RME 6: Opening Keynote - Moving from Disowning to Owning Realistic Mathematics Education


Frank Eade, Ministry of Education, Cayman Islands
David Webb, School of Education, University of Colorado Boulder

Frank Eade
The point of this keynote was to describe Realistic Mathematics Education from the experience of two of its strongest proponents, Frank Eade of the Cayman Islands and David Webb of the United States. With Grand Cayman as a new location for this conference, and many educators coming to an RME conference for the first time, the keynote was designed to describe experiences that any math teacher could relate to.

Frank described his early experiences as a math teacher in the UK. There were influences of New Math from the US, and a sense that things could be changing, but the expectation was to teach traditionally. David's early experiences as a teacher, teaching high school in 1980s Los Angeles, was that instruction was largely teacher-oriented. People talked about group work, but not without much of a sense of purpose. People focused on procedure and problem solving only came at the end after the procedures had been learned.

David Webb
Frank began to learn about dilemmas with students' mathematical experiences. For example, if you ask students to put 0.375, 0.7, and 0.32 in order, many students will say 0.375 is the smallest because it has thousandths, and thousandths are the smallest. Frank read this in research and didn't want to believe it, so he tried it with his own students. They replicated the research, revealing to him that he'd been totally unaware of their misconceptions. He talked to them about why they believed what they did, and got answers like "It's a bit like negative numbers --- the smaller it is, the larger it is." David knew he could organize his entire curriculum around a procedural textbook and hand out worksheets. Students didn't seem to mind and figuring out how to do group work seemed like an uphill battle. Change meant fighting the didactical contract students had formed with their math teachers over many years.

Frank's first experiences with RME came when he visited the University of Amsterdam and some schools there. He observed 7/8 students comparing 2/3 to 3/4. He figured students would answer as his would -- that the fractions were the same because in each fraction you could add one to the numerator to get the denominator. Instead, he saw students drawing pictures and using those pictures to argue their reasoning. The only student he observed who reasoned incorrectly was someone who recently moved there. He started to become a convert. David entered graduate school after 7-8 years of teaching and ended up at Wisconsin with Tom Romberg. His assignment as a grad student was to align every activity in Math in Context to the NCTM Curriculum Standards. There were 30 MiC units, so it was a lot of work, but it gave him a detailed look at RME's approach to curriculum design. Sometimes he was in disbelief about the kind of reasoning that was expected of students before procedural understanding had developed. But as he observed students in classrooms, he became a believer as students approached problems informally but expressed all the reasoning they would need to understand the problem formally.

Frank and David describing models and progressive formalization
Frank's experience with RME curriculum design was part of a MiC pilot. A teacher who was supposed to try a unit for 3 weeks ended up stretching the unit for 10 weeks because kids were so engaged in all the mathematical reasoning -- and because they all had a lot to learn about how to implement and pace a different kind of curriculum. This led to multiple grant-funded projects both to develop new curriculum and to study its implementation. David's experience with design in RME came at the end of the MiC project as focus shifted to implementation and assessment. He had the opportunities to travel the country and offer PD and learn from teachers who needed support to understand the principles of RME and, maybe more importantly, to redefine the roles of teacher and learner in their classrooms.

3A + 2P = $9.20
1A + 2P = $5.20

Frank gave the above problem to a group of students with experience with RME, but they were scared of the symbolic notation. Then one student in the room said, "Wait, it's hats and umbrellas!" (a problem in Math in Context) and the rest of the class caught on and they were able to reason with the mathematics. They had seen the hats and umbrellas problem 2 years before, but the reasoning had stuck with them. David's work with assessment bridged the divides between RME and formative assessment. It was yet another way to rethink what it meant for students to understand mathematics. David began to promote the use of the assessment pyramid, even for teachers to think about how they arrange the bulk of their assessments they use in their classrooms. "Students are capable of solving fascinating problems -- we just have to ask them."

Frank was talking to his wife about math and asked her to explain how she got an answer to a particular fraction problem. She said, "I cheated. I imagined it, but I know I'm supposed to find the common denominator." Here in the Caymans, Frank has worked with students who have struggled but are now seeing the math in their worlds in new ways. RME isn't totally different from other approaches to mathematics, but some distinctions are useful. It's important for teachers to see their role in orchestrating students' mathematical experiences, and not just facilitate. The use of models is fundamental to RME's design, and teachers need to understand that approach to be successful.

When Frank introduced RME to the Caymans, he first visited every classroom to get a sense for the current state of math education on the islands. Staff turnover tends to be high in the Caymans. He found that many children didn't have a sense of shopping and the values of things, so those were opportunities to work those contexts into the curriculum. Students needed more support, so they introduced Mathematics Recovery. Interventions outside the classroom were far more fruitful in helping students because teachers struggled with interventions within the classroom. They employed lesson study to help teachers understand how lessons could be taught, and to address their own experiences with traditional instruction. At the end of primary in 2011, only 25% of students were expected and only 5% were above. At the end of primary in 2018, 62% were expected and 25% were above. That said, Frank says there is great danger in looking at scores like this too much, as we don't want teachers to become too focused on exam success as a measure of achievement.

This Week in Math Ed: February 2, 2018

I think I have a new plan: Since going through news, research, and Colorado events (in addition to daily Twitter updates) are quite a lot of work to do every week, I'll just highlight each one once a month. Something like this:
  • First Friday of the month: Research Report
  • Second Friday of the month: Around the Math Ed Web (events)
  • Third Friday of the month: Math Ed in the News
  • Fourth Friday of the month: Math Ed in Colorado
I might move some things around, but this one-feature-weekly in addition to the Math Ed Said review of Twitter activity seems like a good idea.

Math Ed Said

January 26: James Tanton wrote a wonderful piece on Medium, "Just teach my kid the <explitive> math." I'd guess most math educators have found themselves in these kinds of conversations (even within one's head) and Tanton's version is one of the more articulate versions I've seen.

Shared by: Jenise Sexton, Pam J. Wilson, Chris Hunter, James Tanton, Bridget Dunbar, Katherine Bryant, Jennifer Wilson, Christine Klynen, Cathy Yenca

January 27: Other people must agree with my enjoyment of Tanton's article because more decided to share it on the 27th. Here it is, again: "Just teach my kid the <explitive> math."

Shared by: Scott Leverentz, Barbara Rock, John Colgan, Nerissa Gerodias, Sunil Singh, Ben Blum-Smith, Chris Brownell, Alison Hansel, Bridget Soumeillan, Gregory Taylor, Matthew Oldridge, James Tanton, Laura Kinnel

January 28: How fun! Here in "Equations I Have Known" Joe Schwartz cataloged a number of the ways students use the notation for operations and equality before they've attended to the level of precision we expect from someone who has mastered these conventions. Some are quite common, like run-on equations (1x2=2-1=1), but others might be new to you.

Shared by: Amie Albrecht, Mark Chubb, Heidi Fessenden, Cathy Campbell, Chrissy Newell, Lisa Bejarano, Kit, Chris Kalmbach, Simon Gregg, Marilyn Burns, Jim Doherty, Rene Grimes, Heidi Allum, Joe Schwartz

January 29: As a first-year teacher resisting the mid-'90s feel-good, self-esteem pushes in education, I was convinced that my students (before I'd even had a chance to teach them!) would feel good about math if they were successful at it, and not the other way around. I gradually learned that it wasn't that simple, and new research from Stanford is helping us better understand how a positive attitude toward math predicts math achievement in kids.

Shared by: Laura Wagenman, Jani Nelson, Georgina Rivera, Christopher Rohde, Lara Francisco, Christina Moore, Camsie McAdams, Judy Larsen, Andie Ogden, Rosa Serratore, Kim Webb, Jo Boaler

January 30: More people shared the story, "Positive attitude toward math predicts math achievement in kids."

Shared by: Rosa Serratore, Regina Barrett, Nick Harris, Jennifer Lawler, Chris Brownell, Linda Braddy

January 31: I so appreciate David Bressoud's relentlessness when it comes to calculus education reform. Here he is on The Conversation with "Why colleges must change how they teach calculus." Part of this article talks about the SEMINAL project, which my advisor David Webb works on as part of his role at CU Boulder and on behalf of the Association of Public Land Grand Universities.

Shared by: Eddi Vulić, Robert Cop, David Butler, Jennifer Lawler, Lybrya Kebreab, Heather Johnson, Egan J Chernoff

February 1: More people helped spread the word about the Desmos Teaching Fellowship.

Shared by: Ed Campos Jr, Jocelyn Dagenais, Ron King, Audrey McLaren, Nerissa Gerodias, David Sabol, Julia Finneyfrock, Sadie Estrella, Nanette Johnson, Jennifer Fairbanks, Explore MTBoS, Lisa Bejarano, Emily Campbell, Sara VanDerWerf, Robert Kaplinsky, Molly Daley, Mary Bourassa, Daniel Luevanos, Zack Patterson, Jon Orr, Patty Stephens, Kathy Henderson, Tom Snarsky, Dan Anderson, Lybrya Kebreab, Jedidiah Butler, Matthew Baker, Bob Lochel, Eli Luberoff, Chris Lusto, Bryan Anderson, Jocelyn Dagenais, Desmos.com

Research Report

Educational Studies in Mathematics

ESM published both their January and February issues since my last Research Report on January 5.

Mathematical Thinking and Learning

This issue focuses on mathematics learning and computational thinking.

International Journal of Science and Mathematics Education

Both the January and February issues of IJSME have been published since the last Research Report.
Patricio Herbst presenting at the 2014 NCTM Annual Meeting

Journal of Mathematics Teacher Education

Fields Mathematics Education Journal

Technology Innovations in Statistics Education

This Week in Math Ed: January 26, 2018

Math Ed Said

Graham Fletcher presenting at the 2017 NCTM Annual Meeting
January 19: If you missed it earlier, here again is Graham Fletcher's Fraction Video Series.

Shared by: Shauhna Feitlin, CMC - CA MathCouncil, Nita Cochran, Kit, Devin Anderson

January 20: Robert Kaplinsky did us all a favor and reminded us that a great resource is the Formative Assessment Lessons from the Mathematics Assessment Resource Service (MARS). My recommendation: Do not underestimate how much thought went into these, and be willing to trust them even if you're accustomed to making a lot of adaptations to your usual off-the-shelf lessons.

Shared by: Ed Southall, Shelby Aaberg, Jazmine Falicetti, Gregory White, Emily Campbell, John Faig, Ed Campos Jr, Kimberly Wassmuth, Martin Joyce, Ann Crilley, Nita Cochran, Jennifer Lawler, Robert Kaplinsky

January 21: To show support for Laurie Rubel, the American Association of University Professors are collecting names of people who wish to stand with Laurie.

Shared by: Equity MathEd, Kit, Peg Cagle, Annie Forest, Matt Owen, David Kung, Carrie Diaz Eaton

January 22: With a second contribution to this post in three days, this time of his own design, Robert Kaplinsky shared his Depth of Knowledge Matrix for Secondary Math.

Shared by: Nicol Reiner, Heather Sugrue, Patty Stephens, Bridget Dunbar, Brandi Moore, Robbyn Glinsmann, Kit, Pam J Wilson, Jill Gough, Bob Lochel, Robert Kaplinsky

January 23: Steve Wyborney's "Cube Conversations" is back on this list again after an appearance the first week of the year. I think of these as doing something geometrically to prompt math discourse similar to how arithmetic problems prompt math discourse in number talks.

Shared by: Denise Gaskins, Kit, Donna Boucher, Kat Hendry, Steve Wyborney, Lindel

January 24: Lots of people shared and reshared the announcement of the Desmos Teaching Fellowship.

Shared by: John Berray, Julie Reulbach, Levi Patrick, Heather Kohn, Nolan Doyle, Daniel Luevanos, Eli Luberoff, Melynee Naegele, Audrey McLaren, Ed Campos Jr, Peg Cagle, Shelley Carranza, Cathy Yenca, Jedidiah Butler, Allison Krasnow, Scott Leverentz, Julia Finneyfrock, Bob Lochel, Nerissa Gerodias, Christopher Danielson, Jennifer Fairbanks, Jon Orr, Molly Daley, Annie Perkins, Andrew Busch, Mary Bourassa, Lisa Bejarano, Eric Blask, Kathy Henderson, Dan Anderson, Kristen Fouss, Jocelyn Dagenais, Desmos.com

January 25: More people shared the Desmos Teaching Fellowship announcement.

Shared by: Darren Burris, Norma Gordon, Jennifer Fairbanks, James Cleveland, Sara VanDerWerf, Chris Lusto, Karl Fisch, Clara Maxcy, Christopher Danielson, Patty Stephens, John Colgan, Kit, Matthew Baker, Bridget Dunbar, David Sabol, Imtiaz Damji

Math Ed in Colorado

Conferences: Math on the "Planes", TI's Education Leadership Summit, and MidSchoolMath

There are dwindling number of openings remaining for Math on the "Planes", the two-day math conference hosted by the Colorado Council for Learning Disabilities. The conference will be held February 23-24 in Centennial and will feature Dr. Barbara Dougherty. Register today!

Texas Instruments is hosting a one-day Leadership Summit in Denver on March 22. The title of the summit is "The Leader's Role Implementing STEM and Computer Science Initatives" and the keynote speaker is Peter McLaren, past-president of the Council of State Science Supervisors. Other speakers include Dr. Joanna Bruno of CDE, who will discuss science and computer science in Colorado; Gwen Perea Warniment, who will talk about designing K-12 STEM initatives, and CCTM President Joanie Funderburk, who will focus on mathematics education.

If you're thinking about attending the MidSchoolMath National Conference in Santa Fe, NM, their standard registration deadline is Friday, February 2. The conference features Jo Boaler, Dan Meyer, and Tracy Johnston Zaeger.

Job Openings

Woodland Park is seeking someone to teach Algebra 1 and Algebra 2 beginning February 12. More information can be found at www.wpsdk12.org.

CCTM and CML

The 2018 CCTM Annual Conference will be August 2-3 at the University of Denver. By moving the conference to just before the start of school, you can worry less about your sub plans (and your district won't have sub costs) and you can focus your attention on connecting with your Colorado peers in two days of great math teaching and learning.

CCTM is accepting nominations for five positions on its Board of Directors: President-Elect, Vice President, Secretary, and Regional Representatives for Regions 2 and 5. Nominations close on February 2.

CCTM is also accepting nominations for its teaching and leadership awards. Awardees receive a plaque, one year of CCTM membership, complimentary registration to the CCTM Annual Conference, and a $200 award.

The next meeting of the Colorado Math Leaders will be held at the Instructional Support Facility at 5416 S. Riveria Way in Centennial from 10:00 to 2:00 on Wednesday, February 21. (Note: This is a recent change in location!) If you aren't on the CML mailing list and would like to be, send me an email.

Grant Opportunities

CSEd: Districts have until February 28 to apply for the Computer Science Education grant and receive up to $10,000 for their district to spend on professional development, tuition, books, or other programs and resources to directly support the teaching of computer science in Colorado. Details about the grant and other computer science opportunities can be found on the CDE website.

Lemelson-MIT: If you want to learn more about the $10,000 Lemelson-MIT STEM grants, there is a webinar on Tuesday, February 6 from 4:30-5:30 MT. More information about the program can be found on the Lemelson-MIT website.

Let Your Voice Be Heard!

The Teaching and Learning Conditions in Colorado (TLCC) survey (formerly TELL Colorado) is a key tool to help CDE and other education stakeholders understand the conditions of education around Colorado. The survey should take less than 15 minutes and it closes February 23. Schools and districts that reach the 50 percent participation threshold (and at least five respondents) will be able to access their own data after the survey window closes. This is valuable data that can lead to very rich discussions about improvements for your school and/or district.

COpilot Class: Co-Teaching to Improve Collaboration & Instruction

You may be interested in a class being offered through CEA's COpilot platform that focuses on co-teaching relationships between a general education teacher and a teacher who supports students with disabilities, language learning, or other students from special populations. This class is designed to be taken by both teachers in the relationship. For more information, see the COpilot website.

This Week in Math Ed: January 19, 2018

Math Ed Said

January 12: I think this is the second time I have seen this Desmos graph of the spinning Desmos logo appear as the most-shared thing on Twitter. I don't get it.

Shared by: Pam Harris, Judy Keeney, Karen Gartland, Darren Burris, Cathy Yenca, Kat Hendry, Matt Vaudrey, Scott Leverentz, Jen McAleer, Kyle Pearce, Robbyn Glinsmann, Bob Lochel, Zach Cresswell, Bridget Dunbar, Laura Wheeler

January 13: Laurie Rubel of Brooklyn College got some media attention (and unwanted negative attention from readers of some of those sites) about a story discussing her article, "Equity-Directed Instructional Practices: Beyond the Dominant Perspective" in the Journal of Urban Mathematics Education. As usual, most of the criticism/harassment came from people who didn't read the article, even though it's available via open access.

Shared by: Equity MathEd, Chris Brownell, Kit, Laurie Rubel, Regan Galvan, Brian R Lawler, Geoff Wake

January 14: If you haven't seen SolveMe Mobiles, you should really check them out. These are good examples of how algebra doesn't always have to look like variables and equations, and that students can figure these out as puzzles well before they're in 8th or 9th grade.

Shared by: Heidi Allum, Patrick Honner, Simon Gregg, Emily Stewart, Shelby Aaberg, Jen Overley, Margie Pearse, Dr. Steve Humble

January 15: The original link to this was broken, but I'm pretty sure people were trying to share "Minimizing the 'Matthew Effect'," a blog post by Mark Chubb. By the "Matthew Effect," Mark describes that phenomenon where children who start off a little bit ahead of others get offered more opportunities to engage (in sports, or math, or whatever) and that extra engagement gives them compounding advantages over time — and maybe we should consider that as teachers when structuring participation in the classroom.

Shared by: Jennifer Lawler, Laura Wheeler, Laura Wagenman, Lane Walker, Mark Chubb

Kara Imm leaning in during the 2015 RME Conference.
January 16: First of all, if you're into number talks and haven't been to the Number Strings website, you need to be there like yesterday. Second, take your time and read "Trusting the digits: Developing place value understanding." Are conversations among students happening like this in your classroom? Although unstated, Kara Imm gets the author credit for this post — I recognized her masterful facilitation!

Shared by: Brett Parker, Bridget Dunbar, Marilyn Burns, Erica Litke, Sendhil Revuluri, Kit, Math in the City, Rachel Lambert, Geonz, Rachel Lambert, Kara Imm

January 17: The American Association of University Professors collected names of those wanting to Stand With Dr. Laurie Rubel at Brooklyn College.

Shared by: Spencer Bagley, Annie Perkins, Equity MathEd, Brian R Lawler, Nicole Bannister, Mike Steele, Laurie Rubel, Amanda Jansen, Samuel Otten, LaurieRubel

January 18: Graham Fletcher put together a Fraction Video Series called "The Foundation of Fractions." The videos are presented with teachers as the audience, such as teachers who want to improve the way they teach fractions or the way they want to understand fractions themselves.

Shared by: Amanda Haskell, Kyle Pearce, Shawna Hedgepeth, Jennifer Lawler, Georgina Rivera, Mike Flynn, Emily Campbell, Jill Gough, Pam J. Wilson, Bridget Dunbar, Matthew Oldridge, Chrissy Newell, Graham Fletcher

This Week in Math Ed: January 12, 2018

I don't think I'll need or be able to scour 35+ research journals every week for new articles, so maybe I'll try to do that every other week, or maybe once a month. In any case, here's some of what was popular in the world of math education for the week of January 5th to January 11th, 2018.

Math Ed Said

January 5: This article appeared once here last week, and now it appears twice this week: Sunil Singh's "Six Questions That Math Educators Need To Answer Honestly." My short answer to Question 1, "What is mathematics?": Mathematics is the human activity of reasoning with number and shape. A slightly longer answer would explain that this reasoning produces the artifacts that we associate with mathematics, but particularly as an educator I keep the definition focused on math as a human activity, as Freudenthal did.

Shared by: Eddi Vulić, Richelle Marynowski, Pam J. Wilson, Keith Devlin, Zack Patterson, Dan Meyer

January 6: Again: "Six Questions That Math Educators Need To Answer Honestly."
Shared by: Nita Cochran, Emily Stewart, Dave Martin, Steven Strogatz, Kyle Pearce, Eric Milou, Dan Meyer, Zach Cresswell, Federico Chialvo, Gary Davis, Kat Hendry, Meleia Bridenstine, Alison Hansel, Hilary Kreisberg, Matthew Oldridge

January 7: Heidi Fessenden wrote "Counting Collections: One Nearly-Perfect Answer to Inclusion," a reporting of what's happening with her students when children with and without autism pair up to count collections of objects. Heidi is a thorough storyteller and it's a story I needed to hear, as I've been helping write a state policy document for students with disabilities and stories like this keep my head rooted in practice.

Shared by: Marilyn Burns, Rosa Serratore, Megan Franke, Melinda Knapp, Elham Kazemi, Alison Hansel, Annie Forest, Kassia Wedekind, Jana Sanchez, Andrew Gael, Jamie Garner, Tracy Johnston Zager, Heidi Fessenden

Is your district making segregation worse?
January 8: Want to get lost in some policy, maps, and data? Spend your time on this feature from Vox: "Mapping the imaginary lines we use to segregate our schools." Based heavily on some new work by Berkeley PhD candidate Tomas Monarrez, you can use this tool to explore segregation patterns in districts across the country and understand the policies and historical reasons that shaped them.

Shared by: Jennifer Lawler, Shawna Hedgepeth, Kent Haines, Heidi Fessenden, Michael Soskil, Kit, Thad Domina

January 9: If you missed it on the 8th, here it is again on the 9th: "Mapping the imaginary lines we use to segregate our schools."

Shared by: Morgan Fierst, Kara Imm, Dylan Kane, Elizabeth Self, Samantha Marshall, Karen King, Theodore Chao, Thad Domina, Tyrone Martinez-Black, Heidi Fessenden

January 10: After spending all that time looking at segregation patterns, give yourself a chance to chuckle with Ben Orlin's "Compass Constructions Made Easy."

Shared by: Anna Blinstein, Nathaniel Highstein, Chris Burke, Jo Morgan, Ilona Vashchyshyn, Denise Gaskins, Ryan R Ruff, Jennifer White, Jennifer Lawler, Jen Silverman, Ben Orlin, Michael P Goldenberg

January 11: There's probably a context to this I missed, or maybe this is a new feature: A Desmos graph where the Desmos logo spins around.

Shared by: Sadie Estrella, Bryn Humberstone, Shawna Hedgepeth, Lisa Bejarano, Allison Krasnow, Nerissa Gerodias, Vanessa Cerrahoglu, Kathy Henderson, Heather Sugrue, Andrew Shauver, Jennifer Blinzler, Christopher Danielson, Jocelyn Dagenais, Dan Anderson, Jason Merrill, Shelley Carranza, Jennifer Fairbanks, Mary Bourassa, Jocelyn Dagenais, Eli Luberoff