Is This The End of Childhood As We Know It?

I was having lunch with a friend who works in app design (thank gawd his are only intended for adults) when he described an outfit called “Age of Learning” which puts out a very successful app for young children. I shook my head and opined that any parent who used a tablet as a substitute for person-to-person learning experiences with young children should be referred to child protective services.

Realizing that perhaps I was being a little prejudiced, I decided maybe I should at least look at Age of Learning’s flagship product, “ABCMouse” to get an idea what it was all about, and I will wholeheartedly say this: I was wrong.

Is this the latest form of child abuse?

Is this the latest form of child abuse?

My opinion is that any parent that allows their child to spend more than 5 minutes a day on ABCMouse should be locked in a room for a year and spend all their waking hours in front of this program. He should be forced to forget the smell of a magnolia flower when it opens, the texture of new fallen snow, and the feeling of another person’s skin against his own. She should be denied the experience of a spontaneous laughter, the intimacy of looking someone in the eyes, and the warmth of bunny fur. This parent should eat packaged, pre-digested food and be denied any source of natural light.

I don’t know who bought off the American Association of Pediatrics, but I’m in still in the category of people who believe that putting kids in front of screens instead grassy fields or wooden blocks or finger paints is guilty of child abuse. Oh sure, there are parents out there who claim that that their busy lives do not allow them to spend “quality time” with their kids, but it seems to me that putting your kid in front of a screen for more than a few minutes a day is not the answer at all.

In fact, I would say this: if you buy a tablet, keep it for yourself, and give your kid the packaging to play with. I would wager that there are a lot of interesting things that a child could do with a cardboard box and a pair of safety scissors than anything that will take place on the screen.

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Andrew Hacker, The Math Myth and the Economics of Book Publishing

I’m no fan of how the New York Times covers mathematics education, but every once in  while they find someone who has an interesting voice and even an argument that stands up to even the most minimal scrutiny. Andrew Hacker is back for a repeat performance at the Times, which is okay by my. I enjoyed his previous op-ed about algebra being necessary, and now he’s managed to roll this premise into a full length book. It only took 4 years (his original screed debuted almost 4 years ago), but I guess these things take time. Think of all the trees that had to be grown, pulped, bleached and then inked to get this thing into print!

Even though his book looks like an interesting read, I’m probably going to wait a few months before I pick up a copy on Amazon Marketplace, where it will be available for half the hard cover price of $17.95 (but where it will always sell for $14.95 on the Kindle, for some odd reason…)  Here’s an interesting problem that Hacker could work on: how will the price of his hardcover book decrease over the next few months compared to the ebook price. Here’s my prediction:

Screen Shot 2016-02-29 at 12.48.05 PM

Or better yet, I’ll just download an e-copy from the library and read it for free!


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Learning About Sex Through Porn: The Stock Market Game as Simulation

My 17 year old son goes to a pretty good high school, and for the most part, I have not interfered with the more nefarious activities so far as curriculum goes. It hasn’t been that easy: a few years back I informed my son’s 6th grade math teacher that he was a “total disgrace” based on the math my son was doing in his class, but other than that, I’ve tried to offer honey whenever I could. NYC public school teachers are a resistant lot (I think it’s written into their contracts) and now that both my kids are pretty much finished with the public school system, I have fewer dogs in the fight. I have many bones to pick with the educational-industrial complex, but I try to keep my kids out of the fray.

But this doesn’t stop me from calling out some of the more sacred cows in education, and in this post, I’m going to take aim at something called The Stock Market Game. I have no doubt that the people who invented and run these kinds of investigations have good intentions, but if you know anything about how the stock market works, you would understand that teaching children about finances by investing in the stock market is like teaching sex education by watching  pornography.

Yes, you read that correctly.

While the details of each version of the stock market game are different, the concept is essentially the same: individuals or groups of students are given a cash stake that only an heir to a successful real estate mogul would have access to, invest the money in a portfolio of stocks, and then follow how their “investment” grows or shrinks during the ensuing weeks or months.

Let’s begin with some facts: 22% of all children in the United States live in poverty. It is almost certain that their parents have not and will not ever have any involvement in the stock market, and since most people end up owning stocks through inheritance, the chances that these children will go on to own stocks is pretty small. For many children, playing the stock market game will end up teaching children about an aspect of our economic system that they will never be able to participate in.

Similarly, pornography promotes the same misunderstanding of human sexuality. It presumes that everyone is well endowed (both men and women) and therefore they can engage in this kind of activity. The chances that anybody will engage in sexual behavior of this type if pure fantasy. At best, viewers of pornography should understand that these highly sculpted bodies are not “real,” just as students engage in the stock market game understand it is far, far different from the reality of investing in stocks.

Second, the stock market game promotes all the wrong behaviors when it comes to personal finance. In the game, students are encouraged to invest a huge amount of money all at once, and then buy and sell without the consequences of paying fees or taxes. This promotes “short term” investment behaviors, which time and time again has been shown to be a losing strategy. As any successful investor knows, the optimal way to invest is through slow accumulation, continual reinvestment and holding stocks for the long term.

Similarly, pornography promotes the same kinds of behaviors when it comes to human relationships. The selling point of pornography is to de-emphasize long term intimacy and promote short term gains. In a pornographic film, “success” is defined by obtaining satisfaction from short term encounters, without the consequences of alienation, not to mention the risk of sexually transmitted diseases.

Finally, playing the stock market game gives a completely inaccurate representation of the career opportunities in the financial industry. Anybody who works in investing and finance knows that most stock picking is not done by individual investors, but by highly complex computer algorithms that make decisions and carry out these decisions in a billionth of a second. This kind of behavior leads companies to cheat their investors as well as their customers, all in the interest of supporting their stock prices.

Pornography suffers from exactly the same delusion when applied to sexuality. It proposes that anybody could engage in this kind of life of hedonism, without suffering from any of its consequences. The reality is that pornographic films are highly scripted, the actors are most likely exhausted and poorly paid, and becoming a highly paid “star” is very, very rare. While most viewers probably know that pornography is staged, the reality of working in the adult film industry may not be apparent to everyone.

Okay, so I’m not a fan of the stock market game; am I suggesting that we toss out the whole idea of economic education in schools? No way!  In fact, my daughter took a “personal finance” class during her senior year in school and instead of wasting their time on the stock market game, they actually learned about how to manage the money that they may actually have. This included things like learning about bank fees, tax rates, investing for retirement, credit card interest rates and understanding compound interest and mortgages. That would be a financial education that would be a bazillion times more effective than playing the stock market game.

Unless, of course, we begin to believe our students should be making adult films?

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I’ve got a bone to pick with Jo Boaler…..

Readers of this blog, of which I am certain there are few, will know that I am not an easy mark and that nothing delights me more than taking the sails out of the windbags who pronounce themselves “experts.” At the same time, I do respect those who have dedicated themselves to using actual evidence to advocate for certain policies. So it saddens me to poke huge holes in Jo Boaler’s defense of the Common Core as described in her opinion piece for the Hechinger Report.

This is definitely not Boaler’s best work, and after digesting it, I have reached the conclusion that it was either penned by someone who wants to discredit Boaler’s work, or that Jo herself just wasn’t up to her best form that day (which I imagine happens occasionally….)

Let me just say this: Jo Boaler makes several statements that are majorly incorrect. I’m sorry, but it’s true: they’re wrong. Not that there’s anything wrong with that. And it’s not like I’m trying to start an argument, because, well, it’s not like I’m brilliant or anything. In fact, whenever anybody admits that I am correct, I write it off to the fact that even a broken clock tells the correct time twice a day.

So it’s not that I am right; rather, it is just that many of the statements that Jo Boaler makes are wrong.

The piece, “Memorizers are the lowest achievers and other Common Core math surprises”, which was published last May (I’m sorry, but I’m a little slow to catch up on these things,) appears to make some interesting points, but truth be told, many of Boaler’s assertions lack validity and some of her statements are just illogical. While some of my points may appear to be nitpicky, others are a lot more serious, which makes this all the more difficult to write.

Let’s begin with the opening paragraph: I have encountered a lot of propaganda advocating for the Common Core, but none of it has ever referenced connections between its development and brain or learning sciences; in fact, not a single member of the Common Core working group has any experience with learning or brain sciences. If the authors of the Common Core did incorporate these findings, it must have been imaginary, because as someone who has spent a lot of time investigating educational research and incorporating it into my teaching practice, it appears to me that the CCSS was written without any regard for contemporary findings in educational, psychological, cognitive or neurological research.

CCSS and Educational Research: Dubious Connections?

CCSS and Educational Research: Dubious Connections?

In fact, for many experts, the CCSS are actually contradictory to what neuroscience and learning sciences have to say is “good practice.” According to a joint statement by members of the National Association for the Education of Young Children, the K – 3 standards, which form the bedrock of mathematics education, are wildly out of touch with our understanding of how the young brain learns and understands mathematics. The NAEC statement opposing the CCSS for K – 3 learners was signed by none other than the three past presidents of that organization, as well as noted scholars like Howard Gardner of Harvard University. Jo Boaler, I respect you, but that’s massive opposition to the foundational years of mathematics as envisioned by the authors of the Common Core.

I agree with Boaler (and who wouldn’t, in this day and age?) that children learn math in many different ways, but to say that “mathematics classes of the past decade have valued one type of math learner, one who can memorize well and calculate fast,” is just plain incorrect. I’ve been working in mathematics education for over 30 years, and I can assure Dr. Boaler that this has been an issue since I began my career way back when. In fact, I remember that memorizating and calculation were part of my “mathematics” education growing up in the 60’s, which takes us back a half-century. Stating that this is a problem that just popped up during the last 10 years is more than dismissive, it is pure fantasy.

Moving along to the next paragraph, Boaler cites statistics from the PISA exams as evidence that the lowest achieving students approach math through memorization of steps. First of all, PISA is a terrible source of data from which to draw any kind of interenational comparisons, as it does not take into account references to issues like poverty and educational culture. Should we really be comparing the scores of students in Singapore, a high-performing city-state that has fewer school aged children than the NYC school system, a poverty rate that is 1/3 of the United States and boasts a culture where 97% of all students attend out-of-school programs to boost their academic achievement? I have issues with using the PISA study to reach any conclusions, including the current infatuation with something called “Singapore Math.”

Furthermore, it is not clear which part of the PISA results Boaler used to fabricate her assertion that students who rely on memorization are also the lowest achievers. Her link references the entire PISA report, which includes dozens of pages of Excel worksheets; it is not at all clear which sets of numbers support her conclusion, and whether they are statistically valid or reliable. I would like her to elaborate on which data points she used to reach this very important conclusion (which I hope is true, by the way…..)

Additionally, while there may be a correlation between “memorizers” and low achievement, Dr. Boaler, who I’m sure is quite conversant in statistics, would be the first to admit that this doesn’t necessarily indicate causation. That is, it could be the case that low achieving students struggle with poor memory, rather than memory driven instruction leads to underachieving students.

Finally, as someone who, like me, is suspicious of standardized testing in all its forms, I’m surprised that Boaler herself would rely on PISA scores to support this position. Surely there must be some better data available that proves this case; if not, perhaps we can put some of those Stanford grad students to work on it?

Some of Boaler’s statements are just mathematically illogical. “The past decade has produced a generation of students who are procedurally competent but cannot think their way out of a box” makes no sense – how could ten years of education produce 20 years of students (a.k.a. “a generation”) who are “non thinkers?” With this statement, Boaler is implying that in 2005 some kind of cataclysmic change happened in mathematics education that is only being rectified by the Common Core, which was released in 2010. I was teaching in 2005, and I don’t recall an educational earthquake of the kind Boaler describes. Perhaps I missed it?

Additionally, since the CCSS math standards were released in 2010, and  if we assume it took 2 more years to get them into schools (2012), then the last three years of underachievement (2012 – 2015) which Boaler cites can be blamed on the use of the Standards themselves. Now I know that this is completely wrong, but I think it shows that Boaler’s repeated claim about this issue taking place during the “last decade” is illogical.

Oddly enough, for a piece that trots out the importance of advancement in brain sciences, there is not a lot of factual information about what neuroscience says about learning mathematics. For example, Boaler states that memorizing multiplication facts does not prove that one is “advanced” mathematically, but never states why this is true. Stanislaw Dehaene, in his wonderful book “The Number Sense,” does provide us with an explanation: multiplication facts are stored in the basal ganglia of the brain, which means they are remembered “linguistically.” He goes on to state that this is also the part of the brain which memorizes things like song lyrics, nursery rhymes and prayers. Thus, Boaler is correct when she states that recalling multiplication facts is not in and of itself a mathematical process, but used no evidence to back this up.

At the same time, Boaler seems to imply that there is no need for children to memorize multiplication facts. I have to disagree, and I think most classroom teachers who teach constructivist mathematics would take a similar position: there is nothing more frustrating than watching a 5th grader attempt to analyze the pattern 1, 3, 6, 10, 15 and demanding a calculator to find the difference between the different numbers. Just as we acquire “sight words” so that we don’t have to sound out “or” and “and” every time it comes up in a sentence, we need a certain amount of “sight computation” to reduce the cognitive load to do other kinds of mathematical activities.

I’d like to go on, but I think the evidence I’ve presented above more than makes my case. I invite Dr. Boaler to dispute and/or clarify what she wrote in her piece, because I am, after all, one of her earliest supporters and biggest fans (she even autographed a copy of her book “What’s Math Got To Do With It” for me at an NCTM conference, which I treasure.)  I hope she’ll read this with an eye towards better supporting a set of policies with which I, for the most part, agree.

A disclaimer: I am not aligned with nor am I sympathetic to the views of R. James Milgram and Wayne Bishop, who I believe are the troglodytes of the math education world. I am concerned that Boaler, who I know and admire as a tireless advocate for what “good” math education looks like, is discrediting herself by shilling for the Common Core, and supporting it with information that is both inaccurate or poorly sourced. I hope she’ll retract this misleading piece of writing and replace it with something that is more representative of her mostly excellent work.

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The Death of Success Academy Charter Schools – Why 2016 Was Pivotal

January 1, 2020

With a few tears, the last Success Academy school (SA) closed its remaining campus today. The “SA Trump Institute,” which was formed after the stunning defeat of Donald Trump in the 2016 Presidential Election, was seen as a last ditch effort to revive the struggling charter school brand after numerous setbacks, including the loss of much of its ancillary funding, the opening up of the job market for English majors, and the rejection by parents and students alike for a curriculum that relied on test scores instead of cognitive development and character building.  The untimely retirement of its founder and CEO, Eva Moskowitz, as well as the indictment and incarceration of poster boy, Andrew Cuomo, did nothing to enhance the brand.

Screenshot 2016-01-11 11.59.08

It is widely agreed that the first blow to SA, and the charter school movement as a whole, came with the election of President Bernie Sanders in 2016 and the ensuing 2018 mid-terms which led to the “Sanders’ Minions” taking control of both the House and Senate. With the victory of his “Progressive Power Party,” Sanders undertook economic initiatives which ultimately drained the seemingly endless supply of charter school cash.

Sanders victory led to the “Great Economic Restructuring Act” of 2019, when Janet Yellen was replaced by Ralph Nader as Head of the Federal Reserve. Nader’s first move was to change his title to “Head of Economic Fairness,” and initiated a set of tax reforms which led to the demise of hedge funds, which had been an important source of financial support for SA.  In response, SA founder and CEO Eva Moskowitz was forced to sell off the company’s cappuccino machines and lay off hundreds of teachers whose only educational experience was training baristas at the now-defunct Starbucks coffee chain.

Another blow to SA was the influence of President Sanders’ grandchildren, who exhorted young people to “go beyond cool, teach in a real school!” Heeding their call, throngs of college graduates turned their backs on Teach For America, depriving SA of “fresh meat” to staff its classrooms. When this low-cost source of labor dried up, SA founder and CEO Eva Moskowitz was forced to recruit gullible young technical school graduates from Eastern Europe who were housed in floating dormitories on the East River and prepped for classes with a 3 week intensive English grammar course which focused solely on proper use of the imperative form.

Meanwhile, SA parents grew restless when it was revealed that their children, so focused on preparing for standardized tests, were unable to answer simple questions until they were put into “multiple choice” formats. Reports of parents spending thousands of dollars on “spontaneity institutes,” where their children learned such essential skills as “free thought” and “speaking truth to power,” served to tarnish SA’s luster even further.

The final blow to the SA “mystique” came when it was revealed that SA University, a for-profit secondary institution set up specifically to admit SA graduates who were unprepared for advanced academic study, was using their students to sew backpacks and school uniforms for SA elementary schools, while awarding academic degrees in such majors as Statistics Manipulation, Political Showmanship and Greed Development.

Opening in place of the former SA Trump Institute will be the Michelle Obama School for Nutritional Sciences. Eva Moskowitz, who handed over the reins of SA after a massive e. coli outbreak at Success Academy’s Fast Food Conservatory in 2018, was unavailable for comment. Her patron, the former Governor Andrew Cuomo, now serving time for educational fraud at the newly-opened Alcatraz Federal Prison for Mendacious Political Operatives, refused a request for an interview.

Posted in charter schools, educational malpractice, The New York Times | 1 Comment

You mean I CAN’T improve my brain playing video games????

This just in:

““Brain training” company Lumosity just settled deceptive-advertising charges with the Federal Trade Commission for $2 million. The company will have to notify its customers about the settlement and allow them to easily remove any auto-billing they have on their accounts.”

Well, duh?

I’ve been listening to Lumosity’s commercials on the local public radio stations, and what a better place to advertise but to neurotic white, liberal, New Yorkers, whose greatest fear is losing their mind and what? thinking that Ted Cruz and Donald Trump actually make an iota of sense.

My question: where did Lumosity find the neuroscientists who actually agreed to lend their talents to this nefarious venture?

Just like Bernie Madoff’s regular 8.1% annual investment returns were “too good to be true,” so I was suspicious of Lumosity’s claims to have developed a “brain training program,” that would stave off dementia and Alzheimer’s.

Despite the fact that they paid a huge fine for false advertising, Airborne products are alive and well. I wonder which clinic “proved” they are effective?

I doubt this is going to stop Lumosity from marketing their quackery in the future, just as Airborne payed a $23 million suit over false advertising for convincing gullible buyers that their products don’t actually boost the immune system or prevent colds from happening (Airborne is still alive and well, peddling their remedies in stores all over the country.)

I do workshops on neuroscience and numeracy and unlike the mendacious hucksters who run shops like Lumosity, I actually base my work and suggestions on ACTUAL RESEARCH! This means that I don’t endorse any “do this and it’ll raise test scores overnight” dictates; instead, I make recommendations that will help teachers understand why their students struggle with things like learning the multiplication tables, or why certain language is more conducive to understanding mathematics than others.

Unfortunately, it is companies like Lumosity that puts the “real” findings in neuroscience in a bad light. This is a very young field of study, and we have to be careful about what we accept as “truth” when looking at what is being marketed at us. I figured eventually “Lumosity” would get caught eventually, I just wonder why I had to listen to hundreds of commercials before anybody caught on….


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Why Singapore Math Will Not Put The US At The Top

As mentioned in my previous post, Robert subscribes to a homegrown philosophy of “good” math teaching that relies on some simple principles. His first one, The Vidal Sassoon Principle, focuses on making teachers look good through encouragement and “having their back” if something they try doesn’t go as planned. Today we’ll look at another part of Robert’s coaching philosophy, which he calls “The Sy Syms Principle.”

What did this schmata salesman have to say about math coaching?

Sy Syms, if you were growing up in the New York area in the 70’s, owned a chain of men’s clothing stores called, yes, “Syms.” While Robert never had the opportunity to shop at Syms (he was more of a “Gap” sort of guy, before they went all khaki), he never forgot the tagline on Sy’s commercials: “An educated consumer is our best customer.” This can be reconfigured in several ways to fit what happens in the classroom, including “a well informed educator is our best teacher…..”

So what does this have to do with Singapore Math? First, many schools adopt this curriculum, or its bastardized variants, with the belief that if they slavishly “teach the book,” including the “bar model,” their students will generate the same math scores that puts Singapore at the top of all international comparisons. Sure, this is an admirable goal, but any school that adopts this curriculum based solely on international scores is going to be very, very disappointed in the results.

What the sales people for these curricula are loath to reveal is that Singapore is a very, very different country than the USA, and that unless you are prepared to adopt every single feature of that country’s educational system, you’re not going to get anywhere near the results found in their math program.

Behind the numbers on Singapore's math "achievement."

Behind the numbers on Singapore’s math “achievement.”

There are many elements that skew math scores in favor of Singapore, which includes year round education (students in Singapore go to school in July and August, believe it or not….) and a very low child poverty rate (8.2% versus 22% in the United States), both of which alone would account for much of the disparity, setting aside curriculum.

But there are two nefarious features of education in Singapore that we might want to consider before admiring it as an educational leader. Unlike the US, Singapore only provides limited educational services for students with physical or intellectual impairments, and exempts them from testing (which includes international comparisons.)

Q: What do you think these students in Singapore are doing after school and on weekends? A: Going to more math classes!

But the chief reason for Singapore’s educational success can be placed on the omnipresence of Singapore’s “cram schools,” which are attended by 97% of the school-aged population. These educational centers, which have been termed the “tuition industrial complex,” are highly profitable enterprises which students attend in the evening and on weekend. In fact, if you add up the low rate of child poverty in Singapore, the fact that schooling is year round (and omits those with disabilities) and the influence of “cram schools,” the actual effect of using the Singapore Math is diminished even further.

So what does this all have to do with Sy Syms? As Robert is fond of saying, “an informed math teacher is an effective math teacher,” which means that the next time a teacher is asked why the school does not use Singapore Math, he/she has an answer that is both wise and based on actual information, instead of random test numbers.

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You Are NOT Katherine Gibbs and Math Class is NOT Secretarial School….

I stop in to see K, our sixth grade teacher, who is twelve kinds of awesome and always quick with a witty phrase (it was from her where I learned to say “for all shits and giggles….”) K looked tense. I open with a joke about , but she isn’t having any of it.

She plops open her CMP III tome, and points to the chapter on dividing fractions. I’m thrilled, K not so much.

“I thought I would start by writing the definitions of dividend, divisor and quotient on the board and having them copy it down.”




division vocabulary

A pretty good “do now” to review division vocabulary

I pull out a sheet of scrap paper and scribble down the “do now” shown on the left, explaining that her class learned these words in 3rd, 4th and 5th grade, and she really doesn’t need to use more time “defining” these words.In fact, she could probably lead a pretty lively discussion asking the students to define these terms for themselves.

Which gets me to one of my major issues in teaching: it is 2015 and the era of copying off the board is over. Really over. We should not be writing down definitions on the board and asking kids to copy them down. Repeat: WE ARE NOT RUNNING A SECRETARIAL SCHOOL!

please don't make your students copy these definitions off the board

What not to put on the board…..

This is not to say that definitions have no place from the math class, but the danger is this: if you write down definitions on the board and ask kids to copy them down, you are wasting time on a low level skill, time that could be used to do something much more interesting, like discussing what is the meaning of a “dividend” or what would happen to the quotient if the divisor was larger than the dividend. Here’s something I did with my 5th graders a few weeks back:

division brain teaser

One way to wake up a boring division lesson….

As they say, “you wouldn’t believe what happened next!” A few impulsive students (all boys, as usual) rolled their eyes and called out “35! 35!,” which is exactly what I wanted to happen. However, among all these students who had only a superficial understanding of division, one girl shook her head. “It’s a stupid question!” she explained, “any of those numbers can be a dividend. You could write 7 ÷ 35, 5 ÷ 7, and 35 ÷ 5. The size of the numbers means nothing, so any of the three could be a dividend!”

All of which led into a very interesting about what the “function” of a dividend is, which led to even more insights into the relationship between divisors and quotients.

Later in the day I saw “K” getting lunch. “So, how did it go?” I asked. “Oh, it wasn’t the worst thing I ever did….” she replied.

Wise educators who liked the post above, also put this in their shopping cart:

Division: You're Teaching It Wrong!

Read this and learn 12 more ways to get division right in your classroom!

Posted in Computation, educational malpractice, Language | Tagged , | Leave a comment

A Visit from the Language Police: Diamonds vs. Rhombi

I’m always amused when teachers try to censor correct children’s language, especially when it comes to mathematics. I remember observing a kindergarten teacher working with a child on the names of the pattern block shapes, and the child correctly identifying the orange square and the green triangle without hesitation. When the blue “rhombus” showed up, things took a turn for the worse. The child looked it over and said, “oh, that’s a diamond!” and the teacher said, “no, it’s a rrrr…….” trying to prompt the young man to say the word “rhombus,” which I’m probably going to bet he never heard in his life. The child looked confused and said, “riamond?” The teacher shook her head, and explained, “no, it’s a rhombus. Can you say that?”

Screen Shot 2015-03-09 at 11.01.34 AM

How many rhombi do you see in this picture?

I know that as teachers we like to fulfill our missions by trying to find that “teachable moment” when we can introduce a new word or idea to our students, but as my college art professor, the great Walter Feldman once said, “it’s not what you show, but also what you don’t show that matters.”  In this case, the teacher most likely created a misunderstanding that will stay with the child for many years to come.

The word “rhombus” is a very complicated concept (and yes, nouns can be concepts) and for many years I’ve asked teachers not to introduce this particular term until 4th grade. This may seem like “dumbing down” the curriculum by not introducing a “fancy” word early and often, but in reality, it makes great sense, especially when you consider the development of logical thinking in children.

A rhombus is an example of a shape that has a particular set of characteristics that is not exclusive. A rhombus is a simple closed curve, a polygon, a quadrilateral and a parallelogram. It can be a rectangle and it can be a square. When it is a square, it becomes a type of rectangle, but when it isn’t a square, it remains a “rhombus,” which is not to be confused with a “rhomboid,” which is a parallelogram where the adjacent sides are not equal and where the angles are not right angles.

What we have here is a failure to communicate…

If this is confusing you, then imagine what it must be to a child. Basically, our language for geometric shapes is lacking in that we don’t have exact words for the shapes that include some properties but lack others. For example, a parallelogram describes all quadrilaterals that have 4 sides, where the opposite sides are congruent and parallel. We then have a word for the parallelogram that has 4 right angles: a rectangle. What we don’t have is a name for the parallelogram that does not have 4 right angles. We call it a “parallelogram,” but if we do, then it would exclude the “rectangle.” The best we can do is explain that a “rectangle” is simply a special case of the parallelogram, and go on to admit that there is no word for the parallelogram that is not a rectangle.

Things become considerably more difficult when we discuss the rhombus. A rhombus is a type of parallelogram, for it also has 2 sets of parallel sides which are also congruent. However, a rhombus is another special case of a parallelogram, for it occurs when all 4 sides are equal. Simple, but not so simple….

This is where that nasty shape, “the square,” shows up at the intersection of two different ways to classify shapes: it is linked to the rhombus by having all 4 sides equal, but it is also linked to the rectangle by having 4 right angles. This means it lies at the intersection of two different schemes for classifying quadrilaterals: one that restricts it by the angles, and another that restricts it by the length of its sides. ARGHHHHH!

An example of a statement and its converse.

An example of a statement and its converse.

All of which brings us to these problems of logical thinking having to do with syllogism and bi-conditional logic. One statement would read like this: “all rhombi are parallelograms,” which is true. Its converse, “all parallelograms are rhombi” is not true, for obvious reasons – a parallelogram could also be a rectangle (which could also be a square) or it could be just a plain old parallelogram without right angles, which we call…. a parallelogram. ARGHHHHHH!

Connecting rhombi to squares creates this statement:  “all squares are rhombi, but not all rhombi are squares.”  And what do we call the rhombus that is not a square? “A rhombus!” ARGHHHHHH!

Suffice it to say, all this is very confusing to adults as well as children (I once spent an hour with a supervisor explaining explaining the statement about rhombi and squares, including why we pluralize “rhombus” to “rhombi” instead of the must easier to remember “rhombuses.”) All of which is to say is this: what’s the hurry with introducing the word “rhombus” to children? Why not let them call it a diamond, which is an actual mathematical term? The reasoning behind what makes a rhombus a rhombus is very complicated and highly specific and completely inappropriate for a young child. Think of how much harder it will be if the only rhombus a child has encountered is the blue pattern block? Sure, you can give a long-winded explanation of how the English geometric vocabulary is very complicated and illogical, but why bother? At this age, shouldn’t children be solving puzzles and moving shapes around, instead of learning complicated and irrelevant vocabulary?

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Using “Key Words” to Solve Math Problems: Lame x Lame

I work with a student who attends a school for children with special needs. He’s a very nice kid who is very eager to do well in math, even though it presents many challenges to him. His parents decided to meet with the math teacher with whom he would be working this year, and she mentioned that the students would be doing a unit on problem solving focusing on the “key words” approach to answering word problems.

I was shocked. No, strike that: shocked. I was astonished! I haven’t heard of anybody using this approach for as long as I’ve been working with teachers (which goes back to the previous century), and I had thought that after it had been died a complete death after being thoroughly discredited. How was it possible that this approach had risen to mis-educate a  new generation of students?

One of the “keyword” anchor charts that lurk on Pinterest

I’ll show with just how bad an idea it is to teach problem solving with keywords by using a single example: I had 5 apples in my basket on Monday. On Tuesday I increased the amount of apples so now I have 7 altogether. How many apples did I add on Tuesday?

If the student had been the victim of a teacher who used the “key word” approach, then by following these directions, he would have been absolutely correct to add 5 and 7 to get 12 apples. After all, the “key word” altogether is used in the question, as well as increased and added. The question does not contain any of the subtraction keywords, which includes difference, take away, left, still, minus and take away.

Some teachers might argue that this is a “gotcha” question, but this is not the case. In fact, it is a question that I would hope a 3rd grader who has a grade-level understanding of English would be able to turn into an equation and solve. The “key word” technique is a kind of hunt & peck approach to reading and interpreting word problems, and it results in students performing the wrong operation on anything but the most obvious problems. Am I crazy, or is this a seriously bad way to teach problem solving?

So what should we be doing in the classroom instead of teaching “key words?” The best approach is to do things that actually require thinking, like having the students build models that will help them solve the problem. Mathematicians do this all the time; why not have students? These models could be physical or written, but regardless, they are models and they help students actually “think” about the meaning of a problem.

model2model1Both models to the right can be used to solve the problem described above. The top one uses a “bar model” which is attributed to the Singapore Math program, but was actually developed by W.W. Sawyer over a half-century ago. By comparing the part (the 5 apples I had on Monday) with the whole (the 7 apples I now have on Tuesday), I understand that I am “adding” on to 5 until I get to 7.

The second model does essentially the same thing, but uses manipulatives: the child “acts out” the timeline of the problem by putting 5 beans in the first circle, showing that some apples are being added, and the result is 7.

Please, please don’t download or hang this chart in your classroom.

I don’t know who came up with the idea for using “key words” when teaching children about problem solving. It’s a seriously bad idea that somehow made its into the everyday practice of misguided teachers around everywhere. It substitutes comprehension for shortcuts, and disengages children from the actual practice of what mathematical thinking look like. I can guarantee you that there is not a single economist, biologist, chemist, statistician or anybody working in the field of mathematics who solves a problem using this method. Why would we teach it to our students?



Note: this rant has been brought to you by none other than Robert M. Berkman, proprietor of the SamizdatMath curriculum collective. If you are interested in including visual approaches to problem solving, try out this set of algebra problems which promote algebraic reasoning without the use of nonsensical “key words!”

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