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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No, you can’t “hate math” (even if you say you do)

keep clamI have an online “colleague” who makes no bones about the fact that she hates math. She’s expressed this opinion in numerous message threads on a community board to which we both post. She works in science education, and is about the most civil and respected voice as one is likely to encounter on these kinds of open forums. If there is one vice this person possesses (and I’m quite sure it is only this), it is that she continually professes, quite seriously and earnestly, that she hates math.

I don’t think she’s telling the truth. I believe it is impossible to “hate” math. Saying that you “hate math” is the equivalent of saying “I hate music,” or “I hate food” or “I hate animals.” Okay, everyone dislikes a certain style of music (those 12th century Gregorian Chants are not my favorites, truth be told) and it is possible to have negative reactions to things like okra when it is slimy instead of crispy, and yes, nobody likes lice, but really, a generalized statement declaring a hatred of math is just not possible.

Mathematics is an incredibly diverse field of study, and it encompasses so many different ways of analyzing and solving problems that a blanket statement like “I hate math….” cannot possibly make any sense. In fact, it is so nonsensical that it would be equivalent to declaring a hatred for thinking and feeling.

I can confidently say this, though: there are times when even those of us who know and enjoy mathematics find it either boring or frustrating or some combination of the two, but we also recognize that this this is not unique to mathematics. Whether you are conducting a scientific experiment or producing a blockbuster movie, there will always be extended periods of boredom and frustration. There’s nothing wrong with this, and I can’t imagine that anybody would discredit an entire activity based on this pervasive reality.

You couldn’t possibly hate okra when it’s prepared like this.

So here’s my take: it’s not that my friend “hates math.” She only thinks she hates math. The journey to loving math could begin with something as simple as modifying her blanket contempt for mathematics to something as simple as “I hate math when….” After all, you can hate okra because it’s slimy, but when it’s flash fried in a cornmeal crust, well, that’s an entirely different matter….

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Why Can’t Our Students Conduct Mathematical Research?

library

The Better Living Through Mathematics Library

Every Thursday morning for the past 5 years I’ve been meeting with a 4th grade class to work on a variety of “puzzlas” to stretch their mathematical thinking. I pull these puzzles from a variety of sources, which is not hard as I have a library of math materials that I’ve been collecting for over 30 years. Truthfully, I don’t know if I’ve had an original mathematical thought in my life, as my library provides more than enough inspiration to cover me for several lifetimes.

Every once in a while, an interesting puzzle comes across my desk and this puzzle inevitably leads me to start looking at the problem in a bigger way. That is, the specific puzzle leads to a much bigger question. Such was the case of the  problem below:

Screen Shot 2015-01-14 at 9.54.33 AM

On the face of it, the problem has a single unintuitive answer which requires constructing a set of floor plans for 4 apartments that no one in their right mind would want to live in. However, it does beg a bigger question: suppose we got rid of the kitchens and bathrooms, and just had to divide up the square into 4 equal sections of 9 tiles each where each shape was congruent?

I started with the simplest solution possible: 4 squares of 9 squares each. I then started by moving one tile over from each of the squares:

An approach to solving the 4 way split problem

Moving a single tile, as is done to create tesselations, creates new solutions to this puzzle.

I’ve been working on this puzzle for the past 8 months and have come up with over 40 different solutions, but my fear is that I’ve missed out on some. Here’s what the results of my research looks like:

IMG_2847

….so here’s where you come in, dear teacher: visit my online store and you can download this activity for a (very) small payment! Print it up, try it out with your kids and let’s see if they can come up with variations that I have yet to discover. I’ll publish them here and then submit the results to some mathematics journals to see if we can get it published. Let’s get kids involved in making mathematical discoveries and show them that far from being a dead subject where everything is known, mathematics is alive and quite well.

 

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Dear Amazon: 29 and 40 are not the same….

Everybody loves a story about children and their difficulty with numbers, but who knew it could filter all the way up to the higher echelons of commerce? I’m pointing my finger at you, Amazon! Sure, you can deliver anything the next day, but would it kill you to remind your workers that there is a big difference between something that measures 29″ long and the same item measuring 40″ long? If my calculations are correct, that would be 11″ (which is 11/12 of a foot, if you prefer to see numbers as fractions.)

I recently moved into a new apartment which is on the small size, so paying attention to scale is very important to keeping the place appearing cramped. Hence, a 29″ “soundbar” would fit nicely under my television, while also handling the input from a turntable on the shelf below. I ordered from Amazon and waited for it to be delivered a few days later.

Arriving home today I see a package that most certainly is not 29″, 32″, or even 36″ long: it is looooong. In fact, I knew immediately something was wrong, because Amazon is usually pretty good about packing things in the most efficient way possible: I remember the old days when I would get a huge box holding a tiny item like a paring knife, which made me pretty darned irritated, especially when I had to collapse the box and store it until it could be taken away.

Online shopping companies have gotten much better at this, and since I’ve been ordering a lot of things to equip my apartment, I’ve been impressed by the efficiency of packaging, all of which makes sense: the less room an item takes up, the more items can be put on a truck or plane for shipping (even though shipping prices are measured by weight rather than volume. Wouldn’t it be better to use a combination of the two?)

So I was surprised when I lugged this box inside and laid it atop the tv table. Just so you know, the table is 38″ long:

box

Hmmm, this does not look promising: the box measured 48″ long, and why would Amazon pack a 29″ item (okay, maybe 32″ with packaging) in a box that is over a foot longer?

Unpacking the box, my suspicions were confirmed: this was not a 29″ long unit, but the 40″ model. Now, I appreciate that Amazon was giving me 38% more soundbar for my money, but it was going to look pretty awkward sticking out both ends of the table.

notright

On the dimension of all issues in anybody’s life, this is definitely a “first world problem,” but it seems to me that Amazon has a reputation for being a pretty buttoned-up operation. In fact, I managed to return the item and order a replacement in far less time than it took me to write this post. But really, aren’t the people at the warehouse given enough information to see that something pulled off the shelf is 11″ longer than the one described on the order form?

The next time someone decries the lack of numeracy amount our children, please remember this story and announce to them that at least your students easily recognize that 29″ and 40″ are not the same measurement. Apparently, this is not well known over at Amazon.

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What’s So Elementary About Kindergarten Math?

It was back during the latter days of the Clinton Administration when I was hired for my first position as an elementary level math coordinator. This came after teaching middle school mathematics for the previous 15 years as well as earning an M.S. in elementary mathematics education. The fact that I had not spent a lot of time working with kindergarten through 4th grade children did not seem to deter the hiring committee, perhaps because there were not a lot of people who had expertise working with young children in mathematics. After all, if you were interested in mathematics, why would you want to work with little kids?

Is this all we can expect from elementary school math?

After a few weeks visiting classrooms and observing children at work, I was quickly disabused of the notion that there was not a lot happening in these classrooms. The concepts the students were working on were very complex and, as I spent a considerable portion of my day working with them one on one, there was a lot more that could be done with them beyond counting and making simple patterns. As I worked alongside teachers, I pointed out ways that their mathematics lessons could be richer, more engaging and challenging, while still reaching its intended audience. What I realized is that my previous 15 years of middle and upper school mathematics teaching was not a liability, but an actual advantage: Not only did I know what mathematics students would encounter a year later, but I had the additional foresight to know what these same children would learn 5 – 10 years down the road.

During this period of revelation, I found myself on a car ride with a colleague who taught in the the school’s high school program. As we were driving along, this teacher candidly stated her opinion of my professional work: “I don’t know why we need a lower school math coordinator; after all, aren’t these kids only learning how to add and subtract?”

I held my breath for a few minutes, and then, releasing it slowly, I suppressed my best Dan Akroyd imitation.  I calmly explained to her that the children at our school would learn more about mathematics between the ages of 5 and 10 than the entirety of their lives thereafter. It reminded me of Hillel’s dictum about studying Torah: whatever is abhorrent to yourself, do not do unto others. Everything else is commentary…. 

Everything this man knows about mathematics he learned in kindergarten….

If you think about it, all of advanced math is essentially an elaboration of what a child will learn in elementary school. Algebra is a generalization of arithmetic, while trigonometry builds upon the concepts of ratio and proportion. If you’ve read Dr. Seuss’s “The Cat in the Hat Comes Back,” where Cats A through Z are half the size of each succeeding iteration, then you will understand that he is describing the concept of a limit, from which we get, yes, calculus. Children’s literature is filled with descriptions of advanced mathematics, whether it is the concept of exponential growth described in “One Grain of Rice,” to the “crypto-arithmetic” problems posed in “More Sideways Arithmetic From The Wayside School.”

At the same time, there was plenty that had to be accomplished during the elementary school years when it came to the essentials of reading and writing numbers, as well as solving simple addition and subtraction problems. This does not mean that I had to hold back on doing in-depth investigations that would demonstrate powerful analytic tools while developing important ideas.

All of which lead me to develop an exploration that I conduct with the kindergarten class at my school, which involves an in-depth look at the children’s game, “Crocodile Dentist.”

Can this children's game be a lesson on data, statistics? and patterns

Can this children’s game be a lesson on data, statistics and patterns?

If you are the owner of this game, you know it mixes both surprise and terror: players take turns pushing down one of the lower teeth on this plastic reptile, until the croc “bites” the player who has selected the “dangerous” tooth. The biting tooth changes location each time the game is played, so the surprise comes because the players never know which tooth will bring down the jaw, or whether it will be the first, second or even the last one selected.

The idea of randomization is a fertile ground for exploring mathematics with young children, who are always concerned with the notion of “fairness.” Do all the teeth have the same opportunity to “bite” or do some come up more often than others? Is there a way to predict which tooth will be more dangerous than another, or is the game entirely arbitrary?

I introduced the game by explaining that as someone who is interested in math, this game made me wonder whether a “strategy” could be developed to figure out if some teeth were “safer” to press than others. How would we find out if this was true? This led the class into the concept of “sampling.” It was not enough to play a few games to determine if certain teeth bit more often than others; we would need to play a lot of games, and keep track of which teeth did the biting.

Results of 10 games played by a group of four kindergarteners.

Results of 10 games played by a group of four kindergarteners.

For the next four math sessions, I took a small group of kindergarteners out to play 10 rounds of Crocodile Dentist and recorded the results as a table on a sheet of chart paper:

As we were recording our results, the students learned about some important techniques used in mathematics: we organized the data we collected into a table, and that table is read both vertically and horizontally. They learned about the different kinds of numbers that could be used to describe the game.  For example, ordinal numbers show the “order” of things, like the first through 10th game played, while cardinal numbers record the number of times a specific tooth bit us (for example, tooth #3 bit us 4 times.) The also learned that a number could be used to locate a position, which is called a “nominal number.” In this case, there were 10 teeth on the game, and each one specified a certain location in the crocodile’s mouth.

The entire kindergarten class helped crunch the data from 40 "crocodile dentist" games.

The entire kindergarten class helped crunch the data from 40 “crocodile dentist” games.

The students also learned how to look for patterns in the data they collected: they saw that the same tooth might bite two, three times or even four times in a row, while others may pop up less often or not at all. At the end of each session, each member of the investigating group had to come up with a statement to describe some aspect of the data table we had constructed.

After working with all four groups, we assembled our findings at a “summit” where the 4 posters were displayed, and the class analyzed the results from 40 different games. Looking at the data, the class uncovered specific patterns that could be used to optimize one’s certainty of escaping a bite. For example, tooth #3 came up consistently in all 4 sessions, to the point where in one set of games, it bit us 4 times out of 10 games played. Tooth #2 was even worse: not only did it show up in all 4 samples, but with one group it bit 4 times in a row, for a total of 6 games altogether. This finding developed the idea of a ratios and how they could be compared: tooth #2 was more dangerous than tooth #3, because it came up 6 times out of 10 compared to 4 times out of 10.

The students also found that certain teeth were “safer” than other teeth. For example, tooth #1 was the losing tooth in only 2 out of the 4 sessions, and in those groups, it only bit once each time. Tooth #8 was an even safer bet: it showed up in one of the 4 investigations, and it only bit us once in all 40 games played.

Finally, we saw that while the data could help us make good choices while playing this game, nothing was certain: after discussing our findings, we played a round of Crocodile Dentist using our findings. We avoided teeth like #2 and #3, choosing teeth #10 and #9, which didn’t come up as often in the past. Wouldn’t you know it, but the tooth that bit us during that game was #8, a tooth which had been considered “extremely safe.” 

This investigation models what many would consider “good mathematics,” in that it develops powerful ideas in a context that is provocative and relatable. It motivates children to think analytically about something they might otherwise ignore, and, most of all, it’s just a lot of fun. For a kindergartner, “fun’ is the most important attribute to learning about mathematics (and shouldn’t it be for us all?)

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Have charter schools become a “meme”?

I had to wait a few weeks to post this, because the whole situation made my blood boil, but now that the dog days of August are upon us, and my ire has had a chance to mutate to annoyance, I have reached a conclusion: you know something is wrong with the concept of charter schools when they have become “memes.”

Truthfully, I don’t really know what a “meme”  is, except that they seem to have something to do with the internet and are cultivated and disseminated by young people with lots of time on their hands. Since I have spent a lot of time studying concept development, I utilized my “go to” method to help me understand what exactly a “meme” is: I perform a “search” on that word, and then click on “images” so I can view as many examples as I can. By looking at dozens of examples, I hoped to build a mental prototype of “meme.”

Using this method, I have determined that a “meme” is when you take a photo, place some snarky phrase over it in white using the font “Impact,” and then post it on Facebook, Pinterest or Tumblr.

But this didn’t account for the great interest in memes and their widespread dissemination. I decided I had to do more to investigate, so I searched for definitions. Here’s what I found:

  1. A meme is an element of a culture or system of behavior that may be considered to be passed from one individual to another by nongenetic means, especially imitation.
  2. A meme is a humorous image, video, piece of text, etc. that is copied (often with slight variations) and spread rapidly by Internet users.

Okay, this is beginning to make more sense, especially if I regard a “meme” according to the first sense. If this be the definition, I have reached the following conclusion: charter schools have become “memes.”

Which brings me to my story: I was at my school’s end-of-the-year shindig, which is usually sponsored by our parents’ association and takes place at a somewhat mid-scale but convivial eatery which will supply us with copious amounts of alcohol and fatty food to reward ourselves for a year well done.  There were about 30 of us in attendance, and we spanned a wide range of age, sexual orientation, skin color and height, which is why I love my school: we may be poor in money, but we’re super-rich in diversity.

Nearby was another long table loaded with a much less diverse crowd: it was primarily 20-something caucasian women with a few men of similar age and skin color sprinkled liberally among the crowd. There were one or two “adults” sitting in a clump at the head of the table, and I even spotted one of two people of color, but for the most part it was a pretty homogenous group, and they were clearly enjoying their margaritas. I concluded that this was not a law firm (too few geezers), nor were they celebrating the successful IPO of an internet startup (the restaurant was not fancy enough and there were too many women), and it certainly wasn’t a reunion of the Bethune-Cookman University “Wildcats.” 

I am nosey by nature, so I did a slow recon around the table and confirmed my suspicion: these were indeed the faculty and staff from a school. However, something was not right: the demographics were all off. I’ve worked in numerous private and public schools, and while teaching always attracts a lot of young, white women, even the most upscale private schools have a wide range of ages in their faculty and have embraced hiring policies which encourage diverse and inclusive teaching populations. Whatever this school’s mission was, diversity (at least in faculty recruitment) was not high on its list.

So I was quite miffed by this group: what kind of school has a nearly all white, all 20-something, all female faculty? I approached the maitre-de and offered to buy her a sandwich (my typical bribe) if she would tell me which school was being feted at the table nearby. She looked down at her clipboard and announced, “Oh, that’s the ….. School.”

I pride myself in having my ear to the ground on the goings on when it comes to schools in NYC (I’m a reluctant reader of Chalkface and even follow Beth Fertig on Twitter), so I was somewhat taken aback that one had escaped my notice. I whipped out my iPhone, conducted a search on the school in question and was chagrined by what I read. No, I was beyond chagrined: I was incensed. Here’s all you need to know:

  • Charter school
  • Hedge fund managers on Board of Directors
  • African-American student population

The NYC charter meme may include these uniforms, but it certainly won’t include these girls.

I didn’t read any further: this could have been the webpage for any number of charters that have sprung up over the New York City area. You have to admire the audacity in creating a school which fit the charter school mold so completely (this one was barely 3 years old.) All you have to do is throw in the khaki-pants-with-polo-shirt-with-school-logo uniform and a “no excuses” discipline code and >poof<, you done got a charter school!

This reinforces the saddness I feel whenever I hear the praises undeservedly heaped on the whole “charter school” movement, because it seems to me that the original intent of the charter school laws was to create “new” and “innovative” kinds of schools that “experimented” with different models of learning and instruction. Instead, what has emerged are endless numbers of cookie cutter schools which lack any individual identity or unique personality, where only the desperate and/or misinformed send their children, and where those who do not “make the cut” in the pressure-cooker classes are summarily “counseled out” to neighboring public schools which are obligated to educate those “rejects” (and my guess is that many of those “rejects” are going to emerge as some very brilliant and wonderful adults, if my 30 year tenure in education is any guide….)

Charters are not innovators in any way, shape or form, unless you consider it an “innovation” to underpay and overwork your novice, young faculty, discriminate against English language learners and those requiring special services, and focus relentlessly on high test scores. Of course, you need hedge fund sharks to fund the whole enterprise, and some young, white, primarily female teachers to staff the classrooms, but neither of these seem to be in short supply. Oh, and don’t forget the students…

Coming to a Tumblr blog near you!

Coming to a Tumblr blog near you!

I’m in favor of having all sorts of schools, where informed parents can rest assured that their children are getting a high quality education from a faculty that is professional, diverse and committed. However, I think that charter schools like these are not really schools at all: they offer nothing innovative or interesting; rather, they are memes, which is what seems to attract young people nowadays.

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Do “Americans” really stink at math? Let’s check the numbers.

Last week, journalist and statistical illiterate Elizabeth Green published a highly visible article in the New York Times Magazine that proclaimed that we, the citizens of these United States, “stink” at math. I castigated Green for her misuse of actual data to back up the enticing headline, and while the article had nothing new to say (Stigler and Stevenson had a go at the Asia/US comparison back in 1994), I thought for the most part her comparison of math teaching and the culture of education in Japan and the U.S. held up pretty well.

Now it’s time for my response: Ms. Green, you are wrong – in point of fact, Americans are very good at math, but forget the Japanese, we should be looking at France if we want to see what great math education looks like. How do I know this? I used data!

The Fields Medal is the world’s most prestigious award for mathematics achievement. It is given out every 4 years to a select group of mathematicians around the world for outstanding discoveries in mathematics. It is much more selective than the Nobel Prize,  and while the money is not so great, the distinction is far, far higher.

In its 78 year history, the Fields Medal has been awarded to 54 different individuals from around the world. Here are the standings for the top 10 countries:

  1. The United States of America: 12 winners
  2. France: 10 winners
  3. Russia/Soviet Union: 9 winners
  4. United Kingdom: 6 winners
  5. Japan: 3 winners
  6. Belgium: 2 winners
  7. Germany: 1 winner
  8. Australia: 1 winner
  9. British Hong Kong: 1 winner
  10. Finland: 1 winner

It seems to me if we look at these numbers, the United States looks pretty good – in fact, I would go so far as to say that we totally kick ass! However, before I paint a rosy picture of math achievement in the U.S.A., it should also be known that the last Fields Medal winner from the U.S. was back in, wait for it, 1998! Meanwhile, oh la la, those wacky French and those studious and  irrepressible Russians have been cleaning our clocks: both countries have taken home 4 medals apiece in the last dozen years. And where is Ms. Green’s beloved Japan in this race? Ha, they have only 1 medal, and that was almost a quarter century ago (1990.)

Is this the face of the next United States math education innovation?

What can we conclude from the above data? Well, contrary to Ms. Green’s assertion, and when use factual, measurable and realistic comparison with other countries, it seems like our educational system is doing a very good job at teaching mathematics. If you look at the top 10 institutions that created winners of the Fields Medal, half of them are located in the United States of America! However, I would still keep an eye on the French: three of their institutions were in the top 10, while, wouldn’t you know it, Moscow State University languished at 11th place, and Kyoto University was well at the bottom. The United States may only take up a tiny fraction of the earth’s surface area, but we’ve totally got a complete lock on mathematical research and innovation.

Nowhere in this field do you see the originators of the latest über trendy mathematics program, Singapore. It seems while they are able to get their students to perform highly on the international math tests, they can’t actually spawn a world class mathematician. Frankly, I don’t think this speaks very highly of their educational system. If we’re going to model mathematics education on the practice in any free country, it seems to me we should be following the French. Okay, their food is very fattening and they love Jerry Lewis, but if we want to continue leading the world in mathematics achievement, sacrifices must be made!

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Why the New York Times Stinks at Math

I love it when my hometown rag, The New York Times, attempts to publish something provocative about the state of mathematics and teaching. Yes, the Times has not aged well, and I often wonder if the editors of the various lifestyle sections, including “Style,” “Home,” “Food & Dining,” and “Travel” section live in the same economic climate as the bottom 99% of us, but I do admire the investigative reporting, especially when they decide to do something like pantsing that mendacious asshole, Andrew Cuomo.

So it was with curious interest that I had a look at Elizabeth Green’s article in the July 23rd edition of the New York Times Magazine, “Why Americans Stink At Math.”  Me, I love a good screed disparaging Americans as much as anyone else, although I would like to point out to Ms. Green that “America” is composed of two continents, North and South, and that even if she were referring to North America, we do share this continent with Mexico and Canada. Oh, and then there’s that issue about those people who were here before there was an “America,” but let’s not get technical, okay?

This article is clearly well-researched, and I hope Ms. Green’s book sells well and that “Chalkbeat,” the website where she is the chief executive, gets a bazillion hits. However, it appears that Green herself is pretty poor at math herself, and the Times let her get away with it. Of course, perhaps the Columbia School of Journalism doesn’t require that their students take a basic course in statistics.

Where should we begin? Perhaps the title is a good place: if you’re going to write about how bad “Americans” are at mathematics, it’s probably a good idea to get some reliable and relevant data to back it up. Unfortunately, the “data” that is thrown down in this article is, to put it politely, piss poor.

The first place where Green goes wrong is when she cites “national test results”  about mathematics achievement in the U.S.. First, I wonder which “test results” Green is referencing here (you have to be suspicious when, in the days of the omnipresent interweb, a link is not included to the data supporting this point.)  It may be significant that 2/3 of all 4th and 8th graders are not “proficient” in math, but again, this is a national standard, not an international standard, so this only points to the fact that U.S. children are not achieving according to some standard that was created where, in some dark cave where Dick Cheney and his family reside?

Green goes on to state that half the 4th and 8th graders taking the National Assessment of Educational Progress could not read a thermometer, or that 3/4 of the test takers could not translate a simple word problem into an algebraic expression. Note that this is the National Assessment of Educational Progress – it doesn’t say anything about whether U.S. children are better or worse than anybody else around the globe; for all we know, 7/8 of  the children in Helsinki and 11/13 of the children in Ibaraki couldn’t successfully answer these questions either. Look, I’m not the sharpest pencil in the box, but even I know these numbers are insignificant without a context.

The one mention of an international comparison is that students in Massachusetts, which many of us know is a “state,” lag two years behind their counterparts in Shanghai, China, which if I understand it correctly, is a “city.” Yes, and would you be surprised to learn that the students at the Bronx High School of Science outperformed those in the village of Zhuangjiashan, China? Cherry picking data is never a good idea, except if you can use it to back up a sensational headline (or you are a best selling author.) Oh, and if Ms. Green bothered to do her research, she would have found out that the scores in Shanghai are cherry picked themselves: according to an investigative report by The Guardian, many children in Shanghai are barred from taking these kinds of tests.

Green goes on to cite how the picture does not get better into adulthood (I wonder if she is talking about her own understanding of statistics?) Green uses a 2012 study (which remains unspecified) about how U.S. adults ranked in the bottom 5 of 20 countries in numeracy. Which countries these are, we do not know. What the sample size is, we do not know. How the samples were chosen, we do not know. The only study I know of that was done comparing math proficiency in adults internationally was by the “Program for International Assessment of Adult Competencies” that compared 23 countries and put the U.S. third last.  However, that result had so many methodological flaws that I had to tear it down in a separate blog post that I titled “Is this the most ignorant article about the Common Core ever written?”  If Green has some better data on this, I would love to see it.

Green follows this up with some scary data about miscalculations in the medical profession, including doctors and pharmacists, just to help show how very afraid we should all be. However, this “evidence” is junky as well: does the U.S. rank higher in medical computational errors than other countries or not? Isn’t this an article that is putting forth the thesis that “Americans” are worse at math than everyone else? I wonder if other professions have higher rate of alleged innumeracy, including writers reporting on mathematics education? Ms. Green, would you mind calculating the integral of X squared for me?

I won’t even go on about the A&W burger story and fractions, besides the fact that I was unable to find a reference that it was actually true… the only mention I could find was a case study described in a blog post in 2013, which also, curiously enough, was unreferenced…. It sounds like a good story, but like many other urban legends, this might be one that stands alongside the worms in McDonald’s hamburgers?

Besides the dubious statistics Green uses to promote her cause, I would also like to take her to task by referencing the Louis CK debacle on “Common Core Mathematics.” I followed this story very carefully, and can tell you that the issue being described had nothing to do with the Common Core or mathematics education. What it really was about, in my estimation, is that parents should not really be helping their kids with math homework: it is a subject that is bound up in all sorts of emotions concerning intelligence and anxiety, and since the teaching of it changes every few years, parents inadvertently end up transferring their anxieties to their children. As someone who has worked in the field of teaching mathematics for the past 3 decades, I can tell you right now that the first piece of advice I give parents about promoting their children’s achievement in mathematics is butt out of helping with their homework assignments.

The rest of the article, when it avoided any mention of data, was a good read, and I always enjoy hearing about the work of the wonderful Magdelene Lampert, who is a legend in the world of education. Ms. Green may not know much about statistics, but you can’t beat her narrative ability; my only hope is that she’ll hire a statistician next time to help support her claims.

 

 

Posted in Common Core State Standards, International Assessments, The New York Times | 6 Comments

What Bad Assessment Looks Like, Go Math! Style….

As many of my loyal (and not so loyal) readers know, among the things I’m attempting this year is to help a K – 3 public school in an impoverished area of the Bronx implement an alleged “CCSS aligned” curriculum. I’m not a fan of most math curricula I’ve seen, and even the best require some kind of adaptation to be effective with the classroom population Go Math!, however, is among the worst of the worst I’ve encountered, and in my short tenure attempting to adapt it, I find myself mostly doing to the work of informing teachers what not to do, which includes correcting mistakes in terminology and developmental appropriateness.

In this post, I’m going to be discussing assessment: much of Go Math! relies on “end of chapter” exams that focus on multiple choice questions that are either poorly worded or completely inappropriate. One of the criticisms leveled at Go Math! has been its almost pathologic focus on procedures, without giving adequate coverage of concept development and higher order thinking (despite the fact that the graphic designers sprinkled the workbooks with “HOT Problem” logos that are not very “high” or require much in the way of “thinking.”)

The latest howler came in the 3rd grade chapter on area and perimeter. Most of the multiple choice questions don’t actually require any kind of thinking beyond remembering to count boxes when the word “area” appears in the question, or counting line segments when “perimeter” appears, which includes this task:

Can you figure out why this is a cruddy question?

Can you figure out why this is a cruddy question?

Okay, it’s not bad enough that most of the test asks exactly the same type of question over and over again, but if you really wanted to assess whether a student knows the difference between finding the area of a shape and perimeter, wouldn’t it be a good idea to provide shapes where the area and perimeter are not the same? 

My suspicion began when out of the 40 tests I looked at, each and every student got the question correct. Okay, it’s not a very difficult question to begin with, but when I looked at the work provided, I noticed that half the students had numbered the squares inside, and the other half numbered the line segments around the outside and came up with 22! Perhaps I’m a little misinformed here, but what exactly is this question assessing? (Let’s set aside the fact that Jake’s bedroom is 22 square meters, or 237 square feet, which makes it larger than the Superior Queen Room at the SoHo Grand in NYC.)

This is not to say that multiple choice tests are all evil; however, if you’re going to go through the trouble of wasting a child’s time with these type of assessments, at least make them interesting and meaningful. This is a garbage question that does nothing to help a teacher assess a student’s understanding of perimeter and area, and whatever 20-something year old editor who allowed this to appear should be fired, or confirmed as a replacement for Arne Duncan.

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