Education Week - September 9, 2015 - 11

USING PRIMARY SOURCES
behaviors such as perseverance
and critiquing the reasoning of others. Those behaviors are part of the
Standards for Mathematical Practice, the overarching standards in
the common core that describe the
behaviors of proficient math students and apply to all grades.
Diane Briars, the president of
the National Council of Teachers of
Mathematics, agreed that there is
overlap with the practice standards,
especially one that says students
should construct viable arguments.
"When you're reading these original writings, you're directly building
on that [standard]," she said.
But incorporating these primary
sources into classrooms isn't easy
work, the experts said, and there
are some caveats.
"You read one of these for the first
time and you have no idea what it
says," said Klyve. "The mathematics is explained in vocabulary you
haven't seen before or notation you
haven't seen before." And that can
be hard for all students-but especially English-learners and those
with language or reading deficits.
"What you don't want to have happen is students are so distracted by
the writing style or it's so hard to
make sense of the language used

that you're investing a lot of time in
that rather than the mathematical
ideas," said Briars of NCTM.
For those reasons, the writings
must be carefully selected and excerpted to ensure they're digestible.
"It's very hard to pull an original
source off the shelf and drop it in
class and hope students read it and
do something useful with it," said
Klyve. "We're putting all these projects through peer review, because
teaching the math while being true
to the original source is just hard."
That said, Klyve would "love
high school teachers to try this."
The projects will all be open source
and adaptable, so "I'd easily imagine somebody could modify them to
[explicitly] match the common-core
standards," he said. "I'd do that."

developing strategies to use math
concepts, and critiquing the reasoning of others.
And open problems allow students
to understand how math concepts
relate to each other, rather than
merely understanding how to use
an algorithm the teacher prescribes,
they said. It's not that getting the
correct answer doesn't matter, Montoy-Wilson said. But open problems
emphasize that the process of arriving at the answer matters too, she
said.
"It's pretty scary in terms of what
we want for our future to think of
kids who only know the algorithm
and not why it works," MontoyWilson said. "When you just focus
on getting to the answer, you really
rob kids of grappling and working
on that sweet spot. You don't want to
scaffold or carry the load too heavily
for your kids."
Across subjects, researchers have
found that a teacher's own orientation to learning can affect whether
their students have a fixed mindset
or a growth mindset about their
own abilities.

more likely to have students with
a growth mindset at the end of a
course.
"I think that teaching is really really important," she said. "It's not
just about changing kids' beliefs, it's
about giving them opportunities to
experience it."
Sun also identified strategies
that can help boost students' confidence in the math classroom.
Teachers should encourage multiple attempts at problem solving,
they shouldn't offer unsolicited
help to students, and they should
provide opportunities for students
to resubmit their work, she said.
Some mathematicians have said
professional development for math
teachers should prioritize content
knowledge over pedagogy. Treisman said many math teachers
have the math knowledge to teach
in a more open format that encourages growth mindsets, and they
just need to strengthen the skills
necessary to do so.
Many of today's math teachers
were taught in very traditional
classrooms, and many have not
explored the subject in this way on
their own, he said.
That's why teachers need to practice their own sense-making and
model it for their students, Treisman said. If math were music,
mastering the basic concepts would
be like learning scales and leading
students through discussions of
open problems would be like playing songs, he said.
"Teachers love the idea of mindsets as almost a panacea," Treisman
said, "but they themselves have very
fixed ideas of their own learning.

Views on Math
But a teacher's views on math
also matter, said Kathy Liu Sun, an
assistant professor at Santa Clara
University in Santa Clara, Calif.,
and a former public high school
math teacher.
Sun used surveys to gauge the
mindsets of about 3,400 students
and 40 teachers. She also assessed
teachers' approaches to math,
whether they valued speed and
memorization or "multi-dimensional" problems that allow for multiple strategies and sense making.
Through post surveys and classroom observations, Sun determined that teachers with a "multidimensional" view of math were

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As part of a project being underwritten by the National Science Foundation, students will see
original renderings and writings from early mathematicians, such as the presentation on the left
of binomial coefficients created by Blaise Pascal in about 1654. The drawing, known as Pascal's
triangle, is more likely to be shown in modern textbooks as it is on the right. The modern rendering
is intended to help students see that each number is the sum of the two directly above it.

ORIGINAL PASCAL'S TRIANGLE

1
1
1
1
1
1
1

In every arithmetical
triangle, of two contiguous
cells in the same base, the
upper is to the lower as the
number of cells from the
upper to the top of the base is
to the number of cells from
the lower to the bottom of
the base, inclusive."
BLAISE PASCAL

1
1
1

8
9

1
5

15
35

70
126

1
4

56
84

28
36

6
21

56
126

7
28

1
8

1
9

MODERN PASCAL'S TRIANGLE

Coverage of learning mindsets and skills
is supported in part by a grant from the
Raikes Foundation. Education Week
retains sole editorial control over the
content of this coverage.

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Education Week - September 9, 2015

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