Entry #7

Smith, M. S., Bill, V., & Hughes, E. K. (2008). Successfully implementing high-level tasks. Mathematics Teaching in the Middle School, 14 (3), 132-138.

The authors address an issue resulting from previous research which has shown that cognitively challenging tasks that promote thinking, reasoning and problem solving often decline during implementation. The question becomes why such tasks are so difficult to implement in ways that maintain the rigor of the activity. One reason is that high-level tasks tend to be less intellectually controllable from the teacher's perspective, since a worthwhile task often has more than one specific solution path. Thus the authors proceed to present a plan to control teaching with high-level tasks which they call the TTLP (Thinking Through a Lesson Protocol). The TTLP is divided into 3 parts: Part 1: Selecting and Setting Up a Mathematical Task, Part 2: Supporting Students' Exploration of the Task, and Part 3: Sharing and Discussing the Task. The authors provide the framework of the TTLP (which is an outline of specific questions for teachers to consider) as a figure within the article. In general, Part 1 asks the teacher to identify mathematical goals and expectations of the task. Part 2 focuses on monitoring students as they explore the task and Part 3 focuses on orchestrating whole-group discussion of the task that uses different solution strategies to highlight mathematical ideas. The goal of the TTLP is to prompt teachers to think deeply about a specific lesson that they will be teaching through strategies such as anticipating student responses and creating good questions that will further students' thinking. The TTLP is meant as a reminder and a guide for teachers to gradually mold their teaching around, and not as a daily checklist of questions.

Just as the authors have stated, I too feel as though it is important for teachers to anticipate student responses and develop questions that promote critical thinking. Even with a task that is designed to be thought provoking, much of the value of the task can be lost if teachers do not know how to guide their students through the critical thinking process. I also believe that this is a necessary approach for any task, whether it be considered high-level or not. Even when a set solution path seems like the only approach, there are often many students who reason about things differently. Regardless of how hard the task might appear, I think it is valuable for the teacher to have attempted and thought about the problems for themselves beforehand. I have been in math classes where the teacher admitted they hadn't gone through the problems that they were using as examples because they seemed easy to solve, but in reality the problems had many solution strategies and interesting points that could have been expanded upon if the teacher was better prepared. When teachers are themselves involved in the task they can better guide their students and can build upon alternate strategies that may arise in order to promote deep thinking from the students.

Entry #6

Davis, D., Herron-Thorpe, F. L., & Olson, J. C. (2010). Shrinking your class. Mathematics Teaching in the Middle School, 15 (7), 386-391.

The authors of "Shrinking Your Class" wanted to convey that a math/engineering project that required students to "do" math helped them to retain their learning and collaborate with their classmates, while being genuinely interested in the task at hand. The National Science Foundation Grant brought fifty toys and two engineering graduate students into a middle school math class in an effort to enhance the students' mathematical learning by using an engineering context. The students were learning about scale factors in a unit called Shrinking and Stretching. The students first measured and explored the toys and to discover different scaling factors. Then a classroom discussion about equations involved and the authors note that the students were developing their understanding of ratios, scale factors, fractions, measurement and arithmetic as a result. The students were provided with materials to construct scale models of themselves for one day, and a spend a second day making scale models of objects that would create a cohesive diorama. The authors discuss that the project was successful in creating an envisioning scene for the students. Time was a constraint because some were not finished after the two days. The teacher commented that the students vividly retained what a scale factor was and how to obtain it months later. The students learned to collaborate, felt ownership over their work and enjoyed the task while applying and solidifying mathematical principles that were applicable to engineering design.


Tasks that allow students to "do" math should be frequently used in school classrooms. One reason is that students aren't as focused on learning the mathematical skills themselves as they are completing the task that requires them to use those skills. In baseball and softball, when players are trying to make long throws, they are taught to aim beyond their target so that their throw will actually get to their target. The same principle applies as students who see learning the mathematical skills as the end goal may give up too soon. But when students are motivated to achieve a different goal, using their math skills as a means, they will still learn the math even if their end goal isn't completely achieved. A second reason for "doing" math is that students seem to enjoy their learning more and can use their creativity. In middle school especially, students' minds are still cognitively developing and their curiosity needs to be nurtured. In the article, the authors note that the students became genuinely interested and concerned with successfully participating in the activity. From my own experience, "doing" math fosters a positive and excited outlook toward continued learning which helps me perform better in my classes. A third reason in support of "doing" math is that students retain the information better. Students have a descriptive image of which they can remember what they learned. The teacher in the article points out that her students remembered how and what a scale factor was months later. Thus these types of hands-on mathematical tasks which allow students to "do" math are valuable to students and should be more frequently implemented.