Explorations In Math

Believe it or not

Published on March 29 2012

This week’s guest post is from Ginger Warfield.  Ginger has been a passionate mathematics leader for over 30 years.  She has served on Explorations in Math’s Advisory council for the past 3 years, is a Principal Lecturer Emerita in the Department of Mathematics at the University of Washington and was the former Chair of the Education Committee of the American Women of Mathematics (AWM) and a longtime Education Editor of its Newsletters.  The blog post below is her education column for AWM’s upcoming Newsletter.  If you would like to write a guest post, please contact Dave Gardner.

In the current state of the world, it can be far too easy to focus on the troubles and dangers that beset K-12 education and be drained of energy by that bleak viewing. It was therefore a particular pleasure to me when at a recent conference of WaToToM (Washington Teachers of Teachers of Mathematics) a presentation on the Common Core State Standards (CCSS) permitted me not one but two patches of optimism. In a general effort to spread the sunshine, I decided to present them here.

The first had to do with the Standards themselves. I have been so pleased and excited about the Standards of Mathematical Practice and the way that the CCSS keep them in the foreground, that I missed another key feature. For decades now the phrase “Mile wide, inch deep curriculum” has been so firm a mantra of everyone involved in mathematics education that I have wondered how many household parrots might by now be able to repeat it. It has seemed to me, though, that any efforts to deal with it have simply produced a shuffling of the elements rather than any narrowing or deepening. The writers of the CCSS addressed this problem by standing it on its head: instead of starting by looking at everything a kindergartner should be able to take in and working their way upward, they started at the top and chose a small number of advanced concepts that any educated citizen should have the opportunity to learn. They then worked their way backwards down the levels and produced what they called mathematical progressions that led to these concepts. With those progressions established, they put in some benchmarks for when students need to arrive at specific levels along the route. In doing so, they made strenuous efforts to avoid requiring any topics, even attractive ones, that are not needed for one of the progressions. The goal — plan — hope is that keeping the requirements focused and non-bulky will leave some room for optional topics that teachers choose to teach. In any case, it will provide a structure such that teachers should be able to find out readily which concepts their students should already have available to use and which they are going to need in the next year or two. All of which has at least the potential to keep the curriculum flowing down a narrower, deeper channel. Encouraging.

For the second patch of optimism I need to drop back and throw in a little local history. Back in the 90′s almost all states, my home state of Washington included, produced State Standards. Washington then went farther and was among the relatively few that produced assessments to go with its Standards. The writers of the resulting WASL (Washington Assessment of Student Learning) took note of the absolutely central role of understanding and communicating the ideas behind the (also necessary) mathematical procedures, worked incredibly hard, and produced a test such that if a teacher successfully taught to the test he or she would indeed be teaching what the Standards intended. Unfortunately, the onslaught of No Child Left Behind loaded the WASL with stakes it was not designed to bear, and after a decade or so it wound up essentially eliminated. I was thoroughly disheartened until a conversation with a high school teacher of whom I think a world, in which she said “Yes, but all the teachers I know are teaching very differently and way better as a result of trying to prepare our students for the WASL. We’ve learned a lot!”

Now two large consortia are hard at work producing assessments to correspond to the common Core State Standards. The WaToToM presentation that I mentioned above included some emerging details about the one that Washington is heavily involved in, Smarter Balanced Assessment. Again I was already familiar with a number of key points, such as their determination to de-emphasize summative testing whose results provide information that fails to benefit either students or teachers, and offer formative and interim assessments designed to give teachers information that they can use to improve the learning of the students they have. They also plan to provide a variety of formats of assessments, including some open-ended one or two day projects that will give students an opportunity to demonstrate a very different set of abilities from those required for timed multiple choice or short answer tests (which will also be used, but in moderation.) Smarter Balanced has a long way to go, but what struck me as I listened was that if they succeed in following the path they have laid out for themselves, then they, too, will have created an assessment such that teachers who “teach to the test” will be giving their students exactly what they need, and learning a lot as they do so.

These are harrowing times, and not even my Pollyanna side can maintain that The Solution has been achieved. But despair has very few virtues, and correspondingly I feel enormous gratitude for the incredible efforts the writers of the CCSS and the folks at the Smarter Balanced consortium have put in and are still putting in. I plan to hang fiercely onto the hope they are proffering!

Think About This

Published on March 19 2012

“Mathematics, in the common lay view, is a static discipline based on formulas…But outside the public view, mathematics continues to grow at a rapid rate…the guide to this growth is not calculation and formulas, but an open-ended search for pattern.” -  Lynn Arthur Steen, from On the Shoulders of Giants

The more I think about this quote, the more I like it. Note that Steen isn’t saying that calculation and formulas are not part of math or that they’re not necessary. I’m sure he’d agree that they are a very necessary part of math. But what drives math, he is saying, is the open-ended search for patterns.

The open-ended search for patterns. Two things strike me about this phrase. First, growth in math, whether we’re talking about the growth of the discipline itself or the growth of students learning math (which is what I’m focusing on), occurs when students are encouraged to explore math ideas, play with them, discover the underlying concepts and then apply them in practical situations. Of course they need the calculation and formulas that Steen mentions, but if we confine students only to calculations and formulas, if we don’t encourage divergent thinking and exploration, then our students will be mechanical solvers rather than creative problem solvers.

The other thing I like is his emphasis on patterns. Math has been called the science of patterns and patterns are everywhere in math, often in unexpected places. For example, take this problem I present to third graders:

Kim has 3 hats and 3 jackets. How many different combinations of a

hat and a jacket can she wear?

The simplest way for young children to solve this is with a diagram: connect the hats and coats and then count the connections:

The answer, of course, is 9. But what’s important here, and what hooks students every time, is the discovery of a very cool visual pattern. They begin to understand that math is not a morass of discrete algorithms, procedures, rules, and formulas but, rather, a unified and integrated whole that is logically consistent and makes sense to them.

Another good example of math patterns is Pascal’s Triangle:

In this triangle, if you add the numbers in each row you get consecutive powers of 2. Each row (other than the first) is a multiple of 11. If you look at the diagonals, each one presents us with a pattern: the counting numbers or triangular numbers for example. The diagonal next to the triangular numbers is a pattern where the pattern grows by triangular numbers. And then there’s the hockey stick pattern. (See if you can find it before going online for the answer.)

Finally, type in ‘Math Doodles’ in your search engine and you’ll be rewarded with short videos that explore math patterns in a delightful way. Here’s one of my favorite sites to explore.

Dave Gardner

Mathematician in Residence



Experience the Julia Robinson Festival!

Published on March 12 2012

This week’s guest post is from Dan Finkel from Math for Love. If you would like to write a guest post, please contact Dave Gardner.

Seattle’s first Julia Robinson Mathematics Festival is from 9:00 am to noon on Sunday, March 18, 2012 at the Evergreen School, 15201 Meridian Avenue North, Shoreline, WA. The festival is open to 4th-12th graders and their parents and teachers.

The festival features an abundance of inspiring mathematical activities for students to explore, led by mathematicians, teachers, and volunteers. These activities will range from levels that older elementary students will enjoy, to levels that will challenge the brightest high school students.

At the festival, there will be tables set up, and each table will be staffed by one or two volunteers to introduce the students to a mathematical problem or puzzle that requires creativity and exploration to solve. The activities are designed to engage students for about 20-30 minutes. The students will have an entire morning to explore a number of fascinating, interesting problems in mathematics with the help and encouragement of our volunteers.

The Julia Robinson Festival is challenging but noncompetitive, highlighting rich mathematical problems curated by adults who love math. It’s a perfect place for students who love math, games, or puzzles; for contest-goers looking to see what else math is about; or students who prefer the collaborative, noncompetitive aspects of math to contests.

You can register here.

A New Chapter of Growth for Explorations in Math

Published on March 6 2012

This week’s guest post is from Steve Brugger, Board Chair for Explorations in Math. If you would like to write a guest post, please contact Dave Gardner.

A few months into 2012 and it’s already shaping up to be a big year for Explorations in Math.  My name is Steve Brugger and, after 4+ years on the board of EIM, I’ve been asked to take on the position of Board Chair. During those four years I experienced first-hand the amazing dedication and commitment of Katie Klein and then Laura Larson in the same role, so I did not make the decision to become board chair lightly. But what an amazing time to have this opportunity!

Our most recent newsletter goes to the heart of why I am so excited.  Over  the past 4 years I have watched as the EIM staff have worked tirelessly to improve each of our in-school programs, develop better measurement and effectiveness tools, increase the size and quality of our community events and build our base of corporate and community supporters.  Our goal was to build a solid foundation – a launch platform to take our message of fun, relevant and engaging math programs for elementary students to an ever widening audience.  And now, we are doing just that - growing!

In our own backyard, our success with our Seattle school partners, and on the Eastside with our successful launch into the Lake Washington School District, has gotten enough “buzz” that we are now being approached by other schools and districts that want to implement our programs to build a positive math culture in their schools.  The word is certainly getting out!

And our two pilot programs are just as exciting. In partnership with the Pacific Science Center, we are integrating our programs into their successful science outreach efforts that service communities and schools, and will be able to touch thousands of students and families throughout the state.

Our second pilot program came about in a somewhat serendipitous way.  Last year we invited Greg White, the CEO of LEARN Charter School Network in Chicago, to speak at our annual fundraiser about building a culture of success in inner city schools.  He knew very little of our work in our community but when he arrived here and saw our programs in action, he knew he had to present us to his staff.  Our team immediately won them over and just last month we launched EIM programming in the South Chicago LEARN Campus – and are planning launches in five more!

I am honored to be able to work with the staff, board and all our loyal supporters who believe so strongly in the mission of EIM - to build a sustainable math culture in elementary school communities. Thanks to you, we ARE making a difference and helping our kids to be successful in tomorrow’s world.


My Math Journey

Published on February 23 2012

This week’s guest post is from Tammie Schacher, Explorations in Math’s Executive Director.   If you would like to write a guest post, please contact Dave Gardner.

I wouldn’t call myself a “math geek” although my kids might.  I like math but more importantly I like what knowing math has allowed me.  As with most if not all in the profession, I knew that I wanted to be an architect from a very young age.  Maybe this came partially from being on construction sites with my stepfather and partially from my love of drawing and painting, but once I had decided, I began researching what I needed to do to follow my dream (maybe I have always been a geek).  It didn’t take long to see that math would forever be part of my life.

When I got to high school, I didn’t find too many other girls in my advanced math classes, but I didn’t find many girls in my drafting or computer aided drawing classes either, so I didn’t really think about it.  I concluded that I’d picked a path that not too many girls choose, and that was okay with me at the time.  My friends would often tell me they were jealous that I was in class with all boys, but math? Eww! The ratio of men to women in the architecture field was also obvious and became a concern to me.

When I was practicing architecture, I focused my career on public and affordable housing and had the opportunity to meet with families all over the country.  What struck me as we talked with families each and every visit was the limits kids felt existed.  Those limits almost always were created in their minds or by their environment! I remember seeing a little kid’s face light up when they discovered that they could be an architect too; it had never even crossed their mind.

Sometimes, a spark of inspiration and support is all that a child needs to create a dream, which is what led me to Explorations in Math.  I came here because math opens doors and I want to share with kids and their families that math can help them reach their dreams.  We are improving people’s relationship with math by developing a culture that supports learning and shows that every child can succeed.  That is the spark that inspires me!

What’s Math Got to do with it?

Published on February 15 2012

This isn’t a rhetorical question but, rather, the title of a ‘must read’ math book, whether you’re a parent (or other caregiver) or working in a classroom with students in math. The first thing you need to know about the book is that it is not a dry, scholarly work full of theories and equations. Quite the contrary. What’s Math Got to do with it is a thoroughly enjoyable read. The author, Stanford mathematics education professor Jo Boaler, followed the progress of middle- and high-school students over a number of years, observing teachers who could engage their students in math and just how they did it. She also observed classes where the students were not engaged and in her book points out the differences between those who do engage their classes and those who do not.

What she found should be required reading for all of us. A partial list of chapters is indicative of how much ground she covers: What’s Going Wrong in Classrooms? Stuck in the Slow Lane. Paying the Price for Sugar and Spice. Giving Children the Best Mathematical Start.

She provides many enlightening examples of how children think about math and there are math puzzles and problems (and solutions!) for the reader to solve. For example:

Given a 5-liter jar and a 3-liter jar and an unlimited supply of water, how can you measure out exactly 4 liters of water?

Or this one:

Race to Twenty is a game for two people. Play starts at 0, Player 1 adds either 1 or 2 to 0 and announces the sum. Player 2 adds either 1 or 2 to that sum and announces the new sum. Play continues this way. Whoever gets to 20 is the winner.

Check out the book here.

BTW, don’t be put off by the fact that she followed middle- and high-schoolers. Just about everything in her book applies equally to elementary students as well.

Dave Gardner, Mathematician in Residence

Everyone can do math

Published on January 30 2012

This week’s guest post is from Laura Larson. Laura serves as Explorations in Math’s Board Chair and founded Explorations in Math around a kitchen table 9 years ago.  If you would like write a guest post, please contact Dave Gardner.

Last week I saw an ad for tutoring services that made me laugh out loud.  A teenage son is doing homework and we hear him calling loudly, “Mom, I need help with my math homework”.  Panic stricken, she turns around, runs out of the house and down the street…until at last she reaches the tutoring center, where help is found.  I think that ad tapped into a (seemingly) fundamental parental truth: ask me about anything you want, just don’t ask me to help with your math homework.

When my children were quite young, I read an article discussing an experiment.  Scientists had created a glass floor for babies to crawl across, parent waiting on the other side.  They discovered that if parents were encouraging and confident, the babies would crawl across. If, on the other hand, the parents expressed worry and concern, the babies refused to cross and started to cry.

As a parent you may or may not have the math skills to help your child with their homework, be it 3rd grade or advanced algebra.  But what you do have is the ability to confidently express your belief that your child will be successful.  Our children look to us, more than we realize sometimes, for clues on what to expect.  “Is this something I can do, or not,” they wonder.  Is math for some people, but maybe not me?   Maybe you’re not sure.

But one thing we do know is everyone can do math.  Difficulties and even getting things wrong the first time, are the mental challenges upon which all learning is built.  Your voice, your assurance that yes, this problem has a solution and one way or the other, success will be theirs, can make the difference between a child who gives up on their chances and one who’ll stop at nothing.  If persistence is the key to success, our children must know and believe that they are capable of reaching the other side.  Remind them that the harder the problem, the better they’ll feel once they find the solution.  Sometimes the best help of all is your unwavering belief in them.  Math matters to every child’s future.  And your belief in their ability to succeed at math, matters too.


Cool Math Websites

Published on January 23 2012

We all know there are scores, hundreds, thousands of math-related websites out there. You can spend hours searching and then exploring different sites. Here’s a shortcut: periodically, I’ll be sharing some good websites with you including a summary of the content, whether or not it has ads, whether or not I think it’s worthwhile and, of course, the URL. I have checked them all recently. Meanwhile, here’s the first go-round:


This site lives up to its name: it offers math in fun ways for students, teachers and families. It is aesthetically inviting and user-friendly. You can choose from a variety of math topics from the “Math Menu,” such as Numbers, Algebra or Geometry. After selecting Geometry, for instance, you can choose from a well laid-out, long list of sub-headings (Angles, Plane & Solid Geometry and Using Drafting Tools), which offer explanations, diagrams, formulas, etc. The site includes a comprehensive “Illustrated Math Dictionary,” lots of “Math Puzzles” and cool interactive, online “Math Games,” as well as “Math Worksheets” for students. You can also participate in the “Math Forum,” where users have posted/responded to math questions, math resources, puzzles/games and even age appropriate jokes. This site is a very engaging educational resource for teachers and students alike. You will have to sign up for the Math Forum but that’s quick and simple. The site carries no ads.


Wonderfully accessible resources for teachers, students and families:  clicking on the math index directs the user to math content, where a number of math problems and activities can be accessed.  This website is also helpful because it cites the specific content/math reasoning used for each problem. It’s bright and colorful, and greatly appeals to students. It’s easy to navigate and all information is easily accessible and is completely free of charge.  Additionally, this website can be accessed entirely in Spanish. There is also a “Family Corner” with tips for parents and challenges for the whole family. The site is geared to upper elementary and middle school. This is an NCTM (National Council of Teachers of Mathematics) site. It is ad-free.

You can find additional websites that our team recommends here.  If you have some websites to recommend, please leave that information in the “comments” box for others to read. Thanks!

Dave Gardner

Mathematician in Residenc


Published on January 16 2012

One of my colleagues in the office the other day coined the term “mathitude” to indicate a positive attitude toward math, whether in children or adults. There was a perfect example of that last week in a first grade class at an EIM member school. The students were playing a game and at one point a girl, with a big smile, said to her partner, “I love math!” and her partner, also smiling, nodded in agreement. Clearly, the teacher in this classroom is doing far more than just instilling math competence in her students; she is also instilling math confidence by making sure her students are engaged in math and enjoying the experience.

Contrast that with this situation. We were having dinner at some friends’ the other night. After dinner, we decided to play Apples to Apples®, a game EIM uses and recommends. I’d played it before and told the others that it uses the same kind of mental skills that math requires: flexible thinking and making connections. At the word “math,” one woman blanched and sputtered, “I hate math! Math scares me!”

We need to ask ourselves: Do we want our children, our students, to grow up with this kind of paralyzing feeling about math? Or do we want to instill in them the “mathitude” shown by the first grade girls above? Another first grade boy who, as I was saying my goodbyes to the class after a fun session last year, hastily scrawled a sign and held it up for me to read: “Math Rocks!” I took a photo of him which you can see below.

Let’s hear it for MATHITUDE!

Math Safety Nets

Published on January 9 2012

Our children and our students can be easily intimidated by math and can start developing math anxiety, even math phobia, at a young age. There’s no question that this will seriously hinder their ability to learn, let alone enjoy, math as they go through the grades. One of the things we can do, as parents and as teachers, is to build in safety nets when we work with our children and students. Here are some ideas, things I’ve found to be effective:

• If a child gives a wrong answer, I tell her that that’s OK—I like wrong answers. They’re not as good as correct answers, but here’s why wrong answers are also good: They tell me three good things about her: she’s listening, she’ s thinking and she’s trying, and what more can we ask? Any child who listens, thinks and tries is going to succeed.

• In a similar vein, when a child gives a wrong answer I tell him that it’s OK to be wrong. All of us are wrong sometimes but we learn from our mistakes. I give the example of falling off a bike when learning to ride. Every time you fall off a bike, you’ve done something “wrong,” but you learn from that, get back on and do better.

• Sometimes I’ll ask a question and the student won’t respond. Usually that means he doesn’t know but doesn’t want to say so for fear of appearing “stupid.” In this case, after a few seconds of silence, I simply say, “You know, it’s OK not to know. Nobody knows everything and we’re all learning.”

• Don’t tell your child or your students that math is “hard.” A much better word is “challenging.” In a student’s mind (and for a lot of adults), “hard” is too closely associated with boring and failure. “Challenging,” on the other hand, can be presented as a way of the child testing him or herself. “Challenging” carries it with the possibility of being interesting, rewarding, even fun.

In short, let’s do all we can to instill in our kids the idea that math is, indeed interesting, rewarding, even fun. And, above all, doable.

Dave Gardner

Mathematician in Residence