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.

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