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!

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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

Million Dollar Mathematics of Game Shows from NCTM 2015

At NCTM 2015 I gave a talk on the mathematics of game shows, including Deal or No Deal and several games from The Price Is Right: 1/2 Off, Master Key, and Plinko.

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

The Curriculum Void: Devoid of Proof

by Al Cuoco, EDC  

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

From Al Cuoco, a commentary on formal algebra and bookkeeping

Some interesting thoughts by Al Cuoco about polynomial arithmetic and how Taylor series can be used to solve some interesting problems.

Formal Algebra (PDF)

From Paul Goldenberg, a commentary on “units digit” versus “ones digit”

After a brief conversation, Paul Goldenberg, lead author of EDC’s Think Math! curriculum, had this to say on the debate between the phrases “units digit” and “ones digit”.


Yes, anything can be a unit.  There are lots of arguments one could make either way, I suppose.

A simple case for “ones” is that it is more parallel to “tens” and “tenths” — a specific magnitude named after a specific number.

The number 1 (ok, and –1) is, of course, a “unit” (in the world we care about) but that’s a different use of the word. Unity and oneness are also, though synonymous, slightly different uses than the number-name “one.”  CCSS chooses to call teen numbers “one ten and some extra ones.” It feels slightly weirder to say “one ten and some extra units” but only (again) because of the not-quite-parallel language. And, educationally, “one” is a more familiar word than “unit.”

There!  I’ve defended “one” about as much as I can.  But is that a case for “ones digit” over “units digit”?  Only the “familiarity” argument honestly holds any weight with me, and not much weight.  After all, the idea of choosing a unit and basing everything else on that unit seems valuable, and “1” is the obvious candidate in the integers. When we’re measuring distance between cities (in any sensible country), we definitely choose km and not m as the purpose-specific unit but the entire naming system is based on m as The Unit (km and cm are just modifications of m).

Hmm, so maybe 10 should be the unit in the integers, since our decimal (that’s its name) system is based on powers of 10.  So there! I’ve wound up defending “one” a bit more by showing how slippery “unit” can be.

I’m actually not sure what led to the choice of “units digit” for all of Think Math, but I do know why we chose the symbol u to label that column (because I had to answer it many times, which might well have been the source of the naming we chose): o looks too much like 0.

Is there a strong case to be made against “units digit,” given the various ambiguities we get for either choice?  I’m not convinced there’s anything mathematical in this discussion, since the meanings of any of these terms depends so much on context.  After all, in the numeral 235, the two’s place is in the hundreds’ place. (Oh, should those have apostrophes or not?) Pedagogically, I’m convinced that it doesn’t matter at all.  Oh, and one more argument for “ones digit” (or ones’ digit): if vast majority usage goes one way, then bucking the usage is ornery and unhelpful.  But, sigh, there are so many places to put one’s (or unit’s) energy.  I can’t get worked up much about this unit.