Washington Mathematics
Washington State Mathematics Council
Spring 2019
For the Love of Math....
Contributed by Steve Wyborney: The Duplicator Lab Riddle
The Duplicator Lab Riddle is one of my favorite tasks because it requires the reader to consider some unexpected variables along the way. It also offers a really nice opportunity to rethink the way you approach the task. As the robots multiply in unexpected ways, it will require you to choose an unexpected solution path as well - and discovering that path is part of the fun! Enjoy!
Contributed by Pamela Weber Harris, Author of Building Powerful Numebrcy for Middle and High School Students
One of my favorite “problems” lately is this percent bar page that asks the question:
“What is 88% of 25?”
The page has four separate strategies shown, as if four different people had solved the problem.
Students are asked to study each percent bar and determine the missing information. Then students should describe how a student might have used the relationships to solve 88% of 25 in each of the percent bars.
Before you read on, work through each strategy.
Now, that you’ve looked at three fine strategies for finding 88% of 25 and one really cool strategy for finding 25% of 88, what do you think?
Can you use 25% of 88 to find 88% of 25? Yes! And what a great use of the associative and commutative properties, because .25 x 88 = 25 x (0.01) x 88 = 0.88 x 25.
When I first found this strategy, I was so enamored by it that I just had to write problem strings and this page in my Lessons & Activities for Building Powerful Numeracy book.
Here a couple of other problems that use the same idea: 34% of 10, 87% of 50, and 44% of 75.
Used with permission from the publisher
Contributed by Annie Fetter
I was given the problem Dog’s Mead by a social studies teacher at my high school (who I never actually had as a teacher - but it was a small school, 250 kids in grades 7-12, so you sort of knew everyone). He thought I might like it. I found it fascinating, because it wasn’t simple (like all of my very traditional math schooling had been, and continued to be, through high school), and I couldn’t solve it right away. In fact, I didn’t finish it until after college. I kept it tucked into a book of other cross-number and logic puzzles that my father had given me subsequently (an older edition of this book: https://www.amazon.com/Puzzles-Pleasure-R-Emmet/dp/1566197945). So, for me, it was the first puzzle of that sort I had ever seen, and the first math problem/puzzle I worked on in fits and starts, so it has a special place in my math heart. I found a version of it, as well as some history, here: http://jig.joelpomerantz.com/fun/dogsmead.html
Opportunities to Connect....
2019 Math Olympiad - Registration Now Open (Limited Sites Available)
Registration for the Math Olympiad is now open! However, due to a loss in number of sites this year that nearly caused the cancellation of the event, we are late in opening registration and have changed from a first-come-first-served model to a lottery system. Registration will remain open through March 19. See details at www.wsmc.net/math-olympiad/
Rita Lowe Scholarship - Due March 15th
Do you know an exceptional high school senior or college student interested in becoming a math educator? Are you yourself a math educator seeking to further your education? WSMC awards up to four $1000 scholarships in each of three categories annually: high school seniors interested in math education, college students seeking to become math educators, and current math educators pursuing professional development in math education. Information is available on the WSMC website. Please spread the word!
WSMC Educator Support Fund At Work
The Pre-Service Teachers of Mathematics Club would like to sincerely thank you for your
donation to provide the finances to send our members to the NWMC in Whistler, Canada.
The trip was filled with new experiences and information that we can carry into our future
classrooms. Many take-aways have already been implemented in class activities. A lot of our
participating members have never even traveled out of Washington, so it was a great chance to
see the diversity and accomplishments of all the surrounding region’s teachers. It was an
unforgettable trip that will benefit not only us as future teachers, but also our future students.
Due to some members being unable to attend the conference, the remaining money was used to
send our members, along with other volunteers that are pre-service teachers, to the Colville
Federated Indian Reservation’s elementary/middle school, Nespelum. Here, we led the math
classes at different grade levels in activities designed to strengthen their understanding of their
current subject: ratios. This was a great opportunity for our pre-service teachers to teach in a
diverse classroom, as well as gain perspective into designing and implementing a lesson.
In addition, we have attached several photos taken during the trip to the Northwest Math
Conference and the volunteers during visits to Nespelum at the Colville Indian Reservation.
Member Spotlight... Jana Dean
MAKING MEANING: ORAL LANGUAGE AND CONTEXT MATTER
For six weeks, I have been at the Freudenthal Institute in the Netherlands working on a Fulbright Inquiry project called Math Between Us. When I started this project, my aim was to find new ways to support students in linguistically, culturally and cognitively diverse middle school classrooms. While I haven’t encountered brand new ideas, I am seeing the familiar in a new light. I am understanding more than ever that the combination of oral language and context is extremely important in how we create opportunities for our math students to make meaning for themselves.
As teachers, we use oral language all the time. When we talk, our aim is for students to understand what we mean. Likewise, when students talk about mathematics, we hope to
understand what they mean. More importantly, when students describe what they are learning, it is a way they make meaning for themselves. My experience observing in Dutch classrooms as I learn the language has done a lot to help me understand the ways spoken language helps and gets in the way of figuring out what is going on. I am surprised by how much context in combination with spoken words helps me learn, recognize and remember Dutch, as well as by how difficult it is to learn without context. This happens both in everyday life and in math classes I visit.
When I arrived in the Netherlands, I determined that I would learn as much Dutch as I could as quickly as possible. My flashcard vocabulary is inching up to about eight hundred words. Eight hundred sounds like a lot of words; however, it is a long way from the basic conversational fluency level of about three thousand. My experience around Dutch conversations at this point is that I usually know what people are talking about but I rarely understand what they are saying about it. And the degree to which it is easier for me to listen than to talk is stunning.
I spent last Saturday walking alone around Utrecht intent on meeting my daily goal of five little conversations in Dutch. In the course of shopping for groceries and sundries, thanks to these familiar contexts, I managed far more. As I wound my way home, I gave myself a special challenge of naming what I saw and was doing. Pretty soon I was describing myself walking on the bridge, by the train station, towards the church, and staying out of the bike lane. This was all much easier for me walking about the city than sitting in my apartment with my flashcards. I find that if I am doing or seeing something, the language comes much easier. Context supports my learning.
A few weeks ago, I sat in a high school class as teacher Jos Meens created contexts to support his students. He wrote “rechtevenredig” and “omgekeerd evenredig” on the board. At first I had no idea what he was talking about. As he spoke he represented each idea with an equation, a graph and a table. He did this all in Dutch, while describing what he was doing the whole time. Being able to watch him write and listen at the same time helped me follow his words, but I also was able to understand the nonverbal representations. In that way, the experience itself was a context for understanding. For the students, Mr. Meens also told a story about traveling to school which provided even more illustration for them. Because of the different ways to see and understand the concept, I learned that “rechtevenredig” means proportional and “omgekeerd evenredig” means inversely proportional. The multiple
representations softened the language barrier for me and the math informed my acquisition of Dutch. In other classrooms where I do not have anything contextual to refer to, I find it very difficult to understand any Dutch at all. I infer that the same must be true of our students who struggle with math or with language in our classes.
What Jos Meens was doing for his students was to help them make as many connections between representations as possible on their way to being able to work with formal understandings of ideas. In our work with the Math Education Collaborative’s MEC MSP Leadership Project , we learned to support our students in ‘Navigating the Pentagon.’ Fawn Nguyen also asks students to do this in her Patterns Poster assignment on her blog Finding Ways.
In addition to visiting classrooms, I am also meeting with people who support teacher development here in the Netherlands. This week, I met with Nina Boswinkel who has done a lot of work with contextualizing as it relates to thinking about curriculum for special education students. She coauthored an article called “Beneath the Tip of the Iceberg: Using Representations to Support Student Understanding, published in NCTM’s Teaching Mathematics in the Middle School in September of 2008. The picture below illustrates the iceberg metaphor.
The Iceberg Model, credit: Nina Boswinkel
Only formal representations are above the surface. Like an iceberg, what lies below the surface is more vast than what is above. This depth invites repeated returns to contextualized informal and preformal representations. For example, in building understanding of ¾ with students, a periodic return to apples, pizza and money will support formalized understanding. Nina suggested even staying at “floating capacity” for some students as being more productive for their reasoning than moving to formalization at all. The contextualized ideas give students things that they can describe with their own language and thereby own their understanding.
Alas, rich contexts can sometimes appear to fail. As a Dutch learner, the presence (or my lack of understanding) of a single word can create an enormous barrier for me until I can get help from a friend or use Google to translate. Likewise, my difficulty with nuance means that I am likely to miss relationships between ideas even when I understand all the words. Relationships are often at the heart of understanding math concepts. These experiences may be quite similar to those of our students who struggle to make sense of the mathematics they are learning in spite of the rich and familiar contexts we provide.
A few years ago, I had a very quiet class. More than half of their grade level peers had been accelerated while they had been left behind. Many of them spoke second and third languages at home. About onefourth of them had been identified as needing special education services in reading and writing. Many of these seventh graders told me privately, “I just don’t raise my hand or speak up in math class.” When I gave them familiar or intriguing contexts to build their understanding of math, they often remained so silent that I would impatiently abandon context in the hopes of speeding up the learning. Each time, my race to the top of the iceberg made things worse. They had even less on which to build verbalization and understanding. Their quiet frustration reminded me to return to the contextrich understandings below the surface of the water. Eventually, most of those students grew in their ability to talk about mathematical ideas and much of the class grew two years in mathematical understanding in just one year. Others however, grew far less.
It is for those students that I am learning and describing ways to build bridges between what students can talk about and what they can not. Now that I know more about how essential classroom spoken language is, I will hone in on the ways Dutch teachers build those bridges. I am reading the work of German researcher Susanne Prediger, I have upcoming opportunities in special education classrooms, at a school with a large number of refugee students, in dual
language DutchEnglish middle schools and at a tutoring program in a nearby city for historically marginalized students. I will also continue to meet with teacher educators and experts in both math education and language to learn how they help teachers think about both oral language and contexts to support mathematical development. Stay tuned and follow the story at mathbetweenus.org or email me at jdean@reachone.com.
WSMC Member Supports….
WSMC Educator Support Fund
The Washington State Mathematics Council provides funds to support mathematics-related initiatives, identified by members via an application. Individual awards range from $50 to $500.
Awards
Washington State Mathematics Council supports outstanding math educators by offering three awards. Two WSMC members, each an outstanding mathematics educator, are honored annually. The Hall of Fame Award is given to honor an outstanding contributor to the field of mathematics education, one who has given time and effort over a long period of time (ten years or more).
Rita Lowe Scholarships
WSMC gives annual scholarships in the amount of $1000, awarded to:
- high school students planning to become math teachers
- college students planning to become math teachers, and
- math teachers pursuing further professional growth.
Future Events…
Spring WSMC Board Meeting
May 18, 2019
10am-3pm
Walter Strom MS, Cle Elum, WA
Positions Available!
Email: wsmcnewsletter@wsmc.net
Website: http://www.wsmc.net/
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