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Mastering the Most Challenging Math Standards with Rigorous Instruction

Host:

Mark W. Ellis, Ph.D., NBCT

Professor, College of Education

California State University, Fullerton

Moderator:

Danielle Curran

Executive Math Editor, Curriculum Associates


New Standards for Mathematics


• Understand the concepts behind the calculations




• Flexibility to solve non-routine problems




• Flexibility to solve non-routine problems

• Skill in reasoning and communicating mathematically



Beliefs


Beliefs

• Math makes sense. (And if it does not, ASK!)


Beliefs


• All students can learn to make sense of math.


Beliefs



• Learning should be coherent.


Beliefs



• Learning should be coherent.

• Students learn best through active engagement.

– thinking, reasoning, connecting, communicating, critiquing, revising, and extending knowledge


Guiding Question for Teachers How does my instruction support all students in

developing mathematical authority?


Structure & Sequence for Session



• Discuss most challenging topics by getting at the concepts underpinning them

• Brief Q & A after each section



• Discuss most challenging topics by getting at the concepts underpinning them

• Brief Q & A after each section

Topics

1. Measurement

2. Model with Mathematics

3. Fractions

4. Statistics


Topic: Measurement


Early Concepts & Skills

Conservation

Transitivity

Equal Partitioning

Unit Iteration

Origin

Identify what to measure

Choose unit of measure

Determine number of units

Topic: Measurement


Early Concepts

Conservation

Transitivity

Equal Partitioning

Unit Iteration

Origin

understanding that an object’s length does not change when moved understanding that the measures of two objects can be compared to a third object understanding that an object can be sliced into segments of the same size or unit without changing its overall length understanding that measurement requires using identical copies of a standard length understanding that a measurement of length is a distance from a fixed starting point

Topic: Measurement


The Issue: Skills Without Understanding

From Kamii , C. (2006). How Can We Teach Measurement Better? Teaching Children Mathematics.

Topic: Measurement

http://www.nctm.org/Publications/teaching-children-mathematics/2006/Vol13/Issue3/Measurement-of-Length_-How-Can-We-Teach-It-Better_/


Transitivity

Which is longer, the blue segment or the red segment? How can you be sure?

• What strategies did you use?

From Kamii , C. (2006). How Can We Teach Measurement Better? Teaching Children Mathematics.

Topic: Measurement

http://www.nctm.org/Publications/teaching-children-mathematics/2006/Vol13/Issue3/Measurement-of-Length_-How-Can-We-Teach-It-Better_/


Unit and Unit Iteration

• What is a unit of measure?

– Begin with informal (one’s foot; paperclip)

– Literally place units end-to-end

– “With just 1 copy of a unit, how can I measure?”

• Why standard units of measure?

– Use children’s literature such as How Big is a Foot?

Topic: Measurement

https://aofradkin.wordpress.com/2013/09/11/how-long-is-a-foot/






Origin and Equal Partitioning • What does it mean to “start measuring” at

some point on a ruler?

• How can the same object have two “different” lengths?

• Tasks that spark thinking & discussion – Using just one measuring tool, show 3 ways to

find the length of one segment.

– Measure an object longer than the ruler.

– Estimate lengths (including deciding appropriate units) of objects in photos/videos. Check out Estimation 180.

Topic: Measurement

http://www.estimation180.com/

http://www.estimation180.com/


Measurement: Early Concepts & Skills Grade 1 Grade 2 Grade 3

• Order Objects by Length (1.MD.A.1)

• Compare Lengths (1.MD.A.1)

• Understand Length Measurement (1 MD.A.2)

• Understand Length and Measurement Tools (2.MD.A.1)

• Measure Length (2.MD.A.1) • Understand Measurement

with Different Units (2.MD.A.2)

• Understand How to Estimate with Length (2.MD.A.3)

• Compare Lengths (2.MD.A.4)

• Add and Subtract Lengths (2.MD.B.5)

• Represent Numbers as Lengths on a Number Line (2.MD.B.6)

• Measure Length and Plot Data on Line Plots (3.MD.B.4)

Topic: Measurement


Learning Trajectory • Linear measure (length, perimeter)

Two-dimensional measure (area) (3.MD.C.7b, 3.MD.D.8, 4.MD.A.3, 6.G.A.1)

• More complex problems (compound figures and surface area) (6.G.A.1, 6.G.A.3, 6.G.A.4)

• Line graphs and coordinate grids (6.G.A.1, 6.G.A.3, 6.G.A.4)

– Scaling axes is a form of unitizing

• Transformations and congruence (8.G.A.2)

Topic: Measurement


Q&A on Measurement


Topic: Model with Mathematics


• A mathematical model is a representation of some object, context, or situation.

• “Word problems” should be central to student learning, not peripheral.

• Context matters – draw on students’ interests and experiences.

• Provide strategies that help students create meaningful representations.



Processes of Modeling

• Identify

• Represent

• Mathematize

• Reality Check

• Revise



Supporting Student Modeling in Math

• Provide multiple ways to represent relationships.

– Bar diagrams

– Number lines

– Table of values

• Ask students to write math expressions or relationships based on the representation.

• Require students to explain how their representation and their math expression relates to the context/problem.




From Ready Mathematics Instruction, Grade 4



Freeway Exits Problem


http://robertkaplinsky.com/work/freeway-exits/


Freeway Exits Problem - Modeling


http://exit10a.blogspot.com/2014/01/you-from-jersey-what-exit.html


Middle School

• Which sale offers the best deal? Prove it!

• Think about how this might be represented visually and mathematically.

Related standard(s): 6.RP.A.3, 7.RP.A.3


https://www.illustrativemathematics.org/content-standards/7/RP/A/tasks/114


Two for the price of one Buy one get 25% off the second

Three for the price of two Buy two and get 50% off the second one

Middle School Related standard(s): 6.RP.A.3, 7.RP.A.3


https://www.illustrativemathematics.org/content-standards/7/RP/A/tasks/114


Learning Trajectory

Modeling tasks become less well-defined and mathematical representations move from arithmetic to algebraic, but process remains same.

• Identify

• Represent

• Mathematize

• Reality Check

• Revise



Q&A on Model with Math


Fractions


• Fractions are numbers.

• Fraction operations are logical, not magical.

• Illustrative Mathematics offers a set of short videos aimed at teachers (and other adults) that illustrate how fraction knowledge builds coherently across grade levels.

Fractions

https://www.illustrativemathematics.org/fractions_progression





What are U.S. Students Learning?

3.NF.A.3d, 4.NF.A.2, , 6.NS.C.7

4.NF.C.6, 5.NBT.A.1, 5.NBT.A.3

Topic: Fractions


Examining Student Work

• What does this tell us about the student’s understanding?

• What are some ways to address the misunderstanding?

Standard Reference(s): 5.NF.A.1, 5.NF.A.2

Topic: Fractions


KEY TO SUCCESS—Unitizing (Remember Measurement?)

• Recognize “units” as groups or sets.

• “Whatever can be counted can be added.” Daro,

Mosher, & Cocoran, Learning Trajectories in Mathematics

• Build on idea of units from whole numbers to make

sense of fractions as numbers.

• Build on representations of whole numbers so

fractions are seen as part of a continuum.

Topic: Fractions

http://www.cpre.org/images/stories/cpre_pdfs/learning trajectories in math_ccii report.pdf




Fraction Representations

• Grades 1-2: halves, fourths, and thirds as terms related to geometric shapes

• Grade 3: fractions on number line and with visual models

• Grades 4-6: fraction operations on number line and with visual models to support sense-making and connection to procedures

thirds not thirds

halves

1.G.A.3 2.G.A.2 2.G.A.3

3.G.A.2

Topic: Fractions


Formative Assessment: Error Analysis Task

Topic: Fractions


Connect Fraction and Whole Number Addition

Related standard(s): 3.NF.A.2, 4.NF.B.3a, 4.NF.B.3b

Topic: Fractions

From Ready Mathematics Instruction, Grade 4.


Understanding Fraction Operations

Related standard(s): 3.NF.A.1, 3.NF.A.3, 4.NF.B.4b, 5.NF.A.1

Topic: Fractions


Reasoning about Fraction Division

1. What question might this expression represent?

2. Represent this in a way that makes sense.

• What justifies your method?

3

12

Related standard(s): 5.NF.B.7b, 5.NF.B.7c

Topic: Fractions


Measurement Method: How Many 1/3s are in 2?

63

12

Topic: Fractions


Reasoning about Fraction Division

1. Represent this in a way that makes sense.

2. How is this related mathematically to the previous problem?

• What justifies your method?

3

22

Related standard(s): 5.NF.B.7b, 5.NF.B.7c

Topic: Fractions


Measurement Method: How Many 2/3s are in 2?

33

22

Topic: Fractions


Learning Trajectory

• Unitizing underpins arithmetic operations from whole numbers through to algebra.

(From Daro, Moscher, & Corcoran, p. 42)

Topic: Fractions






Q&A on Fractions


Statistics


Statistics

Imagine a world without variability…

• For example, every 6th grader is 150 cm tall.

– Find the mean, median, mode, and range for the height of a group of 50 sixth-graders.

Without variability, there is no statistics.


Getting to Know Variability ( Related Standard(s): 6.SP.A.1)


Topic: Statistics


Struggle with Statistics: Skills without Concepts

PARCC Grade 6 practice test

Topic: Statistics

http://parcc.pearson.com/resources/Practice_Tests/Grade_6/Math/PC194816-001_6MTHTB_PT.pdf

http://parcc.pearson.com/resources/Practice_Tests/Grade_6/Math/PC194816-001_6MTHTB_PT.pdf


Brooding about Measures of Center

The data below represent the number

of ducklings in several broods in a

neighborhood park.

6, 2, 3, 3, 10, 4

1. Create two different sets of 6 broods with the same

median as the original but a different mean.

2. Create two different sets of 6 broods with the same

mean as the original but a different median.

Related Standard(s): 6.SP.A.2, 6.SP.A.3, 6.SP.B.5, 7.SP.A.1

Topic: Statistics


Characteristics of Mean and Median

3. Write 1 paragraph describing why some

changes in a data set change the mean and

others do not.

4. Write 1 paragraph describing why some

changes in a data set change the median and

others do not.


Topic: Statistics


Application to Real World

From Smith (2011), “Americans and Text Messaging”


Topic: Statistics

http://www.pewinternet.org/files/old-media/Files/Reports/2011/Americans and Text Messaging.pdf

http://www.pewinternet.org/files/old-media/Files/Reports/2011/Americans and Text Messaging.pdf


Learning Trajectory: From Data to Distribution

From Bakker & Gravemeijer, 2004

Topic: Statistics

http://www.researchgate.net/profile/Arthur_Bakker/publication/233864587_Learning_to_reason_about_distribution/links/09e4150ddea80b434c000000.pdf

http://www.researchgate.net/profile/Arthur_Bakker/publication/233864587_Learning_to_reason_about_distribution/links/09e4150ddea80b434c000000.pdf


Q&A on Statistics


Final Q&A

Learn more about the program that helps you master the most challenging math standards:

www.CurriculumAssociates.com/ReadyMath

THANK YOU!

mastering the most challenging math standards with ... · mastering the most challenging math...

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