Hard Arithmetic is Not Deep Mathematics

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Hard Arithmetic is Not Deep Mathematics!

• Teaching for UNDERSTANDING
• Presented by
• Melisa Hancock
• melisa_at_ksu.edu

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History of Mathematics

• 1960s Arithmetic Test A logger cuts and sells
  a truckload of lumber for 100. His cost of
  production is 4/5 of that amount. What is his
  profit?
• 1970s New Math Test A logger exchanges a set
  (L) of lumber for a set (M) of money. The
  cardinality of Set M is 100. The Set C of
  production costs contains 20 fewer points. What
  is the cardinality of Set P of profits?
• 1980s Dumbed Down Math A logger cuts and
  sells a truckload of lumber for 100. His cost
  is 80, his profit is 20. Find and circle the
  number 20.

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History of Math, cont.

• 1990s version An unenlighted logger cuts down
• a beautiful stand of 100 trees in order to make
  a 20 profit. Write an essay explaining how you
  feel about this as a way to make money. Topic
  for discussion How did the forest birds and
  squirrels feel?
• 2004 version A logger sells a truckload of
  lumber for 100. Her cost of production is 120.
  How does Martha Stewart determine that her
  profit margin is 80?
• Teaching in 2010 El hachero vende un camion
  carga por 100. La cuested de production es . .
  . . . . . . . .

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NCTM Standards (1989)

• The Learning Principle makes it very clear that
  learning with UNDERSTANDING is both essential and
  possible. That is, ALL children can and must
  learn mathematics with understanding. It is
  impossible to predict the kinds of problems
  students will face in the future. The Learning
  Principle says that understanding is the only way
  to ensure that students will be able to cope with
  these unknown problems in the future.

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UNDERSTANDING MULTIPLICATION

• Individually, complete the modified Frayer
  Model (Link Sheet)
• Think-Pair-Shair with your neighbor, your ideas
  about multiplication.
• What do your students know about multiplication?
  
• What is important for your students to know and
  be able to do? Mentally?

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Problem . . .

• Since the mid-1800s mathematics books have been
  developed around the SHOW AND TELL method.
  Unfortunately, mathematics books have changed
  very little today. NCTM Standards have caused
  publishers to use appropriate Standards buzz
  words in their texts, but the focus is still on
  procedural knowledge rather than conceptual
  understanding. Lots of WHAT to do and HOW to do
  it.and very little WHEN to do it and WHY you
  do it!

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Challenge For You today

• Regardless of the textbook and you are using . .
• OR the way YOU were taught mathematics . . .TRY
• to incorporate the WHEN and HOW into your \
• teaching of mathematics . . by
• Focus on teaching for understanding application
  of knowledge. (Conceptual Knowledge balanced with
  Procedural Knowledge)
• Incorporate the NCTM Process standards in ALL of
  your lessons.
• Follow the 5-E Instructional Model.

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Teachers at Sunflower and Moonlight Elementary
certainly have this Vision!

• State and National standards are not programs to
  be implemented but are visions for improvement in
  mathematics.

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(Teaching ONLY standard algorithms)If the only
tool you have is a hammer, everything around you
looks like a nail.

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Welcome to Gardner Edgerton!! Founded
1856 Altitude
752 Population 7800 TOTAL
10,408
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Challenging Tasks

• There is no other decision that teachers make
  that has a greater impact on students
  opportunities to learn and on their perceptions
  about what mathematics is than the selection or
  creation of the tasks with which the teacher
  engages the students in studying mathematics.
  (Lappan Briars)

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Remember, Struggling in Mathematics is OK!!!!!!!!

• Struggling in mathematics is no more the enemy
  than sweating is in basketball . . . it just
  means you are
• IN THE GAME!!!!!!!
• (Kimberly Sutton)

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Problem Solving

• Problem solving is the cornerstone of school
  mathematics. Without the ability to solve
  problems, the usefulness and power of mathematics
  ideas, knowledge, and skills are severely
  limited. Students who can efficiently and
  accurately multiply but who cannot identify
  situations that call for multiplication are not
  well prepared.
• Unless students can solve problems, the facts,
  concepts, and procedures they know are of little
  use.

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Mathematical Knowledge

• 1.Conceptual Knowledge (logical relationships,
  representations, an understanding and ability to
  talk, write and give examples of these
  relationships, etc.)
• 2. Procedural Knowledge (knowledge of rules and
  procedures used in carrying out routine
  mathematical tasks and the symbols used to
  represent mathematics)

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We learned something amazing today. All those
things you do with calculators, you can do in
your head, too!
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Problem Broken Calculators

• Imagine that you have a broken calculator on your
  desk. Pretend the division key is broken. Try
  to make .5 without using the division key ( or
  /).
• Try to make 1000 without using either 0 or 1.
  Imagine that the 1 and 0 keys are broken.

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ARE YOU MENTAL?!!!!!
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Procedural Knowledge

• It is generally accepted that procedural rules
  should never be learned in the absence of a
  concept.
• (John A. Van De Walle)

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Where Do We Begin?

• It begins with YOU!
• Becoming an effective teacher unquestionably
  requires a thorough understanding of mathematics
  content.
• (NCTM Content Standards AND Process Standards)
• 2. Focus on Understanding and Application
• Our goal should be to find instructional
  practices to enlighten students about math
  concepts practices that will assist them in
  applying BOTH calculation skills and reasoning
  skills to real math problems, as well as spark
• an interest in further study. (5E Model)

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Reflecting On Our Own Beliefs About
Mathematics

• Misinformed beliefs, early in my career
• Practice Makes Perfect
• Mastering Calculations Is the Ultimate Goal
• Math is About Getting the Right Answer
• Math is a Series of Isolated Skills
• You Must Know Basic Skills Before You Can Learn
  to Solve Problems
• First One Finished Wins
• The Best Mathematicians Can Work Calculations in
  Their Heads
• Teachers Should Tell Students How to Do Math
• Math is Done Just in Math Class
• Some Are Good at Math, Some are Not

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What Does Mean Really Mean?Standard 4 Data
Analysis and Probability

• Sample problem from a typical textbook
• Problem 1 The children in the Hancock family
  are aged 27, 39, 45, 51,
• 33. Find the average age for
  the Hancock family.
• What skills do students need to know in order to
  solve this problem?
• Problem 2 There are five children in a family.
  The mean average age is 39.
• How old might the children be? Use
  your Unifix cubes to solve
• and be ready to share your
  solution.
• What skills are needed to solve this problem?
• (Write-Draw-Talk-Rewrite)

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MEAN

• At our KSU Tailgate party last Saturday I
• purchased the following items
• Hotdogs 5.00
• Coke 2.00
• Buns 1.00
• Chips 4.00
• USING ONLY UNIFIX CUBES SHOW HOW YOU WOULD
• FIND THE MEAN AVERAGE SPENT AT THE TAILGATE
• PARTY. BE READY TO EXPLAIN YOUR REASONING.

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MEAN

• Listed below are the scores from the
• last mathematics test. Using mental
• math, what is the mean score for
• the class? Explain your process.
• 59, 70, 74, 89, 90, 90, 95, 96, 99

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Brainstorming Group Discussion!

• What characteristics would you expect to see in
  your students that would represent number sense?
• When considering your studentswhat do you see
  that is missing with respect to number sense
  development?
• Graphic Organizer My Final Answer

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• A student with Number Sense will look at a
  problem holistically before confronting details.
• Example
• 1 2/3 3/4 1/3

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Addressing Number Sense K-6

• Estimation
• Measurement
• Mental Computation
• Multiple Representation
• Number Relationships
• Relative Size

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Mental Math Activities

• Mental computation is simply an invented strategy
  that is done mentally.

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169 14

• IndividuallySolve the division problem using two
  strategies OTHER THAN the conventional algorithm.
• Unpack the mathematical knowledgeneeded for
  completion of this problem.
• Think-Pair-Sharethe knowledge, concepts,
  connections needed to UNDERSTAND division. Write
  on chart paper.
• Group Sharecomplete the graphic organizer and
  transfer to chart paper, be ready to share your
  package of division knowledge needed for
  UNDERSTANDING (connections among ideas,
  
• concepts, procedures, relationships,
  etc.

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Connections Understanding

• When these connections are made . . .
  mathematics is portrayed as a unified body of
  knowledge rather than as isolated topics or a set
  of procedures to follow.

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What skills are needed BEFORE students can solve
problems mentally?Write-Pair-Share
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Beginning Number Sense in K-1

• Experiencing Numbers in the Early Grades
• Spatial Relationships
• One and Two More/Less Than
• Benchmark Numbers
• Part-Part-Whole Relationships

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Addressing Number Sense in 2-4

• Estimation
• Measurement
• Mental Computation
• Multiple Representation
• Number Relationships
• Relative Size

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Center Activities Resources

• Individually or as a Group
• Do some of the activities, in groups, discuss and
  share.
• Look through resources.
• Whole Group sharing, questions, etc.

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Center Reflections

• Discuss in your group the following questions
• 1. What are the important mathematical ideas
• and understanding that this activity
• promotes?
• 2. What prior knowledge is required in order
• for students to be successful in completing
• and understanding this activity?
• 3. Is this an assessed item? Does it lead
• directly to an assessed item (at a different
• grade level?)
• 4. Where does this fit in your curriculum?

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Recommendations for Addressing Number Sense In
the Classroom

• Conduct a number sense activity every day.
• Address number sense in ALL content strands.
• Incorporate ALL process standards.
• When talking about number sense avoid doing all
  the talking.
• Think-aloud is a good strategy to model a
  difficult task for students.
• When estimating avoid dictating procedures.

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Sure-Fire Rules For Problem Solving

• Most problems are addition.

2. If more than two numbers are given, it has
to be addition.

• When only two numbers are given and they are
  about the same
• subtract.

4. Consider subtraction when money is involved,
particularly if one amount is a round
figure like 50 or 10.00.

• If two numbers are given and one is much larger
  than the other,
• try division.

• Very few problems involve division with a
  remainder. When
• you get a remainder, cross out
  the division and
• multiply instead.

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Sure-Fire Rules For Problem Solving,cont.
7. If you see a fraction, invert it.
8. If you see a decimal, move it.
9. If you see a negative or positive sign, change
it.
10. If the Rules 1-9 do not seem to work, make
one last desperate attempt. Take the set of
numbers in the problem and perform about
two pages of random operations using these
numbers. You should circle about five or
six answers on each page just in case one
of them happens to be the answer. You might get
some partial credit for trying hard.

• Never, never spend too much time solving
  problems. This
• set of rules will get you through
  even the longest
• assignment in no more than 10
  minutes with very little
• thinking!

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Skills are to mathematics what scales are to
music or spelling is to writing. The objective
of learning is to write, to play music, or to
solve problemsnot just to master skills.
Everybody Counts (1989)