It’s not a multiple of 3… or 7… or 13.

So, let’s give you some spoiler space, to try the problem *without any techmology*. Is 2501 prime, or not?

This spoiler space is sponsored by the Illustrative Mathematics Grades 6-8 curriculum, coming very soon! It will be CC-BY, which means you can download and use it free for any purpose, forever.

Enough space yet?

So, having failed to find a simple multiple that divides 2501, I noticed it was one more than a multiple of 4, and a sum of squares:

This is a sign that it might be prime, because all primes that are one more than a multiple of 4 can be written as the sum of two squares (try 17 … 21 … 29). But it’s also a sign that it might *not* be prime, because all primes that are one more than a multiple of 4 can be written as the sum of two squares *in exactly one way* (try 25 … 65).

So the search changed from finding a factor to finding another sum of squares, and the search ended quickly after remembering 2401 was a perfect square:

What’s interesting here is we know from this that 2501 is not prime, but we don’t know what its factors are! Having tested and not finding any prime factors less than 17, this is enough to know that 2501 is the product of exactly two prime factors.

But which ones? To do this, I tried to find a way to write 2501 as a difference of squares. And again, the search ended much more quickly than I expected:

It made me wonder if there are other numbers with the same property, where the number is 1 more than a perfect square, and also has an equal gap to the next square up and next square down… guess what.

26 between 16 and 36

37 between 25 and 49

between and

Let me know if you find anything interesting here, or if you have any favorite numerical calculations like these.

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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)

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- The validity of an idea about mathematics education and the plausibility of that idea are uncorrelated.
- 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.

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

“Ummmmm…”

“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?”

“Seven.”

“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!!”

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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?

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

“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.”

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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!

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

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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.

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We discussed some of the choices made in designing these games and others, and how the most interesting questions about shows are from the producers’ side, not the contestants’ side.

We also played games and I attempted to give away lots of money. Sadly, most of the games were lost.

Below is a PDF with the slides from the talk. I will repost the slides from the previous NCTM game show talks soon.

Thanks! I’ve been asked a few times — yes, I’m happy to present this or any other talk for a school or district, for students or teachers.

Mathematics of Game Shows 2015

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The NPR story, “The Common Core Curriculum Void,” opened with a crucial distinction that both critics and proponents of Common Core often miss: *Standards are not the same as curricula*. Standards lay out a set of proficiency goals—what students should be able to do and understand. Curricula are vehicles for attaining goals that provide teachers with texts, support materials, and tools to help students meet standards. Just as Google Maps provides several route options to reach a given destination, different curricula offer a range of pathways to proficiency. Common Core is a standards document that several curricular designs can support. The NPR piece made this distinction, clearly and eloquently.

The story went on to feature interviews with educators from around the country, all of whom claimed that it’s impossible—with such a short turnaround to publish Common Core-aligned curricula—to support teachers as they help their students meet the standards. I disagree. The Common Core did not appear out of a clear blue sky. It is the endpoint (some would say midpoint) of the evolution of ideas that have a long history and pedigree. It builds on the work of generations of mathematicians and educators, all devoted to closing the gap between mathematics as a school subject and mathematics as a scientific discipline.

I have experienced this disjuncture firsthand. For four decades, making school mathematics more faithful to the real thing has been a centerpiece of my work. Yet I didn’t always feel this way about mathematics.

When I started teaching high school, I thought that mathematics was an ever-growing body of knowledge. Algebra was about equations, geometry was about space, arithmetic was about numbers; every branch of mathematics was about some particular mathematical objects. Gradually, I began to realize that what my students (some of them, anyway) were really taking away from my classes was a *style of work *that manifested itself between the lines of our discussions about triangles and polynomials and sample spaces. I began to see my discipline not only as a collection of results and conjectures, but also as a collection of *habits of mind*.

This became concrete for me when my family and I were building a house at the same time I was researching a problem in number theory. Pounding nails seems nothing like proving theorems, but there is a remarkable similarity between the two projects. It’s not the fact that house-building requires applications of results from elementary mathematics (it does, by the way). Instead, the two projects were alike because of the kinds of thinking they demand. Both theorem-proving and house-building require you to perform thought experiments, to visualize things that don’t (yet) exist, to predict results of experiments that would be impossible to actually carry out, to deal with complexity, and to find similarities among seemingly different phenomena.

For over 20 years, ever since we built that house, I have focused on fostering students’ mathematical ways of thinking in my classes and curriculum writing. I am not alone in infusing mathematical ways of thinking into curricula, and I am convinced that it is one of the most important things students can take away from their mathematics education. For *all *students—whether they eventually build houses, run businesses, use spreadsheets, or prove theorems—the real utility of mathematics is that it provides them with the intellectual schemata necessary to make sense of a world in which the products of mathematical thinking are increasingly pervasive in almost every walk of life. To cut back to the Core: The Common Core standards, particularly the eight Standards for Mathematical Practice, emphasize these same mathematical ways of thinking. As is the case with my own curriculum, and that of colleagues, the Core elevates these mathematical ways of thinking to the same level of importance as the results of that thinking. This is not a new approach. It did not happen overnight. And there are existing curricula—with mathematical ways of thinking baked in—that can help teachers support their students.

The “Common Core curriculum void” is far from the black hole described by the NPR story. In addition to curricula, a variety of resources are available—for example, the nonprofit Achieve The Core has produced a set of criteria for publishers, the Massachusetts Math-Science Advisory Council discussed a list of benchmarks for curriculum adoption, and Illustrative Mathematics contains many tools that can help adoption committees. To “bridge the void” we need to point educators to the right resources, sending them and their students on an efficient route to mathematical proficiency.

*Al Cuoco is a mathematician and researcher at Education Development Center, Inc. (EDC) based in Waltham. A former high school math teacher, he holds a Ph.D. in mathematics and has spent the past 20 years in curriculum development and teacher education. http://mpi.edc.org*

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