Article Discussions

Article Discussion #1
A Model for Understanding Understanding in Mathematics by Edward J. Davis

            This article is about understanding the mathematical process through different kinds of “moves” in teaching mathematics. It is essential that teachers gain knowledge about how students are understanding so it can help guide them for future planning of instruction and assessment of learning. The word “understanding” is a broad term and has various definitions (for example, it may mean being able to state it in your own words or giving examples of it). Also, understanding is a range; students may have full understanding, no understanding, or they could have only a partial understanding. The “moves” that a teacher uses is essentially how the teacher explains the mathematical information to the students. In school mathematics, typically concepts, generalizations, procedures, and numerical facts are the mathematical knowledge taught. Within each of these areas, the article presented two levels. Level 1 was where students are grasping the basic knowledge, such as the “what” (ex. understand “what” the generalization says) whereas Level 2 digs a little deeper and may discuss the “why” (ex. understanding “why” the generalization is true) or “characteristics”  (ex. understanding characteristics of the concept). Also, I like to think of these levels as building blocks. It’s a step-by-step process where students’ understanding gets progressively deeper. For example, when teaching “concepts” (level 1: examples and nonexamples of the concept), students understand a concept when they can: give or identify examples of the concept, defend choices of examples of the concept, give or identify nonexamples of the concept, and defend choices of nonexamples of the concept. You can see that the understanding starts basic and gets more challenging. However, it is important to note that students’ understanding may jump around (they may understand one step or some aspects of one level before understanding another; i.e. they may understand some aspects from Level 2 before Level 1). The article also discussed how elementary teachers’ instruction starts at physical then goes to pictorial and then moves towards symbolic representation. I appreciated how the article then went on to say that when students are at the symbolic level, they should not stop. When they are at this level, they should be able to explain and interpret their work (through pictures, number lines, counting objects, etc.).

            This article was insightful for me as a future elementary teacher because it is critical to know how students are understanding the content. Through the various “moves”, teachers can better determine where certain students are getting stuck in their learning. When teachers understand the moves at which certain students are getting stuck, a few things happen: 1. A teacher can better prepare for future instruction and assessments, 2. A teacher can better diagnose a student’s understanding of a mathematical concept, generalization, procedure, or numerical fact, and moves can help show the student’s knowledge, and 3. A teacher can pick moves and instruct aids or tutors to focus on those specific moves with certain students. I was captivated by how the word “understanding” is so detailed. I learned that there are so many levels to understanding and students will likely be at various levels of understanding, which is important to keep in mind when teaching in order to properly meet the needs of all students. In my future classroom, I think utilizing “moves” will be beneficial so I can gauge where my students strengths and weaknesses are and where I need to reteach certain moves.

Discussion questions:

1.     While this article showed the importance of using “moves” to understand students’ mathematical knowledge/comprehension, what challenges may moves present in the classroom?

2.     If “moves” seem to be successful in mathematics, would a similar process be successful in other subject areas (i.e. language arts, science, etc). Why or why not?

References:

Davis, E.J. (2006). A model for understanding understanding in mathematics. Mathematics Teaching in the Middle School. 12(4). 190-197.

Article Discussion #2
Thinking Through a Lesson: Successfully Implementing High-Level Tasks by Margaret Schwan Smith, Victoria Bill, and Elizabeth K. Hughes

            When teaching mathematics there is an emphasis on providing students with rich, critical thinking tasks where they are given opportunities to use reasoning and problem solving skills. However, when these thought-provoking, often challenging tasks are implemented in the classroom, research has shown that much thinking and reasoning is lost and no meaning is made. Creating and implementing these rich and challenging tasks are often difficult but they are important for students’ mathematical growth. That is where the TTLP, the Thinking Through a Lesson Protocol, comes in. TTLP is a process to help teachers teach high-level tasks and develop a detailed lesson plan prior to teaching. There are three aspects to the TTLP: 1.  Selecting and setting up a mathematical task (teacher identifies the math goals for the lesson and thinks about how students will work through the problem), 2. Exploration of the task (students work individually or in small groups and are asked questions to assess their current knowledge and to guide their learning), and 3. Sharing and discussing the task (class has a whole-group discussion of the different solution strategies they used). While the TTLP may be time-consuming, it is beneficial and effective. For Part 1, selecting and setting up a mathematical task, the first thing a teacher should think about is: What are the mathematical goals for the lesson? For Part 2, exploration of the task, the teacher should create good questions to ask during instruction to help students focus on the main mathematical ideas that are a part of the task they are exploring. The questions created should assess what students understand about the problem and advance them towards the objectives/goals of the math lesson. For Part 3, sharing and discussing of the task, it is important for the teacher to think about the various ways the task can be solved. The teacher should brainstorm the correct and incorrect strategies a student uses to solve the problem. TTLP helps teachers think about what the students are doing and aids teachers in guiding students to make sense of the math content as well as advance their math understanding.

            After reading this article, I learned that while TTLP may be a long and time-consuming process, it is worthwhile. The more teachers use TTLP, the more likely they can start internalizing this framework and plan effective lessons. Furthermore, I learned that anticipating student responses/questions ahead of time and preparing questions prior to the lesson to ask the students is crucial. These prepared questions will help guide the students with the problem at hand while still allowing them to explore the answer for themselves. I can envision myself using the TTLP process in the future because it will allow me to think about how my students may think about the content. It will also help me be more prepared to ask them appropriate questions which still allows them to explore while also guiding them towards the math objective of the task. I believe that with the TTLP process, I will be more prepared and the lessons I teach will be more successful.  

Discussion questions:

1.     This article explains the importance of teachers brainstorming possible correct and incorrect student responses. However, how would you handle a situation where a student’s response is not one you prepared for?

2.     In Part 3, sharing and discussing the task, how will you, as the teacher, lead this class discussion? What do you think is the most effective way? (i.e. Do you think you should have each small group present their strategies? Would you have one student representative from each small group share their strategy? Would you, as the teacher, present strategies and ask if any of the groups did any of the strategies you presented? Would you have students debate which strategy they think is the most accurate?)

References:

Smith, M. S., Bill, V. Hughes, E.K. (n.d.) Thinking through a lesson: Successfully  implementing high-level tasks. Designing and Enacting Rich Instructional Experiences. p. 11-18