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.