Keywords: Algebra, Activities
Ref: Chris1
Author(s): Appelbaum, Elizabeth Berman
Date: 1997
Title: Telephones and Algebra
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 90(2), p. 96-100
Reviewer: Chris
Date of Review: 4/28/99
Appelbaum in this article describes an activity that she used in algebra classes with the specific goal in mind of making the lesson interesting to the students. "Cellular telephones and their rates are a trendy application that may interest many students," she points out. She had two students volunteer to bring their phones in to class and students talked on them to get more interested. An advertisement from a local cell phone company provided four calling plans with different rates based on usage time. Students created piecewise functions to describe the price in terms of time used in a month. One especially challenging part was incorporating the three cent per minute surcharge that appeared in fine print on the ad. These functions were graphed on common axes and easily showed which plan was the cheapest for users in different time ranges. The equations were manipulated to find intersection points where the cheapest plan changes, and the class explored what this ! means when calls are only billed in whole numbers of minutes. An extension of the work was incorporating various taxes as well as optional services like call waiting and voice mail into the formula.
I think Appelbaum’s activity is well thought out and applies many parts of algebra very nicely. It may not be a very good activity for introducing the concepts for the first time since it does span so much of the algebra field. But as a conclusion to a unit on one of the more advanced topics or as a review it would work very well.
I have two hesitations to using students’ personal cell phones for demonstration at the beginning. This anticipatory set seems to introduce at least questions of what would happen if the phones were damaged in the process and unfairly asks those students to pay the bill for class minutes. The other possibly more problematic but less obvious problem I see is that it introduces inequalities in the class. Two or three students will get attention and praise for their phones while many students wouldn’t be able to afford them. However, small modifications in the introduction to the activity should be able to deal with both of these concerns.
Keywords: Activities, Connections, Trigonometry
Ref: Chris2
Author(s): Kim, Hy
Date: 1997
Title: Angled Sunshine, Seasons, and Solar Energy
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 90(7), p. 528-532
Reviewer: Chris
Date of Review: 3/1/99
Kim's article is a good illustration of an integrated unit that uses math concepts to analyze a scientific phenomenon. In it students look at how the angle of sunlight hitting the earth changes with the seasons, and why that is important to changing temperatures. According to the author the unit addresses the NCTM standards of problem solving and making mathematical connections.
In this unit, students first compare the amount of energy received from angled sunlight to vertical sunlight. Using a flashlight and sheets of cardboard at different angles, students first explore this comparison and then use trigonometry to quantify it. To extend this to the earth and sun, a classroom demonstration with a globe moving around a flashlight gives students a spatial picture of what happens in different seasons. Using the tilt angle of the earth and the latitude of specific cities, students then determine the angle of elevation at solar noon on the days of the spring and fall equinoxes and winter and summer solstices.
I like the way that this unit capitalizes on knowledge that students already have. Thinking about why the noon sun is hotter than morning or afternoon sun leads nicely into sun angles at different times of the year. The analysis that uses math is done only after they have a firm grip on the concepts. This is especially necessary because of the three dimensional nature of the problem. Spatial thinking is hard for many students, and adding math too early would just make it harder.
This unit could be extended to include research where students find the latitude of various cities (chart reading skills) or do primary research in taking actual measurements of sun angles.
Keywords: Geometry, Activities
Ref: Chris3
Author(s): Lufkin, Dan
Date: 1996
Title: The Incredible Three-by-Five Card!
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 89(2), p. 96-98
Reviewer: Chris
Date of Review: 3/8/99
Lufkin describes three examinations that a geometry class can do using a 3" x 5" note card. The activities could actually use any paper rectangle, but the note card is sturdy, a convenient size to manipulate, and it makes a connection to students because they have seen a note card before.
The activities all involve three triangles that come out of the note card. The card is first cut into these three right triangles, and Lufkin suggests that they can be named creatively (i.e. Larry, Curley, and Moe rather than just A, B, and C). The first activity is measuring each side in millimeters and angle in degrees. This is probably a review exercise in measurement for most geometry students, but the numbers are used later. The second exercise involves comparing the three angles of the three triangles and then the ratios of the lengths of the sides. This explores what similar triangles are. The third activity uses the ratios and measurements of the previous two activities to derive the Pythagorean theorem.
My favorite thing about Lufkin’s activities are that they are simple, visual, and kinesthetic, but still teach concepts in a strong way. Students actually create and hold in their hands a proof for a theorem that seems complicated. I think this would make a nice lesson for a geometry class either to introduce the Pythagorean theorem or after using it for a couple of days. The measuring activity and similar triangles activity are good reviews of basic skills that are also important.
Keywords: Technology, Geometry
Ref: Chris4
Author(s): Zbiek, Rose Mary
Date: 1996
Title: The Pentagon Problem: Geometric Reasoning with Technology
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 89(2), p. 86-90
Reviewer: Chris
Date of Review: 3/8/99
Zbiek’s article, “The Pentagon Problem” is about much more than pentagons. But it uses this problem as an example for how to use one particular type of software package to explore geometry.
The Pentagon Problem is that if you were to take any pentagon and connect the midpoints of the sides, you get a new smaller pentagon. Repeating the construction gives you a third, and yet smaller, pentagon. The conjecture is that the ratios of area and perimeter from one pentagon to the next smaller one are constant. Students have the task of using technology to decide if the conjecture is true, false, or partially true.
The technology package used is The Geometer’s Sketchpad. It allows the user to draw pentagons and do measurements and calculations on them. It can measure the perimeters and divide them to find the ratio. These are things that could be done by students, and probably should be done once by the students, so they understand what the technology is doing. But the software makes more things possible. For example, the students can use the mouse to drag one or more vertices of the pentagon to change its shape. They can watch as they do this how the area and perimeter of each pentagon change, and how (or if) the ratios between the figures change. Other discussions include the effect of round off error or how ratios would act with a fourth pentagon.
For the recreational mathematician who is attempting the pentagon problem at home, do not read the next sentence. The conjecture does end up being false, but showing it involves some tricky arithmetic. This could end up being an extension for students who are working ahead or as an extra credit problem.
But regardless of the quality of the quality of the pentagon problem, it does illustrate the strength of this type of software in the classroom.
Keywords: Communication, Teaching Strategies
Ref: Chris5
Author(s): Elliott, Wanda Leigh
Date: 1996
Title: Writing: A Necessary Tool for Learning
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 89(2), p. 92-94
Reviewer: Chris
Date of Review: 3/9/99
Elliott begins by mentioning a professional development course that she took on writing which led her to believe that writing is an important part of learning. This includes learning mathematics.
She talks about different times in the class that writing can be used. At the beginning of class a short writing activity can calm and focus the class. It can also review ideas and provides an opportunity for formative assessment. In the course of the class, Elliott might ask her students to write what they understand of a concept. This again gives information to the teacher, but more importantly it gets students to organize their thoughts into at least a form that they can write. Her examples are not polished papers that students spend many hours on, but short journal entries that may or may not have complete sentences or equations. One activity that I especially appreciated was having students predict their test grade and then reflect on it after getting it back. This makes exams into learning experiences as well as assessments. She evens says that some brave teachers “may ask students to evaluate their teaching techniques.” This requires mature students and a mat! ure teacher, but could be a very effective tool.
My hesitation is that writing in class is something that the teacher and students have to make a real commitment to. If it is only done once it seems to me destined to fail. But if a class is in the habit of expressing themselves like this, I think Elliott is right that different learning styles can be addressed very nicely.
Keywords: Standards
Ref: Chris6
Author(s): Frye, Shirley M.
Date: 1989
Title: The NCTM Standards -- Challenges for All Classrooms
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 82, May, p. 312-316
Reviewer: Chris
Date of Review: 3/14/99
This article review is in response to the suggestion to look at a journal article from the late 1980’s which addressed the then brand new NCTM Curriculum and Evaluation Standards for School Mathematics. The author was the president of NCTM in 1989 when the standards were published and she wrote this article. She says that as a central figure in the standards movement, she receives many questions about the standards. This article answers fourteen of the most common ones.
Many of the questions seem to have come from teachers looking for general information. "What are the standards?" and "Why did NCTM think that the standards were necessary?" Frye gives good answers to questions like this, probably duplicating standard responses from other NCTM publications.
Some questions seem almost adversarial. They ask why the standards shouldn't be considered another attempt at "new math," something I know next to nothing about but it looks like it failed. One question asks if a professional organization like NCTM has a rightful place in making curriculum decisions that have been made by more local authorities in the past. Of course, any change brings the question from teachers of "What am I going to have to change." Frye also addresses these questions very diplomatically.
A few questions, though, seem unlikely to have been asked by teachers. One asks what other professional organizations support the standards, and on asks about how teacher education in both preservice and in-service education are being involved. These questions seem to have been put in the article specifically so Frye can answer them.
Frye does do a good job in talking about the issues brought up. But most of what she says is very likely available in other NCTM publications. This is not necessarily a bad thing, however. Teaching is sometimes called creative repetition, and talking about the same thing in journal articles as well as in small group meetings helps to educate people
Keywords: Technology, Teaching Strategies,
Ref: Chris7
Author(s): Everyday Learning Corporation
Date: 2000
Title: Connected Geometry
Journal or Publisher: Everyday Learning Corporation
Volume, Issue, Pages:
Reviewer: Chris
Date of Review: 4/11/99
I spent some time looking at a CD-ROM based textbook, Connected Geometry. The CD comes with Adobe Acrobat, a common software title for viewing documents. Acrobat also allows programmers to insert links to other parts of the document somewhat like a web page on the Internet. But the powerful resources available to the writers of Connected Geometry are mostly left unused.
The program opens with a table of contents that links users to one of six "modules." Each one is of a large theme in Geometry, such as "Investigations in Dissection and Area" and "Pathways to Similarity and Trigonometry." The individual investigations (10 to 30 per module) are mostly activities. They lead students individually through a process and periodically ask questions about what is happening. The questions are often open ended and higher order, and would fit well in a math journal. Physically manipulating figures (such as cutting things out of paper) and using geometry software are common directions throughout the investigations. The lessons are similar to many other activity-based textbooks.
The main problem with the book is that it is hard to use in the Acrobat format. Turning from one page to the next is awkward since an entire page can not be displayed on the screen at once. Also, moving from one part of the book to another can only be done by going backwards to the table of contents and moving in to the new location. The technique of keeping your finger in one page while looking at another is totally lost.
This problem might be justified if the book used the computer format to do things that couldn't be done in a paper book. Examples would be hypertext, (words that link to other parts of the book), or animations. But neither is present in this book. In fact, this book could be produced as a paper textbook with absolutely no changes. In that format it would be an average activity-based text, but in the CD-ROM format it is not very good.
Keywords: Activities, Technology,
Ref: Chris8
Author(s): Career Connection to Teaching with Technology (CCTT) Project
Date: 1999
Title: Activity Collection
Journal or Publisher: Career Connection to Teaching with Technology (CCTT) Project
Volume, Issue, Pages:
http://mainland.cctt.org/mathsummer/
Reviewer: Chris
Date of Review: 4/19/99
The Activity Collection page of the Career Connection to Teaching with Technology (CCTT) Project, located at http://mainland.cctt.org/mathsummer/ is an index of 35 web pages of activities. Although it is not specifically said, my guess is that the pages and activities described were created by students and posted to the Internet by professional web designers. I make this guess because the activities are all quite varied and seem to have been done as a class project, but the graphics and construction are similar in them all.
The pages in the collection are mostly activities that a math class (or individual student) could work through and either discover a new concept of use a familiar one. Many of them are very good, and I plan on using them in my classroom. "Buried Treasure" uses trigonometry on a map to find the location of a buried treasure. Clues to its location are given but the place can't be directly measured because of lakes in the way. Another very good activity is "Single Elimination," which has students schedule a bowling tournament. Two ways of scheduling are proposed and students are to answer questions using probability from each method, and evaluate which one is more fair.
However, there are some activities that need modification or maybe should not be used at all. "Vocabulary" is a detailed list of math terms and their definitions but not an activity. "Football Statistics" has students watch an entire football game and record the gains and losses in yardage on each play. They then sum the yardage statistics and compare them with values from sports sections of newspapers. This is an activity with unclear mathematical goals and very long and monotonous steps to perform. These problems are probably because the projects are likely student made rather than made by professional educators. But if that is the case, some of the projects deserve recognition as good lesson planning.
Keywords: Curriculum, Standards,
Ref: Chris9
Author(s): Ruenzel, David
Date: 1996
Title: Brainstorming
Journal or Publisher: Teacher Magazine
Volume, Issue, Pages: http://www.edweek.org/tm/vol-07/04math.h07
Reviewer: Chris
Date of Review: 5/4/99
I ran across this article on the Internet and found it very interesting. It describes one journalist's quest to find out about IMP, mostly in the Minneapolis school district. Teachers and students using IMP are questioned, as are some non-IMP teachers. He generally comes to the conclusion (which is common in what I have seen) that the people who know about IMP and have used it for more than a few weeks love it. They say that they would never go back to sequential curriculum because it is less interesting for teachers and students and less learning takes place.
One teacher that I talked with at the MCTM conference was quoted in the article. The writer had asked about John Saxon and got an awkward silence. He writes:
There was a moment of silence, as if mentioning Saxon's name had broken some taboo; his drill-oriented textbooks are anathema to everything the IMP teachers are trying to accomplish.
Finally, Jean Stilwell, a teacher at Henry High, spoke up. "If you want to teach students to think, then you have to do all the things John Saxon attacks," she said. "If you want your students to do well on multiple-choice skill-based tests, then use Saxon's texts. But for the world of work, what difference does it make how you do on a multiple-choice test? No one hires you to solve algebra equations--they hire you to solve problems. So Saxon does a great job of teaching you obsolete things. His students learn, review, learn, review, but to what purpose?
"We find that when people use lots of drill and skill, lots of repetition, they can do well for a while, but that it won't last. A few years, and it's gone. That's how I learned calculus, and when I opened the book a few years later, I could remember nothing. I got A's, but did I really learn anything?"
>From what I know about IMP, this article seems to address issues very well. One quote from a non-IMP teacher seems to sum up the views of people against reform: "I'm happy [the IMP teachers] are doing what they're doing," he said. "But I'm more comfortable with telling kids how to do something."
Keywords: Teaching Strategies, ,
Ref: Chris10
Author(s): Toumasis, Charalampos
Date: 1995
Title: Concept Worksheet: An Important Tool for Learning
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 88(2), p. 98-100
Reviewer: Chris
Date of Review: 5/6/99
Toumasis is describing a tool that she uses for complex concepts in her math classes – the concept worksheet. The blank form is filled in by students for each concept that the teacher assigns. On the form are: 1. definition, 2. web of attributes (concept map), 3. two examples (drawn) and why it is an example (mathematical reasons), and 4. two non-examples and why they are not examples. The example given shows "parallelogram" as the concept. The attributes in the concept web are things like "opposite sides are equal," "The opposite sides are parallel," and "The diagonals bisect each other." Examples must be explained mathematically. For the parallelograms, the explanation is AB||DC and AD||BC. This is a good way at getting to see if the students know how to apply the attributes or if they just remember one example that they had seen previously. Finally, the non-examples are supposed to meet some of the attributes but not all of them. For example, a circle would no! t be a good non-example of a parallelogram, but an irregular quadrilateral would.
Toumasis points out the one fault with her idea, one which I had thought of as well. They would take a long time for teachers to read and evaluate. And if a worksheet is done for each concept encountered in a course, the paperwork would seem to have to push something else out of the way. This first issue could be addressed by spot-checking the papers of only randomly selected students or only on randomly selected days. However, students would then not have their worksheets corrected and could let a misconception persist. A way to address the second issue of paperwork could also save teachers time. Only particularly difficult concepts could receive this worksheet analysis. It would probably be excessive to do a worksheet on the rectangle, square, parallelogram, rhombus, kite, and the irregular quadrilateral. But if used limited, I think the Concept worksheet could be a good tool for the math classroom.
Keywords: Assessment, Teaching Strategies,
Ref: Chris11
Author(s): Manon, John Rahn
Date: 1995
Title: The Mathematics Test: A New Role for an Old Friend
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 88(2), p. 138-140
Reviewer: Chris
Date of Review: 5/6/99
Manon writes this article to address a problem he sees in secondary math classrooms. He says that teacher-designed tests have become relied upon too much for assessing how well teachers teach and learners learn. The NCTM's Standards suggest more ways of assessing student learning, but Manon is looking at something different. He suggests more use of math tests for other purposes.
The first suggestion is for test to assess low level thinking like skills. He says tests are efficient for this, but can't really get at comprehension or higher on Bloom's scale like other types of projects can.
Tests can also inform instruction, according to Manon. Here he is talking about formative assessment. The teacher reacts to how students do on the test, rather than using the test as a conclusion to a content area. (Whether content areas should ever really be concluded is another question that should be asked.) Finally, tests can focus students on a particular topic and be used to begin instruction. He talks about using a test for an individual to sort out ideas before working in a collaborative group.
Manon seems to have good ideas here, but some of them are a little strange. On my first reading of the article, I got the impression that he was advocating not less but more paper and pencil tests for math classrooms. I understood that he wanted to use them in non-traditional ways, but more time at a desk looking at a test is not what I think students need to help them learn. On closer reading I no longer think that this is what Manon is saying, but his real points are then pretty shallow.
Keywords: Geometry, Activities, Technology
Ref: Chris12
Author(s): Lanius, Cynthia
Date: 1996-1999
Title: Mathematics of Cartography
Journal or Publisher: Internet
Volume, Issue, Pages: http://math.rice.edu/~lanius/pres/map/
Reviewer: Chris
Date of Review: 5/6/99
This unit plan is one of a few by Lanius on the Internet. Her's are unique because they make such good use of web technology. It is full of graphics, and one of it's strongest points is the number of links to other quality sites it offers. This site can be used by itself with no additional materials or even teacher interaction. However, I think it would be most useful as a resource used in a unit on cartography. It has been a valuable planning resource for me as I base some of my geometry unit on the mathematics of map making.
The unit addresses a few different mathematical content areas. Proportions are used a lot in learning about the scale of a map. Students study coordinate systems through the longitude and latitude systems on the earth. A more advanced section deals with three-dimensional geometry and the problems associated with mapping a sphere to a plane.
This really is a nice way to present a series of lesson plans. I could easily see a class working through Lanius' site over a week or more, but I think more interaction with the teacher would be needed. Possibly part of each day would be spent on this site, and part of the day working away from the computers.
Keywords: Statistics, ,
Ref: Chris13
Author(s):
Date: 1997
Title: Polls: What do the numbers tell us?
Journal or Publisher: The Annenberg/CPB Program
Volume, Issue, Pages: http://www.learner.org/exhibits/statistics/
Reviewer: Chris
Date of Review: 5/8/99
This site is called an interactive learning exhibit on statistics. It traces a fictional mayoral election race with a focus on polling. Polls are followed from one year prior to the race up until election night when votes are tallied.
The site does a good job of explaining polling elements that usually just get mentioned. Random samples and margins of error are two examples. The narrative then goes on to discuss ways that polls get biased by sampling population and sampling methods. Finally, it explores how polls themselves can effect the outcome of the election.
This site would be appropriate for a class studying statistics, especially during an election season. The topics are relevant and accessible.
Keywords: Technology, Teaching Strategies,
Ref: Chris14
Author(s):
Date: 1997-1999
Title: www4teachers
Journal or Publisher: South Central Regional Technology in Education Consortium
Volume, Issue, Pages: http://www.4teachers.org/home/index.shtml
Reviewer: Chris
Date of Review: 5/8/99
This site is for teachers interested in technology in the classroom. It consists almost exclusively of submitted stories from teachers and students on what they are doing in their own schools. This is a very powerful aspect of the site, because visitors know that the techniques proposed here actually have worked somewhere. One section reviews Internet sites created by teachers for use in their own classes, and provides links to them.
I see this site as a good starting place for teachers interested in starting new technology programs or improving existing ones. Another neat aspect of this site is that it is not entirely devoted to mathematics teaching. This means some of its information won't necessarily apply to math teachers, it also means that some articles are interdisciplinary and can encourage integration of other topics into our math classrooms.
Keywords: Teaching Strategies, Standards,
Ref: Chris15
Author(s):
Date:
Title: ENC Online
Journal or Publisher: Eisenhower National Clearinghouse
Volume, Issue, Pages: http://www.enc.org/fr_index.htm
Reviewer: Chris
Date of Review: 5/8/99
This site is a collection of resources for teachers of math and science. The two main sections are "Ideas for Reform" and "Reform in Action." The first section has links to journal articles supporting reform, information about TIMSS, and links to online versions of state framework packages. The second section has examples of things happening in classrooms that other teachers can use. Innovator of the Month profiles educators working in unique ways; Classroom Links provides valuable web sites for educators, and ENC's best page called "The Digital Dozen" is 12 great web sites featured each month.
ENC Online is a wonderful source for math and science teachers. I look at the Digital Dozen every month and have a great time browsing the sites that are interesting to me as an educator but also as a mathematician and scientist. This is definitely a good site for teachers to know about and use.
Keywords: Technology, ,
Ref: Chris16
Author(s): Carl, Iris
Date: 1993
Title: Equal Opportunity: Technology Can be a Bridge to Mathematics Equity
Journal or Publisher: Electronic Learning
Volume, Issue, Pages: 12, p. 60
Reviewer: Chris
Date of Review: 5/9/99
This editorial is a strong statement against the back-to-the-basics themes that have become popular in some circles nationwide. Carl, a past president of NCTM, writes that these movements have taken away opportunities for top students to be challenged while not preparing the majority of students with the mathematical background that they need. One statistic in particular, that "only 6 percent leave [high school] with a 12th-grade quality mathematics education" points out that a small majority of students are being served by traditional methods. This points out why some people see merit in publishers like Saxon where higher order thinking skills must come from the outside because they are not in the text anywhere. But the 94% of students who are not served deserve something better.
Carl supports instead the vision from the NCTM Standards and specifically the use of student centered education. She writes that technology enhances this process by making difficult concepts understandable.
But her article doesn't go much farther than that. She doesn't mention other issues dealing with technology, such as cost or start-up instruction time. The title of her article mentions equal opportunity, so I thought she would mention how underprivileged students can benefit from technology, but she doesn't. She makes some good statements in the editorial, but doesn't back up anything about technology with research or deep analysis. I don't think this editorial would do a very good job of convincing anyone who was not a supporter of technology to believe any different.
Keywords: Manipulatives, Activities,
Ref: Chris17
Author(s): Curcio, Frances R., Sicklick, Francine, and Turkel, Susan B.
Date: 1987
Title: Divide and Conquer: Unit Strips to the Rescue
Journal or Publisher: Arithmetic Teacher
Volume, Issue, Pages: 35(4), p. 6-12
Reviewer: Chris
Date of Review: 5/10/99
This article taught me that not all lesson ideas that I find are good just because they are published. But I'll tell you why later. First, a summary:
The authors are concerned with the way that students learn arithmetic on fractions. They quote Steve Leinwand saying "Yours is not to reason why, just invert and multiply... that's the stuff of sterile instruction." The algorithm of inverting and multiplying to divide a fraction is what they want to do away with. They say that students learn the process but don't learn conceptually what it is doing. Their answer is "unit strips" and a host of activities to go along with them. Unit strips are strips of paper in different sizes and marked corresponding to the fraction of the large, one-unit strip. They work through four activities using the unit strips that get students to think about dividing fractions in a different way than inversion and multiplication.
I love the idea of using unit strips. They are similar in form and function to the unit circle blocks from the manipulative kits we worked with before. Appealing to the kinesthetic learning style is something that all teachers should be working to do. But beyond that, these activities are no good at all. They don't use any real life examples, which we found in class to be extremely helpful in understanding division of fractions. Instead they set up an algorithm using unit strips for changing division problems into multiplication problems. The process they go through is more convoluted and no more demonstrational than the usual "invert and multiply" process.
I do intend to use some manipulative for teaching fractions because I know it is an area that students often struggle with. But because of its presence on the Minnesota Basic Sills Test, we have to figure out how to get our students to understand it. But using these activities is not the way to do it.
Keywords: Assessment, ,
Ref: Chris18
Author(s): Esty, Warren W. and Anne R. Teppo
Date: 1992
Title: Grade Assignment Based on Progressive Improvement
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 85(8) p. 616-618
Reviewer: Chris
Date of Review: 5/10/99
The authors are addressing what they see to be a problem with the way most math teachers assign grades to their students, by what they call the test-score-averaging method. Instead they propose a method of monitoring continual improvement and measuring eventual mastery.
The main problem with taking the average of many tests from throughout the course, according to the authors, is that it penalizes students for lack of prior knowledge instead of focusing on improvement. Recognizing that math courses are by nature cumulative, they say that this grading scheme gives equal weight to mastery of early less complex topics as it does to the actual course objectives which won't be tested until the end of the course. For this reason, they advocate giving tests grades as normal but not counting them toward the course grade. Instead, "only their performance in the final weeks of the class will be counted for assigning grades." But students are still accountable for all course material in those final, cumulative weeks.
I like some parts of this idea. I like where the authors talk about students not being forced into mastery of a topic immediately, and that they have time to tackle more complex situations. However, I question whether students will be motivated to take the first two-thirds of the course seriously if they are not graded for it. We always talk about accountability in the context of cooperative learning, and I think it applies here as well.
Finally, I want to include the summary of the article from the authors, because it sums up their argument better than I feel I have:
We maintain that the primary goal of instruction should be for students to master the material by the end of the course and that class grades should be assigned accordingly. We have found that the progressive-improvement grading system is a pedagogically effective way to assess and facilitate learning. By eliminating the artificial constraint on the material imposed by the test-score-averaging "instant mastery" requirement, the curriculum can be freed to include long-term learning of multistage procedures and broader mathematical concepts. Assessment changes are necessary if we are to implement the curriculum and evaluation standards.
Keywords: Technology, Activities,
Ref: Chris19
Author(s):
Date: 1999
Title: JavaSketchpad DR3 Gallery
Journal or Publisher: Key Curriculum Press
Volume, Issue, Pages: http://www.keypress.com/sketchpad/java_gsp/gallery.html
Reviewer: Chris
Date of Review: 5/13/99
This web site is maintained by Key Curriculum Press, the makers of educational curriculum like the Interactive Math Project and software like The Geometer's Sketchpad. This site is devoted to their Internet based software, Java Sketchpad.
The site has examples of programs written using the web's Java language that can be animated and manipulated by the user. One example is of a six sided polygon that is tessellated in the plane. The user can drag the vertices of the polygon around and the tessellation changes automatically. Most of the demonstrations rely on dynamic changes in the illustration as the user moves points around.
These demonstrations (as well as others that an instructor could make using this software) are wonderful ways of illustrating mathematical concepts. They address the visual learning style better than could ever be done on a chalkboard or paper. They are similar in many ways to the computer illustration that the Japanese teacher had in the videotape that we watched at the beginning of this course, but even better because these are interactive. My guess is that writing programs for demos like these may be a time consuming process, so watching for them on the publisher's web page is a good way to get them.
Keywords: Activities, Algebra,
Ref: Chris20
Author(s): Borlaug, Victoria A.
Date: 1993
Title: From Algebra to Calculus: A Tonka Toy Truck Does the Trick
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 86 (4) p. 282-287
Reviewer: Chris
Date of Review: 5/16/99
In this article Borlaug describes the way that she introduces the concept of slope to her algebra classes. The demonstration can be modified to work very well in a Calculus class. She uses a Tonka Truck on the chalkboard. She drives it vertically on the board next to a vertical number line. She asks the students to remember where the truck was at two seconds, at four seconds, etc. This demonstrated that the location at specific times is lost in a one-dimensional graph. By adding a horizontal axis, these locations can be recorded. As she drives the truck up and down the board, a student plots points on the graph. When they are connected by line segments, they give the approximate location of the truck at any time. The next discussion is of rate and slope, comparing the (pretend) speedometer in the truck to the slope of the graph.
I like this lesson for two reasons. First, it is a discovery lesson. The teacher does not tell the students that two axes are necessary without proving that one didn't work. Second, she created a lasting visual image by driving a Tonka Truck up and down the chalkboard. I think that is an important way of getting students interested and helping them remember what is going on.