Long Descriptions for Chapter Twelve

Figure 12.1: Big Idea Network Map for Grade Three

The graphic illustrates the connections and relationships of some third-grade mathematics concepts. Direct connections include the following:

• Fractions of Shape & Time directly connects to Square Tiles, Fractions as Relationships, Unit Fractions Models, Represent Multivariable Data
• Measuring directly connects to Number Flexibility to 100, Analyze Quadrilaterals, Represent Multivariable Data
• Addition and Subtraction Patterns directly connects to Number Flexibility to 100, Unit Fraction Models, Analyze Quadrilaterals, Represent Multivariable Data
• Square Tiles directly connects to Fractions as Relationships, Number Flexibility to 100, Fractions of Shape & Time
• Fractions as Relationships directly connects to Square Tiles, Fractions of Shape & Time, Unit Fraction Models
• Unit Fraction Models directly connects to Fractions as Relationships, Addition and Subtraction Patterns, Fractions of Shape & Time, Represent Multivariable Data
• Analyze Quadrilaterals directly connects to Number Flexibility to 100, Addition and Subtraction Patterns, Measuring
• Represent Multivariable Data directly connects to Unit Fraction Models, Number Flexibility to 100, Addition and Subtraction Patterns, Measuring, Fractions of Shape & Time
• Number Flexibility to 100 directly connects to Square Tiles, Analyze Quadrilaterals, Represent Multivariable Data, Measuring, Addition and Subtraction Patterns

Figure 12.3: Different Purposes of Assessment Cycles

This image shows the different types of assessments in relation to one another. From left to right the “Student” right arrow points to “Short cycle assessments”: Minute-by-minute; Daily; Weekly; right arrow to “Medium cycle assessments”: Unit and Quarterly; right arrow to “Long-cycle assessments”: Annually; right arrow to “Standards.” Source: adapted from Herman, Joan L., and Margaret Heritage. 2007. Moving from Piecemeal to Effective Formative Assessment Practice: Moving Pictures on the Road to Student Learning. Paper presented at the Council of Chief State School Officers Assessment Conference, Nashville, TN.

Figure 12.9: Sample Mathematical Practice Rubric for SMP.1

Indicating four levels of student proficiency in SMP. 1: Make sense of problems and persevere in solving them.

• Level 1 is “I can show at least one attempt to investigate or solve the task.”
• Level 2 is “I can ask questions to clarify the problem, and I can keep working when things aren’t going well and try again.”
• Level 3 is “I can make sense of problems and persevere in solving them.” (standard reached)
• Level 4 is “I can find a second or third solution and describe how the pathways to these solutions relate.”
  

Figure 12.15: Sample Diagnostic Comments for High Dive Checkpoint 1

The image shows a mathematical task with both student work and teacher diagnostic comments in green. The task set up provides information about the radius of a Ferris wheel, the height above ground of the center of the Ferris wheel, and the time it takes to complete one full rotation of the Ferris wheel. The task asks students to describe how high off the ground a rider (“you”) would be at certain times. Problem one asks, “What is your height off the ground 18 seconds after you pass the 3:00 position?” The student work shows that they begin the problem by calculating how many degrees the wheel rotates each second and determining where the wheel would be in its rotation at 18 seconds. The teacher comments that this initial work is a “good strategy for starting the problem.” The student uses trigonometry to find x and uses x to determine an answer to the question. Pointing to x, the teacher asks, “What does this number represent?” The teacher also notes on the students’ drawing of the Ferris wheel that a drawn triangle “doesn’t look like a right triangle,” subtly questioning the formula the student selected to use to make the calculations.

Problem two asks, “What is your height off the ground 35 seconds after you pass the 3:00 position?” To the student note that “trigonometry works with angles bigger than 90 degrees because of inversion,” the teacher wonders, “what does this mean?” In response to the student calculations, the teacher comments “Thank you for justifying your work!!” In response to the students’ drawing of the Ferris wheel showing a right triangle with one side length labeled and one angle measurement, the teacher comments, “I like your diagram ... Which side of the triangle helps you?”

Figure 12.17: Cycles for Mastery Learning

Cycles for mastery learning process graphic shows how teachers move from instruction with clear Learning Targets (i.e., class lessons and tutoring), to active engagement and practice of the Learning Targets (i.e., class work, homework, extra practice), to assessments and teacher and peer feedback (i.e., tests, exit slips, retakes, observations, projects), to active engagement with feedback (i.e., more practice problems, error analysis, tutoring, etc.).