Effective Mathematics Teaching: Goals in Lesson Planning

Effective Mathematics Teaching

• Establishes clear goals for student learning.
• Situates goals within learning progressions.
• Uses goals to guide instructional decisions.
• Helps decide which tasks fit the learning plan.
• Determines which visuals and math discussions to focus on during lessons.

Goals in Lesson Planning

• Goals should set the course for a lesson.
• Provide support and direction for teachers' instructional decisions.
• The selection of instructional tasks should follow from the stated goals.
• Supply a road map for the lesson.
• Guide teachers in determining which questions to ask.
• Help identify student-generated strategies and ideas to pursue.
• Determine relevant tasks for planned learning progression.
• Determine representations to highlight and focus of mathematical discourse.

Exploring Lesson Goals

• Goal statements for a lesson on exponential functions are compared.
• Goal A focuses on identifying and evaluating exponential functions in the form y = bx.
• Goal B emphasizes understanding exponential growth and the meaning of the exponent and base.

Comparing Goal Statements A and B

• Both address the same math content but expect different types of thinking from students.
• Goal A requires memorization and executing procedures like substituting and evaluating.
• Tasks for Goal A provide values for x and b, asking students to create and evaluate functions.
• Goal A's skills don't invoke conceptual understanding around exponential growth.
• Goal B requires understanding exponential growth and meaning of x (exponent) and b (base) in y = bx.
• Goal B's understanding is essential for recognizing real-world relationships that can be modeled with exponential functions.
• Tasks for Goal B provide scenarios where students model and understand exponential growth based on prior knowledge of linear functions.
• Goals A and B differ because they require different mathematical activity from students, generating differences in student mathematical learning.

Lesson Goals Supporting Teaching and Learning

• Ms. Culver identified goal B as her intention for student learning.
• She chose a task that would help students meet her goals.
• The Pay It Forward task provides context for students to make sense of exponential functions and promoted thinking and reasoning.
• She made tools like graph paper and graphing calculators available.
• She asked questions to help students attend to pattern growth.
• She sequenced presentations to build students' understanding of exponential growth.

Ms. Culver's Lesson and Exponential Growth

• Diagrams, tables, equations, and graphs are used.
• In whole-group discussion, students explain exponential growth using the problem's context.
• Students determine which function (y = 3x or y = 3*) correctly modeled the Pay It Forward situation.
• Goals provided direction for determining tasks, questions, and structuring group discussion.
• Having goals and a task focused instructional decisions on promoting students' understanding of mathematics, rather than procedures or facts without understanding
• Understanding is an essential first step.

Analyzing Graphs

• The classroom of Shalunda Shackelford examines graphs that model the speed of a bike and truck.

The Bicycle and Truck Task

• Involves a bicycle traveling at a steady rate and a truck moving in the same direction.
• Students label graphs, describe movement, and determine which vehicle reaches 300 feet first.
• Students must explain why they agree or disagree with someone's claim that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel.

Graph Inconsistencies

• Graphs may not perfectly model real-life movement.
• For example, vehicles cannot come to an immediate stop.
• Discussion on consistencies and inconsistencies can foster productive mathematical discussions.
• Teachers can ask students to consider how they would change the graph to better model real-life movement.

Ms. Shackelford's Content Goals

• Ms. Shackelford has three content goals for her students that they should understand:
• The language of change and rate of change can describe how two quantities vary together.
• Context helps interpret key features of a graph portraying the relationship between time and distance.
• The average rate of change is the ratio of the change in the dependent variable to the change in the independent variable.
• The Bike and Truck task fits within a sequence of lessons on creating and interpreting functions aligned the mathematics standards.

State and National Standards

• Modeling a relationship between two quantities, interpret key features of graphs and tables.
• Sketch graphs showing key features from a verbal description.
• Relate the domain of a function to its graph and the quantitative relationship.
• Calculate and interpret the average rate of change of a function.

Connecting Standards to the Bike and Truck Task

• Questions 1-3 involve interpreting key graph features portraying the relationship between time and distance.
• Question 3 involved determining which vehicle was first to reach 300 feet.
• Relate domain to the graph and average rate of change.
• Question 1 may prompt discussions of rate of change.
• Question 3 may encourage intuitive discussions to find out who arrived at 300 feet first.

Ms. Shackelford's Process Goals

• Ms. Shackelford worked to facilitate mathematical discussions and target mathematical practices.
• She has process goals for students to engage in mathematical discourse, problem-solving, argumentation, and modeling.
• The 3 part video clip introduces a common misconception from "imaginary friend Chris," draws attention to horizontal portion of the graph, and asks agreement/disagreement.

Analysis of the Discussion

• Ms. Shackelford used content and process goals to focus learning.
• Additional questions address content goals 1, 2, and 3, and maintain students' perseverance.
• She introduced a misconception for students to consider how time and distance vary.
• The instructional moves provided opportunities for students to use the context to interpret key features of the graph and model with math.
• Ms. Shackelford supported students' problem-solving by pressing them to clarify their thinking.
• Ms. Shackelford created an opportunity for students to present and defend opinions and explain the reasoning for their viewpoints and critique with their classmates.
• Ms. Shackelford provided opportunities for engagement in mathematical discourse.
• She asked questions and pressured students for explanations and meaning
• Ms. Shackelford's moves align with her goals for the lesson.
• Clear goals focuses student attention on monitoring their progress toward learning outcomes

Research on Mathematics Goals

• Cases exemplify teachers using goals to inform decisions and focus learning.
• Goals addressed important aspects of students' understanding and aligned with standards.
• Goals and tasks should be embedded within learning progressions and intended to develop students' understanding of important mathematical ideas.
• Goals connected to learning progressions help teachers support students' transitions from prior knowledge.

How Goals Impact Teaching

• Goals guide teachers' instructional decisions and impact the nature and focus of student work.
• They support teachers' decisions, selecting tasks, asking questions, and framing whole-group discussions.
• Teachers should determine which ideas and strategies to pursue for guiding mathematical discussions.
• Goals help teachers select and sequence mathematical work and ideas.
• This equips teachers to support students' learning.

Guiding Instruction

• Research indicates teachers' use of goals to guide instruction supports students' ability to monitor their own learning.
• When teachers explicitly refer to goals during a lesson, students are better able to self-assess and focus their learning.
• Which is an important factor in student achievement.

Promoting Learning

• Teachers' goal use supports students' learning and helps to provide students of all backgrounds to excel.
• High expectations supports for promoting access and equity in learning meaningful mathematics.
• Clear communication, high expectations, and support can lead to significant learning gains in achievement.

Key ideas

• Connecting goals to learning progressions and big mathematical ideas helps to make transitions.
• Rote skills and procedures are not the focus, but instead students should be reasoning, problem solving, and be communicating growth mindset.
• This sets high and clear expectations, and promotes student reasoning, which allows for a meaningful experience.

Effective teaching Practices

• Goals implementation with tasks, questions, and facilitating discourse.
• The connections between the focal practice and other practices are woven throughout the chapter.

Improving Learning

• Tasks provide opportunities to learn and understand concepts necessary.
• Promote student reasoning by implementing tasks that support it.
• Questions used to interact with math ideas aligning to set goals that teachers craft that aligns goals.
• Questions help students focus improving their learning.

Mathematical Discussions

• Goals informs planning around a central frame for eliciting ideas determining a students performance.
• This is done by using meaningful discourse in chapter 7, using a student self assessment.

Key focus areas

• Establishing clear goals to focus learning, explicitly talk about goals to students.
• Promote the understanding, reasoning, and problem solving.
• Create goals that connect learning to what is already understood.
• Goals should be used as focus and the use of those for learning and progress.