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The Algebra Project
What We Teach is as Important as How We Teach: The Critical
Analysis of Mathematical Content
April 22, 2015Pontifica Universidad Catolica del Peru
Bill Crombie, Director of Professional Development, The Algebra Project
The Algebra Project
THE ALGEBRA PROJECT
AbstractFor underserved, underperforming students at both the secondary and tertiary educational levels, the central problem that concerns us is how to accelerate their development in mathematics and related STEM disciplines in order to close a growing opportunity gap in their access to advanced study in the mathematical sciences.
The Algebra Project
THE ALGEBRA PROJECT
We need studies that identify and examine the mathematics content specifically needed for middle and secondary teaching. Such an effort is a prerequisite for design of algebraic structure courses that concentrate on abstract algebra concepts underlying algebra in the middle and secondary school curriculum.
Algebra: Gateway to a Technological Future
Victor Katz
The Present Challenge
The Algebra Project
THE ALGEBRA PROJECT
We recommend supporting initiatives that focus on development of alternative algebraic structures courses/materials for prospective middle and secondary teachers. Such courses can still provide opportunities for students to learn how to construct precise mathematical arguments, but prove to be more beneficial in the preparation of prospective middle and high school mathematics teachers.
Algebra: Gateway to a Technological Future
Victor Katz
The Present Challenge
The Algebra Project
THE ALGEBRA PROJECT
So it is important to develop careful research studies evaluating the efficacy of such courses. All three kinds of efforts – content analysis, design of innovative courses, and study of pre-service teacher learning – are projects in which collaboration among mathematicians, mathematics teacher educators, and mathematics education researchers will be most productive. Furthermore, it will be useful to initiate exploratory studies of ways that other courses in the undergraduate preparation of secondary school teachers could also contribute better to their content preparation.
Algebra: Gateway to a Technological Future
Victor Katz
The Present Challenge
The Algebra Project
THE ALGEBRA PROJECT
Different domains of knowledge, such as science, mathematics, and history, have different organizing properties. It follows, therefore, that to have an in-depth grasp of an area requires knowledge about both the content of the subject and the broader structural organization of the subject.
How People Learn
Bradford
The Present Challenge
The Algebra Project
THE ALGEBRA PROJECT
An understanding of the structure of knowledge provides guidelines for ways to assist learners to acquire a knowledge base effectively and efficiently.
How People Learn
Bradford
The Present Challenge
The Algebra Project
THE ALGEBRA PROJECT
The math education research community has a major focus on examining how we teach (pedagogy) and has approached that focus from a number of broad research agendas; pedagogical content knowledge, PCK, (Shulman, 1986) and mathematical knowledge for teaching, MKT, (Ball, Thanes, Phelps, 2008) are two of the most prominent perspectives. But both of these frameworks take the underlying mathematics as given.
The Present Challenge
The Algebra Project
THE ALGEBRA PROJECT
The research agenda we have initiated and are developing, Mathematical Content Analysis, subjects the underlying mathematics to a process of theoretical reconstruction and critique.
Mathematical Content Analysis (MCA) finds problematic and critically examines what these perspectives take for granted.
The Present Challenge
The Algebra Project
THE ALGEBRA PROJECT
This analysis holds that the conceptual architecture of a knowledge domain can either support or suppress ease of entry as surely as the architecture of a building determines the presence or absence of physical barriers to entry.
The Present Challenge
The Algebra Project
THE ALGEBRA PROJECT
Architecture, whether physical or conceptual, is intentional and subject to design. The conceptual architecture of mathematics is neither predetermined nor immutable. By developing alternative conceptual architectures for critical subjects, such as Calculus and Statistics, Mathematical Content Analysis shifts the border between what is traditionally considered basic versus advanced mathematics.
The Present Challenge
The Algebra Project
THE ALGEBRA PROJECT
Pastuer’s Quadrant
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THE ALGEBRA PROJECT
For students performing below proficiency, and consequently for all students, Mathematical Content Analysis (MCA) has the potential to reset the boundaries on what can be taught within a given content area and consequently what should be taught in that content area.
The Present Challenge
The Algebra Project
THE ALGEBRA PROJECT
To paraphrase Peter Gabriel Bergman in his introduction to Basic Theories of Physics, the crucial task of Mathematical Content Analysis is to help clarify the conceptual framework of the mathematics we teach and to re-forge it from time to time to keep it abreast of societal progress and educational need.
The Present Challenge
THE ALGEBRA PROJECT
The Algebra Project
Everything should be made as simple as possible, but not simpler.
Albert Einstein
The Algebra Project
THE ALGEBRA PROJECT
The study of mathematics is apt to commence in disappointment … The reason for the failure of the science to live up to its reputation is that its fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. Accordingly, the unfortunate learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated by any general conception.
An Introduction to MathematicsAlfred North Whitehead
The Algebra Project
THE ALGEBRA PROJECT
Essential FeaturesThe Calculus is often defined by its central and predominate procedures, by a set of defining techniques. Under this perspective Calculus is synonymous with analysis – the study and application of limits. An alternative is to define a discipline by its constitutive or defining problems. Under this alternative perspective the Calculus is defined by two basic problems – the Tangent Problem and the Area Problem. Such a definition of a knowledge domain makes a clear separation between the problems which the domain addresses and “the technical procedures … invented [for] their exact presentation.”
The Algebra Project
THE ALGEBRA PROJECT
The Tangent Problem
We have to face two problems. One of them is to give the correct geometric idea which allows us to define the tangent to a curve, and the other is to test whether this idea allows us to compute effectively this tangent line when the curve is given by a simple equation with numerical coefficients. It is a remarkable thing that our solution of the first problem will in fact give us a solution to the second.
A First Course in CalculusSerg Lang
The Algebra Project
THE ALGEBRA PROJECT
The Area Problem
Given a polynomial graph and an interval on the x-axis determine the rectangle with area equal to the area between the graph and the x-axis over the length of the given interval.
The area under the Parabola is usually taken as the watershed that separates Algebra and Geometry from the Integral Calculus.
The Algebra Project
THE ALGEBRA PROJECT
Essential FeaturesThe conceptual architecture of a knowledge domain can either facilitate access to the domain or act as a barrier to entry. A re-conceptualization of the Elementary Calculus, without the techniques and procedures of limits, using only the basics of algebra and geometry holds the promise of providing students with an avenue of access to the Calculus with limits and the quantitative sciences which depend upon it.
The Algebra Project
THE ALGEBRA PROJECT
Essential Features
Advanced Knowledge Domain
Central Concepts Advanced Procedures
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THE ALGEBRA PROJECT
The Concept of Speed
Observation shows, in fact, that there exists a basic intuition of speed, independent of any idea of duration and resulting from the primal concept of order … namely the intuition of kinematic overtaking. If a moving object A is behind B at instant T1 and passes in front of moving object B at instant T2, it is judged to be faster, and this holds for all ages: nothing intervenes here except temporal order ( T1 before T2 ) and spatial order ( behind and in front of ), and there is no consideration of duration or space traversed. Speed is therefore initially independent of durations.
Psychology and Epistemology
Jean Piaget
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THE ALGEBRA PROJECT
• A smoothly accelerating car, which we will call the A-Car, is travelling in a given direction on a straight road.
• Consider all possible cars in uniform motion, traveling in parallel lanes on the same road.
• We will call these uniformly moving cars U-Cars. At time T the A-Car and all the cars in uniform motion reach a designated mile marker.
A Thought Experiment
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THE ALGEBRA PROJECT
• We are interested in a particular car in uniform motion which we will call the Transition Car, or T-Car. The velocity of the T-Car has a distinguishing characteristic: it is the velocity which, at time T, separates the U-Cars that overtake the A-Car from the U-Cars that are overtaken by the A-Car.
• On a position vs. time graph the cars in uniform motion correspond to straight lines intersecting the graph of the A-Car at time T.
• Immediately after time T, the U-Car lines passing above the A-Car graph have velocities greater than the velocity of the A-Car and are on one side of the Transition Line. Similarly the U-Car lines passing below the A-Car graph have velocities less than the velocity of the A-Car and are on the other side of the Transition Line.
A Thought Experiment
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THE ALGEBRA PROJECT
A Thought Experiment
Transition Line
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THE ALGEBRA PROJECT
How do you know that the Transition Line even exists?
How do you know if the Transition Line is unique?
Why is the velocity of the A-Car at time T equal to the velocity of the T-Car? The velocity of the A-Car at time T is called its instantaneous velocity.
The Transition Line is the closest line to the graph of the A-Car’s motion at time T. Why?
A Thought Experiment
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THE ALGEBRA PROJECT
In Book III, Proposition 16 of The Elements Euclid defines the tangent to a circle:
The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed.
The Algebra Project
THE ALGEBRA PROJECT
The tangent line through a point on a smooth curve is the closest line to the curve at that point.
Euclidean Tangent Condition
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THE ALGEBRA PROJECT
The Closest Line to a CurveThe Tangent Line at x = 0In this example we will find the closest line to a parabola at the y-intercept, (0,c).
y = ax2 + bx + c
The Algebra Project
THE ALGEBRA PROJECT
The Closest Line to a CurveArithmetic Agreement: The closest line to the parabola requires the closest numerical agreement between the coordinates of the parabola and the coordinates of the line.
Algebraic Agreement: For the coordinates of the parabola and the line to have the closest numerical agreement the polynomials that define the two graphs must maximally agree. The polynomials of the parabola and the line maximally agree when their coefficients are the same to first order.
The Algebra Project
THE ALGEBRA PROJECT
The Closest Line to a Curve
The tangent line at x = 0 is the line that agrees with the parabola at x = 0 to first order. y = ax2 + bx + c yt = bx + c
The Algebra Project
THE ALGEBRA PROJECT
It is a well founded historical generalization that the last thing to be discovered in any science is what the science is really about.
An Introduction to Mathematics
Alfred North Whitehead
The Area Problem
A New Quadrature of the Parabola
y = x2
T
T
T
R = 2T + G
G
(1/2) G
G = 2(1/2 G)
2 G = 4(1/2 G)
2G = G + T
G = T
R = 3T
T = (1/3)R
T = (1/3)b3
The Area Problem
The Odd Case of the Cubic
y = x3
R = 2T + G
G
(1/2) G
G = 2(1/2 G)
(4/3)G = (2/3)R
G = (1/2)R
(1/2)R = 2T
T = (1/4)R
T = (1/4)b4
Differential Calculus
The Algebra and Geometry of the Elementary Calculus
Integral Calculus
This work is part of an on-going project to develop advanced mathematics
from an elementary standpoint.Consider the lines passing through a point on a smooth
graph. Some lines pass from above to below the graph. Some lines pass from below to above the graph. Consider what we will call a Transition Line.A Transition Line is a line that separates the lines that pass above the graph from the lines that pass below the graph at the given point.
Theorem: The Transition Line is uniqueSuppose there are two distinct Transition Lines. If the Transition Lines are distinct and pass through the given point on the graph there is a non-zero angle between them. Consider a line that passes through this angle. By one of the Transition Lines the line within this angle passes above the graph. By the other Transition Line this line passes below the graph. Since the same line can not both pass above and below the graph, the two Transition Lines must coincide.
Theorem: The Transition Line is the Closest Line to the Graph Suppose that there is a line passing between the graph and the Transition Line. If there was such a line, the Transition Line would not separate the line passing above the graph from the lines passing below the graph. Therefore there can be no line passing between the graph and the Transition Line. The Transition Line, the closest line to the graph, is historically referred to as the Tangent Line.
Theorem: The Slope of the Graph at a given point is the Slope of the Transition Line At a given point consider the lines passing above the graph, the lines passing below the graph, and the Transition Line. The lines passing above the graph are changing faster than the graph. The lines passing below the graph are changing slower than the graph. Therefore at the given point the rate of change of the graph is equal to the rate of change of the Transition Line.
The Tangent Problem We have to face two problems. One of them is to give the correct geometric idea which allows us to define the tangent to a curve, and the other is to test whether this idea allows us to compute effectively this tangent line when the curve is given by a simple equation with numerical coefficients. It is a remarkable thing that our solution of the first problem will in fact give us a solution to the second. A First Course in Calculus Serge Lang
The Tangent Line at x = 0In this example we will find the closest line to a parabola at the y-intercept, (0,c).
y = ax2 + bx + c
Arithmetic Agreement: The closest line to the parabola requires the closest numerical agreement between the coordinates of the parabola and the coordinates of the line.
Algebraic Agreement: For the coordinates of the parabola and the line to have the closest numerical agreement the polynomials that define the two graphs must maximally agree. The polynomials of the parabola and the line maximally agree when their coefficients are the same to first order.
The tangent line at x = 0 is the line that agrees with the parabola at x = 0 to first order. y = ax2 + bx + c yt = bx + c
The Tangent Line at x = x0
Consider the closest line toa parabola at the point ( x0 , y0 ).
y = ax2 + bx + c
By a change of coordinates, x = x0 + x´ , the tangent line at x = x0 becomes the tangent line at x´ = 0. y = a( x0 + x´ )2 + b( x0 + x´ ) + c y = a( x´ )2 + ( 2ax0 + b )( x´ ) + ( ax0
2 + bx0 + c ) and from the previous argument the tangent line is given by yt = ( 2ax0 + b )( x´ ) + ( ax0
2 + bx0 + c )
The tangent line at x = x0 is the line that agrees with the parabola at x = x0 to first order. y = ax2 + bx + c yt = ( 2ax0 + b )( x - x0 ) + ( ax0
2 + bx0 + c )
Principle of Agreement: The tangent line to a polynomial graph agrees with the graph to first order.
The Area ProblemGiven a polynomial graph and an interval on the x-axis determine the rectangle with area equal to the area between the graph and the x-axis over the length of the given interval.
Here is a proof without words for the area under the Cubic. Try it.
The Principle of Agreement determines the tangent line to polynomial, rational and algebraic graphs.
The study of mathematics is apt to commence in disappointment … The reason for the failure of the science to live up to its reputation is that its fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. Accordingly, the unfortunate learner finds himself struggling to acquire
a knowledge of a mass of details which are not illuminated by any general conception. An Introduction to Mathematics
Alfred North Whitehead
The Calculus is often defined by its central and predominate procedures, by a set of defining techniques. Under this perspective Calculusis synonymous with analysis – the study and application of limits. An alternative is to define a discipline by its constitutive or defining problems. Under this alternative perspective the Calculus is defined by two basic problems – the Tangent Problem
and the Area Problem. Such a definition of a knowledge domain makes a clear separation between the problems which the domain addresses and “the technical procedures … invented [for] their exact presentation.”
In turn, the conceptual architecture of a knowledge domain can either facilitate access to the domain or act as a barrier to entry. A re-conceptualization of the Elementary Calculus,
without the techniques and procedures of limits, using only the basics of algebra and geometry holds the promise of providing students with an avenue
of access to the Calculus with limits and the quantitative sciences which depend upon it.
In this example we will find the area of the region between the parabola ( y = x2 ) and the positive x-axis within the enclosing unit square. We symbolize the area of this parabolic “triangular” region by P.
The area of the square ( S ) is equal to the area of the two triangles ( 2P ) plus the area of the gap ( G ).
S = 2P + G
The area of the translated figure is equal to half the area of the gap.
( ½ G )
Horizontally scaling the previous figure by a fact of 2 results in a figure with an area twice that of half the gap.
2( ½ G )
Vertically scaling the previous figure by a factor of 2 results in a figure that has twice the area of the gap.
( 2G )
The figure within the previous square is the same size and shape as the original gap and parabolic triangle. It can be derived from the original square containing two triangles and the gap by a reflection about a vertical line located at the midpoint of the base. Since the area of the previous figure was equal to twice the area of the gap, the area of the lower triangle ( P ) must also be equal to the area of the gap ( G ).
G = P
Since the area of the gap ( G ) is equal to the area of the triangle ( P ) the area of the enclosing square is three times the area of the parabolic triangle.
P = ⅓ S
The area of the gap is determined by the difference between its upper and lower boundaries. So the area of the difference between the boundaries is equal to the area of the gap ( G ).
The Algebra Project
•History
•Essential Features
•Expected Student Outcomes
•Program Implementation
THE ALGEBRA PROJECT
•Ball, Thames, Phelps, 2008. Content Knowledge for Teaching: What Makes It Special. Journal of Teacher Education, 59, 389-407.
•Bradford, J. D., Brown, A. L., and Cocking, R. R. (Eds.). (1999) How People Learn: Brain, Mind, Experience, and School. Washington, DC: National Academy Press.
•Grant, M., Crombie, W. (2012). Polynomial Calculus: Rethinking the Role of Architecture and Access to Advanced Study. Paper presented at the International Conference on Math Education XII, Seoul, South Korea.
•Grant, M., Crombie, W., Cobb, N., Tuttle, J., Clawson, A., Hasan, L. (2012). Polynomial Calculus: Rethinking the Role of Calculus in High Schools. Paper presented at the International Conference on Math Education XII, Seoul, South Korea.
•Katz, V. (Ed.). (2007). Algebra: Gateway to a Technological Future, Washington DC: Mathematical Association of America.
•Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational
Researcher, 15(2), 4-14.
