Journal Summaries

A Blizzard of a Value 

(Found in Mathematics Teaching in the Middle School - Vol. 20, No. 6, February 2015)

One of the most important, yet most difficult tasks of creating mathematical problems is making them so that students are engaged, interested, and motivated to solve the problem. As teachers, how do we create effective problems that are mathematically challenging while also relating to students and keeping them engaged in problem-solving? This article was about Ms. Bosetti, a 7th grade math teacher in Ohio. This was a question that she faced regularly, and even more so after participating in over 100 hours of professional development in mathematics. She came to the realization that modeling in mathematics, one of the CCSSM SMP standards, could be realistically achieved by keeping students engaged in real problems that affect their lives, deeply discussing their models and thought processes, and giving and receiving feedback about models. With these concepts in mind, Ms. Bosetti created a realistic problem for all of her students to solve. Her problem asked students which size blizzard at Dairy Queen was the best value. She listed prices for each size, and additionally gave students a cup of each size to work with. Ms. Bosetti knew that this was a challenging task, but felt confident that students could figure it out. She made it clear at the beginning of the school year that her math class would NOT be about solving a problem, finding a solution, and moving on. Instead, students would continually work to find alternate ways of solving problems, justifying their work, and critiquing other's ways of solving problems. Once the DQ problem was presented to students, they self-divided into small groups. Most of the groups began by measuring each cup's height and diameter. From there, many students began to notice that the bases and tops of the cups were not identical in size. Some also questioned exactly how much ice cream was in each blizzard (filled to the top, overflowing, under the rim, etc.). When groups were ready to share their problem-solving methods, Ms. Bosetti asked different groups to explain their findings using different types of displays. One group presented using a graph, another using a table, another using pictures. Once students began presenting, they came to the realization that not all groups came up with the same measurements and numbers. As a class, they discussed that this was due to precision error during measurement. Ultimately, students came to the conclusion that a medium sized blizzard was the best value at DQ. After the problem-solving, Ms. Bosetti asked students to reflect on the activity. For the most part, the students said that they felt more motivated to find the solution because it actually pertained to their lives. They also said that they liked the use of multiple representations (graphs, tables, pictures) to solve the problem. This showed them that multiple strategies could have been used. Additionally, students learned about precision error and how it affected their measurements.

I thought that this was a very interesting article, and an ingenious way to implement effective problem-solving in a middle school mathematics classroom. I think that it is very difficult to make problems that are not only meaningful, but also relatable to students. Ms. Bosetti did an excellent job of this, and even her students noted that they felt more motivated to solve the problem because they ate at DQ frequently. I think that this article will greatly apply to me as a future teacher. Regardless of grade level, I will need to create meaningful problems for my students that they can relate to and feel engaged to solve. I believe that this is a difficult task at some times, because it may be hard to relate to every single student. However, I will need to find out my students' interests, hobbies, likes, etc. to be able to create problems like the one Ms. Bosetti created.

Three Strategies for Opening Curriculum Spaces 

(Found in Teaching Children Mathematics - Vo. 21, No. 6, February 2015)

In most school districts, teachers are provided a mandated textbook or teacher's guide for each subject that is to be taught. In general, teachers are required to go by these textbooks. However, it is not necessary for teachers to follow these guides exactly. In fact, most curriculum materials are created so that teachers have "open space", or additional room, to engage students and relate more to their lives. This article began with a hypothetical question about a teacher reviewing the day's lesson in the teacher's guide, and noticing that the lesson is about subtraction and begins with 10 practice problems followed by 2 word problems. She knows that her students can solve those problems using at least three different strategies. So, how can she use this within the lesson? The CCSSM requires that students utilize multiple strategies when solving problems, in addition to connecting problems to their own lives. This is an issue that many teachers face, and this article provided three suggestions for modifying curriculum materials to promote more learning. First, it was recommended that teacher's rearrange the components of the lesson. Many textbooks have students problem solve at the end of the lesson, but this article suggested that students problem solve throughout. In addition, the teacher should consider getting rid of any parts that include the teacher telling, directing, or showing students how to problem solve. The second suggestion was for teachers to adapt the tasks within the lesson to promote their students' Multiple Mathematical Knowledge Bases, or MMKB. This could be done by adjusting the numbers in word problems for different students, and also encouraging the use of multiple representations and strategies while solving problems. The third suggestion was for teachers to continually look to make authentic connections. Changing word problems to be more relatable to students should be done regularly. In addition, beginning a lesson with problem-solving may be more effective than ending the lesson that way.

This was a very interesting article to read. I have always questioned how we, as teachers, are supposed to make our lessons engaging and relatable to students while also utilizing the provided textbook or teacher guide. After reading this article, I have realized that the teacher guide is supposed to serve as more of an outline of the lesson. Teachers, in turn, are then to modify those outlines to have students more involved and able to connect the lesson to their lives. I took great interest in the second suggestion of adapting the tasks within a lesson for individual students. I have never thought to adjust a single word problem using different numbers for lower and higher level students. This is a wonderful idea, and definitely something that I will do to differentiate mathematics learning in my classroom. In addition, making mathematics relatable to students is extremely important, and ultimately what I think leads to more motivation in solving problems. This is something that I plan to work hard to achieve in my future classroom.

References:

Bostic, J.D. (2015). A blizzard of value. Mathematics Teaching in the Middle School. 20(6). 350-357.

Drake, C. Land, T.J., Bartell, T.G., Aguirre, J.M., Foote, M.Q., McDuffie, A.R., Turner, E.E. Three strategies for opening curriculum spaces. Teaching Children Mathematics. 21(6). 346-353.