# How to do a lot of damage to a kid’s understanding of math in one simple chart….

Robert has been accused of being a serious badass when it comes to mathematics, most likely because he is very adamant about “getting things right.” According to Robert, “there are no shortcuts to understanding,” when it comes to understanding math, but if you look at the materials that are being posted on the web these days, you would believe that manipulating numbers is nothing more than memorizing a dozen or so “rules.” So when this poster showed up on Pinterest (luckily the link to the miscreant was not included), Robert lit’rally blew a gasket:

Would the teacher responsible for this please step forward?

Where does one begin to explain all the problems with this?

The beginning part of this is not all bad: true, multiplying by 2 is also the same as doubling, and yes, you could double a double to multiply by 4. But really, do you think there are a lot of 8 and 9 year olds who can double a double a double?  You can almost see the smoke coming out of little Hudson’s ears as he attempts to double 8 three consecutive times: 8 + 8 = 16, 16 + 16 = 32, 32 + 32…. wait, where was I?

Things only get worse as you get further down the chart: yes, multiplying by 0 always equals 0, but does a child really need to memorize this fact? If the student understands the meaning of multiplication, then reciting “times zero = zero, zilch, nada….” and “times 1 = the other factor” is just nonsense. These are essential features of our number system that require understanding, not memorization: times 0 is the “zero property of multiplication,” and times 1 is the “multiplicative identity.” Yes, the word “multiplicative” is a mouthful for a young child, but this chart totally ignores the totally interesting properties of 0 and 1.

But the real howlers are multiplying by 10 and 100. The “adding zeroes” shortcut is incredibly wrong and will only lead to misconceptions the following year when the student learns about multiplying decimals, when he will end up writing down the following:

Follow the chart and this is what you end up with…

Make no mistake, this error will not go away in 5th, 6th or 7th grade. In fact, the idea that you multiply by powers of 10 (whether that be 10 or 100 or 1,000 or 10,000 and onward….) by “adding zeroes” is a hard one to shake at any age.

Robert says this: STOP TEACHING ADDING ZEROES AS MULTIPLYING BY POWERS OF TEN!

What should we do instead? Show students that multiplying by 10, because we are in a base ten system, “pushes” all the digits over one place to the left, and that zeroes “rush in” to fill those places that are left empty….

The “right” way to multiply by 10s and above…

The beauty of teaching multiplication by powers of ten this way is that it will always work! It will work with whole numbers, negative numbers, decimals, irrationals, rationals, you name it, it will work!

This post has been brought to you by this guide to teaching place value, from which it has been excerpted:

C’mon, you know you want it….

# Enough with fractions, already….

Robert is an occasional reader of our hometown paper, The New York Times, and usually rolls his eyes whenever one of the cub reporters attempts to make sense of mathematics education. Every once in a while, however, the editors manage to slip up and allow someone with an ounce of common sense to write about his chosen field of expertise.

Such was the case when a reporter covered Andrew Hacker and his very convincing case for eliminating the traditional “advanced math” curriculum. We all know this as the algebra-geometry-trigonometry-advanced algebra, yadda, yadda, yadda forced march that college and capital accumulation bound high schoolers must march through. This sequence allegedly prepares the few students who actually understand this stuff for brilliant careers in art history, restaurant management and human resource development. It’s not that Hacker dislikes mathematics or wants to dumb it down; rather, he suggests that maybe a more thoughtful approach focusing on “real life” uses of mathematics would actually achieve the goals we have set out: prepare our students to participate in our quasi-capitalistic economic life and pseudo-democratic political system.

While we’re at it, Robert wishes Hacker would also take a more courageous stand, and push back against the teaching of fractions in the elementary grades. Talk about a waste of time: while fractions were a useful thing to know if you were a working class Sumerian over 5 millennia ago, today they’re just that like that unwelcome uncle who pops up at the Thanksgiving dinner table each year: nobody knows why he’s there, and everybody would digest much easier if he just stayed in his cardboard box.

As Robert tells it, fractions had a good run, and with a 5,000+ year history, they’ve outlived every fad imaginable, including paisley togas and virgin sacrifices. Fractions began their long run with the Egyptians, who utilized them to divide up plots of land and levy taxes. The ancient Sumerians loved fractions so much that they invented the 24 hour day, the 60 minute hour and the 360 degree circle just so they could divide them up into equal pieces really easily: a 24 hour day can be divided into halves, thirds, quarters, sixths, 8ths, 12ths and 24ths (don’t get me started on how easy it is to divide 60 and 360….) You just can’t do that with 10 or 100.

But Robert maintains, fractions are pretty much dead so far as common usage goes, and you can blame that on the French. Yes, those Jerry Lewis and croissant beurre loving eggheads are to blame for the downfall of fractions and the rise of its bitter opponent, the easy and convenient (and mostly sanitary) decimal notation. Quelle barbe!

Typical French citizen holding 300 ml glass of wine with 1 meter long baguette and smoking 100 mm cigarette. He’ll probably live to be 100 years old….

Whatever your feelings for the French, which can be determined by what you call fried potatoes served with ketchup, you’ll have to admit they were on to something by inventing and then adopting the metric system: it’s soooo easy to use! No one who has ever worked with kilograms, meters or liters (as well as newtons, joules and candela) has ever had to struggle finding a common denominator, convert to a mixed number or inverting and multiplying. Is it any wonder that 99.9% of all the countries in the world use decimal units of measurement?

But here in the United States, we love our fractions, which means that it is the only place on earth (well, toss in Liberia and Myanmar) where fractions have to be taken seriously. Open a cookbook and what do you find? Fractions of a pound, fractions of a cup, fractions, fractions, fractions! Robert believes cooking is a pretty pleasurable experience, up there with consensual sex, open-water swimming and playing the concertina, but when you mix in fractions, well, you’re just asking for trouble. Robert wagers that there is a significant fraction of the American population which hates to cook because they’re scared of fumbling with quarter cups and half-pounds.

Sorry, daughter, no cupcakes for your birthday: daddy doesn’t like baking with fractions…

Open a cookbook anywhere else in the world, and there will be nary a fraction in sight: everything will be measured in grams, kilograms and liters, which makes life quite a bit easier. Don’t need an entire kilogram? Just put a decimal point on it, baby! Need to increase a recipe by ten times? Push those digits one place to the left and move on! You just can’t do any of that with half-tablespoons of vanilla…..

And this just has to do with cooking; the prevalence of fractions in American life, from our use of 8 1/2 x 11″ sheets of paper for correspondences to road signs which tell us our exit is 3/4 of a mile away is just so passé. I agree: I know decimals will take some getting used to when it comes to our everyday life, but can’t we give it chance?

My best seller!

Truthfully, Robert loves teaching fractions, and even makes a pretty good living off of them, seeing as how 4 of  the top 5 bestsellers at his online store are about… fractions! Even I think they have some important uses, especially when it comes to algebraic notation, and they can be rather fun when done in a playful and non-threatening way.

I really don’t think this is going to fix the problem…

However, in an era of high-stakes testing and curriculum standardization, maybe it’s time we reconsider the teaching of fractions in elementary school. Fractions prevent our students from solidifying more important concepts and mastering developing skills. They are confusing and have limited uses, yet we persist in teaching them to children who are just not equipped to understand them. Can we agree to delay them until 5th, 6th or 7th grade, when kids are more able to comprehend them, and can be covered in 1/4 the time?

I would wager that 93/95 of the American population would agree with me.

# Geometry: It’s More Than Basetimesheightdividedbytwo….

Robert has been in the teaching game for a long time. A looooong time. Robert has lived through every math fad that has come along, from the “back to basics” movement right through dozens of different math programs including “Chicago Math,” “TERC,’ “Real Math,” “Mathematics in Context,” and numerous others. Let’s put it this way: when he first started teaching, he regularly called Steve Jobs’ secretary for tips on how to install a video card into the Apple IIe….. If you don’t know what the words “secretary,” “video card” and “Apple IIe” mean, you’ll know that was a long time ago….

Believe it or not, Robert keeps some of his old textbooks he used in his teaching library, and every once in a while he pulls one down for either inspiration or, more likely, to roll his eyes and mutter under his breath, “what was I thinking when I taught this?” Here’s one page that especially embarrasses him from a textbook he used in…. wait for it…. 1985!

These kinds of problems make Robert go “arrrrghhh!” and not like a silly pirate….

Robert, older and wiser, looks at this exercise and knows that this is junk, junk, junk.  “Why are there pictures of triangles?” he mutters under his breath. “Don’t the writers know that the students don’t need the illustrations? They’ll just multiply the two numbers together and divide by two to get the answer.”

This is what irks Robert about how math curriculum is designed: the assumption that students are too stupid to actually think through a few steps of their own to get to a solution. So instead of having them figure out what information is important and how it should be used, they just follow a “rule” over and over and over and over again. Of course, this was back in 1985, so surely things are much better now.

Some improvement, but still not there yet….

Well, maybe a little better. Let’s have a look at this example, which appeared in textbook from 1990. Okay, it’s a bit better: the student actually gets to do something besides “plug and chug” a formula four times in a row, and the student does have to think about whether it should be “basetimesheight” or “basetimesheightdividedbytwo.” The student even gets to make some measurements using a centimeter ruler and round the measurements to the nearest tenth. Hey, look ma, mixed practice AND measurement in one activity!

But do you see why this exercise is a fail? Look closely: what are all those dashed lines doing there? Oh, those are the “height lines,” because we all know that asking students to both measure the parts of a geometric figure and apply a simple formula is really pushing them too hard. Let’s just make it easy on them by showing them what to measure!  Damn, Daniel, that’s sad!

Let’s jump ahead 25 years and see how things have changed. Robert enters the terms “practice” and “area of a triangle” into his browser and copies one of the first worksheets to pop up. To protect the guilty, he has eliminated the author’s name (assuming it was an actual person and not a computer program that generated this atrocity.)

It’s 2016, and our students are still doing junk like this?

25 years later! Think about all the changes that has happened, including 6 presidential elections, several wars, the demise of Wilson Philips and the rise of Ariana Grande, and we’re still giving students the exact same dumbed-down tasks? If anything, this one is even more frivolous, because the measurements don’t include decimals, AND the units for the solutions are written out for the student already. Good grief, thinks Robert, what is the point of it all? Basically, these 6 problems are about as relevant to understanding mathematics as using “Grand Theft Auto” would be in a defensive driving course.

This is what Robert does when you use “dumb” worksheets in your class…..

I fix Robert a cup of chamomile tea, pass him his favorite stress toy and wait.

…and wait…

…and wait…

When he’s sufficiently lucid, I ask Robert to tell me what he does when he designs activities involving area of polygons for his students.

“Well, the first thing is that while the formula for finding the area of shapes like triangles and parallelograms involves base and height, the fact is that any part of these shapes can be the base; the height then has to be drawn in or located. These dumb worksheets do all the work for the student: shouldn’t  the students learn for themselves how to identify these parts?”

If you said the base of this triangle was on the bottom, you would be wrong!

That sounds fair: there have been so many times when I’ve looked at problem that my teacher gave me and wondered, “what would happen if I rotated this triangle? Would a different side become the base? And what is the difference between a ‘base’ and a ‘side,’ anyway?”

Robert doesn’t stop there. “What’s going to happen if the student ever has to find the area of a triangle in real life? She’ll be totally lost, because no one told her which side is the base and which part is the height. Then you have kids making very basic mistakes, like believing that the height is just one of the sides, which can be true when the triangle is a right triangle and you use one of the legs (the non-hypotenuse side) as the base. But what if you can’t measure those sides, for some reason? You’re completely stuck, because the problem doesn’t look exactly like the one in the textbook!”

All of which is why Robert is more likely to give the following problem to his students:

Identify the base and height of this triangle, measure to the nearest tenth of a centimeter, and then calculate the area. Do the problem using three different sets of measurements.

“I’ll admit, this is a tricky problem,” Robert explains, “because it doesn’t spoon feed your students all the information needed to solve it. But isn’t that the point? Shouldn’t we get students beyond the lame ‘plug & chug’ arithmetic and have them really think about mathematics?”

And with that, Robert unwraps a new stress toy and invokes a meme….

Readers who took the time to read and understand this post would most likely like to support the ongoing team of therapist who keep Robert under control by purchasing some of these materials to use in their classroom:

# I Scoot, You Scoot, I Suggest You Stop “Scooting”

There’s almost 18,0000 versions of “scoot” games on TpT. Robert says you shouldn’t use any of them.

Robert has been selling materials on TeachersPayTeachers for the past few years (he won’t call them “products,” because that’s debasing his “process,” whatever that means….), and one thing has always mystified him:

What’s the deal with “Scoot?”

To those of you who have been living in a cave alongside Donald Trump, the idea of “scoot” is to number a bunch of “task cards,” put one on each student’s desk, and then have them work on the problem for a few minutes, bang a pot, or use some other signal, and send kids on to the next problem. Teachers claim it’s great because it gets the kids to practice and they can move around the room, yadda yadda yadda.

According to Robert, who has only been working in classrooms since like Neolithic times, this is a seriously bad practice, especially with mathematics.

Here be his three reasons:

Scoot sends the wrong messages about mathematics: If you’re wondering why your students can’t perseverate (yes, that’s a real word) with doing a multi-step math problem, it’s because they’ve been playing too much Scoot. Scoot communicates to children that they have to do math fast, like really fast, and then move on to the next problem. This is not what mathematics is all about. Most people who “know” mathematics understand it is an intense discipline which requires careful thought and attention to details. Scoot reinforces the idea that “good” math is “fast” math. Uh huh….

Scoot makes kids anxious: See those kids who look like they’re having fun? Robert tells me that there are a good number who dread playing this game (contrary to their willingness to “go along”), because the pace is so fast that they never get to finish a problem, and those that they hurry through to finish are often wrong. Meanwhile, all around them they see a friend who has finished their card and is waiting for the pot to be banged. Robert fondly remembers being the last one to finish anything in school, and he recalls nothing is more humiliating than watching the kids next to him finish their multiplication tests before him on a very consistent basis. And you wonder why he calls his career in teaching math “a from of revenge…”

Scoot is dumb: Okay, this is easily open to misinterpretation – when Robert use words like “dumb” and “smart” when referencing math activities, he’s talking about “smart” as giving kids the right amount of time and resources to do different levels of challenges, while “dumb” is giving exactly the same amount of time to do exactly the same thing over and over again. Since the rules of scoot demand that each problem be completed in the same amount of time, the types of problems you can give are pretty limited. That’s “dumb.”

Okay, big mouth, what’s your alternative to “Scoot?”

Robert claims he’s never, every used a “scoot” type game in a classroom, and you know what, his students get just as much practice as yours. Nyeh nyeh nyeh nyeh nyeh. Instead, Robert has his kids work on “Fishbowl Problems.” From what I understand, a “fishbowl” is where you create all different types of math problems related to a subject, making some easy and some hard, mix them up in a “fishbowl” (or a hat, or a large jar, or what have you….) and then circulating around the room letting kids take a problem out of the jar. That actually sounds kind of cool – why didn’t my teachers do that when I was growing up?

Here’s the great part: the kids don’t know what problem the other kids are working on, so if Rahim needs 8 minutes to finish a medium challenge problem, he doesn’t get anxious, because he doesn’t know what type of problem Schwarmala or Dobie or Cookiehead is working on. Moreover, he may finish that problem and then move on to a second problem which is pretty easy, knock it out in 3 minutes and then flex his mojo on a third problem. The point here is this: “fishbowl” is “smart” because there are different kinds of problems to work on and nobody get’s anxious, because there isn’t all this time pressure from the bang of a gong. A kid might select a very challenging problem and feel good that she completed that single problem in 20 minutes, while another kid may do 4 different easier problems during the same amount of time.

This all sounds kind of reasonable to me, but the question is, what should a teacher do with the dozens of “scoot” materials he/she has already purchased? Robert told me that if you add some easier and harder problems, they could easily be turned into “fishbowl” exercises.

So listen to Robert’s admonition and please, PLEASE! Stop “scooting” and start “fishbowling.”

Please?

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# No, Using Singapore Math Will NOT Put Your School On Top Of ANY List…

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.

If these numbers were “true,” then monkeys would come flying out of my butt….

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.

There are many features 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.)

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 evenings and on weekends. 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 role of using a Singapore Math curriculum is negligible. Yet, schools are foolishly spending lots of money on materials and training in the expectation that their scores will rise to the top as well.

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

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.

This post has been brought to you by SamizdatMath, where teachers can learn about and implement good old American methods for effective math teaching in their classroom, including this:

# Robert Makes A Teacher “Walk The Plank” and other tales…

As a math coach, Robert has a very simple philosophy that he developed and now rigidly adheres to. If you are a math coach (or you work with one), you should listen up, because it will be very useful:

• The Vidal Sassoon Principle: If you were watching tv in these United States in the 70s or 80s, you’ll probably remember the commercials for Monsieur Sassoon’s hair products, and his accompanying tagline: “If you don’t look good, we don’t look good.” That same philosophy embodies Robert’s work as a math coach: his job is to make his teachers look as good as possible, which means making sure they have all the materials they need, understand what they are doing, and if anything goes wrong, taking the bullet for any mishaps. If Robert can make a teacher look brilliant, then it must be due to the fact that he is brilliant as well. In short, Robert always has your back.

This principle (one of several which I will be forced to explicated upon in future blog posts, I’ve been told) was put to the test a few weeks back when Newbie Teacher confessed that she was nervous about introducing “long division” to her class. Robert understood: everybody gets nervous teaching division of any kind (buy this and read why…), but there are ways to approach it without getting everybody nervous.

So Robert decided that maybe the best way to approach this was not “head on,” but through the “back door” (please, no salacious jokes about that….) He tossed this out to Newbie Teacher, “Okay, I know this is going to sound crazy, but let’s try this….” and he scribbled the following problem on a piece of paper:

A teacher gave this problem to her students without teaching long division. So sue her!

The teacher looked at it, said “WTF, Robert, how are they going to do this without long division?” Robert’s response? “I don’t know, but let’s find out: you’ve been teaching them all sorts of things about division for the past 3 weeks, I’ll bet they know enough to figure it out.” And so Newbie Teacher, who knew Robert’s “Vidal Sassoon” Principle, “went for it.” She put the class into small groups, showed them the problem they were to solve, handed out large sheets of paper to record their thinking processes, and only stepped in when she saw flaws in logic or explanation. She did not say anything about long division.

Here are a few of the solutions the students came up with:

You can see how the student understands that to figure out the problem, he’ll need to figure out how many times 385 goes into 14,382 (notice that Robert did not make the answer come out “evenly.” That’s because 90% of division problems don’t!) Like the standard algorithm, this student estimated and checked the answer. Unlike the standard algorithm, he didn’t get frustrated and give up: rather, he made another estimate, checked it and then skip counted until he had enough seats.

This student started with 5 planes, saw it was way too low, and then doubled it, then doubled it again, and then used the 10 plane amount and 5 plane number to get to 35 planes, and then counted on to the right number of planes. Notice that he understood that a 38th plane would be needed, even though the answer is technically 37.

When the students had all finished making their posters, Newbie Teacher showed me something cool: she had her students hang up their posters and then read one another’s methods and put “sticky notes” to either praise or question what they had done. Here’s one of the notes below:

As you can see, this is getting to the heart of the standard algorithm for long division: estimate by using the largest “guess” possible, and then add on more groups until you get as close as possible.

This activity was followed by a lesson on “multi-step” division using a “modified algorithm” that Robert prefers to teach to young children, as it is transparent, forgiving and consistent with children’s understanding of division. You can read about that in the product shown below!

Teach division like a pro! Buy this and do it right the first time.

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

Robert stops in to see K, the 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 looks tense. Robert opens with a corny joke , but she isn’t having any of it.

She plops open her CMP III tome, and points to the chapter on dividing fractions. Robert is thrilled and excited, 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.”

Dear Lord, NO!!!!!!

A pretty good “do now” to review division vocabulary

Robert pulls 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 Robert’s 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!

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 Robert did while “guest teaching” a 5th grade class a few weeks back:

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 he 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, Robert bumped into “K” getting lunch. “So, how did it go?” he inquired. “Oh, it wasn’t the worst thing I ever did….” she replied.

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# The Challenge: Design a “Good” Common Core Lesson

Louis C.K. aside, the Common Core State Standards (which Robert dubbed “CoCoStaSta,” because it sounds a lot more fun that “CeeCeeEssEss”) has generated a lot of gosh-awful math lessons that really should never have seen the light of day. But that’s the nature of the beast, isn’t it?  Tell someone they have to learn something, and someone will come along and write a lesson plan that will feature some really bad pedagogy, featuring bad raps , confusing worksheets, and, let’s not forget, lots of graphics dripping with cute.

So Robert I found himself a challenge when a teacher friend from Maryland emailed him a request to help develop a super-boffo lesson plan to impress the grand-high-mystic-rulers administrators at a charter school at which he was applying for a position. Now, there are not a lot of things Robert finds more loathsome than Robert is no fan of charter schools, which he views as a way for private companies to steal public money, but he figured if this guy could get his foot in the door, he would find out how awful the whole sitch is and then skedaddle as soon as he could. Meanwhile, perhaps he could build his chops a bit, and Robert could design something cool under the name of CoCoStaSta.

When asked for the topic, it turned out to be the oddly specific Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends,” which, it turned out, was CoCoStaSta OA 4.2.8.33.PDQ.F#ck, or something of that nature.

Incredibly enough, Robert found this might be a pretty good and even valuable idea for a 2nd grade lesson, so he started thinking about ideas for representing arrays. The first challenge was this: how do I help my students develop the concept of an array? Since Robert is somewhat of an authority on this matter, the challenge wasn’t too great. He would give diverse examples of arrays, as well as “non-examples” of arrays to help my students understand their “critical features.”  Here’s what he came up with:

A concept card which shows diverse examples and non-examples of arrays.

Teachers typically get very little training in concept development, which means that they end up never actually developing these important ideas, resulting in “brittle knowledge” among their students. In fact, dear reader, this has happened to Robert himself. He recalled many, many years ago, when Robert was still learning the art of math teaching, after teaching what he thought was a comprehensive lesson on arrays and multiplication, a student come up to him at the end of class and ask, “Mr. B., what is an array?”

What you’ll notice is that Robert uses diverse assortments of examples, which included shapes, rectangular tables, and a set of eggs. He also made sure that the “superficial features” were also diverse, by making them different colors and changed the color and orientation of the stars and eggs, so that students understood it was the arrangement of the entire set, not the individual members of the set that make up an array. Compare this to the examples that you regularly see published in those cruddy textbooks:

Textbook examples of arrays promotes misconceptions by not showing that the individual elements could differ.

With this out of the way, Robert was ready to create the actual activity. This meant thinking of a “compelling context” that would be familiar to a 2nd grader, without being cute or condescending. If you’ve been reading my posts, you know that Robert hates, hates, hates dislikes materials that go overboard on “cute.” I know there are teachers out there (the vast majority of whom also post selfies of their pets on the Facebook or Instagram) who believe that cute = motivational, but I can assure you that while cute is fun for you, most kids don’t really care. I’m all for creating materials that are motivating, but I can assure you it doesn’t come from big eyes and little noses.

So Robert went with something tangible and familiar: eggs. The context is compelling because it creates opportunities to discuss a larger issue beyond arrays: the fact that eggs are packaged a certain way for practical reasons. That is, in which of these arrays do you usually find eggs sold?

Why are dozens of eggs packaged in some ways and not others?

With my context ready, Robert was set to create the rest of the lesson: begin by defining the concept, giving a compelling and familiar context, then going into the nuts and bolts of the lesson. This meant making sure that to take care of a lot of other matters. For example, there is the matter of how you write an equation. For many children, all equations take the form of “X operation Y equals Z,” which is entirely unfortunate, because it means that many children don’t actually know that an “equals sign” actually means beyond “put the answer over here.” This is why Robert made sure to do things like this:

Children need to see that the equals sign can go on either “side” of the equation in order to develop a strong concept of equivalence.

Again, since concepts and misconceptions are important when designing materials, Robert makes sure to use the lesson as a vehicle to reinforce the idea of equivalence, because while we can assume that our students understand what an equal sign means, it’s not always the case.

It is also the case that many students don’t understand (because they get cruddy examples) that arrays can be read from right to left or top to bottom. Which is why students write 2 equations for their arrays (when possible.) Here’s the example  used in the lesson:

Students should learn that arrays can be read up and down or side to side.

As you can tell by now, this lesson may have seen “easy” to teach, but because it is mathematics, we have to be extra sure that we are paying attention to all the details, and not unknowingly creating misconceptions along the way. Good teaching comes about when we pay attention to these important little things; avoid it, and your students will emerge with even more misunderstandings than when you began.

Another matter that is important is making sure students get to practice a new concept in different ways. This could have just printed up another worksheet with cruddy examples (see above), but you know, Robert just don’t roll that way. Instead, there is a “table” where students had scaffolding examples of different ways to solve array problems:

Yes, task cards are fun, but why not have students solve different kinds of array problems, instead of “count and write?”

With this all done, thought was given to making sure students should had a chance to do some actual problem solving as part of their lesson on arrays. Unfortunately, a lot of what passes for problem solving doesn’t actual qualify, because it tends to focus on routine problems with just a crappy cutesy graphic thrown in. This is why Robert came up with the idea of an “egg-act array mystery” which would actually challenge a second grader. Here’s an example:

Example of actual problem solving task using arrays.

As you can see, the idea is that some information about the array has been provided (there are 20 eggs, and there are 4 in each row; how many rows are there in the array?) and left something for the student to figure out: how many rows would there be in this array? There is also some space for the student to draw the array, as well as prompted them to write two equations which would show the addition problem that emerged from this array. Yes, this is a little more tricky than the typical “draw and count” task, but isn’t the whole goal of this lesson to challenge students to think on a higher level? By the way, did you notice the equations for the arrays was written “20 eggs = _________” instead of “___________ = 20 eggs?” Yeah, that’s what we mean by expanding a concept.

Finally, students were invited to write their own “array mysteries,” because what good is giving students a task if they can’t come up with one of their own? It is also an excellent way to assess whether the student actually understands the task at hand (and believe me, there are lots of students who are winging their way though math class, which makes this step even more important.)

For assessment, students write and solve their own “array mystery” problems.

So there you have it: a CoCoStaSta lesson that develops concepts, avoids fussy or confusing directions, and provides practice and challenge for everyone. Now, that wasn’t so hard, was it?

This rant has been brought to you by the guy who published this “Egg-act Arrays” on TeachersPayTeachers for the low, low price of \$3.95. When you consider how much actual thought (and not cutesy fonts and graphics) went into this, you’ll understand why you are going to buy it….)

# Task Cards: Why They Suck, What You Can Do About Them

Despite what has been said and written about me, I have no problem with the idea of students honing their math skills with repeated practice. Whether you are trying to master a jump shot or perfecting the tuning of your oboe, there is no substitute for practice. As the old lady told the dislocated young man when he inquired as to the best way to get to Carnegie Hall, “practice, young man, practice!”

However, I draw a deep line in the sand when it comes to those really basic “task cards” I see teachers using in their classrooms. The only positive thing I can say about them is that they’re only slightly less evil than those workbooks that they sell in aisle 5 of the local drug store.

Let’s be clear:  just like there are “good” pirates (Jack Sparrow) and “bad” pirates (Somalis in polyester), the same is also true of practice. What does “bad” math practice look like? What frustrates me about many of the “task cards” I see is their almost obsessive focus on routine problems and therefore, very low levels of mathematical thinking. Do you really want your students to practice “regrouping across zeroes” using the problem shown below?

Should your student really be “regrouping across zeroes” to answer this question?

Truthfully, there are much better things you could do with 40 minutes of a school day than practicing this inane skill, but if you feel the need to satisfy whatever CCSS needs to be checked off that day, at least give problems that actually requires the skill to be used. If your students can’t calculate 400 – 398 without regrouping, then their problems with math are probably more serious than what can be accomplished with a task card.

If you feel the need to make a set of task cards that deals with “regrouping across zeroes,” then what you should not be doing is giving them a dozen identical problems that practices the skill over and over again. Believe me, they’ll forget how to do it within a few days anyway, and as Mark Twain said about teaching a pig to sing, it wastes your time and annoys the pig. Except that in this case, you’re wasting the students’ time AND annoying them. Since you probably just downloaded them from some silly website, it’s not like you have any skin in the game.

Okay, so you’ve read this so far and thinking, “okay, this guy is starting to make sense: if I’m going to have my students practice a skill, the least it should be conscientious and interesting. But I’m out of ideas: what should I do?” Here’s where my 30+ years of work in the field of mathematics education has some value: I’m going to give you 3, count them, 3 alternatives to doing the “standard” task card while still focusing on the skill in question (which is really not about regrouping across zeroes, but what you do when there are lots of zeroes in the minuend.)

Sorting task cards by strategy transforms busywork into higher order thinking.

Alternative Task Card Idea #1: Does it have to be so mindless? See that easy-peasy problem above? Why not put a whole bunch of them into the task card deck and ask students to sort the deck into problems that would require them to actually “regroup” from those that can be done through actual reasoning? Do we really want our student mindlessly regrouping problems that they can and should be solving in their heads?

Isn’t this alternative to regrouping across zeroes so much easier?

Alternative Task Card Idea #2: Let’s make it a teaching tool. One of the things that makes me vomit bothers me about task cards is that they don’t help kids see alternatives to the tired and stale thinking that goes into memorizing and practicing an algorithm. If you’ve read my publication about how to teach subtraction “right” , then you’ll know about the importance of having your students understand that if you move both the minuend and subtrahend up or down the same amount, the difference will be the  same. Using this principle, we can create a “twofer” task, which is that the student masters a somewhat important skill (solving a subtraction problem with lots of zeroes in the minuend) while reinforcing an essential concept that most teachers don’t know, understand or apply.

Alternative Task Card Idea #3: Let’s make it an assessment tool. There are different ways to test whether a student understands a concept, and reproducing an algorithm is probably the least comprehensive way to do this. I’ll call it “imitative assessment,” in all we are observing is whether the student can duplicate the exact same task shown to him/her by the teacher. Wouldn’t it be a better idea to assess a student so that we can determine higher levels of understanding? For example, what do you suppose we would learn from a student who completed this?

I’m sure you can appreciate that there is a much higher level of thinking that needs to take place here, far beyond what is captured by mindlessly working out a procedure. The student here has to know something about place value, expanded notation, not to mention negative numbers. They also have to know how to perform rudimentary math calculations in their head (4,000 – 200, and 3,800 – 10.)

# A Visit from the Mathematics Language Squad: 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?”

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.

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?

This post has been brought to you by my authoritative guide on teaching developmental geometry using the educational work of Pierre van Hiele and Diane Geldof.
This 60 page booklet incorporates the theories of Pierre van Hiele and Dina van HIele-Geldorf, better known as the van Hiele Theory, to teach developmental geometry in elementary and middle school. The van Hiele’s were pioneers in the understanding that children go through different levels of cognitive development in their understanding of geometric concepts, and that these levels changed with the age and education of the child. Through their work, we’ve come to understand what kinds of thinking can be developed by children as they grow and develop.
This booklet is based on the idea of using a set of 7 tiles, which I call “van Hiele Tiles,” based on an article published in the journal Teaching Children Mathematics. In the article, Pierre van Hiele describes a method of using “play” to develop children’s thinking about geometry. The booklet describes the Van Hiele Model in details, and then has explicit, self contained activities at each level of thinking.
These activities require the use of a set of 7 tiles, which can be printed and cut out using the template in the book. These tiles are much more versatile than tangrams or pattern blocks, as they contain a much more diverse collection of shapes, including a right scalene triangles, equilateral triangle, isosceles obtuse triangle, isosceles trapezoid, concave trapezoid and oblong rectangle, which can be configured into many different shapes (including larger versions of one another.) You can learn more about these shapes at www.goodnitebob.com .