The article Investigating Mathematics with Technology: Lesson Structures That Encourage a Range of Methods and Solutions by Rebecca McGraw and Maureen Grant compares "two kinds of lessons (we call these Type 1 and Type 2) according to the extent to which they engage students in (1) identifying patterns and searching for relationships, (2) making and investigating mathematical conjectures, and (3) developing and evaluating mathematical arguments" (McGraw & Grant, pg. 303). The authors describe Type 1 lessons by how the instructions "are written to focus students' attention on particular mathematical relationships. Technology is used to help make these relationships transparent to students" (McGraw & Grant, pg. 304). McGraw and Grant claim the difference between Type 1 and Type 2 is that the instruction is different and "the roles of the teachers and students during implementation" (McGraw & Grant, pg. 304).
This difference is immediately apparent when reading through dialogue between the teacher and students for each of the lesson types. In Type 1, the instructor Mr. Jeffries appears to be pulling teeth and really stretching the students for answers about the activity, where students examined equations using their graphing calculators. The lesson seemed to be centered around the teacher and as a result only engaged students who understood the material and had a firm grasp on the concepts being examined.
Type 2 is comparable to lessons that remove the scaffolding and encourage students to arrive at their own conjectures through exploration, which is facilitated by the graphing calculator. In this activity, the students were first presented with graphs and are instructed to group the graphs. The lesson then slowly unravels details about the subject matter by providing supplemental materials in a "reverse" order. Students receive a handout with the equations and are asked to match graphs to equations which leads the students to organize their own thoughts through communication of their conjectures.
While it would be fantastic as an instructor to have every lesson and class period follow Type 2, I do not believe it is practical. However, instead of presenting students with the information up front (i.e. terminology) students can investigate concepts on their own and the teacher can utilize recaps and discussions from Type 2 lessons to incorporate terminology that is necessary for mathematics communication.
McGraw, R., & Grant, M. (2005). Investigating mathematics with technology: Lesson structures that encourage a range of methods and solutions. In W. J. Masalski (Ed.), Technology-supported mathematics learning environments (pp. 303-317). Reston, VA: National Council of Teachers of Mathematics.
Sunday, November 11, 2012
Response to: Deciding When to Use Calculators
The article Deciding When to use Calculators by Anthony Thompson and Stephen Sproule presents a framework that will help "middle school mathematics teachers..." to "...decide when to use calculators with their students" (Thompson & Sproule, pg. 127). Thompson and Sproule present this framework in what they call a "cartesian representation" which includes two dimensions and four categories. The first dimension is student oriented and is divided into the subcategories essential and nonessential. They define essential as "the activity would be too complex for the students to complete without using a calculator" and nonessential as "the mathematics activity can be completed without using a calculator" (Thompson & Sproule, pg, 127). The second dimension of the framework is the goal oriented dimension, which "focuses on two possible pedagogical goals: (1) for the students to find a computational solution and (2) to engage students in problem-solving processes" and thus the two subcategories of the second dimension are product and process (Thompson & Sproule, pg, 128). The authors describe process oriented as "the goal of the activity is for students to understand the processes associated mathematical exploration and problem solving," similarly product oriented means "the goal of the of the activity is for students to determine a computational solution or end product" (Thompson & Sproule, pg, 128).
The framework presented by Thompson and Sproule appears to be a good tool for a quick decision as to whether calculators should be incorporated in a particular activity. However, I believe the decision requires further consideration then the four categories presented in the framework and that there are variables Thompson and Sproule do not address in this brief article. One of which we have not discussed at great length but certainly is worthy of consideration is whether or not students are allowed to utilize advanced calculators on standardized tests. Although this may seem to be a stretch from an article that addresses whether calculators should be used in a classroom, teachers cannot neglect the direct relationship, mainly from a comfort and confidence aspect, that students develop a dependency as a result of using calculators to perform desired tasks. I would argue that Thompson and Sproule failed to consider the dimension of whether a calculator is allowed on gateway standardized tests (i.e. SAT, ACT, AP examinations) to perform specific functions. If it is not, then in my opinion the students should not be exposed to the calculators ability to perform such function to prevent the above mentioned dependency. To clarify, I understand the framework is for middle school teachers but believe that students who form a dependency early may have difficulty on standardized test, where middle school mathematics serves as a foundation.
I agree with the authors on the concept that calculators should be used when the difficulty of a problem lies in the mathematical manipulation, which would consume time and consequently detract from the overall learning goal of the practice problem. However, I would challenge the intent of incorporating such a difficult problem, as it seems somewhat convoluted that we as teachers create problems which detract from the overall goal/task and simply allow students to utilize a tool to eliminate the barrier we just created. Why not just provide simple questions and eliminate the use of a calculators to ensure the students understand the main objective?
Thompson, A. D., & Sproule, S. L. (2000). Deciding when to use calculators. Mathematics teaching in the middle school, 6(2), 126-129.
The framework presented by Thompson and Sproule appears to be a good tool for a quick decision as to whether calculators should be incorporated in a particular activity. However, I believe the decision requires further consideration then the four categories presented in the framework and that there are variables Thompson and Sproule do not address in this brief article. One of which we have not discussed at great length but certainly is worthy of consideration is whether or not students are allowed to utilize advanced calculators on standardized tests. Although this may seem to be a stretch from an article that addresses whether calculators should be used in a classroom, teachers cannot neglect the direct relationship, mainly from a comfort and confidence aspect, that students develop a dependency as a result of using calculators to perform desired tasks. I would argue that Thompson and Sproule failed to consider the dimension of whether a calculator is allowed on gateway standardized tests (i.e. SAT, ACT, AP examinations) to perform specific functions. If it is not, then in my opinion the students should not be exposed to the calculators ability to perform such function to prevent the above mentioned dependency. To clarify, I understand the framework is for middle school teachers but believe that students who form a dependency early may have difficulty on standardized test, where middle school mathematics serves as a foundation.
I agree with the authors on the concept that calculators should be used when the difficulty of a problem lies in the mathematical manipulation, which would consume time and consequently detract from the overall learning goal of the practice problem. However, I would challenge the intent of incorporating such a difficult problem, as it seems somewhat convoluted that we as teachers create problems which detract from the overall goal/task and simply allow students to utilize a tool to eliminate the barrier we just created. Why not just provide simple questions and eliminate the use of a calculators to ensure the students understand the main objective?
Thompson, A. D., & Sproule, S. L. (2000). Deciding when to use calculators. Mathematics teaching in the middle school, 6(2), 126-129.
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