creating and selecting mathematics that support effective ......creating and selecting mathematics...

Creating and Selecting Mathematics Textbooks that Support Effective Teaching and Student Learning

Diane J. Briars, Ph.D.President‐Elect, National Council of

Teachers of [email protected]

Sources

US Education System:Highly Decentralized

Selection of textbooks and other instructional materials is a local responsibility:

• District• School• Teacher• State

US Education System:Highly Decentralized

US has $7 billion K‐12 textbook market• 4 dominant publishers• Internet has allowed emergence of new competitors

Creation• Publishing companies• NSF/IES Funded research projects

GoalCreate and select mathematics textbooks that maximize the mathematics learning of each

student and the effectiveness of each teacher.

Critical Features for Textbook Creation/Selection

• Mathematics Content• Research‐based principles for instructional design

• Other supports: Assessments, resources for differentiation, teacher supports (lesson implementation; professional learning), additional resources

Mathematics Content• Topics (content standards) and sequence

– Learning trajectories and research‐based treatment of specific content

Common Addition and Subtraction Situations

Learning Trajectories and the Common Core State Standards

Learning trajectories are empirically supported hypotheses about the levels or waypoints of thinking, knowledge, and skill in using knowledge, that students are likely to go through as they learn mathematics and, one hopes, reach or exceed the common goals set for their learning. Trajectories involve hypotheses both about the order and nature of the steps in the growth of students’ mathematical understanding, and about the nature of the instructional experiences that might support them in moving step by step toward the goals of school mathematics.



• Rigor: – Balanced development of conceptual understanding, procedural skill

and fluency, and application– Develop mathematical inquiry, problem solving and reasoning (CCSS

mathematical practices)

Grade 66.G. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real‐world and mathematical problems.

What tasks would you use to develop students’ proficiency with this standard?

What Tasks Would You Use?

6.G. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real‐world and mathematical problems. • Compute area of different figures?• Explain the relationship between the areas of different

figures? • Find a missing side of a rectangle or base/height of a

triangle, given the area and another side?

What Tasks Would You Use?

6.G. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real‐world and mathematical problems. • What applications? • “A rectangular carpet is 12 feet long and 9 feet wide.

What is the area of the carpet in square feet?”

County Concerns1. The Jackson County Executive Board is considering a proposal to conduct aerial spraying of insecticides to control the mosquito population. An agricultural organization supports the plan because mosquitoes cause crop damage. An environmental group opposes the plan because of possible food contamination and other medical risks. Here are some facts about the case: • A map of Jackson County is shown here. • All county boundaries are on a S

north– south line or an east–west line. • The estimated annual cost of aerial

spraying is $29 per acre. • There are 640 acres in 1 square mile. • Plan supporters cite a study stating that for every $1 spent on insecticides, farmers would gain

$4 through increased agricultural production.

a. What is the area of Jackson County in square miles? In acres? b. What would be the annual cost to spray the whole county? c. According to plan supporters, how much money would the farmers gain from the spraying

program?

County Concerns2. The sheriff of Adams County and the sheriff of Monroe

County are having an argument. They each believe that their own county is larger than the other county.

Who is right? Write an explanation that would settle the argument.

Understanding a Concept

• Explain it to someone else• Represent it in multiple ways• Apply it to solve simple and complex problems• Reverse givens and unknowns

– E.g., problem posing: Create a situation that can be modeled by 6 ÷1/2 = ?

• Compare and contrast it to other concepts

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the

reasoning of others.4. Model with mathematics. 5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated

reasoning.

SMP 2: Reason abstractly and quantitatively

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: • the ability to decontextualize—to abstract a given situation and represent it

symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and

• the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.

Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.






• Coherence within and across grades• Clarity, accuracy

CCSS Grades 6 & 7Ratios & Proportional Relationships

• Emphasize understanding unit rates associated with ratios.

• Expect students to represent proportional relationships by tables, equations, and graphs, and understand informally that that unit rate indicates the steepness of the graph of the line (informal introduction to slope).

• Expect students to solve problems involving proportional relationships using various methods, such as equivalent ratios and unit rates.

If 2 pounds of beans cost $5, how much will 15 pounds of beans cost?

CCSS Ratio & Proportional Relationships Progression, 9/2011

CCSS de‐emphasizes means/extremes as solution method

2 155 x

2x = 5•15

=

If 2 pounds of beans cost $5, how much will 15 pounds of beans cost?

Coherence

• To what extent is this emphasis on scale factors and unit rates used within grades and across grades?

• To what extent are the ideas of scale factors and unit rate used as the foundation for the development of subsequent mathematical ideas?

• To what extend is a foundation for understanding multiplicative relationships established in previous grades?

CCSS RP Progression, 9/2011

Common Multiplication and Division Situations






• Coherence within and across grades• Clarity, accuracy

Research‐based Principles for Instructional Design

• Engage learners in challenging tasks, with supportive guidance and feedback;

• Use multiple and varied representations, with explicit connections among representations

• Encourage elaboration, questioning and self‐explanation• Access prior knowledge and address students’

misconceptions• Use formative assessment• Prime student motivation by promoting effort‐based views

of learning• Provide opportunities for on‐going review and practice

Which problems are most accessible for beginning algebra students?

U.S. Shirts charges $12 per shirt plus $10 set‐up charge for custom printing.

1. What is the total cost of an order for 3 shirts?



4. A customer spends $70 on T‐shirts. How many shirts did the customer buy?

y = 12x + 101. Solve for y when x = 3, 10, 100.2. Solve 70 = 12x + 10

Introducing Equivalent Algebraic Expressions: Pattern Tasks

• Compute the perimeter for the first four trains.

• Determine the perimeter for the tenth train without constructing it.

• Write a description /expression that could be used to compute the perimeter of any train in the pattern.

• Find as many different ways as you can to represent the perimeter of any train.

Train 1 Train 2 Train 3

Introducing Equivalent Algebraic Expressions: Pattern Tasks

• Explain what each student was thinking to find the perimeter of the nth train.

• Connect your explanation to the picture of the tables.Terri: 1 + 4n + 1 Tim: 1 + 2(2n) + 1Jerry: 5 + 4(n – 2) + 5Linda: 6n – 2(n – 1)

Ongoing Review and Practice

Providing students with periodic opportunities to practice using concepts and skills, along with

feedback about their performance, helps students solidify their knowledge and promotes retention, reflection, generalization, and transfer

of knowledge and skill.

IES Practice Guide, 2007

Ongoing Review and Practice

Students’ Beliefs about Their Intelligence Affect Their Academic Achievement

• Fixed mindset: – Avoid learning situations if they might make mistakes– Try to hide, rather than fix, mistakes or deficiencies– Decrease effort when confronted with challenge

• Growth mindset:– Work to correct mistakes and deficiencies

– View effort as positive; increase effort when challenged

Dweck, 2007

When confronted with challenging school transitions or courses, students with

growth mindsets outperform those with fixed mindsets, even when they enter with

equal skills and knowledge.

Dweck, 2007

Students’ Beliefs about Their Intelligence Affect Their Academic Achievement

Students Can Develop Growth Mindsets

• Teacher praise influences mindsets– Fixed: Praise refers to intelligence– Growth: Praise refers to effort, engagement, perseverance

• Explicit instruction about the brain, its function, and that intellectual development is the result of effort and learning has increased students’ achievement in middle school mathematics.

• Reading stories of struggle by successful individuals can promote a growth mindset

Research‐based Principles for Instructional Design

• Engage learners in challenging tasks, with supportive guidance and feedback;

• Use multiple and varied representations, with explicit connections among representations

• Encourage elaboration, questioning and self‐explanation• Access prior knowledge and address students’

misconceptions• Use formative assessment• Prime student motivation by promoting effort‐based views

of learning• Provide opportunities for on‐going review and practice

What About Technology?

An excellent mathematics program integrates the use of mathematical tools and technology as essential resources to help students learn and make sense of

mathematical ideas, reason mathematically, and communicate their

mathematical thinking. NCTM Principals to Actions, 2014

Prevent Common Misconceptions

Critical Features for Textbook Creation/Selection

• Mathematics Content• Research‐based principles for instructional design

• Other supports: Assessments, resources for differentiation, teacher supports (lesson implementation; professional learning), additional resources

Development Process• Collaborative • Iterative

– Research– Write– Feedback: Reviewers, teachers, observation– Revise

• Refinement rather than rewriting• Formal evaluation

More opportunities for international discussions of design

of effective textbooks and instructional materials would

further increase students learning and teachers’ effectiveness.

Thank [email protected]

creating and selecting mathematics that support effective ......creating and selecting mathematics...

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