Journal: "Teaching K-8." Early Years, Inc. Vol.29, No.4. January, 1999.

I found an article in: "Teaching K-8." It was a cover story written about the President of the National Council of Mathematics (NCTM) visiting a sixth grade classroom. I thought this article would be particularly relevant to this class because we are currently going through the Standards set by the NCTM. Glenda Lappan, President of NCTM found something very unique and special in the class she visited. The article first mentions one of the NCTM’s goals, which is to build a foundation as early as kindergarten that will give students confidence that they can rely on their own thinking to solve math problems. They also want to students to realize that there exist numerous ways of solving problems and reaching the correct solutions. The NCTM feels that the teacher’s role in this endeavor is to summarize for the children and help them put everything together so they can generalize about a problem. The teacher needs to just point the students in the right direction and then they can develop a greater understanding by figuring out the methods on their own. This way of teaching stresses thinking things through instead of just memorizing facts, formulas, or solutions. This is the type of teaching going on in the sixth grade classroom at Western Middle School, Connecticut.

The NCTM also stresses the importance of getting parents involved in what is going on in the classroom. Leslie Paoletti, coordinator of mathematics, science and technology for Greenwich public schools agrees with this importance. "If we can get parents into the classrooms, they’ll see how a calculator is used appropriately – how it develops math concepts, not in lieu of learning math facts which we all believe is quite important" (53). If the parents are supportive of what is happening in the classroom, they will be helping their children by encouraging class work. Parental support is essential and in some cases it is very interesting for them to see the new way things are being taught. If they are taking an interest in the classroom activities, they can then help their children at home succeed in that classroom and in school in general.

Glenda also notes in this article that, "…there’s an important relationship between the workplace and middle school math" (53). The little things that everyone must know for surviving successfully in the world are the foundation laid in the middle school grades. It is important to place emphasis on learning the math skills in the appropriate way, the way the NCTM is setting out to achieve. Math is always changing, therefore, it is imperative that children acquire a grasp of the underlying concepts of mathematical skills to help them throughout their entire lives.

Jennifer Rademacher

 

"What is Standing in the Way of Middle School Mathematics Curriculum Reform?"

Middle School Journal. November 1998, p. 42-48.

I found this article rather interesting because it deals with the changing curriculum of math and refers to everything we’ve been talking about in class lately. What the gist of this article is about is making the necessary changes to improve mathematical learning among all of our students and also improve the way in which we teach and assess those students. The five important forces behind these changes are 1) the NCTM Standards; 2) the Third International Mathematics and Science Study (TIMSS); 3) the national volunteer eighth grade math test; 4) the increasing availability of technology for use in and outside classrooms; and 5) "the feeling by teachers, parents, and administrators that we can and should do a better job of preparing all middle school students in mathematics" (48). People feel that we need to implement changes due to the outcome of conducted tests which showed that U.S. eighth graders score well below the international average in mathematics. It seems as though students are not challenged or stimulated enough to explore new math ideas. It was proven by the TIMSS study that U.S. textbooks cover a wide range of content with too little depth; textbooks are, "a mile wide and an inch deep."

What these forces want to emphasize is the importance of students exploring math through active engagement. Math should also be relevant to topics of today’s world – this makes understanding easier for the students. This material should be new and interesting to students in middle school, not a repeat of what they learned in elementary school. As a result of a study done throughout some schools in Missouri, we see that we need to change, "…from a repetitive, boring, computation-based middle school mathematics program to one that engages, challenges, and prepares students" (43). This is so very important and has been stressed in many of the classes I have taken. It is a total switch in the way some teachers are currently teaching. It will take work on the part of the teachers and students, as well as administrators and parents. Only when the entire group is working together to achieve this change, will it truly happen and be successful.

There is a greater emphasis on technology with these new ideas. It is imperative that children as well as educators, become comfortable with using a variety of forms of technology to aide in the learning of mathematics. This will not take the place of students actually learning mathematical methods, but will be used only to enhance those experiences. Parents, students and educators all need to understand the purpose behind the use of technology to fully appreciate all the help it gives. Besides a greater use of technology, there is also a greater push for more reading and writing in math classes. This has received some negative reactions from students and parents because they do not seem to understand that writing about mathematical concepts will facilitate a child’s understanding of math. I, personally, am not a fan of writing and would rather do numerous math problems over a paper any day. Although, with the way the new mathematical programs are being implemented, it seems necessary to incorporate these other areas into the class as well.

Drastic changes within the mathematical classroom will not happen overnight. They take patience to implement them the correct way and only then will everyone truly benefit from the new programs.

Jennifer Rademacher

http://enc.org/classroom/lessons/docs/104102/nf_4102.htm

This website was very interesting because it suggested ways to make a typical problem more appropriate for middle school students. The typical problem:

"A farmer wants to enclose a pasture using an existing fence and ninety-six meters of fence stored in a shed. What is the largest rectangular pasture that can be made if the farmer uses the existing fence as one side of the rectangle?"

However, the authors suggested that the problem in these words would be meaningless to middle school students. Teachers need to modify the problem so students have responsibility for the situation and the numbers are smaller in order to work with manipulatives easier. The students should not be told how to solve the problem, leaving that task to group decision. After the students have obtained an answer, they need to write their answer up and be able to present the information to the rest of the class. These four steps make the problem much more relevant to middle school students. The students will try to find more information out about the situation. The teacher is at his/her discretion in disclosing more information. The teacher should respond to each group of students question as an isolated case. Using this method, the teacher can respond to the same question differently to two groups thus leading to discussion as to how the original situation affects the answers. The teacher should ask the students to present their information in unique ways, such as using a bar graph. Using many methods, the students can find ties between this problem and other math topics.

This website was very helpful because it dealt with a specific problem and the ways to modify it for a middle school class. I think every math student has seen this problem at least once in their lives. I think the key is to make it appear as a novel situation to the students. I remember trying to figure out problems similar to this one without manipulatives. Then, the problem becomes a routine task. Not giving the students every detail is an effective way to make them completely analyze the problem. Why are certain details important? Right now, I could probably find the biggest area of fencing for the farmer, but I would not understand how it related to the details of the problem.

The idea of making middle school students, or any students for that matter, in charge of the situation is a great way to make the problem real. When the problem begins with "you are," the students feel they are working on an important task. They are playing the role of farmer. I think another way to make this problem real is to have little models of the situation. Instead of drawing the house and fencing on graph paper, maybe groups could actually "build" a model of the house and use some wire as the fence. We could use inches of fencing instead of feet. I think this method would be helpful for students to visualize the precise situation and they would have fun working with something other than pen and paper.

I also liked this website because it gave examples of students’ answers. All to often, I find lesson plans stating what the answer should be, but I like to hear real student examples. Then, I will not be as surprised when groups of students use very different methods and ideas to work out this problem. I think the four steps to making this problem better for middle school students can be applied to any problem.

Kelly Fitzpatrick

 

I found a journal article on the internet that discussed the negative affects of tracking in schools. Although it did make some good points, in the end I think I strongly disagree with their views. One of the main reasons the author of this article had a problem with tracking was because the people who are in the low tracks are the ones who need the most help, but they are often the people who are stuck in the low track forever. These people are most often minority children or children from low income households. I do see this as a problem. I do not think that once a child is in a low track in math or any other subject that he should be confined there. I think the program needs to be designed so that students can move up and down the track as necessary. The article also said that the students in lower tracks are poorly educated and are not cared about as much as the smarter children in the higher tracks. I think this has to do with the attitude of the school district and the teacher. I do not think this should be the case at all, nor do I think it is always the case. If a teacher cares about her students learning it should not matter their ability level. Rather, the teacher should work even harder to make sure that those students who are struggling can learn even more. The article suggested mixed level classrooms where students work together to learn the math. I have many problems with this technique. One of the main problems I have is that when students work in groups that are mixed levels two main things often occur. Either the students who are smarter end up doing all the work and those who are struggling don’t learn anything, or the students who are smarter are held back and not able to reach their full potential. I do not think there is anything wrong with tracking as long as it is done properly. I think tracking is meant to provide extra opportunities to those who succeed in math and extra help to those who need it. The goal of the teacher should be to help those children who are in the lower tracks to understand math and do well in math and hopefully move up to the middle track. The teacher and administrators need to work just as hard to serve the lower tracks as they do to serve the upper. They should strive to teach those lower track children with a goal that they will no longer need those lower tracks because all of those students will be succeeding in math.

Langkamp

The article which I found about middle school teaching and math is "Giving Voice to Middles School Students Through Portfolio Assessment: A Journey to Mathematical Power" from Middle School Journal, Sept 1997, p. 22-29. It talks about a way to help students have a more positive attitude towards mathematics. The teacher helped the students to compile various assignments and projects from each semester/trimester of the year into a final portfolio. A few of the assignments in the first trimester’s portfolio were a student math journal and a math autobiography. The second trimester work included pieces showing their problem solving ability and math improvements, among other things.

The students were able to choose what they felt were their best pieces and could find improvements in their work. They were generally more diverse in their choices for the second trimester’s contents. This gave the impression that they were enjoying a bit of freedom in showing their math skills at work in all areas of their lives and not just book work. The students graded their own portfolios and tended to focus on effort rather than skill level accomplishments.

I think this is a neat idea to use in a math classroom. As a student, I have done portfolios for literature classes and really enjoyed looking through my work to find just the right pieces to fit all the required categories. I think the same enjoyment could be found in doing a math portfolio and I think I may try this when I become a teacher. The students seemed to see their own improvements as did the teacher, so that is the best part of the assessment project.

Peggy Neubert

 

 

For this review, I found an article in Mathematics Teaching in the Middle School, entitled, "On My Mind: Dispelling Myths about Reform in School Mathematics." In this article it states that there is a controversy and debate over the current reform in school mathematics. It seems as though critics of reform efforts have latched onto certain myths in voicing their comments. Myth one is that basic computation is ignored. I agree that this is definitely a myth. Basic facts such as addition and subtraction are still a prerequisite for solving problems that are meaningful, relevant, and interesting to learners. The reform in school mathematics has gone beyond the standard pencil and paper computation and includes estimation and mental computation. Also, if students are going to wise users of technology, they must have a sense of whether computational results are reasonable, rather than just accept computer or calculator output. I personally feel that calculators should not be used for basic skills. Students get too dependent on them and do not understand the underlying rationale behind these basic facts. They did not have calculators in the past, so somewhere along the line, individuals actually had to use mental math and compute basic facts in their head.

The second myth was that answers that are close to correct are good enough. The article stresses that if a problem requires an estimate rather than an exact result, an estimate is good enough. If a problem requires an exact answer, being almost correct is wrong. I agree and disagree with this statement. Yes, answers close to correct are wrong, but there is another side to this issue. In my opinion, process (how an answer is achieved) is more important than the actual answer. Using the correct method to produce the wrong answer is better than using a random method and somehow coming up with the right answer.

A third myth that the article stated was that only one right way exists to teach mathematics. This statement is absolutely false. There are many ways to teach mathematics. The article states that worthwhile mathematics are determined on the basis of teachers’ knowledge of the learning process, learners’ needs and interests, and a firm understanding of the mathematics to be taught. I agree 100%. The degree to which the learner understands the information depends on the teacher. The teaching style that is used by the teacher is determined by the outcome of the learners.

The final myth that was presented was that textbooks identified as "standards-based" support reform efforts. The article states that the rigor, depth, and logic of mathematics are preserved while problem solving, communication, reasoning, and connections within and beyond mathematics are highlighted. It also states that Textbooks that make connections with literature, history, and other subjects across the curriculum, do a disservice to students and to reform efforts.

Winer

 

"Computation in the Elementary Curriculum: Shifting the Emphasis". Teaching

Children Mathematics. December 1998. p.236-241.

In the mathematical classroom, there has been much concern and debate regarding mathematical computational techniques. Teachers are feeling pressure from curriculum guidelines, administrators, parents, their own personal views, state standards and national standards regarding computation. What strategies should be used? When should these strategies be introduced and how much emphasis should be placed on each of the strategies? Teaching Children Mathematics, an educational journal that focuses entirely on teaching mathematics, presented a journal article to address this growing concern of computational methods. "Computation in the Elementary Curriculum: Shifting the Emphasis" portrays the problems that teachers face when teaching computational skills and it even suggests a curriculum plan to help solve this mathematical dilemma.

The first important point in "Computation in the Elementary Curriculum" is the idea that some computational algorithms have "outlived their usefulness" (p. 237). Some traditional computations, such as finding the square root, which were necessary to learn thirty years ago, are just not necessary anymore for today’s growing world. What is also dangerous about this traditional algorithm is that it does not lead the student to understand "why" the square root is achieved from this process. The student will be able to find the right answer but he will not be able to understand what it means. This article presents the idea that it would be more beneficial to the student to find the square root using a calculator with "an emphasis on understanding what a square root is to employ estimation strategies" (p.237). The article is ultimately suggesting at this point that the comprehension of a square root would be more beneficial to a student in more mathematical practices then merely figuring out a square root with out understanding what is happening.

While the idea of traditional computation, such as the example of the square root, is becoming accepted more and more as the wrong path to take for computation, one may ask what the alternatives are. Since traditional, written algorithms decrease the student’s capacity of conceptual understanding and sense making, teachers have begun to turn towards more mental computation and estimation. These two methods encourage the student to think for himself about a particular problem and it also fosters logic problem solving sense. These assets of learning, in turn, create a concrete understanding of a problem and it allows the student to apply this knowledge to a written algorithm and ultimately, understand what a written algorithm means.

"Computation in the Elementary Curriculum" not only addressed these current issues and problems about computation that plague teachers in a classroom, but it offered a designed curriculum to attack the existing written algorithm predicament. The article suggested the following proposal for teachers to use when dealing with computation: 1) mental computation, which involves developing and inventing computational strategies to record work in the younger grades and for whole number and fraction and decimal computation problems in the higher grades, 2) written computation, where in the upper grades students develop written algorithms for problems with the comprehension of why that written algorithm works, 3) estimation where younger students should try to make sense of data using estimation in measurement settings and in higher grades where students produce computational estimates to judge the reasonableness of their answers and finally, 4) calculator, where students use calculators to explore patterns and relationships with numbers and operations and as an efficient tool to do complex computations associated with solving problems (p.239). This proposal was outlined in this manner but it was stressed to teachers to use it openly, making adjustments where they deem it necessary.

Teaching Children’s Mathematics has presented a very useful tools for math teachers in "Computation in the Elementary Curriculum: Shifting the Emphasis" in terms of computational techniques. This article not only presents problems that teachers face with traditional written algorithms but it also provides a solution for tackling this problem. This article is effective because not only does it point out an existing problem, but it also gives alternative advice on how teachers can alter their curriculum to begin to change the traditional computation strategies. This change is important, not only for the benefit of educational journals, but also more importantly, for the benefit of the students.

Erin McKinley

 

In the magazine Mathematics Teaching, I found an interesting article entitled "Developing Maths Trails." This article can be broken down into three major parts namely: the value of a maths trail, planning a maths trail, and lastly implementation of a maths trail. Before I go into these three sections, it is essential that I first define what maths trail is. Maths trail is an activity-based learning, where students follow a predetermined trail, either within the school grounds or in the immediate vicinity. As students follow the trail, they have to undertake a series of tasks or activities that might require drawing, counting, measuring, calculating, estimating, identifying or describing. Whatever the case is, they focus on several aspects and areas of mathematics.

The value of maths trails is tremendous. There are various positive aspects and outcomes of students participating in this type of learning. First, the activities are real and relevant. Second, students will gain a sense of confidence, because they will apply their math knowledge in practical situations. Third, children will enhance social skill, because maths trails promote and work best when students work in pairs. Another positive effect maths trails have is that it enhances children's thinking and communicating skills as they work with one another. Additionally, students learn how to collect and record data accurately, because they have to report back to the class later on. Finally, learning will be fun and enjoyable as students are exposed to a non-classroom, non-restricted environment.

Planning the maths trail can vary teacher-to-teacher and requires the creative side of teachers. The writer, however gives suggestions and advice in planning maths trails that was effective for his sixth grade class. The duration of the trails should be 40 to 45 minutes. Students should be grouped into two’s in order for optimal enjoyment and learning. The materials needed for a pair of students are a clipboard, pencil, and possibly a ruler. The location of the "trails" is completely dependent on the environment of the schools. The teacher can transform anywhere from a playground to the adjacent road to the residential areas into a math "trail."

The implementation of math trails should include the actual running through the activities by students, but should not overlook the discussion with their fellow classmates on their findings and discoveries the following day. Also, once students master the math trails they should consider creating their own trails for fellow classmates to follow. Each pair of students should compose a series of questions pertaining to their trails. For example, a portion of a student-created maths trail is described in the following lines:

What shape is the bottom of the pipe? Walk to the bin, via the white line. Stand at the bin. Look up. How many rectangles can you see that are white? Do a quarter turn to the left. Look at the pipe on your left. What shape is above the pipe? Clue: It has four sides. By the pipe there is a beam that sticks out. Is the beam horizontal or vertical?

Heesundo

 

For my 2nd bibliographic review I chose to take a written response by a man on the web. The location of this excerpt can be found at: http://mathematicallycorrect.com/report.htm

Allen, Frank B. "Repairing School Mathematics in the US." Mathematically Correct. April 3. 1998.

This article was written after the release of the TIMSS results. US Seniors ranked 19th out of 21 countries, and Allen responded to the results by bashing on the (old) NCTM standards. The strange thing

is that Allen was a former president of NCTM and the National Advisor for Mathematically Correct, the people who released this article. Allen puts down all that the standards pushed for like cooperative group learning, constructivism, and other things. It is a harsh article and calls for a revision of the standards, and alas, the standards are being revised.

Sam Hyun

 

Andrews, P. (1998). Peddling the myth. Mathematics in School, 27 (2), 2-4.

This article discusses the myth of why we teach math to children. The myth is that we will use it in our adult lives. However, teachers should not defend math for its usefulness because there are many times when it is not useful(e.g.-finding the volume of a cone or the angle of a triangle). While quick computations are important and useful, math should be taught for enjoyment, wonder and challenge. Andrews makes the statement that, "Our teaching needs to focus on mathematics for what it is and not on its applications". This is definitely an article that should be read by teachers who feel they need to justify their teaching with usefulness.

Gina Krajewski

 

Andrews, P. (1998) Peddling the Myth. Mathematics in School, 27, 2. 2-4.

Paul Andrews addresses the issues of why math is important to teach and why people teach it. He doesn’t feel math is approached as it should be. Everyday situations involve a low ability of math so higher levels of math should be taught because it is interesting and a unique way of viewing what is around us. In the interviews Andrews has conducted, the people say they want to teach math because it was their favorite or best subject in school. Their reasoning should go beyond that and involve the pleasure that they can share with the students. The focus of the math curriculum shouldn’t be its application, rather, it should be on what it is. Hopefully through such a change of the focus we will not be known as being afraid of numbers, as a Hungarian viewed Americans. Whereas, Hungarians think of math as a worthwhile subject that is unique.

Robyn Swieboda

 

Andrews, Paul. (1997, December). A Hungarian Perspective on Mathematics Education. Mathematics Teaching, 161, 14-17.

Many articles I have read comparing English curriculums with other curriculums have praised foreign methods and insulted English methods. This article wisely surmised that teaching math is greatly affected by culture. Therefore, England could not completely adopt Hungarian methods and be successful. While each can learn lessons from the other, complete adoption of the other’s practices would lead to failure.

English curriculum was found to include high quality problem solving activities and investigations. This aspect was not as strong as in Hungary, but problems were taught in greater depth. English students were rarely expected to prove their results. In an example that used Cuisinaire rods to discover the Fibonacci sequence, Hungarian students derived a formula, justified it, and investigated the problem with different combinations of Cuisinaire rods. Relational understanding was taught to Hungarian students.

Varying approaches are also affected by the aim of teaching math. England teaches math to use it to learn science and as a tool for commerce. Hungary teaches math because it is "worthwhile and valued."

Concern for how the children would deal with failure was a big factor for English teachers. They did not want to injure confidence. English students are encouraged to learn for personal satisfaction while the Hungarian approach often leads children to seek approval from teachers and parents. Group work is an area that Hungarian teachers should try to improve.

What both groups gained from the investigation is that the motivation for teaching math directly affects how it is taught. Focus needs to be placed on mathematics much more than its applications, which seem to be the basis of the current English curriculum.

Elizabeth Brands

 

 

Atkins, Sandra. "Listenning to Studnets". Teaching Children Mathematics

January 1999. Vol.5, p.289-295.

This article was basically about the importance of the conversations in a classroom setting.

One math teacher sets up four different settings of conversation with fourth graders in different classes while they are doing math. Children were challenged with questions such as solving for a volume, measuring an angle, and subtracting negative number. She raised many congnitive conflictions and made students to reason and explanin their method.

Some of the dialogs were very helpful for me to understand how the children think when they face such math problems. This article itself was easy to read and very interesting. Things that I learned from reading this article was that conversations between students and conversations among students should be encouraged and performed in math classes because students can really think and reflect upon what they know and why they think that way.

I thought that magazines like this would be very helpful for those who are to become a math teacher one day. I learned something very valuable as I was reading this article today.

hjchang

 

Baxter, Martin. "Failed in number; where can I start?" Mathematics Teaching. June 1998.

This article is very helpful for anyone within the teaching field. As I read it I was enlightened about the alternative ways of teaching students about numbers and basic arithmetic. The article seemed to focus on the progression of number concept in order to learn the multiplication tables. However, there seemed to be a much deeper question called to attention.

If children in the beginning grades never attain a solid grasp on number concepts, how can we ever expect them to truly understand the higher skills such as multiplying and working with negative numbers? The article is written by a man who works very closely with many students suffering from issues such as these. Step by step, he describes the process he uses to reach those children who have this "number failure".

He begins by mastering counting by two’s beyond ten and by ten’s above one hundred. When this is accomplished, they move onto counting by five’s. He says the regularity of the patterns helps the children to understand the concepts they are mastering.

The next big step in the process is to begin doubling numbers. He works with the children on equations such as, 2 = 1 + 1. The children begin to get a clearer picture of exactly what the equal sign represents and their notion of it meaning "the answer is" begins to slip away. When doubling is mastered, they move onto doubling three and then four times, never mentioning that they are forming the "times tables". When this table is created, together, the study the numbers written and they begin to see the multiplication.

Next, the student works on counting by twenty’s, fifty’s, one-hundred’s, and then twenty-five’s. Then, they move up to two hundred’s, two hundred fifty’s, five hundred’s, and one thousand’s. When the student grasps this, they move to counting by decimals. This helps them grasp the concept of 2.19 being a smaller number than 2.2 despite the fact that nineteen is larger than two.

Finally, he concludes with negative numbers. He suggests counting (by any number appropriate) from one to one hundred, then back down passing zero and continuing into the negative numbers. He describes how when this is secure he will use the number line to teach the addition and subtraction of negative and positive numbers.

This process is very helpful to anyone in the education world for two main reasons. First, it shows us a successful way to work with students who seem beyond help. We get to see how we can teach them the number concepts they need to succeed in higher mathematical applications. Secondly, it shows us in a broader sense how we can take a necessary skill and break it down in a number of ways for any student to grasp it. Learning may be the interaction of a student and a teacher, but meaningful learning is the interaction of a student and a dedicated teacher.

Sheila Billups

 

Cantlon, Danise. "Kids + Conjecture = Mathematical Power." Teaching Children

Mathematics. (October 1998) : 108 - 112.

Since I first learned about the NCTM standards, I have always wondered about their practical applications to the classroom. One of the idealistic standards emphasizes the use of conjecturing in the classroom. This article makes the standard a reality. Denise Cantlon, a teacher from a 4^th and 5^th grade classroom reveals the reasons conjectures are essential in her classroom. She believes conjectures empower her students, help them construct mathematical knowledge, and helps them make connections. The students can attain all these goals through collaboration. It is important for the students to discuss ideas among a small group of peers. Then, the students can explain his or her conjecture to the whole class. The class assesses the conjecture by trying to find examples and counterexamples. If the conjecture is accepted, it is posted on a bulletin board in the classroom. The students build on previous conjectures to make new ones.

First, I liked this article because a teacher is explaining a useful method for the mathematics classroom. I would like to promote class discussions in this manner. Empowering children is important because children need to feel they can relate to math. I realize through the process of making conjectures, children can make mathematical connections on their own and through collaboration with their peers. I often wonder if children will make necessary connections in math, but this article proved it can work provided I foster the atmosphere. This article also explained how to incorporate a specific standard in the classroom, a skill I will need to master in the future.

Kelly Fitzpatrick

 

Cole, Karen A. "Walking Around: Getting More From Informal Assessment" Mathematics: Teaching in the

Middle School NCTM Vol. 4, No. 4. January 1999.

An interesting article summarizing a teacher's method of informal assessment: "walking around." She made a good point about the importance of informal assessment. The teacher wrote how you can make a better assessment of a student as you walk around. Paper assessments i.e. quizzes, exams, and homework can and is static. You are only assessing the students for what they know at a specific time. This is important, but there needs to be an aspect of observation over time to see if a student is actually learning and understanding. Two types of assessment or methods for informal assessment is "observe" and "conference." When the teacher "observes," she lets the students discuss amongst themselves. If it is needed, the teacher moves in for a "conference," but this is usually the second form of communication/observation made. The article contains a sample of the "observe" and "conference" There is a brief discussion about how this type of assessment can affect and improve classroom management. It also promotes equity because no one is singled out and everyone is involved with what is going on. Good article. Helpful to improve quality of teaching/assessment.

Sam Hyun

 

 

Crawford, David. Mathematics Teaching. "Learning Probability – Misconceptions and All. June 1997. Pages 23-29.

This article is about how David Crawford introduced probability to his year 7 classroom. He was focusing on why and how learning was taking place. He started off with experiments without mentioning the word probability. There was a great deal of participation from the students but there was not much thought of what the outcomes of the experiments meant. Since this occurred he decided to lead a discussion on the concepts of probability. This would allow the students to start using mathematical terms and language. He concluded his research by posing questions on misconceptions and having them fill out a questionnaire. With the results of these, Crawford was able to say that this group of students are at least at the same level of understanding as his previous students, and this is with him teaching a more theoretically based lesson along with the use of practical work. The best advice he can give from this research is that teachers need to go deeper than focusing on the hows, and start going into the whys of learning.

http://mathforum.org/dr.math/tocs/about.math.middle.html

This web site contains questions from middle school students who are having problems in the math department. It gives great examples of other math web sites to look at for further help and it also tries to answer the children’s questions. There are questions dealing with how to solve problems without paper, who uses calculus, art and math integrated, math in the real world, and making math seem easy. It’s a great place for children to turn if they are faced with a problem at home and have no teacher to turn to. \

Robyn Swieboda

 

 

On My Mind: Dispelling Myths about Reform in School Mathematics by Fran Curcio Mathematics Teaching in the Middle School; February 1999, Volume 4 Number 5

The journal article I read was one I found on the internet from Mathematics Teaching in the Middle School. I found it to be a very useful article because it’s purpose was to rid of the myths held by many about the NCTM standards. One example is that with the new NCTM standards basic concepts is ignored. Curcio states in this article that that is not the case. Children must still learn the basic facts but they can do so through meaningful repetition while playing games and activities rather than by rote memorization. By seeing the information in a variety of contexts they can develop thinking strategies that help support learning the basic facts. Another myth Curcio was trying to end was that answers that are close to correct are good enough. Curcio believes that the new standards support problems that can be solved in more than one way. This does not mean that there is one right or wrong answer, but it does mean that there are answers that are incorrect. However, by offering problems that have many possible answers different students bring different perspectives to solving the problems. A third existing myth is that there is only one right way to teach math. Curcio stresses that students are not "empty vessels" when they come to class. They are products of experiences on which knowledge and skills are built. How to group students, design problems, and present appropriate mathematics are determined based on teacher’s knowledge of their students, the needs of those students, and the math that is being taught. I found this to be a very useful article. It is one I think every math teacher should read because it shows how to use the NCTM standards in the classroom, and it clears up many of the myths that surround those standards.

Langkamp

 

"On My Mind: Dispelling Myths about Reform in School Mathematics", by Fran Curcio, from the February 1999 issue of Mathematics Teaching in the Middle School.

The article that I read, "On My Mind: Dispelling Myths about Reform in School Mathematics" opened my eyes to the way many people view Mathematics teaching. It appalled me to read that often people feel that "basic computation is ignored", "close is good enough", and "there is only one right way to teach mathematics." Throughout my classes here at the University of Illinois, I have formed the opinion that each of these is far from true. The author of this article, Fran Curcio, also holds this opinion and does a good job dispelling any doubt that these "myths" are untrue.

The first of the ideas that many people believe is that "basic computation is ignored" within the standards for the improved teaching techniques. This myth originates from the fact that the new standards tend to down play rote learning and meaningless memorization. Instead, they focus on what they call "meaningful repetition." This process does indeed instill the basic computational abilities in the children. The difference is through "meaningful repetition", teachers hope to foster a better understanding of the facts, formulas, and processes through games and activities rather than simple drill and practice techniques.

The second of the myths that is common is that "answers that are close to correct are good enough." This is far from the truth. Students are now taught the skills to estimation and mental math for uses within the context of their everyday lives. This makes math both more useful and interesting. Very rarely will a student be asked by a friend to divide 500 by 20 using long division, however, they may be asked how many pieces of 20 cent candy they will be able to buy with a 5 dollar bill after school. These are the sort of problems students are learning to solve and it only has one answer that is correct and that is "close enough." This answer is 25. The other focus in classrooms today is open-ended questions that have many different possible answers and this may be where this second myth originated. Often students will be asked to explain their work and how they arrived at an answer. Although their final answer may be incorrect, their work may show great intuitiveness and thus, may be considered a quality answer.

The third myth that must be dispelled is that there is "only one right way to teach mathematics." This is possibly the most destructive of all ideas when it comes to teaching mathematics in the classroom setting. Teaching is by far not a simple "follow these steps" profession. In fact, it is a very "complicated, complex activity." Each classroom is extremely different from year to year, even within the realm of the same teacher. Because a teacher plans for instruction using what they have learned from their students, there are infinite ways to teach mathematics.

Each of these myths is very important to understand and counter. They are each, in their own way, destructive to the success of improved teaching techniques within the standards of today’s classrooms. There is indeed, no one way to teach any group of students. Every student has different needs and ability levels and therefore requires a different teaching technique. It is up to the teacher to open their eyes, ears and minds to each of these differences in order for each student to "acquire the knowledge and tools needed to function in an information-based, technological society."

Sheila Billups

Dickey, Edwin. "Challenges of Mathematics Teaching Today: How Can School Leaders Help?" NASSP Bulletin. February 1997: pp. 1-10.

http://129.252.97.2/dickey/nassp/nassp.html

In his article, "Challenges of Mathematics Teaching Today: How Can School Leaders Help?" Edwin Dickey discusses the idea that students today need more in the mathematics classroom than was needed by students of the past. He states that this will be a significant challenge to math teachers because often they are only familiar with the "old" school setting. Dickey states that teachers need the support of administrators and educational leaders to adopt new curriculum materials and new teaching methods that will advance today’s math students.

First, Dickey introduces various statistics and graphs that depict the rising scores of math students on standardized tests. He also notes the increased enrollment in more "college-prep" math classes such as Algebra I, II and Geometry and less emphasis on remedial math courses. Despite the positive influence on mathematics courses, Dickey states that improvement is still needed.

Dickey also introduces two scenarios representing the "traditional" math class and then the classroom illustrating new strategies and methods. The goal is to implement new strategies and methods in all classrooms that reflect NCTM standards. Although, some schools have already adopted plans, instruction and student learning in mathematics should improve more.

Dickey mentions five challenges that mathematics teachers currently face in today’s classroom. The challenges include: higher expectations (of students), new curriculum materials, high stakes tests not aligned to new curriculums, integration of technology, and block scheduling. Dickey, who also suggests ways that educational leaders can lighten the burden for teachers, discusses each of these challenges in depth.

He concludes that math teachers are some of the most "innovative" in the world. They often are requested to reach out to students in ways that they were never taught. That is why it is imperative that teachers receive the support they need in order to implement better teaching strategies and methods in their classrooms.

Comments:

I came across this article while browsing the web for this assignment. I chose it because Edwin Dickey gives a slightly different perspective than most articles about middle school teaching. He relates the challenges of mathematics teachers through the eyes of an educational leader. In other words, he relates these challenges to what educational leaders can do not only what teachers should do.

Dickey recognizes that many teachers face these challenges, but they cannot overcome them without the help of administrators and other leaders. Often these teachers were not trained with the "new" methods, so it is even harder to implement them in their classrooms. The need to learn technology is high, and often teachers are unprepared. It is imperative that teachers are trained in these areas. Today’s students need advanced instruction to survive in today’s world.

This article brings up some very important points for teachers and administrators alike to keep in mind. Without support and training, teachers cannot improve themselves or their students.

Sarah Ryan

 

Downes, Steve. Women Mathematicians, Male Mathematics. Mathematics in School. May 1997.26.3. pp.26-27.

This article is about teaching grade 8 women the history of Mathematics. It mainly deals with how this one male teacher is trying to teach young women that they have the power to do math and do it well. The concern of the teacher is that there are plenty of male mathematicians to look to as examples. But when it comes to female mathematicians, as the class researched through various means, they were unsuccessful in finding any. This is a dilemma for the teacher because in his effort to empower his female students, he reinforced the idea that math is a male subject. Since the mathematicians they were able to find were all male, this assignment seemed to have hurt the female students more than helping them.

This article is very interesting because it is about a male teacher who is trying to show a broader view of mathematics. His efforts to empower his female students should be applauded. He found through this situation that this is not a lesson that can be taught in isolation, but there needs to be supplemental lessons that would help the female students see that they are more than able to become mathematicians.

jhwang

 

 

Eyles, Alice. Rev. of "Let’s Go Shopping" by John Thomas. Mathematics in School, Jan. 1998: 46.

In her review of John Thomas’ software program, "Let’s Go Shopping," Alice Eyles discusses the details of the program and how it can be used in the classroom. Also included is a current price for the software and the producer. Her review is one of several included in the January 1998 issue of Mathematics in School.

Eyles explains that "Let’s Go Shopping" is basically a virtual supermarket where students can go grocery shopping. The program mimics a real grocery store where students are using skills in selecting and paying for various items. Students who are in late primary to early middle school will benefit most from this program.

In order to make this program beneficial for students at various levels, constraints can be chosen to fit the level of the user. These constraints may include the requirement to give correct change or limiting the user’s spending amount. Eyles adds that other activities are provided in the software package to assist students’ learning including worksheets that represent the different levels that can be used and games that score players according to various criteria.

Eyles rates "Let’s Go Shopping" highly as an accurate representation of a shopping trip. She states that the informative manual is easy to follow, and students will enjoy simulating a shopping trip. She believes that it should not substitute actually participating with concrete materials, though. Using this program as a supplement to lessons that include money-based material would be beneficial to students.

Comments:

Finding review of mathematical materials is quite a simple task. Several journals include sections specifically reserved for review of textbooks and other materials. The Mathematics in School journal reserves the last few pages of each issue for reviews. I looked through several issues before I chose a review that I thought was interesting.

I feel that providing reviews is an excellent idea for teachers. It gives them a resource to consult before purchasing materials and textbooks. It saves them from having to find out the hard way that the product will not be useful for their classroom. I am also glad that these reviews are so easily accessible in various journals.

In addition to finding the reviews very helpful, I am really interested in the product that was reviewed. It sounds like something I would like to include in my classroom. "Let’s Go Shopping" seems like an excellent supplement to learning and discussing money activities in the classroom. Alice Eyles only reinforced my interest in the product.

Sarah Ryan

Faux, Geoff. "What are the big ideas in Mathematics?" Mathematics Teaching June

1998:12-17.

Hewitt, Dave. "Approaching Arithmetic Algebraically" Mathematics Teaching June

1998: 24-5.

Fisher, Alun-Peter. "A number problem" Mathematics Teaching. June 1998: 39.

The journal I have chosen to review is Mathematics Teaching. This periodical had many interesting facts and appealed to me for many reasons. The first article that appealed to me was by Geoff Faux. The article was entitled, "What are the big ideas in Mathematics?" He then listed them: The numbers are ordered and well-structured, Mathematics is shot through with infinity, A lot for a little, Equivalence, Inverse, and Transformation. An example of equivalence that he gave was as follows: He wrote on the board 623-478=? He then asked for another subtraction problem with the same answer. The first one written was 624-479. The second one that was written was 653-508. I thought that this was an excellent way of getting the students to think critically on their own, providing equivalent problems, while enhancing their number sense.

The second article I read was also by him. It listed some big ideas in mathematics education. These were: Only awareness is educable, perception-action-virtual action, and Integration by subordination. The one that I thought was the most important was the first one. This meant that the educator needs to ask the students to contribute what they see has been going on and take the time to step back and reflect/comment on what they have been engaged in. Input from the students is the only way the educator knows what they know and what skills they need to improve.

The third article that I read was "Approaching arithmetic Algebraically" by Dave Hewitt. He stated that finding structure helps carry out arithmetic. For example, 19*16 really equals 19+19+19+…+19 (16 times). Therefore the structure is m*n=m+m+m+…+m (n times). He also brought up algorithms. For instance, in division, an exercise such as 3224/13 is set up in such a way that the student asks, how many times does thirteen go into three? Instead, Hewitt is stating that the real question is how many times does thirteen go into 3224? The answer is 100. If the student keeps up with this and writes each answer above another, he/she will come up with the answer 248, but in a way that is clearer to him/her.

Lastly, I enjoyed a number problem that was towards the end of the journal. It states: Take digits 1 to 9 and form a number using each digit once, say, ABCDEFGHI, such that 1 divides A, 2 divides AB, and so on. This problem involves using divisibility rules, process of elimination, critical thinking, investigation and exploration. All things essential in Mathematics. The answer to this problem was 381 654 729. It is very challenging and incorporates a lot of important concepts in mathematics that children need to know.

This journal is an incredible resource and I encourage every mathematics teacher to use it. It certainly opened my eyes to something different and it thoroughly expanded my horizons in mathematics.

Winer

 

Fennell, Francis. "Mathematics at the Mall" Teaching Children Mathematics, vol. 4, 5 (January 1998): 268-274.

The article I chose to review is called "Mathematics at the Mall" by Francis Fennell and is from the journal, Teaching Children Mathematics. The article was aimed at late elementary or middle school teachers who want to get their students interested in math. The idea behind "Mathematics at the Mall" was to get children to explore math on their own and to realize that math surrounds them. According to Fennell, the tasks described in the article, "invoke problem solving, communication, connections, and reasoning" (268). Fennell also states that to ensure that these tasks are usable, they have been tested in a variety of classrooms.

The article starts out by telling the teachers to ask their students to describe their own experiences at the mall. Teachers should then tell the students that this investigation will deal with things that they have probably never thought about, the amount of space used up at the mall. The first activity is to study the parking available at the mall. The students will have to use percents and estimations, and create both graphs and tables using their data. The article goes on to describe what to do with the students and what to have them do on their own. The second activity deals with the actual shops at the mall. The same principles are applied here. The students will be given certain information that will help them calculate area, and they will be expected to graph their data. They will then be asked to think about what they would need if they were to lease out a certain area of the mall. Enough statistics are given so that the children can have a start at figuring out space, money, and area.

Through this investigation, students will learn that mathematics is not only for the classroom, but is a part of everyday experiences. They will also learn about leasing and renting, and also about what goes in to starting something like a shopping mall. This article gives teachers a step by step guide to doing this investigation yet leaves out enough information so that the teacher can adapt the lesson to his/her own students.

Sarah Lyon

 

There are many curricular areas covered within the series "Exploring Mathematics" by Scott Foresman. One of these areas of teaching is assessment. The series includes a very thorough "Assessment Handbook" along with a "Daily Review Booklet" which could also be used as an assessment resource.

The assessment handbook actually seems to be quite unique. It is full of pages of tests for the students. These are both free response tests and multiple-choice tests. This mixture is very important because every teacher and student is different. The handbook is adapted to suit anyone, be them a teacher who chooses make her students think and understand the math, or a student who learns better through rote methods. The series appears to be quite innovative and open to new forms of assessment and new ways to foster mathematical thinking processes within the minds of students.

If these tests are not enough to cover all possible assessment preferences, the handbook also contains many suggestions for alternative assessment styles. For example, it also includes tests that are completely based on the students understanding and ability to explain their answers in writing. This is very important for those teachers who believe in teaching their students the "why’s" of mathematics rather than the "how’s"

A very detailed description of how to use portfolios in your classroom is also included in the beginning of the handbook. It discusses each part of the portfolio and gives example sheets to be filled out in reference to the student’s work. Some of these sheets are assessment sheets to be kept and used by the teacher for grading, others are for the student to assess and critique their own work and keep in their portfolio.

Of course, these types of assessment can only be used periodically and sometimes it is important to know if your students are grasping new concepts every day. Therefore, the "Daily Review Booklet" is a very useful tool. For the topic of each day, there is a small sheet of review questions to be filled out by each student. This allows the teacher and the student to see which of the concepts have been successfully taught.

Overall, every possible type of assessment, whether long term or short term is supplied in the series "Exploring Mathematics". From tests, to personal progress reports and critiques of a student portfolio, to everyday review sheets, this series has all the bases covered.

Sheila Billups

 

Galindo, Enrique. "Assessing Justification and Proof in Geometry Classes Taught Using Dynamic Software" The Mathematics Teacher 91 (January 1998): 76-82\

The article is about using a Goemeter’s sketchpad or a Cabri Geometry II drawing for making the proofs in Geometry classes more realistic and tangible to students. The author gives three examples of using these "dynamic software" by lessons on exploring loci and deconstructing black boxes. The author also mentions about showing students the difference between drawing and construction using the dynamic software.

Annie Moon

 

 

Graphic Calculators in the Mathematics Classroom"

David Green and Sue Pope, eds. 1993, vii + 124pp., $6.35 paper.

ISBN 0-906588-31-6. The Mathematical Association, 259 London Rd.,

Leicester, LE2 3BE, United Kingdom

The book review tells me both the negative and positive sides of the book.

It suggests what kind of class this book would be most useful. What

helped me the most is that it mentions it's for easier level technology

users.

Annie Moon

 

 

Harris, Mary. "Mathematics for All?" Mathematics Teaching. London: Burlington Press. December 1997: pgs. 3-10.

Mary Harris’ article entitled "Mathematics for All?" discusses the gender issue in mathematics education. She discusses the title of the article in lieu of a conference she had recently attended that suggested that there has always been equality in education. She reviews the history of math education, and its association with the severe gender gap involved in the field of mathematics. In general, Harris states that women have been consciously excluded from learning the same mathematical concepts as males due to their femininity. She discusses three prejudices that have prevented women from advancing in the mathematics field.

First, she introduces the idea that mathematics is traditionally a male enterprise. With this statement, Harris intends to point out that mathematics as a whole is considered a masculine field, and in order to study it, women had to lose some of their femininity. The second topic she discusses refers to the idea that math is an area for only educated men. Harris notes that in the education system there is a prominent tendency in history to associate mathematics with typical gender work. In other words, women supposedly do not need mathematics in the work they do, so it is unnecessary to include it in their formal education. Harris’ final point focuses on the idea that math is traditionally a subject for liberally educated men. She states that math is divided along gender as well as social lines. Those people that are socially elite will most likely receive the best mathematical education.

Harris concludes the article with some statements of her own. She states that often a sense of numeracy stems from literacy. This idea also ties in with the social and gender gaps in mathematics education. Harris believes that mathematics education does not contain the equality the title of the article suggests.

Comments:

This article was not the only interesting one I came across in my search for a journal article. I found several interesting journals and articles. Harris’ article caught my eye because it discussed the gender gap in mathematics. I have always been interested in that aspect of math education. My interest lies in this subject mainly because it applies to me. I am one of few females studying higher level mathematics. It really interests me to learn about society’s views and how they will affect me.

Harris also gave a great review of the history of the gender gap in mathematics and how exactly evolved. I also found her article insightful, and I plan to keep it in mind in how I handle my future classroom.

Sarah Ryan

 

Harris, M. "Mathematics for All?" Mathematics Teaching Cambridge: Burlington Press, December 1997. p 3-10.

In the article, "Mathematics for all?," the author, Mary Harris stresses that mathematics has never been, has never intended to be, and currently may not be taught equally among all students. She focuses on the root of the problem, explaining that mathematics as a whole has always been seen as a male activity, not only in terms of who was allowed to study it, but also in its very nature.

From the beginning of time, men were seen as the mathematicians. In this article, Harris explains that people believed "if God is male, then the opposite that is female embodies the characteristics that contrast with the male strengths and virtues. Femininity is what masculine isn’t" (3). During the nineteenth century, the Church continued to push this belief, remaining the main protagonist in ensuring that the education of girls remained domestic.

In addition, for hundreds of years, the education system saw mathematics only as a male enterprise. "Women could study mathematics, but in doing so had to leave their femaleness behind them and become male" (4). In the 1880s, educated women were referred to as ‘hermaphrodite’ or ‘mongrel’. Even today, one can find many negative stereotypes associated with women mathematician. Unfortunately, these negative stereotypes are discouraging girls from wanting to exceed in math.

Throughout the article, Harris encourages women to stand up against these stereotypes. Women need to join together and begin looking at math through fresh eyes, showing society that mathematics is for all!

This article taught me a lot about the roots of the problem of why girls do not excel in math. Harris does a great job explaining where the problem originated. For over 2000 years, the church and the education system have not given girls equal opportunities to excel in math. I truly to believe to make change, we have to get to the bottom of the problem. We have to admit our mistakes before we can fix them.

Although, progress has been made over the last twenty years or so, unfortunately the problem still exist; girls are still learning at an early age that boys are better at math. Teachers and parents need to be aware of this problem because in many ways they are the problem. Too often girls are told it is okay if you aren’t good in math, that you won’t really need it in life. Studies show that girls fall especially behind in math when they get to junior high. Adolescent females seem to become less confident in math class and are overpowered by the males. Girls need to be told that they can excel in math and they need to be given equal opportunities to do so.

I am hopeful that society will continue to grow and eventually completely erase the horrible notion that math is more for boys than girls. I am proud to say that I am a woman and I love math. I feel that I excel in math more then any other subject. I enjoy math and often like to solve math problems just for fun. Although there are still only a very few women that hold upper hierarchies positions in the field of mathematics, I think in the next few years this number will increase drastically. Women are constantly becoming more confident; therefore, achieving goals that were once only seen attainable by men. As teachers become more aware of the problem, I think more females will be given the same opportunities, and therefore an equal chance to succeed in math.

Amy Breda

 

Higgins, Karen M. & Heglie-King, Mary Ann. (1997). Giving Voice to Middle School Students Through Portfolio Assessment: A Journey to Mathematical Power. Middle School Journal, 29(8), 22-29.

This article is teacher Mary Ann Heglie’s story of how she reformed her math class to change her students’ attitudes about and towards mathematics. With the help of a nearby university professor, she made sure to have records of students’ mathematical perceptions coming into her class, she incorporated the use of math portfolios throughout the year, and finally she investigated her students’ current feelings regarding mathematics.

As you probably suspected, Mary Ann’s eighth grade math students did not enjoy math. They did not know why they were forced to learn math, resented the accepted forms for math assessment, and were apathetic about learning math, never really understanding what was going on with mathematics. They could not recognize any meaning or purpose in mathematics. Mary Ann came up with the idea of incorporating math portfolios into her classroom to give her students a chance to demonstrate their mathematical understanding through a number of different ways. Portfolios would also give students a say in the way they were accessed in math and hopefully change their attitudes regarding math by increasing their self-confidence in their own math abilities.

For the first semester, the teacher set the portfolio conditions. Each student was required to submit at least one piece for each of the following requirements: math in another subject area, individual student math journals, one best piece, one piece from beginning, middle, and end of semester, biography of famous mathematician, and math autobiography. Many of assignments incorporated different subject areas. Reflection sheets were written and attached to each portfolio item, so the student had a chance to explain how their piece fulfilled a specific requirement and why they had chose this particular subject/topic/piece. The teacher reviewed the portfolios and students were later allowed to redo work as their understanding of certain concepts or ideas in mathematics evolved.

The second semester, students were given even more control over their portfolios. The students set the conditions for their portfolios, they developed a grading scale for the portfolios, and each student was involved in grading their portfolio, along with a teacher, and two other students in the class (of which the student had some say in choosing). The contents of the portfolios were to include pieces that demonstrated: problem solving ability, use of math in another subject, improvement in math, use of math in a career, understanding of math content areas covered during semester, use of writing to express understanding of math problems and procedures, and one valuable piece.

The result of a math class which did not follow the standard-teach during class, do problems from the book for homework-had amazing results with the students. Possibly the best improvement was that students started to try to understand math, they began to put time into math study, and they found math to be interesting. Once this had been accomplished, students understanding of concepts and demonstration of math skills dramatically improved along with their self-confidence. They began to see math as a class in which effort could help them succeed, rather than viewing ability as the only possible way to be triumphant in math. Many students began to view themselves as good math students. This is something that each of these students will now carry with them into their future math classes.

I found this article and the results of Mary Ann’s efforts very encouraging. As teachers we can even have an impact on our students’ perception of math, regardless of the fact that students had already reached the middle school level.

Erika Sajpel

 

Howson, Geoffrey. "Some Thoughts on Constructing a Curriculum".

Mathematics December 1998, p. 18-21.

The first important point in this journal article concerned the way in which mathematics works. Mathematics is viewed as working in two directions – a horizontal and a vertical. The horizontal direction deals with the transition between the perceived world and the world of symbols. It involves the use of real problem solutions to solve mathematical problems. The vertical component involves the extension of knowledge with in the world of symbols. It is these two directions that are important to consider and to understand when teaching mathematics.

The studies in this article have found four approaches to teaching that are dominant in a wide variety of books and curriculums. These four approaches include the mechanistic approach, the structuralist approach, the empirical approach and the realistic approach. The mechanistic approach includes the repeated practice of many problems until a pattern is instilled in the brain on how to perform a particular type of problem. The structural approach concentrates on the use of structural apparatus to teach problems in mathematics. The empirical approach concentrates on mental mathematics and because of this, it greatly lacks the comprehension of what is happening (the vertical approach). Finally, the realistic approach is one that most education majors are learning about – the effort to make children gap their informal knowledge of an idea with symbols and text-related problems.

This article also focuses on relating mathematics to "street smart" ideas. The article proposes that educators present real life ideas and occurrences in students’ lives to mathematics in the classroom. For example, children who live in cold climates already have an informal understanding of negative numbers because of below freezing temperatures. Teachers should use experiences like these to reinforce mathematical concepts and to also highlight vertical and horizontal directions of mathematics. This type of learning should also provoke the students to come up with their own observations and ideas. It is important, however, that the teacher modifies and regulates the student’s learning so that he or she can clear up any misconceptions and ultimately lead them to the correct concept.

Erin McKinley

 

Jennings, S., Dunne, R. (1997). Improving the Quality of Teaching and Learning Mathematics. Mathematics Teaching, 158, 34-37.

This article discusses the issue of improving the quality of teaching and learning mathematics. The proposed method is by whole class teaching as opposed to "traditional" methods and individualized learning. Through observing and conducting case studies, they have found that language should play an important role and it can help to develop better understanding and find greater meaning to mathematics. This article provokes thoughts about methods of teaching and how there should be ownership of the classroom even by the means of innovative methods; it must be tried and tested. There is an example of how fractions would be taught: "a fifth" is recognized as an entity without the need to separate the numerator and the denominator. This is taught through a purposeful activity that allows the whole class to participate. This article addresses the way that children learn as a whole is crucial- that though the classroom should be a place where individual needs of learning mathematics, there has to be some common ground for the children, so whole class would benefit in learning mathematics.

jhwang

 

Jones, Keith. "Some Lessons in Mathematics: A Comparison of Mathematics Teaching in Japan and America." Mathematics Teaching. Burlington Press, Cambridge June

1997: 6-9.

In the June edition of the Journal, Mathematics Teaching, there was an article about the differences between mathematics teaching in Japan and in the United States. The author, Keith Jones, did a study involving 131 grade 9 classrooms; 50 in Japan, and 81 in the United States. One complete lesson was taped in each classroom, and copies were made of the classroom’s textbook pages or of any worksheets the class was using.

In Jones’ results, he talks about how different our way of teaching mathematics is compared to Japan’s way. For instance, Japan has no tracking of any kind and the United States has a large tracking program. In this study, 50% of the U.S. classrooms used textbooks as the main source of education, whereas only 2% of Japanese classrooms used textbooks. Generally, in the U.S., educators instruct their students to learn skills rather than the concepts behind them. Japanese classrooms are highly focused on mathematical thinking and understanding concepts. For example, in the schools Jones visited, the students frequently were asked to come up with their own problems, invent solutions to them and explain why they work.

The article also talked about the TIMSS achievement test scores for 13 year-olds. There were 41 countries involved in these tests. Out of these countries, Japan came in 3^rd and the United States came in 28^th. One surprising thing about these results is that overall, students in the United States spend 143 hours per year studying mathematics and Japan only spends 117 hours per year. Teachers in the U.S. also give more homework per week than Japanese teachers. (U.S. gives at least 90 minutes per week and Japan gives less than 1 hour per week.

Generally what this article concluded was that the Japanese are doing something right. In the United States, mathematics teachers focus too much on homework and learning basic skills. We should learn from these classrooms in Japan that math should be about understanding why rather than just learning how to do it.

Sarah Lyon

 

Jones, Keith. Some Lessons in Mathematics: A Comparison of Mathematics Teaching in Japan and America, "Mathematics Teaching 159." June 1997.

For the BR 2, I read an article in " Mathematics Teaching" called, Some Lessons in Mathematics: A Comparison of Mathematics Teaching In Japan and America. This article discussed the results of an international study done on mathematics classrooms in America and Japan. The author of the article is from England and he discusses the results of the study and how they should do a study like this on England’s math classrooms. He says that according to a report England’s performance in math is very poor and could stand some improvement. He goes on to list some factors about why England performs so poorly on math tests which include: less time spent on homework, lots of ability grouping happening, and not knowing how to use calculators properly.

He then says that in order to improve England’s math performance, they need to know about other math practices in the world so he focuses on a study of math classrooms in Japan and America. This study videotaped year nine classrooms in both Japan and America and textbooks they used and worksheets they did, were collected along with the videotape. They found that the US students study math more hours on average than Japanese students do and that Japan has no ability groupings in their classrooms. They also found that US teachers in math spent most of their classroom time on routine procedures whereas Japanese teachers asked their students to invent new solutions or proofs on their own.

The research found that Japanese teachers were more intent on their students understanding the concepts behind the problems. A typical lesson was for the teacher to ask the students to do a problem, and then they discuss the problem and the students perform similar problems. In the US however, the teacher lectures about a concept, then works examples and then assigns homework. They found that the US math teacher rarely lets the students develop the concept. The teachers do the work and lecture while the students listen and answer short questions offered by the teacher to help make the lesson flow more smoothly. They also found that US teachers most often ask students to practice computing skills whereas Japanese teachers ask students to analyze problems and math relationships. They found that on tests Japanese students came in third and US students come in twenty eighth. They reasoned that it is because of how math is taught, quality of the teachers and what the students are assigned for homework that makes the Japanese better at math.

The author makes a final comparison by saying that England is more like the US in it’s math abilities and that they achieve about the same level on tests. The author concludes the article by saying that in order to get to the root of England’s math problems, they need to do studies like this on math classrooms in England so they can learn from other countries.

I found this article to be extremely interesting because of the way the US and Japanese classrooms were compared. It was surprising to see how much more concentrated the Japanese were on problem solving and letting students discover concepts for themselves rather than using the US method of lecture and homework. I think math teachers here in the US could learn a lot from reading about this study and apply some of the Japanese methods to our classrooms.

Jessica Wielgolewski

Lowther, Martha H. "The Square-Patio Problem." Mathematics Teacher. Vol 89: Num 1. pp 4-6.

This article was interesting in the fact that it talked about a project that teachers could do in a classroom and then gave a worksheet to aid the lesson. It hink it is a good article in that it gives a real-life problem for the students to solve which lead them to learning about matrices. Truthfully it was more than this article which impressed me. I really like the format of the journal as a whole. There were more lesson aids and also articles about how teachers can best reach their students. I would like to look through this journal and others like it when I begin teaching. This journal is for secondary teachers, but I know there are many for elementary and middle school teachers.

Peggy Neubert

 

Mewborn, Denise, "The Quarter : An Illustration of NCTM’s Professional Teaching Standards Teaching Children Math Vol. 5 #3, Nov. 1998, p. 160-163.

The National Council of Teacher’s of Mathematics (NCTM) Professional Standards should greatly impact the classroom. Teachers should be changing their classrooms and their teaching styles, posing worthwhile questions/tasks that encourage children to make connection, communicate, problem solve, and reason. (These sure sound familiar!) Teachers should be listening to students, have them clarify and justify their ideas, and decide where further in dpth investigation is needed. Students are active learners; they are to question, reason, explore, and convince. The learning environment should show respect and value for other’s ideas. It should allow for risk taking by questioning and offering one’s own ideas.

The article went to discuss "a teachable moment". The teacher noticed through children’s discussion that there was confusion about the term "quarter" as it applied to different topics: time, money, percentages, and fractions. The class, along with the teachers, created a "lesson" that was dominated by the children. The teacher intervened mainly to facilitate and ask for further explanations from children. The environment of the classroom valued and encouraged mathematical thinking. From this point the teachers then created a "formal" lesson for future classes.

Tina Wiegel

 

Norwood, Karen S. and Glenda Carter. "Journal Writing: An Insight Into Students' Understanding." Teaching Children Mathematics (November 1994):146-48.

Using math journals in the classroom, whether to explore students' knowledge of a topic yet to be introduced, to help students review or deepen their understanding of a concept, or as an alternative form of assessment, is still a fairly new practice. Many students will be surprised to find out that writing will be expected of them in their MATH class. Yet, math journals can be an extremely powerful tool in teaching math to middle school students, as Karen Norwood and Glenda Carter show us in their article " Journal Writing: An Insight Into Students' Understanding."

The article mentions some very strong arguments for why math journals should be used in the classroom, some of which I already mentioned in the introduction. Some other reasons for using math journals include: that they allow a teacher to see how students are understanding a unit in progress and see where more time needs to be spent, students become aware of their thoughts and understandings as they reflect on their math experiences through writing, it allows us to see things that standardized tests do not (the thinking behind the execution of the math problem), and teachers can go back and try to identify where a student who is lost became confused by reading their journals. Journal writing also allows students to use their reading, writing, explaining, and questioning skills in the math class. By writing about math students are more likely to recognize connections between math concepts, as well as, between mathematics and other subject areas. In this way, students are relating their new knowledge to their previous knowledge.

In addition to discussing the importance of math journals in the class, the article also suggests many ideas for journal topics. They include topics which touch on the affective aspects of math, in addition, to expository writings on math. Here are some of my favorite suggestions for math journal entries (ones I felt where the best ideas and unique too), taken directly from this article:

________________________________________________________________________

1. Write a "mathematics autobiography." Describe your earliest experiences in mathematics both in and outside of school.

3. Explain what is most important to understand about (adding fractions with unlike denominators; multiplying decimals).

4. Describe any places you became stuck when solving a problem, and tell what you did to get unstuck.

5. What I like most (or least) about mathematics is: ________________ __________________________________________________________

2. Find a graph in a newspaper or magazine. Write a paragraph about what the graph represents and why the graph is drawn the way it is.

5. Explain to your cousin how multiplication is like addition. You may use pictures or graphs.

7. Explain to your sister or brother in fourth grade how to add 1/2 and 1/3. Be specific. You may use pictures or graphs.

8. How do you use fractions in your life?

________________________________________________________________________

These suggestions should help you develop a picture of a good math journal. By using math journals in the classroom, you are also helping to achieve the NCTM standard of Mathematics as Communication. Students are learning about math by learning to communicate their math knowledge through writing and talking and sharing. I am eager to include math journals in my future classroom and should hope that any other future math teachers would also consider it.

Erika Sajpel

 

Ponza, M. P. "A Role for the History of Mathematics in the Teaching and Learning of Mathematics." Mathematics in School Mathematical Association: Sept. 1998

p.10-13.

In a school with an excessive number of pupils in every course, and where economic scarcity influences the possibility of having text as teaching aids, Maria Ponza found the best way to get her students interested in mathematics is through its history.

In the article, "A Role for the History of Mathematics in the Teaching and Learning of Mathematics," Maria Victoria Ponza, a teacher in Argentina, explains her new, creative way to teach mathematics.

Instead of memorizing math equations, Ponza’s students spend class time investigating the history of important figures in the mathematical past. "The famous mathematician lives and discoveries enabled students to see human aspects of mathematics they have never previously imagined" (p.10). In some of the units the pupils were captivated by the life of certain mathematicians. For instance, when researching the history of equations, one group of students became extremely interested in the life of Evariste Galois. Together, they decided to write a small dramatic piece describing the short life of Galois, his social struggle and his enormous output. Together, the whole class put on the play. The play was a hit, filled with excitement and energy. (Ponza included the a complete layout of the play at the end of the article).

Overall, I was amazed at the work that was accomplished in Ponza’s classroom.

I think it would be very difficult to teach math without any textbooks or manipulatives. I admire Ponza for her determination not to give up. I think it is wonderful that her students are learning so much, but at the same time having fun. Her students are actually interested in what they are learning. Unlike most math classes, the students are actually learning and understanding the development of formulas and equations, as well as why and when they are used. Unfortunately, I have never had an experience like the one described in Maria Ponza’s class. In the United States, math is often taught in the traditional way. Teacher lectures, students take notes, homework is assigned, tests are given. Students are never given time to explore and investigate the history of mathematics.

In addition, Ponza did an excellent job integrating not only math and history, but English as well. Not only did her students learn about the history of mathematics, but they also learned researching, writing, and acting skills. In addition, the students learned to explore and solve problems on their own. Ponza encouraged her class to be active learners. The class worked as a team, helping each other out and always working together.

All told, I was quite impressed by this teaching style. I admire Ponza for her creativity. She made math fun and exciting! I truly believe that is what teaching is all about.

Amy Breda

 

Pugalee, David. "Promoting Mathematical Learning through Writing."

Mathematics in School (January 1998): 20-22.

The Magazine, Mathematics in School, contained an interesting article entitled "Promoting Mathematical Learning through Writing." It seems as though nowadays more teachers are increasingly using interdisciplinary methods of teaching. Interestingly, although the subject matter of math and writing are seemingly distinct, the two areas can merge to enhance learning in individual students. The article clearly presents the advantages of integrating writing into the math curriculum.

Writing should not be easily overlooked as a tool that can heighten learning. In fact, there are several advantages in integrating writing into the math curriculum. First and foremost, it aids students in developing knowledge of mathematics. Students can write summaries of a lesson. They can discuss rules, their importance, and any relevance it has to their lives in their journals as well. As students express in their own words what they learned that day, math concepts, both simple and complex, will definitely be more applicable and useful.

Another important advantage that writing has in math classes in the development of self-monitoring and reflective behaviors. Not only it is crucial that teachers guide students in growing in knowledge and understanding of various subject matters. Another important aspect of teaching is creating an environment that fosters critical and reflective thinking. Writing can fulfill the aspect of nurturing the critical minds of students. Through writing, students can explain what made a problem difficult or easy. They can further explain why an answer is reasonable. They can analyze the quality of one’s own work. The use of writing is seemingly unbounded.

Another advantageous feature of the use of writing in the math curriculum is promoting discourse. In a classroom of twenty-some students, it is really hard to meet everyone’s needs. Hence, it is crucial for students to keep a journal that would be read by the teacher. Such would provide means for each individual student to communicate with the teacher where they are at in terms of level of understanding. The teachers can employ writing as a means of assessment.

The article’s practical and profitable aspects of integrating writing to any math curriculum drew me in. As a future teacher I will not overlook the importance of writing. I intend to incorporate writing and other subject matters as much as possible in order for optimal and meaningful learning.

Heesundo

 

Reys, R., Reys, B., Barnes, D., Beem, J., & Papick, I. (1998). What is Sanding in the Way of Middle School Mathematics Curriculum Reform? Middle School Journal, 30(2), 42-48.

By this point in my education, it is no surprise that the "old" way of teaching math is not going to cut it anymore. This article starts by stating once again what the problem is: that U.S. students drop significantly in international rankings of math in the middle school years because most of what they do in these years are review exercises. The most helpful part of this article is that it answers some really tough questions about what to do about it. For example, the article points out that many parents can be an obstacle in switching to a more standards-based math program, because they have some very traditional, deep rooted beliefs about math. These authors suggest addressing these questions by holding scheduled "parent's night" sessions that focus on the new math curricula. Another issue that the article addresses is that people are concerned about how well students in a standards-based curriculum will fare in a more traditional high school. These authors suggest that a regular means of communication should be established between middle and secondary mathematics teachers through a district wide mathematics committee or through regular professional development activities that include both middle school and secondary teachers.

Joni Anderson

 

Roni Jo Draper. "Active Learning in mathematics: Desktop Teaching" Mathematics Teacher 90 (Nov. 1997): 622-625

This article is about a teaching method "Desktop Teaching" to teach mathematics during the final time. It is an interesting way to prepare for the final by involving students to review the material in a creative way. Students are to make short lessons on given topic to teach to their classmates. The teacher used the idea of "Active learning, in which students take an active role in their learning rather than learn passively through teacher-directed instruction, affords students the opportunity to communicate mathematically and to gain the kind of mathematical power discussed in the Standards document." (pg. 622) Some examples that were mentioned: "One student taught her classmates about the coordinate grid and graphing lines using an "under the sea" theme. Another student taught how to graph inequalities and intervals using string and M&M's."