Keywords: Curriculum, Teaching Strategies
Ref: Casey1 one...
Author(s): Edwards, Barbara S.
Date: December 2000
Title: The Challenges of Implementing Innovation
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 93 number 9, 777-780
Reviewer: Casey
Date of Review: 2-15-04
This was an article about changes in the way a teacher does curriculum. It started out with a story lamenting the fact that the writer had gone to a workshop and learned all kinds of neat new ways to do math, only to find that implementing them was more of a challenge than she expected. She goes on to list things that influence how change is implemented, including a teacher's past experiences, a teacher's beliefs about teaching, and a teacher's knowledge of mathematics and pedagogy. In one example, a teacher was trying to reform from teaching in a very traditional format of memorization. She ended up with a blend of the two which resulted in a good process but a negative outcome, as the kids simply ended up with more memorized facts. Her background was a barrier for her, and a solution has to be more than just a change in curriculum in this case.
The article ends with 'tips for successful change.' The first is to be reflective, which basically means that it is important to process how and whether a new idea worked and what can be done to make it better. This should be done for any new ideas to prevent the trying of one new idea after another without feedback. The second is be patient, as change will take time. Thirdly, look for and provide support. This way a teacher can talk through ideas and events and get input from colleagues. Lastly, turn obstacles into opportunities. For example, many teachers cited the AP exam as an obstacles to trying new curriculum. However, she found that trying a more cooperative learning style with her more as a supervisor than a lecturer.
Overall this was a good article, and this information is useful for
new
teachers as we try to find a way to teach that suits our personal
styles.
Keywords: Activities, Curriculum, Teaching Strategies
Ref: Casey2
Author(s): Gerver, Robert K; Sgroi, Richard J.
Date: January 2003
Title: Creating and Using Guided-Discovery Lessons
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 96, number 1
Reviewer: Casey
Date of Review: 2-18-04
This article is about 'Guided discovery lessons.' The idea behind a guided discovery lesson is that students can follow it on their own and have an 'Aha!' experience about a particular phenomenon or subject of mathematics. The guided discover lesson steers the student on a step by step basis through the subject matter, asking questions to promote the Aha! experience.
Next we learn how to create a guided-discovery lesson. The article first tells us where we can find such lessons, and then walks us through creating one of our own. The first step is to select the content. It should be something that the students do not know so that they can make discoveries of their own. Secondly one is to state the aim of the lesson. This should not give away the main idea to be learned, as that may take away the Aha! component. Next is identifying the prerequisites. We need to make sure that students know enough to complete the excursive with out becoming frustrated. Next we set up the lesson graphically so we can check that transitions will be smooth and the order is logical. Next we actually write the lesson. The article points out that we need to be careful of both extremes of explaining too much and not enough, but to error on the side of too much if anything. Remember that we have a lot more knowledge then the students do. Next we have someone who does not know the material try the activity as a guinea pig. We may then discover pitfalls in its construction. Next we write a follow up activity so that we can check to make sure that students learned and that we hold them accountable for that. Lastly we have field testing and revising. The article notes that it would be helpful to keep a journal in order to record what works and what doesn't.
Keywords: Algebra, Activities, Technology
Ref: Casey3
Author(s): Obrien, Tim
Date: 1997
Title: Algebra 1: Graphing Linear Equations
Journal or Publisher: http://www.terragon.com/tkobrien/algebra/
Volume, Issue, Pages:
Reviewer: Casey
Date of Review: 2-23-04
This is a lesson designed to teach students about graphing equations using technology. It basically reads like a text with a few interactive things to do. When there is an example, instead of assuming that the reader performs the example, the lesson has the steps to perform the example built into the program so that you actually do the example my clicking on things. There are a couple of issues I have with the graphs. When the interactive browser graphs, it leaves the line from the bottom left corner to the top right corner and just changes the axis. This can be confusing for someone just learning to graph. Surely they should learn that one can change axis and do so, but that is probably not the best way to begin graphing. However, it is really nice that the program lets the student build tables of functions on the computer and then graph them, I think that it would be a good way to get them to actually do the examples that would normally be in a text. The author also starts the lesson with an example of water in a pool rising, so that the equation reflects the total level of the water. It is good that he started with a word problem, then taught some math, and then went back to the problem. However, it's probably time for people to come up with some new examples instead of the classics...I appreciate the classics, but we have a new generation of students and they may work better if examples can be more interesting and applicable. Otherwise neat format for a lesson.
Keywords: Standards, Assessment...
Ref: Casey4
Author(s): Romagnano, Lew
Date: January 2001
Title: The Myth of Objectivity in Mathematics Assessment
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 94, Number 1
Reviewer: Casey
Date of Review: 2-25-04
This article is about the fact that the new push towards alternative assessments is mostly subjective, and gives ways to deal with that fact. He states that older ways of assessment, which are focused more on skills, are more objective because they are testing to see if you can do something. The new push, however, is to assess based on a student's mathematical understanding, meaning that at some point the teacher is going to have to make a subjective judgement about the student they are assessing. The author gives a great example of the subjective nature of grading with a problem that gives a typical student answer. The student answers the problem incorrectly because of two sign errors. The author then states that a teacher may assume that the student knows a lot about the problem but has problems with signs, or they may assume that the student attempted to memorize the algorithm for solving the problem and messed it up because they don't have deeper understanding on how to do the problem. This is the subjectivity of looking at mathematical understanding. The author then uses an example from the calculus advanced placement test to demonstrate that a problem may be graded objectively but still may not actually be useful. There is a rubric for the AP problem, yet it is shown that two students could get the same score on the problem and have a completely different understanding of the material. The last example the author uses is the SAT. He states that the score a student gets on the SAT is really only approximations of their 'true scores,' which are the score that they would get if taken a similar test over and over again (similar in content, but not in questions). Their scores would be distributed much like a normal curve, and there is no way of knowing where one particular scoring is on a students curve. Finally, the author gives some tips on how to check for understanding. He asks questions like 'explain the method you used' and asks students to explain hypothetical situations instead of actually performing tasks (how is it possible that there would be no real solutions to a quadratic equation? Explain and give an example.)
Keywords: Research , Teaching Strategies...
>Ref: Casey5
Author(s): D'Ambrosio, Beatriz S., Harkness, Shelly Sheats,
Boone,
William J.
Date: 2004
Title: Planning district-Wide Professional Development:
Insights
Gained From Teachers and Students Regarding Mathematics Teaching in a
Large
Urban District
Journal or Publisher: School Science and Mathematics
Volume, Issue, Pages: Volume 104, Number 1, pp. 5-15
Reviewer: Casey
Date of Review: 3-1-04
This outline was outlining research in the differences between how students and teachers perceive their classrooms, and how that information should be used to better a classroom. The goals stated in the article are to 1) Provide professional development to teachers, focusing curriculum on NCTM standards, 2) Support the development of inquiry, 3) Implement instructional strategies based on best practices in mathematics, and 4) Increase teacher content knowledge of mathematics. The data for this study was collected from over 250 elementary, 400 middle school, and 300 high school mathematics students. The researchers made comparisons of the grade levels to each other in how they answered certain questions about their classroom. For example, according to the students, both elementary and middle school classrooms appeared to be highly textbook centered. However, 51% of elementary and only 29% of middle school students reported the regular use of 'hands on materials.' There are 10 questions that question the use of textbooks, groups, hands-on materials, calculators, computers, explaining work, making up student problems, tests, projects, and worksheets. There was another set of 15 similar questions as well. Each question is listed with the percentage of students who answered "very often' or 'often.'
While the information was very interesting and useful, the article lacks in application and filling out it's self-stated goals. It seems that they are expecting the reader to apply the results for themselves, which surely will be doable, but more ideas from outside one's own box are always good. One possibility is using the particular questions that teachers responded to as well and noting the questions that had major discrepancies in what students and teachers thought. Using this information, we can realize that students don't always know why we are doing what we are doing, and give them more reasons that there is actually a point to everything that we do, a good lesson for any teacher.
Keywords: Activities, Problem Solving, Representations
>Ref: Casey6
Author(s): Mathews, Susann M.
Date: 2004
Title: Mathematical Modeling: Convoying Merchant Ships
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 9, Number 7, pp. 382-391
Reviewer: Casey
Date of Review: 3-2-04
This was an article about a specific project having to do with Mathematical Modeling. The students were presented with a question; should the British convoy their ships or continue to make individual trips across the Atlantic in the context of WWI? Specifically, there were three assertions in opposition to convoying that they were to address using mathematical modeling; 1) The delay caused by assembling a large number of merchant ships would reduce the amount of tonnage shipped. 2) A convoy can travel no faster than its slowest ship. 3) Convoy ships might actually be easier targets for enemy submarines than ships sailing alone. The author of this article wrote about how the students really got into the problem, even to the extent of looking up information on their own that was not required. They then worked on a recursive model first to address the first two questions. They first devolved a model taking into account the number of ships lost for each trip and the time that it took each ship to cross. The convoys took longer, but lost less ships, so does the lower loss rate justify the longer time? They then added shipbuilding into consideration. This got the students to an equation that was above their level, but the teacher minimally explained it and some understood. They found that it indeed made sense with the numbers they had to convoy the ships, as the total tonnage reaching Britain from the US was higher.
Next they used a Geometric model to evaluate the 3rd assertion, that the convoys would make greater targets than the individuals. They used circles to model a ships vulnerability to attacking submarines, and found that since many ships packed together can (in this case) increase the total radius of vulnerability by .2*r, thus with the model from the first part, they were justified in convoying.
This is a great project because the kids learned that there is a concrete, extremely important use for mathematical modeling, and also because this project kept them engaged in their work. The students left the class wanting to implement their work into their social studies classes about wars. This project also spanned disciplines, as the teacher got a Navy submarine officer to speak to the class about the importance of the problem and give them some background information. The author noted that it would be a good project to team up with a social studies teacher on.
Keywords: Standards......
>Ref: Casey7
Author(s): Geddes, Dorothy et el
Date: 1992
Title: Curriculum and Evaluation Standards for School
Mathematics
Journal or Publisher: National Council of Teachers of
Mathematics
Volume, Issue, Pages: Geometry in the Middle Grades
Reviewer: Casey
Date of Review: 3-10-04
In looking at this book, I singled out the first 'cluster' to evaluate. The book states that its general purpose is to supplement a teacher's curriculum with extra activities and lessons designed to fulfill the NCTM standards. Thus in evaluating the book we want to look at two things; which standards a particular activity or lesson address and how well the lesson accomplishes the goal of addressing that standard. The cluster has a list of objectives at the beginning, which for cluster A includes providing hands on experience with construction of two and three dimensional shapes, observing and identifying properties related to these shapes, and comparing and contrasting the properties though examples and non-examples. These and the rest of the objectives definitely seem reasonable in helping students get a firm grasp at basic two and three dimensional concepts.
Looking at the first activity, exploring cubes, the students will be using marshmallows and toothpicks to explore the properties of cubes. My first intuition of this is skeptical, just because marshmallows will inevitably be very tempting for some of the students. Thus at the very least one would want to have an extra bag handy in case the materials start disappearing. On a more serious note, the activity has a section where the students are to draw the cube they created with marshmallows and label the verticies, and then investigate it through a series of questions. The students will have to know some vocabulary, and the teacher will want to make sure they are clear on that before starting the activity. The questions are great, as they essentially pick apart the components of the cube and let the students find out what composes it on their own. Then as they continue the lesson students are asked to create cubes from cube networks, and then to combine cubes to make models of other shapes like buildings.
I find a couple of problems with this activity and some good things as well. One of the problems are that I think there is a possibility that the activity will get boring for them, because with the exception of the marshmallow cubes this is basically a worksheet. The other problem is that the book has not made clear to me what standard is being addressed with this activity. However, I think that with some additional planning, this activity can be used as a framework for something even better by getting the kid's hands on more than some marshmallow cubes. They can cut out or make networks for cubes, use cubes you have to make models of buildings, and the list goes on. Thus based on this one small example I would say that the book should be useful for giving teachers a framework for constructive learning that they can run with.
Keywords: Assessment......
Ref: Casey8
Author(s): Kersaint, Gladis, Chappell, Michaele
Date: 2004
Title: What Do You See? A Case for Examining Students' Work
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 97, Number 2, pp. 102-106
Reviewer: Casey
Date of Review: 3-11-04
This article is about different interpretations of illustrations. The authors had assigned a problem of finding the surface area and volume of a staircase constructed from rectangular three-dimensional rods. When they wrote the problem, they assumed that students would look at surface area as the staircase is floating in the air, so that the bottom and side of the staircase would be included in the calculations. However, some students assumed that the staircase was resting on the ground and/or up against a wall, both reasonable assumptions based on various experiences with staircases. The area that students get obviously depends on which interpretation they subscribe to. Initially the authors were simply marking off points for incorrect answers, when they noticed that some students had correct reasoning with 'wrong' answers. Thus they concluded that the question needed to be clearer in what it stated, and if ambiguity arose, that student interpretations should be taken into account when grading such problems. The authors also briefly brought up an interesting point. When surface area is taught, it is often taught as the area of all sides enclosing a solid. However, this leaves a lot of room for error in a problem like finding the amount of plastic needed to build 1000 straws. Obviously, to us anyway, the base need not be included in surface area calculations. However, often times the surface area of a cylinder is taught as 2*pi*r*h+2*pi*r^2 without any qualification of which term is which. This leads me to conclude that instead of teaching a formula, students should be taught how to derive a formula for surface area, and thus understand more and be able to change formulas for different settings. Another conclusion that the authors drew was that teachers must be careful not to reject answers simply because it is not "the answer." Student thought process as interpreted through their work must also be taken into consideration.
Keywords: Equity/Diversity, Teaching Strategies,
Communication
Ref: Casey9
Author(s): Kitchen, Richard S.
Date: 2004
Title: Challenges Associated with Developing Discursive
Classrooms
in High-Poverty, Rural Schools
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 97, Number 1, pp. 28-31
Reviewer: Casey
Date of Review: 3/16/04
This article is about developing mathematical discourse in cultures where discourse is contrary to what the culture teaches. Some cultures emphasize that children should always respect their elders by paying attention and that it is not acceptable to speak. The specific case that he is talking about is with Navajo students. The author begins the article by talking about two forms of communication in the classroom, univocal and dialogic. Univocal is usually characterized by a lecture format, and classroom participation is limited. In a dialogic classroom, students and teachers have "mathematical conversations" (p. 28) that lead students to deeper understandings.
Next the author wrote about the research he did. He asked teachers a number of questions, including asking them to "name cultural and social issues that challenged them in their efforts to initiate a discursive classroom environment." (p. 29) He found three main themes from the teachers' responses. 1) Students with greater mathematical knowledge talk more. 2) Students who engage in mathematical discourse may not be engaged in higher-order thinking. 3) Students may resist sharing mathematical ideas.
Expanding on 1), the author found that teachers thought it
unrealistic
to expect all students to participate equally was unrealistic. Their
ability was too spread out. The author notes that many of these
teachers came from
areas with few resources including the lack of training. Simple
strategies
such as grouping non-verbal students together were not known to them.
For
2), responses indicated that teachers have seen discourse that does not
involve
higher order thinking. The author notes that often this is a problem of
asking
the wrong questions, such as those that only require a prompt response
rather
than an explanation. For 3), many problems were cited, including
language
barriers and high class sizes. The author points out that when students
habitually
experience low expectations, they have low responses.
Keywords: Representations, Statistics
Ref: Casey10
Author(s): Harper, Suzanne R.
Date: 2004
Title: Student's Interpretations of Misleading Graphs
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 9, Number 6, pp. 340-344
Reviewer: Casey
Date of Review: 3-18-04
This article is about how graphs can be manipulated to present an illusion that is different than the facts the data shows. This can be done in many ways, with a popular one being manipulation of scales so that one of the axes does not start at 0, thus making changes look larger than they really are percentage wise. This is a really important concept to cover because of the way the media and advertisers will purposefully manipulate graphs in these ways to shock people or get them to buy things. Additionally, the article reports that students in 8th grade looking at manipulated graphs compared to more absolute graphs still did not realize the difference; on one question only 2% of students found the difference and on another question only 8% found the difference. This shows that middle school students who should be learning how these changes work are not learning it. The author then tested the same problems on a class who had just completed a unit on statistics, and though all but one of them answered the question correctly, almost half did not give satisfactory reasoning for doing so, according to the author.
The author concluded with a helpful “Implications for Teaching” section at the end of the article. She noted that many students noticed the scaling on the axis as a misleading feature, but they didn’t think of differences in the source of the graph or ‘pictorial embellishments.’ She says that having them study graphs in newspapers will not only help them become more “statistically literate,” but also help them learn of political agendas in media sources. Thus they learn that finding information about the source of the graph can give you information on the graph itself.
Keywords: Geometry, Trigonometry...
Ref: Casey11
Author(s): Hirshfeld, Alan W.
Date: 2004
Title: The Triangles of Aristarchus
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 97, Number 4, pp. 328-331
Reviewer: Casey
Date of Review: 3-30-04
This article is about how Aristarchus used geometry to perform different solar measurements. For the first part, the author talks about how Aristarchus didn't know trigonometry, so he figured out relative distances between the earth and the moon, and the earth and the sun. He found that the sun was 19 times farther away from the earth than the moon was, a far cry from the modern value of 390 times. However, this was because he estimated an angle, and was interested much more in the method of measurement than the measurement itself. This is an important principle for students to learn because they find that sometimes we can calculate something to a much higher accuracy than we can measure it, and must wait until techniques come out that can measure to higher accuracy.
Next Aristarchus used a solar eclipse to estimate the relative diameters of the sun and the moon. Since the moon almost entirely covers the sun in such an eclipse, and the moon is 19 times closer (in his reasoning), the diameter must be 19 times smaller. While the numbers are again off, the logic is correct. He then uses similar triangles and a lunar eclipse to similarly estimate the relative earth and moon diameters. From here he did not take any more steps, as the author suggests that he could easily have made a scale model of the earth, sun, and moon, and from there argue for a heliocentric model. However, the author does note that Aristarchus' methods became the standard for cosmos measurement in the middle ages, not a small feat.
This article is great at laying out a neat way to look at the
geometry
of triangles, but it fails in suggesting a concrete way of
implementation into the classroom. My feeling is that one could fit
this into a number of different lessons, including those on similar
triangles, right triangles, basic trig, or even as a type of summary
project. No matter what, it is clear that Aristarchus' accomplishments
were truly revolutionary for his time.
Keywords: Number and Operation, Technology, Activities
Ref: Casey12
Author(s): Attia, Tamar
Date: 2003
Title: Using School Lunches to Study Proportion
Journal or Publisher: Mathematics Teaching in the Middle
School
Volume, Issue, Pages: Volume 9, Number 1
Reviewer: Casey
Date of Review: 4-01-04
This is an article where a teacher used nutrition and school lunches
to learn about proportions and percentages. The author decided that
since students like food, they would possibly be interested in studying
their dietary habits and thus would be motivated to learn about
proportion. She started out by having them gather data by recording
their lunches for a week. She also had the school nurse come in for a
talk about nutrition. Then the students each calculated their RDA
(recommended dietary allowance) using a website. They researched the
nutrition aspect of the project. Then they set up pie charts for both
their average eating habits (which they calculated for an average day
using Excel) and for their ideal day using their RDA. They then were
able to look at the pie charts and understood proportions based on the
idea that if you reduced the intake of one of the nutrients, the others
increased proportionally even though they did not increase in actual
number. One main misconception that popped up was the students
sometimes didn't recognize the difference between reducing one nutrient
and increasing others. Which one you did depends on the actual numbers
for the nutrients, not the proportions themselves. Overall, the project
incorporated learning about nutrition, technology, and proportions, and
kept the students interested. One drawback that I see to this project
is that you would have to spend a significant amount of time studying
things not related to the goal of the project, so one would have to
decide if the added motivation was worth the time lost.
Keywords: Proof, Teaching Strategies...
Ref: Casey13
Author(s): Cox, Rhonda L.
Date: 2004
Title: Using Conjectures to Teach Students the Role of
Proof
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 97, number 1, pp. 48-52
Reviewer: Casey
Date of Review: 4-06-04
This article is about a non-traditional approach a teacher took in teaching students in geometry how to write proofs. The author states that she had been frustrated in the past of the lack of understanding of writing proofs as well as the lack of motivation for writing proofs. The author begins teaching about proof before her official proof unit. She teaches the students about the basic concept of proof as well as the different styles of proof. She then used a unit on quadrilaterals to teach about proofs. She took the textbooks away from the students in order that their conjectures will not be simply taken from the book. She also notes that this can be a difficult way to teach because the teacher must assume different roles than what is assumed in a traditional classroom, including being a discussion leader rather than lecturer and giving up more control to the students.
The unit basically involves the students making conjectures in groups, and then coming together as a class to discuss the conjectures. Then they choose which conjectures to prove and work on that in groups as well. For example, the teacher could give the students a parallelogram and have them measure the different parts of the parallelogram. Then the students use these measurements to make conjectures about parallelograms. Students eventually learn to plan the order in which conjectures are proven so that they can use information from one proof to prove another conjecture. When critiquing a proof, the teacher places the proof on the overhead projector, and the entire class reviews and critiques the proof. Only when the proof is accepted is the statement accepted as a theorem.
The benefits of this type of unit are that students have shown more
interest in proof and have demonstrated a greater understanding of
proofs and proving techniques. The downside is that this unit takes
around 20-25 class periods, whereas her traditional method only took 10
class periods. However, this was a trade off that she was willing to
take based on the increase in understanding of proofs that was shown.
Additionally, teachers have to be more flexible, as different sections
will choose to prove different conjectures.
Keywords: Representations, Standards, Activities
Ref: Casey14
Author(s): Preston, Ronald V., Amanda, Garner S.
Date: 2003
Title: Spotlight on the Standards; Representation as a
Vehicle for Solving and Communicating
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 9, Number 1, pp. 38-43
Reviewer: Casey
Date of Review: 4-13-04
This article is specifically relating to the Representations standard. It is written to demonstrate a method for fulfilling the standard in a fun and interesting project for students. The students were in ‘committees’ that were supposed to decide what type of class party would be the most economical. The teachers gave the students data, and then basically let them run with it. The teachers then circulated to make sure that students understood the problem and cleared up any misconceptions. The teachers found that the students used many different representations to sort through the data, including tables, line graphs, bar graphs, and pie charts (which were particularly not useful). Students were then able to use these representations to find which party method was the most economical. One side problem that popped up during the class time was that students who graphed the results often did so with problematic axis sizes, such that the graphs were barely distinguishable. Also, the teachers noted that it is somewhat sketch to be using continuous models to represent discrete data. They decided to save that discussion for another day though. The teachers found that students sometimes chose their favorite or most familiar representation instead of what was the most useful. Tailoring problems to specific representations could solve this, but the teachers in this article were more interested in what the students could come up with on their own. Overall, this activity is a great one for implementing the representations standard into the classroom and teaching students the importance and usefulness of different representations.
Keywords: Measurement, Activities, Representations
Ref: Casey15
Author(s): Buhl, David, Oursland, Mark, Finko, Kristen
Date: 2003
Title: The Legend of Paul Bunyan, Lessons in Measurement
Journal or Publisher: Middle School Mathematics Teacher
Volume, Issue, Pages: Volume 8, Number 8, pp. 441-448
Reviewer: Casey
Date of Review: 4-19-04
This is an article about a measurement activity. The activity takes some legends of Paul Bunyan and turns them into problems involving measurement and ratio. The teacher states that the kids are interested in the stories, which helps motivation for doing the problems. In addition, they are not only performing a task, but the result is sometimes evidence for or against the legends. One of the tasks the students performed was to construct a scale model of Lake Superior, where Paul herded whales. One of the things the students find is that in order to use the same scale in all dimensions, the depth of their models will be approximately 1 mm. Thus they learn that it is possible and sometimes advantageous to use different scale factors for different dimensions. Other aspects of Paul's life that are investigated are his skillet and his Ox, Babe. This activity is great in Minnesota, as there are many legends of Paul Bunyan here, but there are also legends of his in other states, as well as other legends that could be used for this same type of purpose. The activities are both meaningful and interesting, and students learn how to use information to create a model to investigate the validity of the information.
Keywords: Communications, Standards, Teaching Strategies
Ref: Casey16
Author(s): Stein, Mary Kay
Date: 2001
Title: Putting Umph into
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 7, No 2, pp. 110-112
Reviewer: Casey
Date of Review: 5-02-04
This article is about stimulating a mathematical discussion in a middle school math class. The argument is that students will learn better if they have time to formulate arguments about which answer they think is true and are forced to defend them. In a classroom setting, they recommend having a student suggest an answer, finding other students to publicly agree with that student, and having the student explain their resoning. Then we look for other ideas about how to solve the problem, and ask for justifications of those and how they relate to the solution that was already presented. Then collectively the class can figure out which of the answers and/or methods is correct. One of the interesting facts about this article is that the research for it was taken in mostly poor urban schools, which are often thought of as not conducive to this type of discourse. Overall, the idea that mathematical discourse can greatly improve mathematical thinking is solid, and the article demonstrates this is a clear and concise way.
Keywords: Assessment, Teaching Strategies...
Ref: Casey17
Author(s): Britton, Kristine L., Johannes, Jennifer L.
Date: 2003
Title: Portfolios and a Backward Approach to Assessment
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol 9 No 2 pp. 70-76
Reviewer: Casey
Date of Review: 5-4-04
This was an article about assessment, and how assessment relates to learning. The initial claim is that assessment is not supposed to be about a grade, but instead is supposed to reflect how much a student has learned. The writers are also concerned with the lack of information going home to parents, especially the information that a student was doing well, as most exchanges occurred only if students needed help. The writers then decided to take the approach to assessment outlined in "Understanding By Design." This meant what they call a 'backward approach' to assessment, starting with a desired result and designing lessons and units based on that. Then they had students make portfolios of their work on triangles, along with self-assessment and teacher assessed handouts. The students then took these portfolios, apparently as an assignment, and discussed them with their parents. The feedback they got was positive, with parents saying that they knew their kids were learning something and they didn't care at this point about letter grades. Thus the experiment was a success, and the teachers concluded by saying that working closely with parents and administration was important in assessment.
Keywords: Activities, Problem Solving, Teaching Strategies
Ref: Casey18
Author(s): Engel, Bill, Schmidt, Diane
Date: 2004
Title: The Galactic Spaceship Tour Challenge
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 97, number 5, pp. 314-319
Reviewer: Casey
Date of Review: 5-12-04
This is an article about a very involved problem that the authors think can be used in high school classes. It really is an example of a class of problems, meant to integrate many types of math and use them toward one goal. There are definite advantages to doing this, as it gets kids motivated because there is a somewhat real life goal, it makes them use a lot of the math and problem solving strategies that they have learned over the years, and it gives them an opportunity to use groups when they actually will need other's help, not just because they are told to work in groups.
The problem itself is called the Galactic Spaceship Tour Challenge. The students are given information regarding a tour to 5 different planets, and have to plan the tour to maximize profit. I thought this sounded really easy until I continued to read about what was involved. There are a lot of choices to make. There are 5 planets, and since we start on one of them there are 4! paths to choose to see all of the planets. The path must be chosen that minimizes costs. Then they have to pick a spaceship based on speed, crew cost, ship capacity, and ship rental fee. Then they have to figure out what to charge for tickets to get the maximum profit, based on an equation given from 'market research.' Overall, it is a great overall type of problem that would be really neat to spend time towards the end of a class on, to wrap things up.
Keywords: Geometry......
Ref: Casey19
Author(s): Soto-Johnson, Hortensia, Bechthold, Dawn
Date: 2004
Title: Tessellating the Sphere with Regular Polygons
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 97, number 3, pp. 165-167
Reviewer: Casey
Date of Review: 5-12-04
This is a short but informative article about tessellating a sphere with regular polygons. This is meant to be a supplement to regular tessellation subjects, probably meant for the higher-grade levels. It is really neat to see that one can start implementing and thinking about non-Euclidian geometry without actually talking about parallel postulates and such. The article starts by showing how we know that only 3 regular polygons tessellate a plane. Then it generalizes this method to look at a sphere in place of the plane. There is some higher order math here, so care should be used as to who looks at this material. Another really cool part of the article is that they include explicit instructions of how to construct the tessellations, and pictures of the tessellations themselves.
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Keywords: Algebra, Curriculum, Teaching Strategies
Ref: Casey20
Author(s): Maida, Paula
Date: 2004
Title: Using Algebra without realizing it
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 9, number 9, pp. 484-488
Reviewer: Casey
Date of Review: 5-19-04
This article is in some ways about how to sneak algebra into a middle school classroom. In doing so it outlines some basic tricks for teaching the concept of unknowns. This is a way for pre-algebra, or even algebra students to be learning how algebra works. One of the ways she accomplished this was to start by having fifth graders use pictorial representations in the place where a letter would go in an algebra class. The author gave an example where the weights of a soccer ball and baseball together are known, a soccer ball and football are known, and a baseball and football are known. Then the student can use pictorials to represent s+b=1.35, s+f=1.9, b+f=1.25. Then by substituting pictures, the student is able to solve for the weights of each, using algebra in the process. Next the author shows that Unifix cubes can be used in the same way to represent variables. Students then can answer questions based on their own logical reasoning. The conclusion is that we can get students to start to think algebraically before they even know what algebra is, and in so doing enhance their mathematical learning.