Keywords: Teaching Strategies
Ref: Jimmy1
Author(s): Leonard, Bill
Date: 1997
Title: Proof: The Power of Persuasion
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 90(3), 202-205
Reviewer: Jimmy
Date of Review: 2/18/99
High school students do not like proofs. I especially did not like the geometry proofs, because they seemed quite abstract. This article identifies some methods that help students grasp the concept of mathematical proof and how it works.
The author states more emphasis should be placed upon convincing others than on an absolutely spotless proof. A proof essentially means to convince someone, and if students do not place this objective first, problems arise. Students must learn to build strong arguments before they learn the exact rules of proofs. Students who simplify arguments or cannot understand when a proof is faulty should observe counterexamples or extremal cases. These should not be meant to discourage the student. Motivation for convincing someone must also exist. Everyone knows how difficult it is to prove something to someone who does not care.
Rather than teach a proof and show examples, ask questions about counterexamples or applications, then follow up with the proof. This way, students can see the importance of the proof more easily. This also builds excitement about the discovery of a rule, instead of the common application of a principle. "A thirst for water must precede enthusiasm for receiving a drink." Students should leave high school more with an understanding of why proofs are necessary, than how to set up perfect proofs. The author states two goals for students: the desire to question statements, and to understand how and why a given proof has a certain form. These qualities provide a deeper understanding, not just of proofs, but also of mathematical thought. The purpose of math proofs, the persuasion of others about a math statement, is that much clearer.
I thought this article had some beneficial examples, concerning angle trisection, consecutive composite integers, and the infinitude of primes. These are obviously helpful insights, the only trouble I can see is setting up enthusiasm for some of the more simpler geometric proofs. I do not think those always lend themselves easily towards student excitement, but maybe?
Keywords: Curriculum, Algebra, Geometry
Ref: Jimmy2
Author(s): Hirschhorn, Daniel B., Thompson, Denisse R., Usiskin, Zalman, Senk, Sharon L.
Date: 1995
Title: Rethinking the First Two Years of High School Mathematics with the UCSMP
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 88(8), 640-647
Reviewer: Jimmy
Date of Review: 2/22/99
This article highlighted some of the successful techniques of the UCSMP (University of Chicago School Mathematics Project) curriculum plans, which began back in 1983. It started due to studies back then which highlighted a need for different approaches to the curriculum. These approaches fit in well with the new standards proposed in the late 1980s, especially those of the NCTM. In particular, this article deals with the curriculum proposed by UCSMP regarding the first two years of high school.
To realize these proposals, certain coursework should be covered in the seventh and eighth grades. UCSMP recommends two courses, one called Transitions, which covers arithmetic, prealgebra, and preogeometry, and Algebra, a first course which incorporates lots of geometric concepts. These courses should cover a bit more than they have in the past; studies have found that under 35% of the content in seventh and eighth grade math textbooks is new material. It is also the belief of the UCSMP that a firm grounding in algebra and its relation to geometry is necessary to high school mathematics. Internationally, many, if not most, nations incorporate this amount of algebra into their middle school curriculum standards.
In high school, freshmen go through wildly new experiences, including those in their math courses. Another study has found that over 85% of the text in freshmen math books is new material. Thus few students without a strong introductory algebra course in middle school do well. Other new concepts adding to the difficulty of meeting high school math the first time are the ability to read math texts for learning, daily homework, and the introduction of new ideas much more frequently.
The courses recommended for freshmen and sophomores are Advanced Algebra and Geometry. Rather than seeing geometry as “the odd man out” subject in high school math curriculum, UCSMP views it as a natural complement to algebraic concepts. This echoes the fact that NCTM highlights geometry in several parts of its Standards. A point on a coordinate axis should be written as P=(x,y), rather than P(x,y); this is one example of UCSMP’s view towards incorporating algebra and geometry more than has been done in the past. Similar relations include area and volume formulae as functions, reflections and their counterparts in functional symmetry, and rotations and sine and cosine functions.
UCSMP also feels geometry should be taught with more generality. Teachers should start with coordinates and translations, which leads to more general definitions of congruence and similarity than is often seen in textbooks. In another part of geometry, proofs should begin simply, with emphasis on step-by-step ideas, then grow longer and more complex.
UCSMP also recommends a SPUR approach to all mathematics lessons. This means teaching students from four viewpoints: Skills, Properties, Uses, and Representations. Skills-oriented students learn procedures and rules for formulae, while Properties students are analysis-oriented: learning why a rule holds. Uses refers to students who apply math to the real world, and Representations students are those who enjoy viewing math concepts. Several examples of incorporating all these approaches in a lesson are given.
This curriculum approach seems quite sound. The SPUR approach particularly intrigues me. I think I will probably have difficulty teaching students who learn in different ways. The introduction of algebra and geometry by the end of sophomore year leaves students open to take a precalculus, discrete math class and a functional, statistics class their remaining two years: a good background for math or science majors. I think there are a lot of good approaches to setting up a math curriculum, it's difficult to decide which is the best. Every one has good ideas.
Keywords: Standards, Curriculum
Ref: Jimmy3
Author(s): Mumme, Judith; Weissglass, Julian
Date: 1989
Title: The Role of the Teacher in Implementing the Standards
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 82(7), 522-526
Reviewer: Jimmy
Date of Review: 3/15/99
The Standards obviously revolve around how a teacher implements them. The new ideas that these standards bring should neither overwhelm nor frighten a teacher. This article highlights ways to avoid some of the frustrations that could accompany a teacher implementing the standards for the first time. The authors answer some of the more common concerns that were voiced when the standards first came out.
The article starts by emphasizing that the standards do not represent a single way or method to teach mathematics, rather they should help teachers reflect upon their teaching styles. Teachers should realize they themselves drive the learning that goes on in schools. Textbooks cannot create new styles or methods of learning, only a teacher can change how a subject is taught.
Teachers just beginning with the standards should also realize that the standards represent a set of reflections that can help make them a better instructor. It can be difficult to try and change the curriculum at a school or teach in a new manner, so teachers should seek out another colleague enthusiastic about the standards. This helps relax the pace; implementing the standards can take a while. Effective new teaching methods do not develop overnight.
Once several teachers start teaching the standards, others will be more willing to join in the effort. Resistant colleagues should not be agitated, but left to observe the example the standards set. Hopefully a leader will emerge who can manage the other teachers' efforts and guide work towards implementing the standards.
One main point a teacher can start with is encouraging small group work. Discussions and writing among students, along with requiring justification of answers, creates learning construction that goes beyond former methods. The questions students raise are excellent tangent points to discuss. Students themselves often have the best ideas for topics to pursue in upcoming classes.
Meanwhile, a teacher can join planning teams and community-school committees to further the standards. The teacher should remember not to make this a top-down decision, however. Teachers at the bottom level should want to implement the standards, rather than feel forced into doing so. Those who do should set an example which allows observers from outside in the community to learn what is happening. Again, a teacher shouldn't worry about making this a quick process: all curriculum changes require time.
I found this article mainly a type of reassurance for teachers. It didn't offer too many breakthrough ideas. Instead it stressed the fact that all change takes time, and that those who rush change often annoy a lot of people along the way. I heartily agree with this. Other than that, the article mainly recommended the key notions of the standards and a few ideas about finding comrades who think similarly. It's too bad I haven't found an article in any math education journals discussing the downsides or wrong way to implement he standards (if there are any). I think that might be interesting.
Keywords: Geometry, Technology,
Ref: Jimmy4
Author(s): Chazan, Daniel
Date: 1990
Title: Students' Microcomputer-Aided Exploration in Geometry
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Novemer, 1990, 628-635 (sorry I can't figure out the volume)
Reviewer: Jimmy
Date of Review: 3/24/99
I have never seen geometry and computers as a natural mix. I never used computer geometry programs in school, and geometry intuitively seems to me to be too proof-based to understand through computer explorations. It appears that many students are now learning their rules and applications through this method, however. I guess I better understand them, eh.
The focus of this article is a piece of software called The Geometric Supposers, a precursor of Geometer's Sketchpad. It features geometric constructions and measurements. Students can repeat procedures thanks to a memory function. With these features, students can attack geometric problems with conjectures of their own, before looking in their textbook. As the Schoenfeld article points out, in this manner students feel as if they are truly discovering a mathematical theorem, instead of regurgitating an old idea just like every other student who has used the textbook.
Several skills will develop through the use of Supposers. Verifying allows students to determine whether a geometric theorem holds. After moving beyond this simple ‘check the answer’ case, conjecturing about new statements will appear more natural. Have students add more segments to a figure or ask them about the central ideas of congruence or similarity if they appear to be stuck. Once these skills are mastered, generalization and the posing of new problems can start. Ask students when a particular theorem holds, or if something is true for a different shape. This is described as the ‘what if not’ strategy from Brown and Walter (1983). Communicating now shows that students can help others understand; presentation of an idea requires a strong grasp of what that idea is. The author recommends lab reports for the teacher. Now comes proof of the constructions, the movement from the given to the conclusion. As a teacher, try to set up constructions in order so that if one is true, so is the next. This highlights for students what they want to prove. Finally, have students expand their insights into as many applications as possible.
With the use of computer applications such as Geometric Supposer, teachers can begin geometry lessons at the verification stage, as computer constructions are typically much faster and more accurate than student drawings. This allows a more logical and interesting approach to proofs in geometry that does not restrict students to repeating tried-and-true constructions found in every math textbook. A good approach, I think.
Keywords: Probability, Curriculum,
Ref: Jimmy5
Author(s): Watson, Jane M.
Date: 1995
Title: Conditional Probability: Its Place in the Mathematics Curriculum
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 88 (1), 12-16
Reviewer: Jimmy
Date of Review: 3/29/99
Conditional probabilities can be taught to high school students, even to students that aren't seniors. This article indicates several benefits and ways to do so. Myself, I never learned much about probability, even less about conditional probabilities in high school. Although NCTM and other standards resources place conditional probabilities late in curriculum development, these ideas frequently appear in the real world. Studying some of these real world applications is the focus of the article.
As early as grades eight through ten, conditional statements, applications in sporting data, notions of independence, and two-way tables with conditional probability can be taught. Examples of conditional statements abound throughout newspapers. "If you can't get to vote on 23 May, vote early" is an example of conditional statements not necessarily tied to mathematics (13). Consider teaching the formal logic of if, then statements; this approach is not absolutely necessary for students in these grade levels, however.
Once students grasp the concepts of P(A) and P(B | A), start off with the classic batting average example. Not just what is the frequency Sammy Sosa gets a hit, but what is the probability he gets a hit at home, on the road, against left-handers, right-handers, and so on. One of the dozens of baseball statistics books could come in handy here. After covering conditional probabilities ( P(getting a hit | against a left-hander) ), move on to independence. When it appears the hitter has no definite advantage for one option or the other, e.g. he hits equally well at home as on the road, identify this as the notion of independence. That is, when P(A) = P(A | B), we have independence.
Finally, consider setting up two-way tables, with two independent characteristics of a probability along the axes. The example used in the article is the number of people infected with the AIDS virus, with homosexual and heterosexual transmission along the vertical and geographical location along the horizontal. Totals appear on the bottom and right hand side, which can be used for various conditional probabilities.
I think this article indicates several useful methods for teaching conditional probability. It hasn't quite convinced me of the need for conditional probability at the levels prescribed, but students can obviously see the impact and bias of certain statistics presented in a conditional manner. Perhaps more newspaper examples might convince them of the usefulness of knowing when a conditional probability can mislead a person.
Keywords: Probability, Issues,
Ref: Jimmy6
Author(s): Schilling, Mark
Date: 1998
Title: Living in a World of Risk
Journal or Publisher: Math Horizons
Volume, Issue, Pages: April 1998, 14-16
Reviewer: Jimmy
Date of Review: 3/31/99
I wish to halt my streak of Mathematics Teacher articles, so I've looked at several other magazines over break. This first review comes from Math Horizons, which has okay articles for teachers. They're mostly literary descriptions of math people or general concepts. This makes most of them difficult to adapt to a lesson, I think. However, the article I found seems to answer some good questions about how to set up a statistical problem and answer it, something high school students could use.
The first example in the text concerns the set-up of a statistical analysis of the hazard of driving while using a cell phone. Critiques of previous analyses are made, along with the effective method used by one of the latest findings. The probability of two celebrities dying by ski accident (which happened in the winter of '97-98) appears next, with an examination of just how extraordinary an event this was. An explanation of why such incredible things sometimes do occur is also included.
The main point of the article, how to set up a real-life probability and why setting them up can be so difficult and variable, is the focus of the final example. Could the December 1997 explosion of TWA Flight 800 have resulted from a meteorite? Several experts give widely different probabilities, and their reasoning is examined.
I found this article fairly useful. The probability of two celebrities dying by ski accident is microscopic, 0.0000065, and I could never understand how such improbable events occurred. The explanation given is fascinating and simple, and I think quite instructive for high school students. Considering the many thousands of 'unlikely occurrences,' the number we can expect to occur in a given year is reasonable. This idea, along with the variable nature of determining real-life probabilities, are key concepts to introduce to high schoolers.
Keywords: Problem Solving, Connections,
Ref: Jimmy7
Author(s): Nakhshin, V.
Date: 1990
Title: Equations Think For You
Journal or Publisher: Quantum
Volume, Issue, Pages: January 1990, 46-48
Reviewer: Jimmy
Date of Review: 4/5/99
Today I looked at the magazine "Quantum," a joint American-Soviet math and physics magazine. It contains several interesting selections of problems, most of which originate from Russia, I believe. There's also a section entitled "at the blackboard," which provides an interesting analysis of the usefulness of math equations in physics. Understanding the numbers we derive after choosing the rules that apply to a situation leads us to a correct understandings of that physical situation.
The first example is the classic two-blocks are connected by a pulley, determine which way they move, given the masses, force of kinetic friction, etc. The author demonstrates a correct derivation and responses to the numbers the equations give. Following a second example of heated water placed in ice, with a goal of determining the final temperature, the usefulness of knowing what our answers represent is driven home. The approach the author takes, why he guesses certain things will happen, is also examined.
A third example of the current moving through a circuit, and a fourth of the time it takes a ball thrown straight up to reach a certain height follow. This last example doesn't quite have an answer; understanding why this problem gives a negative under the root sign of the quadratic equation means understanding that the question assumes the ball will reach a height that it cannot.
These examples subtly suggest that although equations drive certain areas of science, especially in physics, we need to know more than how to do algebra. We also need to understand the implications of our answers in the sciences. Why do they sometimes turn out absolutely wrong? Have we assumed something we should not have? A decent lesson for students to learn.
I also think this magazine contains some interesting insight into how the Russian school system concentrates on math and physics problem solving; different from our American system in several ways.
Keywords: Curriculum, Management,
Ref: Jimmy8
Author(s): NCSM Position Statement
Date: 1998
Title: Focusing the Dialogue: Suggestions for Engaging in Productive Discourse on the Future of School Mathematics
Journal or Publisher: NCSM Newsletter
Volume, Issue, Pages: XXCIII(1), 13-16
Reviewer: Jimmy
Date of Review: 4/5/99
This article comes from "Mathematics Education Leadership," the journal of the National Council of Supervisors of Mathematics. Apparently aimed at department chairs and heads, it seems a lot like an NCTM newsletter. The article I reviewed covered how to set up a community-wide dialogue about a school's math program. It identified simple ideas to stress and things to remember.
Organizing these discussions means respecting all points of view, no matter how diverse; all participants have the best interests of students at mind. A good way to start off such discussions is by pointing out how this issue affects everyone in the community, from students to teachers to college professors to business leaders to parents. Note some facts that everyone accepts: the inadequate national performance of American math students, the desire of all for better math skills for all students, and the ability to creatively solve original, real-world problems.
Once these common points have been established, move on to discussion. Remembering that this discussion is critical to implementing a new school curriculum, talk about what traditional methods to keep, when to use calculators and technology, how much group work and how much tracking should occur, and what assessment strategies to use. Focus upon the students, not what happened to the adults. Do not forget that the key is respecting all points of view: set up a listening atmosphere.
If you happen to be the supervisor or department chair setting up the meeting, keep the dialogue on track with gentle reminders of the main points. Determine the content, methods, and assessment to be used in your department. Outline these points, create common ground for all, and expect more than one dialogue.
This article makes perfect sense to me. I think the key to
maintaining a listening atmosphere lies in the initial invitation to
discussion. Try to keep such an announcement simple and unbiased towards any position. Parents have a tendency to get jumpy when they hear things might change in schools.
Keywords: Standards, Assessment,
Ref: Jimmy9
Author(s): Garet, Michael S.; Mills, Virginia L.
Date: 1995
Title: Changes in Teaching Practices: The Effects of the Curriculum and Evaluation Standards
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 88(5), 380-387
Reviewer: Jimmy
Date of Review: 4/5/99
This article was a good statistical analysis of the impact of the NCTM Standards upon mathematics departments around the country. Conducted in 1991, it measured actual progress since 1986 along with expected changes by 1996. The study concerned curriculum changes in algebra classes from 550 public schools within a 100 mile radius of Chicago; 72% of the schools responded to the survey.
Four areas were studied: content, methods, technology, and assessment. As NCTM recommended, factoring was slowly moving out, and more discrete math and statistics were moving into the content. Cooperative learning methods are rising, but lecture teaching is decreasing at a very slow rate. The most visible change is in technology, where calculator and computer use has and will continue to rapidly increase. As of 1991, few assessment changes had occurred, but by 1996, teachers expected more oral reports and group and partner tests. Short answer tests will still remain highly popular, however.
These trends represented overall changes, but among types of schools there was wide variation. In 1986 there was little variance, but by 1991, suburban schools had responded to the Standards first, with urban areas also making progress; small, rural schools typically were slower in implementing the recommendations.
This survey also evaluated technology, and about half the schools had graphing packages and exploratory packages for computers, but only 15% had some type of statistics package. Again, suburban and urban schools used more technology than rural schools. Textbook use also was observed, and among rural schools, Saxon textbooks were second most popular, even in 1991. Tracking and different levels of placement for students occurred frequently in schools big enough to use such systems.
These numbers are not too surprising; the statistics about technology match what I would have guessed. It does surprise me a little that urban schools have changed at about the same rate as suburban schools. If cooperative learning and different styles of assessment grow as much as this article implies they will, I'm going to be teaching classes in quite a different manner than when I was taught.
Keywords: Algebra, Geometry, Technology
Ref: Jimmy10
Author(s): Lapp, Douglas
Date: 1999
Title: Multiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 92(2), 109-113
Reviewer: Jimmy
Date of Review: 4/25/99
This article features some nice connections between algebra, calculators, and geometry. It shows some nice answers to the summation problems we worked on earlier in class, summation of the first i numbers and summation of the first i square numbers. Using several different techniques can help a student understand the ways that mathematics provides different levels of understanding, from simple memorization to geometrical applications to statistical analyses.
Using sum sequence on TI-82s or TI-92s can give reasonable answers and good points to determine data for a regression fit to a line, but after a certain level on both calculators, this tool becomes worthless as a direct calculation for these summations. On the other hand, once these points are plotted and a quadratic or cubic curve is fit, a student easily learns the fact that the sum of the first i numbers is 0.5i2 + 0.5i. This in turn motivates the students towards giving an actual proof of this formula, which can be done with the old pyramid of blocks or staircase picture.
The summation of squares also has several nice approaches. First using a matrix, this idea can be generalized to the sum of cubes and so on, allowing the students ways to calculate several sums of numbers. Then the real beauty of the article is the geometric proof of the sum of squares.
This was a real short, simple article that had several real short, simple, but powerful math ideas. Again, I say I really liked the geometric proof for the formula for the sum of squares. This author also has some good ideas for both the TI-82 and TI-92, including some limitations for these instruments that might motivate students to look for better ways to calculate things.
Keywords: Connections, Algebra, Geometry
Ref: Jimmy11
Author(s): Laing, David R.; White, Arthur T.
Date: 1991
Title: Exhibiting Connections between Algebra and Geometry
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: December 1991, 703-705
Reviewer: Jimmy
Date of Review: 4/30/99
Another article about a nice connection between algebra and geometry covers the expression 2n/(n-2) and its applications. Everything in the paper could serve as an easily introduced lesson that specifically answers the question of how algebra could ever show up in geometry. It also answers a tessellation problem in a simple manner.
The first use of the expression 2n/(n-2) shows up in the question of when does a rectangle have equal area and perimeter? This becomes an easy question of equating the two formulas for area and perimeter when a person has an m by n dimensional rectangle. Solve it for m and you come up with the expression.
Next the article discusses tessellations of the plane and which regular polygons can cover the plane. Using some geometry facts which might take a minute or two of thinking to check, it is shown that again we use the expression 2n/(n-2); this time whenever the answer is an integer we have an n-gon that will tessellate the plane.
The final part of the article discusses three simple but clever answers as to when 2n/(n-2) will be an integer. One involves asymptotes, another polynomial division, and the third a clever area-perimeter box argument. This third argument is quite good.
If I could write simple articles like this and prove connections between algebra and geometry, my life as a teacher would be so nice. Instead I get to read these articles, and this one is worthwhile. Its explanation of tessellations of the plane by n-gons is one of the briefest I have seen. It took me a little while to understand a few of the concepts, but they are easily accessible to high school students.
Keywords: Algebra, Geometry,
Ref: Jimmy12
Author(s): Horak, Virginia M.; Horak, Willis J.
Date: 1981
Title: Geometric Proofs of Algebraic Identities
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: ?, 212-216
Reviewer: Jimmy
Date of Review: 5/17/99
Here is yet another installment of connections between algebra and geometry. This article gets about as connected as you can get, with straight-out picture identities of algebraic expressions such as (a+b)^2=a^2+2ab+b^2. First there is a short discussion of the ancient traditions of geometric proofs of algebra, and then about seven or eight of these equations are graphically demonstrated.
The author begins by identifying the ancient Greek practices of proving these identities with geometry. They viewed the length of segments as the variables a or b, and thus computation of a product meant computation of an area for them. In particular, square and rectangular areas are used in this article, and the rearrangement or different lengths of the figures is the key. In this case, you really must see the article to view the proofs that go on. Interestingly, these proofs do not extend beyond third-degree equations, since the Greeks had no fourth dimension in which to place such variables.
These demonstrations could provide an excellent approach for students who have studied geometry more in depth than algebra, or for visually oriented learners. This is about as pure a geometric approach to algebra as I can imagine. It also provides practice for students to create proofs. The givens and end results are obvious, and having students make the connections could be interesting, since often there is a long list of givens and facts from which students can choose.
I think if I had time, I would definitely include this lesson in a geometry course. The students would review their algebra, see obvious applications of geometry, and learn how to choose and manipulate pertinent information from a somewhat lengthy list of given facts. For further work along a similar vein, the author mentions a 1940 text by a man named E. S. Loomis which compiles some 370, often geometric, proofs of the Pythagorean theorem.
Keywords: Curriculum, Algebra, Standards
Ref: Jimmy13
Author(s): Steen, Lynn Arthur
Date: 1992
Title: Does Everybody Need to Study Algebra?
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 85(4), 258-260
Reviewer: Jimmy
Date of Review: 5/17/99
I happened to glance upon a familiar name in the Mathematics Teacher, so here’s a short article about the need for math classes for everyone. Starting off with two opposition quotes from well-known editorialists, Lynn Steen proceeds to review the value of teaching algebra to all high school students. Dr. Steen does agree that it’s absurd to teach all students algebra, when they are forced to learn in the current (1992) algebra class styles found in the typical American classroom. We need to shift focus from different content for different students to different levels of depth in the content.
Dr. Steen advocates structuring classes not upon different types of math for students to learn, but different "breadth, depth, and approach" for individual students. Students less inclined towards algebra should not skip the subject, but should receive more focus upon the placement of algebra in real-life situations. As the critics note, symbolic algebra, as is taught in most classrooms, rarely appears in real life. Instead it is couched in "incomplete data, ambiguous graphs, uncertain inferences, and hasty generalizations." We need to improve the abilities of all students to recognize and adapt to such obstacles.
This article identifies a subtle truth: we expect all students to learn English to a simple degree of proficiency, and consequently change our teaching methods so that this happens. But when it comes to parts of math such as algebra, we assume some people just can’t learn the subject, and instead teach them different, easier material. We have to learn to adapt our teaching so that every student has the chance to learn how to apply algebra and why this is important. Otherwise our society will continue to fall further behind in these basic skills.
Keywords: Teaching Strategies, Management,
Ref: Jimmy14
Author(s): Artzt, Alice F.
Date: 1994
Title: Integrating Writing and Cooperative Learning in the Mathematics Class
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 87(2), 80-85
Reviewer: Jimmy
Date of Review: 5/17/99
Having students write about their cooperative learning experiences helps the planning of such groups in many ways. The feedback students receive in their groups also helps them clarify their explanations of their work. These two strategies feed off of each other in a nice manner.
Journal writing about cooperative groups can identify many problems or solutions that a teacher does not normally see. Before setting up cooperative groups, have students write about what kind of group they would like to participate in, and what level of learner they think they are. Having students write about their groups after working with them for several weeks also identifies problems or advantages of the groups. In this article the teacher noted that several groups she thought had great cooperation each had a member who felt out of place but was afraid to mention it to the group. One student complained of feeling "slow," another student claimed to be unable to help as much as she wanted.
Also have students talk about the help they give and receive within the groups. This identifies problems students may have with working cooperatively and ways they can fix them. It also describes the help that individual students give each other and the new methods they see for solving problems. Student explanations of problems for lab reports or other writing often grows clearer when they discuss the responses with their group members. This interchange of ideas benefits all students.
This is an excellent idea that I never would have thought of myself. Having students write about their learning abilities and styles rather than guessing these abilities increases the likelihood of good group work. Of course, their friends could be the people they have the most in common with regarding learning styles, so maybe they will talk a lot, but this discussion is useful. At any rate, the writing allows for teachers to honestly evaluate just how well the groups are working and make useful adjustments.
Keywords: Management
Ref: Jimmy15
Author(s): Chapin, Suzanne H.; Eastman, Kristen E.
Date: 1996
Title: External and Internal Characteristics of Learning Environments
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 89(2), 112-115
Reviewer: Jimmy
Date of Review: 5/17/99
The learning environment within a classroom can affect students in subtle ways. The obvious structure of the seating in the class, the materials found in the room, and the length of the class period all are noticeable, but there is more to designing a class. Students need to feel motivated and confident in their learning, which is affected by the attitudes and ideals of the teacher. Together, these internal and external characteristics combine to produce the total learning environment.
Keep you options open regarding the materials available for use in a classroom. Look for help from the community, local guest speakers from businesses, and see if there is any possibility for field trips. Talk to your department or administrators about the possibility of changing class length for different learners. Discuss with other teachers why they have set up their classroom seating in the manner they have. Try to keep many options open as far as changing the physical appearance of the classroom. Even a small change can influence students and make a class appear more exciting.
Similarly, keep your teaching styles open and aimed towards learning. Create an inviting atmosphere, display student work, and appreciate all forms of learning. Talk with other teachers about assessment, ways to deal with student behaviors, and new lessons to use. Keep a sense of perseverance, sensibility, and excitement in your classes. The main way to develop these mental habits is through experience. When a teacher struggles with a material until it is presented in a new manner, often that teacher realizes the importance of several approaches to a subject. So look for opportunities for growth in the math field, and try as many different, new things as possible.
This article was about as close as I could find to one describing classroom seating. It didn’t cover that too much, but it gave some important, basic tips about learning environments. Teachers need to remember that variety is the spice of life, and that we only find better teaching methods once we find more teaching methods, good or bad. We constantly need to search for new attitudes to develop, new skills to learn, and new ways to set up classes, otherwise we are stuck in the mud.
Keywords: Assessment
Ref: Jimmy16
Author(s): Mayer, Jennifer; Hillman, Susan
Date: 1996
Title: Assessing Students' Thinking through Writing
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 89(5), 428-432
Reviewer: Jimmy
Date of Review: 5/17/99
This article discusses the benefits of several types of writing in the mathematics classroom. Journals, lab reports, and portfolios all add different elements of learning and evaluation opportunities. All writing adds deeper levels of thinking processes to assignments. Students are forced to expound upon their reasoning and show why they chose the methods they did. Further, students begin to see their learning in a positive attitude and start reflection processes.
This appreciation for journals does not happen overnight. Slowly, over the course of a semester, a student starts to understand they can view their learning as something useful and comment upon the way it affects them. Perhaps some will even learn something from this approach and grow a little more motivated. This insight into a student’s learning can help a teacher structure class lessons to more effective means.
Lab reports also highlight the abilities and deficiencies in student thought. When students work through the problems and explain their methods, they again are forced to coherently demonstrate their abilities. Especially in this type of writing the teacher should provide examples of desired material and exemplary work. Students like to know what to produce and how they will be graded.
Portfolios also give much opportunity for genuine assessment. Students show not just what they have done in class, but which work they value and which learning they think they have benefited from. Teachers have the ability to see what type of motivation students have for themselves when it comes to learning concepts. The concepts students identify as difficult, the ones they are proud to have solved, will show up in portfolios. Teachers can take this insight and use it to design more in depth lessons the next time around.
I think that more writing in the classroom is useful for all these reasons. It really helps identify where students are struggling. Perhaps it is more work to read portfolios and journals than simple multiple-choice or short answer exams, but that’s the way it goes. Good teaching takes time, I think that’s what Dr. Wallace and Dr. Holden are trying to tell us.
Keywords: Assessment, Issues,
Ref: Jimmy17
Author(s): Cooney, Thomas J.; Bell, Karen; Fisher-Cauble, Diane; Sanchez, Wendy R.
Date: 1996
Title: The Demands of Alternative Assessment: What Teachers Say
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 89(6), 484-487
Reviewer: Jimmy
Date of Review: 5/17/99
Teachers were interviewed and asked to discuss the alternative assessments they used and the advantages and disadvantages of such methods. The article began with an example from a teacher named Margaret who began using open-ended questions on tests. The questions she used highlighted and displayed several levels of comprehension among three students, which likely would not have appeared upon a multiple-choice exam. The reasons for having students create and justify their answers appear crystal clear in this example. Possible arguments against alternative assessment were then discussed.
It is best to remember that the more alternative assessment is used, followed by instruction, then another short alternative assessment, then instruction, the more the distinction is blurred and the more students view the learning-evaluation process as simple day to day routine with less pressure than tests. Admitted, alternative assessment has less structure than the average multiple-choice exam, since there are more ways for students to adequately answer a problem, but teachers have to learn to deal with this individually if they want to better understand how and what their students learn. Other teachers claim that this lack of structure will slow down the pace of their class. True, students may focus more in depth upon certain topics, but there has never been any proven correlation between alternative assessment and lower test scores according to this article.
Setting up alternative assessment also frightens many teachers. If a teacher starts simple, and maybe assigns portfolios to one class, then they get a feel for a particular type of assessment and can better teach it to more classes later on. This trial run can show time-saving hints to use before sprining the idea on all classes. Also, make sure to budget time to demonstrate what good responses will look like from students. This stops students from claiming unfair grading.
The responses of parents to these assessments often surprised the teachers. At first some of the parents seemed skeptical, but often they displayed interest in the writing aspects of the assessments, and are more easily convinced of the grades their children receive when they had a portfolio to look at. Teachers also were impressed by the assessments and the strong abilities students showed once they were given a chance at them.
The key point I picked up in this article was the hint to begin a new type of alternative assessment with one class only, and see how well I can teach it. This will give me a break if I find the assessment taking up huge amounts of time. I also think it will be easier as a new teacher to deal with the structure issue and my ability to try new things. I’ve certainly had a bunch of different types of classes here at college, so I can accept new methods to teach lessons.
Keywords: Problem Solving, Teaching Strategies,
Ref: Jimmy18
Author(s): Gonzales, Nancy A.; Fernandez, Albert; Knecht, Corine
Date: 1996
Title: Active Participation in the Classroom through Creative Problem Generation
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 89(5), 383-385
Reviewer: Jimmy
Date of Review: 5/17/99
Several teachers developed a method to involve students with mathematics through creative problem construction. This ties the problem into student experience and also allows students the chance to see how data they know and don’t know affects their problem-solving abilities. It also highlights the many applications and possible uses of math.
The teachers here advocated a system called 'pass it along,' where students or groups of students would each add a new statement or fact to a story problem. The last students received the important job of analyzing the information and creating a problem that could hopefully be solved. Then as a group, the class would understand the problem, devise a plan, enact the plan, and look back at their solution; this is a variant of Polya’s problem-solving method. The goals were to have everyone involved and talking, with an abundance of creativity.
Two examples were given of this method. Both times the teacher recorded responses upon the board, and students were generally given a couple minutes to think of a statement to add to the story. The more coherent the teacher desires the story to be, the more time allotted to each student or group. The first teacher had small groups of three give statements that generally were a sentence long. The second teacher had individual students contribute one phrase at a time. Analysis of the given information before the final question was crucial. This is where the problem solving skills came in. Both stories contained a mass of information, only some of which could be used and some of which helped the problem in tricky ways.
I think this is an interesting exercise for students. I don’t know how often I would use this, it seems that after several rounds of this students might keep coming up with the same situations and grow bored. Still, it does involve the students in realistic situations and helps them decide when problems are solvable and when more information is needed.
Keywords: Number Theory, History,
Ref: Jimmy19
Author(s): Bezuszka, Stanley J.; Kenney, Margaret J.
Date: 1997
Title: Even Perfect Numbers: (Update)^2
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 90(8), 628-633
Reviewer: Jimmy
Date of Review: 5/17/99
For those of your students who have an interest in absolutely isolated fields of mathematics that produce strange, difficult results, this article is for them. The perfect numbers represent one of the more inapplicable fields in all of math. After about seven or eight, they start growing incredibly difficult to calculate or comprehend. With a little research, students can produce an interesting report on this topic and the people who have furthered the little knowledge that exists about the subject.
A perfect number being any number that equals the sum of its integer divisors (not including itself), the first four examples 6, 28, 496, and 8128 were known as long ago as AD 100. Until the late 1800s, only twelve or so were known. After this slow start, things began to speed up, with the next 12 appearing in the next 100 years, thanks to the advent of computers. In 1952, five were discovered with the use of Western Automatic Computer of the National Bureau of Standards in LA. The 25th discovery was by two eighteen year olds who had searched for three years. The last two have been found through the World Wide Web, in a program called the Great Internet Mersenne Prime Search (GIMPS). People hook up their personal computers for use together; the last one discovered, in 1997, was by an Englishman whose computer took fifteen days to verify the primality of the 895, 932 digit number. It can also be shown that the factorization of any even perfect number fits the formula !
2^(n-1)*((2^n)-1), where n and (2^n)-1 are prime.
After a brief introduction to this topic, students can use spreadsheets to determine the relationship between 2^(n-1) and 2^n. Using geometric series, they can prove the above mentioned result concerning the formula for even perfect numbers. Logarithms can be used to determine the number of digits found in large perfect numbers. Searches on the World Wide Web using the topic "The Largest Known Primes" reveal a wealth of information.
I think this would make an interesting project for an advanced class. Little high school mathematics deals with number theory, and here is an excellent example that students can study and digest in a couple of nights. I don’t think it would make a good lesson plan; I can envision many students easily growing bored with this topic quickly, but as an independent research it could work.
Keywords: Algebra
Ref: Jimmy20
Author(s): Contino, Michael A.
Date: 1995
Title: Linear Functions with Two Points of Intersection?
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 88(5), 376-378
Reviewer: Jimmy
Date of Review: 5/17/99
Learning about the difference between a continuous function and a step function often seems trivial. Simply chop apart a continuous function into several pieces and you have a step function. Most of the rules of continuous functions apply, so why worry about it? This article demonstrates a case where it does pay to worry about the difference between step functions and continuous functions. We have two linear step functions that intersect each other twice. The example provided in the article deals with comparisons of postage rates between the US postal service and a commercial company such as UPS. The prices given in the article are quite close to real life, so it makes for an interesting example. Plotting the two data sets as linear equations, obviously we arrive at one and only one intersection point. This is where it becomes more profitable to use the UPS company rather than the Post Office. However, two surprises are in store. First, turn the problem into a case of step-functions, since the post office and UPS both deal with each additional ounce being a certain amount, rather than compute fractions of ounces. The first surprise is that this has an effect; rounding the x input before inserting it into the formula rather than rounding the entire answer after the formula is computed changes the cutoff point. The second surprise is that if you look at the step functions more in detail, you realize that after several o! unces, it switches back to Post Office, then after about five more ounces, it switches to UPS being more efficient again. This is a sharp contrast between step functions and continuous functions. I think this offers an excellent example of graphical analysis of functions and algebra and the conclusions students can draw from these observations. Perhaps if students were to write in their journals about what they expect to happen, this could provide an wonderful opportunity for reflection and exploration. I would probably use this in a class, both as a good example of interesting relationships among functions and as a realistic example of mathematics and its use.