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!"

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....

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?

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.....

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!

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 them to the nearest tenth of a centimeter, and then calculate the are. Do the problem using three different sets of measurements.

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….

Screenshot 2016-02-24 11.19.05

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:

Screenshot 2016-02-24 10.06.24Screenshot 2016-02-24 10.08.00

Screenshot 2016-02-24 10.09.16  Screenshot 2016-02-24 10.10.22

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


There's a lot of "scoot" games on TpT. You shouldn't use any of them.

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. 

Screenshot 2016-01-22 17.00.39

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.”


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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....

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.

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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!

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:

airplane solution 3

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.

airplane solution 1

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.

airplane problem with notes

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:

airplane note alone

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!

long division advert

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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!!!!!!

division vocabulary

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!

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

What not to put on the board…..

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

division brain teaser

One way to wake up a boring division lesson….

As they say, “you wouldn’t believe what happened next!” A few impulsive students (all boys, as usual) rolled their eyes and called out “35! 35!,” which is exactly what 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, 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.

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?

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:

Screen Shot 2015-06-05 at 9.02.11 AM

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.

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:

Screen Shot 2015-06-05 at 9.08.28 AM

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.

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.

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?


Buy this or you will never be able to teach arrays properly!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

Screenshot 2015-05-27 12.26.26

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?

Screen Shot 2015-05-27 at 10.11.06 AM

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.

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?

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?

Screenshot 2015-05-27 14.04.58I’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?”

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?


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 .


THIS HAS TO STOP: “Key Words” Approach To Problem Solving

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!”

Sorry, you can’t “hate math,” even if you tried.

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.

How could you possibly hate this dish?

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 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….


Note: this posting has been brought to you by my online store, where you can find this  78 page collection of activities designed to facilitate number sense using different individual coins, as well as in combinations of 2, 3 or 4 coins. The guide also has an extensive preface describing the different types of number sense that needs to be developed at different grade levels.