Keywords: Teaching Strategies, Communication, Curriculum
Ref: Barbara1
Author(s): Ward, D. Cherry
Date: 2001
Title: Under Construction: On Becoming a Constructivist in View
of the Standards
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 9(2), p.94-96
Reviewer: Barbara
Date of Review: 2/15/04
This article discussed the importance of constructivism, which is 'a belief that all knowledge is necessarily a product of our own cognitive acts' (94). Throughout the past decades, teachers have attempted to change their instructional methods. Teachers have focused on teacher-centered instruction, drill and practice, student-centered instruction, or discovery learning. But, ironically, teachers have basically all taught the same way despite all these new instructional methods.
It is important for teachers to use and teach the idea of constructivism. Becoming a constructivist allows for students to build on their previous knowledge, better grasp concepts, and go from knowing the material to understanding the material. Furthermore, knowledge that is constructed by the student is remembered more often while direct instruction may be forgotten.
In order to teach constructivism teachers need to communicate more with their students, understanding each student's construction that may or may not differ from their own. In addition, teachers must offer additional situations to test students' knowledge. Lastly, it is important for the teacher to remember that the process may be more important than the solution. It should be encouraged that students write down and explain their process in addition to giving an answer. That way, the teacher can see the student's process and make assessments based on the overall process and answer, and not solely on the answer.
I found this article very interesting. I have noticed that I have become
more confident in my mathematical abilities when my teacher or professor
has looked at the process by which I approached the problem and not simply
by looking at the answer. If the answer is wrong and there isn't a process
shown, how is a teacher supposed to know why the student got the answer wrong?
By looking at a student's process approached the teacher is able to either
explain why their process may not be a good one or the teacher may see that
their process along with their own process are both correct. Mathematics
is not only about finding a solution to a question; it is also about the
thought process behind the solution. This article has really made me think
about how I want to teach mathematics in my classroom. After reading the
article, constructivism may be the way to go.
Keywords: History, Teaching Strategies
Ref: Barbara2
Author(s): Mendez, Edith P.
Date: 2001
Title: A History of Mathematical Dialogue in Textbooks and Classrooms
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94, 3, p.170-172
Reviewer: Barbara
Date of Review: 2-17-04
This article examined textbooks and classrooms from the time of Socrates up until the nineteenth century in search of mathematical dialogue that existed between teachers and their students. In ancient sources around the time of Plato and Socrates, “student communication was a part of education” (p.170). However, students were led directly, not given time to think or did not have enough time to ask their own questions. The same situation was true during the Medieval European and Renaissance Eras. Throughout the Medieval European Era schools expected students to listen and receive knowledge from the teacher. There was no time for questions. In addition, teachers during the Medieval European time often used lecturing as their most important teaching method. Still, dialogue was evident in mathematical problems, which were given in the form of a question.
Similar to the medieval era, throughout the Renaissance Era, teachers used minimal dialogue. Students during this time were expected to learn mathematics through memorization and drilling. The same method of memorization and drilling was used in the newly established United States during the seventeenth and eighteenth century.
It was not until much later that mathematic teachers started believing dialogue was an efficient and beneficial method of teaching. A seventeenth century Czech education reformer Comenius believed that teaching should be carried out in the form of dialogue because not only does it grab the reader's attention but it also is confidence-inspiring for those who may have difficulties with mathematics. Another individual, Montaigne, in 1965 made a huge leap by saying that teachers “should listen to his pupil speaking in his turn” (p.171). Lastly, in 1821, Warren Colburn developed a more student-centered approach to teaching. He encouraged students to develop their own methods for problem solving and share their methods with the class.
Throughout this article I was expecting to see some spectacular dialogue evident today, but I none of that was evident to me. After reading the article I still did not see huge strides in the communication and dialogue way of teaching. I know that it exists, but it just was not shown to me in this article so that was kind of disappointing. However, it was interesting to look back in time and discover how mathematics was taught. What I did discover, though, did not surprise me. I think that the new idea of student-centered, constructivism, discovery learning are all new ideas and methods of teaching, which teachers are still struggling to teach. As a prospective teacher I hope I am not the type of teacher who uses methods of memorization and drilling. I believe that are more constructive methods out there. After all, mathematics is not just about memorizing definitions and concepts. Mathematics is a problem-solving/discovery process along with much more.
Keywords: Planning, Teaching Strategies...
Ref: Barbara3
Author(s): Ream, David
Date: 1999
Title: http://score.kings.k12.ca.us/lessons/pytha.html
Journal or Publisher: Kings County Office of Education
Volume, Issue, Pages:
Reviewer: Barbara
Date of Review: 2-22-04
The lesson plan that I chose to look at deals with the Pythagorean theorem. In this lesson plan, students will work for three days, and the students will, in the end, know and understand the Pythagorean theorem. The method the teacher chose to explore this topic was to directly explain the Pythagorean theorem to the students, giving them specific examples and going through them as a class. Also, the teacher chose to have the class have partners and explore the Pythagorean theorem by working in the computer lab. In the pairs, students will work together to complete a lab worksheet, which is explained step-by-step. Finally, at the end of day three students will check their work with the teacher and write a lab write-up.
I chose this lesson plan because, in my in-class tutoring experiences, this is the topic we have been working on. Thus, I wanted to see other methods of teaching the Pythagorean theorem. Although I was able to see another way of teaching this idea, I did not think that this lesson plan was very detailed or complete. It was difficult for me to follow what exactly the students would be doing each day, and I did not understand how the teacher would accomplish his or her objective(s). For instance, in the lesson plan it says that the teacher will demonstrate the lab activity. What is the lab activity exactly? Moreover, the lesson plan says that on day one the teacher will give two examples using the Pythagorean theorem, and will then check for understanding. How is the teacher going to check for understanding? Is the teacher going to ask specific questions? In conclusion, I thought I would learn a great, new way of teaching students the Pythagorean theorem, but, instead, I was only confused by the whole lesson plan.
Keywords: Teaching Strategies, Communication...
Ref: Barbara4
Author(s): Brinkmann, Astrid
Date: 2003
Title: Mind Mapping as a Tool in Mathematics Education
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 96 (2), p.96-101
Reviewer: Barbara
Date of Review: 2-24-04
Tony Buzan, a mathematician, psychologist, and brain researcher, invented mind mapping, a special technique for taking notes. Mind mapping is structured similar to a tree seen from an observer in the sky. From the sky a person sees the tree trunk in the middle with branches branching off from the tree trunk, and then smaller branches branching off from those branches. This structure and way of taking notes can be used in a mathematical setting. As we know, mathematics involves ideas and concepts that are all related and intertwined together. Therefore, there may be one central idea and then, from that idea, other concepts come. For instance, volume may be a bigger concept, and, from volume, we can arrive at the different formulas of volume for many three-dimensional geometric shapes. Thus, mind mapping is an effective tool for organizing thoughts and it has been said to improve achievement.
One of the main reasons that mind mapping is beneficial, besides organization, is that mind mapping utilizes both the right and left side of a person's brain, according to Buzan. The right side of the brain performs tasks related to imagination, color and geometry while the left side of the brain is responsible for logic, words, and lists. The mind mapping technique, if used effectively, allows for both sides of the brains to work together, and hopefully increase memory retention and understanding.
In conclusion, mind mapping could be utilized more often in mathematical classrooms. Some teachers have reported that using mind maps has helped frustrated students improve. In addition, mind mapping is a way to involve the students, help them organize their thoughts, and allows for all students to clearly see the connections that exist in mathematics.
I believe that mind mapping seems to be a great tool. In junior high school along with high school some of my English teachers had the students draw a web outline. This outline helped us organize our thoughts around a central idea. So, if English teachers use a similar method, I do not see why a mathematics teacher could not as well. Mind mapping probably would have helped me see the connections for myself, and organize my mathematical thoughts.
Keywords: Issues......
>Ref: Barbara5
Author(s): Reys, Barbara J.; Reys, Robert E.
Date: 2004
Title: Recruiting Mathematics Teachers: Strategies to Consider
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 97, (2), pg. 92-95
Reviewer: Barbara
Date of Review: 3-01-04
The Federal No Child Left Behind Act demands teachers in schools today to be "highly qualified," meaning that teachers must have a bachelors degree and must have the appropriate teaching certification for the state in which they are teaching. However, this does not always seem to be true throughout the United States. Not only are many mathematics teachers under qualified, less than half not having a minor or major in mathematics, but there is also a huge shortage of mathematics teachers in the junior high school and secondary setting.
Of the mathematics teachers we have today, only forty-one percent of those teachers actually have mathematics as their area of study. Furthermore, thirty percent lack the correct state teaching certification. As a result, Glenn Commission report has come up with creative ways to attract new mathematics teachers. This new creative approach involves seven steps ranging from putting a team of teachers together, discussing ways to improve, to having current mathematics teachers recruit current students in becoming math teachers. The most important step, though, would have to be the last step, which involves sustaining the process. The actions outlined in the seven steps need to be continued for years to come. This creative approach will work successfully if continued for many years to come.
After reading this article, I was shocked at the low percentages of teachers that are certified to teach mathematics along with the number of mathematic teachers who actually have majored in mathematics in college. It was sad to hear how many students are not given the opportunity to be taught mathematics by the best-qualified teacher. In addition, it was sad to hear that many individuals today do not want to become math teachers. However, for someone who will be graduating this year and will be looking for a job very soon, it was a very uplifting article to read, knowing that the job market for math teachers is high right now.
Keywords: Geometry, Teaching Strategies...
>Ref: Barbara6
Author(s): Morris, Barbara H.
Date: 2004
Title: The Beauty of Geometry
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 9, 7, pg. 358-361
Reviewer: Barbara
Date of Review: 3-01-04
Stained-glass windows are seen in many places in our world ranging from churches to museums. Not only are stained-glass windows beautiful and aesthetically pleasing, but stained-glass windows are also applicable to the mathematical study of geometry. Many believe that the study of geometry is boring, dull, and requires analytical thought. However, many middle schools are changing this belief into one of interest, excitement, and enthusiasm.
Many middle schools today are taking something evident in the real world such as stained-glass windows and applying it to geometry. This stained-glass window project is used to spark the interests of students while still following the standards of using problem solving, reasoning, communication, connections, and representation. For this project each student is required to complete a list of specific tasks. The first task of a student is to make a glossary of twenty terms including the definition, an example, and an application. Next, the student constructs a rough draft of their stained-glass window utilizing all twenty geometric terms (triangle, complementary angle, etc.). After the teacher carefully looks over the rough draft, recommending changes where needed, the student will complete their final draft. Finally, the student will paint their stained-glass window. When all students have completed their stained-glass windows, all students are to display their window, and every student is to analyze their fellow students work.
Allowing students to analyze their peer's work allows the teacher the chance to assess students' knowledge by observing interaction among students and also by looking over the student's evaluation of one another. Throughout this project, students acquire greater problem solving skills by figuring out for themselves how to include all twenty terms into their picture. Students also are able to make connections to real world, communicate with their teacher and their fellow students, and represent geometric ideas with their resulting stained-glass window.
I thought that this activity was a great example of how to grab student's attention while completing the standards. If I were to teach a geometry class, I would seriously consider including this in my lesson plans. It is interesting, fun, and ties in multiple disciplines such as history and art. In my geometry class in high school we applied terms to exercised, one after the other. With this project, students would be able to apply their knowledge of terms to something more interesting.
Keywords: Problem Solving, Teaching Strategies...
>Ref: Barbara7
Author(s): Martinez, G. R. Joseph
Date: 2001
Title: Thinking and Writing Mathematically: "Achilles and the Tortoise"
as an Algebraic Word Problem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94, 4, pg.248-251
Reviewer: Barbara
Date of Review: 3-07-04
When one usually thinks about mathematical word problems, feelings of hate, disgust, and nervousness usually come to mind. These feelings need to be change. Word problems are essential in teaching mathematics because it engages students in active learning, increases understanding, and encourages students in developing problem solving strategies rather than memorize the mathematics material. Therefore, there may need to be a change in what types of word problems exist in textbooks and ones that are taught in the classroom. Instead of having a word problem dealing with steamboats and sheep, there should be more word problems involving cars and merchandize, which are things that students of today are able to relate to. Word problems are not boring, too hard, and can be applicable to the world we live in today. These attitudes are exactly what Joseph G.R. Martinez and the NCTM tried to change.
Martinez combined "an imaginative context with a thinking and writing exercise" when he applied mathematics to the story of Zeno's famous paradox, "Achilles and the Tortoise" (pg. 249). In this story problem it discusses how the tortoise, given a 100-yard head start, will always remain in the lead over the quick Achilles. The theory behind the problem deals with limits of sequences, and other mathematical ideas. Throughout this problem, Martinez notice that not only did his students find it interesting but also that his students were developing very interesting ways of approaching and solving the problem. Even though it may not make logical sense that Achilles never catches up to the tortoise, it was shown to make mathematical sense applying the mathematical principles known today.
It was not until I reached high school that I didn't feel anxiety when I saw a word problem. I had always hated them when I was younger, thinking that they were too hard and not really interesting. However, with all the ideas and methods people are using today to change that issue, the word problems I see currently are not only y interesting but also relate to the real world. It is very important that word problems are relatable as well as applicable to the mathematics, because otherwise students will become bored and won't understand why it is so important. Also, since our school systems are shifting more towards an integrated method, word problems are being used more often. This helps in reducing the negative feelings, and hopefully encourages students to think mathematically and develop reasonable problem solving skills. It is still a struggle today, but I think we have come a long way.
Keywords: Standards, Curriculum, Number...
>Ref: Barbara8
Author(s): Reys, Barbara
Date: 1991
Title: Developing Number Sense
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: pg.1-14
Reviewer: Barbara
Date of Review: 3-09-04
There are many definitions of number sense. Number sense "refers to an intuitive feeling for numbers and their various uses and interpretations" (pg.3). Number sense "is characterized by a desire to make sense of numerical situations," and is a way of thinking that exists in all areas of mathematics teaching and learning (pg. 4). Presently, the NCTM Curriculum and Evaluation Standards are recommending several changes in the already existing middle school standards, which desires more of a shift towards developing number sense. Currently, the Curriculum and Evaluations Standards characterize number sense in the standards in these ways: "(1) understanding the meanings of numbers, (2) having an awareness of multiple relationships among numbers, (3) recognizing the relative magnitude of numbers, (4) knowing the relative effect of operating on numbers, and (5) possessing referents for measures of common objects and situations in the environment" (pg.4).
There are many ways in which teachers meets these standards each and every day. One of the most essential things a teacher must allow his or her students to do is for them to make logical connections from previous knowledge and apply it to new situations. For instance, looking at the sequence 1 + 3/4 + 2/3 + 1/4 +1/3, a student hopefully will notice that 3/4 +1/4 = 1 and 2/3 +1/3 = 1 and then add on an extra one and the result will be three. In order to know the solution to this sequence students will reorder the numbers and find connections about the numbers from the previous knowledge they learned about fractions.
Another important task that a teacher must carry out is to engage the students in number sense, carrying out purposeful activities that ask students to think about number and their relationships. Students must be a part of the mathematics. They must develop their own techniques and share their ideas with the class. A teacher is there to guide their students and be a moderator instead of being a director. A teacher can become a guider by using process questions, which are those questions that require more than a simple factual response, when having class discussions. Therefore, there are many things that teachers need to do to ensure that their students acquire number sense, because number sense "builds on students' natural insights and convinces them that mathematics makes sense, that it is not just a collection of rules to be applied" (pg. 13).
I never realized that this concept of number sense existed. After reading this workbook I have realized what number sense really is all about. I also did not realize all the factors and details that go into number sense to ensure that a teacher's students understand the representations and connections among numbers. I wish that my teachers in middle school would have used more of a guiding method instead of standing in the front of the room "dumping" all this information right in front of me, leaving me little time to make sense of it all as I was going through it.
Keywords: Teaching Strategies, Communication
Ref: Barbara9
Author(s): Albert, Lillie R.; Mayotte, Gail; Cutlersohn, Sheila
Date: 2002
Title: Making Observations Interactive
Journal or Publisher: Mathematics Teacher in the Middle School
Volume, Issue, Pages: 7, 7, pg.396-401
Reviewer: Barbara
Date of Review: 3-18-04
This article focuses on an integrative observational assessment in student learning of mathematical ideas. Integrative observational assessment is “a pedagogical approach that invites engagement between teacher and student through written dialogue to help students develop their understanding of mathematics” (pg. 396). This approach involves students writing down answers and explanations to a set of assigned questions while, at the same time, the teacher is circulating the room looking at their student's responses. Furthermore, instead of a teacher verbally simply saying 'good job' or 'that is not correct,' the teacher will write down questions on a student's paper either try to veer them in the correct direction or writing questions that challenge the student to think further. The questions that a teacher asks must 'provide insight into a dilemma or problem, have a reference point that relates to a child's existing knowledge and experiences, encourage diverse ways of thinking, and provide reason for thinking beyond skills and facts” (pg. 396-397).
This method of observational assessment encourages and allows for student to stretch their thinking. It allows students to analyze, use reasoning, and make sure they have a clear explanation and understanding of the material being covered. In addition, by writing the question on the student's piece of paper decreases distractions verbal communication may cause, and also decreases the stress and anxiety some students may feel when a teacher openly, in front of the whole class, verbalizes that they are doing the problem wrong. It allows for students to feel comfortable and confident in their work while also allowing students to take more chances and take more risks. Even though this process cannot be used every single day, and a teacher may not be able to write on every one of their students, this approach is still a worthwhile approach that has been shown to be successful in a mathematical setting.
I am always interested in unique, new and interactive ways of teaching
mathematics, and this method introduce in this article seems like one I will
want to store away in my tool box. I do think this assessment is a great
idea. It gives all students the opportunity to think about important mathematical
concepts individually while still having obtaining the guidance, support,
and feedback from their teacher without interrupting the class or without
embarrassing the student. Especially in mathematics I think that many students
have so many questions about math, but they may not be confident enough in
their abilities to verbally ask the teacher in front of their peers. This
strategy eliminates that stress and anxiety. Now, students don't have to
be nervous anymore since the feedback is given to them in a written form
and not a verbal one. Lastly, I believe that this method will stretch students
to think more. Also, it allows for students not only to think of the answer
but why it works and how they came to it.
Keywords: Algebra......
Ref: Barbara10
Author(s): Lannin, John K.
Date: 2003
Title: Developing Algebraic Reasoning Through Generalization
Journal or Publisher: Mathematics Teacher in the Middle School
Volume, Issue, Pages: 8, 7, pg.342-348
Reviewer: Barbara
Date of Review: 3-14-04
The NCTM 2000 standards encourage teachers to help students in the middle grades to become familiar with formal algebra. One of the ways this familiarization can occur is to build a transition where students develop meaning for the algebraic symbols they use and then apply this knowledge to formal algebra. Building connections and having "opportunities to develop understanding of patterns and functions, represent and analyze mathematical situations, develop mathematical models, and analyze change" is essential for students among the middle grades to truly develop the understanding and skill involved in algebra.
John K. Lannin has researched and examined several strategies that students often use as they attempt to generalize a problem/situation and develop a clear explanation. The example problem given in this article is called the Cube Sticker problem. In this problem, students are to count the faces of a given rod (cubes joined together in a row). For instance a rod of length two (so two cubes joined together) will have 10 faces exposed and will need what they call 10 stickers to cover the faces. Then, the student adds on another cube, resulting in a rod of length 3, and then concludes how many stickers will be needed for this rod. Eventually, the student's task is to develop a general statement or rule that can be used to find the number of stickers for any given rod.
Lannin found that most often all the students approached or began the problem the same, namely by counting the faces directly, but that the students conclusions and explanations were very different. Some students used a recursive strategy. They concluded that to go to the next level you take the previous level and add on four. This is a correct solution but may be difficult when students need to find the number of stickers needed for a rod of length 1,000. Other students found that you need to multiply the rod length by four and add two to find the number of stickers on a specified rod. However, students were able to come up with this solution but had difficulty understanding why this solution is correct and why it works.
Therefore, encouraging students to explain their reasoning and having class discussions helps students understand the pattern or the generalization. Coming up with the solution is important, but explaining the solution is even more essential. Otherwise, all the work carried out in the exploration will have been lost, and what was believed to be a great lesson won't be so great anymore.
I never realized how important and challenging it was for students to be able to understand the pattern/solution until I took discrete mathematics here at St. Olaf. I struggled with that class mainly because I could always find the pattern but I had such a hard time truly understanding the pattern. Professor Molnar thought it was neat if you found the pattern but did not consider it anything special compared to if you could explain why the pattern is the way it is. So, that was a huge struggle for me and after that class I have realized how important it is to generalize situations but, moreover, to be able to justify the solution found.
Ref: Barbara12
Author(s): Kenney, Patricia; Zawojewski, Judith S.;
Silver, Edward A.
Date: 1998
Title: Marcy's Dot Pattern
Journal or Publisher: Mathematics Teaching in Middle
School
Volume, Issue, Pages: 3, 7, pg. 474-477
Reviewer: Barbara
Date of Review: 4-05-04
What strategies should a teacher use when helping students analyze a situation/pattern? Will some strategies work better for some students more than others? Lastly, when asking for explanation what are the teacher's expectations? These are important questions for a teacher to ask himself or herself each time they hand out a problem such as the one discussed in this article. This article analyzes the solutions and explanations to what is called the Dot Problem. In the dot problem, patterns of dots are displayed for the first three steps. What is asked of the student is to figure out how many dots will there be for the twentieth step?
After looking at the pattern, some students noticed immediately without sketching out the following steps that the twentieth stage will have 420 dots, namely 20 x 21. On the other hand, some students wrote out each step and then came up with the answer of 420 dots for the twentieth step. So, both methods end up with the same correct solution, but the processes are different. Which process is better or is there a better way at all? For the method that had written out each step, these students may be able to identify the pattern but they "…might not use that pattern in a general way to keep their calculations to a minimum" (pg.476). This suggests to me that the student is not making the connection to the broader generalizations that may be drawn from this problem.
Furthermore, students were also asked to explain their solution. Again, there were many answers. Some students just showed the sketches while others used a "different" looking algebraic approach. All of these explanations, even though they arrived at the right answer, may not be sufficient explanations. This suggests that students may need additional practice that provides more opportunities for written explanations. Some students may need to be guided more in the right direction in order to come up with a suitable explanation. Explanations are important because what may seem like a number problem actually develops into an algebraic expression that can be used to generate the number of dots for every step, not matter how large.
I carried out this dot problem and came up with 420 dots and also
came up with an algebraic generalization of (n x (n + 1)) where n is
the step. I did not have much trouble with this because I have worked
a lot with algebra and I have always loved algebra when I was little.
However, I can understand that something that looks so simple to me is
difficult for many students. Some students have difficulties with
algebra. That is exactly why I think this problem is a great link
between dealing with numbers to start thinking about algebra and how to
express all of these cool patterns, etc. for the nth step instead of
just the 20th.
Ref: Barbara11
Author(s): Bennett Jr., Albert B.; Nelson, L. Ted
Date: 2002
Title: Divisibility Tests: So Right for Discoveries
Journal or Publisher: Mathematics Teaching in the Middle
School
Volume, Issue, Pages: 7, 8, pg. 460-464
Reviewer: Barbara
Date of Review: 4-05-04
Did you know that when you double the tens digits of a two-digit number and then add the ones digit that if the sum is divisible by 8 so is the original number? These methods are called divisibility tests. They are quick ways of looking at the divisibility of a number without the use of a calculator. Many students have discovered these divisibility rules all on their own while others have been taught these quick methods in their classes. No matter how these rules are learned, these rules are important when building and developing a sense of number.
It is essential in schools for students to develop an understanding of numbers, ways of representing numbers, and discovering the relationships among numbers. Using a calculator to divide bigger numbers in later grades, after the long-hand method of division is already taught and retained, is not always the best option. Learning and discovering these quick divisibility tests helps students to think more in-depth about numbers and their underlying operations that can be used. In this article they outlined the methods used for discovery several divisibility rules. All of the methods involve illustrating numbers as blocks of tens and units of ones. Then, the student is to remove a certain amount from each ten and then carry out a simple mathematical concept either with the remaining pieces or with the removed pieces. It sounds confusing but it is actually fairly simple.
I do believe that learning all these rules is useful for students.
It will hopefully shy students away from the calculator, which is
already depended on in many schools today. However, I have a hard time
understanding why these rules work. I see that they do work but I
don't understand why and I did not see a proof of why it does. So, yes
these rules can be useful and may also enhance number sense, but will
it confuse students more or will it really enhance their learning?
Ref: Barbara13
Author(s): Preston, Ronald V.; Garner, Amanda S.
Date: 2003
Title: Representation as a Vehicle for Solving and
Communicating
Journal or Publisher: Mathematics Teacher in the Middle
School
Volume, Issue, Pages: 9, 1, pg. 38-43
Reviewer: Barbara
Date of Review: 4-12-04
Ms. Simpson's seventh grade classes were challenged to use data to decide which party plans option would be the best. The "best" plan was the cheapest plan. Many students approached this data problem in several different ways. While some students began drawing tables and calculating numbers, other students started by making graphs of the data. In both methods, students were analyzing the mathematics behind this problem. Students use different methods of approaching problems often, when given the opportunity to explore a mathematical problem. That is why our standards set forth today have stressed the importance of representations. Representations are important because "they are vehicles for learning and communicating; they support learners of many different styles; and they come in many different forms, allowing students to use combinations of representations to gain more information than would be possible with a single representation" (pg. 39).
Focusing on Ms. Simpson's data problem the different representations that could have been used are tables, linear graphs, bar graphs, and algebraic equations. When students were asked to justify their answer using their mathematical discoveries, many students utilized all of these representations. Students began by discussing the table, then went on to showing the graphs, and, lastly, used the equations of each party plan to sum up the analysis. As you can see, with one problem students have been given the opportunity to learn and utilize essential algebraic concepts taught today. These concepts do not need to be taught separately, they are intertwined and apply to one another in many ways, just as this problem showed.
When I was in middle school I do not remember if I was given the opportunity to use my own representations. I think I was so used to modelling my work behind the teacher's work that I failed to think about how I would represent the problem. In schools today, students need to be given the opportunity to work independently or with a group , coming up with their own procedure, method, and analysis of problems. I believe that students will gain more for that than from modelling their learning behind what the teacher does. Those ideas won't be embedded in their heads, but the ideas they have come up with themselves will.
Keywords: Proof......
Ref: Barbara14
Author(s): Scher, Daniel
Date: 2003
Title: Dynamic Visualization and Proof: A New Approach to a Classic Problem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 96, 6, pg.394-398
Reviewer: Barbara
Date of Review: 4-19-04
In 2001-2002 three teachers joined together with an international collaboration to test U.S.-Russian Interactive Geometry labs in their classroom. This program introduces teachers in the United States to "…effective mathematics teaching practices from abroad" (394). This new approach takes students from visualizing a problem to deductive reasoning. The problem that was analyzed was called the Pirate Problem. In this story problem, a pirate is trying to tell a friend where he has hidden the treasure. His directions state that when you find the only palm tree on the island you are to walk directly to the falcon- shaped rock, turn a quarter-circle to the right, walk the same distance you did from the tree to the rock, and place a stick. Then, go back to the palm tree, walk to the owl-shaped rock, turn a quarter-circle to the left, walk the same distance you did from the tree to this rock, and again place a stick in the ground. Then, find the midpoint between the two sticks and there the treasure will lie. The question that needs to be answered is: "If the two rocks remain but the palm tree has long since died, can the riches still be unearthed?" (394).
The resulting solution to the question is that you can find the treasure without the palm tree. There were several methods used when approaching this problem, but all of the methods focused on using geometry Sketchpad. With the program, students explored multiple cases where each case resulted in the same answer; you don't need the palm tree to find the treasure. Although this problem does seem difficult, students in high school should be able to attempt this problem with the Sketchpad and come to a conclusion. The proof, however, is the trickier part and that is why splitting it up into cases made the process a lot easier.
To tell the truth, I was a little confused by this problem. I understood the steps they took in their process but I had a hard time understanding why you don't need the palm tree. I see that in all cases that, when you drag some points, the position of the treasure does not change, but I don't see why. I guess, I would have liked a better explanation of why instead of showing me what they did, and automatically assume that I understand. I would be nervous to teach this to my geometry class since I don't have a complete understanding.
Keywords: Problem Solving......
Ref: Barbara15
Author(s): Kahan, Jeremy A.; Wyberg, Terry R.
Date: 2003
Title: Problem Solving Can Generate New Approaches to Mathematics: The Case of
Probability
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 96, 5, pg. 328-332
Reviewer: Barbara
Date of Review: 4-19-04
This article discusses a World Series problem, which says: the Yankees and the Mets are in the World Series (the best four-out-of-seven games) and the Mets have won one game while the Yankees have won two games. The winner receives one million dollars if they win, but all of a sudden, after the third game, the umpires decide to go on strike and the remaining games are cancelled. The money is to then be divided up in proportion to the teams' probabilities of winning the series. So, how should the money be split up?
There are many different ways to come to a reasonable mathematical solution to this problem. One of the methods of solving this problem involves simulations. Simulations "help students build evidence, intuition, and insight, which they can use as a basis for further thinking, theory building, and evaluating theory" (328). Another method is to use trees, which essentially look at the possibilities of the problem. For instance, the Yankees could win the fourth game or they could lose the fourth game and from this other outcomes will branch out for the fifth game, etc. The last method discussed in this article deals with generating functions. Some students may feel more apt to looking at algebraic functions, etc. instead of trying to represent the problem using diagrams or trees. Whatever way is carried out, students all came to a similar conclusion, which stated that the Mets have about a 31.5 percent chance of winning the World Series and therefore should receive 31.5 percent of one million dollars.
The only barriers that will be encountered in this problem is that the problem may be viewed as unrealistic by some students, not applicable or interesting to some, and the solution to the problem does not really correspond to exactly what would happen. Teams are not given rewards for winning and the World Series probably would not be called off due to a strike. Still, the problem is interesting because it does relate to a lot of people in our world. On the other hand, some people may not have an interest in baseball. However, this problem could easily be changed to a different topic/subject that is of interest. Lastly, it is hard to predict exactly what will happen in this problem. There is home field advantage, past reputations in the World Series, and the scores of teams who do well or don't do well especially under pressure. Regardless of all of this, the problem is still good to look at and sparks a lot of interesting conversation, discussion, and analysis.
Keywords: Communications......
Ref: Barbara17
Author(s): Lee, Hea-Jin; Jung Woo Sik
Date: 2004
Title: Limited English Proficient Students: Mathematical
Understanding
Journal or Publisher: Mathematics Teaching in the Middle Schools
Volume, Issue, Pages: 9, 5, pg. 269-272
Reviewer: Barbara
Date of Review: 5-02-04
In our diverse, growing society today, there seems to be a lack of communication skills for LEP, Limited English Proficient, students, especially in mathematics. Having English as a second language causes barriers not only in language related courses but also in mathematics courses. It is difficult enough for a student to understand all of the concepts involved in mathematics, and, therefore, is especially hard for students who are learning English in school. That is why it is important that the following steps be taken to encourage and help LEP students in a mathematics environment.
Even though an LEP student is not answering the math questions as in-depth as other does not necessarily imply they do not understand what is going on. The difficult thing is that LEP students have trouble communicating their ideas using the English language. It is difficult for them to understand the meanings of concepts. That is why it is important for teachers to have multiple questioning strategies along with using multiple sources. Teachers should ask students to explain using a picture or teachers should teach a lesson using pictures, diagrams, and body language. That way, LEP students will see these cues and understand the mathematical concept better.
Another way that teachers can help LEP students communicate better is to utilize cooperative learning. Allowing LEP students to be paired with English speaking students will allow for both students to learn from one another. Often it is easier for a student to interact with a peer and learn from them. Another thing that is essential in this communication process is parent involvement. It is important that parents become familiar with their child's mathematics classroom, teacher, and also become familiar with the expectations in the classroom as well as outside the classroom. All of these things are ways to improve the communication of mathematics among LEP students.
Keywords: Connections......
Ref: Barbara18
Author(s): Devlin, Keith
Date: 2002
Title: Numbers in the Garden and Geometry in the Jungle
Journal or Publisher: Mathematics Teaching in Middle Schools
Volume, Issue, Pages: 7, 8, pg. 422-425
Reviewer: Barbara
Date of Review: 5-02-04
Some children find mathematics to be boring and intimidating. However, by connecting mathematics to other areas such as nature will not only make mathematics more fun, but also it will show students just how important mathematics is in our world. In this article, Devlin looks at the connections between the Fibonacci sequence and the number of petals on a flower. He also discusses the connection between mathematics and the coats animals have. Devlin believes that too often the fascinating applications are saved for extra days at the end of a unit or at the end of the year. But, these applications should be used during the units and incorporated into lessons, making connections between the real world and the topic at hand. By doing so, students will become more involved and excited about mathematics.
I thought that this article was rather hard to read. I thought the purpose of the article was a great one, but I had a hard time understanding what exactly they were talking about. I do not understand how I am supposed to apply the idea of counting petals of flowers into my mathematics lessons. I think it would be fascinating if there were some connection between the two, but I just do not see that connection in this article.
Keywords: Probability......
Ref: Barbara16
Author(s): Arbaugh, Fran; Scholten, Carolyn M.; Essex, N. Kathryn
Date: 2001
Title: Date in the Middle Grades: Wed Quest
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 7, 2, pg.90-95
Reviewer: Barbara
Date of Review: 5-02-04
Carolyn Scholten, an eighth-grade teacher in a small community in southern Indiana, had her students involved in a learner-centered activity. This activity was focused on probability, the concepts associated with probability, and being able to determine future events. Scholten decided to have her students use a new Web-based program. In this program, students first were introduced to concepts associated with probability included probability, odds, event, outcome, and equally likely, and students also became associated with a probability scale. After responding to this introduction in their math journals, students were ready to look at data. All of the data they looked at was simulated as a class or carried out individually on the web program. Along with the simulation process, each student was given activity sheets to use as a guide. These sheets were used as a way to organize the student, help the student analyze the data, helped students calculate theoretical probabilities, and helped students make generalizations and predictions. Student's looked at simulations involving tossing a coin and rolling a die. As students built larger sample sizes, they realized that the probability for tossing a coin approaches .50 and the probability for rolling a dice approaches 1/6. After this two-week unit, Scholten believes it would be even more interesting to look at more real-world data such as predicting the outcomes of sporting events. The activities that Scholten had her students engage in not only introduce students to probability but it also helps students move smoothly into high school where probability is a bigger focus under the standards.
I think that probability
should be looked at more in middle school. We make predictions and talk about
probability every day when we think about the chance a team has of winning, the
chance a person has of going to college, and the chance that someone's family
will win the lottery. Probability is very common and therefore, I believe,
would spark the interest of m
any students. Unlike calculus and other areas of mathematics, probability can
be connected to so many aspects of our everyday life in a way that would draw a
student in to wonder what is going to happen in the future? Or, what are the
chances I will win this game on Saturday given we have lost our four past games?
Return to Index
Keywords: Measurement......
Ref: Barbara19
Author(s): Buhl, David; Oursland, Mark; Finco, Kristin
Date: 2003
Title: The Legend of Paul Bunyan: An Exploration in Measurement
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 8, 8, pg. 441-448
Reviewer: Barbara
Date of Review: 5-13-04
This article deals with the idea of using story telling as a method to teach mathematics. The example given in the article deals with the legend of Paul Bunyan. This legend talks about the great size of Babe, the ox, Paul Bunyan's frying pan, and the geographical lakes that Paul Bunyan himself formed. In this exploration students are supposed to use skills including estimation, finding area and volume, setting up proportions, working with scaling factors, and constructing objects out of clay in one, two, and three dimensions. This activity not only has students directly involved in the mathematics at hand but also allows the students to learn something new and interesting about their geographical surroundings.
One thing that I found interesting is that many
students found the idea of estimating nerve racking. "We are always taught to
measure exactly," they said. Thus, having students estimate measurements
encourages important estimation skills. There are many useful skills carried
out and learned in this activity, and this activity gives students the
opportunity to apply mathematics to other subject areas. Everyone loves stories
and applying mathematics to story telling is even better!
Return to Index
Keywords: Technology, Statistics...
Ref: Barbara20
Author(s): Bratton, George N.
Date: 1999
Title: The Role of Technology in Intorductory Statistics Classes
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 92, 8, pg. 666-669
Reviewer: Barbara
Date of Review: 5-13-04
Technology has played a significant role in the modern world of mathematics. Technology is even more useful in statistics. In order to see actual statistics and actual models, which make up the basis of statistics, students will need appropriate technology whether that come from computer applications or from a calculator. Because of cost issues, calculators seem to be the choice of many schools.
An ongoing debate has been going on dealing with the idea that too much work is being done on the calculator. However, technology does three things: "it makes teaching some topics unnecessary, it permits teaching some topics better, and it allows teaching some topic that have never been taught" (666). In statistics courses all of these things hold true. Students need technology in order to be successful in this course. Statistics become too tedious asking students to calculate permutations by hand and having students memorize formulas. Using a calculator and having students develop fundamental statistical concepts is what is more important. Technology is capable of taking students much further in the world of statistics than paper and pencil can.