COMET • Vol. 10, No. 25 – 3 November 2009


The California Association of Mathematics Teacher Educators (CAMTE): New Web Site; Upcoming CMC Conference Sessions


The California Association of Mathematics Teacher Educators (CAMTE) invites you to visit and explore the organization’s new Web site at

You are also invited to attend the CAMTE sessions at the California Mathematics Council (CMC)-South conference in Palm Springs this coming weekend and at the CMC-North conference in Pacific Grove (Asilomar Conference Grounds) next month. Information about the CAMTE sessions can be found at

If you are involved in the professional development and/or preservice education of K-12 mathematics teachers, you are encouraged to join and become an active member of CAMTE! A brochure listing CAMTE’s goals and some of its activities since its inception five years ago is available for download:  A membership form is also available at 



(1) Journals Published by the National Council of Teachers of Mathematics (NCTM)


The National Council of Teachers of Mathematics (NCTM) publishes four journals designed as resources for teachers of mathematics and teacher educators. Teaching Children Mathematics (TCM) includes article on teaching mathematics at the elementary school level. The focus of this journal and Mathematics Teaching in the Middle School (MTMS) “is on intuitive, exploratory investigations that use informal reasoning to help students develop a strong conceptual basis that leads to greater mathematical abstraction” (

“The Mathematics Teacher (MT) is devoted to improving mathematics instruction from grade 8-14 and supporting teacher education programs. It provides a forum for sharing activities and pedagogical strategies, deepening understanding of mathematical ideas, and linking mathematics education research to practice” (

“The Journal for Research in Mathematics Education (JRME)… is devoted to the interests of teachers of mathematics and mathematics education at all levels–preschool through adult. JRME is a forum for disciplined inquiry into the teaching and learning of mathematics” ( JRME includes scholarly reports of mathematics education research.

Free previews of articles in the journals are occasionally provided on the NCTM Web site. (NCTM members receive at least one of the journals with their membership and can access all of the articles in that journal online.) The following three articles are from the November 2009 issues of TCM, MTMS, and MT.


(2) “Early Algebra to Reach the Range of Learners” by Deborah Schifter, Susan Jo Russell, and Virginia Bastable

SourceTeaching Children Mathematics – November 2009
URL (Click on the “Download” button”)

Abstract: Experiences explicitly stating generalizations and finding examples, counterexamples, and proofs–algebraic reasoning–lets young students think more about the principles that underlie their work and can support those who struggle as well as those who excel.

Excerpt (visit the Web site above to download the rest of the article):

Second grader Cecile declares, “I know that seven plus seven equals fourteen, so I took one from one of the sevens and put it on the other seven, so now it is six plus eight, and it’s fourteen.”

In another classroom, third grader Trevor explains, “To solve thirty-nine plus eighteen, I added one to thirty-nine and subtracted one from eighteen to make an easier problem with the same answer: forty plus seventeen equals fifty-seven.”

Many teachers will likely find Cecile’s and Trevor’s comments familiar. After all, these students are employing strategies used frequently by young children who are learning to add. What few teachers may have considered is that implicit in Cecile and Trevor’s strategy is a generalization about the behavior of addition that is worthy of investigation. In fact, such investigations–explicitly articulating generalizations about the behavior of the operations, justifying them, and considering the extent or limits of the generalization–are a central aspect of early algebraic reasoning.

Although Cecile and Trevor are talking about specific numbers, if challenged, they might be able to state a general principle: When two numbers are added, if an amount is subtracted from the first addend and added to the second, the total remains the same. In later years, they will learn to express the same idea in formal notation:
If ab, and are numbers, then = (– x) + (x) …


(3) “Proof: Examples and Beyond” by Eric J. Knuth, Jeffrey M. Choppin, and Kristen N. Bieda

Source Mathematics Teaching in the Middle School – November 2009

Abstract: Asking middle school students to verify the math they do requires them to think about proof. By doing so, students construct arguments in the middle school and are more ready for proof in high school.


Many consider proof to be central to the discipline of mathematics and to the work of mathematicians. However, the role of proof in school mathematics has traditionally been limited to high school geometry.

This absence of proof in school mathematics has not gone unnoticed and, in fact, has been a target of criticism. Sowder and Harel, for example, argue against limiting students’ experiences with proof to geometry: “It seems clear that to delay exposure to reason-giving until the secondary-school geometry course and to expect at that point an instant appreciation for the more sophisticated mathematical justifications is an unreasonable expectation” (1998, p. 674).

Reflecting an awareness of such criticism, as well as embracing the important role that proof plays in learning mathematics, recent reform efforts have called for substantial changes in students’ experiences with proof. In particular, Principles and Standards for School Mathematics recommends that “Reasoning and proof should be a consistent part of students’ mathematical experience in prekindergarten through grade 12” (NCTM 2000, p. 56).

During middle school, it is expected that students will “examine patterns and structures to detect regularities; formulate generalizations and conjectures about observed regularities; evaluate conjectures; [and] construct and evaluate mathematical arguments” (p. 262). These recommendations, however, pose serious challenges for both students and teachers because the recommended practices represent a significant departure from what is normally found in typical middle school mathematics.

The goal of this article is to help teachers both recognize and capitalize on classroom opportunities so that students can be meaningfully engaged in mathematical proof…


(4) “Soft Drinks, Mind Reading, and Number Theory” by Kyle T. Schultz

SourceMathematics Teacher – November 2009


Proof is a central component of mathematicians’ work, used for verification, explanation, discovery, and communication. Unfortunately, high school students’ experiences with proof are often limited to verifying mathematical statements or relationships that are already known to be true. As a result, students often fail to grasp the true nature of what it means to do mathematics.

In response to this deficiency, NCTM (2000) has stated clearly that “reasoning and proof are not special activities reserved for special times or special topics in the curriculum but should be a natural, ongoing part of classroom discussions” (p. 342). This article presents a classroom discussion of a mathematical idea that arose spontaneously but that could be used purposefully as a mechanism to develop students’ conceptions of proof and the nature of mathematical work…


(5) “‘Platooning’ Instruction” by Lucy Hood

Source: Harvard Education Letter – November/December 2009

To platoon or not to platoon? That’s the question facing Irving Hamer, Deputy Superintendent of Academic Operations, Technology and Innovation for the Memphis City Schools. This year for the first time, the state’s achievement test, known as TCAP (Tennessee Comprehensive Assessment Program), will include algebraic concepts on the fifth-grade test. Hamer says Memphis “is bracing for a very heavy downturn in student performance on the exams.”

Hamer’s office has taken a close look at the district’s 351 fifth-grade teachers and found that not one majored in math. “So what that means to the teaching of algebra at grade five is [that] it will most certainly be done by people who don’t have extensive math preparation,” he says. That doesn’t mean they won’t be able to teach what’s required, he says, but “the thinking on the part of this administration is that maybe one way to get higher-order math in the fifth grade would be to departmentalize the fifth grade and to make sure the math is being taught by the most able math teachers in a fifth-grade configuration.”

Hamer’s not alone in his thinking. In the ¬Denver Public Schools–where departmentalizing is called platooning–elementary students as young as six change classrooms, sharing teachers who specialize in only one, two, or three subjects. In the School District of Palm Beach County, departmentalization became a mandate this year for all elementary schools in grades three through five.

In districts across the country, there has also been a noticeable increase in the number of elementary schools that have adopted some form of departmentalization on their own, educators say. Education consultant Steve Peha has noticed the difference. When he began working with schools 15 years ago, presenting workshops in reading, writing, math, assessment, and test preparation at all grade levels, roughly 5 percent of the elementary schools where he worked departmentalized instruction. Now, he says, “it’s more of a normal thing,” and the percentage is closer to 20 percent. “It will continue to grow,” he predicts, “as the need for high scores in tested grades and subjects increases.”

Breaking from Tradition

Platooning (or departmentalization) is nothing new. It’s what middle and high schools have been doing for ages–divvying up instruction according to subject area, with students rotating to different rooms headed up by different teachers for different subjects. What is new is applying that idea to elementary schools, long the bastion of a one-teacher-per-classroom model.

Elementary school teachers are trained to be generalists who spend the entire year with one group of about 25 kids and teach them the gamut of subjects-math, ¬science, social studies, and language arts. The conventional wisdom has been that younger students benefit from the stability and continuity provided by having the same teacher every day all day for the whole year. “In the hierarchy of priorities, keeping the kids together with one teacher is way up there,” comments Molly McCloskey, managing director of the Whole Child Programs at the Association for Supervision and Curriculum Development. “Focusing on the relationships is way up there,” she says. “The more we focus on that as a critical variable in every decision we make, the more we are thinking through the eyes of the children.”

Some educators also say the traditional model allows the generalist teacher to more easily make connections between subjects using a single theme, such as ancient Egypt. “The danger of departmentalization is the creation of silos,” says Katherine Boles, senior lecturer at the Harvard Graduate School of Education. “We have to teach [students] to be critical thinkers across subject areas and [to think] deeply about American history and the connection to literacy and science, instead of isolating it and platooning.”

While testing pressures may be driving the trend to bring platooning to the lower grades, advocates say it has benefits that go beyond simply delivering content by creating more opportunities for teachers to collaborate with adults and to share their enthusiasm for favorite subjects with students.

In Denver, where platooning has been in place on a school-by-school basis for at least nine years, educators say the benefits go beyond increasing test scores, giving teachers the opportunity to collaborate on curriculum and student progress and to share their passions for a subject.

At Denver’s Slavens Elementary School, for example, most grade levels have two teachers. One teaches language arts and another teaches math, science, and social studies. For continuity, all teachers use the same workshop format for reading, math, and science. The main drawback-each teacher has 50 students instead of 25.

That doesn’t bother Michelle DuMoulin, a first-grade math, science, and social studies teacher. “I know them as mathematicians and scientists-all 50 of them,” she says. When students stream into her room, “it’s almost like they are more excited and rejuvenated. The entire room exudes what subject you’re teaching, and I think that’s really cool for kids,” she says, adding, “I feel the theme that ties us together is the thinking and metacognitive strategies we are using to teach kids to be thinkers and to delve deep into units.”

Going back to the traditional generalist model would be very difficult, says DuMoulin. “I am so passionate about math and science. When you have the kids all day, you can extend reading and writing all day if you need to, but sometimes science and math-especially science-go by the wayside. [Now] I don’t feel like I’m skimping on everything. I feel good about it every day.”

Second-grade language arts teacher Barb Smith, who “lives and breathes” reading and writing, says she and her math/science partner have set joint expectations for basic writing in both their rooms.

At parent-teacher conference time, it’s nice to have a partner, too, Smith notes. “When you are sitting there with a parent who has hard news to hear, you have someone to back you up. When you have another teacher say, ‘I see it in the afternoon class, too,’ that helps.”

Parents at Slavens, a relatively affluent school, have also become strong supporters of departmentalized instruction. In fact, only four families signed up when the school had to open a traditional classroom headed by a sole teacher due to a bulge in enrollment in first grade.

The research on the effectiveness of departmentalizing, however, is not clear. “In no area do we have solid research that would tell us that the use of something called a ‘specialist’ improves kids’ learning–at least in part because the notion of what a specialist is can vary so much,” says Deborah Ball, dean of the School of Education at the University of Michigan and a member of the National Mathematics Advisory Panel. Nevertheless, Ball calls the idea “promising.” In 2008, the panel recommended that researchers look into the effectiveness of using specialists, or departmentalized instruction, to teach math, she notes.

“We have a large-scale teacher education problem in this country,” says Ball. When standards are raised, it’s not just the students who are affected; teachers must also acquire new skills in order to teach to those standards, she says. Departmentalizing is a cost-neutral way of upgrading instruction because no additional teachers need to be hired and professional development can instead be focused on fewer teachers, she adds…[See the full article for more about platooning, including “Special Benefits for Special Ed.”]


(6) “Using Stimulus Dollars to Decode Human Number Sense” by Lisa De Nike

SourceThe Gazette – Johns Hopkins University

Rats do it; so do monkeys, pigeons and people: They quickly and intuitively size up the number of objects in their environment. It’s this inborn “approximate number system” that enables, for instance, a skittering rodent to quickly ascertain which garbage heap offers the most delectable food waste, or helps an in-a-rush motorist decide which tollbooth lane will offer the quickest passage.

Though research indicates that we are born with this sense (scientists contend that it probably evolved very early, to help animals and our human ancestors survive in the wild), little is known about how number sense changes and develops throughout the human life cycle and how it affects the way people comprehend mathematics.

That’s why a team of psychologists at The Johns Hopkins University’s Krieger School of Arts and Sciences is using a $1.6 million National Institutes of Health grant, underwritten in part by the federal stimulus package, to finance a multifaceted study aimed at decoding some of the mysteries of the human approximate number system, or ANS. They want to find out, for instance, everything from how it changes from infancy through adulthood to the impact that number sense acuity has on later success (or failure) in academic and higher order mathematics.

Lisa Feigenson is teaming up with her research partner and husband, Justin Halberda, both assistant professors in the Department of Psychological and Brain Sciences, on the project.

“What we are setting out to do is to examine numerical ability in infants, young children and adults, and to document how it changes over the course of their development,” said Feigenson, who also co-directs the Homewood campus-based Laboratory for Childhood Development, where she and Halberda conduct research into everything from how infants keep track of objects to whether babies and children are logical and rational when making decisions (they seem to be).

The team came to Johns Hopkins from Harvard University in summer 2004 to establish the laboratory, a brightly painted, toy-strewn, cheery facility where children ages 3 months to 6 years participate in a variety of studies aimed at shining a light on human development’s many aspects.

Though Feigenson specializes in the study of how infants keep track of and remember objects, and Halberda primarily studies logical reasoning and language, this grant merges the partners’ research interests into one large project that will delve into various aspects of the human approximate number system. For instance, though we know that babies, children and adults can all rapidly estimate, without counting, the number of items put before them, an understanding of how this ability changes over time is lacking, Halberda says.

“And we know even less about how this untrained ability interacts with, and even possibly affects, the formal math that we later learn in school. One of our aims is to track individual children over a period of several years so that we can understand how their approximate number system develops, both prior to and during exposure to school mathematics,” he said.

Though Feigenson and Halberda hope to shed light on how the approximate number system changes from infancy through young adulthood, one of the other key questions they aim to answer is whether individual differences in numerical acuity will accurately predict whether children will end up being good math students. Last year, the team published in the journal Nature a study tentatively linking number sense to math achievement in school. The researchers found that knowing how precisely a high school freshman can estimate the number of objects in a group reflects how well he has done in math as far back as kindergarten.

“We learned that good number sense at age 14 correlates with higher scores on standardized math tests throughout a child’s life, and weaker number sense at 14 predicts lower scores on the same tests,” Halberda said. “What this seems to indicate is that the very basic number sense we humans share with animals has some impact on the formal mathematics we learn in school, and these things may inform and interact with each other throughout our lives.”

Halberda says that he is intrigued by numbers because of his interest in logical and scientific thinking. Feigenson has a different reason.

“Numerical knowledge is a case study for understanding human thinking in general,” she said. “Understanding how we perceive and reason about the world around us, whether in terms of numerical quantities, or in terms of social entities, or in terms of the physical world, begins to tell us what it means to be a human thinker.”

Feigenson says that studying the ANS is important because it provides an example of an ability that is innate–present in animals and humans from birth, without training–but that in humans is also significantly shaped by experiences that she calls “uniquely human.”

“It’s only humans, or a small subset of humans, who ever learn to solve quadratic equations or do long division, for that matter,” she said. “The basis for these abilities may be something evolutionarily ancient that we share with rats and monkeys. Because of this, the ANS is an excellent opportunity to study the interaction of knowledge that is inborn with knowledge that is the product of formal instruction.”

In addition to contributing to basic cognitive science knowledge, the team’s research results may ultimately benefit people with dyscalculia, a learning disability that afflicts about one in every 15 people and brings with it innate difficulty in comprehending mathematics and other number-based skills.

“Our project may have important repercussions not only for children with math-specific cognitive deficits but also for studying impairments in understanding numbers, such as happens after people suffer a stroke or other illness,” Feigenson said.

The team’s study is one of more than 250 stimulus-funded research grants totaling more than $114 million that Johns Hopkins has garnered since Congress passed the American Recovery and Revitalization Act of 2009 (informally known by the acronym ARRA)


Related Web sites:

Lisa Feigenson: www.psy.jhu/fs/faculty/feigenson.htm
Justin Halberda’s Web site: www.psy.jhu/fs/faculty/halberda.htm


(7) “Scientists See Numbers Inside People’s Heads” by Charles Q. Choi

Source: LiveScience – 25 September 2009
URL (Abstract):

By carefully analyzing brain activity, scientists can tell what number a person has just seen, research now reveals.

They can similarly tell how many dots a person was presented with.

Past investigations had uncovered brain cells in monkeys that were linked with numbers. Although scientists had found brain regions linked with numerical tasks in humans–the frontal and parietal lobes, to be exact–until now patterns of brain activity linked with specific numbers had proven elusive.

Scientists had 10 volunteers watch either numerals or dots on a screen while a part of their brain known as the intraparietal cortex was scanned–it’s a region of the parietal lobe especially linked with numbers. They next rigorously analyzed brain activity to decipher which patterns might be linked with the numbers the volunteers had observed.

When it came to small numbers of dots, the researchers found that brain activity patterns changed gradually in a way that reflected the ordered nature of the numbers. For example, one might be able to conclude that the pattern for six is between that for five and seven.

In the case of the numerals, the researchers could not detect this same gradual change. This suggests their methods simply might not be sensitive enough to detect this progression yet, or that these symbols are, in fact, coded as more precise, discrete entities in the brain.

“Activation patterns for numbers of dots seem to be stronger–are more easily discriminated–than those for digits, suggesting that maybe still more neurons encode specifically numbers of objects–the evolutionary older representation–than abstract symbolic numbers,” said researcher Evelyn Eger at the University of Paris-Sud in Orsay, France.

Given that numbers “are in principle infinite, it is very unlikely that the brain can have, or we can detect, a signature for each number,” Eger noted. “There is some hint in our data that smaller numbers have a clearer signature, which may be related to their frequency of occurrence in daily life, but further work would be needed to say something more definite about this and about how the brain deals with larger numbers.”

The methods employed in this research could ultimately help unlock how the brain makes sophisticated calculations and how the brain changes as people learn math, the researchers said.

“We are only beginning to access the most basic building blocks that symbolic math probably relies on,” Eger said. “We still have no clear idea of how these number representations interact and are combined in mathematical operations, but the fact that we can resolve them in humans gives hope that at some point we can come up with paradigms that let us address this.”