Designing a Continuum of Learning to Assess Mathematical Practice NCSM April, 2011

Learning Intentions We Are Learning To … analyze students’ thinking on a Continuum of Student Thinking and Understanding. advance students’ thinking by developing questions and designing adaptations and modifications to move students to the next stage or stages.

Success Criteria We will know we are successful when we connect students’ work samples from a task to the Continuum of Learning and to the appropriate CCSS standards.

Focus in High School Mathematics Six investigations: Chapter 1: Country Data Chapter 2: Old Faithful Chapter 3: The Olympics Chapter 4: Starbucks Customers Chapter 5: Memorizing Words Chapter 6: Soft Drinks / Heart Disease

Focus in High School Mathematics Chapter 3: The Olympics Will Women Run Fast than Men in the Olympics? (And if yes, when?)

The Task … Will Women Run Fast than Men in the Olympics? (And if yes, when?) Open up the task.. What is the story behind the data? See handout.

What’s the story behind the numbers?

Discussion of the Task How might you develop the task? Discuss in small groups how you would use the data to answer the research question. What important ideas surfaced that are connected to the task?

Key Elements and Habits of Mind Key Element: Analyzing Data Habits: Analyzing a problem … Monitoring one’s progress… Seeking and using connections… Reflecting on one’s solutions…

Key Elements and Habits of Mind Key Element: Analyzing Data Habit: Analyzing a problem … Looking for patterns and relationships by describing overall patterns in data looking for hidden structure in the data making preliminary deductions and conjectures

Key Elements and Habits of Mind Key Element: Analyzing Data Monitoring one’s progress… Evaluating a chosen strategy by evaluating the consistency of an observation with a model applying the iterative statistical process to the investigation

Connect to a CCSS High School Conceptual Category and Domain Conceptual Category: Number and Quantity Domain: Quantity (N – Q)

CCSS Cluster … Cluster(s): Reason quantitatively and use units to solve problems. Use units as a way to understand problems; … ; choose and interpret the scale and the origin in graphs and data displays. Define appropriate quantities for the purpose of descriptive modeling. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

What Standards for Mathematical Practices can be developed?

1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. … more

They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. … more

Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. !!

Continuum of Student Learning Part 1: Planning Stage Part 2: Analyzing Students’ Thinking Part 3: Advancing Students’ Thinking

Continuum … Part 1 See template …

Continuum … Part 2 Starting OffEmergingPracticingAdvancing Examine the student work samples. Identify samples that you think would belong to the stages of the continuum. Describe what aspects of the students’ thinking were evident that you think would place the student’s work in the continuum stage.

Continuum … Part 3 Advancing Students’ Thinking Discuss: What questions will I ask? What modifications or adaptations will I do?

Asking Questions Problem Comprehension Can students understand, define, formulate, or explain the problem or task? Can they cope with poorly defined problems? What is the problem about? What can you tell me about it? Would you please explain that in your own words? What do you know about this part? Is there something that can be eliminated or that is missing? What assumptions do you have to make?

Questions … Approaches and Strategies Do students have an organized approach to the problem or task? Where would you find the needed information? What have you tried? What steps did you take? What did not work? How did you organize the information? Do you have a record? Did you have a system? A strategy? A design? Would it help to draw a diagram or make a sketch? How would it look if you used these materials?

Questions … Solutions Do students reach a result? Do they consider other possibilities? Is that the only possible answer? How would you check the steps you have taken, or your answer? Is there anything you have overlooked? Is the solution reasonable, considering the context? How did you know you were finished?

Summary We Are Learning To … analyze students’ thinking on a Continuum of Student Thinking and Understanding. advance students’ thinking by asking good questions and making adaptations and modifications to move students to the next stage or stages. We will know we are successful when we can understand the components of the Continuum of Student Thinking and Understanding and fill in the form after analyzing student work.

Gap Times Female – Male (in sec) What does the above mean? How can we use this data to investigate our research question?

Time differences What trend do we see? If women were to run as fast as men, how would that be represented in this graph? How would a faster time for women be represented?

Regression Line The linear regression represents the gap. Is this line a good summary of what is happening? Why or why not?

Separate Regression Lines If women will run as fast or faster than men, how would it be presented in this graph?

Predicting the Distant Future How does the model use to represent the Olympic times breakdown? What might be the more accurate representation of the times in the future?

An Exponential Model In what way does this model represent the Olympic times now and in the future?