Understanding Math Concepts
As discussed in class, it is
difficult for teachers to look inside students’ minds to know if the students
understand a mathematical concept. Even if students complete a traditional
paper and pencil type of assessment, it does not guarantee that the student
truly understood the concept because he or she could have just memorized a
formula or made logical guesses on the assignment. In order to genuinely know
students understand a concept, the teacher has to lead them through a series of
progressive “moves” . The author of A
Model for Understanding Understanding in Mathematics describes
understanding as a continuum. It is not possible to create a one line
definition of the word but it is possible to list certain characteristics or
evidences of understanding for teachers to use as guidelines for instruction.
Students who have a clear understanding of a concept can do tasks such as
restating the concept in their own words, giving examples of the concept,
recognizing the concept in multiple situations, identifying connections between
the concept and other concepts or ideas, or stating what is opposite or
contradictory to the concept. This definition is still a working definition of
understanding for understanding can be shown in a myriad of ways.
Teachers have to scaffold
instruction in order for students to reach higher levels of understanding.
Students may be able to give examples of the concept in the early learning
stage but most likely would not be able to identify things that are true about
the examples of the concept. Teachers must begin by asking students to display
their understanding of concepts through less complex explanation and
calculation and then slowly build toward the more complicated ideas. Students
have to understand the ideas in the first level of understanding before moving
onto the second level. However, there is not a specific order in which teachers
have to get their students to reach before moving onto the next criteria of
understanding. For example, students can display understanding by providing an
example of the concept while simultaneously identifying a non-example of the
concept in some instances. The structure of the curriculum is entirely
dependent upon the concept being taught, the skill levels of the students, and
the experience of the teacher.
Thinking through a Lesson:
Successfully Implementing High Level Tasks
The TTLP or Thinking Through a
Lesson Protocol is designed for mathematics teachers to implement high level
tasks for their students. The TTLP is a lesson planning process that consists
of three steps: 1) selecting and setting up a mathematical task, 2) supporting
students’ exploration of the task, and 3) sharing and discussing the task. The
first step of the protocol asks teachers to decide on exactly what they want
their students to learn at the conclusion of the lesson. Teachers need to be
clear and concise when creating lesson objectives. Teacher need to consider
student prior knowledge, expectations for when students are working on the
task, challenges some students might face while working on the task, and how to
introduce the task to students. The second step is concerned with how the
teacher will monitor students while they are working on the assigned task. The
teacher needs to consider how to get students started on the task, how to keep
students engaged while working on the task, and how to advance students
mathematical understanding while working on the task. Finally, the third step
to the protocol asks teachers to determine how students will share the
procedures they took to solve the problem, how to ensure every student in the
classroom participates, and how to assess student understanding.
Over time teachers who use the TTLP
method, ask these questions automatically when they are creating their lesson
plans and do not need to complete the entire protocol line by line. The purpose
of the protocol is to change teacher thinking and planning of mathematics lessons
so that they are focusing on advancement of student understanding rather than impromptu
planning. Teachers who use this method have reported that lessons go smoother
and students are able to take more away from lessons in which the teacher can accommodate
all the diverse learning styles of the classroom. This is the result of
teachers anticipating what procedures students are going to be using to solve
the problem in advance.
Thanks Sara:)
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