Sunday, May 11, 2014

May must be Latin for 'Hectic'

May 2014 Update

In this issue:  

  • Fall VCTM Conference Date Set
  • Common Core Statistics Course Offering
  • Failing Schools, Again?  Really?
  • 2048 Update


Mark Your Calendar- the VCTM Fall Conference Date is set;  The Vermont Council of Teachers of Mathematics (VCTM) invites you to our Annual Conference, “Teaching Mathematics Today” to be held on Friday, October 17 at St. Michael’s College in Colchester.

Join us for a day of conversation, presentations, idea sharing, and resource gathering. Network with mathematics educators from across the state and learn about what’s working well in other teachers’ classrooms. If you have ideas to share related to the topics of teaching and assessing the Common Core State Standards in Mathematics; SBAC; Project-based instruction; and Heterogeneous grouping K-12, please keep an eye out for our forthcoming call for presentation proposals. Registration will open in late summer. We hope to see you in October!


Common Core Stats Course Offering

Common Core Statistics for Secondary Education Teachers (61074  EDCI 200 Z1); Meeting Dates & Times: June 23-27 (8 AM - 4 PM)

The latest research on purposes for including probability and statistics in a mathematics curriculum will be reviewed and discussed.  Methods for employing best practices with respect to teaching probability and statistics will be presented throughout this course.  Participants will explore problems that require them to formulate questions, gather, display, summarize, examine and interpret data.  We will model data and use probability models and random processes to discover patterns and deviations from patterns.

Participants will investigate how students’ learning of statistics and probability is different than mathematics.  Teachers in this course will be given various instructional resources, including lesson plans and activities, technology resources, web applets, data sets and appropriate investigative assignments. At the conclusion of the course, teachers will be familiar with the Common Core probability and statistics standards for high school students and develop an appropriate investigation with a scoring guideline to use for assessment.

UVM Summer University registration can be found using this link.  Or contact Stacey Anthony from Essex HS ( if you would like any additional information.


The sky is allegedly falling.  Again.  *Sigh*   
Newsflash:  Recent NAEP data reveals that schools are not improving in math.  Again.  Really?  Are teachers really failing?  Or are students failing? Is everyone failing, or is this a measure that is not in sync with the changes that we have and are making in today's schools?  

What's your opinion about what needs to be measured?  Creativity?  Problem solving?  Basic skills?  Our ability to solve pure math problems against the rest of the world?   What is it that will really show the public that we are doing our jobs well?   [And, if we are really failing, then why are most of the new cutting edge computer companies here in the US?] 

We'd love for you to comment on this!!


2048 Update:  In the last month, 2048 went completely viral and it became cool to tell someone that you reached the 4096 or 8192 level.  Believe it or not, I have a student who reached the 2^17 level- scoring an incredible 2.85 million points!!  Personally, I'm a bit more pedestrian at about 59,000 points- how about you?

In case you are wondering, 2048 is a free app that is way too addictive, but lots of fun.  

Tuesday, April 1, 2014

April 2014

April Update:

Vermont Council of Teachers of Math eNewsletter....   so much is going on in the world of math education, and if you are like me, it all seems a little overwhelming at times.  There are various course and workshop offerings in this posting and being offered by the state.  I encourage you to take the time and go to one or more of them.   The more we know and understand about the changes, the easier they will be in the long run.  

As always, if you have any comments or suggestions, please do not hesitate to send it along to me .  - David Rome, VCTM eNewsletter Editor

In this issue:
  • VCTM Looking to Meet your Needs At a Fall 2014 Conference
  • Interview article:  Looking at math more globally
  • Middle School Free Space!
  • How To Learn Math:  For Parents and Teachers
  • 2048:  The Latest Game Craze 'Crushes' the Competition

VCTM Looking to Meet your Needs At a Fall 2014 Conference.    The VCTM conference committee is looking for your input as to what you would like to hear about at the upcoming fall conference- would you take just one or two minutes and fill out this brief questionnaire?  


Interview article:  Looking at math more globally.  This interview with math professor Steven Strogatz recently appeared in Math Horizons.  It speaks to the needs of the common core quite nicely.  A good read, for sure. 


Middle School Free Space!   A free seminar is available for either May 5, May 15, or May 29th in the comfort of your home from 6:30 to 7:30 EST.  

Engineering (the 'E' in STEM) is a new way to get students engaged in designing their own hands-on projects, and in this webinar you will learn how to make it easier for teachers to do this using everyday materials to keep down that cost. 
Log in from home or from school on your own computer and join this chat-friendly presentation about a satellite engineering hands-on activity (math based and fun for all ages, focused on grades 6-8).  All you need is internet!  The presentation will also focus on best teaching practices aligned with current standards, and provide many NASA resources for hands-on activities for educators K-12.  You receive a certificate for attending plus 1.5 pd hours sent directly to your registered email address.


How To Learn Math:  For Parents and Teachers  Explore the new research ideas on mathematics learning and student mindsets that can transform students' experiences with math.  Whether you are a teacher preparing to implement the new Common Core State Standards, a parent wanting to give your children the best math start in life, and administrator wanting to know ways to encourage math teachers or another helper of math learners, this course will help you.  The sessions are all interactive and include various thinking tasks to promote active engagement- such as reflecting on videos, designing lessons, and discussing ideas with peers.  Taught by Jo Boaler, professor of mathematics education, Stanford University, you will learn useful ideas and practices that you can apply immediately, for example. 
The course will be self-paced, you can start and end the course at any time in the months it is open.

In the first run of the course many school districts in the US gave 16 professional development hours to the teachers who took the course - which means finishing the course and completing most of the assignments. Stanford University makes no representations that participation in the course, including participation leading to a record of completion, will be accepted by any school district or other entity as evidence of professional development. Participants are solely responsible for determining whether participation in the course, including obtaining a record of completion, will be accepted by a school district, or any other entity, as evidence of professional development coursework.  The course costs $125..  No other materials or textbooks need to be purchased.  Sign up or find more information at this site. 


2048:  The Latest Game Craze 'Crushes' the Competition  

If you are like me and have all kinds of time to spend on silly video games (this is sarcasm), you might want to check out the new free app on iPhones and iPads '2048'.  Why, you ask?  Is this not just another Angry Birds or Candy Crush?  Well, yes, from a time-waste perspective.....but this one is mathematical!  Invented recently (in one weekend!) by Gabriele Cirulli, a 19 year-old Italian student, it is sweeping the globe in popularity, so stay ahead of your students and boast your personal best  score!  It takes about 1 minute to learn, and one game to figure out ways to get better.  If you do have strategies for getting better scores, make comments below and we'll see who is the first to get to 2048!

Friday, February 21, 2014

February Update

A couple of things to share with you all this month.  A 'full' posting in Mid March to follow....

1.  A workshop offering in mid-MarchMcGraw Hill

2.  Governors Institute Offering: 
Do you know a high school student who is interested in math, but unsure what a career in math would look like? At the Governor's Institute on Mathematical Sciences, students get to find out! The Mathematical Sciences Institute, held at UVM from June 22-27, gives students hands-on experience with advanced math, the chance to explore mathematical concepts alongside other talented math students, and receive valuable mentorship from leading mathematicians. Throughout the week, students make friends from all across Vermont who share the same academic passions while living on the UVM campus and experiencing college life.
Here is what some recent GIV Mathematical Sciences grads have to say about the Institute:
“The lectures were very insightful and intellectually stimulating, and they also offered several career possibilities if we choose to study math beyond high school. Overall, it was an amazing experience… I enjoy math but I never knew about all the possibilities math has to offer. This camp made me realize that I could make a life out of this subject, something I had never considered before.”
“Having the opportunity to delve into interesting fields of mathematics with equally interested and passionate peers was a rare and precious experience.  It was awesome to be in an environment where everyone loved learning and working with math.”
All motivated Vermont students should apply to the Governor's Institutes. Other Institutes offered by GIV include Arts, Asian Cultures, Information Technology, Engineering, Entrepreneurship, Environmental Science and Technology, and Food, Farms, and Your Future. Students apply by visiting our website at and completing the application method chosen by their school. 
Please encourage interested students to apply!
Valerie Russell
Administrative Coordinator
The Governor's Institutes of Vermont
4049 Williston Rd. #4
South Burlington, VT 05403
Phone:  802-865-4448

Tuesday, January 14, 2014

January 2014 eNewsletter

Vermont Council of Teachers of Math eNewsletter!  This newsletter is now being seen by about 700 math educators all around VT, and many have registered since the first issue last month.  If you know someone who might benefit from this newsletter or blog, please feel free to forward this along to them and have them visit  our blog site 'Math in Vermont'. Our mission is to have Vermont be one interconnected math network, sharing our ideas and strategies, and showing how we should all be cognizant of each others work in all grade levels.  To date, about 40% of those on the mailing list work with students under age 11 and this gives us all an opportunity to appreciate what they are doing in the classroom.  So if you would like to submit an article (as a few members have done in this newsletter) or have comments or suggestions, please send it along to me .  - David Rome, VCTM eNewsletter Editor


In this issue:
  • President's Message- Jason Cushner
  • The Myth of Lagging U.S. Schools by Alfie Kohn
  • A Device to Identify and Calm Math Anxiety
  • Presidential Award Winners and Finalists Announced
  • The Price is Right- Free Mathematicians in Your Classroom
  • The Greater Burlington Math League
  • Book Review:  The Impact of Identity in K-8 Mathematics; Rethinking Equity -Based Practices
  • Profile:  Q&A with VCTM President Jason Cushner
  • Article:  Helping Students Work Through Mathematical Conjectures
  • Special Winter Offer for Girls by the Governors Institute 

President's Message    January 2014

Happy New Year to you all!  As I work with different folks around implementing the Common Core State Standards, I am struck by the gravitational pull towards the content standards instead of the Mathematical Practices.  So much of our curriculum and resources have been focused on content, while part of the shift of Common Core in theory is to reduce the amount of content in a year to have the space to focus on the Math Practices .
Content may be more aligned with what most of us have been doing and is easier to measure.  That being said, it’s not necessarily what is best in preparing our students for their futures.  The question is how to go about emphasizing the Mathematical Practices.  If you have great methods,  please share.
One method I learned about through reading Ben Blumsmith’s Blog is getting students to summarize each others ideas and critique them, as opposed to the teacher jumping in as the expert and saying whether something is right or not.  Having students summarize each others theories both increases skills in mathematical communication and can serve as an easy lead in to mathematical discourse.
Have  a great new year and stay in touch about how we can support you in math excellence for all.

Jason Cushner
VCTM President
Program Coordinator Big Picture South Burlington
Rowland Fellow


We're Number Umpteenth!  The Myth of Lagging U.S. Schools  Copyright 2013 by Alfie Kohn.  Reprinted with the author's permission.  For more, please visit

Attacking the quality of education in America has become a sport that has been spreading for decades.  Educational topic writer Alfie Kohn recently put several of the critics arguments in context, and should give us all a moment to step back and contemplate the larger picture of what is really going on.  To read the full article, go to the posting following this one below and please feel free to comment at the bottom of the article and even get a response back from the author himself.  A brief excerpt from the article:  
"A dedicated group of education experts has been challenging this canard for years, but their writings rarely appear in popular publications, and each of their efforts at debunking typically focuses on just one of the many problems with the claim.  Here, then, is the big picture:  a concise overview of the multiple responses you might offer the next time someone declares that American kids come up short.  (First, though, I’d suggest politely inquiring as to the evidence for his or her statement.  The wholly unsatisfactory reply you’re likely to receive may constitute a rebuttal in its own right.)


A Tool for Taming Fear and Anxiety in Learning Math, Submitted by Sue Abrams, Montpelier High School 

We all know that in the ideal world of math instruction, good lessons are those in which students construct their understanding of math concepts based on experiences with models, engaging real world problems, etc. Good, probing questions by the teacher or classmates can also help a student attain those "aha" moments that are essential for true learning.
However, students arrive at our doors with emotional barriers that can shut down their openness to engaging in the learning process in any meaningful way. Some students have had years of accumulated negative feelings associated with learning math, and others appear successful but are extremely anxious about their math ability nonetheless.

With this in mind, I was prompted to tickle the edge of this math anxiety by trying a new tool that is often used by people in the medical field to help people with anxiety and stress. It's called the Inner Balance Trainer sold by the Institute of HeartMath in Boulder Creek, California. This module attaches to an iPad or iPhone to activate an app which allows a student to monitor in real time their HRV, heart rate variability, as they practice breathing techniques with the aid of a breath pacer. The strength of this is the very visible feedback students get when they see their HRV change from very random jumping around to looking much like a smooth sine wave in a matter of minutes. Furthermore, it takes only about five minutes or so per day to make a significant impact.

With the help of a sizable grant from the Institute of HeartMath and another grant from the Montpelier High School Boosters Club, I was able to purchase a set of 22 Inner Balance Trainers and have begun to introduce them to students in math classes. Our students will have the option to participate in a research project over a few months, while others will drop in during our 15 minute recess to try them out or just use them before assessments or even regular math lessons. Sue Beem, another member of the math department, is a collaborator on the project, and although the results are not yet known, we are excited to get right to the heart of the anxiousness or stress students can feel in math class by potentially helping them feel the inner balance needed to open the door to learning. You are welcome to get in touch with me if you have questions at


Two Vermont Math Teachers Chosen for Presidential Teaching Award for 2012 as 2013 Finalists are Announced
Joy Dobson and Mary Ellis were selected for the 2012 Presidential Award for Excellence in Math and Science Teaching (PAEMST) for teachers in grades K-6 for 2012.  Joy Dobson has served as an educator at Weybridge Elementary in the Addison Central Supervisory Union for 20 years. She has taught single and multigraded classrooms of kindergarten through second grade.   Mary Ellis has been an educator at Enosburg Elementary School for the past 15 years. Currently a fourth grade teacher, Mary teaches science, social studies, and mathematics to her homeroom class and provides mathematics instruction to another fourth grade class each day. She has taught fifth and sixth grade and served as the school’s Mathematics Coach. In her role as coach, she modeled lessons, collected and organized classroom data for analysis, and collaborated with teachers to improve instruction.
Joy provides an exceptionally strong mathematics program and ensures that students have the mathematical language, concepts, skills, and models to approach and apply mathematics with confidence and clear thinking. She is dedicated to not only addressing the needs of all students, but also to ensuring that their learning meets the very highest expectations. This work is founded on her unfailing belief in each child as a capable thinker and learner.  As a teacher leader, Joy is held in high regard throughout the supervisory union. She serves as a district-level Math Leader, collaboratively designing and implementing professional development for teachers, developing curriculum, and serving as a studio teacher. As a mathematics mentor, Joy is known for her open-door policy, encouraging teachers to observe, ask questions, and grow professionally while experiencing a multigraded classroom and differentiated instruction with high academic expectations. She has a B.S. in elementary education from Nyack College and an M.Ed. in education from the University of Vermont. She is certified in elementary education
A lifelong learner, has participated in many mathematics and technology professional development opportunities. She has a deep understanding of elementary mathematics content and research-based instructional strategies. She is very involved in the Franklin Northeast Supervisory Union. She has served on task forces to develop mathematics curriculum and assessments, has facilitated districtwide in-service meetings, and is part of a district committee focused on the implementation of the Common Core State Standards for Mathematics. Mary has a B.A. in psychology from Boston College and an M.Ed. in curriculum and instruction from the University of Vermont through the Vermont Mathematics Initiative. She is certified in elementary education.
Each year, a national committee of distinguished scientists, mathematicians and educators recommends up to 108 teachers to receive PAEMST awards - up to two teachers –mathematics or science- from each state, the District of Columbia, Puerto Rico, the U.S. territories as a group, and the schools operated in the United States and overseas by the Department of Defense Education Activity. The awards program is administered by the National Science Foundation (NSF) on behalf of the White House.   PAEMST awardees receive a $10,000 award, a Presidential certificate and a trip to Washington, DC, for a series of recognition events, professional development activities and an awards ceremony.  Congratulations to Joy and Mary!
Meanwhile, the VCTM has announced the Vermont state finalists for the 2013 PAEMST awards for grades 7-12.  They are Sue Abrams  (Montpelier High School),  Erin Danner (Mill River Union High School), Julie St. Martin (Champlain Valley Union High School), and David Rome (Burlington High School).   All four finalists were recognized at the Association for Teachers of Mathematics of New England fall conference in November.   They are now eligible to receive a PAEMST award, the nation’s highest honor for U.S. mathematics and science teachers for grades K-12. 
The 2014 PAEMST will honor mathematics and science (including computer science) teachers working in grades K-6 and nominations are open right now, but will close on April 1, 2014,  Visit the PAEMST website for more details.


The Price is Right- Free Mathematicians in Your Classroom    from John Devino of the Vermont Math Coalition

As classroom teachers, we are continually looking for ways to make mathematics relevant and meaningful to our students.  This would be a good time to bring a mathematician to your classroom, who is capable of reaching your students.  With the weather being a “hot topic” (sorry!) lately, Janel Hanrahan’s presentation entitled “Mathematics and the Atmosphere” seems like a natural.  

The Vermont State Mathematics Coalition has 26 programs available for you and your students.  It all begins with an e-mail or phone call to one of the presenters.
Go to and click on the link for Expanding Horizons.  If you have questions or comments, contact me or call me.    Phone:  (802) 863-5403


The Greater Burlington Math League  by Janet Soltau, Colchester High School

The Greater Burlington Math League has been competing in the greater Chittenden County area for over 50 years.  The league,  founded by Tony Trono of Burlington High School in 1961 has had a varying number of teams of the years, but currently has 12 schools participating with approximately 240 participants.  Every month teams gather at a different school to compete in 5 different categories: Arithmetic, Geometry, Algebra, Advanced Math, and Team.  Topics within the categories vary from meet to meet, with the team test including topics from each of the categories, which 5 students get to work on together. The majority of the season students do all math with out a calculator (with the exception of the team test), which is a great mental challenge.

In recent history, Essex and South Burlington, have traded the title back and forth. Awards are given to the top students in each category at the end of the season, and the top ten individuals overall. The top team this year will win $150 sponsored by the Mu Alpha Theta, a mathematical honor society.

Current participating high schools are BFA St. Albans, Burlington, Colchester, CVU, Essex, Milton, Missisquoi, Mt. Abe, Mt. Mansfield, Rice, South Burlington, and Vergennes. Past participants have included BFA Fairfax, Middlebury and Winooski.  For more information, contact Janet 

Math league coaches also encourage students to participate in the Vermont Mathematics Coalition Talent Search ( Top students from the talent search, math league, the UVM math test and the AMC math test are recruited to represent the state of Vermont in the American Regional Math League.


The Impact of Identity in K-8 Mathematics; Rethinking Equity   Copyright © 2013 by the National Council of Teachers of Mathematics, Inc.,

In this book by authors Julia Aguirre, Karne Mayfield-ingram, and Danny Bernard Martin, the math community of parents and teachers, rather than just the latter, in the success of the students. Among the highlights of the book: 
  • Mathematics Identity is the dispositions and deeply held beliefs that students develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics in powerful ways across the contexts of their lives.
  •  Mathematics is both the gateway and the gatekeeper to learning opportunities in school and beyond
  • It is critical to focus on finding ways to build on the resilience created by a child’s life conditions rather than on the conditions themselves; focus on learning not labeling.
  • Feedback that students receive on assessments can contribute positively or negatively to a child’s disposition about his/her ability to learn mathematics based on how the feedback is delivered, what information is shared with the student, and  whether or not the students is involved in self-reflection.
  • How a child’s progress is communicated to parents can affect how they see their child as a learner of mathematics.
  • Regardless of family income, education, or cultural background  if parents are involved in their child’s education those students experience higher academic achievements than those whose parents are not involved
  • Additional Authors to consider on this topic: Jo Boaler, Vi Hart, and Robert Q. Berry, III 


Profile:  Q&A with VCTM President Jason Cushner 
In an effort to better know the people involved with math in Vermont, we being offering interviews with VCTM Board members in each issue of our eNewsletter.  

David Rome (DR):  Tell us a  little bit about yourself- where are you from and where have you taught?
Jason Cushner (JC) :I love to find innovative ways to teach math.  I’ve taught math through building a low income health clinic, calculating water flow for the Colorado River, building a boat, and studying discrimination.  I’ve taught in multiple schools in Colorado and Vermont (and even taught English in Turkey for a while).  I also enjoy rock climbing, backpacking, and going on adventures with my kids (2 & 5).  I have been on an international climbing expedition to climb an unclimbed peak in Pakistan.  I met my wife teaching an integrated Calculus and Physics course

DR:  You are also involved with the PAEMST (see above article) - have you been a recipient?
JC:  Yes I won the Presidential Award in 2003 in Colorado and was one of 4 teachers asked to testify for congress on STEM education.  I had no idea how nice you had to dress for all the events until a few days before when they sent me the descriptions and realized that I didn’t own nice enough clothes for the events.  So I went to the thrift store and bought some nice clothes.  Right before leaving for the opening events and possibly meeting the president my wife called me back and realized I had the $4.99 price tag still stapled to my suit.

I also am National Board Certified, Outstanding Mathematics Teacher for the State of Colorado, and a current Rowland Fellow.

DR:  Tell us why you are involved with the VCTM and what  you see as its role for Vermont math teachers.
JC:  I’m involved with VCTM because it’s great professional development for me and I think it’s key to have a support network for engaging and inspiring students in math.  The role of VCTM for Vermont math teachers is to:
·       Connect math teachers to provide professional support
·       Be a hub of resources for math education
·       Support teachers and administrators in improving math education in the state
 DR:  What is the program that you implement at South Burlington HS, and what type of student does it reach?
 JC:  I’m currently program coordinator for the Big Picture program at South Burlington.  It’s where student create individuated curriculum around their interests and internships.  So we have students interning at a Biomedical lab at UVM, elementary school math classes, graphic design studios, farms, computer networking firms, and much more.  Students have an adviser who then helps them create individualized learning plans to deepen learning. 
The program attracts the full spectrum of students. Last year we got students who got a 4.0 GPA in AP classes and we had a student who came in with a 0.4 GPA (not easy to do & that student is a nationally published blogger and has presented at multiple education conferences).

DR:  What do you see as the biggest challenge for teachers in general, with the rolling out of the Common Core?
JC:  Part of the philosophy of the Common Core is a reduction in content so that there is space to work on the Mathematical practices (problem solving, persistence…) and those skills are not as easily measured in content.  I think schools and the media need to give teachers the time to work on how to better integrate the Mathematical Practices into curriculum and share success that are happening in these areas across the state.


Article: Helping Students Work Through Mathematical Conjectures
In this thoughtful article in Teaching Children Mathematics, Amy Hillen and Tad Watanabe (Kennesaw State University) say that an important Common Core math reasoning skill is making conjectures and assessing them based on evidence. They suggest the following 60-minute lesson for elementary students:
-   The teacher displays 1-9 on the board: 1 2 3 4 5 6 7 8 9
-   The teacher picks the number 4 and writes it on another part of the board.
-   The teacher calls on a student to pick another number – 8 is chosen.
-   The teacher writes the 8 by the 4.
-   “With these two numbers, which two-digit numbers can we make?” the teacher asks.
-   48 and 84
-   “Which is larger?” 84. The teacher writes the numbers using vertical notation:  84 - 48
-   The teacher asks students to subtract, and the answer is 36.
-   “Let’s try this with some other numbers.” This time, a student picks the first number, 7. The teacher picks the second number, 3 (making sure the difference is going to be 36).
-   The teacher again sets up the subtraction problem:  73 – 37, and the answer is 36.
-   “It’s the same!” students exclaim. “It’s always going to be 36!”
-   The teacher explains that this is a conjecture and writes it on the board – “If two different numerals are picked randomly to form 2 two-digit numbers, the difference will always be 36.”
-   “I wonder if this will always be true,” says the teacher. “How can we find this out?” Students suggest some ideas, and the teacher has students spend ten minutes working with a partner – just enough time for them to come up with only one or two possible combinations (students were given a template to set up the subtraction problems).
-   Students post their subtraction problems on the board.
-   Everyone looks at the examples (duplicates are removed), and students notice that some problems have answers other than 36. Their initial conjecture was not true.
-   Now students are asked to work in pairs to revise their initial conjecture and formulate new conjectures based on the set of subtraction problems posted on the board.
-   Some students are boggled by the number of problems on the board and are prompted to sort them into groups with the same answer and asked, “What do you notice about the problems whose difference is 36?”
-   Now the whole class discusses different conjectures. One student says that when the numbers chosen are next to each other, the answer is always 9. The teacher has the student clarify what “next to each other” means.
-   Another student says that when the chosen numbers are separated by one number, the answer is always 18.
-   Now students work in pairs again to try to refine the conjecture. Students often notice that the difference will always be a “nine fact” – a multiple of nine; that all the problems that have the same difference involve numerals whose difference is the same; and the difference will always be a product of 9 and the difference between the two.
-   Students are challenged to explain to a skeptic why these conjectures will always be true. The teacher asks, “How do we know that there is not a problem that we don’t have on the board whose difference is not a multiple of 9?” One approach is “proof by exhaustion” – trying all the possible problems. But is there another way? This might lead to organizing examples in a systematic way.
-   The teacher concludes the lesson by pointing out the important reasoning-and-proving work they have done, the importance of perseverance when the initial conjecture proved false, and how that led to a more sophisticated finding. In fact, conjecturing and proving (and disproving) are the essence of doing mathematics.

To see a delightful video of a Japanese teacher teaching this lesson, go to (free registration)

“Mysterious Subtraction” by Amy Hillen and Tad Watanabe in Teaching Children Mathematics, December 2013/January 2014 (Vol. 20, #5 p. 294-301),; the authors can be reached at and


Special Winter Offer for Girls by the Governors Institute 
If you are not on the MathNet mailing list, then you might not have heard about this offering from the Governor's Institute that was circulated recently:  
Hello and Happy New Year!  I’m writing from the Governor’s Institutes of Vermont.  Because we always have so many Math Institute applicants we can’t serve in the summer, we are holding a winter event just for high school advanced math fans.   This residential weekend exploration of math is just for girls and will be held at Goddard College February 7-9, 2014 (Marlboro, VT) with lead faculty member Marlboro professor Julie Rana.  Student applications are being accepted now on a first come-first serve basis. 

Young math lovers (male and female) may also be interested in our brand new Astronomy strand at the second Winter Weekend, where participants will learn how to apply math, computer science, and graphic design to map the stars. We will also offer Engineering and Information Technology topicals.  More information about GIV is available at, and information about Winter Weekend specifically can be found at

The registration process for GIV Winter Weekends is much briefer than our summer programs, since they are weekend-long programs, and as always, scholarships are provided to any students with financial need.  Interested students may apply directly to GIV (rather than through their schools as they do in the summer).  Please encourage your students to register quickly since we do expect the program to fill up!   Applications can be mailed to the address below or emailed to

Karen Taylor Mitchell, MPA, Executive Director Governor's Institutes of Vermont, 4049 Williston Road, #4, South Burlington, VT  05403, 802-865-4GIV,

Sunday, January 12, 2014

We're Number Umpteenth! The Myth of Lagging U.S. Schools, by Alfie Kohn

Copyright 2013 by Alfie Kohn.  Reprinted with the author's permission.  For more, please visit

Beliefs that are debatable or even patently false may be repeated so often that at some point they come to be accepted as fact.  We seem to have crossed that threshold with the claim that U.S. schools are significantly worse than those in most other countries.  Sometimes the person who parrots this line will even insert a number — “We’re only ____th in the world,  you know!” — although, not surprisingly, the number changes with each retelling.
The assertion that our students compare unfavorably to those in other countries has long been heard from politicians and corporate executives whose goal is to justify various “get tough” reforms:  high-stakes testing, a nationalized curriculum (see under:Common Core “State” Standards), more homework, a longer school day or year, and so on.
But by now the premise is so widely accepted that it’s casually repeated by just about everyone — including educators, I’m sorry to say — and in the service of a wide range of prescriptions and agendas, including some that could be classified as progressive.  Recently I’ve seen it used in a documentary arguing for more thoughtful math instruction, a petition to promote teaching the “whole child,” and an article in a popular on-line magazine that calls for the abolition of grades (following a reference to “America’s long steady decline in education”).
Unsurprisingly, this misconception has filtered out to the general public.  According to abrand-new poll, a plurality of Americans — and a majority of college graduates! — believe (incorrectly) that American 15-year-olds are at the bottom when their scores on tests of science knowledge are compared to those of students in other developed countries.[1]
A dedicated group of education experts has been challenging this canard for years, but their writings rarely appear in popular publications, and each of their efforts at debunking typically focuses on just one of the many problems with the claim.  Here, then, is the big picture:  a concise overview of the multiple responses you might offer the next time someone declares that American kids come up short.  (First, though, I’d suggest politely inquiring as to the evidence for his or her statement.  The wholly unsatisfactory reply you’re likely to receive may constitute a rebuttal in its own right.)
1.  Even taking the numbers at face value, the U.S. fares reasonably well. Results will vary depending on the age of the students being tested, the subject matter, which test is involved, and which round of results is being reported.  It’s possible to cherry-pick scores to make just about any country look especially good or bad.  U.S. performance is more impressive when the focus is on younger students, for example — so, predictably, it’s the high school numbers that are most often cited.  When someone reduces our schools to a single number, you can bet it’s the one that casts them in the worst possible light.
But even with older students, there may be less to the bad news than meets the eye.  Asan article in Scientific American noted a few years back, most countries’ science scores were actually pretty similar.[2]  That’s worth keeping in mind whenever a new batch of numbers is released.  If there’s little (or even no) statistically significant difference among, say, the nations placing third through tenth, it would be irresponsible to cite those rankings as if they were meaningful.
Overall, when a pair of researchers carefully reviewed half a dozen different international achievement surveys conducted from 1991 to 2001, they found that “U.S. students have generally performed above average in comparisons with students in other industrialized nations.”[3]  And that still seems to be the case based on the most recent data, which include math and science scores for grade 4, grade 8, and age 15, as well as reading scores for grade 4 and age 15.  Of those eight results, the U.S. scoredabove average in five, average in two, and below average in one.[4]  Not exactly the dire picture that’s typically painted.
2.  What do we really learn from standardized tests?  While there are differences in quality between the most commonly used exams (e.g., PISA, TIMSS), the fact is that any one-shot, pencil-and-paper standardized test — particularly one whose questions are multiple-choice — offers a deeply flawed indicator of learning as compared with authentic classroom-based assessments.[5]   The former taps students’ skill at taking standardized tests, which is a skill unto itself; the latter taps what students have learned, what sense they make of it, and what they can do with it.  A standardized test produces a summary statistic labeled “student achievement,” which is very different from a narrative account of students’ achievements.  Anyone who cites the results of a test is obliged to defend the construction of the test itself, to show that the results are not only statistically valid but meaningful.  Needless to say, very few people who say something like “the U.S. is below average in math” have any idea how math proficiency has been measured.
3.  Are we comparing apples to watermelons?  Even if the tests were good measures of important intellectual proficiencies, the students being tested in different countries aren’t always comparable.  As scholars Iris Rotberg and the late Gerald Bracey have pointed out for years, some countries test groups of students who are unrepresentative with respect to age, family income, or number of years spent studying science and math.  The older, richer, and more academically selective a cohort of students in a given country, the better that country is going to look in international comparisons.[6]
4.  Rich American kids do fine; poor American kids don’t.  It’s ridiculous to offer a summary statistic for all children at a given grade level in light of the enormous variation in scores within this country.  To do so is roughly analogous to proposing an average pollution statistic for the United States that tells us the cleanliness of “American air.”  Test scores are largely a function of socioeconomic status.  Our wealthier students perform very well when compared to other countries; our poorer students do not.  And we have a lot more poor children than do other industrialized nations.  One example,supplied by Linda Darling-Hammond:  “In 2009 U.S. schools with fewer than 10 percent of students in poverty ranked first among all nations on PISA tests in reading, while those serving more than 75 percent of students in poverty scored alongside nations like Serbia, ranking about fiftieth.”[7]
5.  Why treat learning as if it were a competitive sport?  All of these results emphasize rankings more than ratings, which means the question of educational success has been framed in terms of who’s beating whom.  This is troubling for several reasons.
a)  Education ≠ economy.  If our reason for emphasizing students’ relative standing (rather than their absolute achievement) has to do with “competitiveness in the 21st-century global economy” — a phrase that issues from politicians, businesspeople, and journalists with all the thoughtfulness of a sneeze, then we would do well to ask two questions.  The first, based on values, is whether we regard educating children as something that’s primarily justified in terms of corporate profits.
The second question, based on facts, is whether the state of a nation’s economy is meaningfully affected by the test scores of students in that nation.  Various strands of evidence have converged to suggest that the answer is no.  For individual students, school achievement is only weakly related to subsequent workplace performance.  And for nations, there’s little correlation between average test scores and economic vigor, even if you try to connect scores during one period with the economy some years later (when that cohort of students has grown up).[8]  Moreover, Yong Zhao has shown that “PISA scores in reading, math, and sciences are negatively correlated with entrepreneurship indicators in almost every category at statistically significant levels.”[9]
b)  Why is the relative relevant?  Once we’ve refuted the myth that test scores drive economic success, what reason would we have to fret about our country’s standing as measured by those scores?  What sense does it make to focus on relativeperformance?  After all, to say that our students are first or tenth on a list doesn’t tell us whether they’re doing well or poorly; it gives us no useful information about how much they know or how good our schools are.  If all the countries did reasonably well in absolute terms, there would be no shame in being at the bottom.  (Nor would “average” be synonymous with “mediocre.”)  If all the countries did poorly, there would be no glory in being at the top.  Exclamatory headlines about how “our” schools are doing compared to “theirs” suggest that we’re less concerned with the quality of education than with whether we can chant, “We’re Number One!”
c) Hoping foreign kids won’t learn?  To focus on rankings is not only irrational but morally offensive.  If our goal is for American kids to triumph over those who live elsewhere, then the implication is that we want children who live in other countries to fail, at least in relative terms.  We want them not to learn successfully just because they’re not Americans.  That’s built into the notion of “competitiveness” (as opposed to excellence or success), which by definition means that one individual or group can succeed only if others don’t.  This is a troubling way to look at any endeavor, but where children are concerned, it’s indefensible.  And it’s worth pointing out these implications to anyone who cites the results of an international ranking.
Moreover, rather than defending policies designed to help our graduates “compete,” I’d argue that we should make decisions on the basis of what will help them learn tocollaborate effectively.  Educators, too, ought to think in terms of working with – and learning from – their counterparts in other countries so that children everywhere will become more proficient and enthusiastic learners.  But every time we rank “our” kids against “theirs,” that outcome becomes a little less likely.
1.  Pew Research Center for People and the Press, “Public’s Knowledge of Science and Technology,” April 22, 2013.  Available at:
2.  W. Wayt Gibbs and Douglas Fox, “The False Crisis in Science Education,”  Scientific American, October 1999:  87-92.
3.  Erling E. Boe and Sujie Shin, “Is the United States Really Losing the International Horse Race in Academic Achievement?” Phi Delta Kappan, May 2005: 688-695.
4.  National Center for Economic Statistics, Average Performance of U.S. Students Relative to International Peers on the Most Recent International Assessments in Reading, Mathematics, and Science: Results from PIRLS 2006, TIMSS 2007, and PISA 2009, 2011.  Available at:
5.  See, for example, Alfie Kohn, The Case Against Standardized Testing (Heinemann, 2000); or Phillip Harris et al., The Myths of Standardized Tests (Rowman & Littlefield, 2011).
6.  For example, see Iris C. Rotberg, “Interpretation of International Test Score Comparisons,” Science, May 15, 1998: 1030-31.
7.  Linda Darling-Hammond, “Redlining Our Schools,” The Nation, January 30, 2012: 12.  Also see Mel Riddile, “PISA: It’s Poverty Not Stupid,” The Principal Difference[NASSP blog], December 15, 2010 (; and Martin Carnoy and Richard Rothstein, “What Do International Tests Really Show About  U.S. Student Performance?”, Economic Policy Institute report, January 28, 2013 (
8. Keith Baker, “High Test Scores: The Wrong Road to National Economic Success,”Kappa Delta Pi Record, Spring 2011: 116-20; Zalman Usiskin, “Do We Need National Standards with Teeth?” Educational Leadership, November 2007: 40; and Gerald W. Bracey, “Test Scores and Economic Growth,” Phi Delta Kappan, March 2007: 554-56.  “The reason is clear,” says Iris Rotberg.  “Other variables, such as outsourcing to gain access to lower-wage employees, the climate and incentives for innovation, tax rates, health-care and retirement costs, the extent of government subsidies or partnerships, protectionism, intellectual-property enforcement, natural resources, and exchange rates overwhelm mathematics and science scores in predicting economic competitiveness” (“International Test Scores, Irrelevant Policies,” Education Week, September 14, 2001: 32).
9.  Yong Zhao, “Flunking Innovation and Creativity,” Phi Delta Kappan, September 2012: 58.  Emphasis added.