Education Week - October 11, 2023 - 19

only pays benefits, they said, if students have
enough background knowledge in what they're
expected to do.
Xiuwen Wu, an associate professor of special
education at National Louis University in
Wheeling, Ill., talked about a video she shows
to her math education students.
In the video, which depicts an elementary lesson
on perimeter and area, the teacher asks students
to use one-inch tiles to calculate a shape's
perimeter. But the students aren't clear on the difference
between perimeter and area, and some
fill in the shape with tiles and then count those.
In effect, it shows, the students didn't realize that
they were performing the wrong operation.
" If the students within that group lack prior
knowledge about perimeter and area, the definitions,
it's a dangerous struggle because they
would be frustrated easily, " Wu said. That kind
of struggle, without a clear goal, can make kids
feel defeated, and the exercise futile, she added.
" But if the students have enough prior
knowledge, and they can use the activity to
connect to that prior knowledge, I think it
could [be beneficial], " Wu said. " Math is about
reasoning. It's about finding solutions to problems.
If it's reasonably difficult, then they can
benefit from it. "
Brian Bushart, a 4th grade teacher in West
Irondequoit Schools in New York, remembers
learning about these kinds of hands-on, sensemaking
activities in his preservice program in
the early 2000s. To him, it seems like universities
are still " championing a lot of the exact
same things. "
However, some of his own beliefs have since
shifted, Bushart said. Sometimes, these activities
are effective ways to solidify and connect
student knowledge. But other times, he said,
they're " very inefficient, and hard to achieve. "
" I've definitely come around on that there's
some really solid research on explicit instruction,
and the need to make things explicit for
students, " he said.
A solid foundation sets students up for problem-solving
success, he said. " If they get frustrated
by it, if they don't get the thing that you
wanted them to learn from it, that's kind of a
terrifying thought, " he said. " I think there's a
place for problem-solving, and I think it's after
students have some knowledge of something
that might be helpful on the problem. "
Fact fluency: Is it necessary or nice to have?
Another divide among survey respondents
surfaced in fact fluency, the ability to quickly
recall basic arithmetic facts.
Cognitive science researchers say that fact
fluency is important, because it frees up brain
space for students to work through more complicated,
multi-step problems. It's harder for a
student to work out a system of equations, for
example, if they also have to spend time and effort
figuring out 56/7 or 12x9.
But efficiency and reducing students' cognitive
load aren't the only reasons that math facts
are important, said Nicole McNeil, a professor
of cognitive psychology who studies math learning
at the University of Notre Dame.
" I think people don't appreciate, necessarily,
the idea that if you become fluent in the math
facts in a well-organized way, that the connections
made there actually create conceptual
knowledge. "
For example, McNeil said, a lot of K-12 teachers
use partners of 10. They build students' understanding
of 10 using equivalent values: 4+6
= 10, 5+5 =10, so 4+6 = 5+5. That helps students
think about the relationships between different
sets of numbers-a skill that sets them up for
algebraic thinking down the line, McNeil said.
" When you are organizing facts practice in
this way to create connections-such that when
you activate one, the others are also activated-
if you do that beyond the 10s, it starts to create a
network, " she said.
But not all university educators agreed that
fact fluency is essential.
Most math and math education instructors-78
percent-agreed that K-12 teachers
should teach relevant skills and facts at the
same time as they teach problem-solving. But
there was a divide between instructors who felt
that fact fluency was essential, and those who
thought it was helpful, but not essential.
Too much focus on memorizing facts can
often mean less value is placed on engaging
deeply with mathematical concepts, said Kyndall
Brown, the executive director of the California
Mathematics Project, a professional development
network. It can also create a narrow
understanding of what success in the subject
means, he said.
" Who's traditionally been considered 'good at
math' are students who are able to perform computations
and procedures with speed and accuracy.
Those are the ones who tend to get labeled,
'That's a good math student,' " Brown said.
K-12 teachers who reviewed the survey findings
took a middle road-they agreed that fact fluency
was important, in part because it forms a foundation
for more conceptual problem-solving.
Tara Warren, a math instructional coach in the
Santa Monica-Malibu Unified school district,
said she leans more toward thinking that fact fluency
is essential. " You have to have your students
work on things that are conceptual problems, but
it builds " from fact fluency, Warren said.
Instructors' perspectives on curriculum and
course sequencing
Instructional philosophy is just one ingredient
in classroom practice-what materials teachers
use, and how courses are structured and sequenced,
also play a role.
When asked about curriculum materials,
most math education instructors thought
that teachers should have some flexibility in
choosing resources. Fifty-two percent said
that all teachers should use the same core
math instructional program, but pick and
choose parts of it to use as needed, while 31
percent said that teachers should select, assemble,
and/or create their own curriculum.
Seventeen percent said that teachers should
follow a core curriculum closely.
Their preference toward individual choice
aligns with what most K-12 teachers actually
do. Previous surveys have shown that most
teachers use a combination of materials, supplementing
and creating resources when they
feel their district-provided options don't serve
student needs.
But Bushart, the New York state teacher, was
surprised to see that almost a third of instructors
expected teachers to create their own curriculum.
" That is setting your teachers up for failure,
and just a lot of unnecessary work, " he said.
There can be value in starting with a common
program, he said-it provides a foundation for
new teachers, and it makes it more likely that
students will see familiar terms and strategies
used across classes and grade levels.
The survey also asked about course progressions
and ability grouping, a process in which students
are divided by their proficiency level and some are
exposed to material that is more advanced.
When asked about who should take calculus
in high school, 42 percent of math and math
education instructors said that only students
who have taken prerequisite courses and have
earned a certain grade and/or a certain score
on a math test should be able to take the course.
Thirty-seven percent said that anyone who
wants to take the course should be able to, but
that it should not be required.
These findings on ability grouping and calculus
concerned Wiggins, of the Dana Center,
Teacher educators' perspectives
on ability grouping
In your view, when, if ever, should
schools start grouping students by
ability/achievement for math
instruction, with some classes
or groups exposed to
curricula that is more
advanced?
4%
Grades 11-12
Grades 9-10
Grades 6-8
Grades 3-5
Grades K-2
Never-schools should not
group students by ability/
achievement for math
instruction
Do teacher educators think fact fluency is essential?
How important is it for K-12 students to be fluent with math facts (e.g.,
single-digit addition, multiplication, times-tables)?
It's essential for them to work
on higher-order skills or more
conceptual problems
It is helpful for them to work on
higher-order skills or conceptual
problems, but not essential
It doesn't make a difference
44%
56%
0%
11%
23%
18%
16%
27%
Note: Totals may not add up to 100% due to rounding.
*Results show responses from postsecondary instructors teaching either math education courses or math
courses in which at least some of the students were enrolled in teacher education programs.
EdWeek Research Center survey, July 2023
who favors mixed-ability groups.
" Ability grouping is just never a good idea, "
she said. " What usually happens in these
[mixed-ability] classrooms is collaborative discussion
between students, and the different
abilities of students lift each other up. "
She was disappointed to see that many instructors
thought there should be restrictions
on which students can take calculus.
" This is a lot of the reason why you don't see
a lot of African Americans and Latinos and students
experiencing poverty in this field, " Wiggins
said of STEM professions. " They're saying
there should be some way to block you out
of calculus. But calculus is the only way you're
going to get into these fields. "
Not all high schools offer calculus, and the
data on which students have access to the
course show inequities: White and Asian students
are more likely to attend schools that offer
calculus than their Black and Latino peers.
Powell, the special education professor at the
University of Texas at Austin, thought that many
of the survey respondents probably imagined
ability grouping as " high " and " low " classes.
But there are other types of ability grouping. For
example, she said, working with small groups of
students to intervene on discrete topics can have
significant benefits. That kind of grouping does
have a place in schools, she said.
" The thing that we know about intervention is
that earlier support is better, " Powell said.
Asking the right questions
For all the focus on philosophy, some math
educators say they have even more essential
questions about how math is taught.
Yeh would have wanted to see more questions
in the survey that went beyond instructors' perspectives
on teaching methods to some of the
deeper, thornier issues about a changing world.
" What are the key things that our society
needs now, now that we have phones that can
calculate thousands of calculations in a second,
and that we've had a pandemic where data literacy
has become a problem? " she asked.
" What is math? " she continued. " What places
does math take place? What does it mean to be
successful in math? "
ADDITIONAL STORIES: Read more
from the EdWeek Research Center's
survey of instructors in math teacher
preparation. > edweek.org
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Education Week - October 11, 2023

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