Keywords: Equity/Diversity
Ref: Mish1
Author(s): Herzig, Abbe
Year of publication : 2005
Title: Goals for Achieving Diversity in Mathematics
Classrooms
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 99, No. 4, p. 253-259
Reviewer: Mish
Date of Review: February 17, 2008
Goals for Achieving Diversity in Mathematics Classrooms does a very
good job of presenting why teachers need to have an understanding of
the diversity in their classrooms and how it is important to student
learning. Herzig makes a strong presentation of the preconceptions of
learning mathematics and how these need to be changed. I found the
article to be an interesting and thought-provoking read, but I found it
to be a bit thin on math-specific recommendations (they were quite good
generally) for the classroom.
Keywords: Connections, Activities
Ref: Mish2
Author(s): Devlin, Keith.
Year of publication : 2002
Title: Numbers in the Garden and Geometry in the Jungle
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 7, No. 8, pages 422-425
Reviewer: Mish
Date of Review: February 19, 2008
Numbers in the Garden and Geometry in the Jungle begins by observing that while most people may not be widely interested in math, they are probably interested in nature. Devlin goes on to point out where in our gardens we can find patterns, or more specifically, the Fibonacci numbers, in flowers, pineapples, and leaves in trees. He moves on to give a brief but interesting introduction to the geometry of animal coal patterns. Devlin believes that showing children what cool things can be done with mathematics will make them more interested in learning more, analogous to learning a musical instrument or how to skateboard.
Devlin's article makes a good argument for these enrichment
activities. It's relatively easy to convince yourself that
students will have a good time observing these patterns in nature, or
in animal coat patterns. The only question is where would he recommend
tying it in? He argues that enrichment like this is essential, but
doesn't state where he believes it should go. But this is a
small contention. The Fibonacci stuff could go nicely when learning
sequences and series (or earlier with simple patterns), and surely the
geometry of animal coats could merit at least a 5 or 10 minute aside in
a geometry class sometime. Devlin does make one explicit recommendation
in the form of the PBS series, Life by the Numbers, suggesting a
segment at the beginning of class every now and then. Overall, pretty
nice little article, if a bit thin on where to go for direction on
enrichment activities.
Keywords: Connections, Representations, Communications
Ref: Mish3
Author(s): Peterson, Blake E.
Year of publication : 2006
Title: Counting Dots and Measuring Area: Rich Problems from
Japan
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 12, No. 4, pages 214-220
Reviewer: Mish
Date of Review: February 25, 2008
Counting Dots and Measuring Area: Rich Problems from Japan presents two rich problems—one involving representations of ways to count dots forming a square, the other a way to generate and develop the formula for a piecewise graph. Peterson discusses the numerous ways in which the students approached the dot problem, and gives illustrations of the various results. He spends more time discussing the various merits of the second piecewise function question, illustrating the skills and observations necessary to make the problem work. He closes by discussing the importance of choosing rich problems in order to foster students making the desired connections, arguing that students need to work with problems that require them to make connections if they are ever going to learn to do so.
This article presents two slick problems that will allow students to
really get their hands on concepts. It includes a sample worksheet for
the piecewise graph problem, and examples of various student approaches
to the dot-square problem. It’s easily accessible for adaptation for
classroom use, I’m definitely holding on to it!
Keywords: Algebra, Proof
Ref: Mish4
Author(s): Lannin, John K.
Year of publication : 2003
Title: Developing Algebraic Reasoning Through Generalization
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 8(7), p. 342
Reviewer: Mish
Date of Review: February 27, 2008
“Developing Algebraic Reasoning Through Generalization” discusses the importance of developing students’ ability to construct generalization. The article utilizes the Cube Sticker problem (working in a situation involving stickers on the faces of cubes that form rods) to demonstrate several ways students may approach problems. Lannin goes on to show various ways students attempt to justify their work, and discusses the pitfalls of proof-by-example. The article illustrates how to question students in such a way that helps them connect their rules to context.
This article was well-written and informative. We see an example of
a very good problem for students to work on to form generalization
skills, and the several ways in which students may approach the task.
Lannin’s suggestions for how teachers should respond to the various
student approaches are helpful and meaningful in the task of getting
students away from proof-by-example reasoning and towards thinking
generally.
Keywords: Communications
Ref: Mish5
Author(s): Csongor, Julianna; Craig, Carolyn
Year of publication : 2005
Title: Say What You Mean and Mean What You Say
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 99, No.3, 181
Reviewer: Mish
Date of Review: March 5, 2008
Csongor and Craig discuss the importance of strong communication skills in society today. They illustrate the need for teachers to find ways to help their students communicate better, but also to want to communicate better. The trick for them is to make it fun. They give an example in which students are provided with geometric diagrams using polygons, line segments, etc. The activity is done in pairs, with one student as a communicator and another as the artist. The goal is for the artist to draw the figure according the communicator’s specifications. The students are allowed varying degrees of assistance as they become familiar with the activity.
I think the activity would be a great opportunity for students to
employ their geometric vocabulary. The activity can be fun as well,
provided that students are paired appropriately. The varying levels of
assistance also can assist in tiering not only by ability, but in
allowing students to practice and improve. Overall, I think the example
provided by Csongor and Craig is somewhat small, but it’s definitely a
worthwhile activity.
Keywords: Algebra, Number and Operation
Ref: Mish6
Author(s): Davis, Jon D.
Year of publication : 2005
Title: Connecting Procedural and Conceptual Knowledge of
Functions
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 99, No. 1, pgs. 37-39.
Reviewer: Mish
Date of Review: March 10, 2008
Davis’ article addresses the possibility of students having both procedural and conceptual understanding of a topic, but being unable to connect the two. He offers functions as a concrete example, discussing how to approach a simple equation like 2x-9=73. He says that students may try entering 2x-9=y1 into a calculator, and then either tracing the resulting graph until finding the appropriate x value or looking at the table. After this, students should then learn the appropriate symbol manipulation to solve for x. Lastly, students should practice symbol manipulation as a procedure. Approaching functions in this manner, Davis believes, will strengthen the connection between students’ conceptual and procedural knowledge of functions.
I was surprised at the suggestion of solving the equation
graphically or by looking at the table for the function first, but I do
believe that this may be a strong way to help students understand how
functions work. I think there’s a good possibility for a better
understanding of variable as well. I also appreciated the practical
tie-in of technology. This is definitely an instance in which graphing
calculators will be a strong asset.
Keywords: Algebra. Number and Operation
Ref: Mish7
Author(s): Tent, Margaret W.
Year of publication : 2006
Title: Understanding the Properties of Arithmetic: A
Prerequisite for Success in Algebra
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 12, No. 1, pages 22-25
Reviewer: Mish
Date of Review: March 12, 2008
Tent goes in depth in the many ways that the associative, commutative, distributive and identity properties lay the foundation for algebra. “Algebra can be considered a generalization of arithmetic…” She gives many examples of various ways to consider arithmetic problems using the aforementioned properties, as well as student downfalls and ways to avoid them. She also shows how she likes to illustrate the use of these properties using pictures.
At times, the article seems a little simple, but there are some very
good examples of strong uses of the properties of arithmetic. I think
that students’ number sense could be greatly enhanced by approaching
arithmetic in the ways that Tent offers. Also, her methods of
illustrating the properties provides students with a fun, conceptually
sound way to think about the distributive, commutative, and associative
properties.
Keywords: Teaching Strategie
Ref: Mish8
Author(s): Pierce, Rebecca L. and Cheryll M. Adams
Year of publication : 2005
Title: Using Tiered Lessons in Mathematics
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 11(3), p. 144
Reviewer: Mish
Date of Review: March 14, 2008
“Using Tiered Lessons in Mathematics” is an informative article about differentiated instruction in mathematics. The article begins by discussing what differentiated instruction is and what constitutes its components (i.e. content, process, and product). It offers a CIRCLE Model—Creating an Integrated Response for Challenging Learners Equitably, with 4 components: classroom management techniques, anchoring activities, differentiated instruction strategies, and differentiated assessment. Within this framework, the article outlines an 8-step process for “tiering” a lesson. Examples of tiered lessons are included.
While I found Pierce and Adams’ article to be very informative about
differentiated instruction, I thought it was slightly lacking on the
“in Mathematics” part of the title. The intro to DI was excellent, as
well as the discussion of their CIRCLE model and the examples of ways
in which it can be implemented. The 8-step process for developing a
tiered lesson was a great way to get teachers to think about
differentiating their lesson plans, but I though it was a bit shy on
how to effectively tier lessons in math. The examples of lesson plans
provided were a good place to start, but I was left wanting more. This
is a good introductory article to DI, but it could use a little more
emphasis on mathematics.
Keywords: Geometry, Planning
Ref: Mish9
Author(s): Groth, Randall E.
Year of publication : 2005
Title: Linking Theory and Practice in Teaching Geomtery
Journal or Publisher: mathematics Teacher
Volume, Issue, Pages: Vol. 99, No. 1, pg. 27
Reviewer: Mish
Date of Review: March 30, 2008
Groth’s article begins by observing that often, one option for instruction can be emphasized to the point where others are excluded. He proceeds to share his experience employing van Hiele theory in the teaching of geometry. Van Hiele theory outlines 5 levels at which students think about geometry. It comes with a recommendation for instruction to help students learn to operate at the various levels. Groth then shares his experience using this theory to teach about quadrilaterals and the triangle inequality.
While I found the notions that van Hiele theory suggested to be
useful, I actually had a tough time understanding exactly what the
various levels were. The wording was negative in one level (what
students do wrong) and positive (focusing on what they could do) in
higher levels. The suggestions for how to structure instruction where
helpful, but unclear as to whether they should be applied in lessons,
units, or both. The examples of how Groth used the theory were very
helpful, and his reflection helped me see how this material could be
used. Overall, a little fragmented, but the fragments have potential.
Keywords: Probability, Problem Solving
Ref: Mish10
Author(s): Rubel, Laurie H.
Year of publication : 2006
Title: Good Things Always Come in Threes: Three Cards, Three
Prisoners, and Three Doors
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 99, No. 6, pg. 401
Reviewer: Mish
Date of Review: April 7, 2008
Rubel's article discusses three well-known problems in probability: Monty Hall's Three Doors problem, the Three Cards problem, and the Three Prisoners problem. In general, if three outcomes are equally likely, and one of the undesired outcomes is eliminated, does the probability of the desired outcome change. The answer is yes, but it is very counter-intuitive. Rubel examines two common downfalls: the assumption that the added information doesn't change anything, and the assumption that the additional information creates a level playing field for the remaining two outcomes. She also discusses the correct reasoning in thee context of each problem, as well as the applications of Bayes' theorem.
While I found the general premise of the problem to be aggravatingly counter-intuitive, I think there's a good opportunity for some discovery learning in these problems. Rubel's article sets you up well for such an activity because it examines in-depth the most common mistakes and how to correct them. My only difficulty with the article was a lack of diagrams. We're only shown the arithmetic used in calculating the answers via Bayes' theorem and a table of how students answered. Keywords: Problem Solving.
Ref: Mish11
Author(s): Mooney, Edward S.
Year of publication : 2006/2007
Title: Elizabeth's Long Walk
Journal or Publisher: Mathmeatics Teaching in the Middle School
Volume, Issue, Pages: Vol. 12, No. 5, pg. 263
Reviewer: Mish
Date of Review: April 7, 2008
Elizabeth's Long Walk discusses submissions of various students' work on the problem: Elizabeth visits her friend Andrew and then returns home by the same route. She always walks 2 km/h when going uphill, 6 km/h when going downhill, and 3 km/h when on level ground. If her total walking time is 6 hours, then what is the total distance she walks in kilometers? We see guess and check methods, assumptions of equal walk time to and from, and an algebraic strategy that segues nicely into an excel spreadsheet. The inclusions of student work illustrate the article nicely.
As this article examines students' work on the problem, and not
necessarily integrating the problem itself into instruction, it's a
little unclear on when the researchers would recommend this problem be
explored or how it can best be presented. Personally I think it might
go nicely in a rate unit. Regardless, the problem is a rich one, and
offers many possibilities for work on representation and communication,
as well as work in groups using various mediums to convey their
solutions.
Keywords: Problem Solving
Ref: Mish12
Author(s): Metz, Mary Lou
Year of publication : 2003
Title: Making Sense of Percents
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 9, No. 1 pg. 44
Reviewer: Mish
Date of Review: April 13, 2008
“Making Sense of Percents” isn’t so much an article as it is a presentation of a percentage problem. There is the initial question of the price of an already-discounted item, followed by an exploration as to whether or not the price would have been different if the discounts had been taken in a different order. Questions to help the student(s) flesh out the problem are presented first, followed by solutions to all questions offered by Metz.
While I thought it odd that this wasn’t a text-driven article discussing how to approach percents, I still found the problem to be a rich one with many opportunities for teachable moments. Not only do the students work with explicit numbers, they also generate formulas in order to compare the results when the discounts are taken in different orders. Students must also work between percents and decimals (or fractions, if they like), which adds another element of complexity to the problem. I’m going to be doing a unit involving percents, so I’ll definitely keep this problem handy. Keywords: Algebra, Standards
Ref: Mish13
Author(s): Marshall, Eldred
Year of publication : 2001
Title: Functions Made Easy
Journal or Publisher: SCORE Mathematics
Volume, Issue, Pages: http://score.kings.k12.ca.us/lessons/functions.html
Reviewer: Mish
Date of Review: April 23, 2008
This lesson progresses through functions as relationships first as a cause and effect concept using objects other than numbers, for example getting rid of headaches by using aspirin. It moves to the idea of functions as machines, having students list something you would put into a machine, the machine itself, and what you would get from it, i.e. bread, a toaster, and toast. Next it goes to functions as equations, where students list inputs, “the machine”, and outputs. It comes with a wonderful applet, “The Function Machine”, where students put inputs into various formulas and view the respective outputs, as well as a “Mystery Machine” where the students must find the formula. This is followed with a similar worksheet. The lesson finishes with graphs approached in a similar fashion to the earlier concepts.
While I was stymied by the number of worksheets at first, I think that this lesson could have several opportunities for cooperative learning in which students work on the worksheets in different groups. I appreciate the approach to functions as machines; it gives a great context for input and output, as well as provides a strong foundation for function sense. This function sense could be furthered using the Function Machine, a truly wonderful little applet. Keywords: Teaching Strategies, Communications,
Representations
Ref: Mish14
Author(s): Scanlon, Regina M.
Year of publication : 2006
Title: Using Engaging Contexts to Introduce Concepts
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 12(2), p. 123
Reviewer: Mish
Date of Review: April 23, 2008
Scanlon begins by explaining that she has several favorite activities with which she begins new concepts, an idea promoted in PSSM. In her article she offers three activities for the following three concepts: rate, distributive property, and Cartesian coordinates. For rate, she gives the students 30 seconds to trace as many stars as possible on a sheet of paper. They then calculate rates, unit rates, and equivalent rates and discuss the various aspects of their results. For distributive property, the students “go shopping” for twins (where each twin gets his/her own present) and calculate prices, most often using the distributive property. Lastly, a modified version of Battleship is played to familiarize students with the Cartesian coordinate system. All of these activities have options for extension.
This is a great article! I’m using the star activity in my unit plan. It offers many applications to the ideas in my textbook chapter and is conducive to written reflection. I think the distributive activity may be less accessible to extension, but is still an excellent way to conceptualize the distributive property. Finally, the Battleship game is an excellent way to develop not only students’ familiarity with the Cartesian coordinate system, but also their intuition with the thing. EXCELLENT ARTICLE! Keywords: Teaching Strategies
Ref: Mish15
Author(s): Thompson, Charles S.; Bush, William S.
Year of publication :
Title: Improving Middle School Teacher’s Reasoning about
Proportional Reasoning
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 8(8), p. 398
Reviewer: Mish
Date of Review: April 14, 2008
This article examines efforts made in Kentucky to further develop teachers’ approaches to and familiarity with proportional reasoning. The article stresses that proportional reasoning is at the core of middle school mathematics and is a foundation for high school math. Thompson and Bush illustrate the difference non-proportional reasoning and proportional reasoning, as well as the differences between multiplicative and additive reasoning. They also described the series of professional development seminars and summer academies the Kentucky teachers attended and what the teachers had to say afterwards.
The descriptions of the professional development were interesting, but as I would never actually see that program, I found it hard to get engaged with that part of the article. I found the discussion of the various types of reasoning to be very informative; I hadn’t thought about the various types of proportional reasoning before. I also appreciated the examples of ways students could reason using simple numbers or no numbers at all in addition to the more explicit examples. Keywords: Problem Solving
Ref: Mish16
Author(s): Chamberlin, Michelle T.; Zawojewski, Judith
Year of publication : 2006
Title: A Worthwhile Mathematical Task for Students and Their
Teachers
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 12, No. 2 p.82-87
Reviewer: Mish
Date of Review: May 3, 2008
Chamberlin and Zawojewski outline teachers’ experiences with Kid Case Studies, problems in which the students study data, find, and present their solutions to the class. First, the teachers came together to discuss possible student solutions to the case studies and what to expect during the activity. Then the teachers presented the activity in their classrooms. They came together afterwards to discuss what went well, what was challenging, and how they could improve the activity for the future. Teacher consensus about the activity was that it was hard to allow the students to wrestle with the problem, but that any direct assistance led the students away from their own thoughts on the problem.
Overall, the case study discussed in this article, “Departing
On-Time” looked like an excellent problem for students to work and
present on. There were several possible solutions and many ways for the
students to interpret the problem, thus allowing them to take ownership
of their solutions. I can easily see why it was hard for the teachers
to watch the students struggle; there are many ways to approach the
problem. However, I can also see “answerable” and “unanswerable”
questions—logistical versus those that require the teacher to direct
the students in answering the question. At any rate, the problem has
many teaching opportunities as well as several options for how to play
to students learning styles, be they visual, auditory, or otherwise.
Keywords: Connections, Probability
Ref: Mish17
Author(s): McShea, Betsy; Vogel, Judith; Yarnevich, Maureen
Year of publication : 2005
Title: Harry Potter and the Magic of Mathematics
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 10, No. 8
Reviewer: Mish
Date of Review: May 3, 2008
This article discusses the importance of finding mathematical connections to literature as a way of engaging students in their learning of mathematics. In this article, connections to math were found in Harry Potter. First, there were conversion and rate examples using the wizarding money system of galleons, sickles, and knuts. These conversion ideas were then expanded into function and linear modeling by solving a problem involving Harry’s candy purchase on the Hogwarts Express. Lastly, there was a probability connection by asking the students to determine the likelihood of Ron, Harry, and Hermione all being sorted into Gryffindor.
This was a really fun article to read! As a Harry Potter fan, I can
verify that, as far as the series goes, these problems have merit.
They’re not much of a stretch for students to want to determine. The
connections students can make will present good opportunities for
discussion in class, since the problems offered in the article extend
to the “Muggle World” quite easily!
Keywords: Algebra, Technology, Standards
Ref: Mish18
Author(s): NCTM
Year of publication : 1995
Title: Curriculum and Evaluation Standards For School
Mathematics: Addenda Series
Journal or Publisher: NCTM
Volume, Issue, Pages: Algebra in a Technological World
Reviewer: Mish
Date of Review: May 7, 2008
The book starts off by highlighting how mathematics teaching is going to change because of technology. Now, our focus is no longer on how to solve equations by hand, or to generate models by hand, but on understanding what algebra actually is. The book presents many enrichment activities and suggestions for how to teach and assess them.
Firstly, I'd like a chance to sit down and do some of these
activities, because I think I'd benefit from them. The detail is
excellent and you are provided with a wealth of great ideas and
suggestions. If I had anything to say against this book would be that
there are too many suggestions. Not in the sense that one can ever have
too many suggestions, but that the pages of the book are very busy and
full of suggestions. It's a little hard to wade through. But if you do,
you'll find fabulous opportunities for enriching students'
understanding.
Keywords: Probability, Teaching Strategies, Standards
Ref: Mish19
Author(s): NCTM
Year of publication : 2003
Title: Navigations through Probability in Grades 6-8
Journal or Publisher: NCTM
Volume, Issue, Pages: Navigations Series
Reviewer: Mish
Date of Review: May 7, 2008
This book begins by discussing how students' experiences with probability outside the classroom can develop modes of thinking probabilistically that may not be sound. It also mentions that our colloquial uses of phrases such as 'likely' and 'unlikely' may also misinform students. It suggests that even with simple experiments we can develop mathematically sophisticated understanding. Although the book does recommend that technology be integrated, with great benefits.
This book presents many great and relatively simple activities that
set a strong foundation for students' learning in probability. The
activities are laid out a section at a time, with worksheets and
transparencies provided in the back. The directions for the activities
are laid out in a text-driven manner (that is, not so step-by-step),
but the text that teachers are given has many ways that students may
think about the problems, and suggestions for how we can question in
such a way that broadens understanding. I found this book to be more
accessible than the Addenda book, simply because there was a lot less
going on on each page.
Keywords: Algebra, Teaching Strategies
Ref: Mish20
Author(s): Lee, Leslie; 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
Reviewer: Mish
Date of Review: May 12, 2008
Lee and Freiman’s article emphasizes the importance not only of pattern exploration, but pattern exploration as a beginning to thinking about algebra. Often, they say, patterns are only explored at a surface level, far enough to find the one pattern, not enough to find several, or even a general approach. They suggest starting with the first few figures, then working up to the 10th, while looking for a general formula. To motivate students, they propose asking for the 58th level, or the 100th or 201st. They also stress that if students find different general formulas, that they try to determine whether or not they are the same. This will help students to form more algebraic habits of mind.
The “growing T” pattern the authors use as an example is an
excellent model for use in the classroom. Illustrating the function y =
3x + 1, this pattern is simple enough to be approachable for young
students, but mathematically rich enough o teach some good mathematics.
The steps, questions, and samples of students’ work Lee and Freiman
provide make the activity readily available for use in the classroom.
Great article!