June 18th Articles

A Model for Understanding Understanding in Mathematics

Understanding is a hard concept for many teachers and students to grasp. Both Holt and van Engen view understanding as a continuum. Students can have partial understandings. They may be able to answer a question correctly, but they aren't quite sure why they took the steps they did or how to explain their solution. In order for teachers to promote a better understanding for students, they can make "moves" in teaching different kinds of mathematics. Moves will help teachers to identify objectives that students will need to accomplish in order to have a complete understanding of the material. In addition to planning to make moves, teachers can use moves to evaluate and assess the level of understanding of students for a given mathematical knowledge. There are for different types of mathematical knowledges: concepts, generalizations, procedures, and numerical facts. For each of these categories, there are different moves that teachers can make. There are also 2 different levels of understanding. In order to understand a concept, students must be able to give examples and non examples of the concept and also understand different characteristics of the concept. Statements of relationships between concepts are called generalizations. In order to fully understand generalizations, students must know what the generalization is saying and give proof of why it is true.  The next type of mathematical knowledge is procedures. To prove understanding, student must make moves such as: accurately carrying out the procedure, showing another student how to do the procedure, or paraphrasing the procedure step by step. They also need to be able to explain why the procedure works. Many numerical facts are generalizations, so many of the same moves for generalizations apply to number facts. In order to fully understand numerical facts, students must be able to recall the fact and understand what it is saying and also understand why the fact is true and realize its significance. It is important for teacher to implement and rely on these moves to promote students understanding of different mathematical knowledge. Moves in teaching mathematics can serve as a basis for defining understanding of mathematics. I know it is difficult to see if students truly understand a problem, or if they just wrote down the answer. Students need to be able to understand the steps they took to solve the problem and why those steps helped them to come up with a correct answer. 

Davis, E. (2006) A Model of Understanding Understanding in Mathematics. Mathematics Teaching in Middle School.  12 (4). 190-197. Retrieved June 15, 2015.

Thinking Through a Lesson: Successfully Implementing High-Level Tasks

Tasks that give students the opportunity to think critically and use reasoning skills are the most difficult for teachers to implement into the classroom. This is because procedures for solving high-level problems are not usually specified in advance. There are different ways to solve high-level tasks so it is hard for a teacher to present this to the class without confusion. It is important for teachers plan in detail prior to the lesson. Teachers can use TTLP, or Thinking Through a Lesson Protocol to help further the use of cognitively challenging tasks. The TTLP helps to develop lessons that make students use their mathematical thinking as the major ingredient for understanding a concept. There are three different sections in the TTLP, which include: (1) selecting and setting up the mathematical task, (2) supporting students explanation of the task, and (3) sharing and discussing the task. Part 1 is where the teacher should lay the groundwork and identify goals for the lesson and also set expectations for how students will work. Part 2 focuses on monitoring the students during the task. Teachers need to me sure to ask students throughout the lesson to make sure they understand the concept and are on their way to complete the goal for the lesson. Part 3 focuses on a class discussion of the task. Students will need to share their results with the class and explain how they got their answer and justify why they believe it is correct. I believe that the TTLP is a great tool to use in planning for the classroom. Although it can look very overwhelming, I think that it will help in the long run. I always struggle with coming up with questions to ask students throughout a lesson, so brainstorming and having questions ready is something I plan on doing. It also helps not only the students think deeper about a concept, but also the teacher. This is vital because the teacher needs to have a solid understanding of the concept before conveying it to students. I believe the TTLP will be very beneficial and will help promote students problem solving and critical thinking skills while working on high-level tasks. 

Smith, M., Bill, V., Hughes, E. Thinking Through a Lesson: Successfully Implementing High-Level Tasks. Rich and Engaging Mathematical Tasks: Grades 5-9. 11-18. Retrieved June 15, 2015.