Keywords: Geometry, Curriculum, Keyword 3, Optional...
Ref: Nick21
Author(s): Foresman, Scott
Date: 1993
Title: Geometry
Journal or Publisher: The University of Chicago School Mathematics Project
Volume, Issue, Pages: 874 pages
Reviewer: Nick
Date of Review: 5/18/03
This text covers geometry in a high school setting. The University of Chicago School Mathematics Project (or Chicago Math) has been viewed as an intermediate between the old, traditional sequential texts and the new, reform integrated texts. The goal of Chicago Math is to 1) prepare students to use mathematics effectively in today’s world, 2) promote independent thinking and learning, 3) help all students improve their performance, and 4) provide practical support for teachers. The Chicago Math secondary curriculum, including this text, is the first full math curriculum to implement the NCTM Standards. Chicago Math prides itself on providing examples of mathematics in real-world situations. It also is one of the first curricula to focus on the use of technology in the classroom. That is the basic background to Chicago Math.
‘Geometry’ seems to fit nicely into the goals Chicago Math has laid out. This text uses many real-world examples and applications, such as studying reflections on a pond when learning about transformations. This text also integrates history into the study of geometry and blends new and traditional methods and approaches to teaching geometry. Students use geometric software to learn more and gain a deeper understanding geometry.
The chapters in the book are as follows:
1. Points and Lines
2. Definitions and If-then Statements
3. Angles and Lines
4. Reflections
5. Polygons
6. Transformations and Congruence
7. Triangle Congruence
8. Measurement and Formulas
9. Three-Dimensional Figures
10. Surface Area and Volumes
11. Coordinate Geometry
12. Similarity
13. Logic and Indirect Reasoning
14. Trigonometry and Reasoning
15. Further Work with Circles
I have used this book to plan a unit for an education class. I wrote a unit based on this text’s chapter 5, Polygons. I thought that the math in the chapter was pretty covered well, but there were some things that I did not like. I thought that the material could have been presented in a way that would bring more meaning to students. There seemed to be a lot of memorization. Memorization without meaning equals no understanding. But, one the whole, it appears to be a solid book with good coverage of mathematical concepts and ideas. I think I would enjoy teaching a class using this text. I had no trouble inserting my own ideas into the framework of the ideas presented in the book. I think many teachers would enjoy using this book since it is bordering on both the traditional and the integrated theories of teaching math.
Keywords: Geometry, Curriculum, Keyword 3, Optional...
Ref: Nick20
Author(s): Lappan; Fey; Fitzgerald; Friel; Phillips
Date: 1996
Title: Shapes and Designs
Journal or Publisher: Connected Math
Volume, Issue, Pages: 129 pages
Reviewer: Nick
Date of Review: 5/18/03
‘Shapes and Designs’ is one of eight units in a sixth grade integrated math curriculum. This unit deals with the topic of two-dimensional geometry. The goal of this unit is to have students discover patterns and regularities in the relations among edges and angles and basic polygons and to understand how those patterns can be helpful in using polygonal shapes to create interesting designs and useful structures. ‘Shapes and Designs’ is the first unit in the Connected Math sequence to cover any ideas from geometry. The approach of geometry in this unit is unique: the primary focus is on recognition of properties of shapes, not on simple classification and naming of figures. The unit is to be covered in about 19 or 20 days, consisting of 6 investigations, 2 check-ups, a quiz, a self-assessment, and a unit project.
I prepared a unit plan based on ‘Shapes and Designs’. Through my study of this text and unit, I really thought this was great. Great ideas for lessons were given. Students were always discovering, exploring, inquiring. It made teaching this unit sound really exciting and fun. I would love to teach with this text and other Connected Math texts as my guides. I experienced two integrated unit during all my years of school. Those two units still remain the most memorable units I have been a part of. I feel as though students would feel the same way about this ‘Shapes and Designs’ unit. At least it sounds fun and interesting to me.
Keywords: Geometry, Curriculum, Keyword 3, Optional...
Ref: Nick19
Author(s): Foresman, Scott
Date: 1990
Title: Advanced Algebra
Journal or Publisher: The University of Chicago School Mathematics Project
Volume, Issue, Pages: 930 pages
Reviewer: Nick
Date of Review: 5/16/03
This book, Advanced Algebra, covers what traditionally has been called Algebra 2. The University of Chicago School Mathematics Project (or Chicago Math) has been viewed as an intermediate between the old, traditional sequential texts and the new, reform integrated texts. The goal of Chicago Math is to 1) prepare students to use mathematics effectively in today’s world, 2) promote independent thinking and learning, 3) help all students improve their performance, and 4) provide practical support for teachers. The Chicago Math secondary curriculum, including this text, is the first full math curriculum to implement the NCTM Standards. Chicago Math prides itself on providing examples of mathematics in real-world situations. It also is one of the first curricula to focus on the use of technology in the classroom. That is the basic background to Chicago Math.
Advanced Algebra fits nicely into the goals Chicago Math has laid out. This text uses many real-world examples and applications, such as analyzing population trends in major cities. This text also integrates history into the study of Algebra, as well as concepts from Geometry and other areas of math. Students write programs for their calculators and use other forms of technology to help them gain a deeper understanding of Algebra.
The chapters in the book are as follows:
1. The Language of Algebra
2. Variations and Graphs
3. Linear Relations
4. Matrices
5. Systems
6. Parabolas and Quadratic Equations
7. Functions
8. Powers and Roots
9. Exponents and Logarithms
10. Trigonometry
11. Polynomials
12. Quadratic Relations
13. Series, Combinations, and Statistics
14. Dimensions and Space
I have used this book in some depth and have liked it for the most part. I tutored a student who was in a class who used this text. The chapters we spent the most time on Chapters 4, 5, and 9. I thought that the math in the chapters was covered well, but the examples were a bit outdated and sometimes pretty uninteresting in terms of real-world applications. But, it is a solid book with good coverage of mathematical concepts and ideas. The layout of the lessons was good too—real-world example, idea or concept, skills, and exercises. I think I would enjoy teaching a class using this text. It leaves plenty of room for personal interpretation and creativity within each lesson.
Keywords: Proof
Ref: Nick18
Author(s): Hodgson, Ted; Riley, Kate J.
Date: December 2001
Title: Real World Problems as Contexts for Proof
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: v94 i9 p724-728
Reviewer: Nick
Date of Review: 5/13/03
This article displayed an example of how to use real world problems to teach to students the importance of reasoning and proof in mathematics. The example is used to demonstrate the author’s belief that "proof can be a natural outgrowth of exploration of real-world problem situations."
The real world problem in this article deals with playing pool. In the problem, a cue ball is positioned 2 feet away from the side of the pool table and 4 feet away from the ball it is to hit (a.k.a. the "target ball"). The target ball is also 2 feet from the side of the table. Each ball has a radius of 1 1/8 inches. The goal is to hit the cue ball off the wall and then hit the target ball.
The students made conjectures about what path the ball would have to take to hit both the wall and the target ball. All students arrived at the belief that the ball should have to hit the wall at the midpoint between the two balls. This was the conjecture that every student arrived at, but through various routes of thinking. But although all students arrived at the same conclusion, this was not enough evidence to definitively say that their solution was the correct solution. So, they tested their methods and models. And they discovered that their hypotheses were incorrect. They could not prove that their solution was the correct solution.
Therefore, the students went back to the drawing board and tried to remedy the flaws in their previous solution. They determined that the radius of the cue ball needed to be taken into consideration. So, they used geometry and geometric assumptions and lemmas to prove that the correct spot to hit the ball off the wall was just before the half-way point between the two balls. That would take the radius of the ball into account.
The students used proof to solve a real-world problem. They found meaning in their work and in the proofs they were completing. And that is the goal when teaching students about proof and reasoning; meaning must accompany proof. Without meaning, the students will have learned little to nothing. Using real-world problems seems to be an effective way to let students find meaning in their proofs. This article gave one example of a real-world context for proof. And it is an example that I would consider using. But more important than the example in this article is the reasoning behind using real-world examples when teaching proof. And that is what I will remember. I believe that there are infinite chances to turn real-world examples into problems requiring proof. I love proof and I hope to use the ideas expressed in this article in my future classroom.
**********************
Keywords: Assessment
Ref: Nick16
Author(s): Romagnano, Lew
Date: January 2001
Title: The Myth of Objectivity in Mathematics Assessment
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 94, No. 1, p. 31(7)
Reviewer: Nick
Date of Review: 4/24/03
This article makes the claim in regards to mathematics assessment, “Objectivity would be wonderful if we could have it, but it does not exist.” The article states no form of assessment in a mathematics classroom can be completely objective—not even basic skills tests. The “alternative” assessment options (as opposed to traditional quiz-and-test assessment), such as open-ended problems, performance tasks, writing assignments, and portfolios, are especially dependent on subjectivity. The article cites three examples to defend this viewpoint.
1) A TEACHER MADE QUIZ
The teacher-made quiz used in this example included the following task: Solve: x2+x-6=0. It is to be graded on a 5-point scale, with partial credit. Though this task appears as though it would be easy to grade objectively, the difficulties of this are displayed. For example, one student “solved” this incorrectly by factoring x2+x-6=0 into (x-3)(x+2)=0 and thus arriving at the incorrect solution. So, it is possible that the student fully knows how to solve this problem and understands why, but simply had trouble keeping the signs straight . Or, it is also possible that the student memorized the procedure for solving these types of problems, but has no understanding of why it is solved this way. There are other explanations for the student’s mistake(s). But, the teacher can choose only one of these explanations and assign a grade accordingly. Scores of 2, 3, and 4 have been given at almost equal amounts when this example is given to practicing teachers. Therefore, there is a great deal of judgment or subjectivity involved when grading a seemingly straight-forward problem such as this one.
2) THE AP CALCULUS TEST
The AP Calculus test appears to be a more consistent form of assessment than that found above in the teacher-made quiz. This is because there is a scoring rubric used by all graders with very explicit guidelines for assigning points. Therefore, a student’s answer graded by several people will receive the same score from each person, thus making the scoring very consistent. But, because of this, points are often awarded for mathematical ideas that are not concepts of Calculus at all. For example, setting a derivative equal to zero is worth one point in one of the problems on the AP test. This is an Algebra skill, not one of Calculus. Therefore, as the article states, “the specificity required for consistent scoring can reduce the usefulness of the scores themselves.” Though the AP test is graded consistently, it is sometimes not testing appropriate knowledge.
3) THE SAT-I MATHEMATICS TEST
The SAT is a test that is used in so many places by so many people to determine so many things. A person’s score on their SAT can make or break their chances of getting into the college they want to attend. The article poses the question, “If two students take the SAT and one scores a 470 on the math section and the other a 530, which one is smarter?” The obvious answer is the person who scored the 530. And this is most likely true, but not always. A person’s score on their SAT is determined in relation to all other test-takers of the same group. Everything depends on the pool of test takers and on the questions that are chosen for the test. Therefore, there is a large amount of variability associated with SAT test scores. The standard deviation is 30 points. Therefore, the interval estimates of the scores for the two people mentioned above (one scoring 470, the other 530) overlap. So, it is possible that the person who scored 470 is actually smarter than the person who scored 530. There is subjectivity involved in which test items are chosen for the test.
Since objectivity does not exist in regards to assessment, the article claims that there needs to be “agreed-on subjectivity.” This means that teachers need to design classroom assessment tasks that will get the students to give the information desired. They also need to devise, and share with the students, the guidelines for scoring their work. The article suggests that, since objectivity does not exist, teachers should look to make assessment more consistent and useful. In doing this, teachers will more accurately determine student understanding of mathematics.
I thought this was a great article. I have always been fascinated with the topic of assessment, since I have always believed that it can so easily mismeasure understanding. This article helped confirm my thoughts, but also get ideas of ways to improve assessment techniques and methods. The ideas in this article will help me create a classroom in which assessment actually measures understanding.
Keywords: Probability, Problem Solving, Keyword 3, Optional...
Ref: Nick17
Author(s): Teppo, Anne R.; Hodgson, Ted
Date: February 2001
Title: Dinosaurs, Dinosaur Eggs, and Probability
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 4, No. 2, p. 86(7)
Reviewer: Nick
Date of Review: 4/24/03
This article outlines a real-world activity combining paleontology with probability and statistics. The activity is derived from an actual study reported in “Nature”. A clutch, or an area in which dinosaur eggs have been laid, was found containing 22 eggs in Montana. The way the eggs were fossilized gave the paleontologists a chance to research the egg-laying habits of dinosaurs and compare them to their closest living relatives, crocodiles and birds. The arrangement of the eggs in the clutch was unusual—the eggs were in pairs throughout the clutch. The paleontologists wanted to determine if the eggs were laid randomly in pairs, or if this was some method to the egg-laying by the dinosaur. This is where the mathematics lesson is developed.
Lessons for students of all levels can be created from this problem. The article outlines activities that can be used to answer the question of whether the eggs were laid randomly or not. Students can draw clutches on graph paper and in one clutch draw egg pairs and in another draw randomly laid eggs. Then, the average minimum distances between two eggs are found for each clutch and reported. The students then make a histogram of their results. Then, by the histogram, they are able to determine that the eggs must have been laid non-randomly and in some sort of pattern of pairs. Lessons can be developed for all ages by adding more dimensions to the problem, requiring more in-depth data collection, and by incorporating a software-writing assignment for a math/computer science class.
This article presents an activity that is appropriate for all levels. Who doesn’t like dinosaurs? The activity requires the answering of a questions that is both interesting and real. PSSM states that probability should be taught through real-world problems. This does exactly that, and I like this problem. If I teach probability, this lesson will most likely find its way into my class.
Keywords: Number and Operation, Connections,
Ref: Nick15
Author(s): Johnson, Craig M.
Date: Nov. 2001
Title: Functions of Number Theory in Music
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: v94 i8 p. 700(8)
Reviewer: Nick
Date of Review: 4/14/03
This article outlines one of the connections between math and music. In particular, it describes the integration of number theory and pitch classes. The author explains the connections in detail and follows the explanations with examples. He uses congruence classes to explain how octaves can be described and how scales can be determined. For example, 5(mod12) would describe the placement of the note “F” in all octaves, if center “C” was labelled zero. Then, using translations of the basic position sequence {0,2,4,5,7,9,11,12}, every major scale can be found. So, to find the major scale in the key of F, we translate every term in the above sequence forward 5 units, creating {5,7,9,10,12,14,16,17}.
He proceeds to describe how to find any key if we know the number of sharps or flats in the key signature. These equations are: (for n sharps): k(n)=7n(mod12) and (for n flats): k(n)=5n(mod12). Johnson also shows a way to transpose a piece of music into another key by graphing it and translating the whole graph the appropriate distance.
This article gives many ideas of how to incorporate music into teaching math, particularly number theory. It is a great way to help musicians see the connection math has to their passion or interests. Many musicians may already know the music theory behind these ideas, but not know the math theory behind them. This lesson helps them to see both. And those who know neither the music or math theory involved with these activities receive a hands-on way of learning both.
This article has many great ideas in it. When teaching number theory, using the ideas of music, and also playing it, would be a great way to have students learn math while engaging in a fun activity.
These ideas could be used at any level. This is because the math does not need to be very complex, but the applications can be. Therefore, with tweaking, lessons can be created from the integration of number theory with pitch classes from levels K through 12 and beyond. I like the idea of trying this in a classroom.
Keywords: Algebra, Technology, Teaching Strategies
Ref: Nick13
Author(s): Pugalee, David
Date: Summer 2001
Title: Algebra for all: the role of technology and constructivism in an algebra course for at-risk students.
Journal or Publisher: Preventing School Failure
Volume, Issue, Pages: v45 i4 p171(6)
Reviewer: Nick
Date of Review: 4/9/03
This article outlined the positive effects of using technology-based and constructivist-based education in an algebra classroom of at-risk students. Constructivism and technology are two of the “hot topics” in math education today. They seem to be two key components in helping students receive a math education rich with understanding. But, the use of constructivism and technology has often been limited to high potential and sometimes middle level math classrooms. Lower level math classrooms, or classrooms full of students who are at-risk, often have not jumped on the technology and constructivism bandwagon. Direct instruction still rules in these classrooms. This article gives examples of why students in both lower level math classrooms and classrooms full of students who are at-risk greatly benefit from the use of technology and constructivist approaches to learning.
Often, technology can be well incorporated into a constructivism-based lesson. The article gives an example of a lesson in which students explore the relation between forms of equations and slope. They used their calculators in activities guided by the teacher to draw conclusions the relations between various equations and various slopes. The students did not simply “receive” from the teacher the knowledge they needed. They explored and discovered what they needed to know. They questioned and hypothesized. Through the use of technology in a constructivist approach, the students gained a more thorough understanding of slope and how to create equations with desired slopes than they would have in a more traditional direct instruction-based lesson.
Students who are at-risk have often been simply written off as unable to do math. But, the examples described in this article indicate that simply is not the case. Therefore, it is of utmost importance for teachers to do what they can to help students at-risk discover the mathematics they have running through their heads. Constructivism and technology are two very good ways to help achieve this goal.
The use of technology and constructivism are not limited to algebra classrooms however. Though it is easy to incorporate technology into algebra lessons, it can be a helpful tool in every area of math.
I think this article was very good. It is necessary to include the students who are at-risk in the current wave of educational reform. They deserve to be in a present classroom, learning according to the best methods known to date. Algebra is often when tracking starts, so it is often the time when students in low-level classrooms start having less than adequate opportunities to learn. If students who are at-risk in a low-level classroom can receive the same opportunities as the students in the higher-level classrooms, I think teachers will help students learn and understand that math is for all, which is the goal to which we are striving.
Keywords: Measurement..
Ref: Nick14
Author(s): Reece, Charlotte Strange; Kamii, Constance
Date: Nov. 2001
Title: The measurement of volume: why do young children measure inaccurately?
Journal or Publisher: School Science and Mathematics
Volume, Issue, Pages: v101 i7 p356(6)
Reviewer: Nick
Date of Review: 4/10/03
This article deals with elementary students’ understanding of measurement, or lack thereof. The NCTM Measurement Standard expects students to understand units of measurement by grade 2. But, the author claims that the standard is unrealistic for second graders.
Measurement can be broken down into direct and indirect comparisons. Direct comparisons are usually easy to understand for all people, independent of age. In other words, one can compare which of two pencils is longer by simply putting them next to each other and noting which is longer. Indirect comparisons are not easily understandable, especially for younger students.
Indirect comparisons, necessary to understanding measurement because direct comparisons are often impossible, require the ability to make two kinds of mental relationships—transitive reasoning and unit iteration. Transitivity refers to the ability to deduce a third relationship from two (or more) other relationships of equality or inequality. Someone who can reason transitively can mark a ruler to measure one pencil’s length and then compare that mark to the mark created by the second pencil. If the mark is equal, they know the pencils are equal in length. If they are not equal, they can deduce which pencil is longer by noting which mark is farther down the ruler. Unit iteration involves making a part-whole relationship within each whole. Someone who understands unit iteration understands that an inch is part of a foot (1/12 of the whole).
The author provides data suggesting that students’ ability to demonstrate transitivity and unit iteration increases with each increasing grade level. 33% of 2nd graders, 51% of 3rd graders, 66% of 4th graders, and 68% of 5th graders had a high level of understanding of transitivity. But, the number were smaller for understanding of unit iteration. 15% of 2nd graders, 47% of 3rd graders, 56% of 4th graders, and 66% of 5th graders had a high level of understanding of unit iteration. The study showed that every student who understood transitivity also understood unit iteration, but not vice versa. But, since only 33% and 15% of second graders have a high understanding of transitivity and unit iteration, respectively, the author is led to believe that the Measurement Standards for second graders are unreasonable. But, teachers can improve and accelerate understanding of these concepts by inventing activities that stimulate thought and understanding. Teachers must also pose thought-provoking questions and problems.
Though this article relates to elementary students and classrooms, I think the issues discussed reach far beyond elementary school. The statistics show that 66% of 5th graders have a high level of understanding of measurement. That’s only 2/3 of all students. Though this is a majority, there are still 34 students out of 100 who are entering 6th grade without adequate knowledge of measurement. I see this as a problem, especially since the standards claim that 2nd graders should be understanding these concepts. I think that more effective teaching practices at all levels can help remedy the problem or help kids catch up, but I also think that the standards should possibly be rethought and made more realistic. It is something for all teachers at all levels to pay attention to. This is most likely a problem that is not limited to second graders trying to understand measurement. It most likely is present in every area in the standards and at every level in some place or another. Teachers must be on guard for standards that are either unreasonably easy or unreasonably hard for students to reach, and therefore teach accordingly.
Keywords: Activities, Connections
Ref: Nick12
Author(s): Cloke, Gayle; Ewing, Nola; Stevens, Dory
Date: Oct. 2001
Title: The Fine Art of Mathematics
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: v28 i2 p108
Reviewer: Nick
Date of Review: 4/2/03
This article outlines various “Math by the Month” activities for elementary school mathematics. “Math by the Month” activities are activities that can be used with students individually, in small groups, or as Problems of the Week (POWs). The common theme to all the activities is the integration of math and art. All forms of art are explored: dance, music, drama, poetry, literature, and the visual arts. The article describes in detail the activities used for the month of October, 2001.
There are activities for three different grade level categories: K-2, 3-4, and 5-6. I will briefly describe 2 activities from each category that the article illustrates.
K-2
Poetry: The authors suggest using a familiar nursery rhyme or poem, such as Hickory, Dickory, Dock, and then replacing the original phrases with true math statements. This is one they suggest:
Hickory Dickory Dock,
The mouse ran up the clock,
Five and three make eight,
Hurry, don’t be late,
Hickory Dickory Dock.
The rhyming is intended to help students remember mathematical truths.
Number collage: Students first decide on an arithmetic rule, such as “numbers used when counting by 2’s, or 5’s, or 10’s, or some number.” Then, students choose a number. They then look through magazines for the numbers used according to the rule established and then cut them out and make a collage. (So, if the chosen number is 7, students will look for 14, 21, 28, 35, and so on.) If magazines do not have the numbers the students are looking for, they can draw their own. After the collages have been finished, students exchange collages and try to determine the rule used. It is a good exercise for students to become exposed to data analysis and to discover patterns.
3-4
Weave your facts: Students have ten 1/2”x12” strips of both yellow and red paper. The yellow strips represent “ones”; the red, “tens”. Students use the strips to learn about multiplication. If a student wants to find 2*3, he or she would lay down two yellow strips horizontally and then 3 yellow strips vertically, on top of the two strips. They then count all the intersections of the strips to determine the product. For 3*21, the student would lay down three yellow strips horizontally, with 2 red strips and one yellow strip vertically on top of the horizontal strips. They then count all intersections again, but with the red/yellow intersection counting ten each. Therefore, this method of multiplication can work for any number, with say “hundreds” represented by blue. So, yellow/yellow would count one, yellow/red would count ten, yellow/blue and red/red would both count one hundred, red/blue would count one thousand, and blue/blue would count ten thousand. It is a good visual representation of multiplication.
Digit drawing: Each student has a piece of graph paper. They pick a number which a specified number of digits. If five digits was the chosen length, then a student could choose 24351 as their number. They then would start a “digit drawing”--draw a line of length 2, then draw a line 4 units long 90 degrees to the right of the original segment. Then, draw a 3 unit segment at 90 degrees to the right of the 4-unit segment. Continue this process until the drawing starts retracing its paths. Students can color their designs. They then can explore what the designs of 2-, 3-, 4-, and other digit numbers would look like. It is a good exercise to explore patterns and tilings.
5-6
Harmonic vibrations: Students discover the relationship between octaves and vibrations. So, a student will play an A in one octave and A in 2 different octaves and determine what the vibrations per second are for each. Students would be given the data for the vibrations per second. They then find the relationship for A and for other notes. This helps students explore linear relationships.
Poetic patterns: Students write their own Japanese lantern poem. A lantern poem consists of 5 lines--line one with one syllable, line two with two, three with three, four with four, and five with one. (The poem looks like a lantern.) Students then write lantern poems about a mathematical concept or topic. This activity helps students communicate their mathematical knowledge.
These activities seem like great problems for helping students explore various mathematical concepts. I think the integration of art and math is very important, especially in the elementary grades. If students can see a connection to math and other disciplines, especially disciplines that are commonly believed to be unrelated to math, many good things can happen. Students will make meaningful connections, gain deeper understanding, and hopefully reduce their math anxieties. I think the activities outlined in this article, and other like them, are great ways to let kids discover and learn math. If I ever choose to teach in the elementary grades, I am sure I will use exercises such as these.
Keywords: Curriculum
Ref: Nick10
Author(s): Czerniak, Charlene; Weber Jr., William; Sandmann, Alexa; Ahern, John
Date: Dec 1999
Title: A Literature Review of Science and Mathematics Integration
Journal or Publisher: School Science and Mathematics
Volume, Issue, Pages: v99 i8 p421
Reviewer: Nick
Date of Review: 3/18/03
This article focuses mainly on integrated curricula in regards to the integration of mathematics and science. The authors argue the importance of integrated curricula, but also the difficulties encountered when trying to implement it.
The idea of integrated curricula has recently become incredibly popular among educators. But the authors state that its popularity is not the reason to implement integrated curricula. They believe that curriculum integration is only natural and logical. Life is integrated, not made up of clearly divided segments. They also believe that the use of integrated curricula will help schools be geared for the special needs of the students. Curriculum integration also helps develop critical thinking skills and the ability to make connections. It appears to be the way to go.
Other educators, though, question the effectiveness of integrated curricula. They argue that the limited research on the issue is inconclusive. The authors also believe that much of the negativity surrounding the issue of integrated curricula centers around the fact that there exists an unfocused definition of the word “integration”. Lonning and DeFranco (1997) developed a comparable continuum of integration for science and mathematics, ranging from independent mathematics, mathematics focus, balanced mathematics and science, science focus, and independent science.
There are some problems implementing integrated curricula. Time is the biggest issue. The school day is often not structured to accommodate the needs of integrated curricula. Also, standardized tests often measure knowledge from distinct academic disciplines, which is something that integrated curricula do not provide.
Though there are obstacles, the research and testimonials indicate that students learn more being in integrated curricula. I don’t know how I feel about the issue. It sounds great to me. But, the traditional style of education was what I went through, and I loved it and it worked well for me. Maybe integrated curricula would have been even better; I don’t know. But, it is possible that some students thrive in traditional settings and others in integrated settings. Maybe both need to be provided. But, one thing is for sure—life is integrated. Therefore, integrated curricula appear as though they should have a prominent role in education.
Keywords: Geometry, Technology, Teaching Strategies
Ref: Nick11
Author(s): Battista, Michael T.
Date: Feb. 2002
Title: Learning Geometry in a Dynamic Computer Environment
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: v8 i6 p333(7)
Reviewer: Nick
Date of Review: 3/19/03
This article illustrates the powerful role dynamic software can play in teaching and learning mathematics, especially in regards to teaching and learning Geometry. NCTM’s Principles and Standards suggests that interactive geometry software can (and probably should) be used to enhance student learning. The article describes why dynamic geometry software is such a powerful tool.
The article states, “To genuinely understand, appreciate, and use the property-based conceptual system in Geometry, students should actively participate in developing and working with the system, not in memorizing facts that others have established about the system.” Therefore, as the article outlines, inquiry, problem solving, and sense making should be encouraged and supported in the classroom. Students need to invent, test, and refine their ideas to build mathematical meanings. So often Geometry is taught by making students memorize axioms, definitions, and properties. Dynamic geometry software provides students with the opportunity to discover the axioms, definitions, and properties of Geometry themselves.
Dynamic geometry software provides students with objects that can be manipulated on screen. The article describes five “episodes” in which students used dynamic geometry software to make important discoveries about Geometry.
-Episode 1: Three students were investigating properties of squares using dynamic geometry software (I will call this DGS from now on). Each of the three made different observations about squares, then tested their observations/hypotheses by clicking and dragging on the squares on the computer screen. The students then agreed upon which observations were correct and which were not. The DGS allows students who are at different ability and sophistication levels to work together and hear each others’ comments to develop theories.
-Episode 2: The same students as above used DGS to determine that every rhombus is a parallelogram, but not every parallelogram is a rhombus. By manipulating a rhombus and a parallelogram, the students disproved the hypothesis that every rhombus is a parallelogram, and every parallelogram is a rhombus to arrive at the correct understanding of the properties of a rhombus and of a parallelogram.
-Episode 3: Two students used DGS to arrive at the same findings as those found in Episode 2, but by different methods. These two students started by making vague statements such as “[the rhombus and the parallelogram] look exactly like the same shape.” But they ended by understanding that all sides had to be equal to make a rhombus.
-Episode 4: Two students used DGS to make statements contrasting and comparing rectangles and parallelograms. They, by manipulating the objects on screen, developed the accurate definitions of rectangles and parallelograms without being told them.
-Episode 5: During a whole-class discussion, one student made the comment, “Well, you know how a shape maker can only make a shape if it follows the rules; like a square maker can only make squares. So the parallelogram maker can only make parallelograms. And it can make a rectangle, so a rectangle has to be a parallelogram.” This student could reason about the software to draw conclusions about relationships among different shapes.
In each of the five episodes, the students arrived at important findings and discovered major geometric ideas by using DGS. But, the thing that made this type of learning different from traditional Geometry instruction was the fact that the students were arriving at their own conclusions and not simply being told what they needed to know.
Working in this inquiry, constructivist type of atmosphere helped students move to higher level thinking about Geometry. DGS had a key role to play in this type of instruction.
I think that this was a great article for reinforcing the positive influence DGS can have in a Geometry classroom. If I ever end up teaching Geometry, I will definitely use some sort of DGS to aid in instruction. The discoveries students can make using DGS limitless. Though physical models (such as four rods of equal length connected at the ends by string used to represent squares and rhombuses) can be helpful while teaching and learning Geometry, there are many limitations in these kinds of models. DGS allows students to make a hypothesis and quickly and thoroughly test their hypothesis. It is a powerful tool. The article talked about one specific DGS program, but there are others that work as well. I think that a Geometry class these days that does not use DGS is not letting students explore their full higher-level thinking potential.
Keywords: Standards, Activities
Ref: Nick9
Author(s): Zawojewski, Judith S.
Date: 1991
Title: Curriculum and Evaluation Standards For School Mathematics: Addenda Series, Grades 5-8: DEALING WITH DATA AND CHANCE
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: 72 pages
Reviewer: Nick
Date of Review: 3/11/03
This Addenda Series was written to help teachers as they implement the NCTM standards. This book in particular deals with data and chance and provides examples and lessons that teachers can use. The first 5 chapters of the book deal with developing the natural abilities that middle school students have in the area of data and chance. Each of the first five chapters are broken down into 5 themes: data gathering by students, communication, problem solving, reasoning, and connections. Each chapter has activities and illustrations. An illustration found in chapter 1 is named "Is this game fair?" In this illustration, students are asked to predict whether or not they think a game in which 3 points is given to one player if the sum of two dice is seven and 1 point is given to the other player if anything but a 7 is rolled. They then test their hypothesis by either gathering enough data to be convinced and/or by using their understanding of probability and odds to analyze the game mathematically.
I really like this book. I especially like how applicable the examples are and how easily they could be implemented into the classroom. It’s also great that the whole book was written to give teachers ideas of how to meet the standards. This is a very nice thing because then anything that is found in this book can be directly applied to the classroom, and if appropriately applied, standards will be met. I am sure I will use this book often.
Keywords: Geometry, Standards
Ref: Nick8
Author(s): Day, Roger; Kelley, Paul; Krussel, Libby; Lott, Johnny W.; Hirstein, James
Date: 2001
Title: Navigating Through Geometry in Grades 9-12
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: 152 pages
Reviewer: Nick
Date of Review: 3/12/03
"Navigating through Geometry in Grades 9-12" is a guide to help teachers successfully implement the Geometry Standard from the Principles and Standards. This book has four chapters, each dealing with different Geometry concepts. Chapter 1, Transforming Our World, covers the issue geometric transformations. Chapter 2, The Geometry of Position and Map Making, deals with earth measure and its ties to Geometry. Chapter 3, Multiple Dimensions of Similarity, covers the topic of Symmetry. Chapter 4, Visualizing Limits in Our World, uses visuals to understand concepts of infinity.
The book is a great resource for anyone who is serious about implementing the Geometry Standard. Each chapter has many examples of potential lessons for many topics. Each lesson idea has the goals of lesson listed, materials needed, and why the content of the lesson is important. One lesson that I really enjoyed was "The Koch Snowflake Curve: How Big Am I?" found in Chapter 4. This lesson has students explore the Koch Snowflake to investigate the properties of perimeter and area. (I find it fascinating that the perimeter is infinite, but the area is not; I think students would also enjoy discovering this.) This lesson also helps students gain an intuitive sense of limit. Very important themes and concepts are investigated in this activity/lesson, and it looks to be a fun activity. I would enjoy using it in a classroom someday.
I think this is a great book. I will definitely use it and the ideas from it in my classroom to help me implement the Geometry Standard. I recommend it to teachers who also want solid activity and lesson ideas.
Keywords: Equity, Standards, Teaching Strategies
Ref: Nick6
Author(s): Gilbert, Melissa
Date: 2001
Title: Applying the Equity Principle
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 7, No. 1, pp. 18-19, 36
Reviewer: Nick
Date of Review: 3/3/03
This article provides strategies for establishing and maintaining a classroom in which there is "high expectations and support for all students". The article gave many techniques for establishing equity in the classroom:
1)Teachers can randomly call on students for their input rather than calling on just the volunteers. This avoids unbalanced input from classmates.
2)Students can work in "jigsaw groups"-groups in which 4 (or so) students make up each group and each student becomes an expert on a given topic. Then, this group can break up and form a new group with experts from different groups. Then, all the "different experts" can share their information.
3)Teachers should urge students to draw their own conclusions from their own work, so they can reach a deeper level of understanding. If a teacher always tells students how to simply calculate an answer, students will become dependent on the teacher as the "sole bearer of knowledge and truth" and will become less dependent on themselves.
4)Teachers need to combat gender stereotypes by repeatedly disproving the notion that math and science are less important and harder to understand for girls than they are for boys.
5)Teachers must acknowledge students' experiences as vital sources of knowledge. Understanding students helps teachers make connections with their students. Making connections with their students helps a teacher adjust his/her teaching to accommodate the students' backgrounds and needs.
The article refers to more resources available on the Internet: aaus.org/9000/links.html
This article gives many good ideas for ways to meet the Equity Principle outlined in the Principles and Standards. Each teacher needs to find his/her own ways of best establishing equity in the classroom, but the ideas above are good places to start and to build from. I think this article is a good resource for any teacher that is serious about equity in the classroom.
Keywords: Activities, Connections,
Ref: Nick7
Author(s): Ronau, Robert; Karp, Karen
Date: September 2001
Title: Power Over Trash: Integrating Mathematics, Science, and Children's Literature
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 7, No. 1, pp. 27-31
Reviewer: Nick
Date of Review: 03/03/03
This article outlined a multiple day lesson for a sixth grade classroom in which students found connections between math and science by completing a data collection activity. The activity outlined in this article began with the teacher reading the students a book about a man who is determined to collect all the litter in the world. The story continues, but it is read to get the students the mindframe of thinking about litter. Therefore, the students are asked to collect all the litter (and all forms or garbage) from the school grounds. The students were divided into groups and each group given a different region of the grounds to pick up trash. Each group tallied the amount of trash collected in each of several categories (such as Paper, Plastic, Metal, Glass, Cardboard, Wood, etc.), which are determined by the class prior to collection of the garbage. Then, after each group completed the collection, they came back to the classroom and began the visual organizations of the data.
The students, after tallying the total amount of garbage collected, received a strip of cash register tape the length of a little over 2 centimeters per each trash item collected. (For example, if 40 items were collected, a strip of tape 85 centimeters long would be used.) Then, the students divided the register tape into sections by category, each category having a length of (2cm)*(number of items collected in that category). The students discussed what these strips of register tape represented, and then used the strip to aid them in creating pie (or circle) graphs and bar graphs.
After the graphing was completed, students were asked to discuss connections between the various types of graphs created. They then graphed the data using calculators and computers and compared the graphs to their manually-created graphs. And finally, the students searched online for relevant information regarding litter. Some students took this information they found online and the information they gathered from the activity to write a report for the school newspaper.
The goal of the activity was to help students realize that science and math are very closely related. Students were also to gain a better understanding of why representing data graphically is important and often an easy way to interpret the data.
I think this would be a great lesson to use in the middle school. I also think, with some adjustments, that this lesson could be used at the elementary and high school levels. Many kids remarked during the activity, "This doesn't seem like math class." Comments like these indicate that math can be a very fun subject, dynamic and relevant to life. A lesson like this one would be help reach and teach the math-phobic and math-haters. I really like this lesson, and I like the way the article is organized. The article provides teachers many ways to tweak the lesson to the teacher's liking, such as making the lesson shorter (or longer), more in-depth, and so on. This article provides a good example for a activity-based lesson-teacher provides structure and guidance, but lets the students make the discoveries on their own.
Keywords: Teaching Strategies, Standards, Proof
Ref: Nick4
Author(s): Whitenack, Joy; Yackel, Erna
Date: 2002
Title: Making mathematical arguments in the primary grades: the
importance of explaining and justifying ideas. (Principles and Standards).
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: vol. 8, issue 9, pp. 524-527
Reviewer: Nick
Date of Review: 2/19/03
"If students are to learn and make conjectures, experiment with various approaches to solving problems, construct mathematical arguments and respond to others’ arguments, then creating an environment that fosters these kinds of activities is essential" (NCTM 2000, p. 18).
This article referred to a second grade classroom in which an effective teaching strategy was used to get students to learn about "reasoning and proof". The example taken from this classroom describes students engaged in conversation about various approaches to solving a mathematical problem. The students, in the process of discussing, appear to be building and strengthening their mathematical thinking skills and their reasoning skills. The article highlights the importance of creating a classroom in which appropriate and guided discussion takes place. The author believes a classroom such as this one is the kind of classroom the authors of the Principles and Standards had dreamed of. The author also states that there is a dynamic relationship between group discussion and independent discovery in regards to mathematical reasoning—there cannot be one without the other.
The article refers to
the importance of using guided mathematical discussion in a second grade classroom, but I
believe that it is important regardless of the grade level. Guided discussion helps to develop
reasoning skills at the middle school, high school, and college level and beyond. Therefore, this
article can be helpful for teachers of any level. It is a very encouraging article for middle and
high school math teachers, because if an exciting and lively math classrooms can exist at the
second grade level, it surely can exist at higher levels as well.
Keywords: Teaching Strategies
Ref: Nick5
Author(s): Rubenstein, Rheta; Thompson, Denisse
Date: 2001
Title: Learning Mathematical Symbolism: Challenges and Instructional
Strategies
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 94, No. 4, pp. 265-271
Reviewer: Nick
Date of Review: 2/24/03
This article expresses the importance of clearly defining and consistently using mathematical symbols by teachers in the mathematics classroom. Mathematical symbols are the "letters" of the language of mathematics. Therefore, if students cannot interpret the symbols used in the classroom, they will not be able to understand the mathematics used in the classroom. Understanding mathematical symbols is essential to learning mathematics.
This article outlines the uses of mathematical symbols and the parallels between mathematical symbols and language. For example, the phrases "is equal to" and "is a subset of" play the role of verbs in mathematics. The article outlines the analogous mathematical symbols to nouns, operators, and phrases.
Instructional strategies to help students read and use mathematical symbols are also described in this article. It is stated that meaning must precede the understanding of symbols. For example, a student must understand what addition means and how it is used before he can learn how to use the symbol "+". Using mathematical symbols in discussion of mathematics is also important to understanding. Another method for gaining student understanding of mathematical symbols is to let the students read mathematical symbols for themselves. Teachers can then take note of where students read (and then also probably understand) mathematical symbols incorrectly. The article goes on to describe visual strategies and also projects that can be used to help students understand and use mathematical symbols correctly.
This is a very good article, because it covers in-depth an issue that teachers often overlook-the use of mathematical symbols. It is easy for a teacher to think that students will understand what is meant by the use of a particular symbol, simply because the teacher has used the symbol and has understood it for 20 or 30 years. But, it is good and necessary to reminded that students maybe have never seen the symbol before; or worse yet, the students have seen the symbol before, but have used it incorrectly. This article is very helpful and I believe should be reviewed periodically by teachers.
Keywords: Standards, Technology, Activities
Ref: Nick3
Author(s): Arbauch, Fran; Scholten, Carolyn M.; Essex, N. Kathryn
Date: 2001
Title: Data in the Middle Grades: A Probability WebQuest
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 7, No. 2, pp. 90-95
Reviewer: Nick
Date of Review: 2/17/03
This article outlines the WebQuest that Carolyn Scholten designed to meet the Middle School Principles and Standards of "Data Analysis and Probability". The WebQuest is an online activity (available at www.socs.k12.in.us/schools/ovms/cscholten/proability/ProbabilityWQ.htm) in which students in grades 6 to 8 get introduced to the concepts of Probability and Data Analysis. She uses Microsoft Excel spreadsheets to aid the students in producing large sample sizes for experiments such as flipping coins, rolling dice, and so on.
By using the computer to help the students conduct the experiment, it helps the students focus more on the analysis of the data than on the actual collection of the data. Students conduct experiments in which the probability of the possible outcomes are equal (such as flipping a coin), providing a fairly basic example for students to compare theoretical and experimental probability. The spreadsheet helps students more easily see that the larger the number of trials in an experiment, the closer the values of the theoretical and experimental probabilities.
The article also states that further application of to real life probabilities and data analysis, beyond the coin toss and die rolling examples, can be made using this WebQuest. This is important to give students a more complete picture of the usefulness of probability and data analysis.
This WebQuest seems to be a great way to meet the Middle
School Principles and Standards of "Data Analysis and Probability". I think that the online
activities and experiments are great ways to cut out the sometimes tedious process of data
collection, allowing students to focus on the important aspects of the activities and on the
important aspects of Probability and Data Analysis, in general. I would like to use this WebQuest
in my classroom someday.
Keywords: Connections, Technology, Activities
Ref: Nick1
Author(s): Fernandez, Maria
Date: 1999
Title: Making Music with Mathematics
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 2, pp. 90-97
Reviewer: Nick
Date of Review: 02/11/03
This article, "Making Music with Mathematics", gave a great approach to linking a common interest of many, if not all, students-music-with mathematics. This article describes an activity in which the students use 20-oz plastic soft-drink bottles filled with water to create music. The students ultimately make the discovery about the relationship between frequency and musical pitch. It is a great way to introduce or expand on the topic of Frequency.
The materials required (per group) for this activity are as follows: two 20-oz. soft-drink bottles, a Calculator Based Laboratory (CBL), a CBL microphone, and two calculators (at least one TI-82).
Technology plays a huge role in the making this activity work. The technology in this activity calculates, records, and displays the data necessary to determine the relationships between frequency and pitch. Therefore, it would be nearly impossible to complete this activity in a classroom in which the technology required is not accessible.
The activity described in this article can be used to simply touch on the issue of frequency. This activity, however, can also be used to look at the more in-depth analysis of the sine-wave nature of frequency and how specific sine functions can be found given various frequencies.
The activity is best used
at the high school level and can easily be used to parallel topics covered in either music or
physics classes.
Keywords: Technology, Teaching Strategies,
Ref: Nick2
Author(s): Podlesni, James
Date: 1999
Title: A New Breed of Calculators: Do They Change the Way We Teach?
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 2, pp. 88-89
Reviewer: Nick
Date of Review: 2/13/03
In this article, the author argues that the use of calculators may not be as good of educational tools as some may think. He especially believes that the newer calculators, such as the TI-89 and Casio 9970, may be detrimental to the mathematics education. The newer calculators eliminate the need to think. Even in a calculus class, students can just punch an expression into one of these newer calculators, and the expression of the derivative (or anti-derivative) is just one button push away.
Podlesni argues, and I agree with him, that although the use of calculators eliminates the need to do tedious work by hand, it also does not allow for students to have the valuable experience of doing some of this tedious work by hand. Although most people would agree that graphing by calculator is much preferred to graphing by hand, many important connections about slope and other concepts can be made by graphing by hand. The more and more teachers use calculators to teach and to eliminate tedious tasks, are students going to learn math and math's "big picture" less and less? Nobody really knows for sure, but Podlesni (and I) believe that may be the case.
Podlesni also questions the role of these new calculators on standardized tests. Should the calculators be banned from use on the ACT, SAT, AP and other such tests? If they are banned, he claims that it may not be easily policed. TI-89's look too much like TI-83's to be able to tell the difference easily. Therefore, should the tests be changed? If they are, then ALL tests need to be changed, to keep consistency. But, is that what we want, really? Do we want to rewrite the tests to adjust to the high power calculators? Will that really gauge understanding of the material the test is trying to assess? It's hard to say. But, this is a topic and issue that will be around forever, and it will get even more and more interesting (and controversial) as technology advances.
I think this was a great article and I would recommend any teacher read it. I may be biased because I believe that calculators are being overused in math classrooms, and have believed this to be so for many years. The bottom line in this debate is whether or not students learn more easily or more quickly or more in-depth with the use of calculators. Once we get an idea of the answers to those questions, we will know better the role of calculators in the classroom.