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What’s next for Parallel Distributed Processing?Mathematical Cognition and Other New Directions
Core features of the PDP approach to representation and learning
The knowledge is in the connectionsIt’s intrinsically implicitIt is acquired by a blind automatic procedureBy performing gradient descent rather than by making explicit inferences or engaging in any kind of reasoning processIt can approximate systems of rulesWithout ever having anyIt captures the gradual nature of developmental changeAnd emphasizes the importance of the gradual accumulation of small changes
H I N T
/h/ /i/ /n/ /t/
Second and Third Waves ofNeural Networks
Some classical applications of PDP models:
Reading and morphology
Some recent breakthroughs in machine learning:
Surpassing human performance in Atari games
Sentiment Analysis (Socher et al, 2013)
My lab’s new direction:
Why is Math so Hard to Learn?
Late grade-school-aged kids misunderstand equationsWhat goes in the blank: 7 + 3 + 4 = __ + 4Many middle-school-aged kids misunderstand fractionsIs 19/20 closer to 1 or 21?Most Stanford undergraduates don’t understand the rudiments of trigonometryWhich expression below has the same value as cos(-30°)? sin(30°) -sin(30°) cos(30°) -cos(30°)
Failure to attach the appropriate meaning to mathematical expressions
A fraction N/D represents a certain number N of pieces of a unit whole divided into D equal partsAn equation represents an equivalence relation between two quantities, one to the left and one to the right of the equals signThe sine / cosine of an angle θ in degrees representsthe projection of a point on the unit circle specified by θ onto the vertical / horizontal axis through the center of the circle, or equivalently, the coordinates of the point on the circle
Reported Circle Use: “A Lot” “A Little” or “Not at all”
Why are these things hard to learn?
Learning Depends on the Prepared Mind
Algebra for eighth graders?
“though strong math students can benefit from taking algebra in eighth grade, it is "decidedly harmful" for weaker math students to be rushed into advanced math
Failure to appreciate what X/Y means
Setting up the appropriate encoding habits
Failure to rely on the unit circle?
Failure of a module for visuospatial cognition or failure to develop the habit of mapping numbers into
a multi-faceted coordinate
Habits of Mind1
Learning to encode expressions automatically so that their meaning is readily apparent in the mind requires gradual connection adjustments that occurs incrementally over repeated opportunities to learnThis is no different in principle from learning to read words aloudWe quickly loose awareness that we are engaging in these processes – once we understand well, meaning is a habit of mind we cannot readily appreciate that others do not have
Margolis, H. (1987).
U. of Chicago Press.
Case Study in Readiness:The Balance Scale
Training involved more cases in which the weight varied than cases
In which the distance variedNetwork’s task was to activate the unit corresponding to the side thatShould go down, or (if the sides are in balance) to set the activation ofboth output units to .5
Balance Scale Model
Siegler’s Readiness Experiment
Two groups of children
5 year old Rule 1 children
7-8 year old Rule 1 children
Children saw 15 conflict problems with feedback
1/3: the side with greater weight would go down
1/3: the side with greater distance would go down
1/3: the two sides balance
olds progressed to rule 2
olds showed no change or reverted to guessing
Benefit From a Brief Lesson in the Unit Circle Depends on a Prepared Mind
Applying these ideas to Mathematical Cognition
Application 1: Representation of approximate number
, 2012; Zou & McClelland (poster)
Application 2: Learning to correctly
Mickey & McClelland (talk)
Application 3: Incremental improvements in strategies for adding small numbers
Hanson, McKenzie & McClelland (talk)
Application 4: Learning to geometry
Me and anyone who is willing to help me!
The Approximate Number Problem
Must we really imagine that a system for representing number approximately is innate, or can the problem be solved using a generic neural network?Can we account for the developmental improvement in acuity of the approximate number system?Can we understand why our representations of approximate number have the properties that they do?Specifically, why does our sensitivity to approximate numbers approximately conform to Weber’s law?
Why Neural Networks?
Deep (unsupervised) learning can create an invertible internal representation that is driven solely by the goal of capturing the content of its inputsAs Stoianov & Zorzi (2012) showed, this is sufficient to support human level performance in numerosity judgment.Using ‘stochastic gradient descent’ instead of batch learning allows us to explore both the initial state and progressive refinement of representations.Zou & McClelland explore the developmental trajectory, also explored in subsequent work by S & Z.
Errors on Equivalence Problems
Children are reliably incorrect in answers problems
of the form:
a = b + __
They tend to put the sum of a and b in the blank, rather than the correct answer, which is b – a.
When given such equations in a brief presentation, and asked to reproduce them, they tend to reproduce them as
a + b = __
Children’s experiences are biased in ways that are consistent with these errors.
Why a Neural Network?
Gradually learns in a way that depends on statistics of training set
Exhibits ‘pattern completion’
biases that capture both
math errors and problem
Gradually learns it was out of its errors, capturing patterns in the data
These two models are great, but…
When we solve mathematical problems, we often perform a sequence of operations.
These operations are not rigidly structured, so we need flexibility
And as we gain facility, we can (spontaneously) develop more efficient strategies
Strategy change in simple addition5 + 2 = 7
Children appear to gradually progress through a series of alternative ‘strategies’, with strategy choice being probabilistic and with the probabilities changing gradually over age
Children can be induced to change strategies if given problems that give a clear advantage to one strategy
Children’s strategies seem constrained to be consistent with the principles of addition, even though children can’t necessarily articulate such principles.
Incremental, Hierarchical, Supervised Reinforcement Learning in a Neural Network
A strategy is a sequence of steps, and reward only comes at the end
The time it takes as well as the outcome are automatically considerations in reinforcement learning.
The use of a neural network as function
Re-use of number skills previously acquired leads to selection of task-appropriate rather than task inappropriate strategies.
The ‘strategy’ as a whole emerges as an assemblage of strategy chunks, each associated with a component skill relevant to addition.
A key idea is that learning is curriculum based:
The culture and educational system provide early experiences in initial components that then provide the previously acquired skills.
Intuitive Geometry Project: Motivations
Geometrical intuition as developing gradually with age, through a series of ‘levels’.A year’s course in Geometry has no special impact on student’s ‘level’.Lessons learned from presenting students with the Socratic Dialog uncovering the supposed prior understanding of how to create a square with twice the area of a given square.In spite of profession of ‘understanding’ after walking through the dialog, those with many misconceptions can’t demonstrate the solution on a new square.Geometry as grounded in Intuition but ultimately connected to proofCarmenga, Transforming Geometric Proof with Reflections, Rotations and TranslationsHenderson, Experiencing Geomety
Given: ∠A≅∠A’, AC≅A’C’, ∠C≅∠C’Prove: △ABC≅△A’B’C’Idea: translate A to A’rotate △ABC until AC coincides with A’C’reflect over A’C’ if necessary. Then the whole triangle coincides!
Given: ∠A≅∠A’, AC≅A’C’, ∠C≅∠C’Translate △ABC so that A coincides with A’.Rotate △ABC so that ray AC coincides with ray A’C’. Since AC≅A’C’, C coincides with C’.If B and B’ are on different sides of line AC, reflect △ABC over line AC.Since ∠A≅∠A’ and AC and A’C’ coincide and are on the same side of the angle, ∠A coincides with ∠A’.Since the angles coincide, the other rays AB and A’B’ coincide. Similarly, since ∠C≅∠C’ and AC and A’C’ coincide, ∠C coincides with ∠C’ and the other rays CB and C’B’ coincide. Since ray AB coincides with ray A’B’ and ray CB with ray C’B’and two lines intersect in at most one point, B coincides with B’. Since all sides and angles coincide, △ABC≅△A’B’C’.
How Can We Begin to Make Progress on this Ambitious Project?
Create a simulated agent that must carry out tasks in a virtual world
Similar to the
Agent has a few actions it can perform
Change its point of view on its (2-D) world
Move, rotate and flip objects
Measure, copy, and construct objects to instruction using geometry tools
E.g., adjustable straightedge and compass
Demo some time during the conference!
Train the agent using incremental supervised learning
Find named objects, find objects that have the same shape as a given target
Translate, rotate, and flip objects to fit them through shaped wholes
Learn to measure length and angle
Learn to impose alternative frames of reference to identify congruent shapes under rotations and flips
Two Relevant Ideas
Use eye movements to bring objects to the center of gaze, where they can be recognized in canonical position.Plaut, McClelland, & Seidenberg, NCPW 1Mnih et al, 2014Use transforming autoencoders to learn effects of transformations on the way objects look.Hinton et al, 2011
out Euclidean constructions
- Based on given diagrams
- Purely from instruction
. Determine perimeter and area of polygons and circles in given units
. Establish correspondence between figures A and B
complex geometry problems requiring several intermediate inferences and
One Example Problem
Challenges, Open Questions, and Broader Directions
Explicit cognition, metacognitive knowledge, ant their relation to knowledge in connections
Abstract mathematics, proof, and justification
Are they, at least in part, extensions of concrete embodied reasoning
A broader understanding of understanding in embodied terms
We don’t just map mathematical expressions onto learned conceptual structures, we do the same in general when we understand ideas expressed in language as well