The Proactive Role of the Teacher in Mathematics Education

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The Proactive Role of the Teacher in Mathematics
Education

• Koeno Gravemeijer
• Freudenthal Institute for
• Science Mathematics Education
• Langeveld Institute
• Utrecht University
• koeno_at_fi.uu.nl

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ConstructivismPeople construct their own
knowledge
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The Proactive Role of the Mathematics Teacher

• Reconcile the constructive activity and the
  intellectual autonomy of the students with the
  pedagogical agenda of the teacher
• ? Proactive role of the teacher
• Select and introduce tasks anticipating student
  thinking
• Orchestrate classroom discussions
• Establish and cultivate an inquiry classroom
  culture

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Intellectual autonomy (Kamii)

• Teacher authority and student autonomy
• Student autonomy
• Responsibility for being able to justify ones
  claims in the math lesson
• Teacher authority
• What mathematics is
• How is mathematics learned in this classroom
• Goals tasks

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Choosing/designing instructional tasks

• Choosing tasks with an eye on what it might bring
  about ?hypothetical learning trajectory-
  envision the mental activities of the students -
  anticipate how their thinking might help them to
  develop mathematical insights (Simon,
  1995) hypothetical need to check and revise

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Example

• Discussion of Area with student-teachers
• Area length x width ?
• Blind algorithm??

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Rectangles problem 1. Determine how many
rectangles, of size and shape of the rectangle
that you were given, could fit on the top surface
of your table. Rectangles cannot be overlapped,
cannot be cut, nor can they overlap the edges of
the table. Be prepared to describe to the class
how you solved this problem.
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Rectangles problem 2. Bill said, If the table is
13 rectangles long and 9 rectangles wide, and if
I count 1, 2, 3, ..., 9, and then I multiply, 13
x 9, then I have counted the corner rectangle
twice. Respond to Bills comment.
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Rectangles problem 3. I used the turned
rectangles method, and I got 32 for table A, and
22 for table B. Can we now say something about
which table is bigger?
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The stick problem. Two people work together to
measure the size of a rectangular table one
measures the length and the other the width. They
use a stick to measure with. The sticks, however,
are of different lengths. Louisa says, The
length is four of my sticks. Ruiz says, The
width is three of my sticks. What can you say
about the area of the rectangular table?
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Original turned rectangles problem
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Contrast

• The hypothetical learning trajectory is very
  different from the classic Socratic way of
  developing an idea in interaction with the
  students
• Who does the thinking?

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• Socratic lesson

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• Socratic lesson

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• Socratic lesson

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• Socratic lesson

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• Socratic lesson

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• Socratic lesson

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Hypothetical learning trajectory versus Socratic
lesson

• Doubling square (2), in the Socratic lesson, the
  teacher does the thinking teacher checks whether
  the students follows along
• Area Simon (1), with the hypothetical learning
  trajectory, the students do the thinking the
  teacher tries to adapt to what the students are
  thinking.

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Teachers need help

• We cannot expect teachers to develop HLTs from
  thin air.
• We should aim at offering teachers means of
  support for construing and revising HLT's
• By developing prototypical instructional
  sequences and local instruction theories for
  various topics (fractions, long division, )

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Instruction theories(different levels)

• Hypothetical learning trajectory
• Local instruction theory (topic)
• Domain-specific instruction theory
• Guided reinvention
• tasks (routes levels)
• symbolizations informal (arithmetic rack, PT
  meeting)
• conventions
• Emergent modeling
• Didactical phenomenology

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Classroom culture

• Tasks inquiry math students have to engage in a
  problem solving activity
• Resistance of students to teachers attempts to
  implement a problem solving approach (Desforges
  Cockburn Hiebert Stigler)
• Explanation
• That is what they are used to reproducing the
  teachers reasoning and procedures (Didactical
  contract)
• Communication ? expectations
• An everyday-life example

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Communication ? expectations

• Elicitation pattern
• Could you please tell me the way to West Street?

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Communication ? expectations

• Elicitation pattern
• Could you please tell me the way to West Street?
• You take the second on the right, then the first
  on the left, and then you work straight into West
  Street.

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Communication ? expectations

• Elicitation pattern
• Could you please tell me the way to West Street?
• You take the second on the right, then the first
  on the left, and then you work straight into West
  Street.
• OK, the second on the right, then the first on
  the left, and then straight ahead. Perfect, well
  done.
• Now, could you also tell me the way to Central
  Avenue?

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Didactical contract/social norms

• Cobb Yackel emergent perspective
• social ? psychological
• classroom social norms ?individual beliefs
• Beliefs about expectations and obligations,
  shaped by experience
• Socio-mathematical norms
• Mathematical practices

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Social norms school-math

• Elicitation pattern
• No responsibility for the students
• Donna (1)
• Mr. K. How many?
• Donna Eight
• Mr. K. How many?
• Donna Eh, seven(?)

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Social normsInquiry math

• Intellectual autonomy of the students (instead of
  having the teacher or textbook as the authority)
• Act as a learning community
• Learning as a group learning from and with each
  other
• Obligations to explain justify try to
  understand, ask for clarification challenge

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How to change the didactical contract

• Establishing social norms ? experience
• What is valued
• What is rewarded
• Using instances as opportunities to clarify norms
• Example Donna (taken from Erna Yackel)

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• Mr. K. How many?
• Donna Eight
• Mr. K. How many?
• Donna Eh, seven(?)
• Next Mr. K. moves to other students. Later as it
  is
• Established that 8 was the right answer, Donna
• complains
• Donna I said eight but you said I was wrong!
• Mr. K. What is your name?
• Donna Dona
• Mr. K. What is your name?
• Donna Dona
• Mr. K. And if I would ask you again, What is
  your name?, would you say anything else but
  Donna?

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Cultivating an inquiry classroom culture

• Asking for explanations
• Please explain your answer. (Why is that so? How
  do you know?)
• Asking for clarifying questions
• Who has a question for Jim?
• Pass the problem along
• Who can answer Paulas question?
• Asking for a personal judgment
• Ann says that it will cost 16.25, do you agree?
• Promoting that students listen and try to
  understand
• Did you follow what he said, could you explain it
  to me?

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Cultivating an inquiry classroom culture

• Revoicing (for instance to help students to
  follow the argument)
• Modeling favorable behavior
• Showing genuine interest in the students
  thinking
• Building on the input of the students

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Mathematics in the City, CUNY Cathy Fosnot
Maarten Dolk
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Turkey video, Cultivating an inquiry classroom
culture

• Try to explain in such a manner that everybody
  can understand.
• Listen carefully, and see if you understand.
• Who thinks he can explain what Amber and Vicky
  tried to do?
• Do you have something to add?
• Without telling them how many hours could you
  explain to them how they could figure that out?
• Great question, did you understand
• Tell them ..
• Did you hear what he said ?

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Central role of the teacher

• Role of the students ? Participation rules of
  the game
• Genuine interest example Researcher as a
  teacher How do they think?

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• Discussion on two data sets, which show the
  number of serious injuries in accidents with cars
  with airbags and cars without airbags.
• The data are already summarized graph shows the
  extremes and the medians for both sets of data.
• Until now, data sets that had to be compared had
  the same extreme high and low values.

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• Ben You cannot compare them this way, the
  starting points differ. It would have been
  better if they both would have been from 284
  thru 1000
• T1. ??
• Jack If you could change the numbers, which you
  cannot, then you could move this one.
• L2. And what would you know then?
• Jack Nothing, but it would be a better way of
  looking.
• L1. But you said you cannot change the numbers.
  Here the lowest number is 284, and there it is
  400.
• Ben Yes, but I am not talking about the
  original, if they would begin at 284 and end at
  1000, than you could look at the median.

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• Mary He does not use the numbers. He says,
  forget about the numbers for a moment and just
  look at the interval.
• T1. I think, I get it.
• T2. If we would know the size of this interval
  and of this interval, how would that help us?
• Ben Well, than, which be would be the most would
  be the worst car. It would be like taking that
  one, however, there are two different numbers.
  It is no use to take four different numbers and
  then compare the medians.
• T1. Let me see if I understand. Please check
  if I am correct. You say you cannot compare the
  medians, because you have to take the intervals
  into account. It is not enough to know the
  medians. You have to know where the upper half
  of the data is, and where the lower half is.

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Socio-mathematical norms

• Classroom norms specific to mathematics
• What counts as a mathematical problem
• What counts as a mathematical solution
• What counts as a different solution
• What counts as a more sophisticated solution
• Criteria ? intellectual autonomy students

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Socio-mathematical norms

• What counts as a (mathematical) problem/solution
• Mathematical problem ? reality (Verschaffel)
• Jim has 5 planks of 2 meters.
• gt How many planks of 1 meter can he make?
• John has 4 planks of 2½ meters.
• gt How many planks of 1 meter can he make?
• (Width of the cut made by the saw?)

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Socio-mathematical norms Mathematics ? Reality

• Marys friend Ann is staying for diner, now
  there are 5 cheeseburgers for 6 people (father,
  mother, Mary, her brothers and Ann).
• gt How should they share the cheeseburgers?

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Socio-mathematical norms Mathematics ? Reality

• Real-life solutions
• Mary should share with Ann

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Socio-mathematical norms Mathematics ? Reality

• Real-life solutions
• Mary should share with Ann
• Go to the shop and buy one extra

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Socio-mathematical norms Mathematics ? Reality

• Real-life solutions
• Mary should share with Ann
• Go to the shop and buy one extra

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Socio-mathematical norms Mathematics ? Reality

• Mathematically productive solutions
• 5 6 5/6
• 5 6 ½ 1/3

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Socio-mathematical norms Mathematics ? Reality

• Socio-mathematical norms about to what extent
  reality has to be taken into account have to be
  developed in interaction (implicit, experience)
• In general all classroom norms are based on
  experience
• What is valued
• What is rewarded

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Developing mathematical interest

• Need for a shift from solving a problem to
  thinking about the solution process from a
  mathematical perspective
• Is there a better way?
• Does it always work?
• Can I prove that?
• Mathematical progress

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Mathematical practices

• HLT Whos learning trajectory?
• All students???
• Individual learning routes???
• Alternative sequence of mathematical practices
  all students are more or less on the same track
• First students have to explain and justify,
• Later they dont ask for justification anymore?
  mathematical practice

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Toulmins argumentation scheme
data
conclusion
warrant
backing
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Toulmins argumentation scheme
4 stamps of 15 cents
Total price 60 cents
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Toulmins argumentation scheme
4 stamps of 15 cents
Total price 60 cents
4 x 15 60
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Toulmins argumentation scheme
4 stamps of 15 cents
Total price 60 cents
4 x 15 60
2 x 15 30 30 30 60
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Roles of the teacher Variety of roles

• establishing social norms
• presenting instructional tasks
• identifying topics for discussion (mathematical
  issues)
• orchestrating whole class discussions
• keeping order keeping students involved
• explaining conventions

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Summary pro-active role
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• Anticipate
• Make inferences about student knowledge,
  understanding, attitudes, interests
• Select tasks, HLT ? Goals
• Enact
• Introduce instructional tasks
• Analyze student activity
• Frame topics for discussion
• Orchestrate that discussion
• Cultivate classroom culture
• Reflect
• Evaluate the learning trajectory
• Assess student thinking
• Look for potentional starting points

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A Local Instruction Theory as a means of support
for teachers
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Helping students to invent

• If you want students to invent valuable
  mathematics, you have to offer them support

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Helping students to invent

• If you want students to invent valuable
  mathematics, you have to offer them support
• If you want teachers to help students to invent
  valuable mathematics, you have to offer them
  support

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Helping students to invent

• If you want students to invent valuable
  mathematics, you have to offer them support
• If you want teachers to help students to invent
  valuable mathematics, you have to offer them
  support
• If you want to help textbook authors to help
  teachers to help students to invent valuable
  mathematics, you have to offer them support

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What type of support should be offered to
teachers?

• We should aim at offering teachers means of
  support for construing and revising HLT's
• By developing prototypical instructional
  sequences and local instruction theories for
  various topics (fractions, long division, )

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Local Instruction Theory

• Set of exemplary instructional activities
• Rationale (theory) that underpins it
• Learning route
• Means of support
• Instructional activities
• Tools
• Classroom culture

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RME Theory

• RME Design heuristics
• Guided reinvention
• Didactical phenomenology
• Emergent modeling

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Design heuristics

• Guided reinvention through progressive
  mathematization
• A route has to be mapped out that allows the
  students to (re)invent the intended mathematics
  by themselves. (Or experience it as such.)
• history of mathematics
• informal solution procedures

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Design heuristics

• Didactical phenomenology
• Present-day applications starting points
• ? Problem situations that may give rise to
  situation-specific solution procedures
• Phenomenology of mathematics how the thought
  thing (concept, procedure, tool) organizes the
  phenomenon.

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Design heuristics

• Emergent modeling
• Emergent modeling modeling as organizing the
  model and the conception of what is being
  modeled coevolve.
• The model
• overarching model
• a series of symbolizations/tools

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Design heuristics

• Emergent modeling
• At first a model is constituted as a
  context-specific model of acting in a situation,
• The model changes character, it becomes an entity
  of its own,
• and as such it can function as a model for more
  formal mathematical reasoning

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Data Analysis as an example

• Design Experiments Middle School Design
  Collaborative NCISLA (OERI) NSF
• Paul Cobb
• Kay McClain
• Cliff Konold
• Koeno Gravemeijer
• Erna Yackel (Advisor)
• Seventh graders (12-years old) - 12 weeks - 34
  sessions - univariate data
• Eighth graders (13 years-old) - 14 weeks - 41
  sessions - bivariate data

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Potential endpoint of the learning trajectory

• mean, mode, median, quartiles, ...
• as means/characteristics distribution
• distribution as an object-like entity

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Distribution as an object
a) All Dutch, c) Families with parents
lt30 b) All married Dutch d) beds for adults
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Data Aanalysis Minitools

• Minitool 1

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Classroom episodesBattery life span
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• Casey And I was saying, see like theres
  seven green that last longer.
• --------------------------------------------------
  -----------------
• Janice Shes saying that out of ten of the
  batteries that lasted the longest, seven of
  them are green, and thats the most number, so
  the Always Ready batteries are better because
  more of those batteries lasted longer.

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• The next student to explain his reasoning, Brad,
  directed the teacher to place the value tool at
  80.

• Brad See, theres still green ones Always
  Ready behind 80, but all of the Tough Cell is
  above 80. I would rather have a consistent
  battery that I know will get me over 80 hours
  than one that you just try to guess.
• Teacher Why were you picking 80?
• Brad Because most of the Tough Cell batteries
  are all over 80.

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• Barry Like, if youre using them for
  something real important and youre only
  going to have like one or two batteries, then
  I think you need to go with the most constant
  thing. But if youre going like, "Oh well, I
  just have a lot of batteries here to use,"
  then you need to have most of the highest.

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Data Aanalysis Minitools

• Minitool 2

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Classroom episodesSpeed trap
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• Janice If you look at the graphs and look at
  them like hills, then for the before group
  the speeds are spread out and more than 55,
  and if you look at the after graph, then more
  people are bunched up close to the speed
  limit which means that the majority of the
  people slowed down close to the speed
  limit.

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Cascade of inscriptions
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Three interrelated processes

• Model shift from model-of to model-for
• overarching model visual model of a data set
• The constitution of some new mathematical reality
• network of mathematical/statistical relations
  notions of density, shape, spread, skewness
  implicit notions of measure and variable
• a series of symbolizations/tools
• Value bars
• Dotplot
• Four equal groups
• Box plot

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Local Instruction Theory

• Guided Reinvention
• Didactical Phenomenology
• Solving applied problems, which gives rise to
  mathematizing or organizing
• 1. Organizing measurement values ? data points on
  an axis
• 2. Organizing the distribution in of data points
  ? density
• 3. Organizing density ? density function

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Local Instruction Theory

• Emergent Modeling
• Model of a set of measures
• Model for reasoning about a distribution
• 25 25 25 25

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Local Instruction Theory

• Tools and Imagery
• Use of new tools is grounded in imagery of
  earlier activities
• History in the learning process ltgt meaning
• Problematizing tool use as a basis for
  introducing more sophisticated tools experience
  the reinvention process

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Local Instruction Theory

• Set of exemplary instructional activities
• Rationale (theory) that underpins it
• Learning route
• Means of support
• Instructional activities
• Tools
• Classroom culture

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Local Instruction Theory

• Local instruction theory as a framework of
  reference
• The teacher will still need to construe his or
  her own hypothetical learning trajectories
• Adaptations to
• this teacher
• these students
• this moment in time

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• Thank you