Guesses, Induction and Math is All About Patterns, Dept.


There is a video of Prof. Georg Polya, giving a lecture to a group of students. Polya was a mathematics professor at Stanford, known best perhaps for his seminal work on problem solving called “How To Solve It”. In the 60’s, the Mathematical Association of America filmed a lecture of his on the role that guessing plays in mathematics.

It is worthwhile watching the video (linked to above). Polya is very charming and gives a good lecture.  Unfortunately, the lecture tends to get misinterpreted.

In the past, I have offered criticism of the trend in math ed to give students a type of problem called the “growing tile” problem. Students as young as fifth graders are given these type of problems in which they attempt to find the pattern that describes the sequence of a growing number of tiles or dots. These types of problems are given supposedly to develop the “habit of mind” to see a pattern and then to represent it mathematically.

Such problems are reliant on intuition — i.e., the student must be able to recognize a mathematical pattern using inductive reasoning. There is nothing wrong with inductive reasoning, or guessing what a pattern is. In fact, as Polya points out, it is an essential part of solving problems in mathematics. He poses a problem in his lecture and asks his students to guess what the next number (in this case, the number of spaces created by a specific number of planes). He then, at about 25:50 in the video, asks his students if the guesses they have generated prove anything. The answer from the students, and echoed by Polya, is “No.” He goes on to say “Don’t believe in all your guesses.”  And in fact, the problem he poses in the video does not conform to the pattern of guesses that seems so apparent.

This is the part that sometimes gets left out of the “habit of mind” and “growing tiles” approach that is popular in various math programs (CPM being one of many) and advocated by Jo Boaler and others. While some  problems conform to the inductive conjectures that students produce, other problems such as  Polya’s do not. For example, this problem, taken from Moise-Downs “Geometry” textbook, is from a set of problems to reinforce the idea that intuition often does not coincide with deductive reasoning.

Moise Downs regions problem

In the above problem, although 32 is a natural guess from observing the pattern, it is not correct. There will never be 32 regions. In general, there will be 31 regions formed.  If the six points are equally spaced around the circle there will be 30 regions formed.

Presenting growing tile problems in early grades is more of an IQ test than fostering any kind of mathematical thinking. But more important than that is the unintended habit of mind that springs from such inductive type reasoning. Specifically, students learn the habit of jumping to conclusions based on inductive responses. This develops a habit of mind in which once a person thinks they have the pattern, then there is nothing further to be done. Such thinking becomes a hindrance later when working on more complex problems.

Polya’s lecture has become a kind of poster child for this type of “math is about patterns” type of thinking. Polya says guessing and patterns are important for sure; he devotes a whole lecture to this topic. But he also says you ultimately have to prove your guess.  That’s  the part that doesn’t get played up too much by the “math is all about patterns” crowd.




A State to Watch

Arizona, like other states, is in the process of introducing new math and ELA standards in place of Common Core. Whether this is a “rebranding” of CC, as has happened in other states, remains a question, but this part of the news article provides some hope that it won’t be just a renaming:

“Some of the key changes in math standards are as follows:

“All examples that provided guidance on how a standard should be taught were removed.

“Standards involving calculating money and time were added.

“Standards for Algebra I and II, which used to be in category, were divided to provide additional clarity.”

These changes are more than I’ve seen in the states that have simply rebranded CC. In fact, the removal of examples that provide guidance is a significant step. Although examples are just that and do not have to be followed, the interpretation by many school districts, publishers, and profession development vendors is to take the examples as gospel. Removing the examples will create much more flexibility in how the standards are to be taught.

Also, breaking algebra I and II into separate categories is important. Lumping all of algebra into one composite set of standards has resulted in confusion over what concepts to teach when. Now if they would remove statistical concepts from algebra…well, one can’t hope for too many miracles. But this is a state to watch at any rate.

That Which Passeth for Understanding and Education, Dept.

Nova Scotia is part of a growing trend to teach young children “coding”–a 21st century skill which used to be called “programming”. Programming actually used to be taught in high schools in the 60’s, when FORTRAN was the main programming language, and keypunches were still used. It employed understandings of translating mathematical algorithms into computer language. That is not how this generation of “coding” works.

From the article about this phenomenon:

“In a corner of the Nova Scotia legislature, Grade 6 students Bridget Daly and Hannah Harley ponder how to program small, yellow-and-black robots shaped like bumble bees.

“The 11-year-olds from Rockingham Elementary School in Halifax are setting direction and speed for a race between two “beebots” at Province House — an example of what’s coming to all elementary schools this fall as the government expands computer coding in the curriculum.

“The girls said learning computer code is a welcome challenge, and hands-on technology like the beebots will be a help in solving math and science problems.

” “Instead of having to visualize it and thinking about it, you can use it without having any problems,” said Hannah.”

And therein lies the problem.

I just went through a dreadful PD session as part of teacher prep week at the school where I’m teaching. The moderator (a staunch constructivist, full of condemnation for approaches in which students “regurgitate facts” rather than “construct their own knowledge” and other mischaracterizations of traditional teaching) bragged to us how he taught first graders how to code or program using a computer language, to draw a polygon of any number of sides specified.

Later in the PD, he had us work with “coding” using a language that employed pictorial symbols for commands. He seemed to think that it taught programming because the problems, as it were, amounted to drawing figures and it taught you how to “tell the computer” to move forward a specified distance, then turn a specified number of degrees, move forward again, and you could instruct it to repeat that sequence any number of times. As you progressed through the exercises, the program gave you harder problems and added more commands to your arsenal, including “lifting pen up”, “putting pen down” so you could draw lines that were not continuous with what you were drawing. He said it was well-scaffolded and thought such scaffolding was great. I agree scaffolding is good, but all the scaffolding in the world was not going to teach what computer science really is about under this approach.

The high level pictorial language actually was translating the commands into Java script, so you weren’t really learning the base language, any more than working with a spreadsheet is teaching you the underlying machine code that makes the spreadsheet work. Of course he had us work with the programs by ourselves rather than him offering instruction. That’s the constructivist way. It was OK if we received direct instruction from a fellow student, however. Why receiving instruction from a teacher is somehow “impure” but receiving instruction from a fellow classmate is not is beyond me but so are most of the bad practices that pass for education. And make many PD vendors rich.

Articles I Never Finished Reading, Dept.

“Myla Gupta checked the nest of a leghorn chicken inside a coop she helped build at Terra Linda High School, an example of the project-based learning that is an element of Common Core standards.”


And the students who really do get ahead in STEM majors and careers actually know formulas, how to apply them, and solve problems for which the structure and methods transfer to hosts of other problems.

I got as far as here:

“The standards, which emphasize critical thinking over rote memorization, have met with scathing criticism in some states, often because of the way related testing was done. In California, however, Common Core was supported by 95 percent of 1,000 teachers surveyed in 2015 by EdSource, a journalism website.”

I’m not aware of math standards that emphasize rote memorization. It sounds like this reporter had a buzzword generator and used it until it wore out.

It Isn’t What You Think, Dept.

Yong Zhao, a professor at U of Oregon’s ed school has gone on record saying the PISA exam that has been used to rank nations in math ability, is like a beer drinking contest.

Yong states that though the Asian countries do well on such tests, they do not possess the creativity and entrepreneurship of other countries. Such as the USA.

““We should ignore Pisa entirely,” Professor Zhao said. “I don’t think it is of any value. If you look at the so-called high-scoring areas, like Shanghai and all the East Asian countries, they are trying to get away from what has made them high on Pisa [rankings].”

“The academic, who was educated in China, said that the country’s education system was an effective machine that could instil what the government wanted students to learn, but it did not nurture creativity. The result is that China has a population with similar skills on a narrow spectrum, he claimed.”

I had sent Yong a copy of my book “Confessions of a 21st Century Math Teacher” some time ago.  He responded that he enjoyed the “stories” in the book, but didn’t know what the book’s message was. He would keep reading, though, he said.

I don’t know if he’s kept reading, but my message was simple. The quest for moving away from traditional math was hurting, not helping, students. I would add to that statement that the US has been full of creative people for a very long time. US ingenuity has flourished even when math and other subjects were taught in a traditional manner, and continues to flourish as the education has gone over to bad and unproven practices. Our creativity is NOT a product of our education system.

In short, Yong, PISA does provide information of value. Particularly so, given that the test is based on fuzzy precepts and purports to test students’ ability to solve problems they haven’t seen before. And despite US’s current propensity toward teaching “problem solving” as a skill separate from the knowledge needed to solve the basic problem types, the US does worse than those nations that use the practices that the edu-establishment derides.

But keep reading my book Yong. Maybe something will come to you.

If You Say So, Dept.

Education Week with another article that throws gasoline on the fire of “math has never been taught right” by stating that the math that is taught in high school is too “algebra-based”. This one was written by Michael Schmoker whose bonafides, according to his bio, consist of author, speaker, and consultant. All hail and bow down, please. He also writes for ASCD which is famous for it’s rather reform-math bias.

He supports the ideas of Andrew Hacker. Too much algebra apparently. Reminds me of the scene in Amadeus where the emperor tells Mozart that his latest composition had “too many notes”.

Schmoker confides:

I first encountered the incompatibility between school math and real-world math as a school improvement researcher in the 1990s. In interviews, more than a dozen engineers told me that they rarely used algebra, much less geometry, trigonometry, or calculus. Most of their work required simple algebraic concepts they could learn on the job and arithmetic and statistics for more sophisticated tasks.

I couldn’t help but wonder: How many other professional-preparation programs required either too much or the wrong kind of math for prospective workers? And how many students might have become excellent engineers had it not been for the formidable barrier of required high school and college math courses?

Wow–he polled more than a dozen engineers. So n = 12+. How much more than 12, I wonder. Well, OK, I once spoke with an MRI technician who told me he had to take Algebra 2 in order to qualify for his MRI license, and the course he took had a special application of Fourier series–a kind of ready-made plug-in-the-values type of approach rather than the full blown theory which one gets in calculus. He doesn’t use Fourier series in his work–it’s all done by the computer that puts together the MRI images. So one could say, “Yes, everything is done by computers so how much math do you REALLY need?” Granted, he wasn’t an engineer, he was a technician and in Schmoker’s vision, we could do something similar for engineers rather than force them through the dreaded algebra to calculus sequence.

But I imagine some engineers designed the MRI machinery and some wrote the software for the computers to do what they do, and I also imagine that the programmers needed to know what Fourier series were about and how to work with them. Did Mr. Schmoker talk with those folks, I wonder?

I doubt that one could become an “excellent engineer” without knowing something about differential equations, linear algebra, and a good, fluent, working knowledge of algebra. In my discussions with math professors (which numbers a lot more than a dozen), I hear that every year, there are increasing numbers of incoming freshmen taking calculus who don’t have basic algebra skills and have difficulty with the course because of that.

Yes, I know, the Hacker thesis maintains you don’t need calculus so everything is wonderful–all most people need to know is arithmetic according to the late Lynn Arthur Steen who is mentioned in this article:

[Steen] advocated a more practical quantitative literacy, with the selective use of algebra and the opportunity for students to apply elementary skills such as arithmetic, percentages, and ratios to real-world data in order to understand issues like global warming, the price of gas, or college tuition. He also recognized arithmetic’s important role in the fields of science, engineering, and technology.

OK, then.  In arguendo–as lawyers like to say–if all most people need is arithmetic let’s make sure arithmetic is taught correctly in K-6. If it were, Hacker and company would see more students proficient with percents, fractions and ratios. As it is, the number of students proficient in such skills varies and in some cases (I would even say “many”) depends on what they are taught by parents, tutors or learning centers such as Sylvan, Kumon and Huntington and the like.

This argument that less than 25% of Americans actually use math at work is a red herring. The job of teachers is to prepare students for careers they may choose. Algebra is a key course for future STEM majors and professionals. And last time I checked (full disclosure: I didn’t interview more than a dozen engineers), those in STEM careers needed some fluency in basic algebra. Some of them even use differential equations.

Sustaining the Fantasy, Dept.

Education Week reports on how the Common Core math standards are taken as the “scapegoat” and “bogeyman” for the current state of math education.  It starts with a quote from 1972 by a math professor regarding the 60’s new math:

“When persons with advanced math degrees do not agree upon the answer in a 4th grade math book, something is wrong. … I beg of you, please read the math books children are using. … The extended use of set theory is almost obscene. Set theory is a post-graduate exercise, not suitable for children who can not even multiply yet.”

The quote, the article explains, comes from a new book by Matt Larson (current president of NCTM) in which he talks about math education.

Regarding the quote from the math professor, Larson says this:

“In fact, it’s pulled from a 1972 Washington Post article about New Math, the shift in mathematics instruction that began in the late 1950s. New Math emphasized conceptual understanding over rote memorization—not unlike the common core.”

Ignore for the moment the continued mischaracterization of traditionally taught math (i.e., conceptual understanding vs rote memorization). What is interesting is that it suggests that 60’s New Math and Common Core both have their share of Chicken Littles and that in both cases such complaints were/are unsubstantiated. According to Larson (and others) such complaints come about because people don’t like things that are different than how they learned.

Liana Heiten, the reporter for this article quotes Diane Briars, former NCTM president:

” “Any time a district moves to building more conceptual understanding into their mathematics program, and students are coming home with either homework that looks unfamiliar or … with different computational methods than parents have seen before, there’s always going to be questions,” Diane Briars, who preceeded Larson as NCTM president, told me for a 2014 story on how schools are teaching parents about the common core.”

In fact there were some objectionable aspects to the 60’s New Math  so such complaints were not without substance. The 1972 quote from a math professor, while framed by Larson and Education Week as an exaggeration, was echoed by many people at that time. The math taught in the lower grades during the 60’s New Math era relied on a set-theoretical approach that was too formal for many students, not to mention the teachers. Interestingly, NCTM distanced itself from the 60’s New Math when it fell out of fashion in the mid 70’s, though NCTM’s 1989 (and later, 2000) math standards pushed “understanding” over “procedures” as had the 60’s New Math in part.

As for Common Core math standards, although they call for fluency with math facts, and require learning the standard algorithms, its interpretation and implementation –like the NCTM standards which dominated math education philosophies in the edu-establishment for the past two decades or so–also has been implemented along the ideologies of reform math.

The alternative strategies for adding, subtracting, multiplying and dividing  which are suggested in the CC math standards  have generated many complaints and articles in the press.  But they are nothing new; they have been around for years in the textbooks used in the eras of traditionally taught math. But it used to be that the standard algorithms were taught first, as a main dish. The alternative methods were introduced over a period of a few years as side dishes to help students do mental math, and to help clarify what was going on within the standard algorithms. But interpretations of CC have resulted in delaying the teaching  of these standard algorithms until the grade level in which they appear in the CC standards. The standard algorithm for multidigit addition for example, appears in the CC 4th grade standards.  While this means it is to be learned no later than the fourth grade, popular interpretation and implementation of these standards is to delay its teaching until 4th grade, and to teach only the alternative methods in earlier grades.

Jason Zimba one of the lead writers of the CC math standards has gone on record stating that the standard algorithms can be taught earlier than the grade in which it appears in the CC standards, and Zimba even recommends that that be done.  But reform ideologies have prevailed: The reasons for delay go along with reform-oriented ideas, that teaching the standard algorithms first eclipse the conceptual underpinnings of how they work and students will not “understand”–i.e., it is equated with “rote memorization”.

Furthermore, although the CC website claims that the standards do not dictate pedagogy, it says that “shifts” in instructional strategies are necessary.  It calls for coherence in teaching:

“Mathematics is not a list of disconnected topics, tricks, or mnemonics; it is a coherent body of knowledge made up of interconnected concepts. … Learning is carefully connected across grades so that students can build new understanding onto foundations built in previous years.”

The implication is that this has never been done before. In fact, it might be the case that the textbooks and programs of the last 25+ years, in adhering to NCTM’s math-reform flavored standards did away with such coherence in the lower grades, relying instead on more of a “spiral approach”. But current interpretation is that traditionally taught math was a series of unconnected topics. And perhaps this shift in instructional strategy has led to the teaching of standard algorithms later rather than earlier, in the prevalent belief that doing otherwise leads to “tricks” or “mnemonics” and that the procedures cannot be taught with “understanding”.

Looks like NCTM’s current president is doing his part to sustain that fantasy.

Where Have I Heard This Before, Dept.

This article is mostly about Andrew Hacker’s arguments about math education, but I’m not going to get into that. I want to focus on the usual arguments I see in almost every article about math ed that appears these days.  I would like education writers to do a better job in writing about education in general, and math education in particular, which means asking questions of the people they’re interviewing, rather than taking on faith that traditionally taught math is the culprit. Let’s start with this paragraph:

“Jonathan Farley, professor of mathematics at Morgan State University, agrees that standard ways of teaching the subject can miss the mark. “Math is often times taught in a way that makes it appear useless,” he says. Many instructors teach a concept and give practice problems to be worked ad nauseam without ever connecting to the practical application of said concept. If we were to put this into taxonomy terms, math is oft taught to only the knowledge and comprehension level.”

Depends what grade level you’re talking about.  Dr. Farley doesn’t identify those levels but talks in the typical vague style that most critics of math education do. The practical application of some aspects of math isn’t always seen unless you go into engineering or the sciences later, but it does provide the preparation for doing so.  Requiring ALL students to take math up through calculus is another issue but I don’t think that’s what he’s talking about here.

” ‘While the nature of the subject seems to be application based, it’s never fully fleshed out as such in a practical way. The problem is not the content,’ Farley says, ‘but in the inability to reveal the beauty of math. You don’t study math merely to learn how to build bridges; you study mathematics because it is the poetry of the universe.’ “

Uh, I’m having trouble here. He says math should be taught to see how one applies it in a practical way. Then he says practical ways are immaterial since the real reason to study math is because of its poetry. Well, if he thinks math should be taught for math’s sake, then why is he making such a fuss about application?

The article goes on:

“While experts tend to disagree on the utility of abstract math, all seem to agree that the current way it’s taught doesn’t work.”

This sentence seems lifted from the “Big Book of News Stories About Math Education” that most education writers use.  I run across it in many news stories.  Is the writer talking about K-8? Because the way it’s been taught in lower grades the past 25+ years has been in a state of deterioration. I have made this point repeatedly and met with the same criticism of that statement: that constructivist/inquiry-based and student-centered approaches aren’t all that prevalent. Well, they are a LOT more prevalent than they were in previous eras when students entering algebra 1 in high school actually knew their math facts and how to operate with fractions, decimals and percents.

But if the statement is talking about high school math, I would say that algebra 1 and 2 and geometry have been watered down over the years. There is a dearth of good word problems; such problems (termed “real world applications”) are generally tedious and not very challenging. The word problems of the past (distance/rate, mixture, work, number, coin, etc) did provide structure for problem solving that the current cast of problems does not.

So if the current way of teaching doesn’t work, it isn’t because they’re teaching it like they did in previous eras when it actually did.

8 reasons to ditch traditional teaching methods

This from Greg Ashman’s “Filling the Pail”

Filling the pail

I advocate explicit instruction. Explicit instruction takes the traditional or default approach to teaching and modifies it to make it even more explicit and highly interactive.

This method has its origins in research from the 1960s and 1970s into the behaviours of the most effective teachers and it has been verified since then across a range of different study designs and subjects. You can read more here.

Yet you won’t hear much about explicit teaching if you wander into a education school seminar or a professional development workshop. You won’t read much about it on popular websites for teachers. Instead, you are likely to be encouraged to adopt an implicit or ‘child-centered’ approach. These come in many guises but the common ingredient is that the teacher takes a step back and the students are expected to make some key decisions or figure out some of the concepts for themselves.


View original post 1,315 more words