Geometric Thinking and Concepts

Geometric Thinking

• What makes shapes alike and different are determined by geometric properties
• parallel
• perpendicular
• line symmetry
• rotational symmetry
• similar 
• congruent
• Shapes can be moved in a plane or in space
• Shapes can be described in terms of their location in a plane or in space
• Shapes can be seen from different perspectives

Resources and Games

http://www.kidsmathgamesonline.com/geometry.html
http://www.gamequarium.com/geometry.html
www.geogebra.org
http://www.superteacherworksheets.com/geometry.html
http://www.math-aids.com/Geometry/

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7^th ed.). Boston, MA: Pearson.

Decimals and Percents

What's Important?

• Decimals are another way of writing fractions
• Base-ten place-value system extends in two directions; big and small. 
• Decimal point indicates the units position
• Percents are hundredths and another way of writing fractions and decimals

Decimals, Fractions and Percentages are just different ways of showing the same value:

A Half can be written... As a fraction: 1/2 As a decimal: 0.5 As a percentage: 50%

A Quarter can be written... As a fraction: 1/4 As a decimal: 0.25 As a percentage: 25%

Resources, Games, and Worksheets

http://www.purplemath.com/modules/percents.htm

http://www.math-drills.com/decimal.shtml

http://www.math-play.com/Fractions-Decimals-Percents-Jeopardy/fractions-decimals-percents-jeopardy.html

http://www.mathplayground.com/Decention/Decention.html

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7^th ed.). Boston, MA: Pearson.

Algebraic Thinking

What is important?

• Algebra is a tool used to generalize arithmetic and to represent patterns in the world
• Symbolism must be well understood to be successful in Algebra
• We should use generalized methods to compute 
• a+b=b+a
• Patters should be recognized, extended and generalized
• For every input there is a unique output

Resources and Games:

http://www.mathplayground.com/algebraic_reasoning.html

http://insidemathematics.org/index.php/operations-and-algebraic-thinking-oa

http://www.math-play.com/Algebra-Math-Games.html

http://hotmath.com/games.html

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7^th ed.). Boston, MA: Pearson.

Fraction Concepts

Concepts that must be understood:

• Fractions are Part of a Whole
• Ratios
• Division

Three categories of models:

• Area
• Length
• Quantity or Set

Partitioning is dividing a shape into equal – sized pieces.  This can be done with counters (or any other object that are similar) by dividing the total number of counters into smaller groups of equal numbers.  If the students are working with a specific shape they simply divide that shape into equally sized parts.

Iteration means to look at and discover how many parts compare to the whole.  The best way to help students with fractions in this way is to get them use to thinking of fractions as how many of what.  The “what” is the denominator and the “how many” is the numerator.  Iteration can be done with strips of paper.  Give them a strip of paper and tell them what part of a whole it is.  Then have them find other parts of the whole as well as mixed number fractions based on the strip they were given.

Estimating Fractions

Some ways I can help support student’s development of estimation with fractions are asking them daily questions that forces them to look at fractions of the whole class and estimate a part of the class.  I could ask, for example, what fraction of our class is wearing jeans today?  Another way is to draw pictures with parts of the picture shaded.  Then ask the students to estimate what fraction is either shaded or un-shaded.

Equivalent Fractions

Two equivalent fractions are two ways of describing the same amount using different-sized fractional parts.

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7^th ed.). Boston, MA: Pearson.

Whole-Number and Place-Value Concepts

Place-Value Concepts
There are a ton of resources for games to help students place-value concepts.  Here are a few:

http://gamequarium.com/placevalue.html
http://education.jlab.org/placevalue/
http://www.math-play.com/place-value-games.html
http://www.funbrain.com/tens/

• Sets of ten can be perceived as single entities.  
• 3 sets of tens and 2 ones is a way of describing 32
• The position of digits in numbers determine what they represent and which size group they count
• There are patterns to the way numbers are formed. The 1-9 sequence is prevalent in each decade.

Base 10 Concepts

Resources for Base 10:

http://www.learningbox.com/base10/baseten.html

http://www.math-drills.com/baseten.shtml

Mastering Basic Facts

Addition and Subtraction
Counting

• Addition
• Direct modeling
• counting all
• counting on from first
• counting on from larger
• Counting abstractly
• counting all 
• counting on from first 
• counting on from larger
• Subtraction
• Counting objects
• separating from
• separating to 
• adding on
• Counting fingers
• counting down
• counting up
• Counting abstractly
• counting down
• counting up

Reasoning 

• Addition
• Properties
• a+0=a
• a+1=next whole number
• commutative property
• Known-fact derivations
• 5+6=5+5+1
• 7+6=7+7-1
• Redistributed derived facts
• 7+5=7+(2+3)=(7+3)+2=10+2=12
• Subtraction
• Properties
• a-0=a
• a-1=previous whole number
• Inverses.complement of know addition facts
• 12-5 is known because 5+7=12
• Redistributed derived facts
• 12-5=(7+5)-5=7+(5-5)=7

Retrieval

• Addition and Subtraction
• Retrieval from long-term memory

Multiplication and Division

Multiplication

• Facts can be mastered by relating to existing knowledge
• Repeated addition
• Doubles
• 2 means double
• Zeros and Ones
• Any number multiplied by zero is zero
• Any number multiplied by one is itself
• Nifty Nines
• Look for patterns
• sum of digits is always 9
• tens digit of product is one less the then other factor
• Using Known Facts to Derive Other Facts

Division

• Repeated Subtraction
• How can we use multiplication to help with division?

Additional Resources

Multiplication and Division
http://mathwire.com/numbersense/bfacts.html
http://www.math-games-and-activities-at-home.com/basicmathfacts.html

http://www.heinemann.com/shared/onlineresources/E02962/oconnellmultweb.pdf

Building Assessment into Instruction

Why Do We Assess?


Types of Assessment

• Performance-Based Assessments
• Problem based tasks
• Rubrics and Performance Indicators
• Comparing students work to what is expected
• Rubric - a framework that can be changed based on the students or the task
• Performance Indicators - Description of what each task looks like at every level of the rubric
• Observation Tools
• Anecdotal notes
• Observation rubric
• Checklists for individual students
• Check lists for full class
• Writing and Journals
• Teach the value of writing
• Journals
• Writing prompts and ideas
• Self-assessment
• Diagnostic Interviews
• Getting in-depth information on the student's knowledge
• Tests
• Should only be used as one aspect of assessing the students

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7^th ed.). Boston, MA: Pearson.

Developing Meanings for Operations

Addition and Subtraction

Problem Structures

• Join Problems
• starting amount
• change amount
• result

Frank has 10 bananas.  Tom gave him some more. Frank now has 15 bananas.  How many bananas did Tom give Frank?

• Separate Problems
• initial amount is the whole
• separate parts

Sue had 17 dogs. She gave 6 to Sam.  How many dogs does Sue have now?

• Part-Part-Whole Problems
• two parts that are combined into one

Kevin has 6 pennies and Tony has 7 quarters.  How many coins do they have together?

• Compare Problems
• comparison of two quantities

Gina has 12 frogs and Andrea has 8 frogs.  How many more frogs does Gina have?

Multiplication and Division

Problem Structures

• Equal-Group Problems
• When the number and group size is known - multiplication
• When the number of sets or the size of sets is unknown - division

Sara has 4 bags of potatoes.  There are 10 potatoes in each bag.  How many potatoes does she have?

• Comparison Problems
• Multiple copies of the other
• One set is a multiple of the other

Tom found 24 Easter eggs.  He found 6 times as many as Lou.  How many eggs did Lou find?

• Combination Problems
• Counting the number of possible pairings

Kathy bought 10 pairs of pants and 5 shirts. All of the pieces of clothing can be worn together.  How many different outfits does Kathy have?

• Area and Other Product-of-Measures Problems
• The product is a different type of unit from the other two factors

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7^th ed.). Boston, MA: Pearson.

Problem Solving and Planning

Teaching through Problem Solving

• Most math concepts can be taught by problem solving
• Four-step problem-solving process
• Understanding the problem
• What is the problem about?
• Devising a plan
• How can I solve the problem?
• Are there multiple ways to solve the problem?
• Carrying out the plan
• Solve the problem based on your plan
• Looking back
• Does my answer make sense?
• Let the students do the talking
• How did you solve the problem?
• Why did you solve it this way?
• Why do you think your solution is correct?
• Explain how your solution makes sense.

Teaching Resource - Worksheets and Games

http://www.teachingideas.co.uk/maths/contents_problemsolving.htm



Planning a Problem-Based Lesson



Planning for all Learners

• Make accommodations and modifications
• An accommodation is a provision to the lesson but it does not alter the task
• A modification changes the problem or the task
• Use differentiating instruction
• Instruction should support all students
• learning styles
• cultural influences
• language
• Flexible groupings
• Allow students to collaborate
• You may use random groupings or group according to ability
• Encourage the entire group to answer questions of individual students

Example

http://www.exemplars.com/blog/education/2-tips-for-planning-successful-problem-based-learning-in-your-math-classroom

Additional Information

http://www.edutopia.org/project-based-learning

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7^th ed.). Boston, MA: Pearson.

Number Sense

Pre-K and Kindergarten Number Development

• Teach counting using finger, toys, blocks, anything that can be held and moved around
• Compare using less, same and more
• Have students make set of less, same and more
• Use games to help develop counting skills
• Connect the word with the visual number
• Two is 2

Relationships among Numbers 1-10

• Patterned sets: 
• This relationship deals with children recognizing patterns of numbers without having to count them.  I good example are the numbers patterns on dice.  One activity for this would be to make dot plates with patterns of numbers from 1-10 on each plate.  Have the children get familiar with the patterns, once they are then play a game where you hold up a plate for a short period of time and the first child to tell you the number of dots wins.  This could be done in two teams with two children going against each other at once or as an individual activity.
•  One and two more, one and two less: 
• This strategy involves the children being able to recognize the relationships of numbers that are one or two away from each other.  If given the number 4, the child should be able to tell you that 6 is two more and 3 is one less without having to put too much thought into it.  An activity to do to help with this concept is to have the children get into groups.  Each group is given 10 counters, a cup, a deck of cards #’s 3-10, and a deck of cards with 1 less, 2 less, 1 more, 2 more, and zero written on them.  The first child will draw a number card and put that number of counters in the cup.  The second child will draw from the one and two more, one and two less pile.  They will complete the action and then the children will guess how many counters are still in the cup.
• Anchors or “benchmarks” of 5 and 10: 
• The number 10 plays a big role in our numerical system.  Children need to learn the relationship between 5 and 10 and how they anchor the number system.  One activity to help them see this relationship is to use the 5 frame and 10 frames in order to discover relationships between these anchors.  In the 5 frame the children are only allowed to use 5 counters.  When they remove a counter from the frame have them talk about the relationship between the number of counters and the number of frames.  Once they have a grasp on 5 frames they can start to use 10 frames in the same way.

Relationships for numbers 10-20

• The idea of the tens as a set of ten and some more
• In order to develop this concept with children they need to be able to play with the numbers.  I like the idea of having a chart with 20 squares in two rows of ten, then giving the child 20 counters.  Give them numbers to represent on the chart.  See if they can make the connection.  This can be a very guided exercise, but pointing out the sets of tens can help them see that teens are a set of ten plus some.
• Extension of the one-more/one-less concept of the tens
• With this idea the child will need to understand that teens are a set of ten and some more numbers.  You can then show them the connection of if 6 is one more than 5 then 16 is one more than 15. 

Games
http://mathwire.com/games/numbsensegames.html

Worksheets
http://www.math-drills.com/numbersense.shtml

Resource for Additional Information
http://topnotchteaching.com/experts/guide-to-develop-number-sense/



Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7^th ed.). Boston, MA: Pearson

What it Means to Know and Do Mathematics

Traditional Views and New Approaches: Math Makes Sense!

• Most people believe that mathematics is associated with certainty; that knowing mathematics means being able to get the right answer
• Math must make sense

Constructivist Theory - Based on the work of Jean Piaget

• Learners are creators of their own learning
• Assimilation - new concept fits with prior knowledge
• Accommodation - new concept does not fit with prior knowledge
• Every learner will construct ideas differently.
• Give learners freedom to create their own ideas

Sociocultural Theory - Based on the work of Lev Vygotsky

• Learners move ideas into his or her own psychological realm (Forman, 2003)
• Semiotic Mediation - How information moves from a social plane to an individual plane.
• Learning depends on the learner, the classroom environment  the culture in and out of the classroom. 

Implications for Teaching Mathematics

• build new knowledge from prior knowledge
• provide opportunities to talk about mathematics
• build in opportunities for reflective thought
• encourage multiple approaches
• treat errors as opportunities for learning
• scaffold new content
• honor diversity

Understanding Mathematics

• Use multiple methods to solve problems
• Pictures
• Written Symbols
• Manipulative models
• Real-world situations
• Oral language

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7^th ed.). Boston, MA: Pearson

NCTM Standards

NCTM and the three Standards Documents

• NCTM  is the organization that has the most influence on what happens in school mathematics. 

Professional Standards for Teaching Mathematics - 6 Principles

• The Equity Principle 
• There needs to be strong support for all students to have mathematical excellence

• The Curriculum Principle 
• This must be coherent, focused on important mathematics, and expressed across all grade levels. 

• The Teaching Principle 
• Teachers must understand what the students know and what they need to learn.  They must then challenge the students while supporting them to learn the material.

• The Learning Principle 
• The students must understand the mathematics they are learning

• The Assessment Principle 
• Assessment should be done for the students, not just to the students.

• The Technology principle 
• Teachers should use technology as a key component of the mathematical teaching

The Five Process Standards

• Problem Solving
• Reasoning and Proof
• Communication
• Connections
• Representation 

Other Influences on School Mathematics

• Since the early nineties, the pressures influencing school mathematics have become much more complex. 

• The TIMSS data has caused much of the concern with the popular press pointing out that most industrialized countries significantly outperform U.S students in mathematics and science.

• At least 20 countries performed better than the United States which found itself in a group of 14 countries in the middle of the pack. 

• U.S fourth-grade students did much better comparatively than did those at the eighth grade and high school level. 

• The National Assessment of Educational Progress (NAEP) offers us ongoing indications of what American students are learning. 

• This data suggests that while we continue to show improvement, we are not near where we want to be. Both TIMSS and NAEP are referred to at various times throughout the book. 

• Pressures on teachers from state testing programs and the requirements of NCLB are having a significant influence on what is happening in mathematics classrooms. It is difficult to make general statements concerning these influences across states. Students will undoubtedly have heard of NCLB in their general curriculum course. Finally, the power of the textbook being used in the classroom cannot be ignored. The text points out that there are a number of "standards-based curriculums" that have been developed with NSF and other monies. These programs are more in alignment with the NCTM Standards. A listing of the three elementary and five middle school programs can be found at the end of the chapter. You may want to use some of the "excerpts" from either Investigations or Connected Mathematics that can be found in every chapter in section two.

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7^th ed.). Boston, MA: Pearson