Keywords: Research
Ref: Dave1
Author(s): Maida, Paula
Date: 1995
Title: Reading and Note-taking Prior to Introduction
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 88, number 6, pages 470-472
Reviewer: Dave
Date of Review: February 12
The article discussed pre-reading for math classes. This is basically
how all the math courses at Olaf seem to occur. The students read an assignment
and the professor then lectures on that topic the next class. The author
points out that this is no the most common form of teaching. The standard
seems to be for the teacher to introduce the topic and have the class read
the section and then do the homework. The author tested the method of pre-reading
in her community college course, where the students reacted favorably to
the method. She required note-taking on the reading as well. She believes
that reading the assignment before the teacher goes over it improves student's
comprehension and mathematics literacy. This seems to hold true at the
college level and I think it would be an effective way of teaching high
school students. The author does point out that pre-reading is not always
as effective, though. She cites an example where her class used experiments
to discover pi. She believed that this was a more effective than simply
reading about pi. The article is fairly straightforward with results. I
like the form of pre-reading the author recommends. I think it would be
an effective tool to use in a classroom setting, as long as the teacher
does a good job supporting the students and helping them through new material
that poses problems.
Keywords: Statistics, Management,
Ref: Dave2
Author(s): McCombs, Kim Krusen
Date: 2000
Title: Felix Klein and the NTCM's Standards
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 93, Number 8, pages 714-717
Reviewer: Dave
Date of Review: February 13, 2002
Felix Klein would have been 140 years when the first NCTM Standards were published. The author points this out (actually, that he would have been 150 on the tenth anniversary of the Standards). Klein was a professor in Gottingen University back in the early Twentieth century. He was a reformer of mathematical education. The author believed that he would be in favor of the NCTM Standards. Klein believed in teaching for understanding, not just rote memorization. He favored using technology, though not quite the advanced types we have today. He saw the advent of calculating machines and advised the use of them, in 1904. The idea of math education reform is not exactly new. Klein was very ahead of the time. He had ideas similar to Piaget's stages. Klein tried to bridge the gaps between pure and applied mathematics. This is, again, ahead of the times. Klein seems to be a good example for us as teachers to follow. He seems to be full of good ideas, not necessarily to copy verbatim (after all, he had completely different technology to use). His ideas on teaching for understanding is very important.
Keywords: Assessment, Tests,
Ref: Dave3
Author(s): Hancock, C. Lynn
Date: 1995
Title: Enhancing Mathematics Learning with Open-ended Questions
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 88, Number 6
Reviewer: Dave
Date of Review: February 14
The focus of this article is the use of open-ended questions in math
classes. Many teachers do not see the value of using open-ended questions,
which are hard to assess and take a lot of time. The author explains how
open-ended questions can be a valuable part of a course. These questions
can be used to facilitate a higher level of thinking. The "Lightning Strikes
Again!" question is a good example of what can be done. Standard math test
assessment does not allow for more learning, whereas open-ended questions
can. A test done in the Netherlands showed that students learn more by
analyzing their own work, along with their teacher's comments. The students
were given a test, then had it handed back with their teacher's response,
they were then able to work on it more that night. This stretches the time
while students are learning, beyond just instruction into assessment and
scoring. The scoring for these questions can be done using a rubric that
is discussed with students before starting on the question. This seems
like a very good form of assessment. The students are able to learn, even
while being assessed. I think that reviewing work with the help of a teacher
is helpful. That is how our homework in Math 356 (geometry) was done and
it helped a lot to see what we had done and how we could approach the correct
answer. This seems like a good form of assessment to use to empower students
and to use in a constructivist setting.
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Keywords: Planning, ,
Ref: Dave4
Author(s): Quinn, Robert J.
Date: 2000
Title: Clinching First Place: Calculating the Magic Number
Journal or Publisher: NCTM - Mathematics Teaching in the Middle
School
Volume, Issue, Pages: Volume 6, Number 2, pages 86-89
Reviewer: Dave
Date of Review: 2-18
This article focuses on the using sports scenarios to develop math abilities. The author used this lesson in his eighth grade algebra class, though it does not have too much to do with algebra. The students are given hypothetical standings in a sports league. They are then asked which teams would be able to win the championship with different numbers of remaining games and why those teams could still win it all. The author noted that many of the students were unable to completely explain why one team could win and others could not, but they all seemed to be able to figure out who could win. The author then moves the idea up to professional sports. He ends the lesson with variables being used for win-loss records and asks the students what the magic number for this scenario would be (he had explained what a magic number was earlier). A few of the students were able to develop an equation for this scenario, but most were able to prove that they understood what was going on, though, even if they were not able to fully answer the question. This article explains a good lesson. The students are given a relevant situation (or at least, real world enough). They then build their knowledge of the subject until they are able to construct a solution to the problem that is quite abstract. I think it would make for a fairly good lesson in a middle school classroom.
Keywords: Algebra, Standards,
Ref: Dave5
Author(s): Thorton, Stephen J.
Date: 2001
Title: New Approaches to Algebra: Have We Missed the Point?
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 6, Number 7, pages 388-392
Reviewer: Dave
Date of Review: 2-19
This article addresses the question of whether the ideas put forth by
the NCTM are really doing what they are supposed to in the field of early
algebra. The author uses constructivist type models, but then shows how
the lessons are not really teaching understanding, but just another type
of manipulation. The author then gives examples of how these can be taught
without teaching just manipulation, but understanding. He addresses three
approaches to algebra; the matchstick approach, the symbolic approach and
the functions-and-graph approach. With each approach he gives examples.
Each of these approaches is a different way to comprehend algebra that
could spawn understanding with students. Algebra can be difficult to grasp
for many students. It is important to teach to them for understanding,
not just memorization of forms of manipulating expressions. This is the
idea expressed by the NCTM. The author addresses this idea and gives way
to do this. He ends by pointing out that "the power of algebra lies in
its capacity to develop and communicate insight by representing situations
in alternate ways." By teaching algebra for understanding and in varying
ways, teachers can help students to truly learn algebra.
Keywords: Activities, Probability
Ref: Dave6
Author(s): Stor, Marilyn and Briggs, William L.
Date: 1998
Title: Dice and Disease in the Classroom
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 91, Number 6, pages 464-8
Reviewer: Dave
Date of Review: 2-22
This article sets forth a potential problem for students to work on.
The idea is that students have encounters during a certain time during
the class. These encounters can either be risky or not, which is defined
by the roll of the dice. It is meant to show the progression of illness
through a population. With the numbers found in the class analysis and
discussion can occur. Students can graph the results and find regressions
and work with any number of other ideas. It also brings in ideas about
probability and dice. I thought the problem itself was pretty cool. It
is real world enough that students could find it meaningful. I liked the
fact that the idea of a cure was introduces into the mix. The graphing
and computer analysis seemed pretty advanced to be in high school, but
I think with help students could do it. I think this would work really
well if integrated with a health or science course. Students could work
with actual diseases and try to find results having to do with them. This
would make for an interesting and complex project. I think a good lead
in to assignment (the one outlined in the article) would be to work with
dice probability. I think it would be engaging for the students. A potential
problem could be religious condemnations of dice games.
Keywords: Activities, Communication
Ref: Dave7
Author(s): Taylor, Lydotta M. and King, Joanna L.
Date: 1997
Title: A Popcorn Project for All Students
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 90, Number 3, page 194-6
Reviewer: Dave
Date of Review: 3-1
This article discussed another problem for students to work on. The
basic concept of the problem revolves around percentages and volumes. They
compared several brands of popcorn during their work. The students from
a lower level course were partnered with higher level students for the
project. This created a very good learning environment according to the
author. The lower level students were much more motivated than normal.
The students worked on the problem for a week and then presented to the
group. This is a great idea, because it works with the NCTM Standard that
math is a form of communication. The students learn to communicate mathematically.
Another Standard this seems to work with is the idea that problems should
be realistic, which this is. Volume popped from popcorn is real and easy
to see. The students in each group were given certain tasks by their skill
level and this gave the project a wider scope than it otherwise would have
had. The students enjoyed their work, which showed in their assessment
of the project. The teachers had also set out guidelines ahead of time
for student assessment. All-in-all it seemed like a pretty good project.
The only thing that could make it difficult to do is the need for air-poppers
for the popcorn.
Keywords: Statistics
Ref: Dave9
Author(s): Watson, Mary M
Date: 2000
Title: Statistics in Context
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 93, Number 1, 54-8
Reviewer: Dave
Date of Review: 3-11
Statistical analysis is an important part of mathematics. Other subjects
use statistics, too. This article looks at the issue of statistical literacy
and how we, as teachers, can see if our students understand statistics.
The author lays out a three-step hierarchy of understanding and then goes
on to show what each level will look like in context. At the lowest level
a student should understand the basic vocabulary of statistics and be able
to apply them. A student at a higher level will be able to find this in
the context of societal problems. And a student at the highest level will
be able to look at claims and judge their validity. This is the way student’s
knowledge will develop. It is important to make sure that students develop
with time. The claim put forward in the article could appear in many different
high school subjects. If students do not understand statistics then this
claim would just go right by them. The example used in this article is
a good one. The students’ progress is laid out in an easy to follow manner.
The author does not seem to put forth any ways to advance the students’
knowledge. This is the weak part of the article. I think students being
able to understand statistics in social context is very important. We spend
our lives being bombarded with facts and percentages. It is important that
everyone be able to know what this all means. Otherwise we are not creating
an informed populace, which is part of our jobs as math teachers.
Keywords: Curriculum
Ref: Dave10
Author(s): Bedford, Crayton W.
Date: 1998
Title: The Case for Chaos
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 91, Number 4, pages 276-280
Reviewer: Dave
Date of Review: 3-13
The idea that the author promotes is that it is possible to create a
high school course on chaos and fractals. He says that there are a lot
of student out there that complete pre-calculus, but do not end up taking
calculus. The schools offer discrete math, probability, or computer science.
The author thinks that they should instead try offering a chaos/fractals
class. This class would be able to combine aspects from the other three
classes, which would make it wore appealing than just offering one such
course. The author lays out a semester long course that would teach students
about chaos. Currently, teachers that want to teach about fractals have
to cram a unit into their already packed curriculum. They do not have much
opportunity to explore. I would love to have this class offered here at
St. Olaf, but I do not know how well it could be done in a high school
setting. The teacher would have to have a lot of knowledge about fractals
and this does not seem to be something that many people have. Undergraduates
at St. Olaf only have the chance to study fractals for a week in January,
if they take Geometry. This would make it hard to teach a course on the
subject. But, if the teacher was knowledgeable and could teach the subject
well, this would be a great course for students to take. His case seems
really strong and I think teachers should be given addition training in
order to teach a class like this. Students would really enjoy it and get
a lot out of the course.
Keywords: Probability, Problem Solving, Connections
Ref: Dave11
Author(s): Teppo, Anne R. and Hodgson Ted
Date: 2001
Title: Dinosaurs, Dinosaur Eggs and Probability
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 94, Number 2, pages 86-92
Reviewer: Dave
Date of Review: 4-4
This article focuses on a lesson that would promote probability knowledge.
Probability is something that every high school graduate should understand
according to Scheaffer, Watkins and Landwehr. The lesson the author puts
forward would promote understanding of probability. The lesson is based
around the discovery of an intact dinosaur egg find. The eggs appear to
have been laid in a paired formation which would show a relationship between
dinosaur physiology and that of birds and crocodiles. The problem placed
before the students is whether or not the conjecture makes sense. The students
develop a response to the question by using their know of probability and
the fact that probability is the study of random events. They test whether
the layout of the dinosaur nest is chance or if it is laid out in that
way for a reason. This uses a lot of probability and the language associated
with it. I like this problem. It would definitely draw students into the
mathematics of the problem. This problem would work for many different
levels of students. I like this fact, because every student needs exciting
problems, not just the advanced ones. Students would need to be familiar
with how to approach difficult problems, though. This would be too difficult
for students that had been taught only with a ‘standard’ style. They would
have to work for a little while before being able to approach a problem
of this level.
Keywords: Connections
Ref: Dave12
Author(s): Keller, Rod and Davidson, Doris
Date: 2001
Title: The Math Poem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 94, Number 5, pages 342-347
Reviewer: Dave
Date of Review: 4-8
It is important for students to see the connections between mathematics
and other parts of their education. It is easy to connect math and science
or history, but connecting it with English can be a lot harder. The authors
found a way to do it. They had their students write poetry using mathematical
vocabulary. It allowed the students to use their knowledge of vocabulary
and apply it to their daily existence. Most of the students enjoyed the
assignment. They saw the connections between what they were learning in
math and the rest of their lives. The authors were impressed with the vocabulary
the students were able bring into the assignment, outside the words they
had been given and how well they used the mathematical terminology. As
a student, this assignment would have been hard for me. I would not have
thought about it enough. Now, looking at it, I think that it is brilliant.
I am amazed at the complexity the students are able to reach, especially
the way they use math words. This would be a great assignment for students.
It would be important for the teachers to help students like me. This assignment
could easily frustrate poor English students. The teachers would just need
to find a way to motivate them. They would need to find something that
the students felt passionate enough to write about.
Keywords: Geometry, Connections, Standards
Ref: Dave13
Author(s): Moyer, Patricia S. and Hsia, Wei Hsia
Date: 2001
Title: The Archaeological Dig Site: Using Geometry to Reconstruction
the Past
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 94, Number 3, pages 193-7, plus
inserts
Reviewer: Dave
Date of Review: 4-8
Students often see little connection between geometry and the world
around them. They are able to relate facts and formulas about shapes, but
have no clue how they relate to the world around them. The early ideas
children develop are not nurtured in the mathematical experience and the
knowledge they build of geometry in mathematics never grows as it could.
The NCTM says that students need to explore two- and three-dimensional
shapes and work with them. This is where the lesson the authors prescribe
comes from. In the lesson, the students are given a sheet of paper with
the basic setup of an archeological dig and are sent outside to create
this dig and examine what they find in the area. The lesson reinforces
ideas of congruency and similarity. The teacher can work with this lesson
and place objects within the dig site for the students to describe mathematically.
One of the authors had placed broken pieces of circular "artifacts" in
the dig site and had the students find the over all sizes of the pottery.
The lesson seems like it could work pretty well. Students would be able
to use intelligences other than mathematical in order to set up the dig
site. The authors had integrated this lesson in with several other classes,
which could make the lesson that much better. Included in the article are
templates for broken artifacts and other black-line masters.
Keywords: Communication, Activities, Connections
Ref: Dave14
Author(s): Evered, Lisa J.
Date: 2001
Title: Riddles, Puzzles and Paradoxes: Having Fun with Serious
Mathematics
Journal or Publisher: Mathematics Teaching in the Middle Schools
Volume, Issue, Pages: Volume 6, number 8, pages 458-460
Reviewer: Dave
Date of Review: 4-16
Math teachers use riddles and word puzzles in classes all the time.
It is important for students to be able to look at these and find a way
to solve them. These problems have a real mathematical context. Students
need to read what the words say and figure out what the mathematics are.
This is basically where the author is going with this article. The author
also looks at the idea of paradoxes as a part of the math class. The math
in these riddles has been wrapped in a whimsical and interesting shell.
Students see these and are drawn towards them. Teachers that can develop
good riddles can engage their students without the students even realizing
how deep of mathematics they are doing. I like these word puzzles and riddles.
They are a part of discreet mathematics and a part of our culture. The
Movie Die Hard with a Vengeance had a bunch of these little word games
throughout it. It was fun to watch the characters work with problems in
the same ways we had in previous math classes. These problems are everywhere
and contain some good mathematics and are great for drawing students into
math. <BR>
Keywords: Activities, Connections
Ref: Dave15
Author(s): Battista, Michael J
Date: 1993
Title: Mathematics in Baseball
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 86, Number 4, pages 336-342
Reviewer: Dave
Date of Review: 4-17
This article address the NCTM goal of integrating mathematics. The author puts forth a unit on baseball that combines statistics, geometry, and measurement. It all works around baseball, which is a significant theme that the students can explore. This lesson includes batting averages, slugging percentages, standings, the physical layout of the field, a lot about pitching, including answering the question of how long the pitch takes to reach home plate. The unit is quite effective at bringing together many different subjects into one. There is algebra, geometry and statistics all in one. I, personally, like the unit a lot. I think it would be a lot of fun to take part in. The problem is the terminology of the game. The unit starts with the students writing something about baseball. This is great for communication, but could really leave some students out in the cold. I do not like this quite as much. These students might have trouble trying to figure out what some of the words, like at bats, or doubles actually meant. The teacher would have to take time to make sure that everyone knows what is going on before starting this unit.
Keywords: Problem Solving, ,
Ref: Dave16
Author(s): Geenes, Carole, Gregory, John and Seymour, Dale
Date: 1977
Title: Successul Problem Solving TEchniques
Journal or Publisher: Creative Publications
Volume, Issue, Pages:
Reviewer: Dave
Date of Review:
Carol Greenes, John Gregory and Dale Seymour. Successful Problem Solving Techniques. Palo Alto, California: Creative Publications, Inc, 1977.
This is a good book for problem solving. There are a lot of open-ended questions that do not necessarily have one correct way to solve them. The set-up for these problems teaches students a way to approach these problems that they do not know how to solve with an algorithm. This approach is set up for each problem. The problems are pretty easy to understand and work with. A class could do a set of two or three in a day with ease. A teacher could start out with an easy problem that involves only a little thought and move into the more difficult, thoughtful problems.
I like this book because it forces students to write out their thought process, which is important for solving problems. If students do not do this, they will have trouble with following their work on longer problems. I like this part of the book. I found that it seemed to be quite sexist with illustrations, though. This I did not like. Most of the women pictured have stereotypical Barbie doll figures and are not doing the things that the problems are talking about. The men are playing the games or trying to figure things out, where the women are hosting parties. This could be an issue if a teacher did not change some of the names in the problems and get rid of some of the pictures, if they photocopy the book
Keywords: Manipulatives, Keyword 2, Optional..., Keyword 3, Optional...
Ref: Dave17
Author(s): Bell, Garry
Date: 1997
Title: Showig That a-b = -(b-a)
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 90, Number 5, page 394-6
Reviewer: Dave
Date of Review: May 1
How are students taught that a-b = -(b-a)? Usually the teacher will multiply the two numbers by –1 and then flip them around. This is the standard approach, though it doesn’t necessarily teach the right ideas. The author puts forth new ideas on how we can teach subtraction to show this fact. the author found a way to subtract that takes a second to learn and works quite well. You simply find the absolute difference and then subtract each digit at its place value twice (this makes sense if you see it). Using this method students can investigate how a-b = -(b-a).
I really like this article. First of all, the new method of subtraction is really cool. It works well and you do not have to worry about carrying at all. the problem with this would be that students would not learn place values with it, but, it seems students do not learn place value very well anyhow. It all leads to the fact that a if you subtract a number from itself twice you get it’s opposite. This leads to another way to see that a-b’s opposite is b-a. So, then a-b = -(b-a). The alternative form of subtraction is definitely worth checking out, even without this idea.
Keywords: Assessment, Standards, Keyword 3, Optional...
Ref: Dave18
Author(s): Wilson, Linda
Date: 1994
Title: What Gets Graded Is What Gets Valued
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 87, Number 6, pages 412-4
Reviewer: Dave
Date of Review: May 1
This article is about Ms. League, a high school math teacher. She espouses the Standards, but does not seem to be able to teach to them. She tries to get students to do all the parts of the Standards, but they do not seem to do it. The main reason behind this failure is assessment. The students do not do what they are not graded on. Ms. League only grades on tests, quizzes and the occasional homework assignment. This does not inspire the students to do all of the high ideas that she wants them to, as they know that all of the self-assessment she wants them to do will not get graded. Basically, the gist of the article is: if you want students to do the work, evaluate them. Ms. League does not really work with the Standards, despite what she says. Her tests are nothing but rewritten homework problems. This is not good assessment according to the NCTM. They want assessment to continue students’ learning, not just encourage rote memorization. Her students do not do any of the self-assessment about learning, either. Basically, she seems to just be teaching in the old way, thinking that she is following the Standards
Keywords: Calculus, Geometry, Teaching Strategies
Ref: Dave19
Author(s): Morriss, Paul
Date: 1998
Title: Discovering a geometric Volume Relationship in Calculus
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 91, Number 4. Pages 334-6
Reviewer: Dave
Date of Review: May 8
It seems difficult to get students to discover relationships in calculus classes, especially since most students believe that anything that can be done has been done. The author's class was able to discover a connection between the volume of a cylindrical solid and another solid. This relationship developed out of ideas taught in geometry and was built upon while the students learned about solids of rotation. The author led the class discussion such that the students would discover this formula. The students were excited to be able to discover and create theorem. This is what a math class is about: getting students excited to work with mathematics. The author found a good way to do it. I would like to follow his advice in teaching in a calculus class. The students learn to build their knowledge and have their confidence boosted along the way. They learn that they can bring their mathematical knowledge from different classes together (though their classes should be working in this manner anyhow). This is very valuable as students often see their math classes and disconnected, which makes them seem worthless. Also, linking calculus to geometry is important, as geometry tends to be forgotten by students. Reminding them of ideas from geometry later on in their math careers cannot be bad.
Keywords: Probability, Games, Connections
Ref: Dave20
Author(s): Young, Victoria
Date: 2002
Title: A Matter of "Survival"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 95, number 2, pages 102 - 113
Reviewer: Dave
Date of Review: May 8
The Red Cross pushes donating blood in many high schools and communities,
so children know that donating blood is important and that it can help those
in need. They may also know that there are types of blood, but often not
much more. This is something to discuss before getting into the game of
"survival." This game focuses heavily on probability and data analysis.
The students start by playing the game that finds out how long they would
survive with a certain blood type with random donors. This is basic
probability. After the game students have worksheets that allow them to
dive deeper in to probability questions. After this there are surveys the
students can us to create graphs and analyze the data, trying to pull things
out of it. There is good potential for a conversation about probability and
what it means, especially revolving around the class survey of blood types
and its percentages versus the Red Cross' numbers.
This game and subsequent problem set would be very helpful in a
class. The game itself might be a little dull, but the math that comes out
of it is quite valuable. There connections to the real world are strong
enough that students might get very interested. There might be more to do
with this game, by the way of guest speakers or other ideas. The article
itself has nice black line masters that could be used easily by a teacher
and students.