Keywords: Teaching Strategies, Issues,
Ref: John1
Author(s): Brahier, Daniel J.
Date: 2000
Title: Teaching Secondary and Middle School Mathematics,
Chapter
2
Journal or Publisher: Allyn and Bacon
Volume, Issue, Pages: pages 27-49
Reviewer: John
Date of Review: 2/18/04
Chapter 2: Learning Theories and Psychology in Mathematics Education This chapter examines the psychological concerns in the teaching and learning of mathematics.
There are two major types of research in education that guide our decision making-quantitative research and qualitative research. Quantitative research deals with gathering numerical data and analyzing it. For example, a study by Rogers (1998) found that ability grouping is favorable to learning, specifically those who are gifted. Qualitative research involves the collection and analysis of non-numerical data such as videotapes of classroom episodes, scripts of student-teacher conversations, audio recordings of interviews, or written summaries of student journal entries. Such research focuses on the "words" taken from observations and interviews rather than numbers from tests. Furthermore, experimental research is used when you want to prove whether or not one teaching method or assessment is more effective than another. Finally, a descriptive research is conducted when you want to generate statistics and information for discussion, but not necessarily for comparison. Information presented is not intended as a foundation for arguing one position over another but rather a description of what is going on.
Jerome Bruner, 1941 Harvard graduate, theorized that learning passes through three stages of representation-enactive, iconic, and symbolic. His theory led to the widespread use of hands-on-materials or manipulatives. His theory states that learning begins with an action-touching, feeling, and manipulating, hence why it is called the “enactive” stage. The second stage of learning is the “iconic” stage or pictorial phase, one that depends on visuals to summarize and represent concrete situations. Finally, Bruner’s third stage of learning is the “symbolic” stage in which the use of symbols allows a student to organize information in the mind by relating concepts together.
Historically, the lack of conceptual understanding has been a major problem in mathematics education. For many students, mathematics is simply a set of rules and procedures because they have been taught almost entirely on the symbolic level or in a classroom in which getting the answer was valued over making sense of the mathematics. Bruner stated that students with well-developed symbolic systems might be able to bypass the first two stages when studying concepts, but he warned that their teachers take a risk because these students will not possess the visual images on which to fall back if the symbolic approach is not working (Bruner, 1966).
Developed in the Netherlands in the 1950’s, the van Hiele method theorizes that children pass through five stages, or levels, of geometric reasoning and determining which level their students function on can help teachers understand how to meet the needs of their students. The first level (0) is the “visualization phase” in which a child will look at a square, identify it as a square, and give the reason “because it looks like one”. Level 2, “informal deduction”, the students will begin to compare geometric shapes and construct simple proofs, such as why a parallelogram does not necessarily have to be a rhombus. The ability to accept postulates and theorems and write proofs emerges in Level 3, “deduction”. The highest level, Level 4, or “rigor”, has students working in other geometric systems such as non-Euclidean geometry in which all the work is virtually done on an abstract, proof-oriented level. Most high school textbooks assume that a student is ready to function at Level 3. Van Hiele also found that this method was sequential: students could not understand Level 2 if they did not get Level 1 and so forth.
An inquiry lesson is a lesson in which students work through the activity and essentially invent their own mathematical rules. The teacher’s role is not to provide direct instruction but to select a rich task and to guide the students in their exploration of that problem. Such a lesson is an outgrowth of a model called the constructivist model, which states that knowledge cannot be passively transmitted from on individual from another but rather it is built up or constructed from within as we have experiences in our lives.
Inductive reasoning occurs when students think through several examples and then generalize a rule at the end. Deductive reasoning occurs when the teacher states the rule or definition and then expects the student to apply it to a worksheet or a set of problems; a generalization serves as the starting point and specific examples are applied later. Another method is the concept attainment method, in which students inductively create their own definitions as they are presented with a series of yes and no examples and counterexamples (developed by Bruner). The students must reflect on the common characteristics of elements in a set and make generalizations while the teacher must select examples that will allow students to invent their own definitions and rules.
Finally, a student who is motivated to do mathematics will do so. Motivation include the following three components: goal orientations, emotions, and self-confidence. Students are either motivated by ego goals in which they do their work in order to gain favorable judgment by their peers or mastery goals, in which they emphasize the intrinsic value of learning and self-improvement. Furthermore, we say that a student has interest in an academic topic when the individual believes that the study of that topic will be beneficial in some way. Finally, the teacher’s role is to select tasks that challenge and build self-confidence for more difficult concepts later in the semester or school year. The effective teaching of mathematics depends on the appealing to the needs of students in a way that motivates them. The National Council of Teachers of Mathematics has referred to the affective (feeling) side of mathematics as mathematical disposition. One semester, year, or lifetime of undesirable experiences in mathematics courses could produce what is referred to as mathematics anxiety.
In my opinion, this chapter hits on a lot of issues that came up for me this summer during my work at LearningWorks as well as a lot of my beliefs as a teacher. I truly believe that each student learns differently and find Bruner’s and van Hiele’s methods of different levels and aspects of learning not only extremely intuitive but in my opinion, exactly right. In order for students to understand concepts, I believe that everything from formal definitions to visualizations to student-developed ideas must be used in order for the student to not only truly understand the concept but also truly feel as if they “own” it-they came up with it themselves and reasoned through it rather than have it spoon fed the definition to them. Thus, personally I would tend to lean towards inductive reasoning rather than deductive reasoning in my classroom, simply because I truly believe that a student must take their own ownership of an idea before they can truly understand it rather than have it spoon fed it to them. Not to say that deductive reasoning is not useful because at times it is, but I think overall inductive reasoning is the best way for a student to truly understand a concept. In addition, I really liked the chapter’s section on motivation and how they stress that it is not an issue of whether or not some people like mathematics and some don’t, its about making mathematics a subject that appeals to all students and all interests. I think that idea is crucial if you want all of your students to feel individually motivated to learn.
Keywords: Curriculum, Algebra...
Ref: John2
Author(s): Alejandre, Suzanne
Date: 1996
Title: www.mathforum.org, "Traffic Jame Activity"
Journal or Publisher: Math Forum-Math 7
Volume, Issue, Pages: (none)
Reviewer: John
Date of Review: 2/22/04
This lesson in entitled "Traffic Jam" and is being taught as a way to introduce students to the idea of finding patterns in data, forming equations to explain those data patterns, and factoring those equations to make them as simple as possible.
The problem states that there are seven stepping stones and six people. On the three lefthand stones, facing the center, stand three of the people. The other three people stand on the three righthand stones, also facing the center. The center stone is not occupied. Everyone must move so that the people originally standing on the righthand stepping stones are on the lefthand stones, and those originally standing on the lefthand stepping stones are on the righthand stones, with the center stone again unoccupied. The rules of game are as follows: 1. After each move, each person must be standing on a stepping stone. 2. If you start on the left, you may only move to the right. If you start on the right, you may only move to the left. 3. You may "jump" another person if there is an empty stone on the other side. You may not "jump" more than one person. 4. Only one person can move at a time. Students then are placed in groups of six and use 7 sheets of paper as stepping stones on the floor and try and find the minimum number of moves necessary to complete the task. Once this task is complete, students can use a Java applet called "Traffic Jam" that utilizes the same problem but with different degrees of difficulty.
After students work on the computer for a bit, the teacher presents problems to them to think about that will help them find patterns in their answers, such as "What if there are only 2 people and 3 spaces? How many moves does it take for the two people to exchange positions?", etc. with the next questions displaying increasing combinations of the people and spaces. Students can first make a data table using the information gathered so far. There might just be columns for the number of pairs, the number of people, and the first 3 entries for the minimum number of moves. Students are then asked about if they see any patterns or relationships among the numbers in any of the three columns. With n equaling the number of pairs, can we generate any of the numbers in the minimum number of moves column? After looking at the data, students discover that there is a relationship amongst the three columns and that they can represented the three columns in terms of an algebraic equation.
This is an excellent lesson plan for teaching students how to not only use and examine data but also how to write out algebraic expressions as well as why we do so. I really like the hands on work done with the use of paper as stepping stones so the students can physically see the problem at hand as well as the use of the Java applet so students can work on different variations of the same problem. Furthermore, the fact that the Java applet has different levels of difficulty allows for all students to be challenged no matter how quick they are to find the answer. In addition, I really like the way the whole lessons ties so nicely together with each activity building on the last and the final product, in which students not only discover relationships and patterns in their data but also are able to write out those relationships algebraically, could not be done without the previous questions and work done in the earlier activities. The one critique I have is I am not sure whether or not this lesson was designed for one class period because it seems like it would take awhile, especially if this is the first time students are being introduced to the idea of finding patterns in data, but besides that I think this is an excellent way for students to learn how to analyze data as well as use algebra to express mathematical patterns.
Keywords: Standards......
Ref: John3
Author(s): Ball, Deborah Loewenberg
Date: 1991
Title: Implementing the NCTM Standards: Hopes and Hurdles
Journal or Publisher: Aspen Institute
Volume, Issue, Pages: This paper was prepared for a conference
on
Telecommunications as a Tool for Educational Reform
Reviewer: John
Date of Review: 2/25/04
Article Review of "Implementing the NCTM Standards: Hopes and Hurdles" by Deborah Loewenberg Ball
With increasing pressures to improve American students' mathematical competence, mathematics educators are trying to change the practices and outcomes of school mathematics. In her paper, Ball examines some conceptual issues that bear on the very notion of implementation itself.
Because standardization implies sameness, standards are frequently seen as calls for quality via uniformity. Ball sees this as a narrow viewpoint and states that a standard can also be a set of principles about what is valued. If teaching is an art, then standards for teaching may be able to function like standards in other arts. They may articulate standards of taste and judgment which do not determine a specific product or performance but which can guide the process of constructing and assessing that product or performance. In the case of NCTM, the standards documents represent all of these ideas.
Tension arises when trying to define these "standards", as mathematics educators disagree about things as fundamental as how mathematics and the physical or "everyday" world relate. Areas of tension include curriculum and pedagogy.
A second tension arises from direction versus discretion. On one hand, particular change requires rather clear direction and guidance. On the other hand, teaching in context-specific. Teachers are professionals who must make professional judgments based on expertise, insight, and skill. Many argue that in order to improve students' learning we should strive for consensus about the "what" that students should learn-the scope and sequence of topics-but leave the pedagogy of the curriculum up to the teachers. This position has received significant criticism because it separates content, what is taught, from method, how it is taught.
The ambitious vision of the Professional Standards of Teaching Mathematics (NCTM) holds out the hope of a richer mathematical curriculum, a curriculum aimed at developing all student's abilities to reason, solve problems, and communicate mathematically. This reform agenda is not just about reaching new agreements about what should be taught: It is also about what students should learn.
She concludes that if this reform movement is to have any promise, resources and supports of a variety of kinds will be absolutely crucial to working with, toward-and beyond-the ideas represented in Standards". If the Standards are to influence the directions of mathematics education, new standards are needed for the aims and means of implementation. Abandoning an instrumental view of how the standards might be translated into classroom work is crucial. Teachers and teacher educators will be the key agents of change and should be recognized and supported as such. The uncertainty of practice itself, combined with teachers' sense that they do not have authority and power to work for change, means that they may have difficulty working experimentally and responsibly to develop their practices. They may also not know how to take a more experimental approach to their work, for the pressure to appear competent, smooth, and sure of one's methods and results predominates. All in all, teachers will need a variety of opportunities to learn, and their work would be enhanced if there were more accessible ways to connect with teachers and teacher educators in other communities. A second dimension of support is to communicate in educative ways with parents and other community members. If the Standards become something mechanical to be implemented, the initiative will probably fail.
In my opinion, I strongly agree with what Ball expresses in her article. I believe there is a fine line between process and understanding and it is hard to achieve both in such a short amount of time. I fully agree that general math education needs to lean more towards the conceptual learning and away from learning so many processes. Furthermore, I believe practitioners need to continue to work to develop additional and diverse styles of teaching methods and teachers need to be supported more often. I thought this point was one of her best because I think it personally is difficult for a teacher to go out on a limb and try something new, especially in fear of appearing incompetent. Overall, I really enjoyed this article and like where the NCTM is heading, it is just a matter of time before we see whether or not they can succeed. I believe that with what Ball concluded her paper with in terms of what the NCTM needs to do to succeed is all true and if that can be achieved, children's perception and understanding of the subject will rise dramatically.
Keywords: Activities, Algebra, Teaching Strategies
>Ref: John4
Author(s): Keenan, Edward P.
Date: 1998
Title: "Opening Young Minds to Closure Properties"
Journal or Publisher: National Councils of Teachers of
Mathematics
Volume, Issue, Pages: Mathematics Teacher, Volume 95, Number 1,
January
2002
Reviewer: John
Date of Review: 2/29/04
This article presents an activity that can be used with mathematics classes of various levels to help students understand closure properties as they relate to various mathematical systems. The property of closure is defined as "the set S is closed under binary operation * if and only if, for all elements a and b of set S, a*b=c, where c is an element of S." Although this definition is not hard to state, students can have difficulty understanding what it means and how it relates to different mathematical systems. The article describes this game for some basic number systems and then extends it to other mathematical systems.
The teacher begins the lesson by "ushering out all numbers", rational, irrational, whole, natural, etc. This must be done with a lot of enthusiasm, since the numbers obviously are not visible, with the teacher stopping to yell at 1/2 for going slowly or 55 for arguing with him/her. When all numbers have finally left the room, the teacher shuts the door and begins a discussion about closure, giving the students the four operations that must hold true for closure to be true: addition, subtraction, multiplication, and division. After which, the teacher asks if certain types of numbers can be allowed in the room (have the property of closure), for example the number 3 and 7, and has the students perform the four operations to see if their group has closure. For natural numbers, the students see that for subtraction, 7-3=4 is okay but 3-7=-4, which is not a natural number, so natural numbers do not have closure. When students allow other numbers in, they go to the door and open it, possibly giving a greeting or high five to the number. Students then go through different types of groups of numbers, such as rational and irrational, imaginary, etc., and work through the property of closure. Finally, at the end of the period students return back to the original definition of closure and see how it applies to the activity they did for the day.
I think this is an excellent lesson to give to teach students the property of closure. The definition can be confusing at times and through this activity, students are given the task of actually finding out what types of numbers have closure and physically allows them into to allow numbers that do have the property of closure into the enclosed classroom and ones that don't are assisted out of the closed classroom. It really provides a mental image for students and a way to remember how the idea of closure works. Plus, it gives students a chance to learn how to do mathematical proofs in the respect that you only need to prove one thing wrong with a conjecture in order to prove the whole conjecture is false. For example, the conjecture that natural numbers have closure is false because 3-7=-4 which is not a natural number.
Keywords: Standards......
>Ref: John5
Author(s): Halmos, Paul R.
Date: Nov., 1994
Title: "What is teaching?"
Journal or Publisher: The American Mathematical Monthly
Volume, Issue, Pages: Volume 101, Number 9
Reviewer: John
Date of Review: 3/2/04
The article begins with Halmos reflecting on his first experience teaching: he taught freshman algebra in 1935 to graduate students. He goes on to say that he was a bit nervous but never scared. He always wondered why people said on their first day of teaching that they didn't know how to teach: hadn't they been taught by teachers their whole lives?
Halmos states that through his experience, he has learned there are three types of knowledge that we commonly speak of as subjects for teaching or for learning: what, how, and why. To be educated means to remember something, to be able to use it, and to be able to understand it. The best type of education is when all three of these are used at the same time, not separately. Often times many people wrongly associate the word education with memorization, which is farthest from the truth. People think that someone who is educated is like an encyclopedia, they have all the facts, but if they cannot interpret and analyze them for themselves then they are useless. Furthermore, education is sometimes angled towards application and application alone, a mistake that forgets the vital assets of understanding and remembering. A cellist is not considered a musician if they only memorize and apply the notes, but it is the musicianship and feeling they put into those notes that really separates them from the rest.
The way to teach them the "what" splits into two things: (1) tell them the facts and (2) tell them how to get the facts. In terms of teaching the "how", Halmos has two thoughts. One method is to throw them into the water and watch them figure out to swim. If you give them the foundation tools, trust that they can come to the answer with a little of your guidance. On the other hand, one can regard the role of teacher the same as role of coach. If run them through a few practice examples, they will learn how to correctly apply their concepts and formulas taught that day and will succeed in completing such problems on their own. Finally, concerning the "why" part of teaching, Halmos instructs that teachers should do nothing: let their students figure that out on their own. They need to find their own appreciation and understanding for the why on their own; they will never get it from someone telling them why something works. They need to figure it out on their own.
Finally, he goes on to say the best way to keep students motivated and challenged is to give them problems that are hard but just within their reach so that they can accomplish their short-terms goals through some hard work and brain power.
In my opinion, I agree with most of what Halmos is stating. At times, I really think he can be an idealist in terms of classroom management. I think he addresses some important issues but in my opinion, I felt like saying "so you told us the obvious, but how do we get that to work in the classroom?" Since every situation is different and every student and school district is different, I do not expect him to tell us this but I wish he would have given more of his own personal experience to see what worked for him in getting his ideas and beliefs to work in the classroom. I like his pushing for giving students the tools they need and have them figure out the problem, but I think only through classroom experience will a teacher figure out how to keep his/her students motivated everyday in order to achieve this self-motivated learning.
Keywords: Geometry, Planning, Technology
>Ref: John6
Author(s): Contreras, Jose N.
Date: 2003
Title: A Problem Posing Approach to Specializing,
Generalizing,
and Extending Problems with Interactive Geometry Software"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 96, No. 4, pg. 270-275
Reviewer: John
Date of Review: 3/05/04
The article begins by stating some strong mathematical skills students should have, including the ability to pose problems and conjecturing. Teachers should regularly "ask students to formulate interesting problems based on a wide variety of situations, both within and outside mathematics." In addition, instructional programs should provide students not only with experiences in making and investigating conjectures but also with experiences in developing and evaluating mathematical arguments and proofs. This article models how a particular problem situation can become a fertile environment for engaging students in posing problems and making conjectures with the aid of problem-posing.
This problem uses Cabri Geometry 2 software problem but can use a comparable applet or mathematical program. The problem given has students construct a parallelogram and the angle bisectors of its interior angles. Students do this using Cabri so that angles and sides are exact. After this is complete, the students are asked a series of base questions which include What type of figure is formed by the angle bisectors of the interior angles of a parallelogram? and Does a relationship exist between the area of a parallelogram and the area of the figure formed by the angle bisectors of its interior angles? These types of questions are used to identify the three types of information: knowns, unknowns, and restrictions. Furthermore, the article goes on to suggest that investigating the converses of a theorem or problem can be a worthwhile mathematical activity. For example, "The figure formed by the angle bisectors of the interior angles of a quadrilateral is a rectangle. What type of quadrilateral is it?" Teachers can also pose related problems and formulate the corresponding conjectures for special types of parallelograms (rectangles, rhombuses, and squares). Furthermore, teachers can also extend of generalize from parallelograms to such other types of quadrilaterals as kites, trapezoids, and generic quadrilaterals.
Take advantage of the measurement capabilities of the interactive geometry software. Have students measure the parallelograms sides and interior angles and then have them search for a pattern. We can also vary the concept of angle bisector and consider such other specific lines or segments as medians, altitudes, perpendicular bisectors, and maltitudes (a line that contains the midpoint of a side of a quadrilateral and is perpendicular to the opposite side).
In conclusion, this article shows that problem posing can be both a curricular activity and a means of instruction. It shows not only how we can help students become better problem posers but also how the teacher can use a problem-posing approach as an instructional tool to help students specialize, generalize, and extend problems.
In my opinion, I think this is an excellent lesson. It teaches students how to confront different problems in ways that they may not be used to (i.e. figuring out relationships by manipulating quadrilaterals). Furthermore, it is interactive and I remember using Cabri 2 in geometry and finding it extremely fun and interesting to work with because it is so exact and so easy to manipulate and change the shapes of objects and see what is going on in the measurements of both the lengths of its sides and angle measurements. In addition, students have the satisfaction of finding relationships and conjectures out on their own, which will help them remember it through having their own mastery of the concept. My one reservation would be how to do this sort of lesson in a school that does not have these resources available to them, such as Cabri 2 or computers for that matter. You can always use protractors and rulers but that takes a lot more time than with a computer. I am not sure what I would do to teach this lesson if I did not have a program like Cabri available but would look for a way to do it because I really think this lesson is a good one.
Keywords: Standards, Teaching Strategies...
>Ref: John7
Author(s): (none listed)
Date: 1998
Title: "Achieving Excellence in the Teaching Profession"
Journal or Publisher: Promising Practices: New Ways to Improve
Teacher
Quality
Volume, Issue, Pages: taken from
"http://www.ed.gov/pubs/PromPractice/chapter1.html"
Reviewer: John
Date of Review: 3/10/04
This article talks about the growing need for teachers in this country, particularly in urban and rural areas that experience some degree of poverty.
There are teacher shortages throughout the entire country, particularly in states where population growth is high (for example, Florida, California, Nevada, and Texas). The art of teaching is changing as well, with a high school and even a college degree becoming more and more of a qualification for the ever challenging job market, and thus, teachers are experiencing more pressure than ever to succeed. Pressure such as this as well as low pay have caused teaching jobs, specifically in poor areas, to have a lack of qualified applicants. In some areas, teachers might even lack a minor in the core class that they are teaching, a fact that shows how desperate some areas are for teachers. Furthermore, teachers in such areas are hurt by lack of resources and supplies and thus face the added pressure and stress of coming up with lesson plans and homework in a world where computers and technology are starting to become the social norm in classrooms. Also, the added pressures brought from poor areas into the classroom, such as drugs, violence, and domestic disputes, often put a hindrance on learning that is not experienced in wealthier areas. Such pressures as well as a lack of high pay have caused such poor areas to have a major shortage in numbers and qualified teachers. Policy is aiming to fix this, with pay increases for teachers as well as increased federal funding for schools as ideas but it will take awhile before major changes start to occur.
In my opinion, this article really hits on a lot of key issues in the field of education. Coming from a high school where money and resources were in definite lack and teachers taught core subjects without even having a college degree, I see a strong need for change. Whether or not I teach or go into education policy, the current situation needs to change. Teachers are such a vital part in American society, as they shape the lives of every person who is putting forth the work that makes up our economy, and should thus be rewarded as such with pay increases and better resources. More support needs to be given to those working in poorer areas, because as the article correctly says, teachers in such areas have a lot more added outside pressure than those teaching in wealthier areas. Furthermore, with the added pressure of having a high school and/or college degree to be hired for a job, the importance of teachers increases that much more. Being such a vital part of the shaping and success of society, they should start to be treated as one.
Keywords: Curriculum, Teaching Strategies, Algebra
Ref: John8
Author(s): Heid, M. Kathleen
Date: 1995
Title: "Algebra in a Technological World"
Journal or Publisher: Curriculum and Evaluation Standards for
School
Mathematics
Volume, Issue, Pages: Grades 9-12
Reviewer: John
Date of Review: 3/16/04
This book identifies a common core of mathematical topics that all students should have the opportunity to learn for grades 9-12. The traditional strands of algebra, functions, geometry, and trigonometry are balanced with topics from data analysis and statistics, probability, and discrete mathematics. Despite this breadth of topic, the aim of the curriculum is to continue to focus on topics and not just give an overview lecture. Narrow curricular expectations of memorizing isolated facts and procedures and becoming proficient with by-hand calculations and manipulations give way to developing mathematics as a connected whole with an emphasis on conceptual understanding, multiple representations and their linkages, mathematical modeling, and problem solving.
The specific broadening of curricular intent is reflected in the shift in perspective from algebra as skills for transforming, simplifying, and solving symbolic expressions to algebra as a way to express and analyze relationships. In addition, integration across topics at all grade levels is being stressed as well as a stronger focus on real-world applications, matrices, and the use of emerging calculator and computer technologies as tools for problem solving and conceptual development, while topics that are receiving less attention include by-hand methods for simplifying radical and rational expressions, for factoring polynomials, and for evaluating and graphing functions.
Chapter 1 gives an overview of how technological tools such as graphing calculators and computer-algebra systems support new conceptions of algebra that focus on enabling students to explore, describe, and explain quantitative relations to their world. Chapters 3 and 4 elaborate in detail a modeling and functions approach to algebra. Chapter 5 explores connections between the algebra strand and the geometry and discrete mathematics strands by means of matrices. The final chapter talks about distinctions among, and suggest priorities that be given to the development of, symbol sense, symbolic manipulation, and symbolic reasoning in a technological world.
I really liked this book because not only did it explore a new way of approaching algebra but it also gave some great example problems that I think would be extremely useful when I eventually teach in a classroom. I really liked how they combined both computational problems as well as conceptual questions.
Keywords: Problem Solving, Assessment...Author(s): Cohen, Robin
Date: (No year given)
Title:
http://www.enc.org/topics/inquiry/activities/document
Journal or Publisher: ENC Online
Volume, Issue, Pages: (website)
Reviewer: John
Date of Review: 4-4-04
This article talks about one teacher's approach to getting her students to understand ratios and proportions. Understanding of ratio and proportion begins when students measure models and scale them to the real life objects. Previously, Ms. Cohen, a middle school teacher at Eachester Middle School in New York, would have students go out and measure a variety of objects from furniture to toy cars. She really likes using the toy cars because they are so easily available to students and students can compare their measurements to real life-sized cars and calculate their proportions. She has used scales ranging from 1:144 to 1:18 and at times has been fortunate to acquire the same type of car in two different scales. The students measure various parts of the cars and then compare them. They also discuss trends in car sizes over the years so that they can make connections between the data they are obtaining and what it actually means.
Another object Ms. Cohen enjoys using is toy trains. These are particularly useful because toy trains come in all shapes and sizes, so students can compare the different proportions of different model trains to others.
Despite this, Ms. Cohen's best moment in teaching came when she had her students build replica cars. They did it every other class day for a few weeks while finishing their unit on alternate days. Students got really into building the replica cars and were assigned to build their models cars at a ratio of 1/25 to the size of an original car. As a quick assessment of students' understanding, she asked each student individually to tell her what the scale meant. Thus, students were able to learn the idea of proportion and ratio and how it worked in terms of modeling as well as have fun while doing it.
In my opinion, I think this is a great lesson to use and would
contimplate using it as my microteaching lesson. I think student's are
given a real opportunity to really understand what ratios and
proportions really are and through their building of a replica car, are
able to see how it applies to the real world. I also really liked her
use of having students explain how proportions applied to their project
as a quick assessment to see if they understand the concept of ratios
and proportions. Furthermore, it is a great problem-solving project in
the sense that students have to figure out how to construct a car that
is 1/25 the size of a real one. My only setback to it would be that it
would take too long, especially if you are having students work on it
daily in class. I think I would rather assign it to students as a take
home project and have them write a mini report and give a presentation
on what they did and how ratios and proportions apply to it. Overall
though, I think its a great lesson and am glad that I found it in hopes
of being able to use it in the future.
Keywords: Number and Operation, Teaching Strategies...
Ref: John11
Author(s): BBC (British Broadcasting Corporation)
Date: 2001
Title:
http://www.bbc.co.uk/schools/numbertime/index.shtml
Journal or Publisher: BBC
Volume, Issue, Pages: (website)
Reviewer: John
Date of Review: 4-2-04
This site, created by the BBC, offers different types of both in class and computer games that allow students grades 1 and 2 to learn and practice different number operations. Since there are a variety of topics, I will focus on the lesson that introduces students to and teaches them how to add. The site offers a lesson plan for teachers to base their teaching off of. This particular lesson's goals are to help students develop strategies for adding by counting sets to find the total, counting on from the first number, recognizing that numbers can be added in any order, and counting on from the larger number. It also strives to reinforce vocabulary words such as sum, total, add, equal, more than, add, and equals.
The program focuses on developing strategies for addition and subtraction and can be used to develop instant recall of number bonds for 20. This can be reinforced by using 'Target 20 Balloons' and similar activities regularly in the mental/oral part of lessons. The program also uses print style numerals. It is useful to discuss the different ways in which numerals can be written, encouraging the children to recognize the importance of writing numerals clearly for an audience.
One of the activities, named "El Nombre", has students make up their own quizes using the given vocabulary words above. For example, "What is more: the sum of _+_ or the sum of _+_?" or "What do you need to add to_to make_?" After filling in each blank with numbers, they work on the problems they created and put answer on the back of their card. At the end of the lesson, the quiz cards can be collected in and used for a whole class, group or pair quiz. This activity can be used in the plenary session with children selecting a contestant from each group. Alternatively, the quiz questions can be displayed on a folded piece of card with the answers on the inside.
I think this is an interesting way to introduce and teach number
operations and key vocabulary words to first and second grade students.
Instead of having the sentences already written with blanks, I think I
would have students write their own equivalent statements so that I can
not only have them practice reading and interpreting them but also so I
know that they understand the different vocabularly words.
Furthermore, I think I would have students create a mini-quiz as
explained above and then have them switch and take each other so that
they give challenging questions rather then really easy ones. Despite
this though, I really like this lesson because I think it really gives
students an opportunity to correctly use vocabulary words that they
were taught in class and for the teacher to really see whether or not
they understand what was taught. Furthermore, students are given the
opportunity to create their own quiz and challenge their classmates
which is always fun.
Keywords: Equity/Diversity, Teaching Strategies...
Ref: John9
Author(s): Rasmussen, Claudette
Date: 1995
Title:
http://www.ncrel.org/sdrs/areas/issues/content/cntareas/math/ma1
00.htm
Journal or Publisher: North Central Regional Educational
Laboratory
Volume, Issue, Pages: website
Reviewer: John
Date of Review: 4-01-04
This article talks about the ever-growing importance of equity in the classroom. All students, regardless of race, ethnic group, gender, socioeconomic status, geographic location, age, language, disability, or prior mathematics achievement, deserve equitable access to challenging and meaningful mathematics learning and achievement. Studies have found that a disproportionate number of women, minority students, and poor students are leaving school without the adequate knowledge they need to know mathematics at in order to succeed in the real world. A custom of low expectations, changing workforce needs, economic necessity, and shifting demographics call for unprecedented reform in mathematics education. In response to this, an NCTM specialized group called "Reaching All Students through Mathematics" has pushed for programs that ensure equity and excellence for all students. These programs and other exemplary programs implement high standards and often foster cultural and linguistic diversity in an effort to increase the participation and success of underrepresented groups.
Some of the proposed actions include making mathematics more meaningful and relevant to those students who are in the targeted low- performing groups and reexaming grouping systems, such as tracking students and encouraging students of different levels of mathematics to work together. Furthermore, the article stresses the re-examining of assessments to make sure they are equitable and unbiased as well as inviting parents as partners in their child's mathematical education. In addition, Involve community members - particularly members from a variety of cultural backgrounds and experiences - as role models, tutors, career speakers, consultants, and partners in reform.
The article goes on to present a number of different opinions to proposed changes. For starters, many people that factors such as race and wealth have nothing to do with it. Furthermore, parents of gifted children argue that their child's learning will suffer by being in a "lower" track and that they should be allowed to be placed in the higher math classes if they can meet the challenge. Finally, parents of minority students are concerned about new math standards and curricula that deemphasize paper-and-pencil computation. Computation skills often are associated with mathematical competence, and the lack of mastery of these skills has been used to justify denying opportunities to minority students.
In my opinion, this article hits on a lot of key issues in terms of equity in the classroom. I really liked the way they presented both sides to the story, maybe not completely, but at least they presented point and counterpoint. I think this is a really tough issue to resolve. On one hand, you want to help those students who are not grasping concepts as well as others but at the same time you do not want to hold back any of the gifted students. Furthermore, how do you help those social groups that are statistically not doing as well in the subject as other groups? I did an paper last year for my economics class on Public Policy in which I looked at what educational resources improved test scores and one that did not was just pumping more money into the school. I believe it has to start with the teachers and if you want to even out the playing field, I think the government needs to start paying teachers more to work in failing schools; then you can get the better teachers who would like a nice salary boost as well as the teachers who want to work there in the first place but could not afford to do so because of having to support a family, support themselves, etc. I think once you get those better qualified teachers into those schools, test scores will improve and the playing field will begin to become more levelled.
Keywords: Number and Operation, Problem Solving, Assessment
Ref: John12
Author(s): 2004 National Council of Teachers of Mathematics
Date: 2004
Title: http://www.figurethis.org/challenges/c08/challenge.htm#hint
Journal or Publisher: National Council of Teachers of
Mathematics
Volume, Issue, Pages: (website)
Reviewer: John
Date of Review: 4/19/04
"Suppose you found an old roll of 15 cent stamps. Can you use a combination of 33 cent stamps and 15 cent stamps to mail a package for exactly $1.77?" This website discusses a particular problem used to help students practice their number operation techniques. This problem involves using a combination of numbers to make other numbers. Similar techniques are used to develop codes to maintain security in banking and computer access. The problem makes use of the website though, offering icons that you can click on for "small hints", "answer", and "other similar problems". For example, if you click on the "small hints" icon, they offer constructive questions such as "What would happen if you only have 33 cent stamps? 15 cent stamps?" and "What is the closest postage you can get if you use only 33 cent stamps?" Furthermore, the website also offers additional problems for those students that get through the first one offered rather quickly or just when any student is done for additional practice. Among such problems include the following: "Using only 33 cent postage and 15 cent postage, can you make postage equal to $2.77? 4.77? 17.76?" and "Can every whole number greater than 1 be made by adding some combination of twos and threes?" I like these problems because they not only give students practice with larger numbers, testing their ability to apply the number operations they know to harder problems, but also asks students to draw a conjecture from what they have learned, something they might not see right away but eventually will after thinking about the problem for awhile as well as looking back at the problems they just did, which they know are right from the answers given on the website.
Not only does the problem test your knowledge of the different types of number operations but also how and when to apply them. It also gives students practice at problem solving, since the problem does not directly tell students what to do but rather gives them a situation and asks them to use their tools to come to a logical conclusion. Finally, I think the problem is extremely applicable to real-life and students can actually visualize the problem; perhaps even more effective would be for the teacher to bring in little squares of paper that have the numbers 33 and 15 on them so students can work with visual props.
Overall, I really like this problem and think it would be a great way to not only have students practice their number operations but also a nice assessment to see if students not only understand the computation involved in the operations themselves but also how to apply them to real-life situations, as breaking up money (whether in stamps or currency) is something that is done all the time. The only thing I would change was I would have students not only write their answers down on paper but how they got to them too; what were the logical steps they used to get to their final answer so that I know that they understand what is going on and not just using the answer to get done with the assignment quickly.
Keywords: Number and Operation, Problem Solving, Assessment
Ref: John12
Author(s): 2004 National Council of Teachers of Mathematics
Date: 2004
Title: http://www.figurethis.org/challenges/c08/challenge.htm#hint
Journal or Publisher: National Council of Teachers of
Mathematics
Volume, Issue, Pages: (website)
Reviewer: John
Date of Review: 4/19/04
"Suppose you found an old roll of 15 cent stamps. Can you use a combination of 33 cent stamps and 15 cent stamps to mail a package for exactly $1.77?" This website discusses a particular problem used to help students practice their number operation techniques. This problem involves using a combination of numbers to make other numbers. Similar techniques are used to develop codes to maintain security in banking and computer access. The problem makes use of the website though, offering icons that you can click on for "small hints", "answer", and "other similar problems". For example, if you click on the "small hints" icon, they offer constructive questions such as "What would happen if you only have 33 cent stamps? 15 cent stamps?" and "What is the closest postage you can get if you use only 33 cent stamps?" Furthermore, the website also offers additional problems for those students that get through the first one offered rather quickly or just when any student is done for additional practice. Among such problems include the following: "Using only 33 cent postage and 15 cent postage, can you make postage equal to $2.77? 4.77? 17.76?" and "Can every whole number greater than 1 be made by adding some combination of twos and threes?" I like these problems because they not only give students practice with larger numbers, testing their ability to apply the number operations they know to harder problems, but also asks students to draw a conjecture from what they have learned, something they might not see right away but eventually will after thinking about the problem for awhile as well as looking back at the problems they just did, which they know are right from the answers given on the website.
Not only does the problem test your knowledge of the different types of number operations but also how and when to apply them. It also gives students practice at problem solving, since the problem does not directly tell students what to do but rather gives them a situation and asks them to use their tools to come to a logical conclusion. Finally, I think the problem is extremely applicable to real-life and students can actually visualize the problem; perhaps even more effective would be for the teacher to bring in little squares of paper that have the numbers 33 and 15 on them so students can work with visual props.
Overall, I really like this problem and think it would be a great way to not only have students practice their number operations but also a nice assessment to see if students not only understand the computation involved in the operations themselves but also how to apply them to real-life situations, as breaking up money (whether in stamps or currency) is something that is done all the time. The only thing I would change was I would have students not only write their answers down on paper but how they got to them too; what were the logical steps they used to get to their final answer so that I know that they understand what is going on and not just using the answer to get done with the assignment quickly.
Keywords: Proof, Assessment, Problem Solving
Ref: John13
Author(s): Spencer, Philip
Date: 1998
Title: http://www.math.toronto.edu/mathnet/falseProofs/fallacies.html
Journal or Publisher: University of Toronto Mathematics Network
Volume, Issue, Pages: (website)
Reviewer: John
Date of Review: 2/19/04
Targeted towards students grades 7-12, this World Wide Web site, developed by the University of Toronto Mathematics Network, offers fallacious proofs to investigate for reasoning errors. The reader is challenged to find errors in proofs of some startling discoveries, including a conclusive proof that one is equal to two, that every person in Canada is the same age, and that a ladder will fall infinitely fast if you pull on it.
Using the example for the proof that one is equal to two, the page offers each step of the proof (1-8) and a link to each one. The idea is that the student goes through the proof, figures out where the proof is incorrect, and then clicks on the link to see if he/she is correct. If the student is correct in picking the incorrect step, the link congratulates him/her with a further in-depth explanation of why the step is incorrect. If the student is incorrect in picking the correct incorrect step, then the links alerts the student to their mistake and also offers an in-depth explanation as to why the step is correct.
I think this is an excellent website that contains some great problems to help students with the idea of proofs. I remember when I first learned how to write proofs and thought they were really hard, especially in terms of how to think in the right frame of mind for putting together a mathematical proof. I also really liked how the computer not only showed the proof but also how each step worked in an in-depth explanation that really gives the student a great understanding of why it works or doesn't. I thought some of the problems were hard, but thought that there was a wide enough spread of easy to difficult proofs that it could cover everything from basic proof writing skills to 12th grade honors homework. Finally, I like this exercise because it not only teaches students how to write proofs but also how to spot flaws and question what is presented to them rather than just accepting something on blind faith. Not only does questioning help you understand the material better as you have to work through the logical steps to see if they are correct or not but it also gives the student better "ownership" of the idea because they have now worked through it and have proved to themselves how and why it is correct. Thus, this problem provides a strong exercise that helps students strengthen their reasoning and proof writing skills while teaches them the valuable lesson of being able to question facts that are presented to them and not always rely on blind faith.
Keywords: Probability, Representations, Connections
Ref: John14
Author(s): Garfunkel, Solomon; Godbold, Landy; Pollak, Henry
Date: 1998
Title: "Forearmed and Forewarned: Introduction to Collecting
and
Analyzing Data" from Mathematics: Modeling Our World Textbook
Journal or Publisher: South-Western Educational Publishing
Volume, Issue, Pages: pg. 324-327
Reviewer: John
Date of Review: 5-10-04
This lesson provides a great introduction to learning about best-fit lines and least squared regressions. It has students working with scatter plots and thinking about which type of graph would best represent their data and give the clearest results. Students are given different methods to graph their data with and then are asked to decide which one best fits their data and why. Through working with, graphing and analyzing data, this lesson teaches students the skill of organizing and representing data in the most efficient and clear- cut way. It also reinforces their understanding of the concept of a "scatter plot", or a graph of ordered pairs of data, and will further allow them to see how it can be used to display different sets of data.
Furthermore, students learn the skill of how to analyze a graph and come to various conclusions/results based on that graph through a series of leading questions. Specifically, students are given different predictions and must use the graph that they picked "best-fits" their data and decide whether or not their graph validates each prediction. Finally, students will gain the skill of being able to decide which linear equation best represents their data set based on their scatter plot.
I chose this lesson to review because in my opinion, it provides a great review of data analysis and probability because it gives such great lead in questions and ties in various skills from past lessons into it, thus showing students the connections between past material and present material. For example students are asked to analyze slope and take graphs they drew and describe them in equation format. It also has students create scatter plots, something they learned earlier in the unit but did not work with much until now. Students also see how concepts such as slope and equations apply to the real world, as students use such concepts to analyze the relationship between the lengths of their class forearms as compared to their height.
Finally, this lesson also provides students with different methods of graphing/representing their data and gives them an opportunity to really figure out what sort of results they are trying to obtain and how best they can represent those results through their graphs. Up until this point, students have not really had to decide between graphical representations and I think this lesson does a great job of giving students three different methods to analyze and decide which one represents their data best.
NOTE: Even though probability is not tied into this lesson in particular, the article sheet says we must have one lesson on "Data Analysis and Probability" and I would have this under "Data Analysis" except there is no topic in the drop menu.
Keywords: Technology, Teaching Strategies, Manipulatives
Ref: John15
Author(s): Krieger, Terry
Date: 2004
Title: "Inverse Functions, Intersections, and Ubiquity of e"
Journal or Publisher: (none)
Volume, Issue, Pages: talk given at the MCTM Conference in
Duluth, MN
Reviewer: John
Date of Review: 5-11-04
This review will cover one of the lectures I saw this past weekend at the MCTM Spring Conference in Duluth, MN. The lecture was entitled "Inverse Functions, Intersections, and Ubiquity of e" by Terry Krieger, formerly a teacher with Winona State. The talk provided a look at intersections of graphs of inverse functions with surprising results involving the number e using a software program called "Curtis Pro Software".
Krieger worked on finding a base where two intersecting graphs do not intersect anymore. He looked at two types of graphs: the exponential graph y = b^x and the logarithmic graph y = log (x) with base b. Not only can he figure out the answer using this new software program, but he can also make a Quick time movie to post on the web that shows the movement of the graph from intersecting to not and the change of the base as the graph moves. Using Curtis, Krieger ends up finding that the exponential and logarithmic graph intersect 3 times for b in (0, e^-e), 1 time for b in (e^-e,1), twice for b in (1, e^1/e), and 1 time when b = e^1/e.
In terms of the software, you can find a free downloadable demo version of it at www.curvuspro.ch. There are no current versions for Windows and registration for the program is 40 dollars. In terms of his lecture, the only problem brought up was that he did not look at trigonometric functions with restricted domains.
In my opinion, this was a really interesting and informative talk. At first I was a bit skeptical, but I think this is a great program that gets pretty quick results that are easy to show. Most impressive of all was the fact that you could make Quick Time movies on it, which allows the teacher to work on the problem the night before and then show the shifting of the graphs until they hit their point of intersection rather than having to go through the calculations in class. I also liked the fact that he not only said the good points about the program but also the bad. The fact that there is not a version for Windows provides a major problem since a lot of schools do not have Mac's and to bring a laptop to school everyday is not only a risk but you must also have the technology to be able to hook it up to some sort of big screen. Overall though, I think this was an excellent talk that not only provided a nice overview of the Curtis software but also provided a great example of the program in action, exploring different ways in which it can be used.
Keywords: Activities, Teaching Strategies...
Ref: John16
Author(s): Awe, William G.
Date: 2004
Title: "Bingo, Golf, Who has? And other Activities for your
Classroom"
Journal or Publisher: none
Volume, Issue, Pages: talk given at the MCTM Conference in
Duluth, MN
Reviewer: John
Date of Review: 5-11-04
This review will cover one of the lectures I saw this past weekend at the MCTM Spring Conference in Duluth, MN. The lecture was entitled "Bingo, Golf, Who Has? and Other Activities for your Classroom" by William G. Awe, a mathematics teacher at International Falls High School in International Falls, Minnesota. His talk was my favorite talk of the entire conference, specifically because he gave some great activities and general ideas for teaching that could be used by anyone. These activities were especially useful because they were not only fun but provided great assessments to tell whether or not students understood the material taught during the previous unit.
For example, one of his favorite assessment plans is his golf tournament he holds every year entitled the "Awe Open". The game is miniature golf using paper, rulers, pencils, and math mirrors. Students play all of their holes using transformations such as reflections, dilations, rotations, and translations.
Before doing this activity, students should have covered the transformations together in the textbook, doing assignments. They will have used math mirrors for reflections before as well as will have practiced doing rotations and translations by using multiple reflections. He suggests using Geometer's Sketchpad to do a week of lab activities on all 4 transformations prior to the open.
The Awe Open starts out with two days he calls practice rounds where he will have holes drawn for students to navigate and ask for help. The class will review how to do multiple banks and will also discuss the relations between this and three-cushion billiards, regular pool, laser shows, etc. When the Awe Open begins, students must make their holes in one shot, at times having to navigate around water, walls, sand, and other obstructions. All holes must be made using bank shots and students must make shots in the allotted par, or the least number of bank shots Mr. Awe has found. The winner gets a certificate and a shirt that says "Awe Open Champion".
In addition, Awe provides some great ideas for Pi Week, such as creating a Pi Chant to making Pi Day cards at 123greetings.com. He also suggests having students write their own research paper on a famous mathematician three weeks before Pi Week.
Overall, I thought this was not only the most fun but most informative talk all weekend. Mr. Awe was a great speaker and had some great activities to share with his audience. Not only were his ideas great examples of some solid assessment activities but his passion for mathematics was inspirational.
Return to IndexKeywords: Teaching Strategies, Connections...
Ref: John17
Author(s): Bechthold, Dawn; Soto-Johnson, Hortensia
Date: 2004
Title: "Tesselating the Sphere with Regular Polygons"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 97, No. 3, pg. 165-167
Reviewer: John
Date of Review: 5-17-04
This article examines how spherical geometry can be used to draw students into a deeper understanding of Euclidean geometry. With tessellations being as recognizable as a soccer ball or volleyball, this lesson is extremely accessible to high school students as well.
The lesson begins with a review of tessellations in the Euclidean plane in order to connect these with tessellations of the sphere. The article then goes on to talk about using the equations and facts about three regular polygons that tessellate a plane to figure out facts about tessellations of a sphere. Students are then asked to compare and contrast the facts about tessellations of planes and spheres, such as the difference between the spherical angle sum and the Euclidean angle sum of a polygon. Students are also asked to look at such aspects such as the number of faces for the tessellation, the interior angle measure, etc.
In order to explore such aspects as above, students are given plastic spheres that they can write on with marker. They are given certain instructions that help them to construct different spherical polygons in order to analyze their differences. They are also asked to record in a table the number of faces and tessellation name, given the degree of the vertices and the number of sides of the polygon. Students will find that regular polygons with 3,4, or 6 sides will tessellate the plane while regular polygons with 3,4, or 5 sides will tessellate the sphere. The lesson concludes with a final discussion on the comparisons that can be made between Euclidean geometry and spherical geometry.
Overall, I thought this was a good lesson plan to use. I really
like the
idea of using plastic spheres when having students explore the
differences
between Euclidean and spherical geometry. We did this in my Geometry
class in
college and it was extremely helpful in learning the material. The
lesson
itself seemed to get a bit proof stressed, which is not entirely a bad
thing but
might be a little too much for a high school class, but I think there
are some
great ideas you can extract from this lesson to create your own.
Return to Index
Keywords: Teaching Strategies, Connections, Activities
Ref: John18
Author(s): Edwards, Michael Todd
Date: 2004
Title: "Fostering Mathematical Inquiry with Explorations of
Facial
Symmetry"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 97, No. 4, pg.234-241
Reviewer: John
Date of Review: 5-17-04
This article explores the use of facial symmetry to explore concepts of symmetry in mathematics. The article describes two technology-oriented activities that the author has used with entry-level geometry students during their study of symmetry. Image-processing software and interactive geometry tools enable his students to examine symmetry from algebraic and geometric perspectives in a manner that addresses the unique learning needs of each student.
He begins by having students explore geometric transformation. This is practiced through using concrete materials such as origami paper, rules, compasses, and interactive geometric software to construct reflections of abstract geometric shapes. Students learn how certain shapes exhibit reflection symmetry when they can be reflected onto themselves in one-to-one fashion. Students also note that symmetry lines cut shapes into two congruent parts as well as how perpendiculars to the symmetry line were created when you connect points to their reflection. Such examples provide real world applications of symmetry that appeal to entry-level students.
Edwards then talks about having his students see the relevance of symmetry in their lives. He has them read a Newsweek article called "The Biology of Beauty", which connects symmetry with such academic disciplines as biology, psychology, and physiology in a way that is both intriguing and comprehensible to entry-level students. He suggests that reflection symmetry influences the health, sexual activity, and longevity of various organisms, including humans and has students discuss this, specifically facial symmetry. Eager to capitalize on his student's interest, he also designed a series of activities that encourage his students to investigate facial symmetry mathematically. One of the activities involves popular photo-processing software while the other relies on interactive geometry tolls and student-built models. Under the photo-shop lesson, students are able to alter pictures of different faces to see which one physically appeals to them. They are then asked to measure different aspects of that face and see how symmetrical their face is as compared to the original picture. In addition, using the geometry software they also create faces and measure the one that is most appealing using programs such as Cabri. He says that the activities are successful because they appeal to students' fascination with beauty and celebrity, desiring to investigate the symmetry of the faces of well-known celebrities. They also allow students to apply their mathematical knowledge creatively and flexibly.
Overall, I think this is an excellent lesson. I think it really teaches students about symmetry and I love the real-world application aspect of it. Students are eager to learn more because they can explore their favorite celebrities and learn at the same time. Finally, I presented on symmetry and facial beauty in my Geometry class over interim and I had many compliments as to how interesting the topic was so I know for a fact it draws students' attention.
Keywords: Connections, Geometry, Teaching Strategies
Ref: John19
Author(s): Devlin, Keith
Date: 2002
Title: Numbers in the Garden and Geometry in the Jungle
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 7, #8, pg. 422-424
Reviewer: John
Date of Review: 5-22-04
Many children find numbers either boring or intimidating or both, but practically everyone is fascinated by the living world. Devlin argues that a skilled teacher can take advantage of this. He says that a summertime mathematics lesson can be enlivened by a brief excursion into a garden to count petals and leaf patterns on flowers and plants, "...leading to a surprising discovery that even Mother Nature knows her arithmatic".
Devlin goes on to talk about how tabulating the number of petals in various flowers will lead to the following numbers: 3,5,6,8,13,21,34,55,89. Discounting 6, the remaining numbers are the first nontrivial members of the famous Fibonacci sequence. Also, counting spirals in certain flower heads also leads to Fibonacci numbers. For example, counting the number of spirals on a sunflower will give you 21 running clockwise and 34 running counterclockwise-both Fibonacci numbers. Similarly, pinecones have 5 clockwise spirals and 8 counterclockwise spirals, and the pineapple has 8 counterclockwise spirals and 13 going clockwise.
Devlin goes on to state that scientific explanations have been provided to explain why Mother Nature favors Fibonacci numbers-in essence, the reason is that the numbers represent the most efficient way for the various parts of the flower or plant to achieve their purposes. Similarly, the natural world can also be used to spark interest in other parts of mathematics. In the 1950's, British computer pioneer and wartime code breaker Alan Turing proposed that the possible patterns in an animals fur coat are restricted by geometrical rules. In that case, all the DNA does is determine which particular geometric pattern to adopt.
Furthermore, Oxford mathematician James Murray discovered that by simply varying the size and shape of the skin area of an animal, he could obtain all the coat patterns you see in nature. Very small or very large areas gave no pattern while medium-sized, vaguely rectangular areas gave spots. Long, thin areas gave stripes perpendicular to the length of the area. Finer variations in skin dimension gave rise to the different kinds of spots and stripes we see in nature. If this explanation is correct, the coat patters of animals are determined not by biology but by geometry.
In my opinion, this is a great example of solid connnections
between mathematics and the real world. As I continued to Devlin's
article, I became more and more fascinated with what he had to say and
I can only imagine how middle school kids would react. This could
provide great material for examples during a lesson or even a field
trip in which students are able to go out to a park and investigate
such claims about flowers and the fibonacci sequence. Furthermore,
students become more engaged in the material when they can apply it to
real-life situations and Devlin provides some great examples of such
situations. It is also fun to ponder over whether scientests like
Murray are right in their explanation. Could geometry actually explain
the fur coat of animals?
Keywords: Communications, Assessment, Research
Ref: John20
Author(s): Kaystein, Mary
Date: 2001
Title: Putting Umph into Classroom Discussions
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 7, Number 2, pgs. 110-112
Reviewer: John
Date of Review: 5-22-04
Kaystein argues that the most difficult recommendation of the NCTM's Standards to put into practice is orchestrating classroom discussion-moving from a teacher-centered classroom to one that is centered on students thinking and reasoning. While classroom discussion is viewed as encouraging students to construct and evaluate their own knowledge, as well as the ideas of their classmates, few examples or guidelines exist to help teachers orchestrate such discussion.
Kaystein argues that a prerequisite to such eliciting student discussion is a good task that is rich enough to elicit student thinking and discussion. Even then though, many teachers find that they are always asking students for an explanation and students are always responding with "why". Experts argue that discussion does not have to become predictable but rather developing a personal interest in a particular solution or strategy is a way that students can invest in, and take ownership of, the discourse. First, teachers must create a classroom where students feel comfortable to critique the work of others as well as make mistakes themselves. Second, teachers should select instructional tasks that prompt students to take different positions and find different solutions. Finally, they can encourage students to align with a position and to defend that position, focusing on effective communication skills when making their point.
Experts found that the student discussion followed many of the processes of effective argumentation put forth by rhetoricians, such as examining premises, using warrants to back claims, and communicate counterarguments effectively. In addition, they observed that students were more likely than the teacher to initiate explanations, to provide answers or claims backed by appropriate justification, and to evaluate their own and one another's arguments. Also, equally important, they found that the teacher participated by recruiting attention and participation from the class and by aligning students with positions through rephrasing their contributions; highlighting their positions through repetition; and pointing out implicit but important aspects of their explanations through expansion, including reminding students to say "square" millimeters.
The conclusions to this study state that middle school students from low-income, urban neighborhoods can engage in collective argument and teachers, with judicious selection of tasks and coaching to prompt student participation in their solution, can break out of the role of sole evaluator of student thinking and reasoning in the classroom.
In my opinion, Kaystein brings up some good points. Classroom discussion needs to focus more on students discussion rather than sole teacher discussion. Furthermore, teachers need to focus on their student's ability to communicate their thoughts and positions effectively and have them practice debating their answers with the rest of the class. Only then will students gain the confidence in themselves and begin to argue with other classmates over conflicting answers or views, thus elliciting student centered classroom discussion.