As we go into teaching, something that often concerns us is the issue as to how we will adequately assess the progress of our students. Several ideas come up in testing and its adequacy and alignment (and for nice technical names, look back in your Ed Psychology book or in Learning & Teaching). However, even though there are nice new little things like performance assessments and portfolios, this article brings up the fact that more traditional tests will remain in classrooms, especially mathematics classrooms, well into the foreseeable future. The point of the article is that even though traditional exams will remain, they should to some extent involve questions/problems that require several (more than two) steps, are in a real context-somewhat in contradiction to Mr. Halmos's ideas-are open-ended (in that they can contain a number of possible answers), use graphs or diagrams (which in many cases the student will need to create to solve the problem), involve reasoning, and utilize technology. The article consists of several typical math problems that are then altered in a way fitting to these criteria. The article does indeed deal with the issue that having more complex problems does take more time and effort on both the part of the student and the teacher.
For the most part, I agree almost entirely with the authors of the article. I think that it is necessary that we do establish the need to make connections and not just practice of the use of an algorithm. I am a firm believer that problems should often contain a real-world context. However, in both of these cases let me also say that the skill of manipulation is also important, and without the skill, no matter how cleverly worded the problems may be, the test will not meet its goals. As a side note, I caution against excessive use of technology-students must be able to do the tasks rather than relying on the calculator or computer, especially in the earlier stages of math.
Keywords: Standards, Teaching Strategies
Ref: Jacob2
Author(s): Stepanek, Jennifer
Date: 1997
Title: Science and Mathematics Standards in the Classroom
Journal or Publisher: Northwest Regional Educational I
Volume, Issue, Pages: p 2-19
Reviewer: Jacob
Date of Review: 2/11/02
Standards are one of the big things in education today. Some may say that there are too many of them, or attempting to apply them is simply too much work for a teacher to invest in them. Others claim that standards are almost on their way out. One place where this idea really isn't totally fitting is in mathematics. However, the question does remain as to what the intent of the standards movement is? Why is it that teachers should implement these the "standards" style of educating over a more traditional? The article claims that "[standards] set criteria for more challenging classrooms, enriching curriculum content, and expanding access to improved learning. Standards also offer a framework for authentic pedagogy" (3-4). Although this article also deals with science standards, and how to make science education more life-like, and connected with real-world issues, I'll generally confine my comments to that of mathematics standards. The article quotes five national mathematics standards: Learn to value mathematics; become confidant in one's own ability; become a mathematical problem solver; learn to communicate mathematically; and learn to reason mathematically (4-5). So, what sort of things are we supposed to do in order for our students to accomplish these standards. The article mentions six things that teachers who are implementing these standards must do in order to integrate them successfully. Teachers are to pose the tasks that will require the students to use mathematics, orchestrate and promote discussion in the class, encourage and accept the use of technology, create positive and effective learning environments, and engage in ongoing analysis of teaching and learning (7). Of course, a great deal of planning has to go into the very constructivist lessons that are called for under these standards, in order to give students adequate time and approaches to gain a fuller understanding of the content material. Furthermore, element of the standards curriculum that is really emphasized is that the activities should take place in real-world situations. In order for this to work, teachers need to communicate with each other as well as to determine their own effectiveness with self-evaluations.
I have to say that I'm really not a fan of constructivist lessons. Although it is a good idea to work with stuff the students have an interest in-to some degree-in order to keep them interested in the class, I'm really quite against what I still consider to be a vague idea, and one that seems to be much a part of these standards, that the students should have a decision in where the curriculum goes. This is something that I think is really quite unrealistic-if they know where the curriculum should go, why aren't they teaching the class, rather than in it. Along with the Graduation Standards, I really don't think that massive reordering of classroom activities and curriculum are necessary in order to have effective instruction. The standards should be able to be implemented into any well-throughout curriculum without major adjustments to it.
Keywords: Standards, Algebra
Ref: Jacob3
Author(s): Lambdin, Diana; Lynch, Kathleen; McDaniel, Heidi
Date: 2000
Title: Algebra in the Middle Grades
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 6, No. 3, p 195-98
Reviewer: Jacob
Date of Review: 2/13/02
Standards are often something that may be a little bit frightening. However, often this is because they aren't quite clear, or because the task may seem to daunting (like the excessive paper work that is often part of the Graduation Standards). However, this really isn't all of what Standards are for. The NCTM has come up with standards that are, seemingly enough, general enough to be adequately provided in almost any curriculum. This particular article deals with the issue of the NCTM Algebra standards, particularly as it applies to the middle grades. But, before we get into the article really, lets just note the major ideas of the Algebra Standards throughout the grades: "1) understanding patterns, relations and functions; 2) representing and analyzing mathematical situations and structures using algebraic symbols; using mathematical models to represent and understand quantitative relationships; and 4) analyzing change in various contexts" (196). Specifically, the middle grades (5-8)'s algebra standards cover two main areas: patterns and functions, and Algebra. Although the article claims to be about the latter, I think it is just as much about the former. The article starts out by asking the question "How long would it take a person in a bicycle race of 50 miles to finish?" In order to come up with an explanation for this, the students made graphs of how many jumping jacks they could complete in 2 minutes. These were then graphed-with the students picking the intervals, but stressing that the intervals along a particular axis must be equal. This gave them the opportunity to make a correlation between rate and graphs. To do this, the graphs were displayed and students (not those whose graph it was) had to come up with a story, i.e. if the graph is flat, no jumping jacks were done because the jumper was tired. The jumper then, hopefully, confirmed this story. This represented continuous data. The next activity described in the article was one that was discrete. The students were given a "secret" place and had to graph, without scale, the number of people who would be in that place at each hour of the day. Then, after receiving a list of the "secret" places, the students had to decide which graph represented each one. The final activity was a CBL, in which the students could use their body in order to get the calculator to make a graph-this really is a fairly cool little device, as long as its not overused.
I think that the relation between position, rate and acceleration are very important. The can be taught in middle school on very much a qualitative level-this hopefully would cause students when presented with this idea in calculus to not be quite as perplexed. Also, the experimentation part of the lessons did a good job involving students in their learning-they were able to see first hand about the relations with graphs. This is a very informative article, and Mathematics Teaching in the Middle School has a lot of articles dealing with the NCTM standards (it's the best one for that that I found).
Keywords: Geometry
Ref: Jacob4
Author(s): Lornell, Randi;Westerberg,Judy
Date: 1999
Title: Fractals in High School: Exploring New Geometry
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 3, p. 260-65
Reviewer: Jacob
Date of Review: 2/14/02
The topic of this article is something that has recently become very important in geometry classes-although it wasn't to be heard of when I took Geometry in high school eight years ago. This new finagled thing is Fractal Geometry. Fractal Geometry was first begun at the end of the 19th and beginning of the 20th century, by people like Georg Cantor, Helge von Koch, and Gaston Julia. Their work was "scorned by the mathematical community as 'pathologically unlike anything found in nature' and 'monstrous'" (264). However, now do to the advent of the great computing power of computers and the work of Benoit Mandlebrot, fractals have come back to life, and we see that, contrary to being "pathologically unlike anything found in nature" they do a much better job depicting nature than Euclidean objects could ever hope to do. Some of the simpler fractals are those that can trace their way back to the beginning of fractal geometry, and it is these that are the primary focus of this article. The article does a wonderful job of dealing primarily with the perplexing qualities of the Cantor Set and the Koch curve and snowflake. It goes through several activities in which students have the chance to discover the infinite number of line segments in a Cantor Set, but yet with the set having a length that approaches 0. The lesson dealing with the Cantor Set is very much a constructivist lesson as well as one that deals with recursive formulas-the students are asked to find a recursive formula for the length of each segment at a certain iteration, as well as for the whole length of the set. Slightly more complicated than that Cantor Set (and also more beautiful) is the Koch snowflake. In the Koch snowflake, we can easily see the self-similar nature of all fractals (this is also present in the Cantor Set, but it isn't hard to figure out if some dots are really self similar). The activities that they describe dealing with the Koch snowflake are to construct it using some sort of construction blocks and then perform iterations-the Koch snowflake is made by removing the middle third of each side of an equilateral triangle, and then creating a new one in the place of that removed side. After constructing the triangle, to get a physically manipulatable object, the students can explore the mathematical issues concerning the Koch snowflake. Namely, once again coming up with recursive formulas-this time with the aid of their TI-82 or TI-83-for the perimeter and area. Like the Cantor Set, odd things happen. The perimeter is infinite, and the area is bounded. The article concludes by mentioning how pleasing and frustrating this can be to students in high school math.
I'd like to just put in a little blurb about my own fractal explorations. If you would like to see it, go to http://www.stolaf.edu/people/burkman. Of course, this will likely be dead soon, but I'll try to keep the wonderful fractal links alive somewhere. Fractal geometry is something that can be used very effectively in any stage of geometry. In early grades, one can examine such things as a leaf on a maple tree-see how it has all the jagged edges, and often seems to have a "leaf" coming off if its sides? These are fractals. From this, fractals can still be dealt with all the way through advanced algebra dealing with graphs and operations in the complex plane with objects such as the Mandlebrot and Julia sets. Fractal do a great job showing off the world, and can amaze (and occasionally perplex) those just beginning their study, or those who have been studying mathematics and its related issues for years.
Keywords: Statistics, Connections
Ref: Jacob5
Author(s): Watson, Jane
Date: 2000
Title: Statistics in Context
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 1, p. 54-8
Reviewer: Jacob
Date of Review: 2/14/02
The article bases much of itself off of a supposed statistical study entitled "Family Car is killing us, says Tasmanian researcher." This article goes on to make the following statements: "Twenty years of research has convinced Mr. Robinson that motoring is a health hazard. Mr. Robinson has graphs which show quite dramatically an almost perfect relationship between the increase in heart deaths and the increase in motor vehicles. Similar relationships are shown to exist between lung cancer, leukemia, stroke and diabetes" (55). In order to deal with statistical problems like the one posed here, we must decide where they should occur. Clearly statistical studies, and being able to make sense out of them is something that is relevant across the curriculum. You need to understand statistics in social studies courses (polls, economics, history, and others). In this particular case, it may also be relevant in health-is there some sort of statistical downgrade in the health of the human population? It could also be relevant in auto mechanics-is there something in the car that is causing this problem? And, of course, it is relevant for mathematics in the case of data analysis. So, this problem posed in the article is one of many possible statistics that are thrown at the general population on a fairly regular basis. In order to have a learned population who understands at least some of the statistical methods behind the research that is being presented to you. The article mentions three-step hierarchical system for being able to interpret such statistical studies as this one. This hierarchy is "1) A basic understanding of statistical terminology; 2) An understanding of statistical language and concepts when they are embedded in the context of wider social discussion; 3) A questioning attitude that can apply more sophisticated concepts to contradict claims made without proper statistical foundation" (54). In order to measure how a certain group of students (grades 6-11 in Australia, England and Singapore) were adept at the statistical knowledge they need in order to adequately deal with a thing like this study when it came across them, they "were given two tasks: 1) To draw and label a sketch of what one of Mr. Robinson's graphs might have looked like; 2) To consider questions that students might ask about his research" (55). The article goes on to show how various students had or had not obtained a certain level in the hierarchy. Some failed to reach the first step, by instead of drawing a graph, drew a picture of a car or a circle. Some drew little squiggly lines or other "data" that had nothing to do with the stuff in the "study." Finally, they get to those who are asking questions like: "Who did he research, [for example], smokers, the poor, people that live near growing industry? The age of the people? Their eating habits?" (57) since these undoubtedly would shed some light on the very questionable statistical methods used in the study.
I have become somewhat enamored with statistics since taking Math Stats this past fall. In fact, I would, without doubt have been one who wouldn't have obtained the third level of awareness-though I hope I could plot a linear graph. Anyway, one reason why I've come to appreciate statistics so much is that like another one of my favorite classes-Differential Equations-it does an excellent job of mirroring things in the real world, and, unlike Diff Eq, Statistics doesn't require an extraordinary background in Calculus and Linear Algebra. In fact, understanding statistical studies really requires very little math, but it is a vitally important thing for people in this country to be aware how to do, so that they don't get tricked by all of these polls and other statistics that are often flying around in our media.
Keywords: Algebra, Connections,
Ref: Jacob6
Author(s): Worrall, Laura J.; Quinn, Robert J.
Date: 2001
Title: Promoting Conceptual Understanding of Matrices
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 94, No 1, p. 46-49
Reviewer: Jacob
Date of Review: 3/15/02
Many times when we learn Linear Algebra, especially in high school, the operations of matrices is reduced to one short little unit that involves very little other than arithmetic processes performed upon them—there is little conceptual understanding as to why we are doing what we are doing. Basically, we have no conceptual idea as to what matrices are good for rather than for making us perform rather tedious operations. This article comes up with some very important statements about the need to be able to apply matrices to data. The article basically consists of a lesson dealing with the numbers, costs, other production data of toy motorcycles. The students are then asked to used this data, in the form of matrices, in order to determine various quantities. It is decided that they should first perform the operations by hand, in order to get an appreciation of them (in this particular case, that isn’t bad since the numbers aren’t that complicated), especially in how units will cancel to get answers of the desired units. After the initial computations are complete, the students may use graphing calculators to solve them. Another key aspect of the lesson is the need for the student to communicate what it is that they are doing, and why they are doing it.
I think that the article’s initial point that very often matrix problems are very much arithmetic computations. Matrices are one point where I really believe in the necessity of using calculators. The mechanical operations on matrices can become tedious at best, and other than yielding important information about units are largely present solely for the purpose of having to be done. I think that many Linear Algebra type of courses, even those in advanced high school Algebra classes are moving towards the use of matrices as very handy devices for organizing and categorizing data. Personally, I think that the key to learning matrix operations itself should be founded in applications, rather than just a random array of numbers. If this were done, much of the present thoughts of futility involving matrix operations would be reduced or eliminated if students would be able to see right away how they are useful as a means of organization.
Keywords: Standards, Communication
Ref: Jacob7
Author(s): Ward, Cherry D.
Date: 2001
Title: Under Construction: On Becoming a Constructivist in
View of the Standards
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 94, No. 2, p. 94-96
Reviewer: Jacob
Date of Review: 3/15/02
Constructivism is something that is often misunderstood by many people. So, first we must have an idea of what exactly constructivism is. It is, according to Confrey, "a belief that all knowledge is necessarily a product of our own cognitive acts." This article mentions several key ways in which teachers can institute constructivism in their classrooms. First of all, teachers need to recognize that under many situations, students will construct knowledge that may be specific to a certain situation, i.e. not generalizable. It is then the teacher’s responsibility to come up with problems that would cause the students to rethink their constructions of knowledge. One important idea of this is for teachers to recognize that students constructions may differ from their own—this is naturally the case since constructions are built upon previous knowledge. Furthermore, it is the teacher’s responsibility to make sure that the students are encouraged in these alternate constructions as long as they are mathematically sound. The article also mentions that it is very important to have communication (going in both directions) from the student to the teacher. An important part of this is for teachers to emphasize process over arriving at the "correct" answer. The article goes on to cover several examples by the author in her class: using spaghetti noodle triangles to represent sine and cosine functions from a unit circle, and examining the difference between sine and cosine and tangent graphs that have amplitude, period and phase shifts from their parent graph. The article concludes by listing several websights that teachers can go to in order to acquire lessons, ideas, or to "chat" about various constructivist ideas. These websights are www.nctm.org, www.pbs.org, www.forum.swarthmore.edu, and www.ti.com.
Personally, I think that, as stated in this manner, constructivist is a good thing. I still find myself somewhat weary about such ideas, preferring to focus on applications of ideas once the ideas are developed rather than playing bizarre little messy games with spaghetti noodles. However, I think that it is absolutely necessary that teachers and students communicate (so that different representations are not considered "incorrect"). Furthermore, I believe that the thinking process at arriving by an answer is much more important the arriving at the "correct" answer—after all, we can all make occasional arithmetic mistakes, even those who are supposedly proficient at it, and one should not be judged solely on that basis alone. If the error is one in computation rather than understanding it should not be as serious.
Keywords: History
Ref: Jacob8
Author(s): McNeill, Shelia A.
Date: 2001
Title: The Mayan Zeros
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 94, No. 7, p. 590-592
Reviewer: Jacob
Date of Review: 3/17/02
Imagine life without the number zero. We cannot. However, zero was not always a number; as a matter of fact, it appeared somewhat later than most other numbers did. The number zero was developed by both the Hindus and the Mayans. The article deals with the development and usage of zero in the Mayan number systems. First of all, the Mayans had at least two different number systems, the primary ones being the Head-variant glyphs and the dot, bar, shell method. The latter was the simpler form, with the former being used to represent units of time (which the Mayans had a very elaborate setup) which the dot/bar version was not suitable. Both of these two representations had a symbol for 0—some god picture in the glyphs and shells in the dot/bar/shell system. The Mayan system for general counting was strictly base 20. The Mayans, perhaps more so than many prehistoric cultures were very fond of time—they had it down to a fairly strict science, and their number system was primarily created and used for the purpose of this calendar. The system was called the long-count. A bizarre peculiarity about this system is that things start to go a little strange by the third order of units. At this point in time, they used 18x20 rather than the 20x20 that it should have been. The main development out of the Mayan system which involved zero (and still is a major part of our zero containing system today) was the creation of their remarkable written place-value numeration system with a genuine zero.
Zero is something that has become so "there" to us that it would be almost impossible to imagine a world in which it doesn’t exist. However this is something that actually did happen, and it is good to know history. I think that in many cases, history is something that is very much glossed over in mathematics. We tend to be worried about just getting down to business, and not what originally precipitated this business. However, I think that we should make some effort to at least acquaint our students with the fact that mathematics has history involved in it too—even though we no longer believe in doing things with an abacus.
Keywords: Discrete, Teaching Strategies, Calculus
Ref: Jacob9
Author(s): Allen, Lucas G.
Date: 2001
Title: Teaching Mathematical Induction: An Alternate Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 94, No. 6, p. 500-504
Reviewer: Jacob
Date of Review: 3/17/02
How does one teach the topic of mathematical induction? Students run into the problems of not possibly being familiar with sigma notation, and problems dealing with expressions in recursive or closed forms. Another problem that students may well encounter is difficulty attempting to come up with a function (be it in either closed or recursive form) from a list of data. It is necessary to acknowledge that all students will not find these methods easy or difficult in the same way—some may be more comfortable the former while others are more comfortable with the latter. This new approach to induction is based upon that—"to use induction to prove that two formulas, one in recursive form and the other in a closed or explicit form, will always are for whole numbers" (500). The article is structured as a journal around the author’s experience teaching this information in a precalculus class. The article mentions the validity of the idea that functions that come from tables can be represented in a number of different ways. In fact, there is even a connection between the additive term and the derivative. The end result of this little experiment—the author was a researcher, not the actual teacher—was that students were quite good at applying the induction algorithm to solving linear functions, but they seemed to have some difficulty with exponential, and perhaps not as deep of an understanding of what was going on as would have been hoped for by the author. The author concludes with a concern that of this particular study—the problems that the textbook had on induction were not that varied, thus the students failed to get much exposure.
I’ve finally gotten the hang of induction myself just fairly recently, having now encountered it in my forth college math course. On first glance, mathematical induction is something that looks quite simple (and indeed if everything goes right, the proof usually are). I agree with the author that the scope of the problems herein are very short-sighted. Mathematical induction is a much more powerful tool than just determining if two functions taken from a table and expressed in closed and recursive form are indeed identical. It seems to me that it isn’t only the examples of the text—dealing almost exclusively with linear—but also the premise of this "new approach" that seem to be limited, and not getting the close to the full power of mathematical induction. Of course, even if it were just dealing with linear functions, I would have liked to have a little taste of induction before ERA, perhaps with simpler ideas so that the transition into the more complicated and powerful aspects of mathematical induction can be explored. So, basically, I think the idea of identifying the similarity between recursive and closed form functions is a good place to start for induction arguments, it just could go a lot farther given the time and other suitable conditions.
Keywords: Discrete, Problem Solving
Ref: Jacob10
Author(s): Gannon, Gerald E; Martelli, Mario U.
Date: 2001
Title: Discrete Dynamical Systems Meet the Classic Monkey-and-the-Bananas
Problem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 94, No. 4, p. 299-301
Reviewer: Jacob
Date of Review: 3/17/02
I think before we get too far into this problem, it would be a good idea to say just exactly what the classic monkey-and-the-bananas problem is. In sort, the problem says that there are some number of sailors—say 3—who become stranded on this island, and they have nothing. However, they find that there are bananas growing on the island, so they pick them, and put them all in a pile to divide up the next morning. However, these sailors were suspicious people, so, each one of them, in order, woke up during the night, and separated the piles into three equal shares, and hid one of them. He found out that he had an extra banana, so he gave this to a monkey (who was apparently hanging around camp). Each of the sailors did this in turn. So, if there were three sailors, what is the smallest number of bananas that they could have had at the beginning? The answer is 79, which can be arrived at using a either a guess and check method, or else a rather involved situation of five unknowns with four variables (for those of you saying, hey this can’t be done, it can because we know we are looking for the minimum natural number that would satisfy it—we are seeking a unique solution out of the infinite that we have). Ok. So, this works alright with three sailors, is a little worse but still doable with four, and can even be done with five. However beyond this, the process becomes very tedious—we can’t use nice poly solve programs because there isn’t a match in the number of variables to equations. Therefore, it would be nice to come up with a more efficient way of solving the problem—this is where discrete dynamical systems comes in. If we let x0=the original number of bananas, we can denote, in the case of three sailors, x1, the number left by the first sailor as (2/3)(x0-1). This intersects the line x0=x1 at (-2,-2), so we can rewrite the expression as x1=(2/3)(x0+2)-2. We continue making substitutions in this manner until the n+1st sailor. In point effect, what the discrete dynamical systems is doing to this problem is solving it by bringing in a mythical sailor who does one more division of the bananas. This system is generalizable in a very nice way. In fact, it can be shown that the minimum whole number of bananas can be found by x0=nn+1-(n-1) where n is the number of sailors.
At first glance, this problem is relatively hard to solve using systems of equations—primarily because there isn’t a sufficient number of them. And, to be quite honest, the discrete dynamical systems can be somewhat complicated too. My only concern with using this is that it would be absolutely necessary to know how these first few connections are made, otherwise, although it is an interesting fact, knowing that with five sailors the minimum number would be 1021 is quite mathematically useless. However, once we understand what is occurring in the process, it is ver very sound process to use in order to solve this seemingly complex problem, and once we’ve come up with the general solution, we can easily determine the number of bananas necessary for any number of sailors.
Keywords: Technology, History, Research
Ref: Jacob11
Author(s): Hege, Hans-Christian (Ed); Polthier, Konrad (Ed)
Date: 1998
Title: Video Math Festival
Journal or Publisher: Springer-Verlag
Volume, Issue, Pages:
Reviewer: Jacob
Date of Review: 4/15/02
This video primarily explores computer graphics and their relation to high level mathematics, such as complex fractals, knot energies, and fluid dynamics. These are relatively complicated ideas, and the video segments are so short that we have no idea what is going on. For the most part, we are just looking at some pretty computer generated images and wacky music. There are some elements of this video that are realizable to the normal person, and somewhat interesting such as Fibonocci and the Golden Mean, the Story of pi, and the Theorem of Pythagoras. These give interesting little tid-bits of history, but are still very insubstantial. Fibonocci and the Golden Mean is perhaps one of the more historically interesting of the pieces, covering how Fibonocci numbers occur in a wide variety of places, from the biology of a flower to the architecture of the ancient Greeks and Egyptians. The Story of pi deals with excellent means of arriving at the value of pi, like cutting a concentric circles and spreading them into straight lines to create a triangle. Similar move triangles around graphics were used in dealing with the Theorem of Pythagoras. Another piece that was somewhat advanced (but at least they said something) was the one The Shape of Space which posed the question "What is the shape of the Universe?" My memory is failing me, but one of the interesting ideas comes up when we look at a universe that is in a doughnut shape—some of the light we see coming from the night sky may well be that of our own sun that has traveled around the universe.
The video is extremely lacking in intellibible information. I really don't recommend it at all unless you want to look at short little clips of computer graphics where you have no idea of what is going on or what they are representing. In this sense, this video is somewhat like other math/computer graphics videos—there is an overemphasis on what the super computer can draw and a real lack of emphasis on any of the mathematics going on—unless you are so familiar with the topic that you can clearly figure out what is happening without any commentary to go along with the pictures. The history ones were the most informative and useful, but they too left a lot to be desired—can they really successfully cover the ideas in 3 minutes?
Keywords: Number Theory, Statistics
Ref: Jacob12
Author(s): Flashpohler, David C.; Dinkheller, Ann L.
Date: 1999
Title: German Tanks: A Problem in Estimation
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 8, p.724-728
Reviewer: Jacob
Date of Review: 4/18/02
An important concept in the Number and Operations part of NCTM’s Principles and Standards is that the student is able to estimate, and to be able to determine if the estimation is indeed reasonable. The article mentions several problems that in the past have created problems with student’s estimation in statistics courses. The "first of these is that the examples are contrived; the second, the usual estimates discussed are intuitive" (724). The article then goes on to discuss the problem of determining the number of tanks that the German’s had based upon what the Allies were able to capture. It was determined that these tanks were numbered in sequential order, and the problem lists several numbers on the tanks (58, 20, 74, 8), and the students are asked to find out how many tanks the Germans really had. This particular article notes five possible solutions that the students came up with: 1) doubling the mean; 2) doubling the median; 3) doubling the midrange; 4) equal first and last intervals; 5) average interval length. The students are then asked to estimate which of these is the best solution based upon what is reasonable and most effective. We can see that the first two have major problems if the highest tank number is significantly greater than the rest—they’d lead to an answer saying that there were less tanks produced than were known to be produced (since they were numbered sequentially). The article then goes on to discuss random-number generators—this experiment is to make sure that, like in the preceding case, these solution ideas are not contingent on the sample population. The article also deals with maximum likelihood estimators and minimum variance unbiased estimators (the general conclusion is that in this case the MLE is not quite the best solution because the MLE is the maximum observed tank number—I know this distribution has a name, but its just too hot to think right now). The article concludes with the suggestion that students write in their journals to show understanding of what has happened in the discussion regarding choosing estimators.
As I’ve noted a number of times, I like applied statistics problems.
This problem is a great way to deal with estimation in statistics—does
the estimator give reasonable answers as I’ve pointed out above. I think
that the article also points out a problem in statistics that offers connections—any
decent statistic problem should do this. The article also deals with how
this problem can be dealt with at a wide variety of levels which is good.
It can be pared down to finding estimators, and estimating their reasonability,
and it can be extended all the way to finding MLE’s and what they call
"minimum variance unbiased estimators" (I think I remember these by a different
name) which can well be a major topic of an advanced statistics course.
Keywords: Assessment, Communication, Number Theory
Ref: Jacob13
Author(s): Stylianou, Despina A.; Kenney, Patricia Ann; Silver,
Edward A.; Alacaci, Cengiz
Date: 2000
Title: Gaining Insight into Students' Thinking through Assessment
Tasks
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 6, No. 2, p. 136-143
Reviewer: Jacob
Date of Review: 4/18/02
The article raises a point that I never really felt existed in the mathematics classroom—students showing their work on exams. In my experience, the only time that I haven’t had to show any work is on those standardized tests that really have little to do with math, even though they may contain math. The key to this article is how having students show work and give explanations can give teachers a great deal of understanding of where the student is having difficulty—much more than could possibly be gleaned out of having tests where one is looking for just answers. A student may arrive at the correct answer through totally inappropriate means (which is usually quite rare), or, more seriously, a student may have made one minor arithmetic error (if arithmetic is not what your assessing), and otherwise perfectly understood the problem. Of course, the student could have arrived at the correct answer in the proper way, or on the other hand, a totally incorrect answer with a number of errors in computation and understanding. However, no matter where on this continuum the student falls in the particular problem, both the teacher and the student will be better off with the student explaining his/her processes. The teacher is better off because she/he is then able to determine where the student is having success or difficulty. The student is better off because clarifying in writing as a wonderful way in which to solidify understanding—an adequate explanation would also go a long way to giving partial credit on a problem that the student got wrong. The article examines student’s solutions and explanations on two problems. One of these is called Marcy’s Dots—which is a pattern recognition problem within an array. The other is the Bus Problem—how many buses of a certain size (40 passenger) are necessary to transport a certain number of people (532). In all of these problems the article stresses how important student’s written responses—their thought processes are—no matter if the answer is right, wrong, or surprising (which fit either the problem’s given circumstances or intuition—use a van rather than another bus) "the student’s communication gives [teachers] a window into their thinking about mathematics" (143).
As I said earlier, I don’t think that I’ve ever taken a math test where I didn’t have to show my work. However, neither can I really ever recall having to explain my thinking process in words (probably I went through in the middle of this trend—show work but why write). However, I definitely see the advantage to having students write about their answer and how they arrived at it. Even though showing work does give a lot more in-depth look into student’s thinking processes, it s still somewhat unclear—we can see the mathematical path which the took, but may be unsure as to why they took that path, especially if their answers fall into the wrong or surprising category. The idea of having students actually write—even though it is math class—will, I believe, greatly enable teachers to have a greater understanding of what their students are thinking, and thus be able to focus their instruction in ways that will best alleviate these misunderstandings. (Just as an ending note: this is a really good article and applies well beyond middle school to all levels of schooling. It is also humorous.)
Keywords: Algebra, Teaching Strategies
Ref: Jacob14
Author(s): Feigenbaum, Ruth
Date: 2000
Title: Algebra for Students with Learning Disabilities
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 4, p. 270-4
Reviewer: Jacob
Date of Review: 4/18/02
The article deals with how one should accommodate an algebra classroom for students that have disabilities. The initial question that is raised is do students with learning disabilities really need to take algebra? The answer seems, unquestionably yes. The author is an instructor at a community college, and goes over a number of ways to make an algebra classroom more fitting for students with disabilities. I’ll briefly summarize these ideas of hers. Don’t have class periods that are too long. Get the students up and writing on the board—this has the double effect of giving those with ADHD something to do, and giving students with graphomotor problems more space to work problems. The class is small enabling one-on-one attention. Extended time is given for the tests in class. Furthermore technology helps out with an electronic blackboard that can print whatever is on it. She feels that "support and encouragement enable [her] students to believe that they can be successful in an algebra class" (272). She also gives a number of teaching strategies that she uses in the algebra class. The first of these is that she treats algebra is a foreign language. She also emphasizes the distinction between factors and terms, and addition and multiplication, and the importance and use of parenthesis. She makes use of the "imaginary 1"—that is placing a one is as a place holder for the coefficient. She makes a big deal out of translating the language of mathematics into English—even emphasizing where pauses should be placed. She requires a one step at a time process for solving problems. There are also a number of personal learning strategies that she mentions, such as using different colored pencils for various actions, talking out a process to self or others, changing variable names, and using templates. She also emphasizes the fact that skills and continuos review are necessary.
Before I begin my personal response to this article, I’m going to make a little note: I’m in Exceptional Child right now, and based upon this, I have questionable feelings on her approach. The first one is the issue of inclusion. The program which she is advocating is completely anti-inclusion. In fact, it is an exclusionary program, giving instruction to the LD students in a completely different setting than that which the other students receive the instruction. While I believe her point of the students not feeling "like the ‘dummies,’" I find myself saying that this is going against the very model of inclusion—we must be able to come up with means to instruct LD students in the regular classroom, undergoing the regular curriculum in the least restrictive environment in accordance with their IEP. I think that this statement of hers is largely immaterial. The instructional changes would work very well for a regular inclusionary classroom as well—go ahead and get the students on the board since this will give them a chance to be involved (just be cautious not to have just a single student at the board for too long while everyone else is waiting).
My other major issue with her strategy is in claiming that mathematics is a "foreign language with its own alphabet and grammar." In my opinion, this is a very bad move. The large number of LD cases are language based—they have enough trouble dealing with their native language of English, and the idea of foreign languages—be they Spanish, Latin, or Mathematics—may well frighten them to such a stage that their anxiety over past difficulties with foreign languages will completely freeze their mathematical ability. I think that this goes over into her emphasis on "add out" and "reduce" rather than "cancel." In my opinion, having the students, especially LD students, be able to perform the task and understand what they are doing is the important factor—who really gives a damn if they can remember the proper word. Such emphasis on a minor thing like that will undoubtedly cause problems with students whose LD is in the area of language. OK—this so far sounds like a total slam, but I agree with many of her modifications, and the idea that, especially for LD students, the classroom needs to be very structured. So, give it a look, but do so with skepticism since this author is teaching in a community college that obviously doesn’t emphasize inclusion in its curriculum.
Keywords: Problem Solving, Games, Communication
Ref: Jacob15
Author(s): Meyer, Carol; Sallee, Tom
Date: 1983
Title: Make It Simpler: A Practical Guide to Problem Solving
in Mathematics
Journal or Publisher: Addison-Wesley Publishing Company
Volume, Issue, Pages:
Reviewer: Jacob
Date of Review: 4/27/02
This book deals with problems solving strategies, primarily through their emphasis in a number of games. It is a plan that actually encompasses the entire academic year. The students first of all deal with working in their groups-of-four-simply the aspects involved with working in groups. Working in groups (which really have to be of four students-not three or five [what one does when they have an odd number of students I'm not sure] so that no one is "excluded.") There are three rules that govern the groups-of-four: 1) You are responsible for your own behavior; 2) You must be willing to help anyone in your group who asks; 3) You may not ask the teacher for help unless all four of you have the same question (p. 5). Once they have begun to master this (they work on it continually), they move on to dealing with logic problems and games. The ones specifically mentioned in the book are the Digit Place Game, Poison, and The Color Square Game. All of these problems work on developing student's ability to reason deductively. The text ends with techniques in problem solving. The first of these are categorized under the heading of Understanding the Problem. This we acknowledge is necessary because it won't be possible to solve a problem without understanding it. Understanding consists of 4 parts: "1) Restate the Problem; 2) Clarify the question (includes the hidden question); 3) Organize the information by a) getting rid of unneeded information and b) find needed and assumed information; 4) Check your solution with the problem [i.e. make sure that it makes sense]; 5) Evaluate the solution and rethink your methods when necessary" (p. 43). There are also five steps to solving a problem, some of which may work on certain problems and not on others, and some which may be better than others at solving specific problems. "1) Find and solve a subproblem(s) or hidden problem(s); 2) Use a picture, a diagram, manipulatives, or dramatization; 3) Use either numbers, estimation, and/or fewer steps; 4) Find and use patterns; 5) Work Backwards" (p. 43). The book ends with a large number of problems that make use of these strategies in increasing difficulty-there is an elaborate scheme that I don't need to go into. There is also another section on choosing the right strategy to solve a problem if there is sufficient time in the year (it should be specified in the earlier parts which to method to use).
This book is one that I have mixed feelings about. I'll start with my dislike of it, which starts with the very cheesy means by which they insist on working in the groups-of-four (this is a compound word, and must be held together). This causes a semantic problem for me. There is also the issue of what to do if there are an odd number of people-even but not divisible by 4 is solved by creating a group of 2. Although I agree that it may be best to create a situation where a student isn't shut out of the conversation, this is not always a feasible thing to do, and the group construction technique was not discussed. The other problem I have was that their initial discussion of group processes seemed to be along the lines of "oh-we all have to figure out how to get along, and be the best of friends, and have fun" thus leaving the math really as a side affect of the whole thing rather than being the cause of the group formation in the first place. However, once we got out of the group structure issues (which contradict Eunice Peabody's claim that the most efficient group is of 3 people), and into the actual problem solving strategies, I think the book becomes very good. It does a good job of showing how we would expect students to respond, and what to do with their responses. It shows, through its vignettes a teacher who is very skilled at dealing with group management and questioning. Furthermore, the problems are also very interesting (they could perhaps be a little more relevant-a story line going through the text is of a captain of a space ship who lands on this planet populated with furry worms). The concern for me is the level. Many of the problems-once you figure out what to do-are reduced to arithmetic manipulation (you know, adding, subtracting, multiplying). Thus the level of problems would probably be primarily limited to middle level. However, there are some that are relevant beyond this, like the traditional combinatorial problem of how many handshakes would there be if everyone in a room of n people shook hands with everyone else. So, as an overall impression, the book is really not that impressive at creation of groups, but does a wonderful job as a general problem solving text.
Keywords: Geometry, Standards,
Ref: Jacob16
Author(s): Geddes, Dorothy; et. al.
Date: 1992
Title: Geometry in the Middle Grades
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: Addenda Series Grades 9-8
Reviewer: Jacob
Date of Review: 5/2/02
This particular addenda book deals with geometry as it is presented in the middle grades—obviously by the title. The addenda follows a fairly similar path to Principles and Standards of Mathematics (2000), emphasizing the importance of visualization and spatial thinking in the middle grades. This is primarily accomplished through the use of models. Furthermore, they emphasize the need to use physical manipulatives in order to get students involved in a lesson and to build schema. The book also emphasizes the problem solving aspects of geometry, and how geometric representations can be used to solve certain problems. They emphasize the use of cooperative learning and open-ended questions. The "reading" part of the book comes to a close with the discussion of the van Hiele model of geometry. The van Hiele model of geometry claims that students learn geometry through a series of five sequential phases: Visual, Analysis, Informal deduction, deduction, and rigor. Presumably students should attain the first three of these very well by the time that they complete 8th grade. The fourth one is a primary topic of high school geometry, while the fifth is something that very few people ever achieve—they have to be like math majors or something. The remainder of the book contains black line masters for activities. They are set up in three parts that are essentially sequential: 1) Two- and three-dimensional geometric concepts; 2) Relationships among properties of shapes including angle sums; 3) transformation geometry; 4) enrichment topics. These activities are primarily involving physical manipulation of objects created by the student or teacher. Furthermore, the students use technology in several of the activities.
My personal reaction to this is very much like the rest of my reactions to standards like material and stuff that deals with cooperative learning and that sort of stuff. I’m not a discovery learning person, and this is my personal learning style. Therefore, I see many of these activities as not the most efficient way to accomplish learning. Part of this has to do with the physical manipulative idea—it gets students physically involved with is good for a variety of reasons (the most important of which I think is for differentiating learning to those who have different learning styles or are AD/HD and can’t sit around for very long). I’ll admit my bias at being much more of an applied type—I don’t need to "discover" relationships for myself, but rather I need to have some reason as to why it is important. Namely, I think that answering the often asked question of "What am I ever going to need to do this for?" is far more important than discovering something—and still, even though the schema is formed, have no idea as to WHY you learned it. That said, this really isn’t that much of a constructivist text—at least as far as the text part of it goes. It offers some interesting ideas on van Hiele (although the article we read for class goes into it in a lot more depth), and the use of visualization, models and manipulative that I think are very valuable.
Keywords: Issues, Technology, Keyword 3, Optional...
Ref: Jacob17
Author(s): Thomson, Anthony D.; Sproule, Stephen L.
Date: 2000
Title: Deciding When to Use Calculators
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol 6, No. 2, p. 126-129
Reviewer: Jacob
Date of Review: 5/3/02
This article discusses a framework on when calculators should and should not be used in the classroom. For the most part, they advocate that the students always have access to the calculators—whether or not they will choose to use them ends up being very much up to the students. The key points that they have decided along the nature of calculator use is that teachers need to decide on whether to focus on "the needs and abilities of their students or their own goals for introducing a mathematical activity" (127). They then continue and ask the question of whether or not the activity is feasible to do without a calculator—if it is, then perhaps they shouldn’t use a calculator, but if it is, then they should. The key point of this, I think, is when they mention that "using a calculator can free students from laborious calculations and allow them to focus on more challenging mathematics" (127)—even if a calculator is not absolutely essential to the process. The goal-oriented section is broken down into two pedagogical goals: "1) for students to find a computational solution and 2) to engage students in problem-solving processes" (128). In both of these situations, they seem to consider that use of calculators is, if not essential, quite necessary. They fortunately go on to modify their earlier statement which seems to imply a mutual exclusivity between the students’ goals and those of the teacher, in combing the ideas of essential/nonessential and process/product in such examples as decimal expansion, statistics, nines game, and probability/ratio/proportion problems. They finish by saying that it rests with the teacher to decide how much to use the calculator in his/her classroom.
I have some concerns with calculators and technology in general, as I have often stated. The crux of my argument rests upon the facts that students need to know what they are doing—if incorrect information is fed into the calculator, it will spit out an answer that is other than what you wanted. Hence students need to be able to determine the reasonability of the answer given to them by the machine—in other words, they need to know the processes very well. However, I’m a firm believer in the advantageousness of technology. Unless we are, say, learning the skill of long division (which, as with all skills, I think is important), calculators should be used since insisting on people always doing it by hand is both laborious and leaves chance for error. As far as this specific article goes, I see no reason to attempt to compute the decimal expansion of 1/7—it may does have a pattern, but I’m not sure if finding this is worth while in the decimal motif. (An extension of this could possibly lead to the conclusion that 1/7 is not a rational number because the calculator does not show it to have repeating decimals—I’ve really heard this argument for a number that is rational.) So, in point effect, I believe that the calculator’s (and technology in general) job is to make our work easier. This by no means excuses students from understanding what is going on behind the calculator’s screen, and if needed, they should be able to perform the indicated operation as well as the calculator can.
Keywords: History, Standards, Keyword 3, Optional...
Ref: Jacob18
Author(s): McComas, Kim Krusen
Date: 2000
Title: Felix Klein and the NCTM's Standards
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 93, No. 8, p. 714-717
Reviewer: Jacob
Date of Review: 5/3/02
Apparently there was this guy named Felix Klein who lived in Germany and taught at the University of Gottingen in the late nineteenth century and early twentieth century. This guy apparently had all sorts of ideas that were similar to those that are now imbedded in the NCTM’s Principles and Standards. His main importance is that he cared about how mathematics was taught, and was very influential in the field of math education. One area in which Klein is similar to the Standards is that he de-emphasized procedures and algorithms, instead focusing on the students developing an intuitive understanding of what was happening. Furthermore, he was very keen on the idea that concepts that are quite abstract should not be introduced that early in a student’s education. This goes along with the Standard’s idea about the use of manipulatives, especially in the middle and elementary grades. Some of Klein’s major cases of not being too abstract were in models to deal with geometric representation. In fact, there are a number of different ways in which one can deal with geometric representations, especially in algebra concepts. He also considered the coordinate system (in this case a Euclidean plane) to be a pegboard. He also created a means to deal with logarithms (remember they used to have to have huge tables to figure out what the log of some number was). Klein was also very much into connections, both those between mathematics and outside fields (and how they were taught) and between areas of mathematics—especially geometry and algebra. Furthermore, Klein felt that it was very important to have exposure throughout schooling, and to increase depth of understanding as one progressed. He was also into problem solving and the use of technology—although the technology of his day was nothing compared to what we have today.
I’m somewhat skeptical about this article—not the history that it provides in how Mr. Klein was working in all of these wonderfully pedagogical means that were way ahead of his time—but rather that it seems to me to be almost a NCTM give yourself a pat on the back—you’ve come up with the same thing some German mathematician did a century ago. In this sense, the article is almost propaganda since only a few sentences can go by without the author making a wonderful connection between Klein and the NCTM. Personally I would have preferred a little bit more into Klein’s methods, since they obviously seem satisfactory to the author, rather than focusing so much on how the wondrous NCTM has also come up with these great ideas. Just in closing, it is not my intent to bash Mr. Klein, NCTM, or the Standards in this; they each have a proper place, and I would have liked this article to focus more on Mr. Klein and not on how his practices aligned with the standards. It is fascinating to see how a historical figure could be so ahead of his time, and I would have liked to see more of this.
Keywords: Issues, Communication,
Ref: Jacob19
Author(s): Fiore, Greg
Date: 1999
Title: Mat-AbusedStudents:Are we Prepared to Teach Them?
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol.92, No. 5, p. 403-406
Reviewer: Jacob
Date of Review: 5/8/02
The article deals with the issue of math anxiety, and how many people face it. This anxiety can create in them a case very similar to learned helplessness. They become convinced that they are unable to do mathematics even if this may not actually be the case because of some bad experience that they have had at some point in the past. This bad experience has is normally at the hands of either a teacher or parent inflecting verbal or physical abuse on the student for their inability to do mathematical tasks that the elder considers to be "easy." Students are often very terrified of the subject of mathematics because of this-in fact, math anxiety can often happen even with teachers-especially elementary teachers. The article is about this, and examines the cases of two individuals who were in a community college algebra class and were in their forties. The people who were having these problems would basically shut down in the math class. To alleviate this problem, the instructor decided to have them write a paper entitled "Math and Me," in which the class would explore their past relations with mathematics. In the case of one of the students, a third grade teacher had terribly embarrassed her by having her stand at the board from 9:15 until 3:30 in an attempt (futile) to solve a problem (without the teacher helping). The other student had a parent who verbally and physically abused her when she failed to "get it." The primary thing that this article deals with is the immense power teachers have over students, not only grades, but also their self-esteem, confidence, and eventually even areas of interest tend to be towards places where we have had more positive teacher experiences than negative. The author mentions the importance of using encouragement and "positive talk."
Yes, people who feel that math is not their favorite subject area are something we will have to face. However, a thing that I think is very important is determining why this is not the case-the essay that the author had his students write does a good job of this. People are usually more willing to express themselves in writing than they are talking to them, so this enabled him to get to what was bothering his students and affecting their performance his class. Once he was able to figure out what was going on, there was a chance that he could do something to alleviate this problem. Perhaps they would never be able to love math as they might have without the negative experience, but at least they wouldn't be terrified by it. The author also mentions something else that I think is very important, and that is the need to deal with the emotional health of the student, since without solid emotional health, no matter what intellectual aptitude there may be for the subject there will be no learning happening with that student-this goes back to the issue that as teachers we must first address the real needs of the students before they can learn, and this writing project is a good way to get to know what those needs are. This way, we will be able to reduce, hopefully, some of the math anxiety that these students have.
Keywords: Algebra, Probability, Problem Solving
Ref: Jacob20
Author(s): Young, Paula Grafton
Date: 1998
Title: Probability, Matrices, and Bugs in Trees
Journal or Publisher: The Mathemtacs Teacer
Volume, Issue, Pages: Vol. 91, No. 5, p. 402-405
Reviewer: Jacob
Date of Review: 5/8/02
This article deals with a problem using probability and random walks in order to determine the random motion of bugs living in some trees around a lake. There are several levels of depth to the mathematics and the modeling involved in this problem. There is also some technology issue such as necessary to multiply matrices and creating a random number generator. In addition to the probabilistic idea of random walks, the article brings in ideas relating to matrices in the person of Markov chains, and the initial state matrix and the transition matrix. The initial stage matrix is one that shows what percentage of the bugs that were living on each tree at each time. The transition stage matrix shows how, between observations, the bugs have moved (they only have the option of moving one tree clockwise, one counterclockwise, or staying in the same tree). Thus, for the initial model, which consists of three trees, the transition matrix would be a 3x3 matrix showing how the bugs moved between observations. This gives us a mathematical model of the situation. Furthermore, the Markov chain idea is iterative. The resultant for the first observation becomes the new initial state matrix. There are a number of ways that this can turn out depending on the transition state matrix and the first initial stage matrix. We can get into a steady state, in which there will, in this case always be 1/3 of the bugs per tree, or we can get into a case where all the bugs will be on one tree, or we may end up having a situation in which there is no discernable end pattern. The article ends with a set of problems that go along with it.
I'll admit that I'm a little biased when I went looking for this article-I'm trying to fill up my resource file for my unit plan on, basically multiplying matrices. As an applied mathematician, I find that this example of Markov chains to model population dispersion is a great way to use matrix multiplication, and to make it something more valuable than just following an algorithm. This is indeed a really cool problem that can have a great deal of depth. We can alter the initial conditions, to get ourselves into very complicated interactions. I haven't really looked at it, but it wouldn't surprise me if something like classical chaos could come out of this model as well (as it often does in population models with the correct initial restraints. We can also make the problem more complicated by adding in more trees and more complicated movement behaviors of the bugs. Indeed this is a problem that we can explore in a wide range of depths from the very superficial getting a hang of it idea, into one that can be dealt with as a very complicated mathematical model of an ecosystem.
Keywords: Algebra, Technology,
Ref: Jacob21
Author(s): Pagon, Dusan
Date: 1998
Title: Performing Operations with Matrices on Spreadsheets
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 91, No. 4, p.338-41
Reviewer: Jacob
Date of Review: 5/11/02
The articles title is very self explicit. The article deals with performing operations with matrices on spreadsheets. Basically fairly simple operations such as addition, scalar multiplication, multiplication, determinants and inverses of matrices are covered in the article-although it is hinted that more complicated operations may be possible. The article has a fair amount of technical babble like how to enter stuff into an Excel spreadsheet, which I don't think is really worth going through. The real question to ask is why should we use spreadsheets rather than some other means to determine these matrix ideas. The primary reason cited in the article is that "graphing calculators and most specialized mathematics software, although very efficient, usually do not give students this opportunity, since they work like a 'black box'" (338). This is cryptic, so I'll try to interpret what it means based upon the rest of the article, and how you perform the operations in other software, such as those in TI calculators or Maple and the like. First of all, the primary criticism of the latter is that, as Pagon puts it, they perform their operations in a "black box." The reason why I believe that he used this terminology is because when dealing with such pieces of technology, all we do is enter the matrix, and then tell it what to do. For instance, on a TI, if we want to find the inverse of a matrix, we just time A-1, and it spits it out. The problem with this is that we can do this, and have no idea how the machine came up with that answer. (It is the same with all matrix operations-they occur in the recesses of its computer brain.) On the other hand, Excel has no idea what a "matrix" is-it merely performs operations on numbers. Therefore, a command like A &* B (the Maple command for multiplying matrices) would be meaningless to Excel. In order to get the spreadsheet to multiply matrices, we have to "teach" it how to do so, by inputting the expressions for the appropriate cells. Therefore, what this does is that it makes the calculations easier, since Excel is as well versed at simple arithmetic as anything, including Maple or your TI; however, unlike the latter, you need to know the algorithm for computing the desired outcome in order to enter it into Excel since it won't do it for you.
My response to this is but this: Knowing the algorithm is necessary, but
once that is known, why not use something that does the calculation (and
knows how to do it) for you. Ideas for adding, scalar multiplication, and
multiplication are easily generalizable. So, my idea would be to have them
do enough simple ones by hand-it's easier than entering the data into a
spreadsheet-so that they know the algorithm and how to generalize it. Then,
by all means use technology available, if it would make it easier (in some
cases, it is easier and quicker to just to it by hand). A similar situation
occurs with determinants. Here there is a slightly more complicated
algorithm, but one that is also relatively generalizable. I believe that
everyone who is taking a class involving matrices should work with doing
cofactor expansion to find the determinants. However, this to can become a
laborious process once the matrices get to be a fairly decent size if they
don't have a lot of 0's in them. So, once you've experienced cofactor
expansion and can do it if necessary, use the calculator. Determining
inverses is a slightly more complicated idea-especially once the matrix
grows beyond a 2x2 for which a simple algorithm exists. This is where I
think that using a spreadsheet would be most valuable, since you have to
come up with the algorithm (which isn't that easy of a task), and the
numbers can become complicated in a hurry. Hence you would be able to know
what was going on (rather than it being hidden in secrecy), but not be
boiled down in arithmetic.
Keywords: Proof, Connections, Problem Solving
Ref: Jacob22
Author(s): Redmond, Charles; Federici, Michael P.; Platte,
Donald M.
Date: 1998
Title: Proof by Contradiction and the Electoral College
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 91, No. 8, p. 655-58
Reviewer: Jacob
Date of Review: 5/17/02
Mathematical proof tends to be one of the more abstract concepts in mathematics and it is somewhat difficult to find a viable way to apply proof, and pure mathematics in general, to areas outside of mathematics. However, this article comes up with a very good way to try to accomplish this by posing the question "Of all possible collections of states that yield 270 or more electoral votes—enough to win a presidential election—which collection has the smallest geographical area" (655)? Of course, they do sort-of ruin it when they say that this is really quite meaningless both as a civics study and as a truly really meaningful one mathematically—a computer program can just spit it out for you. However, it does offer a means of making dealing with definitions and such things for constructing a proof. There are several definitions that must clarify in order to for the exercise to continue: If a state, A, is "better" than another, B, and B is a subset of O (the desired states), then A must intersect O. The contrapositive of this statement is also true. So, in order to continue with this, it is necessary to get organization. Create a matrix so that the columns are the states increasing in geographic area as we move to the right. The entries in the matrix are the electoral votes from the particular state. We can make use of this chart to show that if a state is in O, then every state to the left and below it is also in, and also if a state is not in O, then every state above and to the right of it is not in O. Some are easy—Alaska definitely out, New York, definitely in, but others are more difficult—Minnesota out (actually a somewhat involved proof), and Texas out (this is actually sort-of surprising since Texas is listed in the "fewest states needed to win the presidency"). Through application of the above definition of better and its contrapositive, it is possible to eliminate all but the following states, which give rise to the smallest geographical area necessary to win the presidency: DC, RI, NJ, DE, MA, MD, NY, HI, PA, OH, NH, FL, IL, CA, IN VT, NC.
Although the debocleness of the Electoral College has subsided since the last presidential election, it is still a valid issue that makes many mathematical connections to civics. My only concern with the article is that they seem to downplay themselves and their activity, which seems to me to be a great way to introduce proof and proof by contradiction since so many times these are such abstract ideas. It seems to me to be a good case of how to use definitions in proof, and to apply the ideas of intersecting sets and all sorts of other wonderful math-type stuff in a field that often seems fairly alienated from mathematics.