How does your Doughnut Measure Up?

Maida, P., & Maida, M. (2005). How does your doughnut measure up?. Mathematics Teaching in the Middle School, 2(5), 212 - 219.

Paula and Michael Maida discuss the importance of children reaching an understanding of a concept on their own through interactive activities. The Maidas developed a worksheet and exercise for students to find geometric dimensions of doughnuts and doughnut boxes. They believe that this activity helps students develop a more sophisticated method of thinking and understanding. The students first estimate the necessary dimensions, then, using a tape measure, measure the actual dimensions. Students were then asked to find the surface area and volume of the two objects. Because the doughnuts had holes in the middle, students did not know at first how to solve for the volume and surface area. This allowed students to discuss as a class with the teacher mediating how to solve. They concluded that both the doughnut and hole were close to the shape of a right circular cylinder and proceeded with that. At the end of the exercise, students were given real world applications for such measurements. One example was asking how many doughnuts it would take to fill up the earth. The Maidas describe that students were more willing to learn because they were immersed in the activity and were actively involved.

Students learn most when they are actively engaged in the lesson, and object lessons (or exercises) are a beneficial way to create the engagement. When students do not have something to focus their attention on they tend to become side tracked and distracted. In math classes I took in middle school, students talked to one another when the teacher simply lectured the entire period. Only a few were able to follow the lecture and pay attention the entire fifty minutes of class. Also, every student learns differently. Having an activity such as the doughnut measurements, allows students to learn with problem solving and with interaction. Some students learn better problem solving while others need hands-on experience. In the article, the Maidas explain that some of their students did not learn from measuring the doughnut and box, but they understood the concept once they began working on the problems at the end of the worksheet. Finally, students need to know why something is important to learn and how they will be able to use it outside of the classroom. When I was in eighth grade, my classmates always asked my teacher when we would need to know how to factor a polynomial in the real world. My teacher then gave us the assignment to find a way that it would be helpful outside of the classroom. Had my teacher given us examples in class as the Maidas did, my classmates would not have asked those questions. By giving students an interactive activity in class and applying the concept to a real world situation, students will better learn and understand the material.

Common Errors in Counting Problems

Annin, S., and Lai, K. (13 Mar 2010) Common Errors in Counting Problems. Mathematics Teacher, 103(6), 403-409.

In their article, Annin and Lai address the issue of the difficulty that most students have with counting problems, and in particular with permutation and combination problems. There are four main categories of counting problems and problems are fairly simple if it falls directly under one of the four categories. However, most problems will not and thus a broader understanding of how to do counting problems is necessary. Students must understand the principles behind permutations and combinations; they must recognize that how to do the problem depends on whether the order of the objects matters or not. Using examples of different problems, the authors present common mistakes that are made and how teachers can overcome these misconceptions. Annin and Lai suggest having a variety of problems for students to solve so they are exposed to the different wording and set-up that problems may have.

Annin and Lai make compelling arguments in hteir article that I believe are relevant to teaching mathematics. When I was first learning permutations and combinations, my classmates and I really struggled to understand the concepts. My teacher gave us more simple problems to do during class and for homework. By the time the test came around, most students had a difficult time doing the problems that were not straight forward, the problems that did not fall directly under one of the four main types. At the end of the section, most people had an instrumental understanding but not a relational understanding. Students need to understand all the concepts of counting to be able to do any problem that they may encounter.

Children and Fractions

In her paper, Warrington discusses some advantages to teaching students in a manner where they learn the material based on their own understanding and not a set of algorithms. The students were willing to solve problems using their prior knowledge and understanding as Warrington illustrates when she first put 1 divided by 1/3 on the board. This kind of teaching also encourages discussion in the classroom. Students debate with one another to discover the error of their reasoning and to come to the correct answer. Warrington's students believed that 1/(2/3) was either 6 or 3/2 and thus discussed the answer until they arrived at the correct answer. One girl solved a problem by doubling the original quantities. Warrington argues that this kind of thinking is not present in students that are told what to do. One student illustrates the autonomy that can be developed when she disagrees with all her peers answers and stands alone.

Although Warrington discusses many advantages to teaching in this manner, she does not discuss the disadvantages. As we saw earlier with Benny, some students who are left on their own to discover the principles do not learn correct principles. This will stunt their ability to learn deeper mathematics later on in life. Also, some students in the classroom will rely on the same students to do the problems and will not be learning at all. When it comes time to take a test they won't know how to proceed. I think it is important in mathematics teaching to teach with both relational and instrumental understanding. That way students know why something works and how it works.

Constructivism

In von Glasersfeld's article about constructivism, he argues how people "construct" knowledge through their experiences. People do not acquire knowledge in a classroom or through lectures. As we live each day, we go through different experiences. From these experiences, we draw conclusions about different subject matters. One does not learn to shoot a basket without having the experience of standing before the basket with a ball and learning the proper technique to shoot the ball. After one has experienced how it feels to stand correctly and shoot, one constructs a knowledge of how to shoot. One also learns how not to shoot, and this also becomes part of the constructed knowledge. Everyone will learn different things from the same or similar situation. Not everyone will draw the same conclusions from their experiences.

In mathematics education, it is important to remember that everyone learns different things from the same situation. When a teacher defines a word or teaches a principle, not every student will understand the word or principle in the same way. Mathematics, as von Glasersfeld states with constructivism, is an individual learning experience. To teach my students, I would instruct them on how to do a certain problem, teaching them why it works and how it works, and then allow them time to do many example problems so they can construct their own understanding of the principle. To ensure they did the problems right, I would review their work thoroughly and give them feedback.

Benny's Math

In his article Benny's Conception of Rules and Answers in IPI Mathematics, S.H. Erlwanger explores the learning and understanding of Benny, a sixth grader who has the most success in his IPI mathematics class. Throughout the article, Erlwanger argues how important a student-teacher relationship is. The IPI program prevents the teacher from getting directly involved in her student's learning. The students work with note cards that give them examples and rules of how to do a problem and then exercises to practice what they have learned. The teacher acts only as a supervisor, not as a teacher. If a student approaches her with a question, the teacher can help the student understand the concept that they are learning. However, she has no other gauge for how her students are doing. This lack of teacher involvement causes students to teach themselves mathematics and to acquire their own relational understanding. As seen through Benny, students can then make up their own rules of mathematics and never learn correct principles. IPI is detrimental to learning because the students in the program do not have the proper foundation for the rest of their mathematical career.

Teachers are very important for a child's learning and they always will be. In classrooms today, I have seen that students learn better when the teacher takes an individual interest in each class member. Instead of teaching to a large classroom, it feels as if he is teaching directly to you. I have seen that these types of teachers are easier to approach than those that don't act like they care about the individual's learning. When grading homework or tests, a teacher can see where her students are still confused or what they understand. However, the IPI program does not have a way to check a students understanding.

Relational and Instrumental Understanding

In his article, Skemp addresses two types of understanding and expounds on them. He details relational and instrumental understanding in respect to mathematics learning. Instrumental understanding is when students are taught how to apply mathematical principles to situations. They are not taught why something works but rather told that it does work. Learning instrumentally can help students understand better when to use a procedure; it teaches them a blueprint to follow when solving certain types of problems. Students may be better equipped to do their homework, but when a problem is slightly changed on the test, they do not know how to solve the problem because they were never taught the why. However, when students learn relational understanding, they are taught both why something works AND how to apply it. Relational understanding encompasses instrumental understanding. Learning relationally can be more challenging because students must comprehend the theory behind the application, but it is more beneficial in the long run. Students will understand why they use y = mx + b for the equation of a line instead of simply using a formula or equation that was given to them during class.

Mathematical Experiences

Math is the study of numbers and their relations to each other. Math is a way of describing the happenings in the world around us. Through a study of mathematics we learn about the wonders of the world.
The best way for me to learn math is hands on experience. I need to work out a couple problems and I usually understand the concepts behind the problem. Math has always come naturally to me. It is almost a second language of sorts that I speak. I do not know yet how my students will best learn mathematics. Every person learns differently; therefore, I must taper my teaching style to the class and the students that I have. I will try to teach the general concept instead of how to do a specific problem so my students will be able to do any problem I give them.
In math classrooms students are encouraged to explore the world of mathematics with their peers. Working with others is a great way to learn and grow. Also, teachers that I have had make an effort to find new, creative ways of teaching an idea that will help the students remember the concept. One thing I find detrimental to mathematics learning is the list of standards that are necessary for students to memorize in order to pass the test at the end of the year. In doing this, students focus on scoring the highest possible on the STAR tests instead of actually learning the principles.