Keywords: Algebra, Teaching Strategies, Proof
Ref: Emily1
Author(s): Lewkowicz, M.L.
Date: 2003
Title: The Use of "Intrigue" to Enhance Mathematical Thinking and
Motivation in Beginning Algebra
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 96, 2, p. 92-95
Reviewer: Emily
Date of Review: February 16, 2004
Lewkowicz teaches with the understanding that mathematics is a creative art. She contends that students will more likely be intrigued by mathematics if teachers try to develop students' conceptual understandings, than if teachers present a series of isolated rules and procedures.
The "intrigue model" in Lewkowicz's classroom uses mathematically intriguing problems, such as the "think of a number - add 3, square the result, subtract 9, divide by your original number - state your result - (and then…surprise!) now I know your original number" problem. From this problem, Lewkowicz introduces how the use of a variable (algebra) can explain how she knew what numbers her students were thinking of. Lewkowicz's article records other similar examples of "intriguing" problems that lead to the presentation of concepts.
The "intrigue model" fits the understanding of mathematics that we've
been discussing in class. By learning the conceptual explanation behind how
the teacher's "trick" works, students would understand that math is not magic.
Also, learners are actively engaged while trying to explain the problems.
This participation encourages interest and attention, and explaining such
problems is meaningful and could provide a feeling of accomplishment. I
recommend this article as a resource of "intrigue" problems, and as an example
of how apply your view of mathematics to your teaching.
Keywords: Problem Solving, Calculus
Ref: Emily2
Author(s): Dodge, W.; Viktora, S.
Date: 2002
Title: Thinking out of the Box ... Problem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 95, 8, p. 568-574
Reviewer: Emily
Date of Review: February 18
In class we talked about how helpful it can be to have extension questions ready when you give students a problem to work on. That way you can challenge students who finish early. Extensions of a problem are also important because they can lead students to discover the concept exemplified in a particular problem. They can make generalizations about their findings, which is a good thing for mathematicians to be able to do. Overall, extensions to problems have the potential to encourage more critical thinking and in depth exploration of a topic.
With this background, I recommend this article because it exemplifies how a common problem can be extended. In the “box problem,” students are asked to give the area that must be cut from each corner of a piece of paper so that when the paper is folded into an open box, the volume of the resulting box is maximized. This problem can be solved in calculus using derivatives, but it would also be solved by looking at the function of a graphing calculator, actually cutting and taping paper into boxes and filling it with popcorn to measure volume and guess-and-check your way to the answer, etc. Extensions of the problem encourage students to discover relationships between different components of the problem.
This article is a great resource if you’re working with this problem in your class, because it includes the actual problems extended from the original and describes their solutions and educational merit. It gives examples of how one problem can be solved in many different ways, which is an important problem solving realization. Also, the article is helpful simply as an example of some ways that you yourself can extend problems when you are a teacher.
Keywords: Algebra, Problem Solving...
Ref: Emily3
Author(s): Martinex, J.G.R.
Date: 2001
Title: Thinking and Writing Mathematically: "Achilles and the Tortoise"
as an Algebraic Word Problem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94, 4, p. 248-251
Reviewer: Emily
Date of Review: Feb 20
Though word-problems are a very important component of math, many students find them to hard or boring or contrived. This article is an example of how to work with word-problems in such a way that incorporates the five process standards in order to improve the experience.
1) Problem Solving: Martinez suggests that to improve the experiences with word-problems, problems should "engage students' imaginations with creative, thought-provoking problems." As an example for the article, Martinez uses a paragraph describing a line of reasoning that appears logical at first but does not hold up mathematically. The problem asks whether a hare will be able to pass the tortoise in a race.
2) Reasoning: Martinez suggests that problems involve the students in evaluating their problem-solving strategies in both thought and writing. In the sample problem, students write down their initial thoughts before discussing and moving on to further investigation.
3) Communication: Students share the variety of strategies they used to arrive at a solution.
4) Connections: This problem relates to algebra lessons.
5) Representation: Martinez shares a few different solution methods that students generated. I was very impressed with the students' logic and reasoning in finding their solutions until I read that the sample solutions presented came from students in a secondary-level mathematics-methods course. I would expect that the answers of many high school students would be less complex than those suggested in the article.
Still, I recommend this article as an example about a specific application of what we have been learning about in class.
Keywords: Measurement......
Ref: Emily4
Author(s): University Corporation for Atmospheric Research
Date: Accessed 2/22/04
Title: Project SkyMath: Making Mathematical Connections
Journal or Publisher:
Volume, Issue, Pages: http://www.unidata.ucar.edu/staff/blynds/Skymath.html
Reviewer: Emily
Date of Review: Feb 23
Project SkyMath incorporates real-time weather data into the middle school curriculum. The hope is that using this data, and emphasizing its dynamic nature, will promote the standards for mathematics education. The website provides a wealth of information, including the SkyMath module with is 16 activities (to be taught over at least 6 weeks) that use data about temperature. Students collect and analyze data, problem-solve, and communicate with each other (even with distant peers via email). The objectives of the lessons lead students to develop different methods of representing change.
I looked at the lesson plan for the second activity of the unit, called "Be a Weather Watcher." The lesson plans are well-written - the objectives and materials are clear. Even ideas for ongoing assessment are suggested.
The lesson plan does not have an exciting introductory activity, though doing something different using technology and real-world data will probably already be some motivation for students. The lesson is strong at tapping into students' prior knowledge. Background information and rational for the activity are also included.
I recommend this web site and its ideas to middle school teachers who are interested in good lesson plans that incorporate real-world data.
Keywords: Geometry, Activities, Standards
Ref: Emily5
Author(s): Colgan, Lynda E. C. Colgan
Date: Feb 2004
Title: It's friezing in here: Tessellations through art, architecture,
and cultural artifacts
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 9, 6, p. 334-338
Reviewer: Emily
Date of Review: Feb 23
The premise of Colgan's article is that the NCTM standards can be implemented in mathematical learning experiences that go beyond simply content so that students "see and model mathematics as an integrated whole and make both thoughtful connections between related mathematical concepts and purposeful applications to other domains."
She presents a unit that integrates transformational geometry, mathematics history, art, and cultural studies. First students design patterns to review different symmetries, reflections, and other patterns. This is in line with a NCTM standard. Next students study images of frieze patterns in the natural world and in the history of different cultures. Then the students reviewed concepts in the visual arts. The culminating activity was an application of all this knowledge - they designed and created a four-celled tile based on the features they had studied. This involved carving plaster, staining, and glazing. Specific plans for five days of class are described in the article so it would be easy to apply the unit to your classroom if you wanted.
I like this activity because it would be fun for many students - different students would be able to excel and support at each other at different stages of the unit. It is integrative and does address the standard with an application activity that makes the concept meaningful and memorable. For the lesson to succeed, it would be important to have good classroom management techniques to ensure that students were on task. Also, the teacher needs to be clear about the objectives and purpose of each activity so that parents (and students) understand the academic benefits of the unit.
Keywords: Geometry......
>Ref: Emily10
Author(s): Soto-Johnson, Hortensia; Bechhold, Dawn
Date: March 2004
Title: Tessellating the sphere with regular polygons
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 97, 3, p. 165-167
Reviewer: Emily
Date of Review: 3/10/04
The premise of this article is that by studying spherical geometry, students can understand Euclidean geometry more deeply. (This idea reminds me of Professor Wallace's story about how one geometry teacher inspired a couple gifted students to make their mark in mathematics by giving encouraging them to study non-Euclidean geometry.)
First, the authors/teachers review tessellation in the Euclidean plane ("tilings") so that they can make connections with tessellations on a sphere. Then, they write about how the spherical regular polygon area formula can be used to determine the spherical regular polygons that tessellate the sphere - the proof is outlined in the article. There are photographs of the different polygons on the sphere which helps me to visualize what's going on.
I definitely think that if I understood what was going on in this article, I would better understand tessellations in a plane. If this could be explained in a way that student's understood, that would be great.
Keywords: Curriculum, Statistics...
>Ref: Emily11
Author(s): Zawojewski, Judith S.
Date: 1991
Title: Dealing with Data and Chance
Journal or Publisher: Curriculum and Evaluation Standards for School
Mathematics Addenda Series Grades 5-8
Volume, Issue, Pages:
Reviewer: Emily
Date of Review: 3-10-04
This book was very helpful! The sample lessons were engaging and definitely hit on the process standards that they intended to illustrate. One example described an interdisciplinary unit about finding the "average" student. The teacher began the unit by reading poems about being average and the students wrote poems and journal entries about that. By the end of the unit, they had a different idea of what it meant to be average (a mathematical definition of average) and could write a poem about that. This unit addressed the real-life issue discussed in the introduction about how most people use faulty reasoning and unmathematical definitions in ordinary life. Applying good reasoning strategies, etc could help make their activities more efficient. This unit engaged students in actually carrying out the skill they were doing (data collection and analysis of survey data). Students worked through all of the stages of this research process so I imagine that became connected with the project. This interdisciplinary unit using surveys is representative of the quality of the lesson examples presented in this book.
Keywords: Curriculum, Number and Operations
>Ref: Emily6
Author(s): JJ Lo, T Watanabe, and JF Cai
Date: March 2004
Title: Developing ratio concepts: An Asian perspective
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 9, 7, p. 362-367
Reviewer: Emily
Date of Review: 3/1/04
I picked this article because in the first video we watched in class of sample lessons, the lesson by "Mr. Area" that we liked better because it fit into the process standards of teaching math was taught in a Japanese classroom. I wondered how the teaching philosophy/lesson structure we saw would compare with the lessons described in this article.
An analysis of Asian curricular materials identified ideas that are emphasized in the lessons introducing ratios. The key ideas really get point to understanding ratios, not just learning how to solve exercises. The emphasis is on multiplicative comparison, the link to measurement division, the identification of base quantity, and the distinction between ratio and non-ratio pairs of quantities. This article elaborates on each of these key ideas, giving examples from math textbooks in China, Taiwan, and Japan in order to increase teachers' awareness about ways that can help students develop ratio concepts.
Here is one example of things that the Asian textbooks point out. The Asian textbooks define a ration as being a multiplicative relationship, and distinguish between ration a:b as the multiplicative relationship between 2 quantities, and the value of the ratio as the quotient a/b. In contrast, typical US textbooks consider a:b and a/b as 2 different ways to represent a ratio.
More examples are given for the key ideas and what the textbooks do. Pictorial representations and multiple methods of solution were used and discussed which helped make the connection between ratios and concepts like measurement, division, and fractions. The textbooks developed the idea of equivalent ratios and real-life applications before moving into the concept of proportion.
This article helped by understanding of rations and keeping these things in mind will help me to teach about ratios and encourage students to fully understand the concept. I recommend this article to teachers who want to improve the teaching of ratios and development of proportional reasoning.
Keywords: Geometry......
>Ref: Emily7
Author(s): BH Morris
Date: March 2004
Title: The Beauty of Geometry
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 9, 7, p. 358-361
Reviewer: Emily
Date of Review: 3/3/04
This article describes the stained-glass-window geometry project - an aesthetically pleasing application of geometry that can increase students' interest in math. First, the students studied stained-glass designs throughout history. A parent did a demonstration of making stained-glass and students found geometric configurations on cards, buildings, wallpaper, etc. Next, students made a glossary of geometric terms. Then, students created original stained-glass-window designs that incorporated 20 geometric figures (ie. acute triangle, interior angles, intersecting lines, trapezoid, etc). The designs were painted onto the glass of a picture frame. Finally, students labelled the 20 required geometric figures in a fellow student's design. The projects were put on display.
My worry is that this project is overkill - if students already know the geometric terms, then they can move on to applications in the next lesson. However, if students are having a hard time remembering geometric terms, I think that this is an awesome project. Every step of the project relates directly to the overall standard goal of developing an understanding of geometric terms. The project is interdisciplinary (with art, history). Students get lots of practice because they analyze both their project and the project of a peer. This project creates motivation because the finished product looks cool. Also, each method of practicing the terms (making a glossary, making the design, analyzing someone else's design) is different and has a purpose.
Keywords: Number and Operation......
>Ref: Emily8
Author(s): Buchholz, Lisa
Date: March 2004
Title: Learning strategies for addition and subtraction facts:
The road to fluency and the license to think
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: 10, 7, p. 362-367
Reviewer: Emily
Date of Review: 3/5/04
This article was written by a first-grade teacher, but I think it relates to our course because we're learning about how students need to develop concepts, not just computation/algorithmic skills. Basic computation skills are necessary and this teacher researched strategies for teaching them, yet the focus, even for learning addition and subtraction facts, can be on strategies and multiple representations, real-world applications and reasoning.
Students look for doubles in the real-world (ie. 2 fingers plus 2 fingers equals 10 fingers) and write their own definition of "doubles" in their math journals and created examples. Then, the class learning "double plus one." They thought of "5+6" as "5+5+1" and this helped them to be able to solve it in their heads. Later, they also learned to think "doubles minus one" so "6+6-1." Other examples of strategies are "make ten" (so "7+4" is "7+3+1"), adding ten (put a one in the tens place), and the commutative property. Strategies were also learned for subtraction like fact families or "thinking addition" (so see "7-5" and think "5+2=7").
Children shared their favorite strategies and made up their own new strategies when doing mental math story problems together each day. (The class named one boy's method "walk all around the world just to cross the street strategy"). I was so impressed reading quotes of the children describing how they solved the problems. I hope that I can teach math like this teacher!
Keywords: Technology
>Ref: Emily9
Author(s): Schultz, JE
Date: March 2004
Title: The constant feature: Spanning K-12 Mathematics
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 97, 3, p. 198-204
Reviewer: Emily
Date of Review: 3/8/04
This article is a helpful reminder of when it is useful to use technology. The "constant feature" is the feature of a calculator that allows the user to add (or subtract, multiply, or divide) using the same number without entering it each time (just press the "=" key). The author has used this feature to enhance mathematics learning across the grades. The examples he gives in this article relate to recursion.
In kindergarten, using the constant feature with "+1" reinforces counting and number sense. Multiplication/division can be illustrated as multiple additions/subtractions (with maybe a remainder at the end). Exponentiation can be shown as repeated multiplication. The constant feature can be used to solve some story problems - for example, typing in "5+12 = = = =" will assist in finding the general formula "5+12x." The constant feature on a graphing calculator can be used to solve problems involving matrix multiplication.
Keywords: Standards, Curriculum...
Ref: Emily12
Author(s): Reys, BJ; Bay-Williams, JM
Date: October 2003
Title: The Role of Textbooks in Implementing the Curriculum Principle
and the Learning Principle
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 9, 2, p. 120-125
Reviewer: Emily
Date of Review: 3-11-04
The Curriculum Principle is that curriculum standards are coherent (integrated and well-organized so students can build on foundational knowledge), focused on important knowledge (to meet the needs of society and for math's own intellectual merit) and well articulated across the grades (so that teachers can build on knowledge gained in previous years). The Learning Principle is 1) "learning mathematics with understanding is essential," and 2) "students can learn mathematics with understanding." Teachers create the environment that encourages these things to happen - an environment that includes student interaction, proposing and evaluating ideas, etc. The point of the article is that because textbooks significantly influence what gets taught in schools, schools should use standards-based textbooks rather than traditional textbooks because they are better at encouraging the desired learning environment. An example of a lesson from each of the two types of textbooks is included. Some questions are listed to help schools choose high-quality textbooks.
I was impressed with the articulation of the Learning Principle in this article. Teachers and students first have to realize the importance of understanding mathematics. Then teachers and students need to believe that all students can achieve that understanding. Teachers must teach in ways that encourage understanding for all students. Besides that articulation, this article just restated that basic ideas we've been thinking about in class. Reading it was not a particularly radical experience.
Keywords: Curriculum, Algebra...
Ref: Emily13
Author(s): Greens, C, Cavanagh, M, Dacey, L, Findell, C, and
Small, M
Date: 2001
Title: Navigating through Algebra in Prekindergarten-Grade 2.
Principles and Standards of School Mathematics Navigations Series.
Journal or Publisher: Carole Greenes (PreK-2 Editor) and Peggy A
House (Navigations Series Editor), The National Council of Teachers of
Mathematics, Inc
Volume, Issue, Pages:
Reviewer: Emily
Date of Review: May 4, 2004
The discovery in these lessons is really guided. The students investigate step-by-step. A lot is probably done as a whole class because the students might not have the reading level to read instructions and follow each step and their own paces.
The activities seemed fun and developmentally appropriate. They definitely work towards the NCTM standards. I'm going to try to remember these ideas for my classroom. I recommend this curriculum.
Keywords: Assessment......
Ref: Emily14
Author(s): Vos, Kenneth
Date: April 30, 2004
Title: Presented at the MCTM Conference
Journal or Publisher:
Volume, Issue, Pages:
Reviewer: Emily
Date of Review: May 4, 2004
If a doctor is diagnosing a medical issues, you can't get your prescription by just coughing into the phone. You have to come in and get examined so the doctor can diagnose you and make the appropriate prescription. Then you can evaluate, hopefully find that you're doing much better, and if not, get examined again.
Teachers are most likely to skip the "up close" diagnosis step. The student makes a mistake so the teacher gives the student more practice problems. 90% of the errors at the procedural level are mistakes with the algorithm, rather than with basic skills or concepts. If the teacher can understand the students thought process and identify the mistake, the "prescription" will be much more effective. This "diagnosis" begins by looking for patterns in the errors that students make.
During the session, we looked at samples of student work and tried to identify (and explain and extend) the error pattern. Once we figured out the pattern of errors for each student, we were able to brainstorm possible interventions. For example, "John" drifted his columns when he worked out long division problems. It would be helpful for him to use graph paper or to work on regular paper turned sideways to help him line things up or to draw a box around each place value in the answer and make sure to fill in every box (even if just with a zero) "George" generalized a rule too far. It would be helpful to look with him at different examples and ask whether the algorithm fit the situation.
Keywords: Teaching Strategies, Research , Algebra
Ref: Emily15
Author(s): T. Carpenter, E. Fennema, M. L. Franke, L. Levi, S.B.
Empson.
Date: September 2000
Title: Cognitively Guided Instruction: A Research-Based Teacher
Professional Development Program for Elementary School Mathematics
Journal or Publisher: Research Report: National Center for
Improving Student Learning and Achievement in Mathematics and Scienc
Volume, Issue, Pages: At the MCTM Conference May 1, 2004, I went to a
CGI session about fractions and about equality and zero properties.
Reviewer: Emily
Date of Review: May 4, 2004
Research shows that young students can learn to make and justify generalizations about the underlying structure and properities of arithmetic if they are given the opportunity to do so. Developing these skills is in line with the NCTM standards, and children's mathematical thinking is the focus of Cognitively-Guided Instruction (CGI). CGI is a professional development program that is designed to help teachers conceptually understand the development of children's mathematical thinking in different areas. A theme of CGI is that children intuitively solve word problems by directly modeling the action and relations described in. Studies also show that students of teachers who knew more about their students' thinking had higher levels of achievement in problem solving than studnets of teachers who had less of that knowledge. Students in CGI classrooms did the same or better as control classes on standardized tests.
In the session, we watched video clips of students explaining their mathematical thinking, and I was very impressed! In the classroom, the teacher primarily uses true/false and open number sentences to elicit generalizations from students. The discussions focused on how the students knew the sentence was true or false.
Students in CGI classes show significant gains in problem solving, reflecting
the emphasis on that in class, and, in spite the decreased emphasis on drill and
practice, no loss in skills.
Return to Index
Keywords: Representations, Research ...
Ref: Emily16
Author(s): Kato, Yasuhiko; Kamii, Constance; Ozaki, Kyoko;
Nagahiro, Mariko Young
Date: Jan 2002
Title: Children's Representations of Groups of Objects: The
Relationship between Abstraction and Representation
Journal or Publisher: Journal for Research in Mathematics
Education
Volume, Issue, Pages: v33 n1 p30-45
Reviewer: Emily
Date of Review: May 15, 2004
The point of this study was to look at the relationship between children's levels of abstraction in the construction of number and their levels of representation. A close relationship was hypothesized. Piaget (1945/1962) said that children represent what they think about reality - not reality itself. Accordingly, Sinclair (1983) are different levels of representation ranging from global representation of quantity (ie. Draw a few lines to represent 2 balls because you have an idea about more than one) to cardinal value and object-kind representation (ie. Writing 4 pencils). In this study, "abstraction" involves making mental relationships (ie. Same, similar, different, two) between objects.
Sixty Japanese children (ages 3-7) were individually interviewed with a task involving conservation of number (to measure their levels of abstraction) and a task asking them to graphically represent small groups of objects (to measure their levels of representation). For the abstraction task, the children used red and blue counters and were asked to make 2 rows wth the same number of counters. If this was successfully completed, while the child was watching, the interviewer spread out the counters of one row to make that row longer. They asked if/how the rows still had the same number of counters. For the representation task, the children were asked to write/draw what the interviewer showed them so that their mother would be able to tell what was shown. The interviewers finally asked children to write the numerals 1-8, requested in random order.
The conclusion of the study was that children can represent at or below their level of abstraction. Also, most of the children who knew how to write numerals did not use them - they drew pictures or wrote numerals with a one-to-one correspondence (ie. 731 to represent 3 objects).
Mathematics education has a history of teaching symbol manipulation without
paying attention to children's thinking. The implication of this research is
that teachers should focus more on the mental relationships students make (their
abstraction) because the meaning children can give to conventional symbols
depends on their level of abstration. Concrete objects can be used at a high or
low level of abstraction (ie. Adults can look at a base ten block and see one
ten and ten ones simultaneously, while most 6 year olds can see ten and ten ones
only at 2 different points in time).
Return to Index
Keywords: Proof, Geometry...
Ref: Emily17
Author(s): Moyer, Patricia S ; Bolyard, Johnna J.
Date: Feb 2003
Title: Classify and Capture: Using Venn Diagrams and Tangrams To
Develop Abilities in Mathematical Reasoning and Proof
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: v8 n6 p325-30
Reviewer: Emily
Date of Review: May 15, 2004
The authors of this article describe a game called "Classify and Capture." Students are asked to construct mathematical arguments for their classification of shapes. To play, you need 3 hoops (set so they are interlocking to form a Venn diagram), tangrams, and cards that list different attributes, both positive and negative (ie, blue, not blue, triangle, not triangle, medium triangle, not medium triangle, parallelogram, not parallelogram. Each player (of a 4 player game) uses 7 tangram pieces of the same color. Player 1 begins by turning over 3 attribute cards and placing one in each of the 3 circles. All four players place their tangrams in the appropriate circles. After the pieces are places, students use mathematical reasoning to argue against any pieces that are placed incorrectly. If you win the argument, you get to take the disputed piece. Play continues until all the attribute label cards have been used. The player with the most pieces wins.
The example of this game was from an 8th grade class teachers can vary the game according to grade level, topic, and student ability. The article gives some examples of variations. For example, in algebra class, students could play the game by classifying subsets of real numbers (ie. Rational, irrational, integer, whole, and natural).
I would like to try this game with a class. The game setting might feel safer
for making arguments in front of their peers.
Return to Index
Keywords: Communications, Connections...
Ref: Emily18
Author(s): McDuffie, Amy M Roth; Young, Terrell A
Date: 2003
Title: Promoting Mathematical Discourse through Children's
Literature
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: v9 n7 p385-89
Reviewer: Emily
Date of Review: May 15, 2004
It can be difficult to begin a mathematical discussion because students are not used to talking about mathematics and it is difficult to create an environment that fosters discourse. Literature can be a natural context to talk about mathematics - teachers and students are used to using books for discussions. Literature is also useful because it gives students another opportunity "to make meaning and build connections between mathematics and their lives." This article gives examples of how a few books (each geared to a different grade level) can be used to begin a discussion. The books could also be used to develop a mathematical activity, but the examples are about how they can be used for mathematical discussions. The article identifies the NCTM standards that each discussion would fit under.
I liked this article because it gave me a list of books to use. The books are:
A Pig is Big by Douglas Florian (K-1: Measurement), Neil's Numberless World by
Lucy Coats (1-3: Number and Operations), Inchworm and a Half by Elinor Pinczes
(2-5: Number and Operations, Measurement, Problem Solving), and One Riddle, One
Answer by Lauren Thompson (4-5: Communication, Reasoning, Number and
Operations). A couple of the ideas are things I might not have thought of to
talk about at first with the book. I plan to use literature to enrich my
teaching of mathematics.
Return to Index
Keywords: Equity/Diversity, Research , Standards
Ref: Emily19
Author(s): Gutstein, Eric
Date: Jan 2003
Title: Teaching and Learning Mathematics for Social Justice in an
Urban, Latino School
Journal or Publisher: Journal for Research in Mathematics
Education
Volume, Issue, Pages: v34 n1 p37-73
Reviewer: Emily
Date of Review: May 15, 2004
For two years, Eric Gutstein, a 7th grade math teacher (who moved up with his class to 8th grade) in the Pilsen neighborhood (where Barb will be doing her student teaching). Drawing heavily on the work of Freire, Gutstein describes a social justice pedagogy in which students themselves are ultimately part of the solution to injustice. For this to happen students need awareness 1) to understand the conditions of their lives and the sociopolitical dynamics of the world, 2) to understand power relations and develop the belief in themselves as conscious actors in the world, and 3) to develop positive social and cultural identities (with teachers valuing their language and culture and helping them to uncover/understand their history).
Mathematics education can help to accomplish these goals. 1) Students begin to read the world using math - to understand complex issues involving justice and equity. 2) Students develop mathematical power. 3) Students change their orientation towards math. Gutstein used a series of real-world projects and a standards-based curriculum (he used Mathematics in Context, a NSF-based curriculum) to create a curriculum and environment that taught social justice. He researched the effects of his pedagogy (in the 3 areas above) by using qualitative practitioner research methodology such as participant observation, open-ended surveys, and textual analysis of documents.
To give you a sense of what projects looked like here are some examples of the real world contexts in which awareness/social justice can be aided by mathematics. Analyze racially disaggregated data on traffic stops. How could you use mathematics to help answer whether racism has anything to do with the house prices in McFadden (a wealthy, neighboring area)? Do world maps accurately represent area? Simulate the distribution of wealth around continents in the world. Simulate the distribution of wealth within the United States. Compare SAT scores between different groups of people. How have the wages for tomato pickers changed over time?
Using standards-based curriculum can theoretically promote equity, but certain
conditions need to exist. The curriculum is good because it helps students to
view mathematics as a useable tool. Students are encouraged to consider
multiple ways of solving problems. Students are "reinventing" significant
mathematics. Curricula can increase equity by raising math achievement to
improve opportunities (ie. Admission to a magnet high school). On the other
hand, the new curricula could also exacerbate differences based on existing
opportunity-to-learn inequalities (ie. Gifted teachers) and language issues.
Return to Index
Keywords: Technology, Activities...
Ref: Emily20
Author(s): Hall, Matthew
Date: 2003
Title: Calculator Cryptography
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: v96 n3 p210-12
Reviewer: Emily
Date of Review: May 15, 2004
Cryptography (the encoding and decoding of messages) is a real-world application of higher forms of mathematics (like matrices and modular arithmetic). It has been used as part of strategies in wars, used to secure information sent over the web, etc. Cryptography can engage many students.
The author has his students research cryptography and then use their TI-83 calculators to do it - they use an elementary form of that the CIA uses. The instructions and tables that he hands out to students are included and described in the article. First, make a chart that assigns numbers to the different letters and punctuations. Then enter the numbers corresponding to the message as a matrix on the calculator. Multiply the key matrix by the message matrix. Students write a little program in their calculators. They decode the message by using the same process but inverting the key matrix before multiplying. The class encrypts and decrypts messages. The class discusses why the method works. He even asks some students to prove that the method works mathematically.
I liked this article because it was clearly written and the application sounds very interesting and engaging and motivating.