Mathematics: An Unseen Force

We recently recommended The Joy of X: A Guided Tour of Math, from One to Infinity by Steven Strogatz.  One of the more interesting essays explains some of the mathematics behind interesting phenomena.  He offers the architecture of New York’s Grand Central Station whispering galleries as one such example. In the whispering galleries two people can stand at two points 40 feet across the hallway from each other. If one person whispers “sweet nothings” the second person can clearly hear each word, but passersby cannot hear a word.

While this phenomenon seems like magic, it’s actually based on mathematics. The gallery is elliptically shaped (or oval shaped) which means there are two focus points on the floor where all sound waves will bounce from the walls.

One of my favorite examples of elliptical shaped architecture is the National Hall of Statues in Washington, DC. This is where the U. S. House of Representatives held their sessions from 1807-1857. There are two bronze plaques on the floor. If two people stand on those plaques across the hall from each other, they can talk to one another in a normal conversational tone and not miss hearing a word.

The first time a tour guide told me about these two points, I knew immediately that the two points were focal points of the elliptically-shaped hall. But the tour guide also shared that John Adams had his desk sitting at one of those bronze plaques on the floor and pretend he was sleeping. Actually, he could hear every word that was said among the other representatives. This is because his desk was at the point where all sound waves would bounce off the walls and over to the focal points. One of those points was where John Adams was sitting.

So what is wrong with this story?  John Adams served as our second president from 1797 to 1801. At the end of his one and only term, he retired and moved back to his home in Massachusetts.  While I love the math that is demonstrated in a beautiful and historic buildings, I also appreciate and know enough about  U.S. history to realize that John Adams did not serve as a representative in the original US capital in Washington, DC.  But I still can’t help but smile when I hear the story, as it so beautifully illustrates how much mathematics influences our world in unseen ways.

The Joy of X: A Guided Tour of Math, from One to Infinity

The Joy of X  by Steven Strogatz is a series of essays that explore the seemingly limitless span and beauty of mathematics underlying so much of our universe.   Strogatz’ exploration begins by considering the counting of the Sesame Street characters and extends to the unique and inviting applications of trigonometry, limits, and fractals.  But don’t let that scare you. His examples yield insights into the way mathematics influences politics, art, and nature. And, of course, he explores the way science and technology rely so heavily on mathematics as well.

The author takes a delightful approach by using concrete examples in order to illustrate an abstract concept that even the inexperienced and the less mathematically inclined can understand. A reader with a mathematical background will respond to Strogartz examples with such reactions of “I never thought of it like that!” or “I didn’t know that!”

Even if you’re not a math educator, you will find the material enlightening and enjoyable.

Coming Soon: The New Quantiles.com Website

MetaMetrics®  is pleased to announce that the redesigned Quantiles.com will be released on March 14, 2013. The site has been given an all-inclusive makeover, complete with a brand new look and feel, improved navigation and tablet and mobile compatibility.

The new Quantiles.com will feature:

  • A slick, crisp, design
  • Tablet and mobile compatibility
  • New content and images
  • Redesigned tools such as the “Math Skills Database” and “Textbook Search” featuring improved functionality

In addition to these new site features, we are excited to announce “The Summer Math Challenge” a six-week, e-mail-based initiative designed to combat summer math loss. The initiative, based on the Common Core State Standards, will target students who have just completed grades 2 through 5. Parents will receive emails with resources and activities designed to help their kids retain the math skills learned during the previous school year.

We’d like to invite you to witness the unveiling of the new Quantiles.com first hand. Join us March 14, from 3 to 4 PM EDT, and you just might win free pie! Three lucky people who participate in our “Happy Pi Day… Introducing the New Quantiles.com” webinar will receive gift certificates for a free pie shipped nationwide from Porch Pies in Los Angeles, CA. For more information about the webinar, click here. Register today!

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Maintaining Student Interest in STEM-based Career Paths

According to a recent article in EdWeek, among 2013 high school Seniors, there has been a 21% increase in interest in STEM-related careers, as compared to 2004.   The most significant differences were by gender. Among 2013 seniors who were interested in STEM careers, 38% were males compared with only 15% who were female.  Unfortunately, surveys of students in future graduating classes indicated an even wider gap between genders.

The most disturbing element of the recent report is the outcome of the surveys of high school Freshmen. Of those students who reported interest in STEM-based careers as freshmen, approximately 57% lose their interest in fields of science, technology, engineering, and mathematics.  The federal government estimates that there will be around 8.7 million positions within STEM-related fields, as compared to the 7.4 million positions that currently exist.  In order to meet the demands of the future, it is vital that our educational system maintains and enhances student interest in the sciences, technology, and mathematics throughout their high school career. If students are expressing an interest in these areas early in their high school career, it is certainly an indication that such interest should be sustained and encouraged by their teachers and administrators.

As a mathematics educator, I am pleased to see the budding enthusiasm of high school students for STEM related areas. The challenge now is how best to encourage, support, maintain, and enhance their studies in the sciences to preserve that zeal and excitement and to ready those same students to develop their potential for success in college and STEM based careers.

The Importance of Math Facts in a Digital World

With the wide availability of calculators on phones, mobile devices, computers, and many other electronic devices, some may wonder if it’s still essential for students to commit math facts to memory.  In a world replete with digital assistance is the memorization of math facts still necessary?

Researchers Daniel Ansari and Gavin Price of the Numerical Cognition Laboratory at the University of Western Ontario in London, Canada, and Michèle M. M. Mazzocco, the director of the Math Skills Development Project at Kennedy Krieger Institute in Baltimore, analyzed the links between students’ math achievement and the way their brains processed the most basic problems. Their study was published this month in the Journal of Neuroscience. Interestingly, the study shows that the process in which students compute single-digit math problems may be indicative of how well they perform on college-readiness exams.   Students that scored higher appeared to recall answers from memory while the students that were lower performing used an area of the brain associated with processing, indicating they were working through the problem.   As Ansari comments, “Perhaps the building of those networks early in development go on to facilitate high-level learning, which in turn allows you to free up working memory”.

This study appears to support the idea that fluency with basic math facts is, in fact, an important skill.  There are multiple ways to support the codification of basic math fluency: asking your child to recite basic math facts while riding in the car, while waiting for meals at restaurants, while waiting at the doctor’s office.  Siblings can even quiz each other – serving to not only practice math skills, but to signal the importance of academic achievement.  All of these passive settings provide clear opportunities to reinforce and codify basic math fluency.

From Counting to Calculus: Weaving Mathematical Connections

Many curricular frameworks for teaching mathematics tend to be only a list of mathematics topics to be learned, with no clear elaboration of key ideas or organizing principles.  Because of this, students may not be taught to integrate mathematical ideas, which causes gaps in their knowledge and limits their understanding.  The Quantile® Framework for Mathematics, used in conjunction with the Common Core State Standards for Mathematics (CCSSM), helps teachers identify key connections and provide ways to ensure that students gain a comprehensive understanding of mathematics.

Mathematics is hierarchical and lends itself to learning progressions.  Development of mathematical concepts depends on a student’s understanding of prerequisite concepts.  Learning progressions are curricular frameworks that provide sequencing and guide teachers on proportional use of instructional time.  The CCSSM lend themselves to the development of learning progressions because they provide critical areas for instruction.  The CCSSM aligns content across K-12 so new material clearly builds upon concepts learned previously.

The Quantile Framework for Mathematics and its taxonomy provides a unique way to support the implementation of the CCSSM and address individual student needs by reporting both student ability and difficulty of concepts on the same scale—the Quantile scale.  The taxonomy of the Quantile Framework comprises approximately five hundred skills and concepts called QTaxons.  Each QTaxon is linked to related QTaxons, and these groupings form a knowledge cluster.  Knowledge clusters form a tightly woven web that encompasses the mathematics learned from kindergarten through high school.  By using information about student mathematical ability, the difficulty of the mathematical concepts, and the relationship among mathematical concepts, teachers can effectively target instruction for their students.

For a more detailed description, be sure to check out our latest white paper: Weaving Mathematical Connections from Counting to Calculus: Knowledge Clusters and The Quantile® Framework for Mathematics

Math Differentiation in a Common Core World

Just last week, I was invited to speak at the CCSSO Rural Chiefs Conference in Kansas City on the topic of “Supporting Math Differentiation in a Common Core World”.  While there is much written and discussed on the idea of differentiated instruction, in practice there are limited tools and resources to support math differentiation, a deficiency well-documented in this recent Ed Week article, ‘’Educators in Search of Common Core Resources”.

A theme permeating much of my presentation was the seemingly benign but pernicious neglect of math in our country.  By almost any measure, e.g. instructional time, professional development, number of assessments ,instructional programs, etc… math runs a distant second to reading in the amount of instructional attention given.  At least part of the challenge we face in addressing our math crisis in k-12 education will require that we remedy this neglect.

In my suggestions for addressing this imbalance I focused on four critical strategies. While the adoption of the CCSS is a huge first step in the right direction, its real success will rest upon how effectively we implement these standards.  Along with the implementation of these standards, it is critical that we recognize that math – like any other skill – can be learned.  Too often we subscribe, consciously and unconsciously, to the notion that math achievement is an inherent ability, as if math achievement was based on a “math gene”.  If we take more of a Carolyn Dwek growth perspective, as opposed to a fixed mind set, we will go a long way toward promoting the idea that math achievement is possible for all of our students.

Secondly, we need to build math tools and resources that support differentiated instruction.  Once, when leading a math workshop for a school district, the head of the math department informed me, tongue in cheek, that all math teachers know how to differentiate instruction:  “We say it louder and we repeat it”. This RV (repetition and volume) model is likely to only work if the student is hearing impaired.  Yet I suspect we have all seen variations of this model, this when we continue to drill a student on a math problem or concept to no avail.  Meaningful differentiated instruction is really only possible when we are able to measure a student’s math level and the difficulty of the math concepts and skills on a common scale.  This possibility is now a reality with the Quantile Framework for Mathematics.  Once you know a student’s Quantile measure you know what math skills they are ready to learn.  And just as importantly, one can make sure that the learner has acquired the necessary pre-requisite skills.  Unfortunately, we often continue to employ the “RV” model and fail to drill down and provide differentiated content and instruction to meet the unique needs of the learner. (more…)

It Adds Up: Introducing Mathematics Early

As this recent article makes clear, parents are well-advised to introduce mathematics to their children as early as possible.  Research has found that students unable to read at grade level by the 3rd grade are likely to struggle throughout their academic careers.  But in a study of “School Readiness and Later Achievement,” Greg Duncan and colleagues found that in comparing math, literacy, and social-emotional skills, that the math concepts, e.g. knowledge of numbers and ordinality, were the most powerful predictors of later learning.  In fact, a student’s math skills upon school entry, was a better predictor of math and reading ability by 2nd and 3rd grade than their reading skills upon school entry.

This study provides an empirical basis for what many math educators have been saying for years: that preschool aged children should have a much greater degree of math exposure before they ever set foot in a school.    Sadly, a recent study out of Vanderbilt University reports that math in preschool classes is given short-shrift and is taught only 2.5 percent of the day. When math instruction was increased from 2 percent to 4 percent significant math gains were noted.    Fortunately, an introduction to mathematics need not mean strict adherence to a specific curriculum.  Exposure to mathematics may take a simpler form and there are many meaningful math activities that can be taught in the context of play, e.g. the strategy game Chutes and Ladders, tic-tac-toe, the geometric objects that have to be placed through the correct shape, puzzles, and many more.  So if you are looking for a way to help your preschooler have a better chance to succeed in the early grades continue to look for math activities, have math conversations, problem solve together, and teach them to have fun with math.

Math Education Made Interesting

As the new Common Core State Standards are being implemented in math classrooms around the U.S., middle school educators are facing two challenges:

1. Keeping middle school students interested in learning

2. Meeting the rigor of the new standards

Here’s a little hope for math teachers: A recent survey of middle school students by Raytheon Co., indicates 7 out of 10 students like math!  The survey also indicated that Math is the third most popular subject just behind gym and art.  That’s good news.    Another finding in the survey sheds light on how students prefer to learn new subjects.  48% of students prefer hands-on learning, while  37% of students report preferring to learn with computers.  Dead last in order of preference is lecture from a textbook. 

Fortunately, the Quantile Framework for Mathematics provides teachers easy access to hands-on, computer based, free resources to help spark student interest in learning mathematics.  These resources are aligned with Common Core State Standards, and all 50 state curriculums and are available in the Math Skill Database and Quantile Teacher Assistant.

Additionally, these easy to use tools offered at no cost to educators allow for differentiating instruction to meet the needs of all students without the difficulty of navigating through endless math websites. Dr. Malbert Smith and Jason Turner, in a recent white paper wrote, “As the rigorous Common Core State Standards in Mathe­matics move from the adoption stage into the implemen­tation stage, it is imperative that classroom educators be given the tools and resources that will allow them to move beyond whole-class instruction and begin to differentiate for math students at every level.”  The Quantile website simplifies teacher efforts to locate and utilize relevant materials because these resources are attached to each Common Core standard.

As the implementation of the Common Core standards becomes a reality the Quantile website can be a vital tool in the classroom.  In addition to the tools mentioned for the teacher, the free tools offer a meaningful way to differentiate math instruction for all learners and to link students to resources in a way that can be engaging and fun.

Math Education: Start Early, Start Now

Because a high number of parents report feeling intimidated by math concepts we can assume that uncertainty and unease translates into a failure to routinely discuss math skills and concepts with their own young children.  Even parents not steeped in the technical details of reading comprehension and literacy development often spend time reading with their children; and informal literacy activities, e.g. asking a young child to sound out a word, look at an illustration for context clues, or pick their favorite book are a regular part of many parent’s nightly routines.  So it’s not surprising that math discussions and activities often get left behind.  As this recent article by Annie Paul makes clear, failure to introduce young minds to mathematical concepts at an early age can have serious impact on the student’s readiness to learn math skills later in life:

But speaking to them about numbers, fractions, and decimals? Not so much. And yet studies show that “number talk” at home is a key predictor of young children’s achievement in math once they get to school. Now a new study provides evidence that gender is part of the equation: Parents speak to their daughters about numbers far less than their sons…

The frequency of number talk in the children’s homes had a big impact on how well the youngsters understood basic mathematical concepts such as the cardinal number principle, which holds that the last number reached when counting a set of objects determines the size of the set (“One, two,
three—three apples in the bowl!”). A subsequent study by Levine found that the kind of number talk that most strongly predicted later knowledge of numbers involved counting or labeling sets of objects that are right there in front of parent and child–especially large sets, containing between four and ten objects.

Paul goes on to offer a set of helpful suggestions for introducing ‘number-talk’ early on in a child’s development and urges that parents attempt to incorporate number talk at least as often as they talk about words and letters:

  • Note numbers on signs when you’re walking or driving with children: speed limits and exit numbers, building addresses, sale prices in store windows.
  • Ask children to count how many toys they’re playing with, how many books they’ve pulled out to read, or how many pieces of food are on their plate.
  • Use numbers when you refer to time, dates, and temperatures: how many hours and minutes until bedtime, how many weeks and days until a holiday, the high and low the weatherman predicts for that day.
  • With older children, math can become a part of talking about sports, science, history, video games, or whatever else they’re interested in.

We couldn’t agree more.  Mathematics has received far less attention than literacy at both school and at home.  It’s our hope that parents will recognize the importance of numeracy and that lessons that important must start at home.

MetaMetrics is an educational measurement organization. Our renowned psychometric team develops scientific measures of student achievement that link assessment with targeted instruction to improve learning.