Differentiating Instruction in
the Mathematics Classroom.

         Presented by
        Tr. Terry Walsh




        1    ...
Monday
Introduction




2
Real World Connections

                         Who
                          is
   P. 190                this
          ...
Who is this guy?

This is who I am . . .
                 by the numbers.
          Terry Walsh:
      35, 3, 54, 50, 4


...
Tr. Terry Walsh
  By the numbers . . . (one # is used twice)
I was born on December 22, 19 __.
My draft number was only _ ...
Three comments about school:
• Don’t Work Harder Than Your Students.”

     -- title of a book published in 2009

• “Ameri...
Meet your neighbor by the numbers…

 • Select 5 numbers that are meaningful to you
   that will help someone understand wh...
Group and Label
• Write each of your numbers on a post it. One
  number per post it.
• Place all of your numbers from your...
Group and Label


          Isosceles               Sphere
           Cylinder               Scalene
           Square    ...
Group and label




 Page
  149




        10    Cleveland Urban Conference
Group and Label




 Page
  152




 11
Group and Label




 Page
  153




 12
Group and Label




 Page
  154




 13
Group and Label




 Page
  155




 14
Terry Walsh

  By the fractions . . .
I have been married for nearly ____of my life.
_______ of my children are male.

___...
Thoughtful Questions:
♦ Why do some students succeed in mathematics
  and others do not? Is it a matter of skill or will?
...
Workshop Assumptions:
• What teachers do and the instructional decisions that
  they make have a significant impact on wha...
Learning Goals:
Participants will learn:
• The characteristics of the four basic mathematical learning
  styles (Mastery, ...
Now…..
What are YOUR personal Learning Goals for this
 workshop?

 Review the Thoughtful Questions, Basic
  Assumptions a...
Critical Vocabulary:
• Look at Page 10 in your
  handout.
• Fill in one number in each row,
  and find your total score fo...
What’s Your Favorite?...



           Read the four teaching activities
            on page 11 of your handout,
 Page    ...
4 P’s: Previewing Before Reading

Preview: Scan the entire text.     Find out as much as you
  can about what you are goin...
Previewing Worksheet

PREVIEW the workshop materials. List four things you
  learned from your preview.

Make two PREDICAT...
Mastery Math Students
   Want to                    Learn practical information and set
                               pr...
Understanding Math Students
   Want to                       Understand why the math they learn works

   Like math prob...
Self-Expressive Math Students
   Want to                      Use their imagination to explore
                          ...
Interpersonal Math Students
   Want to                     Learn math through dialogue, collaboration,
                  ...
A “Paradox”….
A little about two doctors (PhD’s) you should know
  about…..

Carl Jung



Dr. Harvey Silver




          ...
Robert Sternberg, IBM Prof. of Psychology and Education,
                   Yale University.
                 Learning Sty...
Sternberg and his colleagues drew 2 conclusions


        First, whenever students were taught
          in a way that mat...
Page
 8




       ASCD   31
A Mathematical Task Rotation…
2. Write down a “significant year” in your life. Describe it to
   your neighbor using as ma...
ST Mastery Learner:

   Thinking Goal:   REMEMBERING

   Environment:     CLARITY & CONSISTENCY
   Motivation:      SUC...
NT Understanding Learner:

   Thinking Goal: REASONING
   Environment:   CRITICAL THINKING
                     AND CHAL...
NF Self-Expressive Learner:


   Thinking Goal:   REORGANIZING

   Environment:     COLORFUL AND CHOICE

   Motivation:...
SF Interpersonal Learner:
   Thinking Goal:   RELATE PERSONALLY

   Environment:     COOPERATIVE AND CONVERSATION


   ...
What’s Wrong? vs
  Who’s Right?
          Both ask students to find and
        correct errors. Who’s Right (SF)
P 38     ...
What’s wrong with the following
          problem?

32 + 3(2x – 12)   >   5 – (4 + 9x)
   32 + 6x – 36   >   5 – 4 – 9x
  ...
Justify/Explain

» Write one or more valid
  reasons why the man with
  the full cart is not wrong in
  being in the lane ...
Real World
Connections...
             Write ways that
              numbers are
                 used to
P. 190
         ...
What if? ...
           What if the population of the
        United States kept increasing at
         the same percentag...
Cleveland Urban Conference   43
Teaching with Style to Sensing/Thinking Mastery Learners

Guidelines                                Examples
             ...
Teaching with Style to Intuitive/Thinking Understanding Learners

 Guidelines                                  Examples
  ...
Teaching with Style to Sensing/Feeling Interpersonal Learners

Guidelines                                   Examples
     ...
Teaching with Style to Intuitive/Feeling Self-Expressive Learners

 Guidelines                                Examples
   ...
Students work with
        Questions in all four Styles:
 Mastery questions ask         Interpersonal questions
what stude...
Back in My Classroom
 After learning about our learners, what does
  this mean for us as math teachers?
 What questions ...
How much of students’ success in your math classes is
 due to their understanding what they read or write?




           ...
The Four Functions of Style

                                         SENSING
                                          Ph...
Tuesday
       Session 1
Writing & Reading
        in Math




   52
M&M’s (Math Metaphor, p.129)
My favorite math teacher always used to
 say that fractions are like politicians. At
 first I...
Give One, Get One
                    DIRECTIONS

Stand up, partner with one other person, GIVE one of
  yours, GET one of...
Write to Learn
•   The more students write and think in
    mathematics classes, the more they learn.
    Doug Reeves repo...
When SHOULD students write in
       mathematics?
1.       At the beginning of the lesson.
     •     Access prior knowled...
How much writing do your students
do in your mathematics class?


  None   Very Little   Some   Considerable   A Great Dea...
What kinds of writing do you
want your students to do in
your math classes?

Make a list....



                          ...
My Writing List          (should we add any to your lists?)
     Answers in “proper form” (whatever that is!!)
     Show...
WRITE TO LEARN
                         Provisional: Generate ideas, fluency & flexibility.
                          Aud...
M ake a comparison or justify a decision
A ccess prior knowledge
Think About Learning or Feelings
H ypothesize
E xplain or...
Creative writing can take
       many forms.


Before you use mathematical vocabulary, it
might help to use non-math terms...
Next we will be some creative writing using these terms.

Here is an example of what I mean by “creative” writing:
 The pe...
How many words did you use?

Here is my one very run on sentence
using all 12 words:

The three people formed a right tria...
Thinking about Learning
       or Feelings....


Would you rather have a best friend whose
views are congruent to yours, o...
Support or Refute (P. 69)
          Word Problem
   You will have a short time to skim over
    the word problem in the n...
An Atypical Word Problem
   A truck is on its way to three different motorcycle
    dealerships. The truck contains both ...
Support or Refute: Directions & Questions
Directions:
 Write down whether you think each question is true or
   false.
 ...
An Atypical Word Problem
   A truck is on its way to three different motorcycle
    dealerships. The truck contains both ...
Support or Refute (P. 69 in the Math Tools book)
A Geometry Support or Refute
Agree or Disagree with each of these. Then READ Section 5.4 and
    find statements or ideas ...
An Alg2/Trig Support or Refute
                                .
Alg2/Trig Read section 4.4. First write whether you AGREE...
Reading Questions in Styles
Mastery Questions: Read the actual lines
             finding facts, details, or literal meani...
Note taking vs. Note making
                 not in your handout



Students who are taking notes are usually copying
what...
Note makingExample

          32+3(2x – 12) > 5–(4+9x)
What did I do?                Explain why or ask a ?

           32...
To use this idea, turn off auto format for spelling!!
I cdnuolt blveiee that I cluod aulactly uesdnatnrd
what I was rdanie...
The % and decimal below do not obey the “rules”.

Prbabltiioy can hlep us mkae dcsioines. Wtrei the fowlilong
pgaaaprrh cc...
Cleveland Urban Conference   78
Mathematical
Summaries...
                  Write out how
                   we add two
                    fractions
P. 2...
Math Recipe vs
Anchor Walls     ...         (not in book)

               Create a “recipe
               card” or a fill ...
Anchor Walls

In order to add two fractions, the first
thing we do is make sure the
________________. If they are not,
you...
Making Up Is Fun to Do ...
                   My dad’s sister,
                  Sally, used to ask
                    me...
Tuesday
       Session 2
Vocabulary & Assessment
       Strategies




    83
Fist Lists & Spiders
                 Look at P. 32 of your
                     handout….
                Use the spider ...
3 Way Tie
      Write “fractions”, “percents”,
      and “decimals” at the three
      vertices of the triangle in
      y...
Asessment Menus
             Look at the Conics
             Assessment Menu...
             (p. 34 of the handout).
P. 23...
Tic-Tac-Toe (Vocab. Games)

    (P. 35 in handout)
                   The example was
                   written by a HS
 ...
Task Rotations are a way to use all four styles
     in a single strategy. Task Rotations
  are found in pages 222 to 238 ...
Task Rotations
             The Calculus Task
             Rotation (p.36 in
             handout) shows
P. 222       Styl...
Range Finder
             The “Graduated
             Difficulty” example
             (P.37 in handout),
P. 208       sho...
Convergence Mastery
               For when they
               absolutely, positively
               need to know a
   P....
Unit Tests
             Test Worth Taking is
             a test that poses
             questions in all four
P. 244     ...
What are words and how are
they defined?




What words are important to learn?
Key Word Strategy =
               Dictionary Definition
Bicycle                                  (the key word)

A mode o...
Take the word ___________
                               trapezoid


Non-Examples            General Category
            ...
Take the word mathematics

Non-Examples          General Category
                                                   Examp...
Take the word___________

Non-Examples           General Category
                                                  Exampl...
These are polygons:




   These are NOT polygons:




What makes a polygon a polygon?
     List critical attributes.

   ...
What is Mathematical Literacy?

                                                          1
                              ...
What is Mathematical Literacy?
    Literacy in reading means not only
0011 0010 1010 1101 0001 0100 1011
      being able ...
Mathematical Literacy
0011 0010 1010 1101 0001 0100 1011
     The importance of mathematical
       literacy and the need ...
Jobs requiring mathematical and technical
          skills are growing the fastest among the
0011 0010 eight professional...
What is Mathematical Literacy?
0011 0010 1010 1101 0001 0100 1011
     Mastery of procedural               A language to
 ...
Four Reasons to Teach
Vocabular y:

   • Verbal Intelligence

   • Ability to comprehend new
     information: Academic Ac...
Tuesday
      Session 3
Your ideas & closure
      (for now?)




  105
Page
  8
        How Do I Select the
       Right Tool For the Right
         Learning Situation?




           106
Five Ways to Use Math Tools

      See pages 13 through 15 in the
        Math Tools book:
      • Try one out.
 Page
    ...
How Do I Select the Right Tool For
the Right Learning Situation?


       Use the matrices on pp. 18/19;
          64/65; ...
109
1/23-24/06   ASCD   110
How many of us would
                   like our students to…
•   Think more deeply?
•   Take more intellectual risks?
•  ...
Research clearly indicates the impact of each of
          these on student learning:

                         Category  ...
Mathematics
 Workshop:
Four Thought




               ASCD   113
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Urban Conference Walsh Presentation

  1. 1. Differentiating Instruction in the Mathematics Classroom. Presented by Tr. Terry Walsh 1 ASCD
  2. 2. Monday Introduction 2
  3. 3. Real World Connections Who is P. 190 this guy? Cleveland Urban Conference 3
  4. 4. Who is this guy? This is who I am . . . by the numbers. Terry Walsh: 35, 3, 54, 50, 4 Cleveland Urban Conference 4
  5. 5. Tr. Terry Walsh By the numbers . . . (one # is used twice) I was born on December 22, 19 __. My draft number was only _ _, but I did not serve in the military. I have been married for ____ years, and have____children. I enjoyed adapting my teaching strategies to include all ___ learning styles. I retired ___ years ago, after 32 years in two suburban Chicago high schools. 3 4 35 50 54 Cleveland Urban Conference 5
  6. 6. Three comments about school: • Don’t Work Harder Than Your Students.” -- title of a book published in 2009 • “American High Schools are a place where 1500 students go to watch 150 adults work really hard.” --- a Japanese teacher in the late 1970’s, after visiting a several Ohio high schools. • “Teachers never ask “Why?” if your answer is correct.” -- a student in a math class at Niles West H.S. (Illinois); May, 1972 6
  7. 7. Meet your neighbor by the numbers… • Select 5 numbers that are meaningful to you that will help someone understand who you are. • Then write a sentence or question for each number, leaving a blank line where the number should go. • Share you numbers and sentences with your neighbor. See if he or she can match the correct number to the line. For every correct answer you get a point. See who gets more 7 points.
  8. 8. Group and Label • Write each of your numbers on a post it. One number per post it. • Place all of your numbers from your table in the middle and eliminate any duplicates. • Then group your numbers and label them according to some common characteristics. Then turn you labels over. • Visit another table and try to figure out their groupings. (1 pt. each correct ans.) • Discuss how you can use this activity in your own classroom. 8
  9. 9. Group and Label Isosceles Sphere Cylinder Scalene Square Trapezoid Page Right triangle Rectangle 149 Hexagon Decagon Rhombus Pentagon Oval Cone Octagon Cube Circle pyramid 9 Cleveland Urban Conference
  10. 10. Group and label Page 149 10 Cleveland Urban Conference
  11. 11. Group and Label Page 152 11
  12. 12. Group and Label Page 153 12
  13. 13. Group and Label Page 154 13
  14. 14. Group and Label Page 155 14
  15. 15. Terry Walsh By the fractions . . . I have been married for nearly ____of my life. _______ of my children are male. _______ of my children are married. I live in the same state as ____ of my children. ____ of my children have their own children. 0/3 1/3 3/5 2/3 3/3 Cleveland Urban Conference 15
  16. 16. Thoughtful Questions: ♦ Why do some students succeed in mathematics and others do not? Is it a matter of skill or will? ♦ How can we use research-based teaching tools and strategies to address the style of all learners so they succeed in mathematics? ♦ How do we design units of instruction that are meaningful, manageable, and make students as important as standards? Cleveland Urban Conference 16
  17. 17. Workshop Assumptions: • What teachers do and the instructional decisions that they make have a significant impact on what students learn and how they learn to think.   • Different students approach mathematics using different learning styles and need different things to achieve in mathematics. • Style-based mathematics instruction is more than a way to invite a greater number of students into the teaching and learning process; it is, plain and simple, good math—balanced, rigorous, and­­­­ diverse. Cleveland Urban Conference 17
  18. 18. Learning Goals: Participants will learn: • The characteristics of the four basic mathematical learning styles (Mastery, Understanding, Self-Expressive, and Interpersonal), a start on how to assess your own mathematical teaching style, and students’ mathematical learning styles. • How to use a variety of mathematical teaching tools to differentiate instruction and increase student engagement. • How to select mathematical teaching tools to address NCTM process standards, integrate educational “best practices,” and plan Thoughtful lessons or units to meet instructional objectives and the diverse needs of students. Cleveland Urban Conference 18
  19. 19. Now….. What are YOUR personal Learning Goals for this workshop?  Review the Thoughtful Questions, Basic Assumptions and Goals for the workshop.  Reflect upon your own practice.  Record three things you want to take with you as a result of your participation in this workshop. Cleveland Urban Conference 19
  20. 20. Critical Vocabulary: • Look at Page 10 in your handout. • Fill in one number in each row, and find your total score for critical vocabulary for the workshops. • We will revisit this page, so you will have a chance to improve you score. Cleveland Urban Conference 20
  21. 21. What’s Your Favorite?... Read the four teaching activities on page 11 of your handout, Page select the one you like teaching 193 the most. Write out reasons why you chose the one you did. If you have time, which one would be your least favorite activity to teach? g1g1 21
  22. 22. 4 P’s: Previewing Before Reading Preview: Scan the entire text. Find out as much as you can about what you are going to read without actually reading it. Predict: Based on what you learned during your preview, what do you think the text is about? Prior Knowledge: What do you already know about the subject of the text? Purpose: What can you expect to accomplish from reading the text? 22
  23. 23. Previewing Worksheet PREVIEW the workshop materials. List four things you learned from your preview. Make two PREDICATIONS about what you will learn from the workshop. What PRIOR KNOWLEDGE will you use to enhance your learning in this workshop? What is your PURPOSE for participating in this workshop? What can you expect to accomplish. 23
  24. 24. Mastery Math Students  Want to Learn practical information and set procedures Are like problems they have solved before  Like math problems that and that use algorithms to produce 1 solution  Approach problem solving In a step-by-step manner  Experience difficulty when Math becomes too abstract or when faced with non-routine problems  Want a math teacher who Models new skills, allows practice time and builds in feedback and coaching sessions
  25. 25. Understanding Math Students  Want to Understand why the math they learn works  Like math problems that Ask them to explain, prove, or take a position  Approach problem solving Looking for patterns and identifying hidden questions  Experience difficulty when There is a focus on the social environment of the classroom  Want a math teacher who Challenges them to think and who lets them explain their thinking Cleveland Urban Conference 25
  26. 26. Self-Expressive Math Students  Want to Use their imagination to explore mathematical ideas  Like math problems that Are non-routine, project-like in nature, and that allow them to think “outside the box”  Approach problem solving By visualizing the problem, generating possible solutions, and exploring among the alternatives.  Experience difficulty when Math instruction is focused on drill and practice and rote problem solving  Want a math teacher who Invites imagination and creative problem solving into the math classroom 1/23-24/06 ASCD 26
  27. 27. Interpersonal Math Students  Want to Learn math through dialogue, collaboration, and cooperative learning  Like math problems that Focus on real-world applications and on how math helps people  Approach problem solving As an open discussion among a community of problem solvers  Experience difficulty when Instruction focuses on independent seatwork or when what they are learning seems to lack real-world applications  Want a math teacher who Pays attention to their success and struggles in math 1/23-24/06 ASCD 27
  28. 28. A “Paradox”…. A little about two doctors (PhD’s) you should know about….. Carl Jung Dr. Harvey Silver Cleveland Urban Conference 28
  29. 29. Robert Sternberg, IBM Prof. of Psychology and Education, Yale University. Learning Style Research Study Five different ways for teaching mathematics A memory-based approach emphasizing identification and recall of facts and concepts; Page An analytical approach emphasizing critical 7 thinking, evaluation, and comparative analysis; A creative approach emphasizing imagination and invention; A practical approach emphasizing the application of concepts to real-world contexts and situations; and A diverse approach that incorporated all the approaches 29 Cleveland Urban Conference
  30. 30. Sternberg and his colleagues drew 2 conclusions First, whenever students were taught in a way that matched their own style preferences those students Page 8 outperformed students who were mismatched. Second, students who were taught using a diversity of approaches outperformed all other students on both performance assessments and on multiple-choice memory tests. 30 Cleveland Urban Conference
  31. 31. Page 8 ASCD 31
  32. 32. A Mathematical Task Rotation… 2. Write down a “significant year” in your life. Describe it to your neighbor using as many numbers as you can. 3. Write down the year of your birth. • Write down your age as of 12/31/2009 • Write down the number of years since your “significant year”. • Find the sum of your four numbers. • Compare answers with three other people. • Explain what you discover three ways (algebraically, with words, and graphically). Cleveland Urban Conference 32
  33. 33. ST Mastery Learner:  Thinking Goal: REMEMBERING  Environment: CLARITY & CONSISTENCY  Motivation: SUCCESS  Process: STEP-BY-STEP EXERCISE & PRACTICE  Outcome: WHAT? CORRECT ANSWERS 33
  34. 34. NT Understanding Learner:  Thinking Goal: REASONING  Environment: CRITICAL THINKING AND CHALLENGE  Motivation: CURIOSITY  Process: DOUBT-BY-DOUBT EXPLAIN & PROVE  Outcome: WHY? ARGUMENTS 34
  35. 35. NF Self-Expressive Learner:  Thinking Goal: REORGANIZING  Environment: COLORFUL AND CHOICE  Motivation: ORIGINALITY  Process : DREAM-BY-DREAM EXPLORE POSSIBILITIES  Outcome: WHAT IF? CREATIVE ALTERNATIVES 35
  36. 36. SF Interpersonal Learner:  Thinking Goal: RELATE PERSONALLY  Environment: COOPERATIVE AND CONVERSATION  Motivation: RELATIONSHIPS  Process: FRIEND-BY-FRIEND EXPERIENCE & PERSONALIZE  Outcome: SO WHAT? CURRENT & CONNECTED 36
  37. 37. What’s Wrong? vs Who’s Right? Both ask students to find and correct errors. Who’s Right (SF) P 38 uses a personal story to set the P.196 stage for the work, whereas What’s Wrong (ST) does not. Cleveland Urban Conference 37
  38. 38. What’s wrong with the following problem? 32 + 3(2x – 12) > 5 – (4 + 9x) 32 + 6x – 36 > 5 – 4 – 9x - 4 + 6x > 1 – 9x -5 > - 15x x > 1/3 38
  39. 39. Justify/Explain » Write one or more valid reasons why the man with the full cart is not wrong in being in the lane he is in. » You may work in pairs, groups, or by yourself.
  40. 40. Real World Connections... Write ways that numbers are used to P. 190 determine the location of something. This was in an NCTM journal Cleveland Urban Conference 41
  41. 41. What if? ... What if the population of the United States kept increasing at the same percentage that it did between the first census in 1790 P.158 (3.9 mil.) and the second census in 1800 (5.3 mil.)? What would the population have become in the 2000 census? Cleveland Urban Conference 42
  42. 42. Cleveland Urban Conference 43
  43. 43. Teaching with Style to Sensing/Thinking Mastery Learners Guidelines Examples State objectives and outcomes; provide S tart with clear expectations. clear criteria for evaluation. T ell students what they need to Provide a clear model of what know and how to do it step-by-step. students need to know and should be able to do. E stablish opportunities for concrete Provide hands-on materials; use experiences and for exercise and active games, especially with practice. competition; change tasks often. Check for understanding regularly; P rovide speedy feedback on mass and distribute practice over student performance. time. Test for mastery; apply specific S eparate practice from content and skills to concrete performance. projects and activities. Cleveland Urban Conference 44
  44. 44. Teaching with Style to Intuitive/Thinking Understanding Learners Guidelines Examples Provide questions that puzzle Generate questions for understanding; problem-based learning. and data that teases. “Know, need to know, and want Respond to student queries and to know”; establish provide reasons why. purpose/reason for activity. Open opportunities for critical Pattern-finding activities; critical thinking, problem-solving, research thinking strategies: compare/ contrast, decision making, research. projects, and debate. Build in opportunities for Thesis essays, debates, Socratic seminars, editorials; seek alternative explanation and proof using explanations/points of view. objective data and evidence. Self-directed learning; projects and Evaluate content and process. performances that demonstrate understanding. Cleveland Urban Conference 45
  45. 45. Teaching with Style to Sensing/Feeling Interpersonal Learners Guidelines Examples Use personal hooks; give examples Try to personalize the content. from your own life, encourage students to do as well. Reinforce learning through support Build trust in the classroom; provide a pleasant physical setting; encourage and positive feedback. expression of personal feelings. Use the world outside the classroom Find/use real-world applications; for current and personally relevant use emotional contexts; apply to content. current student concerns. Select activities that build upon Empathy work; decision-making; personal experiences and cooperative learning; class cooperative structures. discussions; peer practice. Take time to establish personal Personal reflections; journal goals, encourage reflection, and writing. praise performance. 46 Cleveland Urban Conference
  46. 46. Teaching with Style to Intuitive/Feeling Self-Expressive Learners Guidelines Examples “What if?”questions; metaphorical Inspire use of imagination, expression; visualizing ideas; invent or explore use of alternatives. imagine; creative problem-solving. Model creative work so students Extrapolate structure; generate performance criteria; model creative examine/establish criteria for guidance and assessment. process. Allow student choice of activities Alternative activities and methods; present ideas in a variety of ways; and methods for demonstrating culminating assessment projects. understanding and knowledge. Give feedback, coach, and provide Opportunities for students to share audiences for sharing work. work/receive feedback from an audience; quality circles. Evaluate and assess Holistic and analytic rubrics; performance according to student assessment; self- established criteria. assessment. 47
  47. 47. Students work with Questions in all four Styles: Mastery questions ask Interpersonal questions what students remember. invite students to reflect and share their feelings. What? So What? Understanding questions Self-Expressive questions require explaining and require the use of proving. imagination. Why? What If? Cleveland Urban Conference 48
  48. 48. Back in My Classroom  After learning about our learners, what does this mean for us as math teachers?  What questions do you have?  What solutions do you see that will allow all students to become more effective mathematics learners?  What actions are you ready to take to meet the needs of all your students? 49
  49. 49. How much of students’ success in your math classes is due to their understanding what they read or write? 50
  50. 50. The Four Functions of Style SENSING Physical Facts Details Here & Now Objective Perspiration Subjective THINKING Analyze FEELING Harmonize Logic Likes/Dislikes Truth Tact Procedures People Inspiration Past & Future Ideas Possibilities Patterns INTUITION 1/23-24/06 ASCD 51
  51. 51. Tuesday Session 1 Writing & Reading in Math 52
  52. 52. M&M’s (Math Metaphor, p.129) My favorite math teacher always used to say that fractions are like politicians. At first I thought she was crazy, but then I started to think about the idea, and found that I agreed with her! Write three ways politicians and fractions are alike, and three ways they are different from each other. g1g1 + 2 53
  53. 53. Give One, Get One DIRECTIONS Stand up, partner with one other person, GIVE one of yours, GET one of theirs. If you both have the same, then create a new idea together to add to your lists. Quickly move to a new partner. Give One, Get One. Repeat 4 times for a total of 6 ideas. Remember: work in dyads. NO HUDDLING, NO COPYING OF EACH OTHER’S TOTAL LISTS. 54
  54. 54. Write to Learn • The more students write and think in mathematics classes, the more they learn. Doug Reeves reports that the correlation between writing in mathematics classes and scores on mathematics tests is a positive correlation of 0.93. 55
  55. 55. When SHOULD students write in mathematics? 1. At the beginning of the lesson. • Access prior knowledge • Generate ideas • Review previous lesson 2. During the lesson • Check for understanding • Practice • Respond to a thoughtful question 3. At the end of the lesson • To review what they have learned • To apply what they have learned • To extend what they have learned to other areas 56
  56. 56. How much writing do your students do in your mathematics class? None Very Little Some Considerable A Great Deal Amount 57
  57. 57. What kinds of writing do you want your students to do in your math classes? Make a list.... 58
  58. 58. My Writing List (should we add any to your lists?)  Answers in “proper form” (whatever that is!!)  Showing their work  Good notes ( making, not merely taking notes! – not in the book)  Definitions (NOT merely copying the text definition)  Complete explanations of their answers when asked for them  Summaries of concepts and procedures  What graphs or charts tell them  Research projects  Pre-lab explanations of how to do conduct an experiment or predict the results  Creative writing (stories, poems, cinquains, haikus, etc.)  Examples of how math really exists in the world, not traditional word problems  Error analysis  Creating patterns  Using complete sentences  How they think or feel about a concept  Compare and contrast  Defend a position
  59. 59. WRITE TO LEARN  Provisional: Generate ideas, fluency & flexibility. Audience: Oneself  Readable: Has Purpose & Audience; coherent & clear, concern with content & organization, write on every other line, knee-to-knee conference. •Voice  Polished: Use writing process steps, attention to •Organization mechanics and technique, edited. Reflection of •Interesting verbs/ adjectives one’s best work. Looks Good, Sounds Smart •Correct spelling & mechanics •Establish Big Ideas & Support w/Details  Publishable: Edited and revised several times. Audience is the wider public community. Cleveland Urban Conference 60
  60. 60. M ake a comparison or justify a decision A ccess prior knowledge Think About Learning or Feelings H ypothesize E xplain or define a mathematical concept M ake real world connections A nalyze errors in thinking Take a position I nterpret data and justify a conclusion C reative writing S ummarize (see P. 141 of your book.) 61
  61. 61. Creative writing can take many forms. Before you use mathematical vocabulary, it might help to use non-math terms. Prepare your students by asking them to write 3 sentences using the term “milky way” and have the term mean something different in each of the 3 sentences!
  62. 62. Next we will be some creative writing using these terms. Here is an example of what I mean by “creative” writing: The people who live in Ponent, Illinois call people who move out of town “exPonents”. Use at least 5 terms from the following list of terms to write 5 sentences (using more than one word in a sentence is even more creative). Opposite Adjacent Side Hypotenuse Acute Obtuse Right Angle Sine Cosine Tangent Triangle
  63. 63. How many words did you use? Here is my one very run on sentence using all 12 words: The three people formed a right triangle with the tan gent adjacent to his very acute angel of a girlfriend as he cosined the loan application she had just sined, he noticed that the obtuse pasta chef in the restaurant on the opposite side of the street was putting the high pot in use to boil spaghetti noodles. Of the 12 terms I used which 5 are the most “creative” (cheating) uses? I would also ask the students to compare their use of the term to the actual mathematical definition of the term.
  64. 64. Thinking about Learning or Feelings.... Would you rather have a best friend whose views are congruent to yours, or similar to yours? Explain your choice using vocabulary terms from the unit.
  65. 65. Support or Refute (P. 69) Word Problem  You will have a short time to skim over the word problem in the next slide. You will not have time to read the problem carefully. Next, you will be asked to answer several True or False questions about the word problem.
  66. 66. An Atypical Word Problem  A truck is on its way to three different motorcycle dealerships. The truck contains both mopeds and motorcycles. Maggie Sutton, who owns all three dealerships, receives an invoice which tells her that a total of 150 vehicles are on the truck for her three dealerships. However, the invoice doesn’t tell her how many of her vehicles are motorcycles and how many are mopeds. The invoice does show that the total mass of her vehicles is 34,800 lbs.. It also shows the mopeds weigh 100 lbs. each while motorcycles weigh 320 lbs.. How many mopeds and how many motorcycles are on the truck for Ms.Sutton’s dealerships?
  67. 67. Support or Refute: Directions & Questions Directions:  Write down whether you think each question is true or false.  Reread the problem and look for words that either support your original answer, or refute it.  Solve the problem if you want to or if you need to do so in order to support or refute one of your original answers. Questions: 1. The problem tells us the total number of vehicles on the truck. True or False? 2. The fact that there are three dealerships is critical to solving the problem. True or False? 3. The best way to solve this problem is to set up an equation with two variables. T/F ? 4. Motorcycles have a greater mass than mopeds. T/F? 5. The solution requires two separate answers. T/F? Why would students need experience with Support or Refute before using this exact problem?
  68. 68. An Atypical Word Problem  A truck is on its way to three different motorcycle dealerships. The truck contains both mopeds and motorcycles. Maggie Sutton, who owns all three dealerships, receives an invoice which tells her that a total of 150 vehicles are on the truck for her three dealerships. However, the invoice doesn’t tell her how many of her vehicles are motorcycles and how many are mopeds. The invoice does show that the total mass of her vehicles is 34,800 lbs.. It also shows the mopeds weigh 100 lbs. each while motorcycles weigh 320 lbs.. How many mopeds and how many motorcycles are on the truck for Ms.Sutton’s dealerships?
  69. 69. Support or Refute (P. 69 in the Math Tools book)
  70. 70. A Geometry Support or Refute Agree or Disagree with each of these. Then READ Section 5.4 and find statements or ideas in the reading that support or refute your original response to each statement. Write down some reference to the location (page and position) of something in the section that agrees or disagrees with your original response to each statement. 1. All the points of a polygon must lie in the same plane. 2. A diagonal connects any two vertices of a polygon. . 3. A pentagon has five sides but it has ten diagonals. 4. A rhomboid is a type of quadrilateral. 5. A kite is a geometry term as well as a thing you can go fly. __ __ 6. In rectangle PQRS, RS and PQ are the diagonals. __ __ 7. In parallelogram ABCD, the diagonals are AC and BD. .
  71. 71. An Alg2/Trig Support or Refute . Alg2/Trig Read section 4.4. First write whether you AGREE or DISAGREE with each of these statements. Then, as you read, cite the text to support or refute your original decision. 1. Descartes rule of signs lets us say something about only the positive roots, not the negative roots of a polynomial. 2 If there is only one variation in signs, then there must be exactly one positive or negative real zero. 3 Y = -X3 + X + 1 has either two or no positive real zeros and exactly one negative real zero. 4. For Y = X4 + X2 – 3X – 6, all possible rational zeros are 1, 2, 3, or 6 5. For Y = 5X4 + X2 – 3X – 6, all possible rational zeros are 1, 2, 3, 6, 1/5, 2/5, 3/5, and 6/5 6.The UPPER Bound occurs when the synthetic division work shows all positive values. 7. The LOWER Bound occurs when the synthetic division work shows all negative values
  72. 72. Reading Questions in Styles Mastery Questions: Read the actual lines finding facts, details, or literal meanings Understanding Questions: Read between the lines explaining, inferring, or comparing Self-Expressive Questions: Read beyond the lines connecting things in new ways or looking for new methods or ideas Interpersonal Questions: Reacting to the lines making personal connections, or finding relevance Example: The term “Detour Proofs” make me uncomfortable.
  73. 73. Note taking vs. Note making not in your handout Students who are taking notes are usually copying what the teacher has written or said. They also copy work done by peers at the board or in groups. Students who are making notes are reading text or example problems and writing out their own explanation of the work as well as questions they have about the problem or concept.s and differences
  74. 74. Note makingExample 32+3(2x – 12) > 5–(4+9x) What did I do? Explain why or ask a ? 32 + 6x – 36 > 5 – 4 – 9x - 4 + 6x > 1 – 9x - 4 > 1 - 15x - 5 > - 15x 1/3 < x x > 1/3
  75. 75. To use this idea, turn off auto format for spelling!! I cdnuolt blveiee that I cluod aulactly uesdnatnrd what I was rdanieg aoubt the phaonmneal pweor of the mnid. Aoccdrnig to rscheearch at Cmabrigde Uieinrvtsy, it deons’t mttaer in waht oredr the ltteers in a wrod are, the olny ipomoatnt tinhg is taht the fsirt and lsat ltteer be in the rhgit pclae. The rset can be a taotl mses and you can sitll raed it! This is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the word as a wlohe. Amzanig ins’t it?
  76. 76. The % and decimal below do not obey the “rules”. Prbabltiioy can hlep us mkae dcsioines. Wtrei the fowlilong pgaaaprrh ccorrectly, tehn awnesr the fuor qiosteuns: Wehn trehe is a 2%7 cnache of pcrepititioan, yuor paenrt wlil prboalby dedcie to crray his ulmrebla to wrok. If one of yuor sohcos’l blal pyealrs has a .741 bitntag arvreae, you wluod ecpext taht she is mroe lkiely not to bat in a tmmeatae form scneod bsae. Mnay pborabiitly stiauitnos ivlovne a pfaoyf, scuh as pinots secrod; leivs seavd; or pfiorts eeanrd. The “epxcteed vuale” of a stiiaoutn is waht the pofayf oevr a lrgae nebmur of oeeccruncs wloud be. In tihs uint. We wlil eolrpxe qitsneous ivvoilnng pbbrltiiaoy and eepcxtd vulae. 1. Which parent went to work? 2. How large is the ball? (or what specific sport is involved? 3. One part of one of the paragraphs has two correct possible “translations”, what are they? 4. How would you use the above to define “epxcteed vuale”?
  77. 77. Cleveland Urban Conference 78
  78. 78. Mathematical Summaries... Write out how we add two fractions P. 27 Cleveland Urban Conference 79
  79. 79. Math Recipe vs Anchor Walls ... (not in book) Create a “recipe card” or a fill in the blank P. 132 template. Cleveland Urban Conference 80
  80. 80. Anchor Walls In order to add two fractions, the first thing we do is make sure the ________________. If they are not, you have to get ________________. If they are, then you simply ____________. After adding them, remember to__________________. Cleveland Urban Conference 81
  81. 81. Making Up Is Fun to Do ... My dad’s sister, Sally, used to ask me why math P.160 teachers picked on her, so let’s write a new sentence to replace, “My dear Aunt...”. Cleveland Urban Conference 82
  82. 82. Tuesday Session 2 Vocabulary & Assessment Strategies 83
  83. 83. Fist Lists & Spiders Look at P. 32 of your handout…. Use the spider in your handout. Write a P. 29 concept in the center, and write an important characteristic about it on each leg of the spider. Cleveland Urban Conference 84
  84. 84. 3 Way Tie Write “fractions”, “percents”, and “decimals” at the three vertices of the triangle in your handout (P. 33). Write P.108 a sentence connecting each pair, and a generalization connecting all three in the center. Cleveland Urban Conference 85
  85. 85. Asessment Menus Look at the Conics Assessment Menu... (p. 34 of the handout). P. 239 Students need to complete 4 tasks, one from each each Style and each level of difficulty. Cleveland Urban Conference 86
  86. 86. Tic-Tac-Toe (Vocab. Games) (P. 35 in handout) The example was written by a HS P.213 teacher in Bowling Green KY.. Students need to complete a winning line of tasks. Cleveland Urban Conference 87
  87. 87. Task Rotations are a way to use all four styles in a single strategy. Task Rotations are found in pages 222 to 238 in the book. (The “significant year” activity was a Task Rotation.) Cleveland Urban Conference 88
  88. 88. Task Rotations The Calculus Task Rotation (p.36 in handout) shows P. 222 Styles can be used in all classes, at all levels to improve student learning. Cleveland Urban Conference 89
  89. 89. Range Finder The “Graduated Difficulty” example (P.37 in handout), P. 208 should be Range Finder. It can be used as a formative assessment, to see where students are. Cleveland Urban Conference 90
  90. 90. Convergence Mastery For when they absolutely, positively need to know a P. 40 concept in order to succeed.... (not in your handout). Cleveland Urban Conference 91
  91. 91. Unit Tests Test Worth Taking is a test that poses questions in all four P. 244 Styles. Look at the Geometry Test in your handout. (P.38-42) Cleveland Urban Conference 92
  92. 92. What are words and how are they defined? What words are important to learn?
  93. 93. Key Word Strategy = Dictionary Definition Bicycle (the key word) A mode of transportation (the bigger idea) With two wheels, a pedal and chain (essential characteristics) system, with energy supplied by the rider Types of bicycles: mountain bikes, dirt bikes, 10 speeds (examples) Distinguished from: motor cycles, unicycles, and scooters (non-examples) Cleveland Urban Conference 94
  94. 94. Take the word ___________ trapezoid Non-Examples General Category Examples rectangle quadrilateral square Key Word trapezoid parallelogram Essential Characteristics Plane figure, four sides, exactly one pair of sides parallel, 95
  95. 95. Take the word mathematics Non-Examples General Category Examples Key Word mathematics Essential Characteristics 96 Cleveland Urban Conference
  96. 96. Take the word___________ Non-Examples General Category Examples 1 Numbers 2 51 Key Word 7 Prime number 91 41 Essential Characteristics A number that has only two multiples one of which is itself 97
  97. 97. These are polygons: These are NOT polygons: What makes a polygon a polygon? List critical attributes. Cleveland Urban Conference 98
  98. 98. What is Mathematical Literacy? 1 2 4 0011 0010 1010 1101 0001 0100 1011 Cleveland Urban Conference 99
  99. 99. What is Mathematical Literacy? Literacy in reading means not only 0011 0010 1010 1101 0001 0100 1011 being able to pronounce and decode words, but also being able to read 2 and comprehend what one reads. Mathematical literacy means the same 1 4 thing--having procedural and computational skills as well as conceptual understanding. Cleveland Urban Conference 100
  100. 100. Mathematical Literacy 0011 0010 1010 1101 0001 0100 1011 The importance of mathematical literacy and the need to understand 2 and be able to use mathematics in everyday life and in the workplace 1 4 have never been greater and will continue to increase. (National Commission on Mathematics and Science for the 21st Century) Cleveland Urban Conference 101
  101. 101. Jobs requiring mathematical and technical skills are growing the fastest among the 0011 0010 eight professional and related 1010 1101 0001 0100 1011 occupations. 2 60% of all new jobs beginning in the 21st 1 century require skills that are possessed 4 by only 20% of the current work force. Cleveland Urban Conference 102
  102. 102. What is Mathematical Literacy? 0011 0010 1010 1101 0001 0100 1011 Mastery of procedural A language to and conceptual communicate and solve 2 knowledge real-world problems 1 4 Understanding of logical Application of strategies reasoning to explain and to formulate and solve prove a solution problems Cleveland Urban Conference 103
  103. 103. Four Reasons to Teach Vocabular y: • Verbal Intelligence • Ability to comprehend new information: Academic Achievement • One’s level of income • Self-confidence and self-image 104 Cleveland Urban Conference
  104. 104. Tuesday Session 3 Your ideas & closure (for now?) 105
  105. 105. Page 8 How Do I Select the Right Tool For the Right Learning Situation? 106
  106. 106. Five Ways to Use Math Tools See pages 13 through 15 in the Math Tools book: • Try one out. Page • Use tools to help you meet a 13 particular standard or objective. • Individualize instruction. • Differentiate instruction for the entire class. • Design more powerful lessons, assessments, and units. 107 Cleveland Urban Conference
  107. 107. How Do I Select the Right Tool For the Right Learning Situation? Use the matrices on pp. 18/19; 64/65; 122/123; & 168/169 See pages 9 through 13 in the Tools book: • Title and Flash Summary • NCTM Process Standards • Educational Research Base • Instructional Objectives. 108 Cleveland Urban Conference
  108. 108. 109
  109. 109. 1/23-24/06 ASCD 110
  110. 110. How many of us would like our students to… • Think more deeply? • Take more intellectual risks? • Recognize that there are many ways to learn? • Develop greater confidence in their ability to learn and improve self-esteem? • Develop better relationships with your and their peers? • Have greater respect for others and their differences? • Take more responsibility for their learning? • Develop a deeper understanding of the connection between what they learn and how they learn? 111 Clevelaerencend Urban Conf
  111. 111. Research clearly indicates the impact of each of these on student learning: Category %ile Gain Identifying Similarities & Differences 45 Summarizing & Note-taking 34 Reinforcing Effort & Providing Recognition 29 Homework & Practice 28 Non-Linguistic Representation 27 Cooperative Learning 27 Setting Objectives & Providing Feedback 23 Generating & Testing Hypotheses 23 Questions, Cues, and Advance Organizers 22 112
  112. 112. Mathematics Workshop: Four Thought ASCD 113

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