Class Notes: From PowerPoint
2004
If you would like to read any of these books in class, please let me know.
Math Curse by Jon Scieszka & Lane Smith
How Big Is A Foot? by Rolf Myller
Frog and Toad Are Friends by Arnold Lobel
The Button Box by Margarette S. Reid
The Greedy Triangle by Marilyn Burns
Peter's Pockets by Eve Rice
The Doorbell Rang by Pat Hutchins
What Comes in 2's, 3's, and 4's? by Suzanne Aker
Rooster's Off to See the World by Olga Torres
A Three Hat Day by Laura Geringer's
Mathematicians Are People, Too by Luetta Reimer & Wilmer Reimer
Pigs Will Be Pigs by Amy Axelrod
One Grain of Rice by Demi
Read and Discuss: Math Curse by Jon Scieszka & Lane Smith
1. Keep a journal for a day recording all the math problems you encounter.
2. Bring in cut-out examples from magazines and newspapers.
3. Communicate using the vocabulary of mathematical terms and symbols used in the book.
4. Discuss why the teacher is named "Mrs. Fibonnaci."
5. Model and solve some of the problems in the book and determine which ones are simply nonsense.
6. Investigate the mathematical conversions, tables, measures, and terms on the end papers of the book.
Equity Curriculum Teaching Learning Assessment Technology
Equity means:having high expectations and worthwhile opportunities for all students. accommodating differences to help everyone learn mathematics. providing resources and support for all classrooms and all students.
A mathematics curriculum is a collection of topics and activities that is coherent, focuses on important mathematics, and well articulated across the grades.
Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. Knowing Mathematics: Knowing mathematics means understanding mathematics and being able to do mathematics. Verbs for doing mathematics: explore, investigate, conjecture, justify, discover, verify, explain, predict, describe, use, solve, represent, develop, etc.
Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.
Assessment should support the learning of important mathematics and furnish useful information to both teachers and students. Assessment should enhance students' learning. Assessment is a valuable tool for making instructional decisions. Assessment should reflect the mathematics that all students need to know and be able to do, and it should focus on students' understanding as well as their procedural skills.
Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning.
Content Standards
Number and Operations
Algebra
Geometry
Measurement
Data Analysis and Probability
Process Standards
Problem solving
Reasoning and proof
Communication
Connections
Representations
Create an effective learning environment Consider student characteristics Use motivational strategies Teach for understanding Engage students actively Encourage communication & reflection Promote social interaction Provide appropriate activities and practice
High expectation & strong support
Safe to take risks
Time to wrestle with important mathematics
Time for students make connections
Prior knowledge
Intellectual strengths
Special learning needs
Cultural/language differences
Curiosity and interests
Use culturally relevant materials
Tell stories
Read books
Play games
Select problems that are realistic and interesting
Understanding is a measure of how well ideas are integrated with or connected to other existing ideas.
Connections
Representations
Understanding We can't understand for our students. No matter how clearly and patiently we explain things. We must give students time to wrestle with problems and ideas to construct their understanding.
Contrast knowing an idea with understanding and idea. What does it mean to say that understanding exists on a continuum?
Give an example, and explain how the idea might be understood at different places along this continuum.
It is intrinsically rewarding
It enhances memory
There is less to remember
It helps with learning new concepts and procedures
It improves problem-solving ability
It is self-generating
It improves attitude and beliefs
Models were described as "thinker toys", "tester toys", and "talker toys." Explain how models can help a child develop a concept.
Engage Students Actively
Explore, Conjecture, Investigate, Solve, Justify, Represent, Formulate,Construct
Verify, Explain, Predict, Develop, Use, Discover
Encourage Communication and Reflection
Provide learning activities that give students opportunities to:
Talk about, write about, describe, explain, clarify, and justify mathematical ideas and solutions.
Promote active listening
Listen to, respond to, and question the teacher and one another.
Promote Social Interaction
Provide learning activities that give students opportunities to
associate with each other
learn from each other
gain respect for each other
Provide Appropriate Practice
Avoid uniformly assigning skill-oriented tasks.
Skills should be practiced at a level appropriate for each student.
Practice is boring for students beyond the need for practice in a particular skill
Practice reinforces a sense of inadequacy for students not ready for practice in a particular skill
Students need a wide variety of
experiences with the concepts:
Shape
Size
Symmetry
Congruence
Similarity
Transformation
Square: 10 cm x 10 cm
Square Triangle Right Isosceles Congruent Similar Midpoint Trapezoid Parallel Parallelogram Quadrilateral Symmetry
Contrast knowing an idea with understanding an idea.
Examine the seven benefits of relational understanding. Select and describe the ones that you think are most important.
Explain how models can help a child develop a concept.
Instructional programs should enable all students to: build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving.
Understand the problem
Devise a plan
Carry out the plan
Look back
Look for a pattern
Construct a table
Make an organized list
Act it out
Draw a picture
Use objects
Guess and check
Work backward
Write an open equation
Solve a simpler problem
Change your point of view
Make a model
A problem is any task or activity:
for which the students have no prescribed or memorized rules or methods,
nor is there a perception by the students that there is a specific "correct" solution (Hiebert et al., 1997).
It must be new.
Students must develop:
an interest in finding solutions to problems;
the confidence to try various strategies;
a willingness to risk being wrong;
the ability to accept frustration that comes from not knowing;
a willingness to persevere when solutions are not immediate; and
an understanding of the difference between not knowing the answer and not having found it yet.
Describe what is mean by tasks or problems that can be used for teaching mathematics.
Be sure to include the three important features that are required to make this effective.
It must begin were the students are.
The engaging aspect of the problem must be do to the mathematics that the students is to learn.
It must require justification and explanations for answers and methods.
A woman bought a horse for $50 and sold it for $60. She then bought the horse back for $70 and sold it again for $80.
What do you think was the financial outcome of these transactions? The woman . . .
Lost $10 Earned $10
Lost $20 Earned $20
Came out even Other (Describe)
Explain your reasoning.
Find all the ways to write the counting numbers from 1 to 25 as the sum of consecutive counting numbers. Some of the numbers are impossible. See if you can find a pattern of those numbers. Then search for other patterns and relationships.
You can combine different denominations of stamps to get the proper postage for a letter or package. If you only had stamps in 3¢ and 5¢ denominations, what amount of postage would be impossible to make? Is there a largest "impossible" amount? Explain your answer.
I have 3 pennies and 3 nickels in my pocket. If I take two coins out of my pocket, what is the probability that the coins are both pennies?
Both nickels? One is a penny and the other is a nickel?
The city park is in the shape of a triangle. The city can afford to install only one light post.
Where should they put the light post if they want to illuminate each corner of the park equally?
The city park is in the shape of a triangle. The city want to put the largest possible circular pond inside the park. Where should they put the center of the pond and how big can the pond be?
Sorting and Classifying Seriating Patterning
Involves focusing on an attribute or characteristic of a set of objects and then grouping the objects accordingly. Sorting Activities Sorting toys and clothes. Sorting objects--junk, buttons, keys, jar lids, nuts and bolts, postcards, shells, rocks, seeds, candy, attribute blocks, etc. Sorting people--using attributes such as length of hair, color of pants, type of shirt, etc. Mystery sorting--guess what is in a box by asking yes or no questions about its attributes. Integrating sorting to other subject areas&endash;Science, Social Studies, Language Arts, etc. The Button Box by Margarette S. Reid The Button Box is a wonderful invitation to the pleasures of a button collection. Discussion Who has a button collection at home? Can anyone describe the container that holds your buttons? Is anyone wearing buttons today?
Dear Parents,
After discussing the book The Button Box in class, the children have become interested and curious about buttons.
I've asked the children each to bring a button to school for our collection. We will be doing a variety of math activities
that will engage the children in sorting, comparing, and counting buttons. Please sent a button you no longer need.
Thank you,
I wonder if any are the same.
How are they the same?
How are they different?
With a partner, take turns telling something that is the same or different about your buttons.
Sort a hand full (10-15) into groups. Record your sort with a picture.
Sort 1: Take a hand full of buttons (10-15). Put your buttons into groups. Draw or write how you put your buttons into groups.
Sort 2: Mix your buttons up. Now sort your buttons by the number of holes. Place your buttons on the graph paper, and record the buttons. Write a sentence telling about your graph.
Sort 3: Draw a picture of Toads jacket. Place your buttons on Toad's jacket to make a pattern. Draw the pattern on the jacket. Write a sentence telling about your pattern.
Toad's Jacket
The process of focusing on an attribute and then ordering a set of objects according to that attribute.
Attribute Objects to be Ordered
Length Sets of pencils, nails, baseball bats, pieces of rope or string
Size Mittens, socks, containers, playground balls
Capacity Measuring cups and spoons, jars, boxes
Mass Sugar, rice, flour, beans, etc.
Height Children, potted plants, buildings
Copy
Describe
Extend
Create
Represent (tables, graphs, symbols), p. 423
Types of Patterns
Repeating Patterns, p. 418
Growing Patterns, p. 420
Pattern by color, shape, size, texture, and many other attributes
Pattern objects by position and quantity
People patterns
Rhythmic patterns (clapping, snapping, tapping)
Physical patterns (hopping, skipping, jumping, etc.)
Patterning with objects (toys, unifix cubes, pattern blocks, buttons, keys, coins, tiles, stickers, quilt patterns, flowers, candy, fruit, etc.)
Story patterns
A cardinal number tells how many are in the set.
An ordinal number indicates the relative position of an object in an ordered set.
A nominal number is used to name something.
A number as a measure is used to indicate an amount obtained by measuring a continuous dimension.
One-to-one--Each object is assigned only one number name. Stable order--The number names must be used in the same order every time one counts. Order Irrelevance--The order in which the objects are counted doesn't matter. Cardinality--The last number name used gives the number of objects.
Counting all
Counting on
Counting back
Skip counting
1. Understanding: connections
meanings of operations
multiple representations
properties
relationships
2. Computational Strategies
basic facts
mental computation
paper-and pencil computation
calculator computation
Addition and Subtraction Meanings
Properties
Identity
Commutative
Associative
Distributive
Relationships
Addition and Subtraction
Multiplication and Division
Addition and Multiplication
Subtraction and Division
Multiplication/Division and Area
Join: Elements are being added or joined to a set. Separate: Elements are being removed from the set. Part-part-whole: There is no action. Part-part-whole problems focus on the relationship between a set and its subsets. Compare: No action. Compare problems involve comparisons between two different sets.
Equal grouping Comparison Combinations
1. Develop understanding first: connections
meanings of operations
multiple representations
properties and relationships
2. Develop thinking strategies through practice
3. Provide appropriate drill
Students should be able to recall single-digit addition facts and the counterparts for subtraction, multiplication, and division.
A child can give a quick response (in less than 3 seconds) without resorting to non-efficient means, such as counting.
Usually begins in grade 1 with more practice required in grades 2 and 3.
Usually begins in grade 3 with more practice required in grades 4 and 5.
Present facts in small groups
Organize facts in families
Organize facts in thinking strategies
One more-Than and Two More-Than
Facts with Zero
Doubles
Near-Doubles (Doubles +1)
Make-Ten Facts
Counting On
Ten Frame Facts
Children should attempt to memorize facts only after understanding is attained and thinking strategies are practiced.
Drill lessons should be short (5-10 minutes) and should be given almost daily.
Children should try to memorize only a few facts at a time and should constantly review previously memorized facts.
Drill activities should be varied, interesting, challenging, and presented with enthusiasm.
8 12 8 9
- 0 - 1 - 5 - 4
Subtracting 0 or 1
Counting Back
Counting On
8 12 14 15
- 5 - 6 - 9 - 6
Think-Addition (Sums 10 or less)
Fact Families for Addition and Subtraction
Doubles
Build Up Through 10
Back Down Through 10
Make a die labeled 1, 2, 1, 2, 1, 2 Make another die labeled 4, 5, 6, 7, 8, 9 After each roll the children should say the complete fact: "Four plus two is six."
Make a die labeled 7, 8, 9, 7, 8, 9 Make another die labeled 4, 5, 6, 7, 8, 9 After each roll the children should say the complete fact: "Seven plus five is 12."
Roll a single die with numerals or dots. Double the result. After each roll the children should say the complete double fact: "Four plus four is eight."
To generate addition facts, cover the sum, the number marked with an asterisk. To generate subtraction facts, cover the either number not marked with an asterisk.
The goal of the game is to remove all 20 cards
in a 5 x 4 array of playing cards.
Remove the face cards (Use A's-10's only)
Make five rows with four in each row.
Roll the dice to get a target number.
Use +, -, x, or ÷ to get your target number.
You may use only the last four cards in each column at a time.
Doubles Fives Facts Zeros and Ones Nifty Nines (7 x 9, 1 less than 7 is 6 and 6 + 3 = 9) Helping Facts
Think Multiplication (fact families)
0 Visualization
Children view a geometric shape as a whole. They do not describe properties of the shape.
It is the appearance of the shape that defines it. A rectangle is a rectangle "because it looks like a rectangle."
1 Analysis
Children recognize the properties of figures. They do not see the relationship between them.
May be able to list all properties of squares, rectangles and parallelograms but not see that these are all subclasses of one another.
2 Informal Deduction
Students see relationships between properties of a class of shapes.
If a figure is a square, then it must be a rectangle.
Four congruent sides and at least one right angle is sufficient to define a square.
3 Formal Deduction
Creates formal deductive proofs.
4 Rigor
Rigorously compare different axiomatic systems.
Not age-dependent
Geometric experience is the greatest single factor influencing advancement through the levels.
A third grader or a high school student could be at level 0.
Reasonable for all children K-2 to be at level 0.
van Hiele Implications for Instruction
Most activities can be designed to begin with the assumption of a particular level and then be raised or lower by means of the types of questioning and guidance provided by the teacher.
Extending Children's Learning 1. Act out parts of the story. 2. Take a walk to look for shapes in the real world. 3. Art activity. Cut out an assortment of polygons in several colors. Have children choose a shape and think about what it might be. Then glue the shape to a piece of white drawing paper and draw a picture around it.
What is the area of a plane figure?
The area of a plane figure is the measure of the region enclosed by the figure.
Area is measured by the number of units (square units) that can be arranged to completely fill the figure.
For some figures, the units may have to be cut up and rearranged.
Area = 1 square unit
Activity: Area of Two
On your geoboard, make as many different polygons as you can with an area of 2 square units.
Record each figure on geodot paper.
Choose one polygon. Copy it to large size geodot paper. Cut it out. Post it. After everyone has posted at least one polygon, post other polygons that are different from those already posted.
In this book, How Big Is A Foot?, Rolf Myller tells the story of a King who wants to give his wife, the queen, a special birthday present. But what do you give to someone who has everything?
Letter to the Apprentice
Write a letter to the apprentice and offer him advice. Tell the apprentice what the problem is and what he could do to fix it.
1. Select an attribute to measure.
2. Choose an appropriate unit of measurement.
3. Use a measuring tool to determine the number of units.
1. Avoid Premature Drill Understanding comes first
connections
meanings of operations
multiple representations
properties and relationships
2. Develop thinking strategies
3. Provide appropriate practice
Students should be able to recall single-digit addition facts and the counterparts for subtraction, multiplication, and division.
A child can give a quick response (in less than 3 seconds) without resorting to non-efficient means, such as counting.
Usually begins in grade 1 with more practice required in grades 2 and 3.
Usually begins in grade 3 with more practice required in grades 4 and 5.
Present facts in small groups
Organize facts in families
Organize facts in thinking strategies
Commutativity Adding Zero or One Doubles Near Doubles (Doubles +1) Counting On (One-/Two-/Three-More-Than) Making Ten
8 12 8 9
- 0 - 1 - 5 - 4
Subtracting 0 or 1
Counting Back
Counting On
8 12 14 15
- 5 - 6 - 9 - 6
Think-Addition (Sums 10 or less)
Fact Families for Addition and Subtraction
Doubles
Build Up Through 10
Back Down Through 10
Basic Fact Activities
One-/Two-More-Than Dice
Making Ten Dice
Double Die
Double Die plus One
Fact Triangles
One-/Two-More-Than Dice
Make a die labeled 1, 2, 1, 2, 1, 2
Make another die labeled 4, 5, 6, 7, 8, 9
After each roll the children should say the complete fact: "Four plus two is six."
Hungry Bug Die
Make a die labeled 7, 8, 9, 7, 8, 9
Make another die labeled 4, 5, 6, 7, 8, 9
After each roll the children should say the complete fact: "Seven plus five is 12."
Ten-Frame Facts
5+1 to 5 +9
Combinations that make 10
Double Die
Roll a single die with numerals or dots.
Double the result.
After each roll the children should say the complete double fact: "Four plus four is eight."
To generate addition facts, cover the sum, the number marked with an asterisk. To generate subtraction facts, cover the either number not marked with an asterisk.
Commutativity
Skip Counting
Repeated Addition
Splitting the Product into Known Facts
Using 0 and 1
Nifty Nines (7 x 9, 1 less than 7 is 6 and 6 + 3 = 9)
Strategies for Division Facts
Think Multiplication (fact families)
Direct Modeling Invented Strategies Intermediate Algorithms Standard Algorithms
How to multiply and divide by 10, powers of ten, and multiples of powers of ten. How to apply the distributive property. How multiplication and division are related.
How many colored Easter eggs will we have if each of the 6 students in the group colors 24 eggs? Use direct modeling Use an invented strategy Use the intermediate algorithm Use the standard algorithm
Jumbo the elephant loves peanuts. If he eats 12 bags of peanuts each day, how many bags will he eat in 14 days? Use direct modeling Use an invented strategy Use the intermediate algorithm Use the standard algorithm
Draw an array model and explain each step in the intermediate algorithm.
14
x 26
24
60
80
200
364
The bag contains 583 jelly beans, and Megan and her three friends want to share them equally. How many jelly beans will Megan and each of her friends get? Use direct modeling Use an invented strategy Use the intermediate algorithm Use the standard algorithm
Write a story problem that follows two rules: (1) It must end in a question. (2) The question must be one that is possible to answer by doing the division: 763 ÷ 5.
Use direct modeling
Use an invented strategy
Use the intermediate algorithm
Use the standard algorithm
Base-ten concepts are enhanced. Invented strategies are built on student understanding. Students make fewer errors with invented strategies. Invented strategies promote mathematical thinking. Invented strategies serve students at least as well on standard tests.
Invented strategies are number-oriented rather than digit-oriented.
Invented strategies are usually left-handed rather than right-handed.
Invented strategies are flexible rather than rigid.
Understanding Fractions Connections meanings of operations multiple representations properties and relationships Computational Strategies discovering patterns
Part of a Whole -- Region, length, and set. Quotient -- 1/3 may mean 1 ÷ 3. Ratio -- 2/5 may mean the ratio of 2 to 5.
To interpret the meaning of a fraction, a/b we must: agree on the unit; understand that the unit is subdivided into b parts of equal size; and understand that we are considering a of the parts of the unit.
Region or area models Rectangular regions, geoboards, paper folding, base-ten blocks, grid paper, circular disks, pattern blocks, tangrams. Length or measurement models Number lines, Cuisenaire rods, paper strips, rulers, yard sticks, meter sticks, tape measure. Set models Counters, money, drawing Xs and Os.
Each student needs: Five 3-by-18-inch strips of construction paper. Fraction die (labeled 1/2, 1/4, 1/8, & 1/16) Cut, label, and initial.
Cover Up Uncover Recording Cover-Up Fraction Sentences Comparing. Give pairs of fractions and have students write <, >, or = to make a true sentence Equivalence. Have students write another fraction equivalent to a given fraction Operations. Write one fraction to complete a sentence
The ability to:
Represent fractions using words, models, diagrams, and symbols and make connections among the various representations.
Give other names for fractions and justify the procedures used to generate the equivalent forms.
Describe the relative magnitude of fractions by comparing fractions to common benchmarks, giving simple estimations, ordering a set of fractions, and finding a fraction between two numbers.
Closest to 0, 1/2, or 1
Give students fractions that are less then 1: 3/8, 1/6, or 7/9
Decide if the fraction is closest to 0, 1/2, or 1.
Ordering Fractions
Which is larger 5/8 or 5/10? Why?
Put these fractions in order from smallest to largest. 1/2 2/3 1/4 3/8 5/12
Find a fraction between 1/4 and 1/3.
Pattern Block Activities
Use pattern blocks to make a figure.
Let the whole figure be 1.
Tell what fraction of the whole figure each piece is.
Pattern Block Activities
Use pattern blocks to make a figure.
Decide on a unit. What is 1?
Tell what fraction is represented by the figure.
#1 Adding of Fractions
Build a model or draw a picture to show 2/3 + 1/4.
Tell a real-world situation or story to show the meaning of 2/3 + 1/4.
Compute 2/3 + 1/4.
#1 Subtracting Fractions
Build a model or draw a picture to show the meaning of 2/3 - 1/4.
Tell a real-world situation or story to show the meaning of 2/3 - 1/4.
Compute 2/3 - 1/4.
#2 Adding of Fractions
Build a model or draw a picture to show 1/2 + 5/6.
Tell a real-world situation or story to show the meaning of 1/2 + 5/6.
Compute 1/2 + 5/6.
#2 Subtracting of Fractions
Build a model or draw a picture to show the meaning of 5/6 - 1/2.
Tell a real-world situation or story to show the meaning of 5/6 - 1/2.
Compute 5/6 - 1/2.
#1 Multiplying of Fractions
Build a model or draw a picture to show the meaning of 3/4 x 1/2.
Tell a real-world situation or story to show the meaning of 3/4 x 1/2.
Compute 3/4 x 1/2.
#1 Dividing Fractions
Build a model or draw a picture to show the meaning of 1 1/2 ÷ 3/4.
Tell a real-world situation or story to show the meaning of 1 1/2 ÷ 3/4.
Compute 1 1/2 ÷ 3/4.
#2 Multiplying of Fractions
Build a model or draw a picture to show the meaning of 2/3 x 3/4.
Tell a real-world situation or story to show the meaning of 2/3 x 3/4.
Compute 2/3 x 3/4.
#1 Dividing Fractions
Build a model or draw a picture to show the meaning of 3/4 ÷ 1/2.
Tell a real-world situation or story to show the meaning of 3/4 ÷ 1/2.
Compute 3/4 ÷ 1/2.
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Knowing and Understanding