Keywords: Algebra, Representations, Connections
Ref: Mark1
Author(s): Van Dyke, Frances; White, Alexander
Date: 2004
Title: Examining Students' Reluctance to Use Graphs
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98, 2, 110-117
Reviewer: Mark
Date of Review: 15-February-2005
“Examining Students’ Reluctance to Use Graphs” studies the connection between graphs and equations as representations of a function. The authors point out that the NCTM Standards place high value on these ideas of algebra, starting already in the 3-5 grade band with “represent and analyze patterns and functions, using words, tables, and graphs.” However, students seem to miss this connection, and often see graphs and equations as unconnected ideas (they lack an understanding of the Cartesian Connection).
The authors discuss one study that looked at a first-year undergraduate calculus class. Students were given problems that focused on underlying concepts, rather than procedural skill. The authors were surprised these students “did not do better.” After each example problem, there is a short discussion of the common wrong answers. For example, when given a graph of the function y = x, students said that the function increases and decreases. This may be due to notational problems—there were arrows on the ends of the y = x line. Maybe students do not have a clear idea of change in mathematics. Whatever the case, there appears to be little connection between this function and its graph.
Concluding the article, the authors suggest that the algebra curriculum be changed. Teachers should introduce the notion of a function using qualitative graphs. Qualitative graphs naturally progress to quantitative graphs, then tables, and finally equations. Teaching algebra this way will improve understanding of graphs, equations, and the broader idea of functions.
This article was valuable in two ways. First, it presented a clear argument for changing the way functions are taught. This backwards way of teaching graphs seems right. Second, the common misconceptions in understanding functions were interesting as a future teacher. These were easy to understand, going through them myself, and recognizing them now, I can hope to prevent them in my teaching.
Keywords: Connections, Communication, Representations
Ref: Mark2
Author(s): Farmer, Jeff D; Neumann, Andrew M.
Date: 2004
Title: Patterns in Perfect Squares: An Activity for
Exploring Mathematical Connections
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98, 4, 260-265
Reviewer: Mark
Date of Review: 17 February 2005
This article promotes an activity designed to address the Connections principle. This activity is basically wading through the perfect squares to find patterns. First, the authors discuss the importance of making mathematical connections. Activities that focus on these connections tend to address the process standards well, especially reasoning, representation, and communication.
This is one of those rich activities. Students are given a list of the perfect squares through to 30*30, or maybe some geometric representations of the perfect squares, and are then asked to find patterns. Through this, the teacher is circulating, asking and answering questions. Once the students have searched for patterns alone for a few minutes, they are to get in groups and compile a list of patterns. Each group shares one of their patterns with the entire class. The idea is that, by having groups explain their patterns, communication skills with improve through the ensuing discussion. The authors predict this activity may take from one to three hours. There was no grade level specified—allegedly, I could give it to classes in grades 5-12, with some discretion.
“This activity is ideal in a classroom where mathematical reasoning and communication are highly valued,” and I agree. Without a strong place in the curriculum, this lesson would be nearly worthless. There must be some connection to other learning, and there must be a direction students can take this learning. I especially like the ease with which this activity could be adapted to different ability levels. It could be a great way to start off the year, since there is little prerequisite knowledge required—a nice way to set the tone for a school year of high-quality mathematics.
Keywords: Statistics, Problem Solving, Activities
Ref: Mark3
Author(s): Mahoney, John F.
Date: 2004
Title: How Many Votes are Needed to be Elected President?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98, 3, 154-157
Reviewer: Mark
Date of Review: 22 February 2005
There is no real surprise with this title—the article discusses using the question ‘how many votes are needed to be elected president?’ in the mathematics classroom. Rephrased, the question is ‘how few votes are needed to be elected president?’ Specifically, this problem looks at the Electoral College, and the odd situation of a candidate losing the popular vote but winning the electoral vote, as Benjamin Harrison did in 1888 and as George W. Bush did in 2000. The author posed this question to high school sophomores and juniors in honors mathematics classes and sent them to work in groups to find an answer. Students had to wade through census data, e.g., each state’s population, and decide on the validity of certain pieces of data. They also had to learn about the Electoral College, e.g., the District of Columbia was granted 3 electoral votes by the Twenty-Third Amendment to the Constitution. They then had to consider ways to attack this problem, come up with possible solutions and justify them with concise and organized explanations. They were given a week.
In the article is a summary of a piece of exemplary student work. This group of students presented 4 different models for finding an answer to the problem, presented with explanations of assumptions made for each. Their starting point was to simplify the problem to only two candidates. They then followed a procedure that exploited the fact that the number of electoral votes per state is not directly proportional to the population of the state. In other words, some states have more ‘voting power’ than others. The students organized the ratios of voters per elector for each state, and then kept a running sum until the electoral vote reached 270. There were 4 models, one assuming that all citizens vote, one assuming that all registered voters vote, another using previous voter turnout data, and the final looking at swing states versus red or blue states. According to their work, a candidate could feasibly win with 8.11% of the population.
This problem is great for the classroom for many reasons. First,
students need to collect _and evaluate_ real data. There is nothing
contrived about this activity. Second, it is group work, more
specifically, it’s group problem solving. There could be lots of
fruitful discussion on the merits of cooperation and problem solving.
Third, there are many correct answers to this question. The author
wrote that he graded the projects on the quality of the writing first,
and then on the quality of the mathematics. Students are required to
work hard on the mathematics, but focus more on explaining their work
in a concise and organized way. Fourth, this is relevant, interesting,
and likely to provoke conversation, that is, real thought. I foresee
students talking to their parents about ‘how crazy it is that you only
need 8.11% of the people in America to vote for you to become
president.’ Furthermore, there could be discussion on the pros and cons
of the Electoral College. Professor Zorn had an essay describing
different methods of voting for many different situations (no longer on
his website), which could be interesting to look at as well. There are
plenty of other positive aspects to using this project, but one
negative side is unfortunately noteworthy as well. I wonder at using
this project in classes of all ages and abilities. The author used the
project in honors classes. Would it be as effective without a motivated
group of students?
Keywords: Connections, Communication,
Representations
Ref: Mark2
Author(s): Farmer, Jeff D; Neumann, Andrew M.
Date: 2004
Title: Patterns in Perfect Squares: An Activity for
Exploring Mathematical Connections
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98, 4, 260-265
Reviewer: Mark
Date of Review: 17 February 2005
This article promotes an activity designed to address the Connections principle. This activity is basically wading through the perfect squares to find patterns. First, the authors discuss the importance of making mathematical connections. Activities that focus on these connections tend to address the process standards well, especially reasoning, representation, and communication.
This is one of those rich activities. Students are given a list of the perfect squares through to 30*30, or maybe some geometric representations of the perfect squares, and are then asked to find patterns. Through this, the teacher is circulating, asking and answering questions. Once the students have searched for patterns alone for a few minutes, they are to get in groups and compile a list of patterns. Each group shares one of their patterns with the entire class. The idea is that, by having groups explain their patterns, communication skills with improve through the ensuing discussion. The authors predict this activity may take from one to three hours. There was no grade level specified—allegedly, I could give it to classes in grades 5-12, with some discretion.
“This activity is ideal in a classroom where
mathematical reasoning and communication are highly valued,” and I
agree. Without a strong place in the curriculum, this lesson would be
nearly worthless. There must be some connection to other learning, and
there must be a direction students can take this learning. I especially
like the ease with which this activity could be adapted to different
ability levels. It could be a great way to start off the year, since
there is little prerequisite knowledge required—a nice way to set the
tone for a school year of high-quality mathematics.
Keywords: Statistics, Problem Solving,
Activities
Ref: Mark3
Author(s): Mahoney, John F.
Date: 2004
Title: How Many Votes are Needed to be Elected President?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98, 3, 154-157
Reviewer: Mark
Date of Review: 22 February 2005
There is no real surprise with this title—the article discusses using the question ‘how many votes are needed to be elected president?’ in the mathematics classroom. Rephrased, the question is ‘how few votes are needed to be elected president?’ Specifically, this problem looks at the Electoral College, and the odd situation of a candidate losing the popular vote but winning the electoral vote, as Benjamin Harrison did in 1888 and as George W. Bush did in 2000. The author posed this question to high school sophomores and juniors in honors mathematics classes and sent them to work in groups to find an answer. Students had to wade through census data, e.g., each state’s population, and decide on the validity of certain pieces of data. They also had to learn about the Electoral College, e.g., the District of Columbia was granted 3 electoral votes by the Twenty-Third Amendment to the Constitution. They then had to consider ways to attack this problem, come up with possible solutions and justify them with concise and organized explanations. They were given a week.
In the article is a summary of a piece of exemplary student work. This group of students presented 4 different models for finding an answer to the problem, presented with explanations of assumptions made for each. Their starting point was to simplify the problem to only two candidates. They then followed a procedure that exploited the fact that the number of electoral votes per state is not directly proportional to the population of the state. In other words, some states have more ‘voting power’ than others. The students organized the ratios of voters per elector for each state, and then kept a running sum until the electoral vote reached 270. There were 4 models, one assuming that all citizens vote, one assuming that all registered voters vote, another using previous voter turnout data, and the final looking at swing states versus red or blue states. According to their work, a candidate could feasibly win with 8.11% of the population.
This problem is great for the classroom for many
reasons. First, students need to collect _and evaluate_ real data.
There is nothing contrived about this activity. Second, it is group
work, more specifically, it’s group problem solving. There could be
lots of fruitful discussion on the merits of cooperation and problem
solving. Third, there are many correct answers to this question. The
author wrote that he graded the projects on the quality of the writing
first, and then on the quality of the mathematics. Students are
required to work hard on the mathematics, but focus more on explaining
their work in a concise and organized way. Fourth, this is relevant,
interesting, and likely to provoke conversation, that is, real thought.
I foresee students talking to their parents about ‘how crazy it is that
you only need 8.11% of the people in America to vote for you to become
president.’ Furthermore, there could be discussion on the pros and cons
of the Electoral College. Professor Zorn had an essay describing
different methods of voting for many different situations (no longer on
his website), which could be interesting to look at as well. There are
plenty of other positive aspects to using this project, but one
negative side is unfortunately noteworthy as well. I wonder at using
this project in classes of all ages and abilities. The author used the
project in honors classes. Would it be as effective without a motivated
group of students?
Keywords: Statistics, Algebra...
Ref: Mark4
Author(s): Edwards, Michael Todd
Date: 2005
Title: Promoting Understanding of Linear Equations with the Median-Slope Algorithm
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98, 6, 414-423
Reviewer: Mark
Date of Review: 28 February 2005
Article discusses using the median-slope method of linear regression rather than the more common least-squares regression in entry-level classrooms. The author claims that students must understand the underlying mathematical concepts involved in least-squares regression lines before reaching for the calculator. To calculate a least-squares line, students must be able to minimize the sum of the squares, which is a calculus problem. So, students who haven’t yet taken calculus will not be able to grasp the ‘underlying mathematical concepts.’ The median-slope method goes as follows: say you have a list of points in the cartesian plane. To find the median-slope, repeatedly calculate the slopes of lines between each pair of points. Order the slopes, and find the median. Do the same for y-intercepts of lines between each pair of points. The median-slope and the median-y-intercept yield a rough fit line for the data. This method is easy to do by hand for a small number of points, but for a large sample size, it’s better to use a calculator program. This is not “black box” mathematics, however, because students understand what the calculator is doing for them. Also, when finding lines by hand, they have excellent practice finding slopes and y-intercepts.
Oddly enough, this was an article that was said to focus on ‘algebraic thinking.’ I am not so sure. The claim was that this technique would promote understanding of linear equations, but I do not see how. In fact, the only algebra I read about in the article was solving for ‘m’ or ‘b’, the slope and the intercept for each pair of points. As far as I’m concerned, repeatedly finding slopes and y-intercepts is a good practice with skills, but it’s not very rewarding. It’s better than skills practice without some end goal in mind, however.
In addition, it seems questionable to use this median-slope technique when the least-squares regression line is the gold standard in linear regression. The author mentions more than once that the median-slope method is not mathematically rigorous. I wonder at the value of teaching something like median-slope regression, pedagogically sound but sacrificing sound mathematics. Do students need to do all the mathematics before they understand? I do not think so. For instance, there are some fantastic java applets out there that represent the idea of least-squares regression and minimizing the sum of squares without a shred of calculus. Odds are that my students will know something about calculators and computers before they enter my classroom--might as well take advantage of that knowledge and stick with least-squares. The one suggestion from this article I will take to heart is learning some basic programming for the TI calculators. It would be nice to be able to write programs for students, or help them write their own.
Keywords: Teaching Strategies, Activities...
Ref: Mark5
Author(s):
Munakata, Mika
Date: 2005
Title:
Constructing Cooperative Logic Problems
Journal
or Publisher: Mathematics Teacher
Volume, Issue, Pages:
98, 6, 386-389
Reviewer: Mark
Date of
Review: 2 February 2005
This article suggests that teachers make students write their own problems, believe it or not. The activity described begins this way: Students are put in groups of 4-6 where they each receive a card. On these cards are necessary clues the group needs to solve a logic problem (good example problem: find the area of a certain geometric figure). The only rule is that students must not show their cards to the others. This rule forces all students to participate—since each clue is needed to solve the problem and students are not allowed to simply show their card to the group, everybody talks. This prevents the problem from being solved by one or two students while everybody else watches from afar. Once they have done this part of the activity, they are given their homework assignments. The assignment is to write their own problems, each with 4-6 clue cards. The following day, the groups work on the problems each group member created, talk about how they solved them, and discuss what was confusing, too easy, poorly worded, imprecise, etc. Students turn in their problems (with answers) at the end of the week.
Students learn a lot from this activity. They begin to appreciate the care needed to make good problems and critique not-so-good ones, they think creatively, and they use many of the mathematical skills teachers want to instill in their students. Best of all, they work collaboratively. They learn to effectively communicate with each other, tactfully critique others’ problems, receive criticism of their own, and (hopefully) recognize the value of group work. I really like the activity described by this article. It is very adaptable and fun, and most importantly, it is good mathematics. It promotes respect and student motivation as well, important character qualities I want to instill in my students.
The article briefly mentions George Polya and the book How to Solve It. He seems like a great mathematician and math teacher. I’d like to learn more about him.
Keywords: Activities, Management, Games
Ref: Mark9
Author(s): Beckmann, Charlene E.
Date: 2004
Title: Team Building in Mathematics Classrooms
Journal or Publisher: Key College Publishing
Volume, Issue, Pages: Rubinstein CD 1.1
Reviewer: Mark
Date of Review: 15 March 2005
This article described a few activities for bringing cooperative learning groups together initially, and one activity designed for use later in the school-year. There were some great ideas, especially if they are debriefed and discussed well afterwards. For example, the Noses Following Hands, students are placed in pairs. The leader extends her hand while the follower tries to keep his nose within two inches of her palm. Reverse roles after a minute. Now try in groups of three, reverse roles so that each group member has a turn at each role. This should lead to discussions of leadership roles, and the how others ought to be treated, 'what goes around comes around' type of idea. I really liked the Friendly Facts activity, where students write comments/compliments about each other and the teacher organizes them and gives them to each student without the others' names. An excellent way to brighten a day.
As optimistic and touchy-feely as these sound, I really like them. This is something a lot like what I do at summer camp with campers just meeting each other for the first time: get to know you games. They really set the mood for the entire canoe trip, and could set the mood for an entire school year. It's great that there are plans to maintain esteem throughout the period of time the group works together. I am slightly skeptical of some of these ideas in a high school classroom, simply because it isn't done very often. It could be seen as supremely dorky. If I take it seriously, and I present it artfully to the students, it could be a real benefit. Not only is it good for classroom management issues (they begin to police themselves) but it's good for pedagogical reasons (they take responsibility for others' learning, and ultimately their own).
Keywords: Statistics, Standards...
Ref: Mark10
Author(s): Hirsch, Christian R. (ed.)
Date: 1992
Title: Curriculum and Evaluation Standards for School Mathematics
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: Data Analysis and Statistics Across the Curriculum, Grades 9-12
Reviewer: Mark
Date of Review: 8 April 2005
The NCTM Data Analysis and Statistics Addenda book for grades 9-12 explains the purpose of statistical education and how statistical learning ought to look. Rather than simply learning statistical procedures, the focus should be on having students develop statistical thinking. This is best accomplished through answering a question or solving a problem because it mimics the work of statisticians. Students should design experiments, collect and organize data, represent the data with visuals and summary statistics, analyze the data (which may include using regression models for prediction), and clearly communicate the results of this analysis. Two major questions teachers should pose to students are "What does 'sampling' mean?" and "What roles do significance and confidence play in statistical analysis?" This approach to teaching statistics makes it easy to cross topics within mathematics as well as connect learning from different disciplines.
The importance of communication in statistical thinking is evident in the 18 different activities presented in the book. One of the activities asks students to find examples of misleading statistics found in newspapers or magazines. They are to explain what is deceptive, suggest (if appropriate) possible motivations for presenting the data in that way, and how it could be changed to be more accurate. Through it all, the teacher demands thorough explanations of answers, leading to classroom discussions among critical-thinking students. Assessment changes as a result of teaching statistics in this way. Rather than counting the number of correct answers on a test, students "show what they know." The book gives examples of assessment rubrics that help students organize their answers as well as make grades less subjective.
This book is a great resource for teachers of mathematics-one I would like to have on my shelf. The discussion of broad concepts in statistics and education was very helpful. The activities were exciting and easily adaptable to different situations, and the explanations of the mathematics they develop were excellent. In my opinion, the best activities were projects where students come up with their own questions to answer or problems to solve. Examples had students rating new cars, correlating athletes and their performance statistics, and judging soda preference (Coke or Pepsi?). I have one unanswered question, however. The book advocated moving away from the use of combinatorial counting problems found in probability theory as a precursor to studying statistics. Why? Is this approach incompatible with the problem-based approach?
Keywords: Statistics, Standards...
Ref: Mark11
Author(s): House, Peggy A. (ed.)
Date: 2003
Title: Principles and Standards for School Mathematics Navigations Series
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: Navigating Through Data Analysis in Grades 9-12
Reviewer: Mark
Date of Review: 8 April 2005
Navigating through Data Analysis clarifies the PSSM Data Analysis and Probability standard and provides many activities or projects to promote statistical understanding. In general, the Navigations series for grades 9-12 seems more advanced and better connected to earlier grades than the Addenda series, which seems to imply that students are learning more upper level statistics today than they were ten years ago (although one of the activities, the newspaper article examination from chapter 5, was nearly identical to one from the Addenda book).
The introduction section gives a broad-scope overview of statistics, which is nice for teachers to give students an idea of the "big picture" of statistics. The activities in the book fill in this conceptual framework with projects that increase understanding of basic ideas such as variability, or sampling. Data for these answering the questions posed by these activities are to be collected by the students or taken from 'real world' sources. There is a strong emphasis on data analysis and communication of conclusions, with obvious connections for technology use. Chapter four specifically focuses on how design study affects the conclusions that can be drawn (causation vs. correlation, and generalizability).
I prefer the Navigations series to the Addenda, not just because they are newer, but because the activities seem more engaging to students. They are easily modified, e.g., sampling rectangles from chapter 1 look a lot like Beth Chance's Gettysburg Address activity that I am such a big fan of, but they are rich enough that not much modification is needed. There are blackline pages to copy for direct use in the classroom, like the Addenda had. There is also a CD-ROM with readings and java applets in addition to the worksheets. This is a great resource-one that will go on my shelf.
Keywords: Problem Solving, Proof,
Ref: Mark6
Author(s): Polya, George
Date: 1957
Title: How to Solve It
Journal or Publisher: Princeton University Press
Volume, Issue, Pages: second edition
Reviewer: Mark
Date of Review: 17 April 2005
How to Solve It is a famous book by a prominent mathematician, George Polya. In the book, Polya looks at heuristic, "the study of the methods and rules of discovery and invention." He distills problem solving into a simple four-step process, carefully explaining each. The first step is "Understanding the Problem." The questions a problem-solver thinks at this stage are "What is the unknown? What are the data? What is the condition (how the unknown is connected to the information)?" Second step, and questions, is "Devising a Plan: Do you know a related problem? Look at the unknown! Here is a problem related to yours and solved before-can you use it?" The third step is "Carrying Out the Plan." The requirement here is to carefully do what was set out to be done, checking each step along the way. Last, but not least, and often overlooked, "Looking Back" is the fourth step, which asks the problem solver to check the solution, whether it is accurate, whether the problem could be solved differently, and how the solution could be extended for use on some other problems.
The first part of the book is called "In the Classroom." It addresses teachers on how to help students solve problems, teaching the problem solving process along the way so that the students learn to solve problems on their own. Polya shows his knowledge of educational theory, especially of scaffolding and inquiry/discovery learning, by suggesting three basic rules of helping students (page 1). "The teacher should help, but not too much and not too little, so that the student shall have a _reasonable share of the work_...The teacher should help the student discreetly, _unobtrusively_...The teacher should put himself in the student's place, he should see the student's case, he should try to understand what is going on in the student's mind, and ask a question or indicate a step that _could have occurred to the student himself_." Then, to actually teach the four steps listed above, the teacher must model good problem solving practices, help students through problem solving on their own, motivate students' interest in solving problems, and give plenty of opportunities for problem solving. This section is short, a scant 32 pages with examples, but is very worthwhile for teachers. The section on questioning (page 20) is fantastic and should be read by lots of people, in my opinion.
The last part of the book is much larger, and is called a "Short Dictionary of Heuristic." This section clarified much of what was said in the first section, even though it seemed quite repetitive, but it was very, very dense-not to be read straight through. There are some great example problems that could be used in the classroom, and the concept explanations are quite lucid. Some of the sections talk about how 'intelligent' mathematicians should read textbooks, for example. These could be quite helpful.
Other than the "Short Dictionary" section, my other suggestion about reading this book is to remember when it was written. There are many blatantly sexist remarks Polya makes. For example, on page 71 he writes, "Wishing to satisfy himself that his proposition is true, the conscientious mathematician tries to see it intuitively and to give formal proof…he acts in this respect like the lady who is a conscientious shopper. Wishing to satisfy herself of the quality of a fabric, she wants to see it and to touch it." He also mentions "pregnant" notation, which is something I do not completely understand (page 139).
Keeping the copyright date in mind, this is a fantastic book on problem solving. It is a difficult read, but well worth it for future teachers. It might even be useful for students to read. Problem solving is clearly an important part of mathematics (indeed, it is a PSSM process standard), but it is rarely taught in such a way that students think about the act of solving problems, rather than just about the problems themselves.
Keywords: Games, Probability, Discrete
Ref: Mark7
Author(s): Markel, William D.
Date: 2005
Title: Cribbage: An Excellent Exercise in Combinatorial Thinking
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98, 8, 519-524
Reviewer: Mark
Date of Review: 17 April 2005
This article suggested using the scoring scheme in cribbage to teach combinatorics. The beginning of the article outlines some basic counting rules, e.g., the multiplication principle for two or more events, permutations versus combinations. The author then goes on to explain the student activity, which could be as simple as 'find the number of points in a hand' to 'how many ways can a certain score be obtained?' There were many examples of scoring hands and finding the probability of getting such a hand using the counting rules. The author also wrote a True Basic computer program that computed the scores for all 356 possible hands, displaying the output of the program. It was interesting to read that, when players say they have 19 points, it actually means they have no points, since such a combination is impossible!
While this article was quite interesting to me, I can see how it would not appeal to everyone. I am a big of cribbage, so I would love to teach others how to play. However, scoring a cribbage hand is quite arbitrary, and the rules are quite specific. It seems like this would take a while to teach. Plus, some people really dislike the game. Poker or blackjack seem like more practical games for teaching combinatorial counting rules and probability in card games. Since cribbage is rather socially benign (less gambling involved), it might be a better option for teaching combinatorics through playing cards. On a side note, reading about the author's computer program made me want to learn the basics of computer programming.
Keywords: Connections, Activities,
Ref: Mark8
Author(s): Zambo, Ron
Date: 2005
Title: The Power of Two: Linking Mathematics and Literature
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 10, 8, 394-399
Reviewer: Mark
Date of Review: 18 April 2005
This article described the benefits of connecting mathematics and literature. There are some purely pedagogical reasons for connecting the two, like the fact that literature can provide a 'real-world' context for understanding mathematical concepts. Other benefits center on how students perceive mathematics. Telling and listening to stories is enjoyable and can reduce math anxiety. Furthermore, literature can easily connect not only to mathematics, but also science, social studies, art, etc. After explaining some of the advantages of teaching mathematics through literature, the author gives the Grain of Rice example.
The book _A Grain of Rice_ is a variation on the grains of wheat on a chessboard problem. Set in ancient China, it is the story of a smart peasant who falls in love with a princess. She loves him back, but her father, the king, is completely opposed. The peasant saves the king's life somehow. As a reward, the peasant may have whatever he desires...except the princess. He asks for a single grain of rice. The king protests, saying he should ask for more. So, the peasant asks for two grains of rice the following day, four on the next, then 8, and so on, growing exponentially. The king agrees, but realizes before too long that he has been duped, and eventually allows the peasant and the princess to marry. For activities, students compute how many grains of rice they peasant would get on day 10, on day 40, on day 100, and how many Olympic-size swimming pools would be required to contain all that rice, which I thought sounded especially fun to figure out.
I am looking forward to using literature to teach mathematics. Making connections among disciplines is very exciting for me. This is not only a great way for students to learn mathematics, but it increases student understanding across the board, and would improve attitude towards learning. The community aspect of the school would improve as well. As a high school student, I really appreciated it when math teachers knew what was going on in science, or English, or history and was able to associate mathematics with those subjects. It showed me how much they care about learning, and made me feel more connected to the school.
Keywords: Problem Solving, Activities, Gifted
Ref: Mark12
Author(s): Gardner, Martin
Date: 1978
Title: aha! Insight
Journal or Publisher: Scientific American
Volume, Issue, Pages:
Reviewer: Mark
Date of Review: 24 April 2005
Aha! Insight is a book of problems or puzzles by Martin Gardner, and came highly recommended by David Molnar. The problems are separated by what branch of mathematics best describes the "aha" insight needed to solve them (combinatorics, geometry, number theory, logic puzzles, procedural games, and a section on word play). There are ten types of problems within each section, which increase in difficulty. For each type of problem, there is an introductory puzzle with an accompanying cartoon. Examples of problems include cutting solid objects, like spheres or cylinders, and remainder problems, like the one found in the children's book A Remainder of One. After each introductory puzzle is an explanation section that includes the solution to the puzzle, an explanation or proof why the solution works, and suggestions of ways to enrich or enhance the problem.
This book is a great resource to have on the shelf for at least two reasons. First, the problems are very rich, so it is easy to tailor them to individual students' ability levels. This gives gifted students opportunities to push themselves without penalizing other students. The problems could also be used for a 'problem of the week' type activity, as an co-curricular or extra credit option. Second, these problems and puzzles are fun for me, the teacher. Working them keeps my mathematical mind sharp, and helps me stay interested in mathematics and interesting as a teacher of mathematics.
Keywords: Teaching Strategies, Activities, Connections
Ref: Mark13
Author(s): McShea, Betsy; Vogel, Judith; Yarnevich, Maureen
Date: 2005
Title: Harry Potter and the Magic of Mathematics
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 10,8,408-414
Reviewer: Mark
Date of Review: 27 April 2005
Article offers three activities connected to quotations from J. K. Rowling's Harry Potter series of children's literature exploring different themes from middle school mathematics. Because of their popularity, these texts offer an excellent opportunity to connect popular literature with mathematics, which stimulates students' interest in the subject while teaching important concepts at the same time. The first activity is a simple exploration of the monetary system of the magic world, working with conversions and expressing measurements with units. The second activity extends this exploration to functions and linear modeling. The article describes a situation in which Harry wants to buy as much of two types of candy (with different prices) as he can with a certain amount of money. After a time of semi-unguided exploration, the teacher guides students to discover a linear equation of the candy situation. Since the equation describes discrete amounts of candy, only whole-number solutions are appropriate—these are easily found using the table function of a graphing calculator. The third activity investigates the connection between the sorting of Hogwarts students into four houses and ideas from probability. In the books, Harry and his friends, Ron and Hermione, are all placed in Gryffindor where they become fast friends—students are given the problem to find the chances of this happening. There are many ways to enrich this problem, such as compiling an organized list of the 64 different ways to sort three students into four houses, discovering conditional probabilities (what changes when information is given?), etc.
This seems like a great way to give mathematics education a real context, even
though the context is based on a fictional story. I had a few qualms about
using these particular books for such a purpose. Since the Harry Potter series
is so popular, they have also attracted some controversy. There may be some
parents who are opposed to their children 'learning about witchcraft,' which is
a concern that must be taken seriously. In addition, the monetary system the
books use is based on rather arbitrary numbers, e.g., seventeen silver Sickles
per Galleon and twenty-nine Knuts for each Sickle. It is useful to learn
conversions and work with changing units of measure, but this case is so far
from a familiar (and logical) base ten monetary system that it could easily
confuse students. Overall, though, I am a fan of the books and I like the idea
of using them to teach mathematics.
Keywords: Equity/Diversity, Issues...
Ref: Mark14
Author(s): Bol, Linda; Berry, Robert Q. III
Date: 2005
Title: Secondary Mathematics Teachers' Perceptions of the
Achievement Gap
Journal or Publisher: The High School Journal
Volume, Issue, Pages: 88(4), 32-45
Reviewer: Mark
Date of Review: 27 April 2005
This article surveyed a group of NCTM members on their perceptions of the causes of the achievement gap between non-white and white students, and ways to reduce the gap. The main findings were that teachers mostly attribute the achievement gap to student characteristics, i.e., intellectual ability, motivation levels and work ethic, attitude towards mathematics, peer pressure, etc. In addition, teachers from schools with more white students were more likely to credit the achievement gap to student characteristics than teachers in schools with higher percentages of minority students. On the other hand, education supervisors (school and district administrators) and university faculty were more likely to explain the achievement gap as an instructional problem or as a problem with the curriculum. Suggestions to close the gap varied widely, from implementing national standards that focus on understanding, like the NCTM standards, to a back to the basics, "back to Saxon style texts," as one teacher responded.
Reading this article seemed to confirm some suspicions I have about education
and diversity. There is a vicious cycle, rife with latent racism, of how
students are perceived and how well they are taught. Students perceived as 'low
potential' will be taught with equally low expectations. These low expectations
lead directly to low performance, which was the reason for the prejudiced
perceptions of potential to begin with. In short, "teachers' expectations,
perceptions, and behaviors sustain and even expand the gap in achievement." As
a future teacher, I see it as my duty to the students, and as a reflection of my
belief in the transformational potential of education, to hold them to high
standards and to do my best for all students. After reading this article, I
would add 'work to have my colleagues to do the same.'
Keywords: Algebra, Statistics, Activities
Ref: Mark15
Author(s): Korteum, Kristi; Molesky, Jason
Date: 2005
Title: CSM--Crime Scene Mathematics: Newton's Law of Cooling
"Crime Scene Lab"
Journal or Publisher:
Volume, Issue, Pages: Session given at MCTM meeting in Duluth, 29-30
April 2005
Reviewer: Mark
Date of Review: 5 May 2005
This presentation was about an excellent classroom activity which has students working with "real world" applications of algebra, with fun logic puzzles thrown in and the potential for a statistics enrichment activity. In short, students come to class only to find a "dead body," which is actually a CPR doll with a warm glass of water next to it. Students measure the body's temperature (the water) when the arrive in class, and then 5-10 minutes later. These data points are used to solve for the constants in Newton's Law of Cooling, which is then used to extrapolate to find the time of death. With this information, students solve logic problems to find out the killer's identity, the motive, and the weapon used. It is sort of like a big game of Clue. The statistics enrichment activity involves collecting more data points and finding a nonlinear regression model for the data. Students first look at a linear model, examining residual plots which suggest the nonlinear relationship between time and temperature of the cooling body. This is also a pretty neat use of the Texas Instruments CBL.
If it weren't obvious from the above, this lesson is very exciting for me. Here is a hands-on, real-world, algebra and statistics-centered lesson that students will almost certainly find interesting. Not only that, but Jason Molesky is my host teacher at Lakeville South High School where I will be student teaching next fall. If this activity and his and Kristi's presentation are any indication of the quality of the experience I will have at LSHS, I am in for a treat.
Note: The document file for this activity is available by emailing either me at
kingsbur@stolaf.edu or Jason Molesky at jmmolesky@isd194.k12.mn.us
Keywords: Calculus, Assessment...
Ref: Mark16
Author(s): Gundacker, Rose; Trier, Kathy
Date: 2005
Title: Calculus Problems & Reader Hints
Journal or Publisher:
Volume, Issue, Pages: Session given at MCTM meeting in Duluth, 29-30
April 2005
Reviewer: Mark
Date of Review: 5 May 2005
Ms. Gundacker and Ms. Trier are AP Calculus teachers extraordinaire. They are also exam readers, which means they grade the tests. Needless to say, they know their stuff about teaching Advanced Placement. In their session, they discussed common misunderstandings among students about big ideas and small details in calculus, like forgetting the constant of integration, especially with initial value problems. Trier and Gundacker also mentioned changes taking place in the way the test is graded. The most recent big change is that "sign charts" are no longer sufficient justification for finding local or absolute extrema of a function. The problem is that, on their own, sign charts do not show a student understands the conclusion they justify.
The sign chart is one example of the many test-specific topics brought up by Gundacker and Trier. I have the knee-jerk reaction to assume that standardized tests generally cannot test for understanding. From what I heard in this session and my own experiences in the AP program, this seems to be not the case with AP. Overall, I am impressed with the ability of such a product-focused class as AP Calculus to focus on undestanding and not simply test-taking strategies. Interesting note: I learned Kathy Trier and I go to the same church!
Some resources they mentioned that are worth repeating are:
The AP Central website at apcentral/collegeboard.com There are forums to ask questions, tips for studying, and general suggestions for teachers and students.
"A Watched Cup Never Boils" by Ellen Kamischke
Old tests—teacher John F. Mahoney has itemized the 2003 AB multiple choice
problems by their primary topic of focus.
Keywords: Technology, Manipulatives...
Ref: Mark17
Author(s): Don Kuusinen
Date: 2005
Title: Let's Do Geometry on a TI-84
Journal or Publisher:
Volume, Issue, Pages: Session given at MCTM meeting in Duluth, 29-30
April 2005
Reviewer: Mark
Date of Review: 5 May 2005
This workshop, held at the MCTM conference in Duluth 29-30 April 2005, focused on the Cabri Jr. program for the TI-83 and TI-84. Anyone familiar with the Cabri program for the PC will recognize its usefulness exploring geometric shapes, constructions, and theorems. Cabri allows the student to 'draw' shapes on the computer screen and then manipulate certain parts of the shape to see the effect of this manipulation on other characteristics of the shape, e.g., increase twofold the length of a side of a square and 'see' the area increase by a factor of 2^2. It is quite a slick program, although using it on the PC might be difficult to do either in terms of logistics, spending time during the school day, or the expense of buying a site license and putting it on up-to-date computers. This is why Cabri Jr. on the calculator is so exciting. Students usually buy their own calculators, or the mathematics department owns a classroom set. Their ubiquity combined with their increasing usefulness makes the line between calculator and computer much fuzzier.
Like all technology, I believe it must be used _well_ if it is to be used at
all. Simply giving students calculators is not effective. They must augment
and improve instruction. They also change what is taught. If there is, for
example, an equation solver function on students' calculators, then it is silly
to assume students won't use it on a test to solve equations. Calculators are
not effective if teachers are not familiar with what they can do. That being
said, calculators are doing much, much more than I remember them doing four
years ago. Alongside the Cabri Jr. program, there are calculator based
Geoboards, spreadsheet tools, probability simulators, programs explaining area
formulas…if you can think it up, there's probably a program. It is exciting,
and very important, for me as a future teacher to explore the opportunities made
possible through technology.
Keywords: Equity/Diversity, Research , Issues
Ref: Mark18
Author(s): Britton, Edward; Raizen, Senta; Kaser, Joyce; Porter,
Andrew
Date: 2002
Title: Open Questions in Mathematics Education.
Journal or Publisher: ERIC, The Education Resources Information
Center
Volume, Issue, Pages: ERIC Identifier: ED478719
Reviewer: Mark
Date of Review: 6 May 2005
This article provides a good overview of the questions not yet answered definitively by research in mathematics education. By 'overview of questions,' I mean literally the questions. This article is an organized list of unresolved issues relating to equity and diversity in mathematics education. There is a section on equitable access to courses and teachers. Another section focuses on culturally-informed content, instruction, and assessment. There are sections on understanding and scaling up effective programs, teacher preparation and professional development, and improving the method of research and expanding the pool of researchers. Two rather large discussion sections, one on gender inequity and the other on the achievement gap, end the article. The gender section makes note of the uncertainty surrounding the gender gap (Does it exist? If so, how does it manifest itself? What are its causes? What are solutions?). Discussion of the connections to technology use in the classroom and the type of motivational practices used (competitive or cooperative?) lead to the need for research on how teachers' beliefs affect how students learn. This segues to the section on the achievement gap. The main areas for research here are in studies of instructional practice, what students do outside the classroom, e.g., what types of homework are assigned, and a more clear picture of what 'quality teaching' means.
As a future teacher, I see this article as valuable for at least three reasons.
First, keeping these questions in mind as I teach will affect how I teach. By
striving to answer these questions, I will stay informed on current research,
evaluate my own actions, and reflect on what works for me. Second, these
questions inform me of my beliefs about equity and diversity. Are they integral
to what I do and who I am, or are they separate 'problems' to be solved? Third,
I can easily see using these questions in discussion with other teachers. The
main goal of this is to improve equity and diversity in my school, with the
added benefits of improving communication and feelings of collegiality/community
among the staff.
Keywords: Research , Equity, Teaching Strategies
Ref: Mark19
Author(s): Haury, David L.; Milbourne, Linda A.
Date: 1999
Title: Should Students be Tracked in Math or Science?
Journal or Publisher: ERIC, The Education Resources Information
Center
Volume, Issue, Pages: ERIC Identifier: ED433217
Reviewer: Mark
Date of Review: 6 May 2005
Contrary to the wording of the title of this article, the goal is not an either/or answer to the question, "should students be tracked in math or science?" It is a yes/no question, and a difficult one at that. This article strives to clearly outline research on this question, looking at it from different angles, before coming to a resounding but provisional 'no.' According to the literature, tracking may boost the achievement of students placed in higher-ability courses, but these gains are offset by the worsening of student achievement in lower-ability tracks. Even worse, removing tracking shows improvements for lower-ability groups, but at the cost of bringing down the test scores of higher- and average-level tracks. This is corroborated by the NAEP tests, which show a dramatic achievement gap, with the average senior in high school at a 7th grad level of understanding. TIMSS results are very similar.
It comes down to how teaching occurs in high-level and low-level mathematics classes. Students in higher-ability tracks focus on problem solving, inquiry- based learning activities, and writing to explain their reasoning. Students in lower-ability tracks spent more time reading the textbook or writing worksheets. There are clear reasons to get rid of tracking, but is its removal possible, and if so, what would that look like? Answers to these questions are still unclear.
When I was in Norway during the Fall 2004, I did some research on the role of
equity in the Norwegian education system. There is no tracking in Norway,
although some officials have begun to push for it. The TIMSS data show an
interesting trend. Young students in Norway tend to have low levels of
achievement compared to the international average, but there is steady
improvement through the grades. The opposite is true for the US. Even so, both
countries are dissatisfied with their students average achievement levels and
want to see them improve. It seems to me like both countries could learn
something from the other.
Keywords: Probability, Statistics, Activities
Ref: Mark20
Author(s): Lee, Hea-Jin
Date: 1999
Title: Resources for Teaching and Learning about Probability and
Statistics.
Journal or Publisher: ERIC, The Education Resources Information
Center
Volume, Issue, Pages: ERIC identifier: ED433219
Reviewer: Mark
Date of Review: 7 May 2005
"Resources for Teaching and Learning about Probability and Statistics" is an ERIC article which describes the need for increased and improved teaching in probability and statistics, the major obstacles to achieving this goal, and recommendations to overcome these barriers. The major obstacles include a lack of available courses, a shortage of teachers prepared to teach probability and statistics, and student misconceptions, beliefs and attitudes toward the subject. Suggestions offered by the article focus on changing the curriculum to include more probability and statistic, focusing on procedural _and_ conceptual knowledge, and increasing the pool of well-trained teachers. The article ends with a long list of websites, many of which no longer exist, and an equally long list of articles in journals like "Mathematics Teacher" describing classroom activities that focus on probability and statistics.
The ideas presented in the beginning of this article were anything but new to me. However, had I not known much about the state of probability and statistics education, this article would have been a clear and concise explanation of the problem and excellent solution suggestions. This is true of all ERIC articles I have read. What I found specifically helpful about this article was the list of activities. It was as though ERIC had waded through the literature for me and fished out the best for me. Quite nice. I assume there are other "Resources for Teaching and Learning" articles on ERIC—these will be valuable to me as a teacher.