All-New Mathematics of Game Shows (NCTM 2016)

Thanks to those who attended my talk at NCTM 2016 at 8am Saturday! Yow, that’s early. At the same time as many other great talks, it was an honor to have a good audience.

Here are the slides from my presentation. Hopefully they make sense to those who weren’t there, but if you have any questions, please let me know.

Slides from NCTM 2016 (PDF)

Ed Begle’s Laws of Mathematics Education

This post from Dan Meyer discussed Ed Begle’s two laws of mathematics education, which feel as relevant today as they did during his presentation in October 1970.

  1. The validity of an idea about mathematics education and the plausibility of that idea are uncorrelated.
  2. Mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected.

You can read all of Begle’s remarks beginning at page 27 of this document, which also includes many other presentations and papers from the same conference.

Another conversation about The Arrow Way

A little more with my second-grade son, age 7, about how he does addition and subtraction, compared with “the Arrow Way” from EngageNY.Screen Shot 2015-11-17 at 6.46.53 PM

“Would you rather do an addition problem or a subtraction problem?”


“Alright I’ll pick one of each. Tell me how to do 39 + 23.”

“It equals 62.”

How did you do it?”

“By adding the 9 and the 3 first, and then adding the tens.”

“How many tens?”

“Six tens.”

“But I only see five tens.”

“That’s because 9 plus 1 equals 10.”

“Oh, that’s really cool. Hey, how would you do 62 – 23?”

“Ummmm… 62 minus 23… 62 minus 23… it equals (whisper) 39.”

How did you do it?”

“Just by … well, I just knew the way I did it before, and minused it.”

“I don’t know what you mean. Tell me more about how you did it!”

“I just counted down.”

“So, you counted down one at a time? Like 62, 61, 60, 59…?”

“Yeah yeah!”

“Really, you counted all 23 numbers down? You didn’t skip any? I didn’t hear you count all those numbers. I think you used a shortcut.”

“No, I just counted them in my brain. All the way down.”

“Ok, what would you do for 62 – 51?”

(Very fast.) “Uh, 11.”

“So, how did you do that so fast? You didn’t count all the numbers, right?”

“Yeah I counted them really fast!!”

“Come on, what did you really do?”

“Well, I counted it down really fast. 62 minus 50 equals 20, minus 1 equals 11. Wait wait wait, 62 minus 50 is 12, not 20! Silly.” (He often mixes up “twelve” and “twenty” in speech, but writes it correctly.)

“Could you do 62 – 51 using the arrow way?”

“No, you can’t. It’s impossible. You would get bored. Minusing 10s is boring.”

“Well, what about writing a minus 50 arrow?”

“You can’t do that! It’s not in the rule.”

“What if I told you that you could change the rule and use whatever numbers you want on the arrows?”

“Nah, you can’t do that. The rule stays as a rule. You can’t change it.”

“Let’s try one more subtraction, then we’re done. But you have to tell me out loud everything you do. Ok? It’s 100 minus 37.”

“Ok. 100 minus 30, then minus 7, equals 77.”

“What do you think?”

“77 is the answer.”

“How did you get that?”

“100 minus 30 is 70, then … no no no wait a second … it’s 63 instead of 77!”

“How did you get that?”

“Wah wah wah wahhhhhh.”

“Really?? Ok you got two different answers. 77 and 63. Are any of those the right answer?”

“63 is the real right answer.”

“What would you say to someone who got 77, to help them get the right answer?”

“Dude, just minus 7 and you’ll get the right answer.”

“What would this problem look like with arrows?”

“100 arrow -10 90 arrow -10 80 arrow -10 70 arrow -1 69 arrow -1 68 arrow -1 …”

“Okay wait. How many arrow -1s are there going to be?”


“So why not just make one arrow -7 instead of seven arrow -1s?”

“No. You just can’t do that. It’s not in the rule. Boom bum bum!”

“Ok let me show you a choice. 100 arrow -30 70 arrow -7 63.”

“That’s not the way we do it in math class!!”

The Arrow Way

On the way home tonight, my second-grade son Aaron starting talking about something he’s learning called The Arrow Way. It’s part of the EngageNY curriculum.

Screen Shot 2015-11-17 at 6.46.53 PM

There’s a lot of positives in this representation, especially when it’s also coupled with number lines. But why couldn’t the problem above just skip the words “using the arrow way”? Or in problems like this, why require students to use this method?

Screen Shot 2015-11-17 at 6.55.46 PM

The result is that some kids think this is a “rule” or “law” instead of a representation/aid/solving method. This can cause misconceptions as kids focus on weird details like what is and isn’t allowed in the representation, rather than on tying the representation to existing knowledge and problems. The visual is nice, but once it’s named, it becomes as important as the other named things kids know in math, and at this point there aren’t many.

Here’s the conversation with Aaron (age 7).

“We have this new rule, it’s called The Arrow Way. The arrow way isn’t a story, it’s a math law.”

“What is it?”

“I don’t know.” (laughing)

“Come on, tell me for real.”

“The rule is +1, -1, +10, -10, and you gotta put one of these above the arrow.”

“So if I started with 50, what could I do?”

“Add 10 to make 60!” (He asked me to make the 60 bigger, “like I’m shouting.”)

“Okay. What else could you do, starting from 50?”

“Minus 10 to make 40! Or you could minus 1 to make 49! And there’s nothing else.”

“Ok. Could you start with 50 and make 70?”

“No. You can’t add 20, because it’s not in the rule.”

“Could you add 100?”

“No! That’s not in the rule!”

“Could you use more than one arrow to make 70?”

“No! Not at all.” (laughing)

“Can you use more than one arrow together?”

“I don’t know, but you can only start from zero, in the rule.”

“Ok. Is that true? Why can you only start from zero?”

“Because that’s a math law, also.”


On speed and fluency

I really enjoyed this article by former NCTM president Cathy Seeley, “Faster Isn’t Smarter”. It’s been interesting over the years to get to talk to many mathematicians that did not feel like they were good at math as children, because of their relatively slow speed.

Fluency matters, but it’s not real surprising that students expect all math problems to be solvable within seconds if they are given thousands of problems that are solved within seconds, and rewarded for how quickly or how many they can do within a time limit. Seeley also discusses issues that emerge in a timed, high-stress environment:

“Some students respond well to competitive and timed situations, thriving on the pressure to bring out their best; others have quite a different reaction. This particular boy received a clear message that some students are good at math and some are not—and he knew which group he was in. He also was prevented from finishing the test, something that causes some students tremendous frustration.”

The article offers some simple suggestions for changing this dynamic in favor of fluency without focusing on speed. Thanks!

From Paul Goldenberg, on mathematics learning and languages

I had a conversation with Grace Chen and Christopher Danielson, among others, about how languages can affect students’ early learning of mathematics. For example, in Mandarin the phrasing for twelve is equivalent to “one ten two” in English.

Paul Goldenberg sent me this and I got his permission to publish it here. Thanks Paul!

I don’t know of any language that uses an essentially spelling pronunciation of the numerals (e.g., 23 as “two three”) although that is essentially what common (out-of-school) practice does with decimals. Only U.S. elementary schools insist that 3.1416 be pronounced “three and one-thousand sixteen ten thousandths.” (It is, or at least was in 2006, required of school teachers in Austin, Texas.)

But I did say that there were many variants on fully or nearly place-value pronunciations.  French ranks near 0 (with near regularity in low numbers not beginning until 16 and succumbing to IE’s general love of toes later on; 91 is four twenties eleven), Danish is slowly reforming from a 5 to a 6, English a 6 or 7.  But, for example, Kinyarwanda does 11 (etc.) as “ten and one” and has only minor anomalies in the naming of numbers (20, as in French, is not built the way larger multiples of 10 are built, though it is far more recognizably two-ish than the French is).

The use of numbers for counting, and for counting things, differs in some languages where agreement with noun gender (or other class) is required, but people are so used to accounting for the affixes that they’re almost invisible (inaudible?) anyway.

But the regularity in structure seems to be more important than regularity in application.  For example, Kinyarwanda’s regularity in structure (and, I suppose, the fact that tiny kids handle money in the hundreds and thousands) gets kids really good really early. Their first graders are better at counting by ones and tens (the latter even if we start at 4 or 7 and proceed to 14 or 17, etc.) than many kids here.  And that’s despite the fact that number-noun agreement requires some eight (I think I’m getting that right) different grammatical forms. (Bleh!)

What I do know firsthand is how differently children respond to print than to spoken (mental) calculations set in some (not necessarily “concrete”) context.  I don’t think it has anything to do with abstract vs. concrete, but rather about some existing semantics.  For example, adding 9 in the context of adding 10 is a no-brainer for kids, but it takes them much longer to look at 23 + 9, especially if it’s set up vertically, and think of doing anything other than what they were taught to do—recalling a specific fact and going through specific notational moves—and they can easily blow steps that have no semantic sense and only syntax (rules and order) to guide them.

From Al Cuoco, examples of the Standards for Mathematical Practice in algebra

Here’s a short piece by EDC’s Al Cuoco on the use of Common Core’s 8 Mathematical Practices as applied to algebra.

Download PDF


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