StandardsBased Teaching: Changing Perspectives

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Standards-Based TeachingChanging Perspectives

• Wamego, Kansas
• August 16, 2004
• David S. Allen Melisa Hancock,
• Kansas State University

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Math Standards

• Content Standards
• Number and Operations
• Algebra
• Geometry
• Measurement
• Data Analysis and Probability

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

Three Types of PS
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No matter how lucidly and patiently teachers
explain to their students, they cannot understand
for their students.
Schifter Fosnot (1993)
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Constructivism Basic-Tenets

• Knowledge is actively created or invented by the
  child
• Children create new mathematical knowledge by
  reflecting on their physical and mental actions.
• No true reality exists, only individual
  interpretations of the world. (most controversial
  tenet)
• Learning is a social process in which children
  grow into the intellectual life of those around
  them.
• When a teacher demands that students use set
  mathematical methods, the sense-making activity
  of students is seriously curtailed.

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Constructivism Two Major Goals

• Students should develop mathematical structures
  that are more complex, abstract, and powerful
  than the ones they currently possess so that they
  are increasingly capable of solving a wide
  variety of meaningful problems.
• Students should become autonomous and
  self-motivated in their mathematical activity.

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Consider This Problem
Consider the task of learning the multiplication
combination7 X 8 56
Talk with your partner to come up with as many
good ways as you can to think about the answer. A
good way is one that is simple enough to become
completely mental (no counting or finger tricks)
and should be mathematical.
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Instrumental Understanding
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Relational Understanding
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Consider This Problem
Memorize the following string of numbers2 5 8
1 1 1 4 1 7 2 0 2 3
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Implications for Teaching and LearningWhat does
a Constructivist teacher do?

• Create a mathematical environment
• Pose worthwhile mathematical tasks
• Use cooperative learning groups
• Use models and calculators as thinking tools
• Require justification of student responses
• Encourage discourse and writing
• Listen actively

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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)
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Problem Solving Goals and Objectives

• Familiarize Students with P.S.
• The four step process
• Strategies for solving problems
• Communicating thinking involved in solution
  process
• Teacher Behaviors
• Knowledge of the problem
• Knowledge of potential solutions
• Engaging students in the P.S. process
• Guiding the process
• Assessing the students P.S. process

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Problem Solving Goals and Objectives

• 3. Student Behaviors
• Involvement in the PS process
• Knowledge of and identification of potential
  strategies for solving specific types of problems
• Application of knowledge and skills needed to
  find a solution
• Communicating and justifying the solution as well
  as the process used to arrive at the solution
• Appropriate use of knowledge and experience
  gained for future problem solving

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Problem Solving Setting the Stage

• Persistence
• Knowing when to change directions
• Knowing what questions to ask when obstacles are
  met
• Problem Selection (Enthusiasm)
• Relevant problems create enthusiasm for problem
  solving
• Enthusiasm on the part of a teacher translates to
  a positive disposition for students.
• Now thats an interesting problem, I wonder how
  we can find the answer.

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Problem Solving Setting the Stage
3. Strategies a) Acquiring a variety of
strategies is essential to experiencing
success. b) Students should be provided with
instruction and practice in using a wide range of
strategies that they can draw upon. c) When
students are presented with a problem that does
not fit into the context of what they already
know, they need to know how to develop strategies
based on previously learned skills and concepts.
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Problem Solving Sample
How much time have you spent watching TV in
your lifetime? Estimate the amount of
time you have spent watching television in your
life. Include video games played on the
television screen, too. Write a report that
explains how you approached the problem, the
figuring you did, and how you came up with your
estimate. What benefits are there to using
this type of problem with your students?
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Problem Solving
Problem solving is at the core of any
mathematics curriculum it is integral to all
mathematical activity. As such, it should
permeate the entire mathematics program. Students
who are consistently presented with challenging
problems learn to develop and apply new
strategies. When they are also given
opportunities to communicate their strategies
with others and reflect on their thinking, their
problem solving abilities are further enhanced.
(Fennell et al. 2000)
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Reasoning and Proof

• Reasoning is a state of mind that causes students
  to explore, to justify, and to validate.
• Students are reasoning when they interpret data,
  when they solve problems, and when they view
  geometric patterns and shapes.
• When students are presented new problems, they
  use reasoning skills to apply previously acquired
  information and to test the validity of their
  solutions.
• Reasoning is the process by which students make
  sense of mathematics.

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Reasoning and Proof

• Reasoning begins with intuition.
• Intuition is used by even the youngest children
  as they begin to make sense of mathematics and
  should be encouraged at all levels.
• Reasoning and proof can be incorporated in the
  mathematics classroom through questioning.
• How did you get your answer? Tell me how you
  thought about that. Why does your solution work?
  Do you think that strategy will always work?

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Reasoning and Proof (Example)

• What is 75 of 80?
• Find the solution to this problem.
• Share your answer with your neighbor.
• Did you get the same solution?
• Did you use the same solution strategy?
• Share solution strategies.
• Process Standards

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Communication

• The communication skills of reading, writing,
  listening, and speaking provide the means for
  sharing ideas and promoting mathematical
  understanding.
• Provides students the opportunity to clarify
  their thinking and reinforce their comprehension
  of the concepts they are working with.
• By listening to their peers students are exposed
  to ideas they may not have thought of.
• Piaget believed that to develop their reasoning
  students must engage in social interaction.

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Communication

• In middle school, mathematics begins to become
  more abstract. New concepts need to be introduced
  conceptually, but students need to move from
  concrete representations to symbolic notation
  more quickly than in elementary school. Effective
  communication of ideas becomes even more
  important.
• Putting ideas down on paper is another means of
  helping students organize their thinking.
• Written reflection can be an important tool for
  teachers in assessing their students
  mathematical understanding. Process Standards

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Connections

• There are three aspects to making connections in
  mathematics.
• Connections are made when one mathematical idea
  is used to build another.
• How is counting related to addition, addition to
  subtraction, addition to multiplication,
  multiplication to area?
• Connections are made among different mathematical
  ideas.
• Teachers need to know what mathematics students
  learned previously in order to build on that
  knowledge. Teachers should also be aware of what
  their students will be studying in subsequent
  grades.

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Connections

• c) Connections are made between mathematics
  and contexts outside the field of mathematics.
• Mathematics permeates other curriculum areas
  and is found in the everyday experience outside
  of school. The use of shapes and patterns is
  prevalent in art and architecture measurement
  skills and classification skills are important in
  science measurement skills and knowledge of
  fractions are utilized in cooking and in building
  models and measurement skills, data gathering,
  and statistics are applied in the social
  sciences.
• (Fennell et al. 2000)

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Representation

• Representations provide vehicles for expressing
  and internalizing mathematical thought.
• Include physical objects, pictures, symbols
• Mental images, words, and ideas
• Formal/Informal representations-
• Conventional symbols, graphs, diagrams
• Informal forms are often invented by students as
  a way of making sense of mathematical ideas and
  communicating those ideas to classmates or the
  teacher.

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Representation

• How can you communicate the idea that
  adding more salt to the popcorn at the movie
  theatre increases drink sales?
• Students do not always see the
  mathematics they way we see it. Our formal
  education has caused us to see mathematics in a
  traditionally abstract or symbolic manner. We
  need to help students access mathematics as a
  product of their environment.
• Process Standards

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Problem Solving Defined!
Problem Solving means engaging in a task for
which the solution method is not known in
advance. (NCTM, 2000) A problem is a
situation in which a person is seeking some goal
and for which a suitable course of action is not
immediately apparent. (Marilyn Burns, 2001)
Solving problems takes place when students think
flexibly, creatively, and analytically to define,
examine, diagnose, and unravel complicated
problems. There must be some blockage on the part
of the potential problem solver. That is a
mathematical task is a problem only if the
problem solver reaches a point where he or she
does not know how to proceed.
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Three Problem Solving Approaches

• Teaching for problem solving.
• Teaching about problem solving.
• Teaching via problem solving.

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Teaching for Problem Solving

• Uses real-life problems as a setting in which
  students can apply and practice recently taught
  concepts and skills.
• The painter mixed the green paint using a 2 to 5
  ratio of blue to yellow paint. How much yellow
  paint is there in 1 gallon of green paint.
• Traditional problem-solving experiences familiar
  to most adults.

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Teaching About Problem Solving

• Refers to instruction that focuses on strategies
  for solving problems
• Polya, 1954
• Four Step Method
• Heuristics
• Process vs. Procedure
• Critical Thinking
• Examples

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Pedagogical Approach to Problem Solving

• Four Step
• Process
• Understanding the problem
• Choosing a strategy
• 3. Implementing a strategy
• 4. Finding and reporting a solution

• Blooms
• Taxonomy
• Knowledge
• Comprehension
• Application
• Analysis
• Synthesis
• Evaluation

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The value of a tool is in its usefulness.
Being able to do pencil-paper computation will
not serve students without the ability to
interpret a problem, analyze what needs to be
done, and evaluate the solution.
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If the only tool you have is a hammer,
everything around you looks like a nail.

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Tug-O-War
Acrobats, Grandmas, and Ivan Round 1 On one side
are four acrobats, each of equal strength. On the
other side are five neighborhood grandmas, each
of equal strength. The result is dead even. Round
2 On one side is Ivan, a dog. Ivan is pitted
against two of the grandmas and one acrobat.
Again its a draw. Round 3 Ivan and three
grandmas are on one side, and the four acrobats
are on the other. Who will win the third round?
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Frozen Yogurt A Manager who operates a
frozen yogurt shop has a vending machine with
three buttons. When these buttons are pushed,
each one dispenses a different flavor of frozen
yogurt. One of the buttons dispenses chocolate,
the second button vanilla, and the third buttons
dispenses todays special. When the shop operator
is extremely rushed, the manager uses chocolate
or vanilla as the special. Suppose the it is a
hectic day, all the buttons are mislabeled, and
the manager offers you a free carton of yogurt if
you can push one button and determine how to
correct all the labels. Which button would you
push?
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• Kansas Resources
• Mr. Allens Kansas History Class was
  working in groups to do research on Kansas
  history. The numbers below reflect the number of
  resources each group used.
• 5, 7, 10, 5, 14, 12, 7, 13, 5, 12
• Which statement is true for the given set of
  data?
• The mean is less than the median.
• The mode is less than the median.
• The mode equals the mean.
• The range equals the sum.

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Recommendations for Teaching About Problem Solving

• Heuristics
• Strategies taught in isolation are not meaningful
  to students.
• Allow students to identify or create meaningful
  solution strategies.
• Post strategies and refer to them often.
• Demonstrate the need to draw upon a wide variety
  of solutions strategies.
• Be selective with problems less is more
  quality over quantity

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Teaching via Problem Solving

• Most, if not all, important mathematics concepts
  and procedures can best be taught through problem
  solving.
• Synonymous with inquiry-based teaching in
  science.
• Teaching with problems is difficult.
• Tasks must be designed or selected each day.
• Difficult to plan more than a few days in
  advance.
• Traditional textbooks do not cater to this
  method.
• Requires well developed mathematical
  understanding on the part of the teacher.

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Teaching via Problem Solving

• Think about the number 6 broken into two
  different amounts. Draw a picture to show a way
  that six things can be in two parts. Think up a
  story to go with your picture. (K-2) (Van
  deWalle, 2004)

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Teaching via Problem Solving
2. Place an X on the number line about where
11/8 would be. Explain why you put your X where
you did. Perhaps you will want to draw and label
other points on the line to help explain your
answer. (3-6) (Van deWalle, 2004)
0
2
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Teaching via Problem Solving
3. I used two identical shapes to make a
rectangle. What might they have been? (Baker
Baker, 1991)
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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
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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Standards-Based TeachingChanging Perspectives
Wamego, Kansas August 16, 2004 David S. Allen
Melisa Hancock, Kansas State University