Keywords: Assessment, Curriculum, Standards
Ref: Stacey1
Author(s): Hancock, C. Lynn
Date: 1995
Title: Enhancing Mathematics Learning with Open-Ended Questions
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 88, 6, pp. 496-498
Reviewer: Stacy
Date of Review: February 17, 1999
Abstract: This article is resourceful and promotes the vision of instilling communication and self-sufficiency in the mathematics classroom. The way we teach mathematics is changing for the better by giving more power to the students by challenging them to think for themselves and focus more on the process versus obtaining correct answers (without obtaining meaningfulness).
Assessment may take longer, but instills character in the students since they are allowed to
think for themselves and conjecture why something happens the way it does. However, one can take a
different perspective about the learning process in regards to assessment. This article advocates
student learning even through the final assessment process; learning does not stop at giving an answer
or explanation. One's learning is enhanced when students are given the opportunity to revise their
written work and learn from their mistakes, with the help of teacher counsel. The implementing of
student involvement in how work is assessed provides a well-balanced learning opportunity in that it
permeates the understanding of what constitutes good mathematics.
Keywords: Assessment, Curriculum, Standards
Ref: Stacey2
Author(s): Hancock, C. Lynn
Date: 1995
Title: Enhancing Mathematics Learning with Open-Ended Questions
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 88, 6, pp. 496-498
Reviewer: Stacey
Date of Review: March 3, 1999
Abstract: Like the standards, I desire to ask students questions that promote thinking for themselves and that emphasize not only getting the right answer, but also the process of getting that right answer. This article addresses this exact issue and provides a suggestion for teachers to better gage the understanding of one's students: use open-ended questions in assessment. Assessment may take longer, as the article reveals, but I think it is worth it to teach quality (along with quantity) in our classrooms.
One way of implementing this type of assessment is to extend lessons that have students solve non-traditional mathematical problems, such as explaining the process of how one would go about getting the correct answer(s). When these non-traditional problems are incorporated into the lessons, learning does not necessarily have to stop after a solution or explanation has been given, but may continue in the evaluation stage. Developing a specific outline -or rubric-- of desired tasks for the students to complete with the problem at hand allows for new student insight to be gained. This rubric is explicit in what is desired for scoring purposes and given to the students before the problem is assigned. Then students themselves evaluate other student work according to the rubric or are given feedback from the teacher on their own work. Having the students evaluate another's work pushes for further understanding and allows students to see how they could have improved thei!
r own explanations. Another option for students to better understand mathematical problems and at the same time provide clarity in teacher evaluation is to allow students to revise their work after receiving feedback. This technique also challenges the students to think on a more deeper academic level. I want to be able to incorporate these strategies into my mathematics lessons for it is beneficial to both the students and myself in searching for true understanding.
Keywords: Algebra, Communication
Ref: Stacey3
Author(s): Stacey, Kaye; MacGregor, Mollie
Date: 1997
Title: Ideas About Symbolism that Students Bring to Algebra
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 90, 2, pp. 110-113
Reviewer: Stacey
Date of Review: February 19, 1999
Abstract: When students first encounter algebra in junior high or senior high school, they bring with them many complex versions of using symbols such as 5 1/2, 53, VIII, and English words, sentences, and abbreviations as a structured language system. They know that the placement of symbols -- both numeric and alphabetical -- is important for determining content and specific meaning. This article gives a good reminder for those that teach algebra to emphasize that each algebraic letter represents a number. Australia believes that algebra should become a somewhat familiar concept to those even in the primary grades and frankly I think this is something we should implement in the United States as well.
Some suggestions I have taken to heart from this article that will better aid students to sort out this specialized language we call algebra include using algebraic notation more frequently, avoid words such as "makes" or "gives" for what is meant by "equals" since students tend to link only parts of the equation and do not consider all the operations that play an intricate part in the whole equation, reiterate that algebraic symbols are representations of numbers, clearly establish the differences between products and exponents, ask my students to explain relationships, and emphasize that letters do not represent particular alphabetical positions as in a code.
If we are aware of our particular word choice in explaining mathematical concepts, we can help provide a sense of clarity among our students and we as a nation can improve mathematical understanding in more of our students.
Keywords: Planning, History
Ref: Stacey4
Author(s): Meserve, Bruce E.
Date: 1989
Title: Looking Ahead in Teacher Preparation: A Personal Perspective on NCTM-MAA Cooperation
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 82,7, pp. 564-570
Reviewer: Stacey
Date of Review: March 1, 1999
Abstract: Seminars (offered by colleges) came into play for elementary and secondary educators as a chance to enhance and update professional preparation, share ideas and concerns about teaching, relax, and make connections with other mathematics educators. This article points out that we as a society no longer can prepare teachers at the college level for the next 35-40 years of their teaching careers. Meserve writes, "To be professionals, we must continue our professional growth throughout our careers"(564). I plan to take advantage of these seminars whenever possible to enhance my teaching techniques, better acquaint myself with the latest technology, and converse with other educators on social and professional levels.
Just as teachers need frequent updates in their backgrounds, guidelines for MAA and NCTM need frequent update for us to achieve national mathematical excellence in our schools. The Mathematical Sciences Education Board (MSEB) was established to provide national leadership for mathematics and science education for children from K-college levels. Programs under this branch include:
I am enthusiastic about this recognition for change nation-wide and want to stay informed on these issues since education is always in a state of flux and directly affects me and my classroom.
Keywords: Teaching Strategies, ,
Abstract: This article not only talks about a cooperative classroom, but a cooperative school, and school district. If we want our students to take risks to develop their intellectual potentials to the greatest extent, we need to foster a whole school community in the same respects. Teachers, just as students, of all grade levels and subject areas need to have support from their peers and a common theme of accountability incorporated into the structure of the school and the classroom. Everyone needs this in order to excel and take risks in enhancing one's personal knowledge and expertise. Johnson and Johnson, leaders of educational research at the University of Minnesota, suggest that an ideal teaching situation is one that is team-taught, where the teachers of all academic interests share in the opportunity to teach the same particular group of students for several years. According to the article, "This serves to strengthen positive interdependence among teac!
hers, heighten shared accountability, and provide purpose for helping and supporting one another in continuously improving instructional expertise". I plan to use cooperative learning for both my students and myself; social support is necessary for any individual's success. I look forward to sharing ideas about continuous improvement for instruction, school-based decisions, and departmental meetings with my colleagues.
This article has some links for possible training sessions/classes through the University of Minnesota and how to encourage students to be peacemakers.
Keywords: Geometry, ,
Abstract: I read this article hoping to use it in a seventh grade classroom where I could choose any topic. I wanted to get the students involved with some hands on activities and at the same time introduce something normally not introduced until higher mathematics levels. I actually carried out the making of a fractal card with some simple folds and cuts. In this type of activity, students can take ownership of their work and admire the beauty and simplicity of their fractal card they have created themselves. For seventh graders, an appropriate extension to this "art project" includes looking at the growth pattern of each iteration of cuts and folds. Each new stage of the fractal card is explicitly evident in matters of its size and number of cells fabricated. I would have also asked them to predict growth at various stages.
This lesson is also applicable to high school students, for those of you teaching at that level. Students could give a general formula for the number of cells at any stage of the growth sequence. The change in surface area can also be a question you would want students to answer, which uses some discrete mathematics. The fractal card can serve as a visual model of a convergent series. There is something even for your calculus students in this lesson, calculating the limit of the sum representing the surface area of each cell on the fractal card.
Keywords: Algebra, ,
<p> Spreadsheets are a comfortable transition for students from arithmetic to algebra. The benefits of using spreadsheets include student familiarity, freedom from calculations, algebraic concepts are a necessity versus a seemingly arbitrary one, and freedom in choosing methodology to solve problems. This particular lesson incorporates number sequences and geometric generation patterns. If differences of formulas for finding the nth digit or the amount of blocks at the nth generation occur, the concept of proving by counterexample can be brought into the discussion.
I like this idea of using the spreadsheets in a mathematics classroom for some additional reasons mentioned in this article (and highlighted above). Any computer with Microsoft Office will have Excel, which is usually standard for both PC's and Macintosh computers used by schools nationwide; this program may be more readily accessible versus a specific mathematics computer program. Another reason is the fact that spreadsheets are nice organizational tools and are used in presenting information all over the place, such as in newspaper articles, company reports, and athletic funding proposals.
Keywords: Assessment, Teaching Strategies,
Abstract: What a great concept - getting students to take responsibility for their own learning and retention of learning! Having students generate their own questions for tests and reviewing for tests facilitates this type of learning. Evaluation is the highest level of Bloom's taxonomy and we should guide students as much as possible to think at this level, where mastery of concepts is highly encouraged. Odafe, author of this article, includes cooperative groups for students to collaborate ideas about what kinds of questions would suffice for coverage of all material on a unit test. This exercise is used as a review for the students prior to the test and the teacher guarantees to use at least one question fabricated by the students on the test, which makes this exercise legitimate in their minds. Questions are edited for ambiguity and improper wording. Students do not have to have anxiety about their particular problem on the test because with proper use of!
cooperative learning techniques employed each person will have contributed in writing the question and making sure they each understand how it is to be solved before moving on to the next question to be generated. After the tests are scored and returned, the teacher asks the students to get together with their learning family where they reflect on their specific questions they designed and log other possible adjustments and extensions they would consider in doing another question of this type. Students are on the look out all through out the unit for possible test questions. It empowers students because they are a part of the process and feel better prepared for tests. The one thing that is problematic with this type of assessment is the length of time it requires away from class time, but ownership for one's own learning and giving students mathematical power is irreplaceable and to me time well spent. I think I would not use this method for every single test so that!
students could get used to this type of higher learning at a less shocking pace from which they are used to.
Keywords: Technology, ,
Abstract: This article is short, but raises an important issue. It is a dialogue between a teacher and a student who at first does poorly in mathematics and then turns that around when accepting the responsibility to take charge of his own learning. For him, technology was the key to his understanding. He says that in using the graphing calculator it was giving him an idea of the complete puzzle. Graphically, it did not make sense to have no slope of line if one let the independent variable be time and the dependent variable be distance, which would mean going somewhere in no length of time. This particular student was not able to put that piece together without the calculator. He has a learning disability and puts in a great deal of time, but cannot put the pieces together without the visual context. We need to have students make those connections on a continual basis. Different students have different learning styles and we need to have students see the !
bigger picture. This student got an A at the end of his algebra experience; it is possible for all students to do mathematics and be good at it.
Keywords: Geometry, Technology,
Although, Shilgalis, author of this article and teacher at Illinois State University, advocates the use of technology, he does emphasize the point that technology is not enough evidence for a rigorous proof. In this article there are two problems that revolve around a story of buried treasure. In these problems, students use Geometer's Sketchpad, a mathematics computer program, to visualize the independence or dependence of particular points of interest in the story such as a tree or a road in relation to where the buried treasure is located. The exercise can be done by pencil and paper, but the accuracy will not be as extensive and various points and lines cannot be rotated or manipulated as easily as it is with technology. This particular lesson includes concepts of complete quadrangle, cross-ratio of four collinear points, and harmonic conjugates - all terms used in studying projective geometry.
This particular lesson will be a bit too advanced for seventh graders, but I like the idea of having students explore things using technology and creative problems, such as the buried treasure idea. I agree with Shilgalis when he says that there has to be some additional connections made by the students in the things behind the scenes of technology. Shilgalis writes, "Indeed, the thinking required for a problem is just as important as it has always been, but the opportunity to investigate properties that remain unchanged as the parameters of a problem are varied adds greatly too the users' enjoyment and to their acquisition of insights and problem-solving skills" (165).
Keywords: Equity, Issues,
This article encourages teachers to make sure they understand how students feel in a particular class or subject area is dependent upon childhood experiences within the classroom and receiving help from parents. Two students attending college level mathematics classes in order to get into the fields of education and nursing each had a overwhelming experience paired with their mathematical learning, which has scarred them for life. They were so frightened of doing mathematics even up through their college experiences because of incidents that embarrassed them and made them feel stupid back in third grade. Negative experiences, unfortunately, are embedded into one's memory for quite some time, which is a defense mechanism we use to protect ourselves from further hurt. This creates anxiety, and teachers need to be aware of this unnecessary, but quite prevalent, baggage students bring with them to the classroom and how they are motivated.
One of the reasons I wanted to go into education is to remedy this as best as I could with my students. I want to give my students a positive mathematics experience. I have chosen mathematics for my area of teaching because I myself have struggled with mathematical thinking and feel that I can be sensitive to this issue. I believe to be a good teacher, the teacher must learn with the students and both parties must enhance their understanding: the teacher in the areas of facilitating good questions and explanations and the students in facilitating good processes of thought to come up with solutions and questions. But because of that struggle, I can appreciate the beauty of mathematics and want to pass on that satisfaction of mathematics with my students.
Keywords: Standards, Assessment,
Abstract: This article was a surprise to me: although most teachers see the need to have students write within the mathematics curriculum, many still do not use this type of learning in their classrooms. Elementary teachers at least talk about the option with their colleagues, whereas senior high school teachers do not tend to do that as much as the elementary teachers. Only about 20% of the 117 respondents to the survey include writing as a way to have students explore mathematical ideas on a regular basis. The research recommended by NCTM has been public for at least forty years and still teachers do not use these techniques to steer away from mere memorization and to facilitate development of the meanings of mathematical words. The survey reveals that "younger women teaching at the elementary level make the greatest use of discovery methods and expressive writing assignments in their mathematics classes. They are twice as senior high school teachers to di!
scuss these methods with their colleagues, despite the fact that a higher percentage of high school teachers have actually attended writing-to-learn (WTL) workshops" (389). Why is such a method that is viewed generally of some benefit to the students still not catching on in the actual teaching agenda? What are some ways I can better educate myself about using student written work to facilitate better understanding among my students? I myself am familiar with such a notion in the mathematics classroom, but only at the college level. I am going to continue my research on this issue.
Keywords: Games, Geometry,
This article is a great anticipatory set activity for introducing proofs in Euclidean geometry. The activities in this article are familiar games, such as basketball or monopoly, that have a particular set of rules and terminology in assessing if the rules are met in playing the game, which are analogous to undefined terms, definitions, postulates, and theorems in a deductive system. Three worksheets are included in this lesson I could use as model problems in a geometry curriculum. This type of learning stresses the importance of learning concepts, where generalizations are applicable to new situations. Plus, students learn better if they can start thinking through a similar, but familiar process.
Keywords: History, Equity, Gifted
This article gives a brief history of 5 female mathematicians. Kelley sums it up best when she writes, "Too much mathematical knowledge has been lost because of the exclusion of so many good minds from its pursuit. As mathematics educators, we must make sure that we lose no more" (596).
Keywords: Assessment, Teaching Strategies,
This article gives some examples of student written journal entries. The gist of this article is that journals are a helpful tool to give teachers a better clue about the actual thinking processes of their students working through problems. Other things such as the need for help is better expressed in written form than the student asking for help verbally. Also, the level of thinking and to what degree extensions are being addressed by a given student can be pinpointed in writing exercises. This type of assessment allows the teacher to adjust the instruction before it is too late. It also allows students to become reflective learners when asked about the preparation for a test and how they might change their strategies given their results. This journal can be seen as a conversation book that enhances learning for both teacher and students.
Keywords: Connections, ,
This article includes a copy of a published article found in the Toronto Star, July 16, 1997 talking about the problem with computers for the year 2000. It also includes supplementary questions that could be used to discuss scientific notation and expressions in the form of base 14, for example. The media clip included emphasizes that if we had 14 fingures instead of 10, we would be more accustomed to counting by 14's and the computer problems would not have occurred until much later than the year 2000. The problems included get students thinking about why this is so and the concept of constants. This is a great way to make connections with current events and mathematics. This would be a great extension or anticipatory set for a lesson on scientific notation and different ways to represent a number.
Ref: Stacey10
Author(s):
Date:
Title: http://www.clcrc.com/pages/cs.html
Journal or Publisher: University of Minnesota
Volume, Issue, Pages:
Reviewer: Stacey
Date of Review: May 12, 1999
Ref: Stacey11
Author(s): Simmt, Elaine, Davis, Brent
Date: 1998
Title: A Space for Exploration in Geometry and Discrete Mathematics
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 2, pp. 102-108
Reviewer: Stacey
Date of Review: May 12, 1999
Ref: Stacey12
Author(s): Friedlander, Alex
Date: 1998
Title: An EXCELlent Bridge to Algebra
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 5, pp. 382-383
Reviewer: Stacey
Date of Review: May 12, 1999
Ref: Stacey13
Author(s): Odafe, Victor U.
Date: 1998
Title: Students Generating Test Items: A Teaching and Assessment Strategy
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 3, pp. 198-202
Reviewer: Stacey
Date of Review: May 12, 1999
Ref: Stacey14
Author(s): Bethell, Sandra Callis, Miller, Nicolas B.
Date: 1998
Title: From an E to An A in First-Year Algebra with the Help of a Graphing Calculator
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 2, pp. 118-119
Reviewer: Stacey
Date of Review: May 13, 1999
Ref: Stacey15
Author(s): Shilgalis, Thomas W.
Date: 1998
Title: Finding Buried Treasures -- An Application of the Geometer's Sketchpad
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 2, pp. 162-165
Reviewer: Stacey
Date of Review: May 13, 1999
Ref: Stacey16
Author(s): Fiore, Greg
Date: 1999
Title: Math-Abused Students: Are we Prepared to Teach Them?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92, 5, pp. 403-405
Reviewer: Stacey
Date of Review: May 13, 1999
Ref: Stacey17
Author(s): Silver, Jennifer Williams
Date: 1999
Title: A Survey on the Use of Writing-to-Learn in Mathematics Classes
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92, 5, pp. 388-389
Reviewer: Stacey
Date of Review: May 13, 1999
Ref: Stacey18
Author(s): Gernes, Don
Date: 1999
Title: The Rules of the Game
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92, 5, pp. 424-429
Reviewer: Stacey
Date of Review: May 13, 1999
Ref: Stacey18
Author(s): Kelley, Loretta
Date: 1996
Title: Why Were so Few Mathematicians Female?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 89, 7, pp. 592-596
Reviewer: Stacey
Date of Review: May 13, 1999
Ref: Stacey19
Author(s): Chapman, Kathleen L.
Date: 1996
Title: Journals: Pathways to Thinking in Second-Year Algebra
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 89, 7, pp. 588-590
Reviewer: Stacey
Date of Review: May 13, 1999
Ref: Stacey20
Author(s): Wood, Dorothy (ed.)
Date: 1998
Title: Filling Gates' Shoes
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91, 3, pp.218-219
Reviewer: Stacey
Date of Review: May 13, 1999