel confident in the new curriculum and teaching methods.

Right. Always leave room in politics and in reporting for blaming it on the teachers who just aren’t doing reform math correctly.

Brave New World of Education Reporting, Dept.

I should go easy on this article because it’s written by a student from the School of Journalism at Michigan State University.  Nevertheless, since education writing is usually about agreeing with those who advocate for ineffective practices, with little or no investigation of the other side, this article is emblematic of how reporters are trained in education writing.

We start with the standard quote from a teacher about how it used to be and how it is now:

Mindy Willis, a curriculum consultant for Pinckney Community Schools, says the role of a teacher is evolving. When she was in grade school, the teacher taught in a direct manner, the students took notes and then were later tested on it, but she says that’s not how the classroom looks now. Today, teachers are managing their classrooms in what’s called an inquiry-based instruction.

“With inquiry-based instruction, you design learning for students where they’re actually going through a process of figuring things out themselves,” said Willis. “So basically, the kids are constructing their own knowledge based on experiences that they’re having and they’re driving, and the teacher is a facilitator to guide them through that process as opposed to just spitting it at them and then regurgitating that information.”

Notice that the “how we teach now” is contrasted with past practices by denigrating them: e.g., teachers “spitting” information at students who then “regurgitate” the information.  Such mischaracterization is a key feature of education reporting and it looks like MSU’s School of Journalism is leading the way.

There’s even a table comparing how it used to be to what we’re moving toward:

There’s plenty to notice in this table, but one thing that grabbed my eye was the description in the “Moving from” column regarding lesson structure: “Lessons contain low-level content, concepts mentioned; lessons not coherently organized.”  I’ve gone through many older math textbooks and do not agree with the caricature presented.  Topics are presented and built upon in a very coherent fashion.  One has only to look at some of today’s modern “spiral process” textbooks (e.g., Everyday Math; Investigations in Number, Data and Space) to see that lessons are anything but coherent.  And in the “Moving toward” column, we see that lessons focus on “high-level and basic” content.  Hard to know how you can do both at the same time, but what I think they mean is they present a “top down” approach, with concept first, and procedure last–and no teaching the procedure until students “understand” the conceptual underpinning. And needless to say, but I’ll say it, such approach has had disastrous results.

And of course no story on education would be complete without talking about project-based learning. This article does not disappoint in that regard:

Another widespread idea across the country is project based learning. According to the Buck Institute for Education, project based learning is about students creating projects that solve real world problems, then students present their finished project to an audience. By doing this repeatedly, students learn key skills like critical thinking, communication, collaboration and creativity.

Yes, the 4C’s: Critical thinking, communication, collaboration and creativity.  Why bother with learning what they need to know in order to do a project–they learn it on the job in a “just in time” manner.  And in so doing, learn the 4C’s which are more important than content knowledge.

The authors of this article even cite research that supports the idea that such skills are key for students to succeed in the 21st Century.

In a research study conducted by the Buck Institute for Education that compared various organizations to their opinions on “21st century skills,” most organizations found critical thinking, collaboration and communication as extremely important attributes in an employee. You can access their research in the link here.

Of course, content knowledge is important–for the teacher, that is:

“A lot of people don’t realize just how much content area preparation secondary teachers have to go through to get their degrees, to become certified and to become teachers,” says Bieda.

Well, it’s good that somebody knows the content. I wonder how the teachers obtained that contact. Through direct instruction or Project Based Learning?  In any event, that content is being kept a deep dark secret and providing students such information would be “spitting it at them” only to have them “regurgitate it back”.  And we can’t have that.  Because the new method is working just great isn’t it?

Don’t hold your breath, by the way, for a follow-up story of how students who wind up majoring in STEM fields are getting the education they need.

Stop Me If You’ve Heard This, Dept.

This article talks about a book titled “Systems for Instructional Improvement”   coauthored by the dean of the University of Southern California ed school.  It is described as  “dedicated to improving math instruction in the U.S.”

Why is it that just about every book, article, tweet, and Linked-In polemic that purports to put math education back on track starts from the following assumption:

“For the past 25 years or so, there’s been a growing recognition that students at the middle-school level, in particular, aren’t developing a deep understanding of mathematics,” said Thomas Smith, dean of UC Riverside’s Graduate School of Education. “A big piece of that is because of the way students in the U.S. are taught; current math instruction tends to be highly procedural — as in ‘use these steps to solve these types of problems’ — instead of allowing students to investigate real-life problems and experiment with different types of solution strategies.”

The same catchwords end up in this catechism of edu-wisdom: “Deep understanding of mathematics”.  Just how deep of an understanding of math do you want middle school students to have? Even in freshman calculus courses, students learn a short-cut version of what limits and continuity are in order to get to the powerful applications of same; namely derivatives and integrals and the (yes) procedures for taking derivatives and finding anti-derivatives.

Students who go further in math will master the delta-epsilon definitions of limits and continuity, and learn about least upper bounds, greatest lower bounds,  limit points, cluster points, open and closed sets and compactness. Such deep understanding of calculus is largely inappropriate at the freshman level–one has to start somewhere in order to go deeper. And so it is the same with middle school math. One starts with procedures and in subsequent math courses, one goes deeper.

And it isn’t as if teachers and textbooks have not provided the conceptual understanding that underlie the procedures that math reformers seem to find so “rote like”. It’s just that students like to know how to do things and as has been stated in various research studies, procedural fluency and understanding often work in tandem. Sometimes conceptual understanding comes first, and sometimes the procedural understanding. But there is a stated resistance to learning the procedure first as if it is handing it to the student and taking the easy way out.  Teachers are made to feel guilty when, seeing that their students are unable to solve the problems assigned, take to breaking down the procedures step by step.

Also, the problems that middle school students have had to solve over the years are really not as far-removed from reality as the spin-meisters would have us believe. K-8 math is largely “applied math” and in earlier times was an essential part of everyday life.  A glance at the textbooks used in Singapore (the Primary Math series is a good example) has very straightforward problems that allow students to apply the concepts of things like decimal, fraction and percent operations.  An example:  “Alex spent 1/3 of his pocket money on a toy airplane and 2/3 of the remainder on a toy robot. He had $20 left.  How much did he spend altogether?”

Singapore has boasted high scores on international tests for years, but the problems that students solve there may be held in disdain by math reform types. Perhaps they think they are not relevant to students concerns. If students are not given proper instruction on how to solve such problems, and are expected to discover “strategies” for solving, they will tend to ask “When am I ever going to use this in real life?”   If given proper instruction with scaffolded problems that are variations on the initial worked example, students generally will tackle such problems. The “When will I ever use this” question is generally an expression of frustration.

Math reformers want kids to use elaborate techniques to get simple answers and call that “understanding”. Or even “deep understanding”.  What they think is understanding is visualization which is not what mathematicians mean by understanding.

But the standard talking points that pervade books about education continue.  The article ends with this chestnut:

“When it comes to teaching, people are often wedded to the ways from which they learned,” Smith added. “But the reality is those ways don’t always help all students learn. Our goal is to support teachers and coaches, as well as school and district leaders, to improve teaching and learning for all students.”

Interestingly, over the past 30 years, math has been increasingly taught in the manner these reformers want to see it done (in the lower grades primarily).  If you point this out to them, they will generally say one of two statements: “No, that isn’t true, but I wish it were” or “They aren’t doing reform correctly.”

In the meantime, many books like “Systems for Instructional Improvement” have been written and not much has changed.  Oh, right. It’s because they aren’t implementing the recommendations. Or they’re doing them wrong.  Or (insert your own excuse here _________________).

Discovery, authenticity and understanding, Part I.

(Originally published at Nonpartisan Education Review in different form. This is an update and revision of same).

By way of introduction, I am a math teacher, but not a mathematician.  I majored in math and have used it throughout my life including my pre-teaching career in which I worked in the field of environmental protection. My facility with math is due to good teaching and good textbooks. The teachers I had in primary and secondary school provided explicit instruction and answered students’ questions; they also posed challenging problems that required us to apply what we had learned. The textbooks I used also contained explanations of the material with examples that showed every step of the problem solving process.

I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools.

Optimistically believing that I could make a difference in at least a few students’ lives, I decided to teach math when I retired. I enrolled in education school and obtained my certification in secondary math teaching, which spans grades 6-12. Although I had a fairly good idea of what I was in for with respect to educational theories, I was still dismayed at what I found in my mathematics education courses.

In class after class, I have heard that when students discover material for themselves, they supposedly learn it more deeply than when it is taught directly. Similarly, I have heard that although direct instruction is effective in helping students learn and use algorithms, it is allegedly ineffective in helping students develop mathematical thinking. Throughout these courses, a general belief has prevailed that answering students’ questions and providing explicit instruction are “handing it to the student” and preventing them from “constructing their own knowledge”—to use the appropriate terminology. Overall, however, I have found that there is general confusion about what “discovery learning” actually means. I hope to make clear in this article what it means, and to identify effective and ineffective methods to foster learning through discovery.

To set this in context, it is important to understand an underlying belief espoused in my school of education: i.e., there is a difference between problem solving and exercises. This view holds that “exercises” are what students do when applying algorithms or routines they know and the term can apply even to word problems. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation. Moreover, we future teachers are told that students’ difficulty in solving problems in new contexts is evidence that the use of “mere exercises” or “procedures” is ineffective and they are overused in classrooms.

As someone who learned math largely though mere exercises and who has creatively applied math in my work, I have to question this thinking. I believe that students’ difficulty in solving new problems is more likely to be because they are novices, not experts.  They have neither the experience, nor the requisite knowledge and/or mastery of skills to allow solving widely varying problems—not because they were given explicit instruction and homework exercises.

Those who make such a differentiation and champion “true” problem solving espouse a belief in having students construct their own knowledge by forcing them to make connections with skills and concepts that they may not have mastered. But, with skills and concepts still gelling students are not likely to be able to apply them to new and unknown situations. Nevertheless, the belief prevails that having students work on such problems fosters a discovery process which the purveyors of this theory view as “authentic work” and the key to “real learning.” ( This concern about “authentic” versus “inauthentic” work comes from progressive education reformers who believe that it’s best for students’ school work to be as realistic as possible, that is, for it to be focused on
learning about and trying to solve “real world” problems. )

One teacher with whom I spoke summed up this philosophy with the following questions: “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure? How do we teach students to apply mathematical thinking in creative ways to solve complex, novel problems? What happens when we get off the ‘script’?” 

Those are important questions, but I will argue in this article the following points: 1) “Aha” experiences and discoveries can and do occur when students are given explicit instructions as well as when working exercises; and 2) Procedural fluency does not exclude conceptual knowledge—it leads ultimately to conceptual understanding and the two are key for applying mathematics to complex problems.

      I’m not against asking students to discover solutions to novel and challenging problems—the experience can be quite powerful, but only under the right conditions. A quick analogy may be useful here. Suppose a person who knows how to drive automatic transmission cars travels to a city and is forced to rent a car with a standard transmission—stick shift with clutch. The person in charge of rentals gives our hero a basic 15 minute course, but he has no opportunity to practice before heading out. In addition to this lack of skill in driving a standard transmission, the city is new to him, so he needs to rely on a map to get to where he needs to go. The attention he must pay to street names and road signs is now eclipsed by the more immediate task of learning how to operate the vehicle. In fact, he would be wise to take a taxi in order to avoid a serious accident. But now suppose that prior to his trip he is told that he will need to drive a standard transmission because where he is going, rental car companies don’t rent out automatic transmission cars. With proper training and guidance, he can start off on quiet streets to get the feel of how to coordinate clutch with shifting, working up to more challenging situations like stopping and starting on hills. Over time, as he accumulates the necessary knowledge, and practice, he’ll need less and less support and will be able to drive solo. There will still be problems that he has to figure out, like driving in traffic jams that require starting, slowing, downshifting, and so forth, but eventually, he will be able to handle new situations with ease. Now, having already achieved driving mastery of the vehicle that will take him where he needs to, the task of driving in a strange city although challenging is more manageable.  He will be able to focus all of his attention on navigating through new streets.

      Whether in driving, math, or any other undertaking that requires knowledge and skill, the more expertise one accumulates, the more one can depart from the script and successfully take on novel problems. It’s essential that at each step, students have the tools, guidance, and opportunities to practice what they learn. It is also essential that problems be well posed. Open-ended, vague, and/or ill-posed problems do not lend themselves to any particular mathematical approach or solution, nor do they generalize to other, future problems. As a result, the challenge is in figuring out what they mean—not in figuring out the math. Well-posed problems that push students to apply their knowledge to novel situations would do much more to develop their mathematical thinking.


To be continued.

NCSM/NCTM Annual Conf, Dept.

Every year the National Council of Teachers of Mathematics has its annual conference, complete with celeb speakers, vendor booths, instructional seminars, and the usual array of topics that pass for effective practices.

From what I hear from a friend who teaches high school math, this year’s was no different.  Her report follows below:

I signed up for a pre-conference workshop on teaching math for social justice.  They made the accusation that colleges of education exacerbate the problem of achievement gap for minorities.  I asked an ed-school professor (maybe from Connecticut) what her school did to alleviate this problem.  Her answer sounded awfully general, so I asked her to give me one, explicit example of a topic they teach that would work toward alleviating social injustice in schools. Her example was that, oh, you can teach students that a comma can mean the same thing as a decimal point in other countries. Of course, this is not what they are talking about at all, so she missed the point. 
They made us do this activity where three siblings were going to give a party for their father’s 70th birthday.  One made something like $20,000 a month; another made $6,000 a month, and the third — a single mother with two children — made $3000 a month.  The sibling making $20,000 a month thought they should split the $4500 cost of the party equally.  Another sibling suggested amounts that are proportional to their salaries.  We were supposed to converse (in groups, of course!!) what is “fair.”  Of course, it launched into a huge discussion about missing information, such as the one who makes $20,000 a month may have a spouse with a disease that requires  a $5000 shot each month, so in other words, were weren’t told about their disposable incomes or other circumstances.  After about 20 minutes, we still hadn’t settled in on anything other than the proportional one.
My issue is this:  I think that MANY kids don’t know how to compute what would be proportional to the incomes in the first place, so why impose all of that drama on it? 
I also attended a session of which Phil Daro was a co-presenter, but had to leave right before he was on stage.  His partner, Kyle Pearce, didn’t know beans about math. Their big thing was about this photo of 5 reams of copy paper stacked against a concrete block wall, and how many reams would it take to reach the ceiling.  They gave the height of the ceiling and the height of the stack of 5 reams of paper.  I divided that height by 5, and then divided the height of the ceiling by that quotient.  I did not set up a proportion at all.  One can say that I used proportional reasoning, but I didn’t need to formally set up a proportion.  I didn’t like the way they presented the solution of the problem.  Clearly, it was designed for teachers who are at middle school or lower level.
The bottom line is that it was all pretty bad.  My school district paid a few thousand dollars to send my colleague and me to this stuff.  We were there for the last day of NCSM and the first day of NCTM.  Needless to say, I was glad to get home!!

Teachers are born not made, Dept.

I get fan mail from time to time and invitations to speak that most of the time never come to fruition.  One such invitation came from the treasurer of a Catholic school in the Los Angeles area.  He had read my book “Math Education in the US: Still Crazy After All These Years” and liked it so much that he ordered ten copies for various teachers in the school.

About two years ago he asked me if I would speak in late August at his school.  I was just starting my teaching job at my current school and had to report the week before school started–which coincided with when he wanted me to speak.  I said I could not given the circumstances, but maybe we could look at doing it in April since I got two weeks off, and surely one of those weeks his school would be in session.  He demonstrated a great amount of inflexibility and said August was the only time.  He then suggested the same time in August a year from then.

I said I couldn’t think that far ahead and let the subject drop. We continued in a back and forth conversation in which he was constantly buttering me up and saying things like “You are a national treasure but you probably don’t realize it.”  He would ask for my opinion on various things and I would give it to him.

One time, however, he said that he thought teachers are born not made, and wondered what my opinion was on the matter.  I said I disagreed and that I had learned a lot about teaching techniques from articles I’ve read from reliable sources, (and including talks I heard at a researchED conference that I attended). One can always improve one’s teaching if one has the inclination–there is always something to be learned.  He apparently didn’t like this, and I never heard from him again.

I’ve thought about this from time to time because I hear others saying it also.  Teachers are no more born than virtuoso musicians are born, or award winning writers or actors. Aside from the few prodigies who may exist (the mathematician Ramanujan comes to mind) in general it takes hard work and much practice and learning. (Even Ramanujan had to learn how to do proofs for what he felt were obvious statements that needed none.)

But the myth prevails, and there is a sub-culture of teachers who look at teaching as a journey. In their world, teachers are ninjas and superheroes in a world of unicorns They attend ed camps not to learn new things but to reinforce their misguided notions about ineffective practices being effective and to be among those who speak the same group-think. The slightest indication of going against the group-think will cast such person to wander in the desert–even those who are national treasures who may not realize who they are.

Clarification and Amplification, Dept.

In my last missive (a pretentious word, I admit, but I dislike the word “post” and I absolutely detest the phrase “smart and thoughtful post”, so please tell people who use such phrase to shut the hell up) I said the following:

“It’s a brave new educational world we live in. I want no part of it, nor any of the damned PD that comes with it either.”

Someone applied an interpretation I didn’t intend and tweeted: “These are the teachers who are retiring in droves.”

What I meant was that I want no part of the cheap rhetoric that passes as educational wisdom. Let me assure my faithful readers and followers that I have no intention of retiring. I will continue to teach despite any allegations, accusations or allusions that I’m doing it all wrong.

Of importance, however, is the fact that there are teachers who are retiring because 1) they can and/or 2) they’ve had it with being told, e.g.,  to not stand at the front of the room and to not teach but facilitate, and many other injurious and ineffective practices that are accepted (and mandated) by the edu-status quo.

Professional Development, Dept.

Teachers are routinely expected to attend professional development (PD) to supposedly help them in their teaching.  I recall one such PD I was forced to attend when I started teaching at my current school.  It was held at the school during the week before school started.  It was called “How to write rock star lesson plans” and it seemed to be all about collaboration, with the general message that writing lesson plans was a waste of time. Teaching should be organic and student-centered.

His ice-breaker was to go around the room asking everyone to name their “super power”. This is typical at PD sessions that seem to abound in references to unicorns, super heroes, Ninjas, rock stars, and other like-minded crap as if teachers are a special breed who must be spoken down to. The teachers at the session complied and he always had some witty comeback or conversation talking about the super power they named, except when he came to me and I said that my super power was “card magic”.  This left him speechless probably because I didn’t say anything about unicorns or something clever, so he went on to the next person.

The day proceeded along those lines.  His basic premise was that “constructivism” was the way to go in teaching.  He  started with a quote from Gary Stager, a known constructivist who publishes and gives talks and is generally well known by fellow constructivists.  I forget what the quote was and it doesn’t matter. It told me I really didn’t want to be in this room for 6 hours. (Yes, 6 hours.)

I thought he was on the right track when he talked about how in California schools, students in early grades, when learning about the California Missions, construct a mission out of sugar cubes.  “That only teaches you how to build a mission out of sugar cubes. Does it teach you anything about the history of missions?”  OK, so far.  But then he talked about a better alternative (wait for it):  Minecraft!

Again, I realized I had 6 more hours of this crap.  In retrospect I could have left and no one would have noticed. But I stayed til the bitter end which included him showing two pictures, one of a class where the seats were in rows and the teacher was at the front, and (one was supposed to assume) the students were bored and disengaged. The second picture was one with whiteboards all around the room with students up and about looking at the various math problems, supposedly engaging in meaningful dialogue about each problem, a la Jo Boaler.  “Now in which class do you think the students are more engaged?” he asked.  Reminds me of a question on a true-false test I had in Social Studies in second grade in which the question was “The fireman is my friend.”  How do I know? Depends on the fireman. And in the case of the classrooms, maybe the students in the first class were engaged. And if they weren’t, maybe the teacher wasn’t that great.

Finally, at the end, he had us write a program in some programming language. Of course, he called it “coding”–no one calls it programming anymore.  “Coding” is an important part of education, he said, though what it had to do with writing “rock star” lesson plans is anyone’s guess, other than one doesn’t have to write a lesson plan to have kids code. Just tell them “Find a way to draw a square using this coding program.”  The coding was the typical “Logo-like” program that allows one to draw line segments, skip around, rotate, etc. In short, it teaches nothing about programming other than how to get things to move on the screen to create shapes–the program has already been written.  I didn’t know how to proceed so I asked the teacher next to me how to do it and he showed me–he had worked with the program before.

Similarly the PE teacher a few seats away was expressing frustration and got mad at the PD leader for not providing instruction.  “You can figure it out,” he told her.  She too received instruction from the teacher next to me, and then she proceeded to tell other teachers how to work with it. The moderator was delighted with this and said “Look; a few minutes ago she didn’t know how to write code, and now she’s telling others”. As if this was proof that students learn better from each other than from a teacher.

I related this tale to a hard-fast educationist some months later and she responded: “Well, everyone knows that students learn more from hearing it from a classmate than from a teacher.”

It’s a brave new educational world we live in.  I want no part of it, nor any of the damned PD that comes with it either.