The Difference it Makes

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The Difference it Makes

• Phil Daro

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Catching Up

• Students with history of going slower are not
  going to catch up without spending more time and
  getting more attention.
• Who teaches whom.
• Change the metaphor not a gap but a knowledge
  debt and need for know-how. The knowledge and
  know-how needed are concrete, the stepping stones
  to algebra.

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Action

• What is your plan to change the way you invest
  student and teacher time?
• What additional resources are you adding to the
  base (time)?
• How are you making the teaching of students who
  are behind the most exciting professional work in
  your district?

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System or Sieve?

• A system of interventions that catch students
  that need a little help and gives it
• Then catches those that need a little more and
  gives it
• Then those who need even more and gives it
• By layering interventions, minimize the number
  who fall through to most expensive

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Intensification
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Dylan Wiliam on Instructional Assessment

• Long-cycle
• Span across units, terms
• Length four weeks to one year
• Medium-cycle
• Span within and between teaching units
• Length one to four weeks
• Short-cycle
• Span within and between lessons
• Length
• day-by-day 24 to 48 hours
• minute-by-minute 5 seconds to 2 hours

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Strategies for increasing instructional
assessment (Wiliam)
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Intensification
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Why do students struggle?

• Misconceptions
• Bugs in procedural knowledge
• Mathematics language learning
• Meta-cognitive lapses
• Lack of knowledge (gaps)
• Disposition, belief, and motivation (see AYD)

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Why do students have to do math. problems?

• to get answers because Homeland Security needs
  them, pronto
• I had to, why shouldnt they?
• so they will listen in class
• to learn mathematics

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Why give students problems to solve?

• To learn mathematics.
• Answers are part of the process, they are not the
  product.
• The product is the students mathematical
  knowledge and know-how.
• The correctness of answers is also part of the
  process. Yes, an important part.

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Wrong Answers

• Are part of the process, too
• What was the student thinking?
• Was it an error of haste or a stubborn
  misconception?

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Three Responses to a Math Problem

• Answer getting
• Making sense of the problem situation
• Making sense of the mathematics you can learn
  from working on the problem

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Answers are a black holehard to escape the pull

• Answer getting short circuits mathematics, making
  mathematical sense
• Very habituated in US teachers versus Japanese
  teachers
• Devised methods for slowing down, postponing
  answer getting

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Answer getting vs. learning mathematics

• USA
• How can I teach my kids to get the answer to this
  problem?
• Use mathematics they already know. Easy,
  reliable, works with bottom half, good for
  classroom management.
• Japanese
• How can I use this problem to teach mathematics
  they dont already know?

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Teaching against the test

• 3 5
• 3 8
• 5 8
• 8 - 3 5
• 8 - 5 3

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• Anna bought 3 bags of red gumballs and 5 bags
• of white gumballs. Each bag of gumballs had
• 7 pieces in it. Which expression could Anna use
• to find the total number of gumballs she bought?
• A (7 X 3) 5
• B (7 X 5) 3
• C 7 X (5 3)
• D 7 (5 X 3)

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• An input-output table is shown below.
• Input (A) Output (B)
• 7 14
• 12 19
• 20 27
• Which of the following could be the rule for the
  input-output table?
• A. A 2 B
• B. A 7 B
• C. A 5 B
• D. A 8 B
• SOURCE Massachusetts Department of Education,
  Massachusetts Comprehensive Assessment System,
  Grade 4, 39, 2006.

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Butterfly method
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Use butterflies on this TIMSS item

• 1/2 1/3 1/4

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Foil FOIL

• Use the distributive property
• It works for trinomials and polynomials in
  general
• What is a polynomial?
• Sum of products product of sums
• This IS the distributive property when a is a
  sum

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Answer Getting

• Getting the answer one way or another and then
  stopping
• Learning a specific method for solving a specific
  kind of problem (100 kinds a year)

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Answer Getting Talk

• Wadja get?
• Howdja do it?
• Do you remember how to do these?
• Here is an easy way to remember how to do these
• Should you divide or multiply?
• Oh yeah, this is a proportion problem. Lets set
  up a proportion?

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Canceling

• x5/x2 xx xxx / xx
• x5/x5 xx xxx / xx xxx

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Misconceptions

• where they come from and how to fix them

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Misconceptions about misconceptions

• They werent listening when they were told
• They have been getting these kinds of problems
  wrong from day 1
• They forgot
• The other side in the math wars did this to the
  students on purpose

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More misconceptions about the cause of
misconceptions

• In the old days, students didnt make these
  mistakes
• They were taught procedures
• They were taught rich problems
• Not enough practice

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Maybe

• Teachers misconceptions perpetuated to another
  generation (where did the teachers get the
  misconceptions? How far back does this go?)
• Mile wide inch deep curriculum causes haste and
  waste
• Some concepts are hard to learn

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Whatever the Cause

• When students reach your class they are not blank
  slates
• They are full of knowledge
• Their knowledge will be flawed and faulty, half
  baked and immature but to them it is knowledge
• This prior knowledge is an asset and an
  interference to new learning

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Second grade

• When you add or subtract, line the numbers up on
  the right, like this
• 23
• 9
• Not like this
• 23
• 9

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Third Grade

• 3.24 2.1 ?
• If you Line the numbers up on the right like
  you spent all last year learning, you get this
• 3.2 4
• 2.1
• You get the wrong answer doing what you learned
  last year. You dont know why.
• Teach line up decimal point.
• Continue developing place value concepts

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Fourth and Fifth Grade

• Time to understand the concept of place value as
  powers of 10.
• You are lining up the units places, the 10s
  places, the 100s places, the tenths places, the
  hundredths places

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Stubborn Misconceptions

• Misconceptions are often prior knowledge applied
  where it does not work
• To the student, it is not a misconception, it is
  a concept they learned correctly
• They dont know why they are getting the wrong
  answer

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Research on Retention of Learning Shell Center
Swan et al
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A whole in the head
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A whole in the whose head?


4/7
3/4 1/3
4/7
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The Unit oneon the Number Line
0 1 2 3
4
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Between 0 and 1
0 1/4 3/4 1 2
3 4
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Adding on the ruler
1/3 2/3 1
0 1/4 2/4 3/4 1
2 3
4
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Differentiating lesson by lesson

• The arc of the lesson

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The arc of the lesson

• Diagnostic make differences visible what are
  the differences in mathematics that different
  students bring to the problem
• All understand the thinking of each from least
  to most mathematically mature
• Converge on grade -level mathematics pull
  students together through the differences in
  their thinking

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Next lesson

• Start all over again
• Each day brings its differences, they never go
  away

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Lesson Structure

• Pose problem whole class (3-5 min)
• Start work solo (1 min)
• Solve problem pair (10 min)
• Prepare to present pair (5 min)
• Selected presents whole cls (15 min)
• Close whole cls (5 min)

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Posing the problem

• Whole class pose problem, make sure students
  understand the language, no hints at solution
• Focus students on the problem situation, not the
  question/answer game. Hide question and ask them
  to formulate questions that make situation into a
  word problem
• Ask 3-6 questions about the same problem
  situation ramp questions up toward key
  mathematics that transfers to other problems

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What problem to use?

• Problems that draw thinking toward the
  mathematics you want to teach. NOT too routine,
  right after learning how to solve
• Ask about a chapter what is the most important
  mathematics students should take with them? Find
  a problem that draws attention to this
  mathematics
• Begin chapter with this problem (from lesson 5
  thru 10, or chapter test). This has diagnostic
  power. Also shows you where time has to go.
• Near end of chapter, external problems needed,
  e.g. Shell Centre

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Solo-pair work

• Solo honors thinking which is solo
• 1 minute is manageable for all, 2 minutes creates
  classroom management issues that arent worth it.
  
• An unfinished problem has more mind on it than a
  solved problem
• Pairs maximize accountability no place to hide
• Pairs optimize eartime everyone is listened to
• You want divergance diagnostic make differences
  visible

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Presentations

• All pairs prepare presentation
• Select 3-5 that show the spread, the differences
  in approaches from least to most mature
• Interact with presenters, engage whole class in
  questions
• Object and focus is for all to understand
  thinking of each, including approaches that
  didnt work slow presenters down to make
  thinking explicit
• Go from least to most mature, draw with marker
  correspondences across approaches
• Converge on mathematical target of lesson

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Close

• Use student work across presentations to state
  and explain the key mathematical ideas of lesson
• Applaud the adaptive problem solving techniques
  that come up, the dispositional behaviors you
  value, the success in understanding each others
  thinking (name the thought)

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The arc of a unit

• Early diagnostic, organize to make differences
  visible
• Pair like students to maximize differences
  between pairs
• Middle spend time where diagnostic lessons show
  needs.
• Late converge on target mathematics
• Pair strong with weak students to minimize
  differences, maximize tutoring

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Each lesson teaches the whole chapter

• Each lesson covers 3-4 weeks in 1-2 days
• Lessons build content by
• increasing the resolution of details
• Developing additional technical know-how
• Generalizing range and complexity of problem
  situations
• Fitting content into student reasoning
• This is not spiraling, this is depth and
  thoroughness for durable learning

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making sense of math. problems
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Word Problem from popular textbook

• The upper Angel Falls, the highest waterfall on
  Earth, are 750 m higher than Niagara Falls. If
  each of the falls were 7 m lower, the upper Angel
  Falls would be 16 times as high as Niagara Falls.
  How high is each waterfall?

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Imagine the Waterfalls Draw
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Diagram it
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The Height of Waterfalls
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Heights
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Height or Waterfalls?
750 m.
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Heights we know
750 m.
7 m.
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Heights we know and dont
750 m.
d
d
7 m.
7 m.
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Heights we know and dont
750 m.
d
d
7 m.
7 m.
Angel 750 d 7 Niagara d 7
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Same height referred to in 2 ways
16d 750 d
750 m.
16d
d
d
7 m.
7 m.
Angel 750 d 7 Niagara d 7
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d ?
16d 750 d 15d 750 d 50
750 m.
16d
d
d
7 m.
7 m.
Angel 750 50 7 807 Niagara 50 7 57
Angel 750 d 7 Niagara d 7
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Activate prior knowledge

• What knowledge?
• Have you ever seen a waterfall?
• What does water look like when it falls?

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What is this problem about?
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What is this problem about?

• HEIGHT!
• Delete waterfalls and it does change the
  problem at all. Replace waterfall with flagpoles,
  buildings, hot air balloonsit doesnt matter.
• The prior knowledge that needs to be activated is
  knowledge of height.

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Bad Advice

• eliminate irrelevant information
• Before you have made sense of the situation, how
  would you know what is relevant?
• After you have made sense, you are already past
  the point of worrying about relevance.

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What mathematics do we want students to learn
from work on this problem

• Sasha went 45 miles at 12 mph. How long did it
  take?

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that they can use on this problem?

• Xavier went 85 miles in two and a half hours.
  Going at the same speed, how long would it take
  for Xavier to go 12 miles.

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Teaching to diagram

• Teaching student to create a diagram about the
  relationships of the quantities in the problem
  that helps them create a mental mathematical
  model of that situation.
• Teach diagramming to one student at a time then
  to partners then to larger groups
• As a group

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Specific techniques

• What does that phrase mean? (pointing to a phrase
  that refers to a quantity).
• Play the naïve student who doesnt understand the
  situation.
• This is what Harold did is he right?
• Can you show me in a diagram?
• Explain your diagram to me
• Where is (quantity) in your diagram?
• Can you label you diagram?
• What are the quantities and how are they related?

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Water Tank

• We are pouring water into a water tank. 5/6
  liter of water is being poured every 2/3 minute.
• Draw a diagram of this situation
• Make up a question that makes this a word problem
  

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Test item

• We are pouring water into a water tank. 5/6
  liter of water is being poured every 2/3 minute.
  How many liters of water will have been poured
  after one minute?

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Where are the numbers going to come from?

• Not from water tanks. You can change to gas
  tanks, swimming pools, or catfish ponds without
  changing the meaning of the word problem.

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Numbers given, implied or asked about

• The number of liters poured
• The number of minutes spent pouring
• The rate of pouring (which relates liters to
  minutes)

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Diagrams are reasoning tools

• A diagram should show where each of these numbers
  come from. Show liters and show minutes.
• The diagram should help us reason about the
  relationship between liters and minutes in this
  situation.

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• The examples range in abstractness. The least
  abstract is not a good reasoning tool because it
  fails to show where the numbers come from. The
  more abstract are easier to reason with, if the
  student can make sense of them.

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goal

• Our goal is to teach students to make sense of ,
  produce and reason with abstract diagrams that
  show all the numbers, their relationships.

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• A good sense making practice is to first make a
  more concrete diagram in early sense making, then
  revise it to a more abstract diagram for
  reasoning purposes.
• A good teaching practice is to have students
  compare and discuss different diagrams for the
  same problem.

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Word problems are a genre of text

• Read poems differently than we read novels or
  instructions or a movie review or a recipe for
  corn fritters
• Genre are contracts between writer and reader
  writer makes assumptions about how the reader
  will read reader needs to make the right
  assumptions
• Genre knowledge must be taught and learned

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Word problem genre assumptions

• The context of word problems is this
• I am reading this for math class
• This is about numbers of who cares what
• Focus language effort on
• Where are the numbers coming from in this
  situation? Domains numbers of, units
• Numbers expressed as phrases in text correspond
  to mathematical phrases (expressions)
• The verb is equals (,- are conjunctions)
• Can I find or formulate two different phrases
  (expressions) that refer to the same number?

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early stages in making sense of the problem
situation

• Focus on the domain of meaning from which the
  numbers come, what are the quantities in the
  situation?
• Imagine the quantities referred to in the problem
  by words, numbers, letters, phrases.
• Represent the relationships among quantities in
  diagrams, tables, words.
• Work out what happens each time by increasing x
  by 1 and figuring what happens to y relate each
  time to the rows of a table and static diagram.

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late stages making sense of situation

• Imagine and define the domain mathematically,
  what are the sets of numbers referred to by the
  variables?
• Animated understanding from each time to any
  time. Generalized grasp of the mathematical
  relationship in the problem situation expressed
  as an equation y for any x.
• Equations and graphs
• Can interpret transformed equations (that express
  x in terms of y, for example) in terms of the
  problem situation

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Making Sense of the mathematics in the problem

• How does the table correspond to the graph? Where
  is this table row on the graph?
• What does a point (a region) on the graph refer
  to in the problem situation?
• How does the equation correspond to the situation
  (what do the letters refer to, what do the
  operations and the say about the situation?)?
• How do the equation, graph, table, diagram
  correspond to each other feature by feature?
• What kind of equation(s) is it?

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Make up a word problem for which the following
equation is the answer

• y .03x 1

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What have we learned?
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Forget ideologies

• Be skeptical so you can see and hear what is
  really going on
• Be pragmatic
• Peoples core beliefs are changed by their
  experiences and their companionships, not by
  authorities or speakers. Be patient with the
  beliefs of others except the belief that
  intelligence is fixed it is not, learning
  changes intelligence!

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Who teaches whom

• Students taught by first year teachers,
  substitutes, etc. do not benefit from last years
  PD
• The worse off a student is, the less likely they
  are to be taught by someone who benefitted from
  PD
• Change who teaches whom for a really big effect

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Each Day Teachers solve their problems in
priority (theirs) order

• Students engaged in some proper activity
• Teacher impresses students that he/she can handle
  being in charge
• Students respect (make be exchanged for other
  positive attitude toward) teacher
• - Enjoy class (optional)
• - Work hard
• - Learn

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After 2 or 3 years of teaching

• They have found their way of getting 1, 2, 3 or
  else they leave or hideout in the profession
• Its a bad deal for a teacher to trade 1, 2,3 for
  a pig in the poke
• Before they will try your 4,5,6 , You have to
  offer a 4,5,6 that either fits their 1,2,3 or
  you have to give them an alternative 1,2,3 that
  seems attractive to them
• Note They have to deal with students as persons,
  so AYD will appeal to them if it isnt all on them

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1,2,3s that scaffold 4,5,6s that work

• Major design challenge
• When you hear, a good program, but hard to
  implement it usually has design problems of this
  sortwrong to blame it on teachers beliefs

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Malcolm Swan example navigator

• Goldilocks problems that lead to concepts through
  work on misconceptions (faulty prior knowledge)
• Discussion craftily scaffolded
• Instructional assessment on all cycles,
  especially within lesson
• Tasks easy as possible to engage as activities
  that also hook straightaway to questions that
  lead to concept
• encouraged uncertainties at the door of insights

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Pedagogy

• Make conceptions and misconceptions visible to
  the student
• Design problems that elicit misconceptions so
  they can be dealt with
• Students need to be listened to and responded to
• Partner work
• Revise conceptions
• Debug processes
• Meta-cognitive skills

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Diagnostic Learning

• Revise conceptions
• Debug processes
• Meta-cognitive learning skills
• Social Learning skills you have to know how to
  help each other with math. homework

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Content Foundations

• Uses simple algebra to repair and strengthen
  students arithmetic foundations
• Extensive use number line to revisit number
  concepts and transfer to knowledge of coordinate
  graphs
• Explicit use of number properties and properties
  of equality in reasoning about arithmetic and
  transfer to reasoning with letters and
  expressions that represent numbers in algebra
• Use good problems to teach path from concrete
  reasoning to symbolic expressions of algebra
• Program takes aim at algebra anticipates
  learning difficulties and prepares students to
  handle difficulties
• Skills routinely exercised AND non-routine
  problems

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Key Features of a good ramp up curriculum

• Built from Algebra down, rather than from the
  deficiencies up
• Acknowledges that students have unreliable, but
  real knowledge about mathematics
• Balance and coherence
• Skills, problem-solving, and conceptual
  understanding with a coherence that revolves
  around optimizing the conceptual understanding

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What do we mean by conceptual coherence?

• Apply one important concept in 100 situations
  rather than memorizing 100 procedures that do not
  transfer to other situations.
• Curriculum is a mile deep instead of a mile
  wide
• Typical practice is to opt for short-term
  efficiencies, rather than teach for general
  application throughout mathematics.
• Result typical students can get Bs on chapter
  tests, but dont remember what they learned
  later when they need to learn more mathematics
• Use basic rules of arithmetic (same as algebra)
  instead of clutter of specific named methods

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Teach from misconceptions

• Most common misconceptions consist of applying a
  correctly learned procedure to an inappropriate
  situation.
• Lessons are designed to surface and deal with the
  most common misconceptions
• Create cognitive conflict to help students
  revise misconceptions
• Misconceptions interfere with initial teaching
  and thats why repeated initial teaching doest
  work

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Key features of the a well designed intervention

• Lean and clean lessons that are simple and
  focused on the math to be learned
• Rituals and Routines that maximize student
  interaction with the mathematics
• Emphasis on students, student work, and student
  discourse
• Teaches and motivates how to be a good math
  student
• Assessment that is ongoing and instrumental in
  promoting student learning
• Support for teachers
• The mathematics
• Student work with commentary, and guidance on
  getting the full power of the mathematics from
  the workshop

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Bottom up

• Schools fill interventions from the bottom up,
  not from Algebra down.
• Ready for Algebra
• 1 double-period year away from ready
• area, fractions with like denominators, single
  digit addition and multiplication facts, read at
  4th grade level,
• More than 1 double-period year away

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Plan for it

• Need a plan from the bottom up so students get
  what they need

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Malcolm Swan example

• Goldilocks problems that lead to concepts through
  work on misconceptions (faulty prior knowledge)
• Discussion craftily scaffolded
• Instructional assessment on all cycles,
  especially within lesson
• Tasks easy as possible to engage as activities
  that also hook straightaway to questions that
  lead to concept
• encouraged uncertainties at the door of insights

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Tiered Levels of Intervention
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About Tasks Interpreting Multiple Representations
Student Materials, Classroom Routines, and Tasks
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About Tasks Making Posters
Student Materials, Classroom Routines, and Tasks
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Social and meta-cognitive skills have to be
taught by design

• Beliefs about ones own mathematical intelligence
• good at math vs. learning makes me smarter
• Meta-cognitive engagement modeled and prompted
• Does this make sense?
• What did I do wrong?
• Social skills learning how to help and be helped
  with math work gt basic skill for algebra do
  homework together, study for test together

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Maritas homework
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Diagnostic Teaching

• Goal is to surface and make students aware of
  their misconceptions
• Begin with a problem or activity that surfaces
  the various ways students may think about the
  math.
• Engage in reflective discussion (challenging for
  teachers but research shows that it develops
  long-term learning)
• Reference Bell, A. Principles for the Design of
  Teaching Educational Studies in Mathematics. 24
  5-34, 1993

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Operations and Word Problems
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Operations and Word Problems
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Operations and Word Problems
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Knowing Fractions
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Knowing Fractions
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Knowing Fractions
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Knowing Fractions
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Understanding Fractions
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Understanding Fractions
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