MATH 2303/2304/3305/3308 Workshop Aug 20, 2013 Bell Hall 130A, UTEP.

MATH 2303/2304/3305/3308

Workshop

Aug 20, 2013Bell Hall 130A, UTEP

Let’s Get to Know Each Other!Introduction

UTEP Math Sciences• Art Duval• Gregory Allison• Jeremy Rameriz• Kien Lim• Vodene Schultz

UTEP Teacher Education• Joyce Cashman• Song An

EPCC Mathematics• Eduardo Urquidi• Fernando Falcon• Ruben

Carrizales• Ruth Ordaz

Pearson Publisher• Diana Baniak

Agenda (Proposed)

9:00 am Breakfast

9:30 am Introduction & Challenges

9:40 am Effective Use of Textbook

11:15 am My Math Lab & Online Resources

12:30 pm Lunch

1:30 pm Timeline

2:00 pm Assessment & Collaboration

3:30 pm End

Agenda (Actual)

9:00 am Breakfast

9:35 am Introduction & Challenges

10:15 am Core Ideas in Section 6.4

11:15 am My Math Lab & Online Resources

12:20 pm Lunch

1:00 pm Question 4 on Activity 6M

2:55 pm Objectives of Activity 6NResources for Section 6.4

3:35 pm End

Objectives

Enhance our pre-service teachers’ conceptual understanding and mathematical thinking

• Get to know each other

• Know more about textbook and its resources

• Share ideas (e.g., how to use textbook optimally; what and how to assess)

• Collaborate and share resources

Introduction

What Challenges Do We Face as Instructors of These Courses?

Discuss!

What Challenges Do We Face as Instructors of These Courses?

• Math anxiety• Lack of content mastery• Procedure oriented• Don’t see the point of learning the content• Don’t want to know the conceptual underpinnings• Pass without understanding• How do they get so far?• Last math course was taken many years ago (no

math course in senior year)• Too much emphasis on testing. No time for deep

learning.• Non-thinkers• Irrelevant

One Big Challenge!

“Doing mathematics means following rules laid down by the teacher,

knowing mathematics means remembering and applying the correct rule when the teacher asks a question, and

mathematical truth is determined when the answer is ratified by the teacher.” (Lampert, 1990, p. 31)

Existing Beliefs

Pedagogical Content

Knowledge

Common Content

Knowledge (CCK)

Specialized Content Knowledge (SCK)

Knowledge of

Content and Students

(KCS)

Knowledge of Content

and Teaching

(KCT)

Subject Matter

Knowledge

Knowledge at the

mathematical horizon

Knowledge of

curriculum

Mathematical Knowledge for Teaching

Notes about the Beckmann Text

• Explaining Why

• Core Concepts

• CCSSM Standards for Mathematical Practices

• Arithmetic Operations

• Visual Representations

• In-class Activities

• Practice Exercises vs. Problems

• Chapter Summaries

• IMAP Videos

Standards for Mathematical Practice in Common Core State Standards in Mathematics (CCSSI, 2010, p. 6-8)

• Make sense of problems and persevere in solving them

• Reason abstractly and quantitatively

• Construct viable arguments and critique the reasoning of others

• Model with mathematics

• Use appropriate tools strategically

• Attend to precision

• Look for and make use of structure

• Look for and express regularity in repeated reasoning

One More Thing about Teaching

• Teaching for Conceptual Understanding

• Teaching for Mathematical Thinking• Teaching with Grace

The Lesson of Grace in Teaching by Francis Su

Effective Use of Textbook

Discuss one section of the text (Section 6.4 for M1351/2303/3305/3308)

• Core ideas

• In-class Activities

• Homework for students (problems to assigned, pages to read, etc.)

• Resources

Core Ideas for Section 6.4Discussion

1. What important math ideas can students learn from Sec6.4?

• Two conceptualization of fraction division: repeated-subtraction model in 6.4 & sharing equally in 6.5

• Different techniques for dividing fractions:o Common denominator (divide numerator)o Divide across (divide numerator, divide

denominator)o Invert-multiply

• Division as inverse multiplication

• Common denominator strategy is meant for the “how many groups” perspective (it’s easier to see how many times the divisor fits into the dividend if we have common unit fractions; e.g., 2 one-sixths goes into 3 one-sixths 1½ times)

Core Ideas for Section 6.4Discussion

2. What habits of mind can we foster?

• To solve the problem we first need to manipulate (e.g., make the denominators the same)

• See the structural similarity with division involving whole numbers (see Fig. 6.13 on page 250 in Beckmann)

• Have a sense of magnitude (on a number line), and not just numbers/symbols (i.e., foster number sense)

• Thinking reversibility (inverse operations)3. What ideas can we learn from analyzing Section 6.4?• Making students uncomfortable is necessary

• Experiencing the need for common denominator

MyMathLab & Online ResourcesWe went over the following:

• Signing in at http://www.mymathlab.com/

• Creating and downloading “Student Registration Handout” for students to enroll in your course (the pdf contains your specific Course ID)

• Creating a course, creating homework, managing gradebook

• Tools for Success (e.g., e-manipulatives, IMAP videos, downloads, handouts, internet resources for each chapter)

• Practice/Homework/Test items in mymathlab tend to be procedure-oriented whereas problems in the text tend to be conceptual

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LUNCH

Work on Item 4 in Section 6.4Individual Work & Group Work

Why is this question challenging?(we skipped the part on writing a word problem)

Group Presentations

• The divisor is a fraction

• The divisor is larger than the dividend (“how many group” view doesn’t really work because there is not even 1 group)

What solutions were presented?

Group Presentations

• Common denominator strategy: 1/3 ÷ 3/4 = 4/12 ÷ 9/12 = 4/9It was conceptually difficult to explain why answer is 4/9, even with the support of a diagram.

• Invert-multiply strategy: 1/3 ÷ 3/4 = 1/3 × 4/3 = 4/9This strategy relies on a standard procedure without really explaining the meaning of “1/3 ÷ 3/4” or how many ¾ are in 1/3.

What solutions were presented? (con’t)

Group Presentations

• Make the divisor a whole number and use sharing-equally model: 1/3 ÷ 3/4 = 4/3 ÷ 3 = 4/9This avoids the meaning of 1/3 ÷ 3/4. • Proportional reasoning:

3/4 cups make 1 batch1/3 cups make x BatchScaling both quantities by a factor of 4/9.This strategy transforms a fraction division problem into a missing-value problem involving a proportion.

What solutions were presented? (con’t)

Group Presentations

• Division as inverse-multiplication: 1/3 ÷ 3/4 = A A x 3/4 = 1/3This view of division is most suitable for division fraction where the divisor is greater than the dividend because what the question is really asking is “what fraction of ¾ are in 1/3” instead of “how many times of ¾ fit into 1/3”.

How could we synthesize all these solutions?

(i.e., how can a teacher provide closure)One strategy is to ask students to share what they have learned; the teacher records their ideas, elaborating or providing additional ideas as needed.

• Multiple ways to solve the problemo Using the invert-multiply procedureo Changing it to a missing-factor problem in a

multiplication o Changing it to a proportion-type problem

(may use double number line or a table)

o Common denominatoro Making the divisor a whole numbero Making both divisor and dividend whole numbers:

a/b ÷c/d = ad ÷ bc

How could we synthesize all these solutions?

(i.e., how can a teacher provide closure)Continue to elicit from students/participants:

• “How many groups” view is hard when the divisor is greater than dividend.

• Extending basic concept (“how many groups” view like 8 ÷ 2) into a non-intuitive situation (like 1/3 ÷ 3/4).

Class Activities 6M & 6N (Skipped)

1. What do you think are the learning objectives for 6M?

2. What do you think are the learning objectives for 6N?

3. Which activities should we use? Why?a. 6Mb. 6Nc. Both 6M and 6Nd. Neither 6M and 6N

What is the purpose of activities?Activity 6N

• Q4 is to show student the equivalence between the invert-multiply form, 5/7 × 3/4, and the division form 5/7 ÷ 3/4.

• Q2 is to explain why we can divide across by interpreting the division problem as a missing-factor problem.

Activity 6M

• Q2 is to highlight the importance of attending to the referent. 1 ÷ 2/3 = 1 unit of 2/3 (divisor) with a remainder of 1/6 of the unit of 1 (dividend).

Class Activities 6M & 6NQuestions related to Specialized Content Knowledge

a. Why is it more difficult for students to model 7 ¾ than to model 3/4 ?

b. Why is the dividing-across procedure valid, i.e., 6/20 3/4 = (6 3)/(20

Homework for Section 6.4 (Skipped)

1. Which pages should we assign students to read?

2. Which items should we assign?

a. Practice Exercises for Section 6.4

b. Problems for Section 6.4

3. What else should/can we assign for homework to substantiate student conceptual understanding and/or mathematical thinking?

Resources for Section 6.4What resources can we use to enhance student learning of the materials in Section 6.4?• IMAP Videos – Students’ Reasoning with

Fractionso Elliot: Explaining his reasoning for 1½ ÷ 1/3

http://media.pearsoncmg.com/cmg/pmmg_mml_shared/flash_video_player/player.html?video=aw/IMAP/video/0321

o Trina: Learning the invert-multiply procedure http://media.pearsoncmg.com/cmg/pmmg_mml_shared/flash_video_player/player.html?video=aw/IMAP/video/0380

• TIMSS Videos – Japanese Lesson Studyo After learning dividing across 9/20 ÷ 3/5 = (9÷3)/(20÷5),

students struggled with 9/30 ÷3/7http://hrd.apec.org/index.php/Lesson_Study_Video_for_Let%27s_Think_About_How_To_Multiply_and_Divide_Fractions:_Students%27_Presentation_%282_of_5%29

• Online Manipulatives

Alignment with TExES Competencies

• We can administer a pre-test with TExES items so that students get a sense of TExES and start preparing for it by taking math courses with the goal of doing well in the TExES exam.

• The pre-tests were given out. They can also be downloaded from our wiki at http://t-t-t.wikispaces.com/Resources-2303-1350

.

• This PowerPoint presentation is posted at http://t-t-t.wikispaces.com/Workshop+Materials.

THE END