Basic importent topic

18, 2013 — When early elementary math
teachers ask students to explain their problem-
solving strategies and then tailor instruction to
address specific gaps in their understanding,
students learn significantly more than those
taught using a more traditional approach. This
was the conclusion of a yearlong study of
nearly 5,000 kindergarten and first-grade
students conducted by researchers at Florida
State University.
The researchers found that “formative
assessment,” or the use of ongoing evaluation
of student understanding to inform targeted
instruction, increased students’ mastery of
foundational math concepts that are known to
be essential to later achievement in
mathematics and science.
Their results corroborated those of two earlier
pilot projects indicating that implementation of
the Mathematics Formative Assessment System
(MFAS) can markedly improve academic
performance in mathematics. The findings
further suggested that MFAS may help close
the gender gap that often develops by third
grade.
“The results of the most recent study
conducted in schools across Florida are
exciting,” said Laura Lang, principal
investigator who directed development and
testing of MFAS. “The randomized field trial
showed that students in K-3 classes where
teachers used MFAS were well ahead of other
students taught by teachers using more
traditional approaches. As one of the
elementary principals of a participating school
put it, MFAS is a real ‘game changer’ in terms
of student engagement and success in math.”
MFAS was created through the efforts of
researchers at the Florida Center for Research
in Science, Technology, Engineering and
Mathematics (FCR–STEM) who received $2.9
million in competitively awarded grant funds
from the Florida Department of Education’s
Race to the Top program to pursue the
project. MFAS is fully aligned with the Common
Core State Standards adopted in Florida and
many other states.
The randomized field trial was conducted in
partnership with 31 schools and 301 teachers
in three Florida districts across the state —
one urban, one suburban and one rural.
Schools were randomly assigned to either the
MFAS treatment group or to a group that used
a more typical approach to math instruction.
Comparing average annual gains in math on
nationally normed tests to the results, learning
was accelerated when teachers integrated MFAS
in their day-to-day instruction.
“In kindergarten, we can infer that students
learned at a rate equivalent to an extra six
weeks of instruction,” Lang said. “In first
grade, the gains were even greater — two
months of extra instruction. It was as if we
extended the school year without actually
adding any more days to it.”
In constructing MFAS, Lang and her team drew
upon research demonstrating that the learning
of mathematics is facilitated when teachers
gain deeper insights into what their students
already know and are able to do as well as
what students do not know and are unable to
do. Teachers gather these insights through
careful observation and by engaging students
in discussions of their mathematical thinking.
“Formative assessment is a process, not a
test,” Lang said, “and feedback is a key
element.”
The approach enables teachers to address each
child’s instructional needs. Teachers can avoid
holding back those who are ready to advance,
while efficiently helping those who are
struggling. This contrasts sharply with current
practice in many elementary classrooms.
“Based on our classroom observations over the
past four years, teachers typically rely heavily
on a math textbook to guide the planning of
day-to-day instruction and often provide
students feedback only on whether their
answers are correct,” Lang said. “Teachers
integrating formative assessment in instruction
not only ask students to do math tasks but
also to explain their reasoning and to justify
their solutions. As a result, teachers are better
equipped to identify misconceptions,
determine gaps in understanding and adjust
their instruction accordingly.”
Students play a key role in the formative
assessment process. MFAS actively engages
students, encouraging them to monitor and
regulate their own learning. Students also
evaluate each other’s work and provide
productive feedback, working as a team.
MFAS also has potential long-term effects on
closing the gender gap in mathematics, Lang
said. Studies show that even though both boys
and girls enter school with a fundamental
number sense, by the third grade boys tend to
do better in mathematics.
The results of a pilot study conducted in
second- and third-grade classrooms suggest
that, in classrooms where MFAS was used, by
third grade the girls showed no statistically
significant difference in mathematics
achievement from boys, according to Mark
LaVenia, methodologist on the MFAS team.
However, in classrooms with more
conventional instruction, girls continued to lag
behind boys in math achievement.

Story Source:

The above story is based on materials provided
by Florida State University, via Newswise.

Unlocking Biology With Math

Oct. 7, 2013 — Scientists at USC have created
a mathematical model that explains and
predicts the biological process that creates
antibody diversity -- the phenomenon that
keeps us healthy by generating robust immune
systems through hypermutation.
The work is a collaboration between Myron
Goodman, professor of biological sciences and
chemistry at the USC Dornsife College of
Letters, Arts and Sciences; and Chi Mak,
professor of chemistry at USC Dornsife.
"To me, it was the holy grail," Goodman said.
"We can now predict the motion of a key
enzyme that initiates hypermutations in
immunoglobulin (Ig) genes."

Goodman first described the process that
creates antibody diversity two years ago. In
short, an enzyme called "activation-induced
deoxycytidine deaminase" (or AID) moves up
and down single-stranded DNA that encodes
the pattern for antibodies and sporadically
alters the strand by converting one nitrogen
base to another, which is called "deamination."

The change creates DNA with a different
pattern -- a mutation.
These mutations, which AID creates a million-
fold times more often than would otherwise
occur, generate antibodies of all different sorts
-- giving you protection against germs that
your body hasn't even seen yet.
"It's why when I sneeze, you don't die,"
Goodman said.
In studying the seemingly random motion of
AID up and down DNA, Goodman wanted to
understand why it moved how it did, and why
it deaminated in some places much more than
others.

"We looked at the raw data and asked what the
enzyme was doing to create that," Goodman
said. He and his team were able to develop
statistical models whose probabilities roughly
matched the data well, and were even able to
trace individual enzymes visually and watch
them work.

But they were all just
approximations, albeit reasonable ones.
Collaborating with Mak, however, offered
something better: a rigorous mathematical
model that describes the enzyme's motion and
interaction with the DNA and an algorithm for
directly reading out AID's dynamics from the
mutation patterns.
At the time, Mak was working on the
mathematics of quantum mechanics. Using
similar techniques, Mak was able to help
generate the model, which has been shown
through testing to be accurate.

"Mathematics is the universal language behind
physical science, but its central role in
interpreting biology is just beginning to be
recognized," Mak said. Goodman and Mak
collaborated on the research with Phuong
Pham, assistant research professor, and Samir
Afif, a graduate student at USC Dornsife. An
article on their work, which will appear in
print in the Journal of Biological Chemistry on
October 11, was selected by the journal as a
"paper of the week."

Next, the team will generalize the
mathematical model to study the "real life"
action of AID as it initiates mutations during
the transcription of Ig variable and constant
regions, which is the process needed to
generate immunodiversity in human B-cells.

Non-Traditional Mathematics Curriculum Results in Higher Standardized Test Scores

Sept16, 2013 — For many years, studies have
shown that American students score
significantly lower than students worldwide in
mathematics achievement, ranking 25 th among
34 countries. Now, researchers from
theUniversity of Missouri have found high
school students in the United States achieve
higher scores on a standardized mathematics
test if they study from a curriculum known as
integrated mathematics.
James Tarr, a professor in the MU College of
Education, and Doug Grouws, a professor
emeritus from MU, studied more than 3,000
high school students around the country to
determine whether there is a difference in
achievement when students study from an
integrated mathematics program or a more
traditional curriculum. Integrated mathematics
is a curriculum that combines several
mathematic topics, such as algebra, geometry
and statistics, into single courses. Many
countries that currently perform higher than
the U.S. in mathematics achievement use a
more integrated curriculum. Traditional U.S.
mathematics curricula typically organize the
content into year-long courses, so that a 9 th
grade student may take Algebra I, followed by
Geometry, followed by Algebra II before a pre-
Calculus course.
Tarr and Grouws found that students who
studied from an integrated mathematics
program scored significantly higher on
standardized tests administered to all
participating students, after controlling for
many teacher and student attributes. Tarr says
these findings may challenge some long-
standing views on mathematics education in
the U.S.
"Many educators in America have strong views
that a more traditional approach to math
education is the best way to educate high
school students," Tarr said. "Results of our
study simply do not support such impassioned
views, especially when discussing high-
achieving students. We found students with
higher prior achievement scores benefitted
more from the integrated mathematics
program than students who studied from the
traditional curriculum."

Tarr and Grouws' papers, which were recently
published in the Journal for Research in
Mathematics Education, come from a three-
year study measuring educational outcomes
for students studying from different types of
mathematics curricula. Tarr says improving
American mathematics education is vital for
the future of the country
.
"Many countries that the U.S. competes with
economically are outpacing us in many fields,
particularly in mathematics and science," Tarr
said. "It is crucial that we re-evaluate our
school mathematics curricula and how it is
implemented if we hope to remain competitive
on a global stage."
Tarr and Grouws' longitudinal study is funded
by grant of more than $2 million from the
National Science Foundation.

Story Source:

The above story is based on materials provided
by University of Missouri-Columbia.

Journal Reference:

1. James E. Tarr, Douglas A. Grouws, Óscar
Chávez, and Victor M. Soria. The Effects of
Content Organization and Curriculum
Implementation on Students’ Mathematics
Learning in Second-Year High School
Courses .
Journal for Research in Mathematics.

Photo maths

Some maths articles :  only image,  just like topic.

Arresting Model Stops Cars

Sep. 5, 2013 — Researchers in China have
developed a mathematical model that could
help engineers design a flexible vehicle-arrest
system for stopping cars involved in criminal
activity or terrorism, such as suspect car
bombers attempting break through a check
point, without wrecking the car or killing the
occupants.

Writing in a forthcoming issue of the
International Journal of Vehicle Design, Pak Kin
Wong and colleagues in the Department of
Electromechanical Engineering at the
University of Macau, in Taipa, Macao, explain
how common vehicle-arrest systems used by
law enforcement, the military and in anti-
terrorism activities, usually cause serious
damage to the vehicle and maim or kill the
occupants. A more positive system for bringing
a car chase to a halt or stopping a car-bomber
in their tracks is needed if perpetrators,
witnesses and evidence are to be protected.

A flexible system would increase the stopping
distance of a vehicle involved in criminal or
terrorist activity and allow its kinetic energy to
be dissipated without the complete destruction
of the vehicle as otherwise occurs with solid,
immovable barriers and equipment currently
used. The team's mathematical model of
vehicle arrest with different flexible materials
and designs bears up to theoretical and
experimental scrutiny and offers engineers a
new set of variables to embed in their design
program in the development of new, effect
vehicle arrest systems. Moreover, the system
could allow the design of an "intelligent"
vehicle-arrest system for roadblocks and
checkpoints that could respond differently
depending on vehicle speed and type and allow
for greater control in bringing a vehicle to a
stop

Story Source:

The above story is based on materials provided
by Inderscience Publishers , via EurekAlert!, a
service of AAAS.
And ( science daily magazine ) .

Note: Materials may be edited for content and
length. For further information, please contact
the source cited above.

Journal Reference:

1. Pak Kin Wong et al. Modelling and testing of
arresting process in flexible vehicle
arresting systems. Int. J. Vehicle Design ,
2013, 64, 1-25

Generosity Leads to Evolutionary Success, Biologists Show

Sep. 2, 2013 — With new insights into the
classical game theory match-up known as the
"Prisoner's Dilemma," University of
Pennsylvania biologists offer a mathematically
based explanation for why cooperation and
generosity have evolved in nature.
Their work builds upon the seminal findings of
economist John Nash, who advanced the field
of game theory in the 1950s, as well as those
of computational biologist William Press and
physicist-mathematician Freeman Dyson, who
last year identified a new class of strategies for
succeeding in the Prisoner's Dilemma.
Postdoctoral researcher Alexander J. Stewart
and associate professor Joshua B. Plotkin, both
of Penn's Department of Biology in the School
of Arts and Sciences, examined the outcome of
the Prisoner's Dilemma as played repeatedly by
a large, evolving population of players. While
other researchers have previously suggested
that cooperative strategies can be successful in
such a scenario, Stewart and Plotkin offer
mathematical proof that the only strategies
that succeed in the long term are generous
ones. They report their findings in the
Proceedings of the National Academy of
Sciences the week of Sept. 2.
"Ever since Darwin," Plotkin said, "biologists
have been puzzled about why there is so much
apparent cooperation, and even flat-out
generosity and altruism, in nature. The
literature on game theory has worked to
explain why generosity arises. Our paper
provides such an explanation for why we see
so much generosity in front of us."
The Prisoner's Dilemma is a way of studying
how individuals choose whether or not to
cooperate. In the game, if both players
cooperate, they both receive a payoff. If one
cooperates and the other does not, the
cooperating player receives the smallest
possible payoff, and the defecting player the
largest. If both players do not cooperate, they
receive a payoff, but it is less than what they
would gain if both had cooperated. In other
words, it pays to cooperate, but it can pay
even more to be selfish.
In the Iterated Prisoner's Dilemma, two players
repeatedly face off against one another and can
employ different strategies to beat their
opponent. In 2012, Press and Dyson "shocked
the world of game theory," Plotkin said, by
identifying a group of strategies for playing
this version of the game. They called this class
of approaches "zero determinant" strategies
because the score of one player is related
linearly to the other. What's more, they
focused on a subset of zero determinant
approaches they deemed to be extortion
strategies. If a player employed an extortion
strategy against an unwitting opponent, that
player could force the opponent into receiving
a lower score or payoff.
Stewart and Plotkin became intrigued with this
finding, and last year wrote a commentary in
PNAS about the Press and Dyson work. They
began to explore a different approach to the
Prisoner's Dilemma. Instead of a head-to-head
competition, they envisioned a population of
players matching up against one another, as
might occur in a human or animal society in
nature. The most successful players would get
to "reproduce" more, passing on their
strategies to the next generation of players.
It quickly became clear to the Penn biologists
that extortion strategies wouldn't do well if
played within a large, evolving population
because an extortion strategy doesn't succeed
if played against itself.
"The fact that there are extortion strategies
immediately suggests that, at the other end of
the scale, there might also be generous
strategies," Stewart said. "You might think
being generous would be a stupid thing to do,
and it is if there are only two players in the
game, but, if there are many players and they
all play generously, they all benefit from each
other's generosity."

In generous strategies, which are essentially
the opposite of extortion strategies, players
tend to cooperate with their opponents, but, if
they don't, they suffer more than their
opponents do over the long term.
"Forgiveness" is also a feature of these
strategies. A player who encounters a defector
may punish the defector a bit but after a time
may cooperate with the defector again.
Stewart noticed the first of these generous
approaches among the zero determinant
strategies that Press and Dyson had defined.

After simulating how some generous strategies
would fare in an evolving population, he and
Plotkin crafted a mathematical proof showing
that, not only can generous strategies succeed
in the evolutionary version of the Prisoner's
Dilemma, in fact these are the only approaches
that resist defectors over the long term.
"Our paper shows that no selfish strategies will
succeed in evolution," Plotkin said. "The only
strategies that are evolutionarily robust are
generous ones."
The discovery, while abstract, helps explain the
presence of generosity in nature, an inclination
that can sometimes seem counter to the
Darwinian notion of survival of the fittest.

"When people act generously they feel it is
almost instinctual, and indeed a large
literature in evolutionary psychology shows
that people derive happiness from being
generous," Plotkin said. "It's not just in
humans. Of course social insects behave this
way, but even bacteria and viruses share gene
products and behave in ways that can't be
described as anything but generous."
"We find that in evolution, a population that
encourages cooperation does well," Stewart
said. "To maintain cooperation over the long
term, it is best to be generous."

Story Source:

The above story is based on materials provided
by University of Pennsylvania , via
EurekAlert!, a service of AAAS.

How Vegetation Competes for Rainfall in Dry Regions

Aug. 30, 2013 — The greater the plant density
in a given area, the greater the amount of
rainwater that seeps into the ground. This is
due to a higher presence of dense roots and
organic matter in the soil. Since water is a
limited resource in many dry ecosystems, such
as semi-arid environments and semi-deserts,
there is a benefit to vegetation to adapt by
forming closer networks with little space
between plants.

Hence, vegetation in semi-arid environments
(or regions with low rainfall) self-organizes into
patterns or "bands." The pattern formation
occurs where stripes of vegetation run parallel
to the contours of a hill, and are interlaid with
stripes of bare ground. Banded vegetation is
common where there is low rainfall. In a paper
published last month in the SIAM Journal on
Applied Mathematics, author Jonathan A.
Sherratt uses a mathematical model to
determine the levels of precipitation within
which such pattern formation occurs.

"Vegetation patterns are a common feature in
semi-arid environments, occurring in Africa,
Australia and North America," explains
Sherratt. "Field studies of these ecosystems are
extremely difficult because of their remoteness
and physical harshness; moreover there are no
laboratory replicates. Therefore mathematical
modeling has the potential to be an extremely
valuable tool, enabling prediction of how
pattern vegetation will respond to changes in
external conditions."
Several mathematical models have attempted
to address banded vegetation in semi-arid
environments, of which the oldest and most
established is a system of partial differential
equations, called the Klausmeier model.
The Klausmeier model is based on a water
redistribution hypothesis, which assumes that
rain falling on bare ground infiltrates only
slightly; most of it runs downhill in the
direction of the next vegetation band. It is
here that rain water seeps into the soil and
promotes growth of new foliage. This implies
that moisture levels are higher on the uphill
edge of the bands. Hence, as plants compete
for water, bands move uphill with each
generation. This uphill migration of bands
occurs as new vegetation grows upslope of the
bands and old vegetation dies on the
downslope edge.

In this paper, the author uses the Klausmeier
model, which is a system of reaction-diffusion-
advection equations, to determine the critical
rainfall level needed for pattern formation
based on a variety of ecological parameters,
such as rainfall, evaporation, plant uptake,
downhill flow, and plant loss. He also
investigates the uphill migration speeds of the
bands. "My research focuses on the way in
which patterns change as annual rainfall varies.
In particular, I predict an abrupt shift in
pattern formation as rainfall is decreased,
which dramatically affects ecosystems," says
Sherratt. "The mathematical analysis enables
me to derive a formula for the minimum level
of annual rainfall for which banded vegetation
is viable; below this, there is a transition to
complete desert."

The model has value in making resource
decisions and addressing environmental
concerns. "Since many semi-arid regions with
banded vegetation are used for grazing and/or
timber, this prediction has significant
implications for land management," Sherratt
says. "Another issue for which mathematical
modeling can be of value is the resilience of
patterned vegetation to environmental change.

This type of conclusion raises the possibility of
using mathematical models as an early warning
system that catastrophic changes in the
ecosystem are imminent, enabling appropriate
action (such as reduced grazing)."
The simplicity of the model allows the author
to make detailed predictions, but more
realistic models are required to further this
work. "All mathematical models are a
compromise between the complexity needed to
adequately reflect real-world phenomena, and
the simplicity that enables the application of
mathematical methods.

My paper concerns a
relatively simple model for vegetation
patterning, and I have been able to exploit this
simplicity to obtain detailed mathematical
predictions," explains Sherratt. "A number of
other researchers have proposed more realistic
(and more complex) models, and
corresponding study of these models is an
important area for future work. The
mathematical challenges are considerable, but
the rewards would be great, with the potential
to predict things such as critical levels of
annual rainfall with a high degree of
quantitative accuracy."

Story Source:

The above story is based on materials provided
by Society for Industrial and Applied
Mathematics.
And ( sciencedaily magzine  ).

Journal Reference:

1. Jonathan A. Sherratt. Pattern Solutions of the
Klausmeier Model for Banded Vegetation in
Semiarid Environments V: The Transition
from Patterns to Desert. SIAM Journal on
Applied Mathematics, 2013; 73 (4): 1347 DOI:
10.1137/120899510