Stanford neuroscience -- Using symmetry to learn about numb# NextGeneration - 我爱宝宝
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Stanford neuroscience research identifies more effective way to teach
abstract math concepts to children
A new study shows that students who use symmetry to learn about numbers tap
into critical brain circuits. The approach appears promising in improving
math skills in general.
By Edmund L. Andrews
(Courtesy [email protected])
Building on new discoveries about how the brain grapples with abstract
mathematics, researchers at Stanford Graduate School of Education (https://
ed.stanford.edu/) have developed a classroom strategy for teaching children
the often baffling concepts surrounding negative numbers.
The new strategy recruits the brain's use of visual symmetry to make sense
of the physical world, and it could have profound implications for the way
elementary schools teach math.
Using symmetry appears to have helped not just in teaching children about
negative numbers but in improving their ability to solve higher-level math
problems they haven't seen before.
"Learning about negative numbers is one of the first times that kids learn
about abstract numbers - it's a gateway to more abstract learning,'' said
Jessica Tsang, a Stanford researcher and lead author of a new study with
Daniel Schwartz and Kristen Blair of Stanford and Laura Bofferding of Purdue
University.
The study marks the latest and most concrete result of a five-year project
that bridges the gap between new insights from neuroscience and the testing
of new classroom teaching tools for fourth-graders in the San Francisco Bay
Area. It is published in the current issue of the journal Cognition &
Instruction. [SEE http://www.tandfonline.com/doi/abs/10.1080/07370008.2015.1038539?journalCode=hcgi20#.VaF1ECgQiS0 ]
Researchers have long thought that the brain harnesses and adapts "perceptuo
-motor" capacities to make sense of abstract ideas.
For the purposes of making sense of math, the most widely studied of these
capacities has been the ability to compare physical magnitudes such as size.
Over the past 15 years, neuroscientists have confirmed that the same region
of the brain that assesses physical magnitudes is also key to comparing the
magnitudes represented symbolically by numerals.
Three years ago, the Stanford researchers proposed that the brain actually
re-purposes several additional capabilities, such as sequencing and ordering
, to solve math problems. In this project, the researchers focused on the
brain's ability to process visual symmetry.
In 2012 the researchers found that most adults identified the midpoint
between a negative number and a positive number more easily if the integers
were more symmetric about zero.
For example, people were faster computing the mid-point of 6 and -4 than 8
and -2, even though both pairs of digits were displayed in the same
locations on a video screen. Moreover, using functional magnetic resonance
imaging, the authors found that visual symmetry-coding regions of the brain
became more active for the more symmetric pairs.
Given that the adult brain recruits symmetry circuits for conceptual math
problems, the researchers hypothesized that helping young students engage
this native ability would improve their mathematics learning.
For the new study, the authors created a guide and tools that incorporate
ideas about using symmetry to teach negative numbers. As intuitive as it
might seem for people to rely on symmetry as a way to make sense of the
world, few curricular materials now used for teaching math make explicit use
of it.
The researchers then tested the tools with students and compared the results
to those of other students who were taught with the existing approaches.
One of the new classroom teaching tools was a specially designed hands-on
manipulative device. Students worked with a magnetic plastic strip that was
numbered. To solve the problem 3 + -2, students attached three magnetized
blocks to the right of zero and two blocks to the left of zero. The
manipulative further included a hinge at zero, the point of integer symmetry
. Students folded the two sides together, and the number of extra blocks on
either side gave the answer, in this case +1. The hinge at zero helped
students recruit their native abilities with symmetry, and the numbers on
the little platform helped them coordinate the sense of symmetry with the
symbolic digits.
After four hours of instruction spread over three weeks, the researchers
documented encouraging results.
First, the children quickly applied the hands-on lessons with symmetry to
build new strategies for solving abstract problems involving symbolic digits
. When the symmetry students were asked to add up a string of negative and
positive numbers, for example, many used a balancing strategy to simplify
the problem.
Some paired up negative and positive numbers that would add up to zero and
cancel each other out. Others clustered the positive and negative integers
on separate sides, which made the balance between the positives and
negatives more apparent.
Second, students were likely to incorporate symmetry as an almost automatic
part of their thinking. That's important, said Tsang, because many skills -
such as decoding words in reading - are more effective when they become
instantaneous and reflexive.
But the biggest surprise was on what educational researchers call "
generativity" - the tendency of students to apply the ideas of symmetry on
their own to problems they haven't encountered before.
As it turned out, students who learned to rely on symmetry didn't simply do
better than other students on the material they had just been taught. They
also did better on topics that they hadn't yet studied, such as making sense
of negative fractions and solving pre-algebraic problems.
"The big difference was that the symmetry instruction enabled students to
solve novel problems and to continue learning without explicit instruction,"
said Schwartz, who holds the Nomellini & Olivier Professorship in
Educational Technology at Stanford. "By untangling how the brain comes to
know mathematics, we helped with a major goal of education - putting
children on an upward trajectory of future learning."
---------------------------
SIDEBAR PHOTO: As part of a four-hour curriculum on integers, fourth-grade
students played games using a special number line that could be folded in
half at the zero point, allowing the symmetry between positive and negative
quantities to stand out.
--------------------------
This research was funded by the National Science Foundation and the Marcus
and Marianne Wallenberg Foundation.
--------------------------
abstract math concepts to children
A new study shows that students who use symmetry to learn about numbers tap
into critical brain circuits. The approach appears promising in improving
math skills in general.
By Edmund L. Andrews
(Courtesy [email protected])
Building on new discoveries about how the brain grapples with abstract
mathematics, researchers at Stanford Graduate School of Education (https://
ed.stanford.edu/) have developed a classroom strategy for teaching children
the often baffling concepts surrounding negative numbers.
The new strategy recruits the brain's use of visual symmetry to make sense
of the physical world, and it could have profound implications for the way
elementary schools teach math.
Using symmetry appears to have helped not just in teaching children about
negative numbers but in improving their ability to solve higher-level math
problems they haven't seen before.
"Learning about negative numbers is one of the first times that kids learn
about abstract numbers - it's a gateway to more abstract learning,'' said
Jessica Tsang, a Stanford researcher and lead author of a new study with
Daniel Schwartz and Kristen Blair of Stanford and Laura Bofferding of Purdue
University.
The study marks the latest and most concrete result of a five-year project
that bridges the gap between new insights from neuroscience and the testing
of new classroom teaching tools for fourth-graders in the San Francisco Bay
Area. It is published in the current issue of the journal Cognition &
Instruction. [SEE http://www.tandfonline.com/doi/abs/10.1080/07370008.2015.1038539?journalCode=hcgi20#.VaF1ECgQiS0 ]
Researchers have long thought that the brain harnesses and adapts "perceptuo
-motor" capacities to make sense of abstract ideas.
For the purposes of making sense of math, the most widely studied of these
capacities has been the ability to compare physical magnitudes such as size.
Over the past 15 years, neuroscientists have confirmed that the same region
of the brain that assesses physical magnitudes is also key to comparing the
magnitudes represented symbolically by numerals.
Three years ago, the Stanford researchers proposed that the brain actually
re-purposes several additional capabilities, such as sequencing and ordering
, to solve math problems. In this project, the researchers focused on the
brain's ability to process visual symmetry.
In 2012 the researchers found that most adults identified the midpoint
between a negative number and a positive number more easily if the integers
were more symmetric about zero.
For example, people were faster computing the mid-point of 6 and -4 than 8
and -2, even though both pairs of digits were displayed in the same
locations on a video screen. Moreover, using functional magnetic resonance
imaging, the authors found that visual symmetry-coding regions of the brain
became more active for the more symmetric pairs.
Given that the adult brain recruits symmetry circuits for conceptual math
problems, the researchers hypothesized that helping young students engage
this native ability would improve their mathematics learning.
For the new study, the authors created a guide and tools that incorporate
ideas about using symmetry to teach negative numbers. As intuitive as it
might seem for people to rely on symmetry as a way to make sense of the
world, few curricular materials now used for teaching math make explicit use
of it.
The researchers then tested the tools with students and compared the results
to those of other students who were taught with the existing approaches.
One of the new classroom teaching tools was a specially designed hands-on
manipulative device. Students worked with a magnetic plastic strip that was
numbered. To solve the problem 3 + -2, students attached three magnetized
blocks to the right of zero and two blocks to the left of zero. The
manipulative further included a hinge at zero, the point of integer symmetry
. Students folded the two sides together, and the number of extra blocks on
either side gave the answer, in this case +1. The hinge at zero helped
students recruit their native abilities with symmetry, and the numbers on
the little platform helped them coordinate the sense of symmetry with the
symbolic digits.
After four hours of instruction spread over three weeks, the researchers
documented encouraging results.
First, the children quickly applied the hands-on lessons with symmetry to
build new strategies for solving abstract problems involving symbolic digits
. When the symmetry students were asked to add up a string of negative and
positive numbers, for example, many used a balancing strategy to simplify
the problem.
Some paired up negative and positive numbers that would add up to zero and
cancel each other out. Others clustered the positive and negative integers
on separate sides, which made the balance between the positives and
negatives more apparent.
Second, students were likely to incorporate symmetry as an almost automatic
part of their thinking. That's important, said Tsang, because many skills -
such as decoding words in reading - are more effective when they become
instantaneous and reflexive.
But the biggest surprise was on what educational researchers call "
generativity" - the tendency of students to apply the ideas of symmetry on
their own to problems they haven't encountered before.
As it turned out, students who learned to rely on symmetry didn't simply do
better than other students on the material they had just been taught. They
also did better on topics that they hadn't yet studied, such as making sense
of negative fractions and solving pre-algebraic problems.
"The big difference was that the symmetry instruction enabled students to
solve novel problems and to continue learning without explicit instruction,"
said Schwartz, who holds the Nomellini & Olivier Professorship in
Educational Technology at Stanford. "By untangling how the brain comes to
know mathematics, we helped with a major goal of education - putting
children on an upward trajectory of future learning."
---------------------------
SIDEBAR PHOTO: As part of a four-hour curriculum on integers, fourth-grade
students played games using a special number line that could be folded in
half at the zero point, allowing the symmetry between positive and negative
quantities to stand out.
--------------------------
This research was funded by the National Science Foundation and the Marcus
and Marianne Wallenberg Foundation.
--------------------------