Keywords:
Number and Operation, Activities, Technology
Ref: Taylor1
Author(s): Olson, Melfried
Year of publication : 2009
Title: Exploring Equivalent Fractions with the Graphing
Calculator
Journal or Publisher: Mathematics teaching in the Middle school
Volume, Issue, Pages: Vol. 14, No. 6, February 2009
Reviewer: Taylor
Date of Review: February 14, 2009
This
article addressed the common
problem of teaching students to identify equivalent fractions. Often,
students
have trouble determining whether or not a fraction like (12/15) is
reduced
completely, or if it is equivalent to another, simpler fraction like
(4/5). The
author, a fifth-grade math teacher, assessed her students’
understanding of
equivalent fractions and utilized the numberline program on the TI-73
graphing
calculator to promote discussion amongst her pupils.
Ms.
Olson found that her students
had several misconceptions about reducing fractions appropriately.
While some
students understood the strategy of finding common factors for the
numerator
and denominator of a fraction, others simply divided both by ‘2,’ and
became
confused shortly thereafter when they found denominators that were
non-whole
numbers. The teacher used the TI calculator program to show that the
resulting
faction was still equivalent, but pointed out that fractions generally
need to
be written using whole numbers.
While
I might argue that falling
back on technology like a TI calculator can often lead to student
dependency on
technology, this teacher suggested that the TI calculator program, when
used
effectively and properly, enabled students to explore and discuss
equivalence
of fractions in a more fluid manor. After reading this article, I can
see some
merit in utilizing the TI technology in a classroom, so long as it is
used
effectively.
Return to Index
Keywords:
Equity/Diversity, Discrete
Ref: Taylor2
Author(s): Robichaux, Rebecca; Rodrigue, Paulette
Year of publication : 2006
Title: Discovering Euler Circuits and Paths through a
Culturally
Relevant Lesson
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 11, No. 7, March 2006
Reviewer: Taylor
Date of Review: February 22, 2009
Two
educational researchers, Rebecca
Robichaux and Paulette Rodrigue, write this article about utilizing
culturally
relevant examples in their math classrooms to explore topics in
discrete
mathematics and graph theory. Their classrooms, located in 'the heart
of
southern Louisiana's Cajuan Country,' is home to many children whose
parents
work in the shrimping and fishing industries. By bringing in problems
related
to shrimping, Robichaux and Rodrigue were able to increase student
participation and interest in the lesson at hand.
The
lesson that they taught posed a
seemingly simple question to students: given a series of vertices
('jumping-off
points from which the boats could go'), if they desired to catch shrimp
along
every path created by these connected vertices, and only travel down
each path
once, could they be successful? This problem is famous in graph theory,
and
similar concepts can be applied to many traveling / path problems in
the world.
The authors supply instructions for teaching this lesson, and a list of
materials required. They suggest presenting this problem to students,
giving
the students a brief background on odd and even vertices, and then
letting the
students loose to explore in groups of four.
The
lesson outlined in this article
appeared to be very effective; students got to work quickly, and were
very interested
in the problem at hand. If possible, I would like to adapt this lesson
to fit
my own repertoire. In their conclusion, the authors discuss the
possibility of
applying their lesson plan to other parts of the nation. They point
out, after
all, that food industries exist in many places in the US, and similar
graph
theory problems could be presented to orange farmers' children in
Georgia, corn
farmers' children in Iowa, etc.
Keywords:
Connections, Discrete,
Ref: Taylor3
Author(s): Petras, Richard T.
Year of publication : 2001
Title: Privacy for the Twenty-First Century: Cryptography
Journal or Publisher: MATHEMATICS TEACHER
Volume, Issue, Pages: Vol. 94, No. 8, 689-92
Reviewer: Taylor
Date of Review: February 25, 2009
The
article that I read discussed
the applications of high-level RSA cryptography on internet and
wireless
communication security. Internet security, as the article points out,
is a very
hot topic, and much desired by the general public. Internet privacy is
very
important to all computer users, so cryptography is necessary.
The
mathematics behind RSA
cryptography is complex, though relatively accessible on a low-level to
many
high school students. When teaching basic cryptography, we can make
connections
to many academic fields, mathematical or otherwise. A decent knowledge
of
modular arithmetic is necessary to understand RSA; this form of
mathematics
stems from early-primary school's 'division and remainders.'
Connections drawn
from early (perhaps 4th grade) mathematics can be beneficial for
students to
learn modular arithmetic, and then cryptography.
Addressing
the topic of RSA can also
connect to mathematical topics of functions. RSA relies on a function
that is
essentially impossible to invert. That is, f(x) can be computed easily,
but
without some requisite information (an RSA key), f^-1(x) cannot be
computed
efficiently (or at all). Students in a classroom would be able to
discuss
connections between RSA and other functions that are one-way, or not
bijections
(the absolute value function, f(x)=x^2, etc).
If
there is interest in the
classroom to talk about computing, privacy, or politics, the topic of
RSA
cryptography also connects to these fields. General ethics on human
privacy vs.
the politics of allowing criminals to hide files efficiently were
discussed in
this article, and could be addressed in the classroom.
Keywords:
Representations, Geometry
Ref: Taylor4
Author(s): Pagni, David L.
Year of publication : 2007
Title: Finding Areas on Dot Paper
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: VOL. 12, NO. 5 . DECEMBER 2006/JANUARY
2007
Reviewer: Taylor
Date of Review: March 1, 2009
This
article presented a method by
which students can learn about and derive formulas for computing the
area of
various geometric shapes. Many textbooks present formulas to find the
area of
standard platonic shapes (for example, the area of the triangle is
often listed
as (length) x (height) / 2), but many students are unaware of where the
methods
required to derive these formulas. David Pagni, who teaches at the
California
State University, suggests in this article that the use of 'dot paper'
can
assist students in understanding and deriving area formulas.
Dot
paper is essentially graph, with
a dot paced at each intersection of the grid. Some dot paper ("square
dot
paper") is drawn in a slanted manor for convenience. Polygons like
triangles, rectangles, or trapezoids are then drawn using the dots as
vertices,
and volumes are then computed. After several volumes of each type of
shape are
computed, students conjecture about the relationships between polygon
side
lengths and volumes. The article is accompanied by several sample
activity
sheets that teachers can complete with their students.
This
lesson appears very sound, and
is something that I would like to pursue, were I to teach geometry.
This lesson
allows students to visit representations of geometric figures in
multiple ways,
allowing them to understand fully the ways by which areas of geometric
figures
are computed.
Keywords:
Problem Solving, Proof, Calculus
Ref: Taylor5
Author(s): Perrin, John R.
Year of publication : 2008
Title: Developing Reasoning Through Proof in High School
Calculus
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: VOL. 102, NO. 5, 341-348
Reviewer: Taylor
Date of Review: March 1, 2009
Students
are often expected,
sometimes unrealistically, to understand and be able to write
mathematical
proofs by the time that they enter post-secondary school. While this is
a
desirable goal, the author notes that this is not often the case.
Students often
struggle with proof concepts. This article focused on one teacher, John
Perrin,
and his work with high school students in an AP Calculus course.
Perrin
found that students often
struggle with the requirements of a proper proofs. Students will supply
examples
that indicate that a statement is true, but do not provide general
enough
arguments to prove that a given statement is true for all cases. After
being
reminded that proofs need to be general enough for all cases, some
students are
able to complete the easier proofs. For harder proofs, though, Perrin
suggests
guided practice. Instead of assigning proofs, he goes through proofs in
class,
seeking input concerning the 'next step' from students. In this way
students
think critically about the proof, but are not bogged down by the formal
logic
with which they may not be familiar.
Perrin
completes the article by discussing various proofs
that could be done in class, and the steps required to complete each
proof.
This article could prove to be a very good resource for teaching a
lesson on a
difficult proof, and it outlines many tried-and-true methods for
assisting
students through difficult proofs and concepts.
Keywords:
Communications
Ref: Taylor6
Author(s): Whitin, Phyllis; Whitin, David
Year of publication : 2002
Title: Promoting Communication in the Mathematics Classroom
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: December 2002, Pages 205-11
Reviewer: Taylor
Date of Review: March 4, 2009
Authors
David Whitin, who teaches
mathematics, and Phyllis Whitin, who teaches language arts, address a
common
problem in mathematics curriculum. Students, even those who are well
versed in
mathematics and very good at manipulating equations, often find it
difficult to
express and explain mathematical concepts in plain English. This
article
suggested several techniques that teachers can use to foster 'literate
mathematicians.' The authors often use journals in their math classroom
to
allow students to express their thought processes and develop creative
ideas to
solve problems. The students share their ideas with one another,
refining their
own ideas as they discuss.
In
addition to in-class
communication exercises, this article suggests assigning homework that
involves
student communication with their parents. This way, students can gain
insight
into what their family members know about mathematics and can then
either learn
from their parents, or explain new mathematical concepts to them,
thereby
solidifying their own knowledge.
The
authors found that using
journals and discussion in their math classroom and at home seemed to
increase
their students' ability to communicate their thoughts as they solved
problems.
It also helped to solidify or clarify student understanding of
difficult
mathematical concepts.
Keywords:
Number and Operation
Ref: Taylor7
Author(s): Bay-Williams, Jennifer M.; Martinie, Sherri L.
Year of publication : 2003
Title: Thinking Rationally about Number and Operations in
the Middle
School
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 8, No. 6., pp. 282-286; February
2003
Reviewer: Taylor
Date of Review: March 8, 2009
As
the title suggests, the authors
of this article wrote on teaching students to think rationally about
numbers
and operations. In particular, authors Jennifer Bay-Williams and Sherri
Martinie observed student responses to learning about ratios and
fractions.
Students often compare and order fractions in many ways. Some graph the
fractions on a number line, while others convert fractions to their
decimal
equivalents and compare those. While both of these methods are correct,
the
authors made a few notes about appropriately teaching this concept:
First, they
suggested that teachers should allow students to think critically about
comparing fractions, instead of giving them an algorithm outline that
students
have to follow. Second, teachers are encouraged to disallow calculator
use when
first teaching students to compare fractions. Mental math that involves
fractions is essential to survival in our society, and students will
not always
have a calculator handy at all times.
The
authors also addressed student
ability to relate fractions to real-world scenarios. According to the
article,
U.S. elementary school teachers had a tremendously difficult time
coming up
with example exercises for students to do that involve division of
fractions.
While some of these teachers could remember an algorithm for dividing
fractions, they could not create decent examples that used the concept.
This
was a bit disconcerting, considering the fact that we want not only our
teachers, but our students to be able to create and understand this
material.
The
article wrapped up by talking
about some interesting activities that could be used when teaching
proportionality. The book If you Hopped Like a Frog by David
Schwartz
provides some great, pertinent material for students to learn about
proportions
and ratios, and the authors highly suggested bringing this book into
any middle
school classroom where proportions is being taught.
Keywords:
Statistics
Ref: Taylor8
Author(s): Fox, Thomas B.
Year of publication : 2005
Title: Transformations on Data Sets and Their Effects on
Descriptive
Statistics
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 99, No. 3, p. 208-218
Reviewer: Taylor
Date of Review: March 10, 2009
For
this article, Thomas Fox
developed a lesson and activity that addressed data transformations and
descriptive statistics. The activity asks students to examine the
effects of either
translation or scale factor transformation on a data set, and to
conjecture
about and defend their results. After students have examined their data
sets
after a transformation and have defended or reworked their conjectures
about
the effects of these transformations, students are asked to make
generalizations about data transformations. Students found that
descriptive
statistics like mean, quartiles, mode, etc. were shifted by h under a
translation of h, and were scaled by A when their data changed by a
scale
factor of A. Descriptive statistics like variance, standard deviation,
etc.
were effected in different ways by scale factor transformations, and
were not
effected by translation transformations.
This
lesson, according to the
author, was very effective in teaching students about data
transformations, and
also reinforced student understanding of descriptive statistics.
Complete
worksheets were attached to this article.
Keywords:
Algebra, Representations
Ref: Taylor9
Author(s): Lee, Lesley; Freiman, Viktor
Year of publication : 2006
Title: Developing Algebraic Thinking Through Pattern
Exploration
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 11, No. 9, p.428-433
Reviewer: Taylor
Date of Review: March 15, 2009
In
this article authors Lee and
Freiman examine student learning of algebra through pattern
exploration.
Students often struggle with concepts of variables and equations, and
they are
often more receptive to visual representations of new material. Here,
Lee and
Frreiman introduced their students to algebra through a visual puzzle:
Start
with one star. Then, add a star to the left, right, and bottom of that
star.
Continue to add three stars, one to each the left, right, and bottom,
and
develop a larger and larger 'T'. Students examine how each T is related
to
previous T's. They are asked how many stars make up a particular 'T',
or how
big a T a certain number of stars can make.
Students
witness several graphical or visual representations
of this problem and similar problems, but are not introduced to
equations until
later. As the authors point out, many students were capable of
representing the
problem as an equation (3n + 1), where 'n' represents the number of
star-adding
steps they took, but most students were not able to identify the
significance
of n. Many students simply stated "n stands for a big number."
Keywords:
History
Ref: Taylor10
Author(s): Liu, Po-Hung
Year of publication : 2003
Title: Do Teachers Need to Incorporate the History of
Mathematics in
Their Teaching?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 96, No. 6, p. 416-421
Reviewer: Taylor
Date of Review: March 15, 2009
In
his article, teacher Po-Hung Liu
argues for the inclusion of history in a mathematics classroom,
highlighting
the benefits of introducing students to the evolution of mathematics.
He builds
off of Fauvel's research from 1991 that addressed the cognitive,
affective, and
sociocultural effects of incorporating history in the classroom, and
attempts
to pinpoint how history can positively affect curriculum and high
school
teaching.
Liu
notes that mathematics is often
considered a "dull drill" subject, and students attitudes towards
mathematics are not always positive. If history is included in a math
classroom, it is often presented in a neat, polished way that does not
accurately reflect the struggle that famous mathematicians faced. Liu
notes
that when students actually read about the struggles of ancient great
mathematicians, they feel more comfortable with the fact that they are
struggling with the same concepts.
Liu
suggests many ways to present
history in a classroom. One way, which he utilizes frequently, is to
give
students a difficult problem. After students present their solutions or
struggles to the rest of the class, Liu presents solutions to the
problem as
proposed by ancient Greek, Arabic, or Chinese mathematicians. This
allows
students to see mathematics not as a fixed subject, but as being
flexible,
relative, and humanistic.
It is
noted in this article that no
empirical study has been done to indicate that teaching history
actually
affects student performance on traditional tests. From personal
experience,
though, Liu notes that he has observed the benefits of incorporating
mathematics in the curriculum.
Keywords:
Gifted
Ref: Taylor11
Author(s): Chval, Kathryn B.; Davis, Jane A.
Year of publication : 2008
Title: The Gifted Student
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 14, Number 5, p. 267-274
Reviewer: Taylor
Date of Review: March 31, 2009
In
this article, authors Kathryn
Chval and Jane Davis argue the importance of differentiating
instruction for
gifted students, and they examine several ways that teachers can
incorporate
differentiated instruction in their classroom. They note that many
classrooms
have one or more 'gifted students,' and teachers are often either
under-prepared to teach these students, or they feel that these
students will
be fine on their own, and that their attention needs to be on the other
students. In either case, gifted students do not get sufficient
attention.
Some
schools have created 'pull out'
programs, which is where their gifted students are pulled out of there
normal
classroom two days each week and are given more direct attention and
more
challenging school work to complete. This strategy as been shown to be
very
effective at keeping gifted students' attention and encouraging
learning in
gifted students. The authors also suggest methods by which teachers can
update
existing curriculum and assignments, so that they are differentiated.
Keywords:
Technology
Ref: Taylor12
Author(s): Driskell, Shannon
Year of publication : 2005
Title: Teachers' Technology Class Continues Discussion of
Pitfalls
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 99, No. 5, Page 340-342
Reviewer: Taylor
Date of Review: April 5, 2009
Here
is this article, author Shannon
Driskell discusses a technology seminar that she conducted with several
math
teachers. Their goal through this class / seminar was to identify the
positive
and negative aspects of many mathematical technologies including
Internet,
Excel, TI Calculators, Minitab, Mathematica, and The Geometer's
Sketchpad.
They
first identified several
pitfalls in the TI-83 line of calculator. Students were often confused
when the
TI calculator returned '0' as the answer for sin(2*pi), but returned
-2e-13 as
the answer for sin(4*pi). Many students believed the calculator's
non-zero
output to the sin(4*pi) query. The TI calculator also returned a
derivative for
the corner of an absolute value function; this confused many students.
Confusing syntax for the linear regression function, among others, were
also
discussed. Similar confusing pitfalls were found in Excel and
Mathematica.
The
author concludes her article by
suggesting that teachers remain cautious of these pitfalls, but do not
abandon
technology because of a few quarks. Technology, despite its pitfalls,
can still
be useful for teachers. Teachers just need to be aware of these
pitfalls so
they can anticipate student difficulty with technology.
Keywords:
Probability, Algebra, Geometry
Ref: Taylor13
Author(s): Edwards, Michael T; Phelps, Steve
Year of publication : 2008
Title: Can You Fathom This? Connecting Data Analysis,
Algebra, and
Geometry with Probability Simulation
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 102, Issue 3, Page 210-216
Reviewer: Taylor
Date of Review: April 7, 2009
In
this article, authors Michael
Edwards and Steve Phelps explore the Fathom software package and
attempt to
connect the fields of algebra, geometry, data analysis, and
probability. They
note that students live in an increasingly data-driven world, and that
understanding data analysis is crucial to the education of this
generation of
students. Fathom does a fantastic job of generating and displaying
data, and
the authors recommend this software in any algebra, data analysis, or
probability classroom.
Probability
problems that can be
represented by Fathom are numerous, and the representations created in
Fathom
can be used to greatly assist student learning. The authors provide
several
examples of how they used Fathom in class to solve problems, then
related the
solution to geometry and algebra. For example, one problem presented
read:
“What is the probability that random numbers between 0 and 1 will be a
distance
less than 0.1 away from one another?” Random numbers were generated,
and pairs
of values that were less than 0.1 away from one another were
identified. These
ordered pairs were plotted on a graph, and students got to see which
section of
numbers represents the “less than 0.1 difference” set. Fathom provided
a ratio
of numbers within this group to numbers outside the group, and students
verified this number through use of geometry and algebra. Several other
examples were presented in this article.
The
authors of this article created
a unique and innovative way to use Fathom to connect algebra, geometry,
data
analysis and probability simulation. The methods in this article could
be
applied to a variety of courses, and I am now convinced that Fathom
would
complement any mathematics classroom very well.
Keywords:
Problem Solving, Manipulatives, Puzzles
Ref: Taylor14
Author(s): Madden, Sean P.; Diaz, Ricardo
Year of publication : 2008
Title: Spatial Reasoning and Polya's Five Planes Problem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 102, No. 2, p. 128-135
Reviewer: Taylor
Date of Review: April 13, 2009
In
this article, authors Madden and
Diaz present information about teaching spatial reasoning in the
mathematics
classroom. According to psychologist Herbert Simon's
information-processing
theory, the mind consists of several components: a sensory register,
long-term
memory, short-term (working) memory, and an executive editor,
responsible for
decision making. All of these components of the mind are activated, at
least in
part, when students are presented with spatial reasoning problems.
Spatial
reasoning problems require a lot of thought and organization to solve,
so many parts
of the brain are activated.
One
such spacial reasoning problem
is Polya's "Five Planes Problem," which is stated as: "How many
regions of space (maximum) are created by five randomly intersecting
planes?" Solving this problem requires students to visualize
three-dimensional space, organize data in a data table, recognize
recursive
patterns, find algebraic formulas, and use computers. These
requirements cause
students to use several parts of their minds, and challenges students
to
utilize critical thinking and reasoning skills. Madden and Diaz found
that
their students used a variety of ways to solve this problem, from
brute-force
techniques to pattern recognition.
Keywords:
Algebra, Representations
Ref: Taylor15
Author(s): Arcavi, Abraham
Year of publication : 1994
Title: Symbol Sense: Informal Sense-making in Formal
Mathematics
Journal or Publisher: For the Learning of Mathematics
Volume, Issue, Pages: Volume 14, Number 3, Page 24-35
Reviewer: Taylor
Date of Review: April 15, 2009
In
this paper, author Abraham Arcavi
describes a notion that he titles “symbol sense.” In his introduction,
he notes
that many students become proficient in algebra and are good at
manipulating
equations, but many of these same students often struggle when they are
told to
relate algebra to mathematical models and situations. Symbol sense,
generally,
is the ability of one to recognize, interpret, and manipulate algebraic
symbols
in meaningful ways. This includes being able to analyze a situation and
apply
symbolic notation to that situation accordingly.
Arcavi
provides two great examples
where symbolic notation can be used to solve seemingly difficult
problems.
First, he suggests that students should be able to apply algebraic
symbols and
concepts to ‘magic square’ problems. He next suggests that symbols
could be
used to relate areas of squares and rectangles with similar parameters.
In each
of these cases, Arcavi notes that students struggled to apply algebraic
symbols
to mathematical problems, and ultimately became stuck or provided the
incorrect
answer. He goes on to talk about the importance of ‘reading symbols,’
and
determining exactly what a symbol represents.
Arcavi
finishes his article with
several instructional suggestions for teachers. First, Arcavi
acknowledges the
weaknesses of current techniques of teaching algebraic manipulation. He
suggests that teachers need to find meaningful examples for their
students that
require algebraic manipulation so that the students internalize the
manipulations better. He also suggests that students need to interpret
and
justify their answers; simply shouting out correct answers is not
sufficient to
show understanding.
Keywords:
Management
Ref: Taylor16
Author(s): Caldwell, Marion
Year of publication : 2003
Title: Transition Points
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Volume 10, Issue 4, Page 218-223
Reviewer: Taylor
Date of Review: April 15, 2009
This
article’s author, Marion
Caldwell, was not comfortable with classroom management when she first
started
teaching. She ran into problems where students would become unruly when
they
transitioned from activity to activity, and she felt it difficult to
control
her class at those times. In this article Caldwell talks about what she
has
learned about the transitional times of her daily schedule, and
suggests a few
ways to prevent unruly behavior.
Her
first idea to keep order in her
class was similar to a modified rewards system. Each time her class
cooperated
when switching topics, she put a few cents into a jar. When the class
had $2.00
in the jar they were rewarded with treats. This concept gradually grew
into a
point system. Classes were rewarded points if they behaved
appropriately when
they went to the library or assemblies, arrived in the classroom on
time, and
cleaned up after themselves.
She
modified this “transition
points” system further to incorporate decimals, thereby incorporating a
math
lesson into the rewards system. Some tasks (cleaning desks, etc) were
worth
half of a point. Others were worth 1/3, 1/4, and 1/8 points. In order
to
achieve certain point goals (e.g. have the class room get 20 points),
students
were required to add fractions to determine point totals.
This
strategy of classroom
management worked seemingly well in third through fifth grade
classrooms. I am
skeptical of the effectiveness of the program in high school
classrooms, but
perhaps a modified form of the program would be appropriate.
Keywords:
Problem Solving, Teaching Strategies
Ref: Taylor17
Author(s): Williams, Kenneth M.
Year of publication : 2003
Title: Writing About the Problem Solving Process to Improve
Problem
Solving Performance
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 96, Issue 3, Page 185-7
Reviewer: Taylor
Date of Review: April 26, 2009
In
this article, teacher Kenneth
Williams addresses methods by which students can become competent and
confident
mathematical problem solvers. According to Williams (and educational
psychologist George Polya), there are four executive processes used by
mathematicians
when solving problems:
Students
often stop at the first
process when they do not understand a problem, or are not familiar with
the
structure of a problem. Others who do understand the problem often
struggle
with how to devise a plan to tackle the problem. Students are not
familiar
enough with the executive process require to solve problems.
Williams
suggests that a combination
of developing good writing skills in students and presenting students
with
challenging problems of the type that they have not previously
encountered
helps students to develop good mathematical problem-solving skills. In
his
experiment, Williams assigned students some problems that were similar
to those
seen in class each night, along with a few challenging problems with
which the
students had no familiarity. He also had students work out problems in
paragraph form, instead of writing equations haphazardly. As a result,
students
were able to verbalize (in written form) their ideas about solving a
problem,
and the challenges that they faced.
Keywords:
Problem Solving, Activities
Ref: Taylor18
Author(s): Cofman, Judita
Year of publication : 1990
Title: What to Solve? Problems and Suggestions for Young
Mathematicians
Journal or Publisher: Oxford Science Publications
Volume, Issue, Pages:
Reviewer: Taylor
Date of Review: April 27, 2009
In
her book, author Judita Cofman
presents a series of challenging mathematical problems that can be used
in the
classroom to assist students to become better mathematical problems
solvers.
Cofman has a history of leading ‘problem seminars’ at international
camps for
young mathematicians. The problems in this book are presented in the
same manor
that problems are presented at her camps. Students are led step-by-step
through
four sages of problem solving:
Problems
in this book are, indeed, presented in a logical,
ordered manor that encourages independent investigation by students,
demonstrates appropriate problems solving approaches, and examines
significant
past and current mathematical problems. Problems presented in this book
could
easily be incorporated into any mathematics classroom. A wide range of
topics
are addressed, and problems are indexed by topic.
Keywords:
Teaching Strategies, Assessment, Management
Ref: Taylor19
Author(s): Mundahl, Stephanie; Nelson, Nicole; Singer-Towns,
Tim;
Wyberg, Terry
Year of publication : 2009
Title: Surviving Your Beginning Years – Preparing to Teach
Journal or Publisher:
Volume, Issue, Pages: MCTM Spring Conference. Duluth Minnesota.
May,
2009.
Reviewer: Taylor
Date of Review: May 1, 2009
As a
new teacher, I felt compelled
to attend a session on survival tactics and preparation for joining the
work
force. This session, lead by three fairly new teachers and one veteran
teacher,
proved to be very informative, and I gained some great tips and
insights into
creating lesson plans, motivating student engagement, and assessing
student
knowledge.
First,
each fairly new teacher
talked about setting unit and lesson plans. They each suggested that we
stick
to the book our first year: we will have enough to worry about without
having
to design a whole curriculum from scratch. Over the next two years,
though, we
can start to experiment with different curriculum strategies. One
teacher noted
that she had done chapters 1,2,3,4,5,6,7, and 8 her first year, then
chapters
1,2,7,3,4,8,5,6 her second year, and had since omitted chapter 3 from
her
curriculum, in favor of other activities her subsequent years. I feel
comfortable with suggestion, and intend to generally stick with the
book, but
incorporate fun activities or historical artifacts and anecdotes
whenever relevant.
The teachers also suggested differentiating between material that is of
'dire
importance,' and that which is 'icing on the cake.' It may be nice to
teach
students everything in your unit plan, but we as teachers must decide
when we
can simply move on, o! mitting part of a lesson in favor of completing
the
unit.
Next,
the teachers talked about
opening activities and student engagement. I liked Singer-Town's
suggestion of
having four review 'opener' questions each day: he has a '24-7-30'
activity,
where students do two review problems from 24 hours prior (the previous
lesson), one problem from a week prior, and one problem from a month
prior.
This way, students will be constantly reviewing or at least seeing old
material. To gain student engagement, the presenters suggested that we
'make
the students feel like they know something.' That is, they suggest that
we
remind students of what they already know, then add to that knowledge
when we
introduce new topics. This strategy has proven to help students relax
and feel
more in control. They also suggest setting up a routine (like openers,
etc.)
and keeping to that routine. Often, students are missing routine in
their home
lives, so it is important that they have routine at school.
The
session wrapped up with the teachers talking about
checks for understanding. They suggested standard things, like 'exit
slips,'
and informal checks around the room. These, according to the speakers,
are
proven methods that allow teachers to check their students'
comprehension of
the course material.
Keywords:
Technology, Manipulatives
Ref: Taylor20
Author(s): Reiners, Mike
Year of publication : 2009
Title: Dynamic Graphing: How Dynamic Technology Teaches
Transformations
Journal or Publisher:
Volume, Issue, Pages: MCTM Spring Conference. Duluth Minnesota.
May,
2009.
Reviewer: Taylor
Date of Review: May 1, 2009
Mike
Reiners, who is a current high
school teacher and competitive math team coach, presented on 'dynamic
graphing
in the classroom.' The problem with traditional teaching of graphing,
he notes,
is that it is a slow process to graph a parabola or a line. Tables of
values
must be written up, and dots must be drawn and then connected. 'Static
graphing,' as he calls it, is a slow affair, and does not allow
students to see
how changes in values of an equation change the graphs of the
equations. While
it is important for students to learn how to graph equations, it is
also
important for them to recognize when a graph will be inverted,
stretched, or
otherwise changed with changes to constants within the equation.
The
tool that Reiners used to show
'dynamic graphing' was a Casio graphing calculator. Similar to Cabri,
the new
Casio graphing calculators are able to animate a graph. For example,
the
equation Y=AX^2 can be animated by allowing A to change gradually
between -5
and 5. Doing this, students who are presented an animation with
information
about the values of A can make inferences about how A effects the look
of a
graph. This tool allows students to discover for themselves that a
negative
sign in front of an equation flips the equation upside down, and larger
values
of a scale factor (A) stretch the graph in certain ways.
Mr.
Reiners then showed us another function, Y=A*B^X, and
allowed the value of B to change from -5 to 5. The results of this
demonstration were stunning: participants (the high school teachers)
began to
debate the validity of this 'function,' since it was undefined at many
points,
and not continuous at all when B was negative. Though not perhaps at
the level
at which it was discussed in our session, I could definitely see
students
arguing similar arguments about the function in high school classrooms.
Keywords:
Manipulatives, Activities
Ref: Taylor21
Author(s): Droogsma, Larry
Year of publication : 2009
Title: 3-D Graphing for the Kinesthetic Learner
Journal or Publisher:
Volume, Issue, Pages: MCTM Spring Conference. Duluth Minnesota.
May,
2009.
Reviewer: Taylor
Date of Review: May 1, 2009
In
this session, Droogsma presented
a new way to teach three dimensional graphing that he has used, and
found to be
successful in his classroom. Rather than having students graph three
dimensional objects on two dimensional paper, he turns his classroom
into a
three dimensional space, connecting strings from wall-to-wall and from
ceiling
to floor. Each string represents a different axis (X,Y, and Z), and the
intersections of the strings represent the origin. Students are then
given
tennis balls, string, and bed sheets to represent points, lines, and
planes
respectively.
Droogsma
had several observations to share about this
activity. First, he noted that it takes students a while to orient
themselves
to the 3D coordinate system, especially since X and Y usually represent
a plane
that is perpendicular to a student's eye sight, and is now located on
the
floor, below the students. The addition of the Z axis definitely takes
students
a while to get used to. Next, students have trouble reorienting
themselves if
they have to move desks. After a quarter ended and new seats were
given,
students struggled to identify the direction in which they could find a
positive X value. Finally, Droogsma suggested that if we are to mimic
his
material, we should ask the IT / shop teachers for supplies and lesson
materials. He had a series of elaborate cylinders, boxes, and prisms
that were
made of metal, and he said that the shop teachers are always open to
the idea
of making more materials for his class.
Keywords:
Algebra, Activities
Ref: Taylor22
Author(s): Phillips, Elizabeth
Year of publication : 1991
Title: Graphs and Functions as Patterns
Journal or Publisher: Curriculum and Evaluation Standards for
School
Mathematics; NCTM
Volume, Issue, Pages: Patterns and Functions (5-8): Page 55-70
Reviewer: Taylor
Date of Review: May 2, 2009
In
this NCTM-produced text, author
Elizabeth Phillips provides resources for teachers to teach various
lessons on
graphs, functions, and patterns. The text contains several lessons, and
each
one includes an entire lesson plan (opener, exploration, summary,
etc.).
Similar to many modern text books, this book seems to promote student
exploration, (as opposed to traditional lecture teaching). As such,
each lesson
includes extensive exercises that students can complete with a
teacher’s
guidance.
The
first lesson takes a question
about racing (one runner starts earlier, but the other runs faster),
and asks
the question: who will win the race? This exploration question requires
students work in groups and use their knowledge of y-intercepts, slope,
and
graphing to solve the problem. Next, students are asked to make
relationships
about the volume and height of various water bottles. At first,
students find
linear relationships with cylindrical bottles. Then, they are
introduced to
irregular-shaped bottles, and were forced to refine their knowledge of
slope to
account for the new information presented.
This
text seems very reasonable, and
it encompasses a lot of material. If students finish with the various
projects
early, they have the option to work on addition, more challenging
problems that
are also included in this book.
Keywords:
Trigonometry, Number and Operation
Ref: Taylor23
Author(s): Burke, Maurice J.; Kehle, Paul E.; Kennedy, Paul A.;
St.
John, Dennis
Year of publication : 2006
Title: Navigating Through Number and Operations in Grades
9-12
Journal or Publisher: NCTM
Volume, Issue, Pages: Pages 23-26
Reviewer: Taylor
Date of Review: May 2, 2009
Similar
to the 1991 NCTM Addenda
books for School Mathematics, this more recent NCTM publication
provides
teachers with several lessons, each of which include opening
activities,
material lists, activity sheets, goals, and closing activities. The
particular
lesson that I examined includes an activity sheet (Trigonometry Target
Practice), and covers exploration of various values of the tangent
function.
The goal of this lesson is to have students discover the values at
which the
tangent function is irrational or zero.
To
accomplish this goal, students
must first become aware of how the tangent function is computed. After
they
have reviewed these skills, students are given graphing calculators to
use as
they explore the tangent function for various angle values. After these
explorations are complete, students examine lattice points on the graph
(the
points that have integer values for (x,y)), and how they can ‘hit’
these
lattice points with a tangent function. This lesson seems appropriate
for a
high-school trigonometry class, and I can definitely see myself
utilizing this
lesson in the future.
Keywords:
Statistics, Algebra
Ref: Taylor24
Author(s):
Year of publication : 2008
Title: Barbie Bungee
Journal or Publisher: Illuminations
Volume, Issue, Pages: NCTM
Reviewer: Taylor
Date of Review: May 11, 2009
I
found a lesson plan on the
Illuminations website titled 'Barbie Bungee.' This lesson combines data
accumulation and analysis with algebra in a fun, exciting, and well
thought-out
manner. After some instruction, students are given one Barbie doll and
a series
of rubber bands. They collect data on how far a Barbie falls given
various
numbers of rubber bands. After collecting this data, students plot
their
coordinates, find a line of best fit, and apply a linear equation to
the
relationship between number of rubber bands and fall distance. Then,
students
are expected to predict the maximum number of rubber bands that could
be used
to safely allow Barbie to bungee jump from a height of two stories.
Attached
to this lesson plan were
learning objectives, materials, an instructional plan, questions
for students, assessment options,
and extensions to the problem. Everything that I would possibly need is
attached to this lesson, and the lesson plan appears carefully thought
out and
executed.