Keywords: Curriculum, Planning, Teaching Strategies
Ref: Michael1
Author(s): Tarr, James E.; Reys, Barbara J.; Barker, David D.;
Billstein, Rick
Year of Publication: 2006
Title: Selecting High-Quality Mathematics Textbooks
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 12, No. 1, p. 50-54
Reviewer: Michael
Date of Review: February 12, 2007
For content emphasis, the authors note that a wide range of topics are included in many textbooks in order to align with diverse state and district curriculum requirements. However, they note the importance of selecting texts that continue to develop skills, rather than being redundant, and provide students with contextual purposes for learning mathematics.
For instructional focus, the authors discuss the importance of a textbook's providing of problems, activities, and investigations that can engage students and lead them to seek out mathematical ends to these problems. Quality textbooks should also provide the means to connect new ideas to prior knowledge.
For teacher support, the authors highlight the importance of textbooks that offer teachers insight into engaging students in mathematics, as well as providing a clear educational/instructional path. Quality textbooks should also provide ideas for applying activities to a diverse student population and give the teacher appropriate assessment resources.
I found this article to be both valuable and interesting because I
have wondered what the process is that a teacher goes through in
choosing a textbook. Therefore, through this article I now have a guide
for how these decisions should be made. Were I able to cut apart this
issue, I would keep the purple summary boxes for future reference, as
they provide sets of questions that a teacher should ask him- or
herself during this selection process. A textbook that is strong in
these three aforementioned areas should alleviate some stress from a
teacher, as it provides a strong support for both the teacher and
student.
Keywords: Teaching Strategies, Planning, Communications
Ref: Michael3
Author(s): Reinhart, Steven C.
Year of publication : 2000
Title: Never Say Anything a Kid Can Say!
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 5, No. 8, p. 478-483
Reviewer: Michael
Date of Review: February 28, 2007
What should be taught? How should it be taught? Should the classroom be teacher-centered or student-centered? These sorts of questions posed themselves throughout the article. As the title of the article states, teachers should “never say anything a kid can say.” We should let the thoughts and words that are floating around be the students’ thoughts and words whenever possible. Students that verbally question and reason are students that are showing engagement and understanding. By being more of a questioner, the teacher puts students in the driver’s seat in the classroom. By asking for more than just “correct answers,” a teacher forces his or her students to try to gain an understanding of their level of understanding.
Participation is the key to the student-centered classroom that Mr. Reinhart is attempting to build. His “think, pair, share” strategy is based on trying not to overwhelm students (especially middle school children) with the pressure of sharing their thoughts with a room full of their peers. It encourages students to work at a comfortable pace, and it allows them to build support behind any presentations they make. Beyond this strategy, however, Reinhart highlights several other quality teaching tactics that encourage participation. From having students use hand signals during large group discussions, to taking the pencil out of his hand during individual assistance, this article is laced with tips that math teachers, both current and future, should take to heart.
I found Mr. Reinhart’s article to be valuable because of its
practicality. As a future teacher, I often find myself asking the
questions that Reinhart has posed answers to. While he notes that it is
nearly impossible to use everything he suggests at all times, his ideas
lend perspective, and potential guidelines, to the idea of developing
as a teacher, which is a development that hopefully continues through
all of our careers.
Keywords: Assessment, Connections
Ref: Michael4
Author(s): Cramer, Kathleen; Wyberg, Terry
Year of publication : 200?
Title: When Getting the Right Answer is Not Always Enough:
Connecting How Students Order Fractions and Estimate Sums and
Differences
Journal or Publisher: The Learning of Mathematics
Volume, Issue, Pages: p. 205-220
Reviewer: Michael
Date of Review: March 7, 2007
When comparing fractions, it is possible for students to use either conceptual or procedural strategies. A student’s choice of method may play a role in how he or she thinks about the size of fractions, which therefore affects his or her way of operating on these fractions. The nice thing about the three students mentioned is that they all offer a different perspective on how others may approach working with fractions.
Kevin showed a preference for finding common denominators when ordering fractions. When doing fraction estimation, his common denominator method produced exact answers, but defeated the purpose of estimation. When pushed to estimate, he struggled, reverting to whole number strategies.
Ben used a percent strategy to order the fractions, converting via calculator. When the calculator was removed, he used a difference perspective, which is far too inaccurate. He also reverts to a whole number strategy for sums of fractions.
Natalie approached the ordering problems by visualizing “pieces” that connect fractions to a more concrete model. This would qualify as a conceptual strategy. Natalie was able to use this understanding to make sounder sum estimates than Kevin and Ben.
Perhaps the most interesting/valuable part of the article was the
set of questions at the end. While we can identify student preferences,
as above, and their strengths and weaknesses, we do not get anywhere
with this if we don’t ask and attempt to answer these questions. We
notice that Natalie showed the strongest grasp of fraction values. We
also notice, however, that there are times and places for Kevin and
Ben’s methods, and the best case scenario would be students who are
equipped with all of these fraction tools.
Keywords: Algebra, Activities
Ref: Michael5
Author(s):
Year of publication :
Title: Trout Pond Population
Journal or Publisher: NCTM
Volume, Issue, Pages: illuminations.nctm.org/LessonDetail.aspx?ID=L476
Reviewer: Michael
Date of Review: March 16, 2007
There is an activity sheet that goes along with the lesson that I believe I would encourage the students to use if this was their first contact with recursion and iteration. On this sheet there is a table that asks the students to find the number of trout for each year, 1-25, hoping they can thus see the pattern. I think this table would be useful in getting students to see that a table would be a very logical way to organize (and attack) this problem, and hopefully as this topic was discussed further, or reviewed later on, the students would come back to the idea of using a table to help them solve recursion and iteration problems.
There is also a second part to this lesson that brings up the question of, "What would happen if we changed the above parameters for the trout pond?" By parameters, we mean the initial number of fish, the population decrease rate and the restocking number. In order to comfortably manage this exploratory lesson, I think I would (in a class of roughly 24-25) have the students work in pairs, so therefore each student has someone else to bat ideas around with if they're having any difficulty.
Also, when we move into the second phase (changing parameters),
these pairings will be helpful because I can assign two pairs each to
one of the six parameter change situations - lowering or raising each
of the three parameters - and then have them meet in that group of
about four to get their ideas together and perhaps even present what
they believe would be the effects of their respective parameter
changes.
Keywords: Number and Operation, Representations
Ref: Michael6
Author(s): Lappan, Glenda; Fey, James T.; Fitzgerald, William
M.; Friel, Susan N.; Phillips, Elizabeth Difanis
Year of publication :
Title:
Journal or Publisher:
Volume, Issue, Pages:
Reviewer: Michael
Date of Review: Select month Select day of month, 2007
The approach they take to gaining this understanding is one that is immersed in models. Number lines, chip boards, thermometers and graphs are all excellent tools for helping students to fully grasp operating on integers. One or more of these models can be used to express the answers to each of the above questions. Looking to page 1f, we see the mathematical and problem-solving goals of the "Accentuate" book. Among them are the questions I have raised, along with many other goals. The most important thing to see with these goals, however, is how they are all related to one another, all building on the same concepts.
I really enjoyed this overview of the "Accentuate" book, not only
for the questions and goals it brings forth, but also for the examples
that exist in this pre-text text. Some of these examples include:
relating adding a negative and subtracting a positive integer, number
line understanding of the comparative value of negative integers,
opposite chips (+1 + -1 = 0), and understanding the multiplication of
two negatives by looking at the pattern as one of the numbers decreases
from a positive to a negative.
Keywords: Statistics, Probability
Ref: Michael7
Author(s): Schielack, Jr., Vincent P.
Year of publication : 1995
Title: Baseball Cards, Collecting, and Mathematics
Journal or Publisher: Connecting Mathematics Across the
Curriculum (NCTM)
Volume, Issue, Pages: p. 210-218
Reviewer: Michael
Date of Review: April 4, 2007
Why statistics? Well, on the back of most baseball cards you can find a player's career (and year-by-year) statistics in a variety of categories, including at-bats, hits, and batting average. Of course there are many other categories that are kept track of, but the author chose these three to analyze because of the fact that batting average is found by using a function (hits/at-bats) of the other two. Baseball cards can invite students to analyze calculations and create statistics of their own.
Why probability? Well, in collecting, many people attempt to get complete sets of a type of baseball cards (or any other item), or they attempt to get individual players. Since the cards that one buys in a pack can be considered random, things such as the Monte Carlo techniques and Expected Values become applicable. For example, we can determine the expected number of packs of cards needed to attain a complete set of x cards.
I found this article interesting because I spent a great deal of my
youth earnings on sports cards. More seriously, the article appealed to
me because I am always keeping my eyes open for real-life applications
and interesting connections to use in my future classroom. This article
highlighted the value of baseball and baseball cards as statistics and
probability tools. Those are two more tools for my future teaching. (By
the way, the Twins now lead, 1-0 through 1.5 innings... is there any
mathematical value we can get from this??)
Keywords: Representations, Issues
Ref: Michael8
Author(s): Carpenter, Thomas; Franke, Megan Loef; Levi, Linda
Year of publication : 2003
Title: Thinking Mathematically. Chapter 2: Equality
Journal or Publisher: Heinemann Books
Volume, Issue, Pages: pages 8-24
Reviewer: Michael
Date of Review: April 11, 2007
Even when these mistakes are pointed out, children seem to have a great deal of difficulty departing from the early conceptions they have formed for what "=" means. What the authors of this article suggest is that, in order to break these habits (or to stop them from forming in the first place) teachers should invite their students to explore the meaning of = through true/false number sentences that are eventually replaced with open number sentences. Four benchmarks are set. The first is that students are simply able to discuss their thoughts on the meaning of =. Second, students are able to "accept as true some number sentence that is not of the form a+b=c." Third, students are able to recognize the equal relationship between the left and right sides of the equal sign. Fourth, is the comparison of mathematical expressions without actually performing the calculations (19).
I like any ideas about how to strengthen students conceptions of
important mathematical ideas, and the equal sign is definitely one of
those important ideas. There are too many student who struggle
understanding the manipulation that takes place in algebra because they
don't see that what you do to one side of an equation you must do to
the other, since the equality of the sides must be preserved.
Keywords: Algebra
Ref: Michael9
Author(s): Usiskin, Zalman
Year of publication :
Title: Conceptions of School Algebra and Uses of Variables
Journal or Publisher: Algebraic Thinking, Grades K-12
Volume, Issue, Pages: p. 7-13
Reviewer: Michael
Date of Review: April 21, 2007
The author looks at four different conceptions of algebra. The first, algebra as generalized arithmetic, is a means of looking at patterns. The second, algebra as a study of procedures, involves the use of variables in holding a place, either for unknowns or constants, that are to be solved for. The third, algebra as the study of relationships among quantities, looks at things like area, where L and W affect A, and all other functions (e.g. f(x)) where there is a relationship between an input and output. The fourth conception is algebra as the study of structures, which involves things like groups and rings and how the properties of algebra carry over to these structures.
Our concerns, as future teachers, lie in two fundamental algebra
issues. The first is the extent to which algebraic manipulation by hand
needs to be known. The advent of computer technology is the factor
coming into play here. The second issue is about the timing of the
introduction of the function to a student's mathematical world. Some
see functions as a major vehicle of all algebra learning, and some see
them as too advanced and confusing for students just beginning the
study of algebraic ideas.
Keywords: Activities, Problem Solving
Ref: Michael10
Author(s):
Year of publication :
Title: Lesson 2: Multiplying Matrices
Journal or Publisher:
Volume, Issue, Pages: p. 26-35
Reviewer: Michael
Date of Review: April 30, 2007
What I really liked about this lesson is that it was very thorough. There were something like 40 different questions asked throughout the lesson, and each of them probed a valuable area of understanding matrices. Because the questions were often introduced before the cold, hard process, the students were thus forced to bring in prior thoughts and understandings and apply them to the given problems. Matrix multiplication serves as a simplification, or a mathematical representation of the common strain of thought. Thus, it's easy to like the fact that the students were given the opportunity to "discover" matrix multiplication and how matrix dimensions relate.
One of the things that I didn't like about the lesson is that it is
definitely too long to comfortably present in a regular (non-block)
class period. With forty-some questions, there obviously wouldn't be a
whole lot of time to really stop and think. However, over a two-day
period, this lesson might be fine. The one other thing that I was
somewhat uncomfortable with was the lack of description for a "rule"
for matrix multiplication. I understand that this whole exploration
creates the rule(s) for the students, but looking at some of the
problems they were presented with, I could see being forced to take
your exploration and apply it to unlabeled matrices as a little
overwhelming. Some students, I'm sure, would like to have something a
little more succinct in front of them as they are getting used to this
new tool.
Keywords: Algebra, Representations,
Ref: Michael11
Author(s): Steen, Lynn Arthur; Herbert,
Kristen; Rosnick, Peter
Year of publication : 1999
Title: Algebraic Thinking
Journal or Publisher: NCTM
Volume, Issue, Pages: p. 49-51,
p.123-128, p. 313-315
Reviewer: Michael
Date of Review: May 2, 2007
The second article I chose is titled, "Patterns as Tools for Algebraic Reasoning." This article was written with a problem as its framework, where there is a group of people needing to cross a river with a single boat, where the question is, "How many trips will it take to get everyone across?" This question is meant to probe student understanding of what is going on, and lead to a generalization for a group of X number of people. The author describes the students' investigation as a three-step process: pattern seeking, pattern recognition, and then generalization. It is the generalization that shows the students the power of algebraic thinking. Students learn to perceive patterns will obviously grow in their confidence of their own mathematical abilities.
The third article was titled, "Some
Misconceptions concerning the Concept of
Variable." The article discussed the "path of
increasing abstraction," that a mathematics
curriculum naturally follows (313). As things
become more abstract, what is lost in
translation is often what our symbols are
actually being used to stand for. The idea of a
reversed equation is discussed at great length,
where we see student misconceptions about how to
translate sentences into mathematical phrases
(314). What we need to do is to make sure we
protect the distinction between different ideas,
and attempt to foster a greater understanding of
what variables and equations are in our
classrooms.