Keywords: Activities, Teaching
Strategies, Communication
Ref: Nick1
Author(s): Koellner-Clark, Karen; Stallings, L.
Lynn; Hoover, Sue A.
Date: 2002
Title:
Socratic Seminars for Mathematics
Journal or
Publisher: Mathematics Teacher
Volume, Issue, Pages:
95(9); pp. 682-687
Reviewer: Nick
Date of
Review: 2/13/04
This article discusses the nature and effectiveness of ‘Socratic seminars’ for teaching mathematics in the classroom. The defining characteristics of a Socratic seminar are that the teacher takes a passive role in the learning process of the students, and that the students lead and maintain discussion of the topic. Leonard Nelson, a German philosopher and contemporary of Hilbert has said that the teacher’s role in the Socratic seminar is to ensure “a genuine mutual understanding among the students, the concentration on the respective question to prevent digression, and the preservation of the good ideas that had come up in the course of the discussion.” The authors of the article offer a few guidelines for establishing a respectful atmosphere for the Socratic seminar. These include that participants should respect one another’s ideas, that participants do not interrupt one another, that participants settle disagreements among one another (without the teacher’s input), and that the discussion does not digress. The seminars should center on a mathematical problem that is both thought-provoking and challenging, and the goals of such a seminar should be high levels of interest and involvement from the student.
The authors test the effectiveness of the Socratic seminar by observing a handful of such seminars meant to explore and clarify the students’ understanding of the definition of function. Students are asked to discuss and determine, given certain examples like a child swinging on a swing, which of four graphs best illustrates the relationship between time and the motion of the object. Dialogue from the sessions is included, allowing us to see how the sessions progress. The authors conclude that the method of Socratic seminars is very effective in encouraging students to assume responsibility for reasoning and communicating mathematical ideas. Student and teacher feedback shows that all people involved thought the seminars were fun, and students involved in the seminar later showed an understanding of concepts that other students did not.
This article is helpful in explaining and
promoting the positive outcomes of the Socratic seminar method,
making it an excellent resource for teachers who are seeking to
expand their repertoire of teaching techniques. The authors
provide enough dialogue from actual sessions and reflection on
those sessions to give the reader a good feeling for how the
session might play out in their classroom, and what their benefits
and limitations might be. One caveat that the authors fail to
emphasize is that Socratic seminars rely both on the choice of a
well-suited problem, and on the availability of a good chunk of
time (45 minutes or so). Also, it is important that the students
involved feel comfortable with one another. Still, the article is
a good introduction to the method of Socratic seminars in teaching
mathematics – a productive and fun way to enrich students’
reasoning and communication abilities.
Keywords: Research , Issues, Teaching Strategies
Ref: Nick2
Author(s): Brahier, Daniel J.
Date: 2000
Title: Learning Theories and Psychology in Mathematics Education
Journal or Publisher: Allyn and Bacon
Volume, Issue, Pages: Chapter 2 from the book Teaching Secondary and Middle School Mathematics
Reviewer: Nick
Date of Review: 2/17/04
The second chapter of Daniel Brahier’s Teaching Secondary and Middle School Mathematics is entitled “Learning Theories and Psychology in Mathematics Education”, and it addresses issues in researching mathematics education and discusses a few different learning and teaching theories. First, the chapter discusses the importance of research in mathematics education, and the different ways in which such research can be carried out. Two major types of research guide decision-making in the field of math education: quantitative research, which deals with gathering and analyzing numerical data from tests and surveys, and qualitative research, which involves the collection and study of non-numerical data such as videotapes of classroom episodes, transcriptions of student-teacher interactions, and summaries of students’ journal entries. Further, research can be either experimental or descriptive: the former refers to research that is done in an attempt to prove that one method is better than another, and the later refers to research that is carried out simply to generate statistics and information for discussion.
The chapter then goes on to discuss some theories on learning mathematics that are heavily rooted in psychology. Jerome Bruner theorized that learning passes sequentially through three stages of representation – enactive, iconic, and symbolic stages. In the first stage, learning is done though concrete physical interaction with that which is to be learned about. In the second, physical interaction is replaced by visual representations of the physical situation, and in the third stage we reach total abstraction as words and symbols representing information replace visual representations. It is important to note that each of these stages relies on the ones previous, and that if confusion appears at one stage, it is essential to return to the earlier stages of learning as a basis.
The van Hiele Model is a cognitive theory about geometric learning development that postulates that students pass through five levels of geometric reasoning. These are: the visualization phase, where students learn how to identify objects; the analysis phase, where students become able to recognize the attributes of a geometric object; the informal deduction phase, where the student will begin to compare geometric objects and produce simple proofs; the deduction phase, where students begin to understand theorems and write proofs, and; the rigor phase, where students become capable of working on an abstract, proof-oriented level. Research shows that students must pass though these levels sequentially, and that often geometry is taught at too high a level early on. We must make sure that we build the foundations of the lower levels before expecting students to perform adequately at the higher levels.
Methods of teaching have been devised which reflect upon these learning theories. The chapter gives a good example of what is called an inquiry lesson – a lesson in which students work through an activity (often in groups) and essentially invent their own mathematical definitions and rules. This type of lesson fits squarely within the constructivist model of teaching and learning, which holds that knowledge cannot be passively transmitted from one individual to another, but must instead be built up from within from our own experiences. Children create knowledge by both by doing and reflecting upon what they have done. While the constructivist method makes sense on many levels, practicing it requires a significant amount of skill, both in selecting appropriate activities and in guiding students, and it is advisable that such methods also include standard teaching practices such as individual work, pencil-and-paper tests, and lectures.
The chapter goes on to articulate the distinction between inductive and deductive teaching, giving examples of both. Inductive teaching involves starting with several concrete examples and moving towards a general conclusion, allowing students to see the conceptual steps that are taken. Deductive teaching, a more traditional method, entails having the teacher state the rule or definition at the beginning, and then expecting the student to apply that rule or definition to a set of problems. While the inductive method can often be challenging and fun for students, it also requires more time and creativity on the part of the teacher than the deductive method. Even though this is the case, the author stresses that the teacher must be willing to examine what is best for the student when making decisions about teaching methods, and not simple gravitate to what is easiest for the teacher.
Important to a student’s level of engagement in a classroom activity is the student’s motivation, which can be broken down into three different aspects – goal orientations, emotions, and self-confidence. Students are motivated by two types of goal orientations: ego goal orientations, in which students are motivated to succeed in order to gain favorable judgments from others, and mastery goal orientations, in which students are driven to learn based on their belief in the intrinsic value of learning. One challenge that teachers face is trying to develop in their students a strong master goal orientation. The author provides a few examples of how one might go about doing this, as well as a few examples of what not to do. Motivation also comes from a student’s emotions– their interest and curiosity toward the subject. Oftentimes teachers have a great ability to effect these emotions in their students, whether by example or by creating captivating lessons. Finally, self-confidence plays an important role in a student’s motivation, and so teachers are responsible for selecting tasks that will challenge and build self-confidence for more difficult concepts later on.
Related to and encompassing this discussion is the concept of a student’s disposition towards mathematics. This refers not only to the attitudes a student has but also his or her tendency to act in positive way towards the subject. It is important to monitor and assess students’ dispositions towards mathematics in order to develop new ways of teaching the subject effectively, and in order to avoid the creation of mathematics anxiety in students. Many studies show that students’ dispositions often model that of the teacher, and so it is important that the teacher always be enthusiastic and positive about the topics they are presenting.
I found this chapter to be highly informative and helpful as a background to the general theories behind teaching mathematics. It frames nicely the ways in which students often learn, and thoughtfully discusses the many methods teachers can use to engage and challenge their students. I particularly found the numerous classroom examples enlightening, as they truly paint the picture of how the general principles discussed in the text can take shape in the classroom. I definitely see the positive aspects of the constructivist model and the methods of inductive teaching – they certainly seem like natural ways to teach not only the subject content of mathematics but also how to think mathematically. I am glad, however, that the author takes care in his discussion to outline some of the drawbacks of these methods, offering a solution that works as a compromise between older approaches and newer theories. The chapter also got me interested in the research methods needed to develop the theories pertaining to mathematics education. The area of mathematics education research seems to be a very vital and interesting area of study, and one with many nuances and pitfalls.
Keywords: Representations, Research , Teaching Strategies
Ref: Nick3
Author(s): Aspinwall, Leslie; Shaw, Kenneth
Date: 2002
Title: When Visualization Is a Barrier to Mathematical Understanding
Journal or Publisher: Mathematics Teacher Magazine
Volume, Issue, Pages: 95(9): 714-717
Reviewer: Nick
Date of Review: 2/24/04
In this article, the authors examine the role of visualization in mathematical understanding. Almost unanimously, teachers tend to believe that the ability of students to come up with vivid and dynamic visualizations for the mathematical objects they are dealing with is a good thing, and that those visualizations increase the amount of mathematical understanding the students have. By looking at a case dealing with one student's inability to draw coherent conclusions about a graph and its derivative due to the student's vivid and unalterable visualization of the graph, the authors argue that visualization is not always a good thing for mathematical understanding, and that sometimes it can even be a hindrance.
The case the authors study deals with a young student's impressions of the graph of a parabola, and his visual representation of the derivative of that parabola. The student decided that the derivative would look somewhat like a graph of y = x^3, because he thought the graph of the parabola would eventually increase asymptotically to infinity at some point. When asked if the derivative could just be a straight line, the student said that would be impossible. When told that the symbolic expression for the parabola would probably be y = x^2, the student realized that the derivative should be linear, and that the parabola would not increase asymptotically. The student had reached understanding symbolically, but then his visualization again got in the way. Looking at the graphs again, he said that there must be some contradiction, because according to the picture, the derivative could not be linear and the parabola would increase asymptotically. The authors argue that his visualization was too powerful and uncontrollable to let his symbolic insights lead the way to understanding. Using this one example, the authors postulate that visualization can sometimes be a problem, and that other methods of 'seeing' mathematical objects (such as symbolically) must also be stressed as important.
This article is helpful because it brings to our attention something that our intuition does not always reveal to us - that having a picture in mind can sometimes be misleading and keep us from reaching understanding. The authors' careful examination of the case presented in the article is important in illustrating and validating this important claim. It is important to remember that visualization should not be considered the end of mathematical learning, but should instead be taken as one of many ways of seeing a mathematical object. Symbol sense, which is often played down, should be considered equally important for its ability to clarify mathematical concepts, and both symbolic and visual methods should be practiced in teaching and learning mathematics.
Keywords: Planning, Activities, Management
Ref: Nick4
Author(s): Basden, Jon
Date:
Title: The Area of a Parallelogram
Journal or Publisher:
Volume, Issue, Pages: http://www.highland.madison.k12.il.us/jbasden/lessons/llgramarea.html
Reviewer: Nick
Date of Review: 2/22/04
This lesson plan outlines an activity for finding the area of a parallelogram by cutting and rearranging it so that it becomes a rectangle, and then using what is known about rectangles and area. The lesson is meant for 7th graders, and should take approximately forty minutes.
The lesson plan states that before the lesson is started, students should have a working knowledge of area (specifically in relation to rectangles) and the basic properties of parallelograms. Three objectives are stated, involving the ability to solve practical computational problems involving rational numbers, the ability to apply the concepts of length and area in practical situations, and the ability to use concrete models and formulas to find the areas of two-dimensional regions. The lesson requires a number of listed materials, including specially designed worksheets, scissors, tape, and an overhead transparency of the activity.
Students are to receive their materials and the instructor states the need to find a general way to find the area of a parallelogram. The student is told to cut out the parallelogram on their handout, and also to cut the parallelogram into two trapezoids. The instructor asks if the students can find a way to re-arrange the two trapezoids into a shape for which an area formula is already known, and students start to work on taping their pieces of paper back together. Once students discover that they can form a rectangle from the two pieces they have, they are asked how to find the area of that new rectangle. With the instructor's help, the students postulate that the area of a parallelogram with base b and height h should have area A = b*h, just like with a rectangle. Students are then to do various problems from the textbook, and the instructor assigns problems that are applications of the newly discovered formula.
A few methods of assessment are included in the lesson plan. The instructor should be able to informally assess the comprehension of the students during the class lectures and discussion, and as the students work on in-class problems. Finally, formal assessment will come as the students complete an independent assignment.
Overall, this lesson plan gives a decent sense of how the lesson will unfold in the classroom. The worksheets that accompany the lesson plan are very helpful for facilitating the lesson, and the objectives of the lesson are clearly stated and well addressed during the lesson. However, there are many points throughout the lesson plan that could use some more detail. The prerequisites for the lesson are left rather vague (what does it mean to be 'familiar' with the concept of area, for instance), and no greater goal for the lesson is mentioned. No motivation is given for the lesson, and so I worry that the lesson, as outlined, could leave students uninterested and disengaged. The lesson procedure, while giving a good sense of the general sequence of the lesson, lacks transition statements and leaves holes here and there which could be confusing. For example, how is the instructor supposed to 'help' students to 'generalize' the formula of a rectangle to that of the parallelogram? What questions will the instructor ask? Later, the lesson plan mentions that students should try to determine the area of the trapezoid in figure #4 on the overhead transparency. There is no apparent trapezoid in this figure, however, and I wonder if the person who wrote the plan actually meant 'parallelogram' instead of 'trapezoid'. This could be confusing for a substitute teacher. Also, the lesson plan gives no mention to exactly which problems will be assigned to the students, or what the 'independent assignment' on which student assessment will be based is. The lesson plan should probably be more explicit here, and also list some problems or topics that can be discussed if there is extra time available. Finally, the closure of the lesson plan lacks merit because it cannot be tied back to earlier motivations for the discussion (since there were none), and this leaves the instructor with less insight into the effectiveness of the lesson. All in all, I'd say that with a little more detail and a good motivation, this lesson p plan could be much better than it is now.
Keywords: Activities, Statistics...
Ref: Nick5
Author(s): Cushner, Jason
Date: 2003
Title: Problem Solving the Problems of Society
Journal or Publisher: Mathematics Teacher Magazine
Volume, Issue, Pages: 96(5): pp. 320-23
Reviewer: Nick
Date of Review: 2/26/04
This article describes the experience of a class whose assignment it was to collect and analyze data from a nonprofit organization, the Snowboard Outreach Society (SOS). Students analyzed the budget of SOS, projecting its future needs, and analyzed program evaluations from SOS participants. When first encountering the data, many students hoped for formal ways to go about their analysis, and they decided to enter the data into spreadsheets and make graphs. The students learned some basic statistics and performed regressions to predict growth and how to obtain resources to support that growth. These predictions sparked a discussion on how growth should be measured, allowing students to communicate and reason. The students also learned a little linear programming, and how to maximize income given certain constraints. Oftentimes, students realized that the methods they had weren't sufficient in studying the data, and this was good motivation for learning new technique. No student ever asked why they needed to learn the material, and as they moved forward in completing the project, the students began to trust their own mathematical abilities. At the end of the project, the students wrote business reports and gave presentations. The author stresses the success of the project, noting that the connections it exposed between different areas of study was key to making it interesting and relevant. While learning problem solving skills, statistics, and the ins and outs budgeting and finances, the students also had the opportunity to integrate mathematics with social studies, writing, science, and service.
I enjoyed reading this article because the project it describes is really awesome. I like that the project allowed mathematical ideas to be presented in the context of a real world problem, and even more so, a real world problem that has to do with helping people and making a difference in society. The project is a great way both to interest students in the processes of mathematical thinking and problem solving and also to expose the students to the power of mathematics in the real world. The author really expresses this point, especially by quoting many of the students involved. This is nice, because though it's good to hear what the teacher has to say about the idea, it's very important to hear what the students have to say. The article is also helpful because it explains what planning was required in creating such a project. Clearly, the project took a lot of work on the part of the teacher, but it definitely seems worth it.
Keywords: Research , Technology...
>Ref: Nick6
Author(s): Heid, M. Kathleen; Blume, Glendon W.; Hollebrands, Karen; Piez, Cynthia
Date: 2002
Title: Computer Algebra Systems in Mathematical Instruction: Implications from Research
Journal or Publisher: Mathematics Teacher Magazine
Volume, Issue, Pages: 95(8): 586-591
Reviewer: Nick
Date of Review: 2/29/04
This article examines the effects of incorporating computer algebra systems (CAS) into secondary mathematics curricula. On question that must be taken into account is whether the regular use of CAS has a negative effect on the ability of students to do by-hand symbolic computations and manipulations. The article states that, according to research conducted with both seventh, eighth, and ninth graders and with college freshmen, these abilities are not altered in negative ways by the use of CAS in the classroom. In actuality, studies show that many of the students who regularly used CAS performed better on standardized tests.
But the fact that CAS does not hinder students' abilities to do symbolic manipulation is not a compelling reason for the use of CAS. Other studies have probed the question of whether CAS help to increase the overall conceptual understanding of students. These studies have shown that students do tend to have the same or better understanding of the general underlying concepts of mathematics, as well as more specific concepts, such as function and variable. The authors argue that, not having to spend time doing routine calculation, students can spend more time coming to an understanding of the mathematics, and that since CAS often produces results in forms that the student does not expect, the students learn to identify and interpret new forms of expressions. CAS also lends to students the ability to make and test conjectures, important in exploratory mathematics lessons. The authors conclude that CAS opens a door to all kinds of possibilities in teaching mathematics, but that more research needs to done, especially qualitative research about how students using CAS learn to reason symbolically.
Implementing any kind of new teaching tool, such as CAS, can be scary because the teacher has little idea as to how the new tool might effect the students learning. I liked this article because it gives teachers an idea of what to expect when using CAS in the classroom. The article outlines some of the major concerns about using CAS and then presents research pertaining to those concerns in a systematic and conclusive way. The authors' consistent and numerous references give the article strength and validity. I also like the way the research ties into the process standards we have studied in class. CAS provide a good way for students to work with different representations of mathematical ideas, and increase their reasoning skills though problem solving. Students also have to learn how to analyze and communicate the results the CAS gives them in more understandable terms. This analysis of CAS in terms of the process standards makes this article particularly relevant to our class and to teachers in the classroom.
Keywords: Activities, Algebra, Standards
>Ref: Nick7
Author(s): Underwood Gregg, Diana
Date: 2002
Title: Algebra for All: Building Students' Sense of Linear Relationships by Stacking Cubes
Journal or Publisher: Mathematics Teacher Magazine
Volume, Issue, Pages: 95(5): 330-35
Reviewer: Nick
Date of Review: 3/1/04
In this article, the author addresses the difficulties of teaching a conceptual understanding of basic algebra, and she gives some examples of activities she developed to help students make sense of lines and linear equations. At first, her attempts to create effective activities circled around the rule of three - using graphical, numerical and symbolic methods. Finding that these methods did not often lead to understanding, she developed a set of activities involving stacking cubes, noticing patterns, and graphing data.
The first activity involves, given a pattern made by stacks of cubes of increasing heights, finding a pattern and determining the heights of further stacks not shown. This allows students to explore rates of change between consecutive towers and come up with expressions for the number of cubes in a stack at a particular point in the pattern. The second activity involves moving toward the Cartesian plan and toward the equation of a line. Buildings were represented by 'sticks' or vertical lines instead of by cubes. Here students discovered that they could determine a formula for a sequence by first determining the change in height between consecutive towers, working back to find the height of the zeroth tower, and putting those things together in a form resembling y=mx+b. Thus when students finally encounter that formula, they have a reference from which to give it meaning.
The third and fourth activities involve determining rates of change between non-consecutive towers and then determining the heights of other towers in the sequence, building towards an understanding of the Cartesian plane and the use of ordered pairs. The fifth activity involved finding the heights of fractional or 'in between' buildings, leading up to the use of a line to represent the heights of buildings of any possible number. Then questions were asked about lines and slopes, further connecting latter activities to the current one and solidifying students' understanding of the concepts.
I like this article because it describes an activity that starts with understanding a concrete situation, and then slowly altering it until an understanding of more abstract concepts, still referring back to the more concrete examples, is reached. This ties in well with many theories on learning that postulate that students learn best when presented with a concrete physical situation, and using their understanding of that to move to more abstract formulations of the setting. This way, students can use their previous understanding of more concrete situations to build an understanding on a more abstract level.
The activities also embody many of the process standards we have talked about in class. Students are asked to make connections between what they have observed and learned in one setting with what they are currently learning. Linear equations are represented in different ways, first as stacks of blocks, then as vertical 'sticks', and finally as lines in the Cartesian plane. Reasoning skills are exercised and the communication of ideas is also involved.
This activity would be great to use in the classroom, and I'm sure that similar activities could be created in a similar way. It is also helpful to hear the author's own story about teaching this material, her difficulties and eventual success. She inspires us to be creative in our teaching methods and in the activities we create, and to constantly be aware of students' needs and concerns.
Keywords: Problem Solving, Connections, Activities
>Ref: Nick8
Author(s): Yoshinobu, Stan T.
Date: 2003
Title: Mathematics, Politics, and Greenhouse Gas Intensity: An Example of Using Polya's Problem-Solving Strategy
Journal or Publisher: Mathematics Teacher Magazine
Volume, Issue, Pages: 96(9): pp. 646-648
Reviewer: Nick
Date of Review: 3/8/04
In this article, the author discusses a problem he devised and used in the classroom based on an article he read in the newspaper about new environmental protection legislature. The news article he read stated that new legislature would require a cut in greenhouse gas intensity by 18% in the next ten years. Greenhouse gas intensity, not defined in the news article, was later found to be defined as the volume of greenhouse gas emissions divided by the GDP of the country. The question the author posed in class was, "If the GDP of the US is increasing at 2% a year, will the initiative have a positive effect on the environment?"
The author explains how the problem can be dealt with on many different mathematical levels, depending on the level of the students. He had his students get into groups of two or three and work on the problem in class, finding that many students were frustrated by the open-ended nature of the problem. Many students asked for more numbers, and many had only a faint idea as to the kind of answer they were looking for.
The author explains how he used Polya's four steps to problems solving in his class to get students focused on the problem. These steps are: 1) understand the problem, 2) devise a plan, 3) carry out the plan, and 4) look back at the solution. The author comments on how students usually skip all but the third step when working on problems, and then goes on to show how discussing these four steps in class fostered a discussion that led to the students coming up with a common goal. As far as numbers went, many students used variables, while others assigned arbitrary numbers to the problem. Students had to compute the estimated GDP of the US in ten years in order to find the amount of greenhouse gas emissions the initiative would allow. Finally, this amount would have to be compared to current emission levels.
The author goes on to explain how students went about doing the problem, outlining Polya's four steps. The author emphasizes the 'looking back' step, telling us that it is a good way to find errors in reasoning and learn more about mathematics, and it also allows the students to consider other scenarios. Finally, the author discusses the role of the teacher in considering the appropriateness of the mathematical content of the problems they assign for their students, as well as other issues to take into account, such as time management and enrichment opportunities.
I found this article to be very good. I especially liked how the author provides a good example of how teachers can cull interesting and challenging mathematics problems from important news articles and current events. In this way, mathematics teachers can help students to better understand the world they live in, and prepare them for more informed and insightful political and social discussions. It's great to see a teacher really thinking about his class and being creative in his curriculum.
I also liked many of the reflections he had on teaching the problem, especially on the importance of first understanding the problem, and finally looking back on solutions once the problem is solved. The author provides a lot of good insight into why these steps are important and should be stressed in problem-solving. It was also good how the author voiced the concerns and fears of his students at the open-endedness of the problem, and how the class overcame those fears together by discussing the problem.
It was good to see a teacher implementing a classic problem solving strategy such as Polya's. I would have liked a little more detail in the description of each of the steps of the problem, and this is perhaps the one weak area of the article. If the author had gone into more depth about how this particular problem solving technique works and what its strengths and weaknesses are, the article would have been slightly more compelling. But still, I think it was a very refreshing and interesting article to read, with many good ideas throughout.
Keywords: Curriculum, Geometry, Standards
>Ref: Nick9
Author(s): Coxford, Arthur F.
Date: 1992
Title: Geometry from Multiple Perspectives
Journal or Publisher: The National Council of Teachers of Mathematics
Volume, Issue, Pages: Curriculum and Evaluation Standards for School Mathematics, Agenda Series, Grades 9-12
Reviewer: Nick
Date of Review: 3/9/04
This book treats the standard 9-12 geometry curriculum while reflecting new methodologies supporting new curricular goals, and linking the content proposed in the "Curriculum and Evaluation Standards" to that of current programs. It approaches geometry from both the synthetic perspective (where geometric objects are described by segments, points and angles) and the analytic perspective (where geometric objects are coordinatized). The book gives a short introduction to the relevance and power of geometry, linking the subject to the real world and real-life problems. The use of technology is emphasized as a means to exploration and understanding, and material is presented with a view to the five process standards - reasoning, problem solving, connections, representations, and communication.
The book includes many activities that explore certain concepts more in depth, often asking students to 'discuss' their observations and make conjectures. While the activities do not normally root themselves in real-world situations, they seem successful in presenting mathematics as exploration and problem solving. Other features of the book are also quite helpful. The 'Try This' sections provide the teacher with other interesting problems and areas of exploration within the material. The 'Teaching Matters' sections keep teachers tuned into what students should be learning and what level they should be working at. The 'Assessment Matters' sections alert teachers to possible difficulties students might have as certain points, and give them ideas as to how these difficulties might be arising, and how they can be rectified.
All in all, "Geometry from Multiple Perspectives" is a valuable tool for teachers who want to present geometry to their students in a refreshing and interactive way. The book covers a very wide array of topics within the geometry curriculum and provides new approaches. The inclusion of special features such as the 'Try This' and 'Assessment Matters' sections greatly increase the desirability of the text, making it a fairly useful and novel resource.
Keywords: Assessment, Standards...
Ref: Nick10
Author(s):
Kastberg, Signe E.
Date: 2003
Title:
Using Bloom's Taxonomy as a Framework for Classroom
Assessment
Journal or Publisher: Mathematics
Teacher Magazine
Volume, Issue, Pages: 96(6): pp.
402-405
Reviewer: Nick
Date of Review:
3/13/04
In this article, the author seeks a way to improve her assessment of students' knowledge and abilities through the tests she creates for them. Troubled by the feeling that many of the tests she writes only assess students haphazardly, the author looked to Benjamin Bloom's "Taxonomy of Educational Objectives" as a framework for future assessment. In the taxonomy, six processes are emphasized and put into hierarchy: knowledge, comprehension, application, analysis, synthesis, and evaluation. The author defines each of these processes and gives examples of the kind of test questions that would properly assess each of them.
In order to use the Taxonomy as a framework for assessment, the author created what she called a Content-by-Process Matrix. This is essentially a table with the six processes along the top, and the content that needs to be assessed along the side. Each cell of the matrix corresponds to one aspect of the content and one of the processes from the taxonomy. The teacher can use this matrix to analyze that content areas are being assessed by a test, and through what processes they are being assessed. The matrix also makes it easier to see what areas of assessment are being neglected. Furthermore, the Content-by-Process Matrix can help align what is taught with what is being assessed, so that students aren't assessed on things they didn't learn, and so that they are assessed on those things they did learn.
The author introduces a powerful method for analyzing assessment and aligning what is being taught with what is being assessed. She does this in an almost mathematical way, by creating a structure through which to view information (the matrix) and then analyzing the results. The practice that she describes seems as though it could be helpful to any teacher in making their assessment of students more effective and productive. It is also an ingenious for the teacher to assess his/her own modes of assessment. For these reasons I find this article extremely interesting and helpful. I will certainly keep the author's ideas in mind as I plan lessons and write tests and formulate other modes of assessment.
Keywords: Number and Operation, Activities, Standards
Ref: Nick11
Author(s): Bay-Williams, Jennifer M.; Martinie, Sherri L.
Date: 2003
Title: Thinking Rationally about Number and Operations in the
Middle School
Journal or Publisher: Mathematics Teaching in the Middle School
Magazine
Volume, Issue, Pages: 8(6): pp. 282-287
Reviewer: Nick
Date of Review: 5/5/05
This article focuses on the NCTM standard of Number and Operation in middle school by considering three problems and how students have worked on those problems. By examining these problems in the classroom, the authors hope to illustrate some of the middle school content expectations within the Number and Operations standard.
The first problem deals with comparing the values of a set of fractions, and ordering them from least to greatest. This problems corresponds to one content expectations that the NCTM sets forth – that students should be able to compare and order fractions, decimals and percents, and find their approximate locations on the number line. The problem is introduced to the class in a physically concrete way, and students reason through it in many different ways. Some use benchmarks like 1/4, 1/2, 3/4, and 1 to decide how the numbers are arranged on the number line. Other groups converted all fractions into decimals. Others divided the number line with various tick marks, using those as a guide to pinpointing where the fractions belonged on the number line. Here it is important that students learn how to move between various representations of rational numbers and can decide which forms are best in different situations.
The second problem involves cutting three yards of ribbon into sections of 5/12 yards to make bows and determining how many bows can be made, introducing the idea of dividing a whole number by a fraction. This problem works toward the content expectation that students should understand the meaning and effects of arithmetic operations on rational numbers, and should be able to develop and analyze algorithms when computing rational numbers. Some students converted 3 into 36/12 and worked from there, while others worked more concretely, adding 5/12 until they reached 3. The goal of such problem is to illuminate the reason why dividing by a fraction is equivalent to multiplying by its reciprocal – an algorithm that few understand, and even less can really explain. The hope is that by connecting a context to such an algorithm, students will be more able to reason through and remember the algorithm.
The third problem involves figuring out how much you could lift if your strength was proportional to that of an ant, invoking the idea of proportionality, which is a key mathematical idea permeating the curriculum. Some students did the algebra in a fairly straightforward manner, and others found how many times its body weight an ant can lift, and then multiplied this by their own weight. Again, using the concrete example of the ant makes the math fun and interesting, and it allows students imaginations move freely.
Embedding the mathematical ideas surrounding rational numbers in real world contexts seems to make them more memorable and interesting, and it allows the student to use his/her own reasoning to come to conclusions about things. I find this an important thing to remember when deciding how to present such concepts, and I appreciate the way in which this article emphasizes this point. I like the way in which the article concerns itself with the NCTM standards, and is constantly relating the problems it covers to those standards. I also find it helpful that the authors include many of the students' responses to the problems. Frankly, I also enjoy just reading and pondering the problems that the authors present, all of which seem to be excellent ways to get into the ins and outs of rational numbers and operations involving them. I would say that this is a very good article to read – it is informative and interesting, and provides some good teaching ideas.
Keywords: Geometry, Activities, Manipulatives
Ref: Nick12
Author(s): Lege, Steve
Date: 1999
Title: "Why Not Three Dimensions?"
Journal or Publisher: Mathematics Teacher Magazine
Volume, Issue, Pages: 92(7): 560-563
Reviewer: Nick
Date of Review: 4/16/04
This article describes a three year sequence of projects that can help students develop their spatial skills. The author introduces this article by reviewing his high school curriculum, and noticing that students did not have enough practice during algebra and pre-calculus with situations that required analysis in three dimensions. Consequently, when those students reached calculus, they lacked experience necessary to visualize solids created by rotations.
The first project is meant for second-year algebra students, and deals with conic sections. Students are to create models of ellipses given a cylinder and an angle at which the section of the cylinder is to be taken. The project is tied to real world experience by using the context of a building construction project. Working in groups, each student would work on their own individual problem, and share ideas and techniques.
The second project deals with quadric surfaces, which are generated when conic sections are rotated through space. In this project, students are given a set of xyz-axes, created by wooden rods, and an equation for a quadric surface. Students work in groups and each student in a group works on a different quadric surface. Students use cross-sections and other methods to model their surface spatially, and learn how equations in three variables and three-dimensional surfaces are related.
The third project deals with students' difficulties with visualizing solids of revolution, as seen often in calculus. Working in groups, students are given materials to construct three-dimensional solids of rotation from a given two-dimensional shape. The author claims that students became much more comfortable with understanding and visualizing solids of revolution in problems, and would therefore be better prepared for the AP Calculus exam.
The author concludes by stating that students' understanding of three-dimensional situations can be improved by hands-on model building. Sadly, these kinds of activities are lacking in current curricula and textbooks, and so teachers should use projects like those described in the article to augment their students' learning.
Overall I thought this was a good article. The author gets fairly detailed about the projects he has created, and he provides worksheets for the projects in the article. This makes it easy to use such projects in our own classrooms. The idea of getting the students to physically model the math they are doing appeals to educational psychology research which says that spatial experience and important requisite for abstract understanding, and as the author claims, it works. I would have liked to hear more of the students' comments about the projects, as well as seen some possible avenues of assessment, but in general I think this is a strong article.
Keywords: Proof, Activities...
Ref: Nick13
Author(s): Fidler, Mark
Date: 1999
Title: "Chipping Away At Proofs: A Cooperative Approach"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(7): 565-567
Reviewer: Nick
Date of Review: 4/16/2004
In this article, the author grapples with the best way to get kids interested in doing proofs and reasoning through geometric problems. The author realizes that doing proofs can be exciting to kids when a student rushes into class one morning exclaiming that she had finally figured out a proof while collaborating with another student. The fact that the proof was done collaboratively led the author to ponder the possibility of getting students to work more cooperatively on doing proofs. When students worked individually, the author noticed that many students would give up if they weren't able to reason through a proof in when stroke of insight. After many attempts to arrange a cooperative proof-doing experience, which would get students excited about doing proofs and cause some kind of improvement in students' reasoning skills, the author finally found the right formula.
It took a few tries before students became comfortable working in groups on quizzes and tests that involved proofs, but once they did, the teacher noticed much dialogue and discussion between students, and the work they were turning in was much improved. The author found it important to pick problems of a suitable difficulty. It is important to pick some proofs that will present a challenge to the students (while still do- able), as well as some proofs that are easier, so that students feel confident in their abilities. It is also important, as the author notes, that students be arranged in groups by ability. The author arranged her students this way, having students who usually get A's work in pairs, those who earn B's work in threes, and those who generally work at the C level work in groups of four. This way, each group was happy with itself, either in the abilities or the number of brains at work on the problems.
The author concludes that now his students enjoy and look forward to working on proofs in groups, and also like to take group quizzes and tests. This is a testament to the fact that proofs and doing mathematics can be exciting and fun.
I really like the way in which the author tells the story of his struggle with making proofs and mathematical reasoning interesting and fun for his students. It shows us that having a good idea is not enough in teaching, but that one must constantly refine their methods until success is reached. I also liked the amount of student feedback the author presents, because it really sells the idea that he is trying to present: if the students like it and are doing well, then it must be good. It's also good to see a teacher that is committed to engaging his students and working on his own teaching methods – it is an inspiration. But specifically, the article is helpful because it provides us with some concrete ideas for how we might also engage our students in proof writing.
Keywords: Probability, Communication, Problem Solving
Ref: Nick14
Author(s): Lesser, Lawrence M.
Date: 1999
Title: Exploring the Birthday Problem with Spreadsheets
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(5) : 407-411
Reviewer: Nick
Date of Review: 4/16/2004
The birthday problem illustrates the tendency for people to underestimate the probability of something happening more than once. The problem asks how many people would have to be in a room for the probability that two of them share a birthday is 50%. Using Excel, the author uses a recursion method on the probability of a group of size 1, 2, 3, …, k to solve the problem. Whereas the idea of recursion can be hard to grasp as a student, the author finds that for most student, the idea of one number mathematically following from the one above it in a spreadsheet is fairly natural.
Although the previous approach yields an answer, it does not illuminate why the number of people needed in the room (23) is so much lower than most predictions. A second method compares the number of people in the room with the number of opportunities for matching birthdays. This comparison yields a approximating function that grows steeply, and so students can see why the probability increases so quickly. The author then applies calculus to this method to show that such an approximation is a good one.
Studies show that simulations are a good way for students to learn about mathematics. The author recommends using Excel as a way of doing simulations like the birthday problem quickly and naturally. Using the methods the author explains, students can start a discussion about the virtues of each method. The first is more accurate, but it does not tell us _why_. The author also explains the value of having students give a prediction before investigating the results of the birthday problem. This links students more closely to the problem, and make it easier to address misconceptions.
I like this activity and the way the author presents the mathematics and teaching issues surrounding it, because it ties into so many different things. First of all, it's a great way to get students thinking about probability and data organization, as well as a good way to get students familiar with computers and Excel. Secondly, the problem allows room for a mathematical discussion, and helps students to understand the process of mathematical modeling. It also shows students that most math problems can be solved in many different ways, and that different techniques have their own pros and cons. Mathematics is much more qualitative that we are usually led to believe, and I like that this article turns that misconception on its head.
Keywords: Communications, Issues, Teaching Strategies
Ref: Nick15
Author(s): Krussel, Libby
Date: 1998
Title: Teaching the Language of Mathematics
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91(5) : 436 – 441
Reviewer: Nick
Date of Review: 4/24/04
In this article, the author discusses the role of mathematics as a language. She starts her discussion first by asking the question of whether we can actually consider mathematics a language. She provides a few different takes on this question, but finally comes to the conclusion that math is indeed a true language, if only in metaphorical, and not literal, sense. Not only that, she also argues that in order to make mathematics education more effective, we must view it as a language.
Recognizing math as a language, we are now concerned not only with teaching math as it has traditionally been taught, but also the question of how to teach the language of mathematics. There are two general ways in which this can be done: explicitly, through the study of its structure using books dealing with grammar, vocabulary, and syntax; or implicitly, by immersing the student in written, oral, visual, and aural conversations. The author considers these two methods of teaching, deciding that it might be best to teach implicitly at the younger grade-levels, and then move into more explicit territory later.
In coordinating mathematics teaching with the teaching of mathematics, the author is able to make some new claims about math education. First, that illiteracy has unfortunate consequences. Secondly, that students are able to learn multiple languages, and therefore mathematics is not only for the mathematically inclined or talented student. Thirdly, we must realize that vocabulary is not enough, but also that students have to learn the grammar of mathematics to be fluent in it. Of course, we should not overemphasize the study of grammar, either: students also need practice listening to and speaking the language clearly and precisely. These claims, along with many others, are made by the author in light of her understanding of mathematics as a language. She feels that math education can be bettered if we view it in this light, and many new approaches can be made apparent.
I found this article fascinating. There is clearly much to be gleaned from viewing mathematics as a language, and the author makes this quite clear. Her claims about how mathematics can and should be taught as a language are powerful, and speak to many of the current issues in mathematics. By teaching math as a language, teachers can hope to create fluency in their students, allowing them to communicate mathematics clearly and precisely, and leading them to deeper understanding.
Keywords: Geometry, Activities, Proof
Ref: Nick16
Author(s): Boswell, Laurie
Date: 2004
Title: Activities to Develop Reasoning in Geometry
Journal or Publisher: MCTM Annual Conference, Duluth
Volume, Issue, Pages: April 30
Reviewer: Nick
Date of Review: 5/5/04
This presentation, given at the annual MCTM Conference in Duluth, dealt primarily with ways in which teachers develop reasoning abilities in their students through geometry activities. The first point that the presenter made was that, in most geometry activities and worksheets, pictures are provided for the student, and therefore the student gets little practice in coming up with their own picture given certain information, a very important step in the process of geometric reasoning. The presenter therefore urged those in attendance to give their students the opportunity to practice such skills. For example, given information like "Y is between X and Z", students should be able to draw the corresponding picture – a line with three points on it, with Y in the middle. It was noted that oftentimes students will draw Y as the midpoint of X and Z, and that it is important to emphasize that this need not be the case. Students need to recognize where they might be making extraneous assumptions about a situation, and when their drawing of a situation might be misleading.
On top of this, the presenter argued that students should be given practice at, after drawing a picture to represent a situation, coming to the proper conclusions. For example, given the situation, "angle T and angle S are complimentary angles", students should be able to represent the situation graphically, and come to the conclusion that m(T) + m(S) = 90. The presenter noted here the possibility of many different drawings: the angles S and T could make up a right triangle, or they might be represented as the two angles adjacent to the hypotenuse of a right triangle, or they might be drawn completely apart from one another. Each of these representations is correct, but they carry with them different assumptions and different possible insights.
The point of such activities is to give our students familiarity with the building blocks out of which proofs are made. The standard two-column proof is constructed out of a number of singular observations and conclusions, all tied together to make a logical whole. If students become comfortable with making single conclusions about a situation, they have already made much progress towards the ability to write full proofs. All that remains is for the students to put each of the observations together, to connect the dots.
To strengthen this ability, the presenter provided one final activity, which consisted of a given situation, the conclusion to be made, and a two line proof for the conclusion that has been cut up into individual steps, with each statement separated from its justification. The activity involves re- constructing the proof in such a way that it becomes logically coherent, an activity well-suited for cooperative groups, and which encourages communication while fostering the growth of proof-writing skills.
The last activity the presenter showed to us involved guessing a quadrilateral given certain hints about its properties. Hints about the shape are revealed one-by-one, and students discuss, after each hint, what they can infer about the shape. At some point, a student will become certain about what shape is being hinted at, and at that point we ask the question, "do we have enough information? How do we know? What if we didn't know this particular hint to be true?" In this way, students learn about drawing conclusions given certain information, and about logical necessity – what is necessary in order to know something.
I found this presentation to be very interesting and helpful. First, the presenter provided us with a number of great ideas for class activities that would help students with geometric reasoning, and she provided handouts containing each of them. Not only that; she also explained the purpose of each activity and how it could be taught, and why it would be effective. All of her activities were based in the well-researched proposition that students learn better in increments, a little bit at a time. It is hard for students to get their minds around proofs if they are not warmed up to the idea first, so getting kids accustomed with the building blocks of proofs is a great way to ease students into doing full proofs.
The presenter spoke clearly and effectively, addressing all her points in turn and giving good instructions. She was very easy to listen to and seemed very comfortable with her audience, and this made the presentation fun and interesting. This rounded out the overall quality of the session, and made it that much more worthwhile. I would heartily suggest going to this session, if offered again, to anyone interested – I am certainly glad I went.
Keywords: Equity/Diversity, Issues...
Ref: Nick17
Author(s): Levi, Linda
Date: 2000
Title: Gender Equity in Mathematics Education
Journal or Publisher: 7(2): 101
Volume, Issue, Pages: 7(2): 101
Reviewer: Nick
Date of Review: 5/6/04
Although male and female students generally take the same math classes and achieve similar scores on standardized tests, there is still a significant difference between the number of males and females participating in mathematics after high school. This discrepancy leads us to believe that more needs to be done in school to foster gender equity in mathematics. This article considers the role of teachers in the struggle for greater gender equity in mathematics education, identifying three general roles and…
The first role that is identified is that of the teacher as a provider of equal opportunities, respectful of differences. Teachers who assume this role offer the same opportunities for students, but are not concerned in boys excel in mathematics and girls choose other areas of interest. They tend to believe that the biggest problem with gender inequity is that males' interests are more valued than females. Participation in mathematics is seen as important because it is primarily a man's activity, and the subjects that women are interested in are seen less important. These teachers tend to think that we should value other areas in which women excel as much as we value areas in which men excel. Such teachers usually worked to teach their students that all interests and talents should be valued, and that students shouldn't be forced into things they are not interested in. One criticism of such a role is that it is too idealistic. As the author points out, while gender equity is improving in society, it does not seem likely that child care workers will be paid as well as engineers anytime soon, and so we need to find other ways to empower females.
The second role describes the teacher as ensuring that boys and girls are the same, and that they have the same experiences. Such teacher make sure that all students are called on equally and given the same attention, and they don't provide choices for math activities – both boys and girls must experience the same activities. They also work hard to make sure that they don't view boys and girls differently in the classroom. One critique of this role is that even though studies have shown that boys and girls do tend to get very similar experiences in the math classroom, but males still vastly outnumber girls in math-related careers. Therefore it is argued that more needs to be done to truly make a difference.
The third role that teachers assume is to treat girls and boys differently in order to compensate for gender inequities in society. Such teachers work to promote girls' interest in mathematics and related activities, and they work to de-emphasize standard gender roles. Thus they treat both girls and boys specially, often helping boys with working cooperatively and in other activities that are viewed as typically female. Some teachers find this role problematic because they view differential treatment as inherently unfair.
The author notes that there is no best solution to the problem of gender inequity in the classroom. Each of these roles has its strengths and weaknesses, and none of them are set in stone. Further, it is not known of more exposure to mathematics has an overall positive effect on the lives of all women. The author is convinced that teachers need to use their best problem solving abilities to attack this problem – defining the problem, reflecting on our decisions, and examining their influence. The author also presents some ideas for activities that teachers can do together to stimulate conversation about gender equity in the classroom, and she provides a worksheet that can be used to foster such discussions.
I found this article very interesting, and a good way to arouse my thoughts on
gender equity in the math classroom. Listing and describing a few different
roles which teachers can take was helpful because it allowed me to evaluate my
own beliefs and potential practices against those roles, and it helped me to
think about what practices might be problematic, and the messages that certain
ways of teaching can give to students. Ultimately, it seems that teachers need
to pick and choose aspects from all three of these roles – those that best suit
their attitudes towards the issue. Moreover, we need to constantly be reviewing
our attitudes and practices and seeking out current research on the issue. I
like that the author provided some activities for teachers to do with each other
in order to encourage discussion on the issue. This is another key component to
addressing gender inequity in the classroom. Overall, I'd say that this article
is very helpful, and should be considered by all math educators.
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Keywords: Activities, Teaching Strategies, Assessment
Ref: Nick18
Author(s): Boyer, Kimberly R.
Date: 2002
Title: Using Active Learning Strategies to Motivate Students
Journal or Publisher: Mathematics Teaching in the Middle School
Magazine
Volume, Issue, Pages: 8(1): 48-52
Reviewer: Nick
Date of Review: 5/15/2004
In this article, the author showcases the ways in which she has used active learning in the classroom to combat the apathy and behavioral problems of her eighth-grade students. She relies on Harmin's "Strategies to Inspire Active Learning" as framework for designing and implementing her techniques. Harmin argues that actively engaging students in learning involves building a strong classroom community that promotes "dignity, energy, self-management, community, and awareness." The author addresses each of these aspects of active learning in turn throughout the article.
The author finds it important for students to be confident and to feel valuable to the class - in short, to have dignity. Thus she works to build character in her classroom, by putting up positive posters and quotes around her room, having students share good things they have done to others, and openly rewarding students for doing good deeds, and she reports that many students enjoyed such activities.
To energize her classroom, the author often engaged her students in constructive lessons, thereby making them active learners. She gives an example of how she has used constructive lessons in the classroom, and observes that students showed a good understanding of concepts after such lessons. In order to encourage self-management skills in her students, the author had them do self- assessments and discuss what they could do to improve their performance in class. This seemed to help a lot, with students doing much better on subsequent tests.
To facilitate a strong community in her classroom, the author designed activities for the students to get to know each other better. She had them make a mural on the back wall of the class about their experiences with mathematics in the class, and every day one student was picked to share his/her contribution. This activity created an air of positivity in the classroom, allowing students to work better together and minimizing behavior problems. Finally, to promote awareness in the classroom, the author designed lessons that would relate mathematics to the real world, making students more aware of the mathematics in the world around them.
Putting these strategies into practice for a few years, the author noticed a marked improvement in the performance of her students, less problems with behavior and negativity, and a virtual absence of questions like, "when are we going to use this stuff in real life?". The author emphasizes the importance of making the classroom an active learning environment, and also that, in order to be a top-notch teacher, teachers must do more than plan fun activities for the class. They must strive to make their classroom a caring environment where students can feel comfortable and valued
All of the author's suggestions and ideas seem like great ways to encourage student motivation through active learning and cooperation. Not only do they increase motivation, her ideas seem to decrease students' boredom in class, increase their ability to succeed in mathematics, and strengthen their ability to cooperate and get along with others. Her methods make the classroom a positive environment, and not one where students feel intimidated, isolated, or insignificant. What a great way to change the classroom for the better. I especially like that the author included her own students' responses, because this really gives us a sense of the validity and efficacy of her methods. From what students say, the author's classroom is much improved, and students feel more free to share their ideas with the class. This is the foundation of constructive and cooperative learning - that students feel free to express their ideas - and the author has found a great way to zero in on it.
Keywords: History, Teaching Strategies, Activities
Ref: Nick19
Author(s): McDaniel, Michael
Date: 2003
Title:
Not Just Another Theorem: A Cultural and Historical Event
Journal or Publisher: Mathematics Teacher Magazine
Volume, Issue, Pages: 96(4): 282-284
Reviewer: Nick
Date of Review: 5/7/2004
In this article, the author stresses the importance of teaching major theorems, such as the Pythagorean theorem or the quadratic equation, as important historical and cultural events. The author suggests that doing so will encourage the students' interest in mathematics not only as a modern science, but as an important part of our heritage. The author makes many comparisons between mathematics and history, saying that teaching an important theorem in mathematics is like bringing an actual historical artifact to the classroom. In history this is quite rare, but in mathematics it happens quite often. Additionally, the historical gems of mathematics are imbued with pure logical truth, and are therefore even more important and awe-inspiring.
Since theorems are essentially important historical artifacts, the author argues that the teacher and students should treat it with reverence and awe. He suggests that, on the day the theorem is to be presented, the teacher should hold class in a different, more special setting, and bring other things to class to set the mood, such as old math books in foreign languages, maps of the world from the time of the proof, and food characteristic of the time and place. The teacher should also create an elaborate presentation for the theorem, and create activities for the students involving doing a proof of the theorem. He also notes that students should dress up nicely for such days, since they are essentially celebrating an important moment of history.
The author concedes that some teacher might find all this commotion over the presentation to be a conceit. Indeed, the author agrees with this view, but judging from student responses (which are only vaguely alluded to in the article) he argues that they are well worth the extra effort. Furthermore, the author notes that, after a few times presenting a proof in such an involved way, it becomes much easier for the teacher to prepare.
I thought this was a very interesting article. I totally admire the author's reverence and belief in the importance and significance of mathematics to the history of humankind. It is undoubtedly very important for teacher to view and teach mathematics not only as a science but also as a humanity – a historically significant and ongoing project and a testament to humankind's aspirations for truth, order, and beauty.
On the other hand, I think the extent to which the author feels he should emphasize this point (and the ways he suggest we emphasize it) seem almost ridiculous at times. Some of his suggestions seem too difficult to coordinate for the amount of enrichment they provide – like finding a special room to use for that day's class. This might be hard to organize, and it does not seem like it would really help all that much. Bringing in old books also seems a little over-the-top, and the author even acknowledges that some kids to not find this to be helpful. However, his suggestion that the teacher provide a timeline or list some other significant historical events of the time of the proof does seem like a really good idea to me, and these things are rather easy to do. Perhaps the strangest thing that the author suggests is that the students dress up nice for such days in class. I would personally feel a little awkward making such a request to my students, and again, I don't think that this really helps the lesson all that much.
Again, I really admire the author's enthusiasm and belief of the importance of the material he is teaching, and I agree that these attitudes need to come across to all students as much as possible. I do, however, think there are easier and more natural ways to do this, and I'll probably stick with those.
Keywords: Problem Solving, Curriculum...
Ref: Nick20
Author(s): Schettino, Carmel
Date: 2003
Title: Transition to a Problem-Solving Curriculum
Journal or Publisher: Mathematics Teacher Magazine
Volume, Issue, Pages: 96(8): 534-537
Reviewer: Nick
Date of Review: 5/10/2004
In this article, the author discusses the reasons for and ramifications of implementing a 'problem-solving curriculum' in a school where more traditional methods of teaching are the norm. The author bases her article on personal experience transitioning a class to more problem-based learning from a standard curriculum.
For the author, a problem-solving curriculum entails teaching students that they have the freedom to use their knowledge and skills to solve problems and add to their mathematical toolkit, often when students assume that they can't do anything without the teacher first telling them how to do it. She cautions that, in order to commit to the problem-solving method of teaching, educators must first believe that the major goal of mathematics education is to help students develop their ability to solve problems independently, before any content or curricular priorities. The author cautions that educators need to be determined to move away from the 'spoon-feeding' method of teaching mathematics.
The author holds that teaching topics concurrently is one of the most important features of a problem-solving curriculum, but she warns that shifting immediately into such a method of teaching could be too jarring for students. She suggests that teachers hold on to their old textbooks and have students use them when they want, and that teachers still follow the basic outline of their course. While doing this, however, teachers should be introducing more motivational problems to students, and homework assignments should be rooted in problem-solving and in-class discussions. The author gives an example of an in- class session and corresponding homework assignment to illustrate her ideas.
The author stresses that there are many hurdles to overcome when implementing a problem-solving curriculum. One of these hurdles involves the need to readjust assessment techniques so that they are more in line with the new curriculum. New tests and ways of assessment need to be created (including reflective journals and oral assessments), and the way that the teacher grades student work has to change in order to reflect the value of problem-solving skills. Another hurdle that teachers might encounter is their own comfort level with the mathematics they are teaching. In order to teach creative problem-solving methods, one has to truly understand the ins and outs of the mathematics they are dealing with, and this can take some time. Finally, teachers need to be confident enough to let class sessions lead to where they may, and they must learn to take full advantage of teachable moments when they arise.
Along with the numerous advantages of a problem-solving curriculum there comes a few disadvantages. Since class sessions lack rigid structure, it is easy for them to sometimes devolve into chaos when they are not well-planned, and so the author explains some ways around this. Another problem is that students lose out on explicit opportunities to practice new math skills they have encountered, since a problem-solving curriculum is broadly framed and quite dynamic. The author notes that if the curriculum is well designed, then students should be encountering the same ideas often enough to reinforce their mathematical skills. In conclusion, the author gives some words of encouragement to teachers considering the implementation of a problem solving curriculum, and she shares a success story.
I found this article to be very good on all levels. The author introduces the topic of the article very well, both illustrating the meaning of a problem- solving curriculum and explaining why teachers might want to transition into one. Her treatment of the entire issue is comprehensive, and she covers all possible aspects of her subject – everything from how to initiate a problem- solving curriculum to the disadvantages of such a curriculum and how to deal with them. Though she generally advocates the use of a problem-solving curriculum, she also does a good job in explaining the reasons why one might want to avoid it, and what factors influence its success. I found all of this extremely helpful in my own reflections on the possibility of a problem-solving curriculum, and I now feel much more educated on the fine points of the issue. I especially like that the author provided an example of a success story in the article, because it brings to life the positive outcomes of her method and gives the reader a concrete idea of how the problem-solving curriculum should look in the classroom. I would recommend this article to any teacher who sees limitations in their current curriculum and seeks something new and interesting to try. I would even recommend it to those teachers who are set in their ways, for this article may even open a new door for them.