Schemata, Intuition, and Problem Solving

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Schemata, Intuition,andProblem Solving

• Sara Hershkovitz
• Center for Educational Technology ISRAEL

SEMT09 - International Symposium Elementary
Mathematics Teaching Prague, Czech Republic,
August 2009
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Schema
Kant (1724 1804)The link between perceiving
real world objects and categories of pure
understanding
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Schema
Kant distinguished
schema of substance, schema of cause, schema of
community, schema of possibility, schema of
reality, schema of necessity, schema of
quality, schema of relation, schema of modality
and its categories.
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Schema
Kant
A schema is constructed according to the
necessary conditions of the unity of reason the
schema of a thing in general, which is useful
towards the production of the highest degree of
systematic unity in the empirical exercise of
reason .it merely indicates how, under the
guidance of the idea, we ought to investigate the
constitution and the relations of objects in the
world of experience.
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Schema
Kant
Schemata "all possible objects following the
arrangements of the categories"
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Schema
Kant
Category A category is an attribute, property,
quality, or characteristic that can be predicated
of a thing. Kant called them "ontological
predicates. Aristotle claimed that the
following ten predicates or categories could be
asserted of anything in general substance,
quantity, quality, relation, action, affection
(passivity), place, time (date), position, and
state.
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Schema
Piaget (1896 -1980) A schema of an action
consists of those aspects which arerepeatable,
transposable, or generalisable
(1980). Schemata develop by two mechanisms
assimilation and accommodation. .
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Schema
Rumelhart
The building blocks of cognition Rumelhart
discussed schemata while taking into account
different notionsframes, scripts,
retrieving information, organizing actions,
allocating sources, and guiding the flow of
processingfunctional relationships
characteristic of an object. (1980)
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Schema
Rumelhart and Norman
Schemata Data structures for representing the
generic concepts are stored in memory.
generalized concepts underlying objects,
situation, events, sequences of events, action
and sequences Schemata represent the
stereotypes of concepts. Schemata are like
models of the outside world. (1985)
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Schema
Rumelhart and Norman
To process information with the use of schemata
is to determine which model best fits the
incoming information. (1985)
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Schema
Schank Abelson (1977)Anderson (1980) Howard
(1987)
Semantic nets expressing relations Scripts of
behavior SchemaA mental representation of
some aspect of the world
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Schema
Features
Schemata have variables Schemata can be
embedded, one within another Schemata represent
knowledge at all levels of abstraction
Schemata represent knowledge rather than
definitions Schemata are active recognition
devices.
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Schema
Howard (1987) following Rumelhart (1980) A
schema is like a sorting device(allows to
determine that some stimuli are instantiations
and others are not). A schema is like a play
(the parts of a play relate to each other as
specified by scripts, performance) A schema is
like a filter.(it allows us some information in
but not all)
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Schema
Uses of Schemata (Howard 1987) Perception
Providing means of recognising patterns,
analyzing and interpreting new data. Comprehensio
n To understand something is to assimilate it to
something we know.
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Schema
Example Bransford Johnson (1973)
The procedure is actually quite simple. First
you arrange things into different groups. It is
important not to overdo things. That is, it is
better to do too few things at once than too
many. In the short run, this may not seem
important but complications can easily arise.
A mistake can be expensive as well.
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Schema
Example Bransford and Johnson (1973)
At first the whole procedure will seem
complicated. Soon, however, it will become just
another facet of life. After the procedure is
completed one arranges the materials into
different groups again. Then they can be put in
their appropriate places. Eventually they will be
used once more and the whole cycle will then have
to be repeated.
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Schema
Example Bransford and Johnson (1973)
?????
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Schema
Reasons for failing to comprehend (Howard 1987,
Rumelhart 1980) Not knowing appropriate
schema(and cannot readily construct one) Not
comprehending the appropriate given clues to
elicit a schema (even if a schema is
known) Applying a different schema to given
stimuli
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Schema
Nesher (1986) Schema is a strategy of solving a
certain class of problems. A plan for action.
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Schema
Fischbein (1920 - 1998)
A kind of condensed, simplified representation
of a class of objects or events
Adaptive behavior of an organism achieved by
assimilation and accommodation
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Schema
Fischbein (1999) A schema is a program
whichenables the individual to
(a) record, process, control, and mentally
integrate information,
(b) react meaningfully and efficiently to
environmental stimuli.
Related termsFrame Script FrameworkMental
structure
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Knowledge Schema

Schema Knowledge


How many cubes are there?
John had 6 marbles. He lost 2 marbles. How many
marbles does John have now?
Counting
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Knowledge Schema
Schema Knowledge



There are 7 fruits on the plate. 3 of them are
apples and the others are peaches.How many
peaches are there on the plate?
Counting ?
Class inclusion
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Knowledge Schema
Schema Knowledge



Dan had some marbles. He found 5 more
marbles. Now he has 8 marbles. How many marbles
did he have to start with?
Class inclusion ?
Part Part Whole If ABC then C-AB
C-BA
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Knowledge Schema
Schema Knowledge



Dan had some marbles. He lost 5 of them. Now he
has 3 marbles. How many marbles did he have to
start with?
Part Part Whole ?
Part Part Whole Reversibility
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Knowledge Schema
Schema Knowledge



Is the set of whole numbers equivalent to the set
of even numbers?
Part Part Wholereversibility ?
Part Part Whole Infinity
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Knowledge Schema

Counting Class inclusion Part-Part-Whole Rever
sibilityInfinity
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schema
Open Ended Problem

Look at the following numbers 23,
20, 15, 25, which number does not belong? Why?
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schema
Look at the following numbers 23, 20,
15, 25,which number does not belong? Why?
Solution no. 1

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schema
Look at the following numbers 23, 20,
15, 25,which number does not belong? Why?
Solution no. 2

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schema
Look at the following numbers 23, 20,
15, 25,which number does not belong? Why?
Solution no. 3

15 - It is in the 2nd ten and the rest are in
the 3rd ten 20 - The only round number
- This number has more factors 23 - Not a
multiple of 5 - The only prime number 25 -
The sum of its digits is the largest - The
only square number
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schema

Look at the following numbers 23, 20,
15, 25,which number does not belong? Why?
The more schemata
the more solutions
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Schema
To summarize

Schema is the means by which similar experiences
are assimilated and aggregated to a whole.
Schema enables to store and generalize ideas
Schema links together many different kinds of
knowledge
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Intuition
Descartes (1596 -1650)Intuition occurs either
after or concomitantly with analysis. Methods
consist of a set of rules or procedures for using
the natural capacities and operations of the mind
correctly.

Intuition - deduction
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Intuition
Olscamp (2001) analysis-intuition-deduction Pl
ato related to intuition ascontinuing analysis
and preceding synthesis

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intuition
Kant A way in which objectsare directly
grasped. Intuition remains related tosensorial
knowledge (intellectual intuition does not
exist)

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intuition
Fischbein Intuitive understanding Formal
understanding Procedural understanding

Related termsInsight Common sense
Interpretation Inspiration Naïve
reasoning Piaget self-evidence
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intuition
Fischbein Intuitive understanding



Affirmatory intuitions
Anticipatory intuitions

• Direct, self-evident
• Intrinsic certainty
• Coerciveness
• Extrapolativeness
• Globality

Intuition the effect of compression if a
structural schema lies behind this cognition
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intuition
Fischbein Intuitive understanding

Affirmatory intuitions


Anticipatory intuitions

• Grasping the problem
• Distinguishing between the given
  information and the question
• Searching for strategies
• Finding a schema for solving

• Direct, self-evident
• Intrinsic certainty
• Coerciveness
• Extrapolativeness
• Globality

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intuition
Example

Solving without having a specific schema.
Divide 21 matches into two groups so that one
group will be twice as big as the other group.
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Schema - intuition
Intuition Schema







Behavior Cognitive ability
Cognitive ability

Appears unplanned
Plan for action
Global
Built of components
Coercive Immediate Self-evident
Flexible Examined Adjusted
Global Extrapolative
Analytical Logical
Direct, Self-evident
Develops Can be adapted
Intrinsic certainty
Assimilation Accommodation
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Is it possible to teach using schemata?

• Does using schemata promote problem solving ?
• Additive Problems
• Two-step Problems

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Simple additive problems (Nesher 1982)

Mathematicalcompetence Examples Schema

1,2,3,4,5,
6,7
1. There are 5 apples and 2 bananas in the
bag. How many pieces of fruit are
there in the bag?
Building setof objects and counting
2 . There were 5 apples in the bag. Dan
took 2 apples out of the bag. How many
apples are there in the bag?
1,2,3,
4,5
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Simple additive schema
part
part

whole
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Simple additive schema
1. There are 5 apples and 2 bananas in the
bag. How many pieces of fruit are there in
the bag?
5 apples
2 bananas

? fruit
5 2 7
? fruit
2 bananas
5 apples
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Simple additive schema
2. There were 5 apples in the bag. Dan took 2
apples out of the bag. How many apples are
there in the bag?
2 apples
? apples

5 apples
5 2 3
5 apples
? apples
2 apples
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Simple additive problems
Mathematicalcompetence Examples Schema


3. Roni had 3 marbles. Then Tom gave him
some more marbles. Now Roni has 8
marbles. How many marbles did Tom give
Roni?
3 8 11
X
Part-Part-Whole Change
3 ? 8(3 5 8)
V
8 - 3 5
V
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Simple additive problems

Mathematicalcompetence Examples Schema

4. Tom had some marbles. He found 5 more
marbles. Now he has 8 marbles. How many
marbles did he start with?
Part-Part-Whole Reversibility
5 8 13
X
? 5 8 (3 5 8)
V
8 - 5 3
V
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Without Schema Without Schema With Schema With Schema Problem Schema
Post Pre Post Pre
100 77.8 100 73.4 Combine 1 Building sets Counting
100 100 95.7 87 Change 1 Building sets Counting
88.9 77.8 91.3 87 Change 2 Building sets Counting
44.4 33.3 82.6 47.8 Combine 2 Part Part-Whole Change
66.7 55.6 91.3 43.5 Change 3 Part Part-Whole Change
55.6 44.4 87 60.9 Change 4 Part Part-Whole Change
33.3 22.2 73.9 60.9 Change 5 Part-Part-Whole Reversibility
33.3 33.3 52.2 39.1 Change 6 Part-Part-Whole Reversibility

Ogonovski Nesher (2009)
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Without Schema Without Schema With Schema With Schema Problem Schema
Post Pre Post Pre
100 77.8 100 73.4 Combine 1 Building sets Counting
100 100 95.7 87 Change 1 Building sets Counting
88.9 77.8 91.3 87 Change 2 Building sets Counting
44.4 33.3 82.6 47.8 Combine 2 Part Part-Whole Change
66.7 55.6 91.3 43.5 Change 3 Part Part-Whole Change
55.6 44.4 87 60.9 Change 4 Part Part-Whole Change
33.3 22.2 73.9 60.9 Change 5 Part-Part-Whole Reversibility
33.3 33.3 52.2 39.1 Change 6 Part-Part-Whole Reversibility

Ogonovski Nesher (2009)
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Without Schema Without Schema With Schema With Schema Problem Schema
Post Pre Post Pre
100 77.8 100 73.4 Combine 1 Building sets Counting
100 100 95.7 87 Change 1 Building sets Counting
88.9 77.8 91.3 87 Change 2 Building sets Counting
44.4 33.3 82.6 47.8 Combine 2 Part Part-Whole Change
66.7 55.6 91.3 43.5 Change 3 Part Part-Whole Change
55.6 44.4 87 60.9 Change 4 Part Part-Whole Change
33.3 22.2 73.9 60.9 Change 5 Part-Part-Whole Reversibility
33.3 33.3 52.2 39.1 Change 6 Part-Part-Whole Reversibility

Ogonovski Nesher (2009)
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The Building Blocks of word problems

Additive Schema
Multiplicative Schema
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Two-step Word Problems
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Hierarchical Schema
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6 plates
5 oranges on each plate
Dina has 15 bananas and 6 plates with 5 oranges
on each. How many pieces of fruit does Dina
have?

X
? oranges
? oranges
15 bananas

? fruits
6 X 5 15 45
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? plates
5 oranges on each plate
Dina has 45 pieces of fruit. 15 of them are
bananas, and the rest are oranges. She put the
oranges on plates, 5 oranges on each plate. How
many plates are there?

X
? oranges
? oranges
15 bananas

45 fruits
(45 15) 5 6
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6 plates
5 oranges on each plate
Dina has 45 pieces of fruit. She put them in 6
plates on each of which there are 5 oranges. The
rest of the fruit are bananas. How many bananas
does Dina have?

X
? oranges
? oranges
? bananas

45 fruits
45 6 X 5 15
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Shared Part Schema
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Shared Whole Schema
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Hershkovitz Nesher (1992 1994 1996 1998)
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Learning with or without schemata

PopulationTwo groups of 6th gradersTwice a
week during 4 months
SPA Schema for Problem Analysis AP Algebraic
Proposer J. Schwartz
2 easy problems 2 difficult problems
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Learning with or without schematagraded 0-2

Difficult Problems Easy problems Program Students
1.74 1.78 SPA All
0.97 1.72 AP All
1.56 1.67 SPA Low achievers
0.53 1.40 AP Low achievers
1.86 1.86 SPA High achievers
1.35 2.00 AP High achievers
SPA Schema for Problem Analysis AP Algebraic
Proposer
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Learning with or without SchemaLow achievers

Difficult Problems Easy problems Program Students
1.74 1.78 SPA All
0.97 1.72 AP All
1.56 1.67 SPA Low achievers
0.53 1.40 AP Low achievers
1.86 1.86 SPA High achievers
1.35 2.00 AP High achievers
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Learning with or without schemataDifficult
Problems

Difficult Problems Easy problems Program Students
1.74 1.78 SPA All
0.97 1.72 AP All
1.56 1.67 SPA Low achievers
0.53 1.40 AP Low achievers
1.86 1.86 SPA High achievers
1.35 2.00 AP High achievers
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Pathway between Text and Solution of Word
Problems

Solution
Understanding the given text
Word Problem
Constructing a representation
Finding an appropriate schema
Applying the schema to given information
Constructing the math model
Solving the problem
Hershkovitz Nesher (2001)
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Hershkovitz Nesher (2001)

Word problems 3 two-step word problems Typical
of problems taught in math classes      Population
49 fifth and sixth grade Israeli students were
individually interviewed in a single 45-minute
session
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Word Problems   Problem No. 1 I have a book of
320 pages. I already read 80 pages. How many days
are needed to finish reading the book if I read
60 pages each day?   Problem No. 2 In the
morning the flower seller distributed the roses
equally into 6 vases. How many roses did he place
in each vase if during the day he sold 120 roses
and at the end of the day 60 roses were
left?   Problem No. 3 Lunch boxes were prepared
for all participants. Each lunch box had 5 pieces
of fruit of which2 were apples and the rest were
plums. In preparing the lunch boxes 240 plums
were used. How many participants received lunch
boxes?

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Interviews   Every student was asked, for each
of the three problems, to   Read aloud the text
(original text) of the word problem   Retell it
(first retelling)   Solve it.   After solving the
problem, the student was asked to retell the
story again (second retelling).

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Analysis of   Deviation from the original text
in the retelling (a) Changing the wording
without changing the schema, usually by
adding details to the description of the
situation (episode), which were taken from
general world knowledge, and were not
mentioned in the text. (b) Changing the order of
the text. (c ) Retelling the original text
exactly. (d) Changing the schema of the text
Changes were made in the text to fit the
erroneous solution

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Examples  (a) Changing the wording without
changing the schema 

Yael retold problem no. 2 Original problem no. 2
There was a seller. He received roses and equally distributed them in vases. During the day a lot of people arrived and bought a lot of roses. Then he found out that 60 roses were left? In the morning the flower seller distributed roses equally into 6 vases. How many roses did he place in each vase if during the day he sold 120 roses and at the end of the day 60 roses were left?
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Examples  (b) Changing the order of the text  

Michal retold problem no. 2 Original problem no. 2
A flower seller sold 120 flowers and 60 flowers were left. The flowers (those sold and those left) were in 6 vases. How many flowers were there in each vase? In the morning the flower seller distributed roses equally into 6 vases. How many roses did he place in each vase if during the day he sold 120 roses and at the end of the day 60 roses were left?
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Using both categories (a) Changing the wording
without changing the schema (b) Changing the
order of the text

Original problem no. 3
Lunch boxes were prepared for all participants.Each lunch box had 5 pieces of fruit of which2 were apples and the rest were plums. In preparing the lunch boxes 240 plums were used. How many participants received lunch boxes?
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Johnny's repetition was
A member of the entertainment committee, or somebody else, I dont know exactly who, prepared the lunch boxes for the trip the committee organized. In each lunch box they put 5 pieces of fruit of which there were 2 apples and 3 plums. 240 plums were needed to prepare all the lunch boxes. How many children got lunch boxes? While solving the problem he wrote 2 math expressions as follows 3 2 5 and 240 3 80 plums apples fruits He summarized 80 children will get lunch boxes. He continued and said Now I can find out how many apples were needed as well (80X2160).

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(d) Changing the schema

Shay retold problem no.3
Lunch boxes were prepared. There were 5 fruits in each lunch box, of which 3 were plums and 2 were apples.   Shay continued to speak aloud while solving 240548 and said 240 are all the fruit. Each child received 5 fruits.
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Findings
Not Included Incorrect Solutions Correct Solutions Correct Solutions Correct Solutions Problem
(d) (c) (b) (a)
14 6 8 29 43 1
6 26 4 29-38 35 2
4 33 8 29-33 26 3

(a) Changing the wording without changing the
schema (b) Changing the order (c) Retelling
exactly (d) Changing the text into different
schema.
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• Almost all students who correctly solved the
• problems elaborated to some extent Some
  by using their world knowledge, while others
  began with the mathematical solution.
• All students who failed to solve the problems
  changed the text into another schema, usually to
  a simpler one. These changes related to changing
  the text so that it described different
  mathematical structures.
• The second retelling (after solving the
  problems) was consistent with the already
  incorrectly solved problem.

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Anderson et al. (1983) claim that The content
schema embodies the readers existing knowledge
of real and imaginary worlds. What the reader
already believes about the topic helps to
structure the interpretation of new messages
about the topic.  

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We found that all students in our sample, who
constructed a richer text by adding detailed
information, did so because it was useful for
them in order to construct a complete
understanding of the text, find the appropriate
schema, and then solve the problems correctly.  

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Non Routine Problems (N.R.P)
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N.R.P Sharing Pizza
Schema ?? Intuition ??

There were 40 children in the summer camp. For a
dinner some large pizzas and some small pizzas
were ordered. Each large pizza was divide equally
among the children, and each small pizza was
divided equally between the children. How many
pizzas were ordered? Offer some possibilities.
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N.R.P Sharing Pizza

N.R.POpen-ended-ProblemIntuitively - general
world knowledge (direct,
coercive, global) Mathematical schemata -
sharing, dividing,
fraction? (only
after the teachers example)
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Final Notes

1. Intuitions and schemata are two
complementary factors needed for problem
solving in mathematics.
2. Descartes presented the process of
"analysis-intuition-deduction" as a way to
achieve certain knowledge.
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Final Notes
3. Feferman (2000) stressed The ubiquity of
intuition in the common experience of
teaching and learning mathematics is
essential for motivation of notions and
results and to guide one's conceptions via
tacit or explicit analogies in transfer
from familiar ground to unfamiliar terrain
... intuition is necessary for the
understanding of mathematics.

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4. The more schemata a person acquires, the
more intuition he has. (Fischbein 1999)

5. The educational challenge is to enable
children to develop rich repertoire of
mathematical schemata leading to more
intuitions for solving mathematical
problems.
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N.R.P Birthday Cake

Intuition ??? Analysis-intuition-synthesis M
athematical Schemata Factorization,
division with
remainder
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N.R.P Birthday Cakes solution
Length of pattern Solution No.
1 1
2 2
3 3
4 4
5 5
6 6
10 7
12 8
15 9
20 10
30 11
60 12

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N.R.P Birthday Cakes solution
Remainder Length of pattern Solution No.
17 2 8 (R 1) 2 2
17 3 5 (R 2) 3 3
17 4 4 (R 1) 4 4
17 5 3 (R 2) 5 5
17 6 2 (R 5) 6 6
17 10 1 (R 7) 10 7
17 12 1 (R 5) 12 8
17 15 1 (R 2) 15 9
17 20 0 (R 17) 20 10
17 30 0 (R 17) 30 11
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N.R.P Birthday Cakes solution
Ron 3rd grade High Achiever Ron Do I have to
use all the candies? Teacher No, you can use the
same color more than once in the same pattern.
You have to choose a pattern so that the 17th
candy will be purple. Ron placed 5 different
candies in this order blue, purple, orange,
green, and red, and duplicated them.T How did
you know?.

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N.R.P Birthday Cakes solution
R 17 divided by 5 gives the pattern 3 times and
the remainder is 2, so the purple has to be the
second. T Great. Do you think there are more
solutions? R Thought for a while I'll try with
4 candies.T And? R The purple will be the
first.T How?

91
N.R.P Birthday Cakes solution
R 17 divided by 4 gives 4, remainder 1. The
remainder is 1, so this is the purple candy. T
Do you have an idea for "the rule" of the
game? Ron tried patterns with 3 candies and with
2 candies.R The place of the purple candy is
the remainder of the division exercise.

92
Avivah (A teacher)
 Generalization The number of options is 529
since the largest pattern possible is 30. I
knew that from the requirement of the task which
requires to fill 60 candies in a pattern.The
pattern is actually all the factors of the number
since they must repeat. The factors of 60 are 1,
2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 The factor
60 is not interesting, since it will not make a
repeating pattern,

93
Avivah (A teacher)
  therefore I stay with the factor 30 as the
largest pattern. In this pattern I need to fill
only 29 candies, since the 30th candy (in the
17th place) is filled and fixed according to the
problem's requirement. For each of the
remaining 29 candies I have to choose among 5
colors, therefore this situation includes all the
other options and gives the answer I wrote above.

94
Nethanel (6th grade)

Documentation of the solution process by Aviva
First trial puts purple. Smiles.
"I'm sure it will work".
Success Second trial puts purple-blue.
"The same".
Success Nethanel says "Each color
instead of the blue will work. Each sequence of 2
, as long as the purple is first. Third trial
puts red-purple-blue. Success Nethanel
says "6 will also work". Continues and says "9
will also work". Fourth trial puts 9 No
success
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Nethanel (6th grade)
Nethanel says "Wait, why does 6 work and 9
doesn't work?" Waits for a minute. Looks
disturbed.Aha! I got it! 603, 606, 60 is not
divisible by 9. So 4 would also work. I ask
"Where will the purple be?" Nethanel answers
second. No, wait, not second. When you put five
it will be second, no matter what the others will
be. I insist So with 4? Where is the
purple?" Nethanel answers "Same as with 2.
First." Continues "So it can also be 10, as long
as the second and the seventh will be purple."

96
Nethanel (6th grade)

I ask "Second and seventh? You must have
both?" Nethanel answers Yes. No. Wait." Checks
the options with the computer. Second and
seventh Success Second only No
Success Seventh only Success Says Seventh is
enough". I ask "So, does it end?" Nethanel "I
don't know. I think there's no end." I ask
"Why?"
97
Nethanel (6th grade)

Nethanel "I don't know." I ask "What else can
you do?" Nethanel "Every number that 60 is
divisible by, and you have to check where the
purple will be." I ask "So there is an
end?" Nethanel "There is an end to the numbers
that 60 is divisible by, but I also have the 17th
candy and I also have five colors. I don't
know."
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Thank you
Sarah_at_cet.ac.il