LAGRANGIAN AND BERNOULLI RELATIONS

ABSTRACT

Geodesics Using Mathematica

Jacob Lewis*

Columbia University

Abstract: We describe surfaces and geodesics without assuming prior knowledge of differential geometry. This involves selecting and presenting basic definitions and theorems. Included in this discussion are definitions of surface, coordinate patch, geodesic, etc. This summary closes with a proof of the length-minimizing properties of geodesics. Examples of surfaces are given and plotted in Mathematica. We also describe geodesics on these surfaces and plot select examples. The surfaces chosen include some with Clairaut patches, some without, and some surfaces in R3 and some not in R3.

A well-known adage is, “The shortest distance between two points is a straight line.” This is certainly true on the plane, but on other surfaces the adage proves to be false. For example, assume the earth to be a sphere. New York City and Madrid, Spain are both at latitudes of about 40°N. Yet an airplane taking the shortest distance between the two does not follow the 40th parallel. Rather, it arcs north, following the great circle (i.e., circle centered at the sphere’s center) between the two cities.

To generalize the adage–and along the way to explain why planes travel this way–we will introduce a special class of curves on surfaces, called geodesics. Geodesics have

1 INTRODUCTION

Pi/180].sphere[-3.70 Pi/180, 40.43 Pi/180])/(r^2)]

This gives = 0.904385. The radius of the earth can be estimated as 6378 km (3963 miles). So the distance from New York to Madrid along the shortest path is 5768.17 km (3589.50 miles). The radius of the circle around the fortieth parallel is = 2443.48 km; the center of this circle is (0, 0, 4165.83). So the angle between the vector from the center of this circle to New York and the vector from the center of this circle to Madrid (assuming for simplicity both are on the 40th parallel) can be found in Mathematica as angle=ArcCos[(earth[-73.97 Pi/180, 40 Pi/180]-{0,0,4165.83}).(earth[-3.70 Pi/180, 40 Pi/180]-{0,0,4165.83}))/(rho^2)], where rho is the radius of the 40th parallel. angle = 1.22644, so the distance from New York to Madrid along the 40th parallel is 5992.2 km. So the great circle method is significantly shorter.

The Cylinder

The cylinder can be parameterized by x(u, v) = (r cos(u), r sin(u), v). xu●xu = r2, xu●xv = 0, and xv●xv = 1, so this is clearly a Clairaut patch. Solving the Clairaut integral for v in terms of u gives v = ku. Now, unit-speed implies 1 = r2 u’2 + v’2 = (r2 + k2) u’2. So , for some constants k and c. So a geodesic on a cylinder has the form . This is the equation of a helix. Two interesting degenerate cases are where the initial velocity is horizontal (k=0)—in which case the geodesic is a circle going around the cylinder—and where the initial velocity is vertical (k=∞)—in which case the geodesic is a vertical line up the side of the cylinder. These cases and one non-degenerate helical case are shown, using the Mathematica procedure, in Figure 5.

Figure 5

2 FORMULATION

by

theta = ArcCos[(sphere[-73.97 Pi/180, 40.78

The Torus of Revolution

The torus of revolution is the surface formed by revolving a circle in the y-z plane, which does not touch the z-axis, about the z-axis. The surface looks like a doughnut. A patch for the torus of revolution is x(u, v) = ((a + rcosv) cosu, (a + rcosv) sinu, rsinv). r here represents the radius of the circle being revolved; a represents the distance from the circle to the z-axis. The case r=1, a=2 is shown in Figure 6.

Figure 6

The patch here is a v-Clairaut patch. In fact, all surfaces of revolution admit a Clairaut patch. If the curve being revolved is (g(v), h(v)), the patch is x(u,v) = (h(v)cosu, h(v)sinu, g(v)). In the case of the torus, E = xu • xu = (a + rcosv)2, and G= xv • xv = r2. Yet the Clairaut integral cannot be solved explicitly. Here is where the implicit approach of the Mathematica procedure is useful. A few examples may be instructive. If the initial direction is /2, the geodesic goes around the “doughnut hole” as in Figure 7

Figure 7

If the initial point is above the largest circle around the outside of the torus, and the initial direction is 0 (i.e. horizontal), then the geodesic turns downward to the bottom of the torus, then turns upward to the top, then downward, and so on. Figure 8 is an example.

Figure 8

The “Two-Banana” Surface

As we saw with the torus of revolution, there are cases were with a Clairaut patch no explicit equations for the geodesics can be found. If the patch we are working with for a surface is not Clairaut, it becomes even less likely that a search for explicit geodesics will be successful. For example, consider the surface parameterized by x[u,v]=((2+cos(u))cos(v), cos(v)sin(u), sin(v)). We call this surface the “Two-Banana” surface because it looks like two bananas meeting at their ends. This surface is shown in Figure 9.

Figure 9

For this patch, G= xv • xv = cos2v + (5 + 4cosu)sin2v, so the patch is not Clairaut or v-Clairaut. We can use the Mathematica procedure geo to get an idea of what the geodesic equations look like in this case. The results, in Figure 10, are not pretty.

Figure 10

Mathematica, however, can still graph the geodesics implicitly. If the geodesic begins horizontally along the widest circle of a banana, it will go around that circle as in Figure 11. If it begins vertically along the outermost part of a banana, it will loop around both bananas as in Figure 12. In general, however, the geodesics stay on one banana and have complicated shapes. For example, Figure 13 shows the geodesic starting at u=0, v=0 and with initial direction /4.

Figure 11

LikeLike

]]>with

INTERPOLATION FOR LAGRANGIAN

ABSTRACT

11. Derive the Quadratic Formula by Completing the Square (coming soon: cubics, quartics!)

12. The Locker Problem (coming soon: Fundamental Thm of Arithmetic, number theory)

13. Prove the Binomial Theorem (by induction!) (coming soon: more number theory!)

a. Hint: Using construction method of Pascal’s Triangle, find recursive defn of nCk

b. Application: prove: nC0 + nC1 + … + nCn= 2n

Summer Math Series: Week 3

14. Pascal’s Triangle and Pascal’s Binomial Theorem

a. nCk = kth value in nth row of Pascal’s Triangle! (Proof by induction)

b. Rows of Pascal’s Triangle == Coefficients in (x + a)n. That is:

15. The Circle Problem and Pascal’s Triangle

a. How many intersections of chords connecting N vertices?

b. How does this relate to Pascal’s Triangle?

16. Patterns in Pascal’s Triangle (see http://www.kosbie.net/lessonPlans/pascalsTriangle/)

a. Simple Patterns

i. Natural Numbers (1,2,3,4…)

ii. Triangular Numbers (1,3,6,10,…)

iii. Binomial Coefficients (nCk) Pascal’s Binomial Theorem

iv. Tetrahedral Numbers (1,4,10,20,…)

v. Pentatope Numbers (1,5,15,35,70…)

b. More Challenging Patterns

i. Powers of 2 (2,4,8,16,…)

ii. Hexagonal Numbers (1,6,15,28,…)

iii. Fibonacci Numbers (1,1,2,3,5,8,…) Prove This!

iv. Sierpinski’s Triangle

v. Catalan Numbers (1,2,5,14,42,…) Prove This!

vi. Powers of 11 (11, 121, 1331, 14641,…)

17. Applications of the Binomial Theorem

a. Find the coefficient of x3 in (x + 5) 3

b. Prove: nC0 + nC1 + … + nCn= 2n (Hint: 2 = 1+1, so what does 2n = ?)

1 . INTRODUCTION

14. Archimedes’ Approximation of π

a. Inscribe (2n)-gons

15. Newton’s Binomial Theorem

a. Generalization to negative integer powers:

b. (P + PQ)m/n = P m/n + (m/n)AQ + (m-n)/(2n) BQ + (m-2n)/(3n) CQ + …

where A,B,C,… represent the immediately preceding terms

so B = (m/n)AQ, C = (m-n)/(2n) BQ, …

c. After some algebra:

(1 + Q) m/n = 1 + (m/n)Q + (m/n)(m/n – 1)/2 Q2 + (m/n)(m/n – 1)(m/n – 2)/(3*2) Q3 + …

d. That is:

16. Applications of Newton’s Binomial Theorem

a. 1 / (1 + x)3 = 1 – 3x + 6×2 – 10×3 + 15×4 – …

b. Sqrt(1 – x) = 1 – (1/2)x – (1/8)x2 – (1/16)x3 – (5/128)x4 – …

c. So, sqrt(7) = 3 sqrt(1 – 2/9) fast approximation for square roots!

d. Also cube roots, etc, since (1 – x)1/3 can be expanded this way, too…

Summer Math Series: Week 5

17. Newton’s Calculus (“Fluxions” from De Analysi; see Dunham pp. 171-3)

a. f(x) = x2 f’(x) = 2x

i. What this means graphically (max/min of f(x) = x2)

b. f(x) = a g(x) f’(x) = a g’(x)

c. f(x) = g(x) + h(x) f’(x) = g’(x) + h’(x)

d. f(x) = xa f’(x) = a x(a-1)

e. General derivative of a polynomial:

f(x) = a0x0 + … + anxn f’(x) = a1x0 + 2a2x1 + 3a3x2… + nanx(n-1)

f. Homework:

i. Prove the Product Rule:

f(x) = g(x) h(x) f’(x) = g’(x)h(x) + h’(x)g(x)

ii. Prove the Chain Rule:

f(x) = g(h(x)) f’(x) = g’(h(x)) h’(x)

iii. Prove the Quotient Rule:

f(x) = g(x) / h(x) f’(x) = (g’(x) h(x) – h’(x) g(x)) / h(x)2

Hint: rewrite as f(x) = g(x) h(x)-1 and use the Product and Chain Rules.

g. Some other useful derivatives:

i. f(x) = cos(x) f’(x) = -sin(x)

ii. f(x) = sin(x) f’(x) = cos(x)

iii. f(x) = ex f’(x) = ex

iv. Homework: Find the derivatives of tan(x), cot(x), sec(x), csc(x)

h. Integral calculus and Newton’s Physics

i. Constant acceleration: a(t) = a0 (9.8 m/s2)

ii. v’(t) = a(t) v(t) = a0t + v0

1. Why did Newton assume that v’(t) = a(t)

2. Why does this imply that v(t) = a0t + v0?

3. How fast is a free-falling object moving after 5 seconds?

iii. s’(t) = v(t) s(t) = ½ a0t2 + v0t + s0

1. Why did Newton assume that s’(t) = v(t)

2. Why does this imply that s(t) = ½ a0t2 + v0t + s0?

3. How far did that free-falling object travel in 5 seconds?

i. Find loca

LikeLike

]]>and ROOTS OF LAGRANGIAN

ABSTRACT

To compete successfully in the worldwide economy, today’s students must have a high degree of comprehension in mathematics. For too long schools have suffered from the notion that success in mathematics is the province of a talented few. Instead, a new expectation is needed: all students will attain California’s mathematics academic content standards, and many will be inspired to achieve far beyond the minimum standards.

These content standards establish what every student in California can and needs to learn in mathematics. They are comparable to the standards of the most academically demanding nations, including Japan and Singapore – two high-performing countries in the Third International Mathematics and Science Study (TIMSS). Mathematics is critical for all students, not only those who will have careers that demand advanced mathematical preparation but all citizens who will be living in the twenty-first century. These standards are based on the premise that all students are capable of learning rigorous mathematics and learning it well, and all are capable of learning far more than is currently expected. Proficiency in most of mathematics is not an innate characteristic; it is achieved through persistence, effort, and practice on the part of students and rigorous and effective instruction on the part of teachers. Parents and teachers must provide support and encouragement.

The standards focus on essential content for all students and prepare students for the study of advanced mathematics, science and technical careers, and postsecondary study in all content areas. All students are required to grapple with solving problems; develop abstract, analytic thinking skills; learn to deal effectively and comfortably with variables and equations; and use mathematical notation effectively to model situations. The goal in mathematics education is for students to:

Develop fluency in basic computational skills.

Develop an understanding of mathematical concepts.

Become mathematical problem solvers who can recognize and solve routine problems readily and can find ways to reach a solution or goal where no routine path is apparent.

Communicate precisely about quantities, logical relationships, and unknown values through the use of signs, symbols, models, graphs, and mathematical terms.

Reason mathematically by gathering data, analyzing evidence, and building arguments to support or refute hypotheses.

Make connections among mathematical ideas and between mathematics and other disciplines.

The standards identify what all students in California public schools should know and be able to do at each grade level. Nevertheless, local flexibility is maintained with these standards. Topics may be introduced and taught at one or two grade levels before mastery is expected. Decisions about how best to teach the standards are left to teachers, schools, and school districts.

The standards emphasize computational and procedural skills, conceptual understanding, and

Adopted by the California State Board of Education December 1997

1 INTRODUCTION

Spherical:

Transformations

Differential length vectors

Del Operator:

Green’s Theorem

Divergence Theorem

Stoke’s Theorem

Dielectric Material Properties:

Magnetic Material Properties:

Displacement Field:

Magnetic Field Intensity:

LikeLike

]]>LikeLike

]]>LikeLike

]]>DEFINITION AND NEWTONIAN LAGRANGIAN

ABSTRACT The general goal in this discipline is for students to learn the techniques of matrix manipulation so that they can solve systems of linear equations in any number of variables. Linear algebra is most often combined with another subject, such as trigonometry, mathematical analysis, or precalculus.

1.0 Students solve linear equations in any number of variables by using Gauss-Jordan elimination.

2.0 Students interpret linear systems as coefficient matrices and the Gauss-Jordan method as row operations on the coefficient matrix.

3.0 Students reduce rectangular matrices to row echelon form.

4.0 Students perform addition on matrices and vectors.

5.0 Students perform matrix multiplication and multiply vectors by matrices and by scalars.

6.0 Students demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions.

7.0 Students demonstrate an understanding of the geometric interpretation of vectors and vector addition (by means of parallelograms) in the plane and in three-dimensional space.

8.0 Students interpret geometrically the solution sets of systems of equations. For example, the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersection of a pair of lines in the plane.

9.0 Students demonstrate an understanding of the notion of the inverse to a square matrix and apply that concept to solve systems of linear equations.

10.0 Students compute the determinants of 2 x 2 and 3 x 3 matrices and are familiar with their geometric interpretations as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces.

11.0 Students know that a square matrix is invertible if, and only if, its determinant is nonzero. They can compute the inverse to 2 x 2 and 3 x 3 matrices using row reduction methods or Cramer’s rule.

12.0 Students compute the scalar (dot) product of two vectors in n- dimensional space and know that perpendicular vectors have zero dot product.

Pro

1 INTRODUCTION

1.0 Students solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probability distributions and understand the simplifications that arise with independent events.

2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.

3.0 Students demonstrate an understanding of the notion of discrete random variables by using this concept to solve for the probabilities of outcomes, such as the probability of the occurrence of five or fewer heads in 14 coin tosses.

4.0 Students understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable.

5.0 Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable.

6.0 Students know the definition of the variance of a discrete random variable and can determine the variance for a particular discrete random variable.

7.0 Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families.

8.0 Students determine the mean and the standard deviation of a normally distributed random variable.

9.0 Students know the central limit theorem and can use it to obtain approximations for probabilities in problems of finite sample spaces in which the probabilities are distributed binomially.

10.0 Students know the definitions of the mean, median, and mode of distribution of data and can compute each of them in particular situations.

11.0 Students compute the variance and the standard deviation of a distribution of data.

12.0 Students find the line of best fit to a given distribution of data by using least squares regression.

13.0 Students know what the correlation coefficient of two variables means and are familiar with the coefficient’s properties.

14.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

15.0 Students are familiar with the notions of a statistic of a distribution of values, of the sampling distribution of a statistic, and of the variability of a statistic.

16.0 Students know basic facts concerning the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution.

17.0 Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error.

18.0 Students determine the P- value for a statistic for a simple random sample from a normal distribution.

19.0 Students are familiar with the chi- square distribution and chi- square test and understand their uses.

LikeLike

]]>with

DEFINITION AND NOTIONS OF LAGRANGIAN

ABSTRACT

1. Area of a Triangle = bh/2

2. Pythagorean Theorem: Euclid’s Windmill Proof

3. Pythagorean Theorem: Chinese Proof (or perhaps the Indian mathematician Bhaskara’s)

4. Pythagorean Theorem: President Garfield’s Trapezoid Proof

5. The Distance Formula: derived from Pythagorean Theorem!

6. Fermat’s Last Theorem: xn + yn = zn has no positive integral solutions for n>2.

Proven recently by Andrew Wiles (omitted here for lack of room in the margin).

7. Hypotenuses in a “square root spiral” are of length sqrt 2, sqrt 3, sqrt 4, sqrt 5,…

(Inductive proof)

8. The square root of 2 is irrational.

a. (p/q) 2 = 2 p2 = 2q2, then apply Fundamental Theorem of Arithmetic lhs has even # of prime factors, rhs has odd #, QED.

b. (p/q) 2 = 2 p2 = 2q2 p is even … q is even, QED.

9. “Nearly all” real numbers are irrational!

a. The integers are countable (as are evens, primes, powers of 10, …)

b. Integer pairs – Z2 – are countable (dovetailing!)

c. Integer triplets, etc – Z3 , Z4,… – are countable.

d. Rationals are countable.

e. Algebraics are countable.

f. Reals are not countable (diagonalization!)

g. Thus, “nearly all” reals are irrational (even non-algebraic, hence transcendental!)

7. The Circle Problem and Pascal’s Triangle

a. How many intersections of chords connecting N vertices?

b. How does this relate to Pascal’s Triangle?

8. Patterns in Pascal’s Triangle (see http://www.kosbie.net/lessonPlans/pascalsTriangle/)

a. Simple Patterns

i. Natural Numbers (1,2,3,4…)

ii. Triangular Numbers (1,3,6,10,…)

iii. Binomial Coefficients (nCk) Pascal’s Binomial Theorem

iv. Tetrahedral Numbers (1,4,10,20,…)

v. Pentatope Numbers (1,5,15,35,70…)

b. More Challenging Patterns

i. Powers of 2 (2,4,8,16,…)

ii. Hexagonal Numbers (1,6,15,28,…)

iii. Fibonacci Numbers (1,1,2,3,5,8,…) Prove This!

iv. Sierpinski’s Triangle

v. Catalan Numbers (1,2,5,14,42,…) Prove This!

vi. Powers of 11 (11, 121, 1331, 14641,…)

9. Applications of the Binomial Theorem

a. Find the coefficient of x3 in (x + 5) 3

b. Prove: nC0 + nC1 + … + nCn= 2n (Hint: 2 = 1+1, so what does 2n = ?)

Summer Math Series: Week 4

10. π = C/D (By observation! Since Babylonian times, where π =~ 3.125)

11. Area of a Polygon = ½ hQ (1/2 * apothem * perimeter)

12. Archimedes’ Proof that A = πr2

a. Approximate circle with inscribed (2n)-gons

b. Rephrasing of argument on page 93:

i. Apolygon = ½ hQ, but:

1. As sides infinity, h r (apothem radius)

2. As sides infinity, Q C (perimeter circumference)

ii. So:

As sides infinity, Apolygon ½ r C Acircle

(area of polygon area of circle)

c. Last step (p. 96): combine:

i. A = ½ r C

1 . INTRODUCTION

A high-quality mathematics program is essential for all students and provides every student with the opportunity to choose among the full range of future career paths. Mathematics, when taught well, is a subject of beauty and elegance, exciting in its logic and coherence. It trains the mind to be analytic – providing the foundation for intelligent and precise thinking.

To compete successfully in the worldwide economy, today’s students must have a high degree of comprehension in mathematics. For too long schools have suffered from the notion that success in mathematics is the province of a talented few. Instead, a new expectation is needed: all students will attain California’s mathematics academic content standards, and many will be inspired to achieve far beyond the minimum standards.

These content standards establish what every student in California can and needs to learn in mathematics. They are comparable to the standards of the most academically demanding nations, including Japan and Singapore – two high-performing countries in the Third International Mathematics and Science Study (TIMSS). Mathematics is critical for all students, not only those who will have careers that demand advanced mathematical preparation but all citizens who will be living in the twenty-first century. These standards are based on the premise that all students are capable of learning rigorous mathematics and learning it well, and all are capable of learning far more than is currently expected. Proficiency in most of mathematics is not an innate characteristic; it is achieved through persistence, effort, and practice on the part of students and rigorous and effective instruction on the part of teachers. Parents and teachers must provide support and encouragement.

The standards focus on essential content for all students and prepare students for the study of advanced mathematics, science and technical careers, and postsecondary study in all content areas. All students are required to grapple with solving problems; develop abstract, analytic thinking skills; learn to deal effectively and comfortably with variables and equations; and use mathematical notation effectively to model situations. The goal in mathematics education is for students to:

Develop fluency in basic computational skills.

Develop an understanding of mathematical concepts.

Become mathematical problem solvers who can recognize and solve routine problems readily and can find ways to reach a solution or goal where no routine path is apparent.

Communicate precisely about quantities, logical relationships, and unknown values through the use of signs, symbols, models, graphs, and mathematical terms.

Reason mathematically by gathering data, analyzing evidence, and building arguments to support or refute hypotheses.

Make connections among mathematical ideas and between mathematics and other disciplines.

The standards identify what all students in California public schools should know and be able to do at each grade level. Nevertheless, local flexibility is maintained with these standards. Topics may be introduced and taught at one or two grade levels before mastery is expected. Decisions about how best to teach the standards are left to teachers, schools, and school districts.

The standards emphasize computational and procedural skills, conceptual understanding, and

Adopted by the California State Board of Education December 1997

problem solving. These three components of mathematics instruction and learning are not

separate from each other; instead, they are intertwined and mutually reinforcing.

Basic, or computational and procedural, skills are those skills that all students should learn to use routinely and automatically. Students should practice basic skills sufficiently and frequently enough to commit them to memory.

Mathematics makes sense to students who have a conceptual understanding of the domain. They know not only how to apply skills but also when to apply them and why they should apply them. They understand the structure and logic of mathematics and use the concepts flexibly, effectively, and appropriately. In seeing the big picture and in understanding the concepts, they are in a stronger position to apply their knowledge to situations and problems they may not have encountered before and readily recognize when they have made procedural errors.

The mathematical reasoning standards are different from the other standards in that they do not represent a content domain. Mathematical reasoning is involved in all strands.

The standards do not specify how the curriculum should be delivered. Teachers may use direct instruction, explicit teaching, knowledge-based, discovery-learning, investigatory, inquiry based, problem solving-based, guided discovery, set-theory-based, traditional, progressive, or other methods to teach students the subject matter set forth in these standards. At the middle and high school levels, schools can use the standards with an integrated program or with the traditional course sequence of algebra I, geometry, algebra II, and so forth.

Schools that uti

2 . FORMULATION

The standards for grades eight through twelve are organized differently from those for kindergarten through grade seven. Strands are not used for organizational purposes because the mathematics studied in grades eight through twelve falls naturally under the discipline headings algebra, geometry, and so forth. Many schools teach this material in traditional courses; others teach it in an integrated program. To allow local educational agencies and teachers flexibility, the standards for grades eight through twelve do not mandate that a particular discipline be initiated and completed in a single grade. The content of these disciplines must be covered, and students enrolled in these disciplines are expected to achieve the standards regardless of the sequence of the disciplines.

Mathematics Standards and Technology

As rigorous mathematics standards are implemented for all students, the appropriate role of technology in the standards must be clearly understood. The following considerations may be used by schools and teachers to guide their decisions regarding mathematics and technology:

Students require a strong foundation in basic skills. Technology does not replace the need for all students to learn and master basic mathematics skills. All students must be able to add, subtract, multiply, and divide easily without the use of calculators or other electronic tools. In addition, all students need direct work and practice with the concepts and skills underlying the rigorous content described in the Mathematics Content Standards for California Public Schools so that they develop an understanding of quantitative concepts and relationships. The students’ use of technology must build on these skills and understandings; it is not a substitute for them.

Technology should be used to promote mathematics learning. Technology can help promote students’ understanding of mathematical concepts, quantitative reasoning, and achievement when used as a tool for solving problems, testing conjectures, accessing data, and verifying solutions. When students use electronic tools, databases, programming language, and simulations, they have opportunities to extend their comprehension, reasoning, and problem-solving skills beyond what is possible with traditional print resources. For example, graphing calculators allow students to see instantly the graphs of complex functions and to explore the impact of changes. Computer-based geometry construction tools allow students to see figures in three-dimensional space and experiment with the effects of transformations. Spreadsheet programs and databases allow students to key in data and produce various graphs as well as compile statistics. Students can determine the most appropriate ways to display data and quickly and easily make and test conjectures about the impact of change on the data set. In addition, students can exchange ideas and test hypotheses with a far wider audience through the Internet. Technology may also be used to reinforce basic skills through computer-assisted instruction, tutoring systems, and drill-and-practice software.

The focus must be on mathematics content. The focus must be on learning mathematics, using technology as a tool rather than as an end in itself. Technology makes more mathematics accessible and allows one to solve mathematical problems with speed and efficiency. However, technological tools cannot be used effectively without an understanding of mathematical skills, concepts, and relationships. As students learn to use electronic tools, they must also develop the quantitative reasoning necessary to make full use of those tools. They must also have opportunities to reinforce their estimation and mental math skills and the concept of place value so that they can quickly check their calculations for reasonableness and accuracy.

Technology is a powerful tool in mathematics. When used appropriately, technology may help students develop the skills, knowledge, and insight necessary to meet rigorous content standards in mathematics and make a successful transition to the world beyond school. The challenge for educators, parents,

LikeLike

]]>DEFINITION FOR LAGRANGIAN

ABSTRACT

In considering the future of teacher education at the present time, I believe that it is relevant to consider the wider social and political context in which schools and institutions of teacher education are placed at this time. In particular I wish to draw attention to what Prime Minister Tony Blair had to say in his speech to the 1998 Labour Party Conference, where he argued that:

The centre-left may have lost in the battle of ideas in the 1980s, but we are winning now. And we have won a bigger battle today: the battle of values. The challenge we face has to be met by us together: one nation, one community.

When a young black student, filled with talent, is murdered by racist thugs, and Stephen Lawrence becomes a household name not because of the trial into his murder but because of inquiry into why his murderers are walking free, it isn’t just wrong: it weakens the very bonds of decency and respect we need to make our country strong. We stand stronger together.

But where is Mr. Blair’s vision of ‘the battle of values’ when it comes to education policy? Whilst accepting a need to improve levels of achievement, I want to argue that there is a lack of vision in relation to ‘values’ in education at the present time. Further I propose that the reasons for this lie in part in the recent social, political and historical context in relation to the National Curriculum and also, in the perspective of some of the key government agencies, such as the TTA. In particular there are problems about the way in which the National Standards for teacher education have been prescribed.

A SHORT OVERVIEW OF THE RECENT HISTORICAL CONTEXT (1)

Consideration of the recent changes that have taken place in teacher education cannot be made in isolation from those happening in the National Curriculum for schools in England and Wales, just as any future changes to the school curriculum will imply corresponding changes for teacher education. Following a systematically orchestrated campaign from right wing pressure groups throughout the 1980s, political intervention in the school curriculum reached a high point at the Conservative Party conference in 1988, with the famous statement from Prime Minister Margaret Thatcher:

Children who needed to count and multiply were learning anti-racist mathematics – whatever that might be.

It was in such a climate that the proposals were put to the Secretary of State for Education, Kenneth Baker, on the composition of the mathematics curriculum. These proposals stated that it was unnecessary to include any ‘multicultural’ aspects in any of the attainment targets. This position was supported by arguing that that those proposing such an approach with a view to raising the self-esteem of ethnic minority pupils and to improving mutual understanding and tolerance between races, were afflicted with an attitude that was ‘misconceived and patronising’. Tooley’s (1990) support for such a position and his associated critique of arguments put by those in the mathematics education community he labelled as ‘multiculturalists’ is both misleading and flawed in several respects. He misleads by his mischievous suggestion that the ‘multiculturalists’ wished to dictate to teachers: e.g. he asserts that ‘the failure to ‘compel “multicultural” examples’ is not ‘a great handicap’ of the National Curriculum. In fact the pressure at that time was in precisely the opposite direction and compulsion was never part of the agenda of the so-called ‘multiculturalists’ in the first place. In his reflections on this context Woodrow (1996) has pointed out that the concerns of teachers, following the Swann Report and the tragedy leading up to the MacDonald report, have been dissipated as a result of the introduction of a National Curriculum in which there is ‘no internationalism … no celebration of a pluralist culture and no sense of diversity’. Tooley also creates a false dichotomy between those teachers ‘who prefer to raise the political consciousness of their pupils, rather than their mathematical attainment’. Can it not be the case that teachers of mathematics can do both? Is there not a case to be made for considering the contribution teachers of mathematics might make in terms of citizenship and democracy?

Set against this background it is not surprising that the debate around issues of social justice and equal opportunities in the classroom came to whither on the vine during the last decade in England and Wales. Further it is not surprising that schools are now seen, by the African and Caribbean Network of Science and Technology (ACNST), to be failing black pupils in the ‘status and power’ subjects of science, mathematics and technology (Ghouri, 1998b). It does seem that Tooley’s (1990) expectation that the National Curriculum proposals ‘have the potential to tackle that problem’ of underachievement has proved to be unfounded. Rather the ACNST research points towards the lack of role models for young black people e.g. ‘black British scientists’. It does seem that Tooley is also mistaken in his view that the use of exotic stereotypes such as the San people of the Kalahari desert is an appropriate and sufficient level of response to this problem.

THE PROBLEM (1)

My starting point in the debate is to agree with those critics such as Sir Herman Ousley and the ACNST that indeed there is a problem with both the National Curriculum for schools and with the principles and practices underpinning the system of teacher education at this time. In particular, I argue that the heart of the problem is the lack of a shared sense of purpose about the aims of education in this country, which has given rise to conflicting interpretations by government agencies that appear to contradict each other.

It would seem that the issue of the lack of a shared sense of purpose is recognised by the Qualifications and Curriculum Authority (QCA) in identifying working practices on policy formulation as an area in need of reform and in drawing attention to the wider international context. In relation to the nat

1 INTRODUCTION

The lack of a shared sense of purpose in our education system stands in sharp contrast to the sense of consensus that can be seen in other educational systems. For example with regard to the notion of Didaktik in the German and Scandinavian traditions the overall aim of the education system is that of ‘Gebildete’ which can be broadly translated as ‘educated personality’. This means, for example, fostering a sense of egalitarianism and having a curriculum that relates to the central problems of living and is relevant to the key problems of society. There is an emphasis here on attitudes and values, which seems to be singularly lacking from the UK context. This aspect has been highlighted by Moon (1998) who argues that ‘standards do not exist in a vacuum’ and that the imposition of standards without values ‘can easily become standardisation, the very process that a vibrant and dynamic culture has to avoid’. He highlights the educational systems of Scotland, Germany, France, USA and South Africa to illustrate the willingness in these countries to ‘link the education of teachers to a values system’. So for example, the Scottish model, which was developed following extensive consultation, includes in its guidelines ‘a set of attitudes that have particular power in that they are communicated to those being taught’. Included in these is:

• a commitment to views of fairness and equality of opportunity as expressed in multi-cultural and other non-discriminatory policies.

In the South African context the task of reconstructing the education system was preceded by widespread consultation over values, which led to the identification of five core ‘socio-political’ values and five core ‘pedagogical’ values. The former consist of ‘democracy, liberty, equality, justice and peace’ and the latter are made up of ‘relevance, learner-centredness, professionalism, co-operation and collegiality and innovation’. With regard to equality and justice in particular there is specific reference to equity, redress, affirmative action and the removal of gender and racial bias. The contrast with the situation in England and Wales at the present time is stark.

THE WAY FORWARD (1)

In my view the issues that need to be addressed for the future fall under three categories of need:

• for the development a shared sense of purpose about the aims and values of education, as these relate to both schools and teacher education

• to reform working practices between the various stakeholders in the way in which policy is developed

• to reconceptualise the notion of teacher competence as currently set out in the National Standards

The first of these has been the main focus of this paper. However the starting point for such a project would be a key factor in developing such a shared sense of purpose. A rightful concern would be around the question of ‘Whose values?’ and also of the threat of the imposition of an authoritarian agenda. However one might look to the communitarian agenda for a starting point and in particular to Etzioni (1995) who argues that we might start with those values that are widely shared. These include that ‘the dignity of all persons ought to be respected, that tolerance is a virtue and discrimination abhorrent and that peaceful resolution of conflicts is superior to violence’.

LikeLike

]]>NEWTONIAN AND LAGRANGIAN

ABSTRACT Binomial coefficients (N choose K): The number of ways in which you can choose K elements from a set of N elements. This equals n! / ( k! (n-k)! ).

Catalan numbers (1, 2, 5, 14, 42, …): The number of ways you can divide a polygon with N sides into triangles, using non-intersecting diagonals (a triangle has 1 way, a rectangle has 2 ways, a pentagon has 5 ways, a hexagon has 14 ways, and so on). The Catalan numbers can be computed using the formula:

Fibonacci numbers (1, 1, 2, 3, 5, 8, …): A series in which the first two numbers are 1 and each subsequent number is the sum of the preceding two numbers.

Hexagonal numbers (1, 6, 15, 28, …): Numbers that can be represented as the number of points on the perimeter of a hexagon with a constant number of points on each edge. These are given by the formula N * (2N-1), and can be seen in the following figure:

Pentatope numbers (1, 5, 15, 35, 70, …) A figurate number (a number that can be represented by a regular geometric arrangement of equally spaced points) given by:

Ptopn = (1/4)Tn(n+3) = (1/24) n (n+1) (n+2) (n+3)

for tetrahedral number Tn. Note: pentatopes are 4-dimensional analogs of tetrahedra.

Sierpinski’s triangle: a famous fractal formed by connecting triangle midpoints as such:

Tetrahedral numbers (1, 4, 10, 20, …): a figurate number formed by placing discrete points in a tetrahedron (triangular base pyramid). The formula is given by: n(n+1)(n+2)/6.

Triangular numbers (1, 3, 6, 10, …): The number of dots you need to form a triangle:

In any forms ar

]]>LAGRANGIAN AND DEFINITION

ABSTRACT

It was in such a climate that the proposals were put to the Secretary of State for Education, Kenneth Baker, on the composition of the mathematics curriculum. These proposals stated that it was unnecessary to include any ‘multicultural’ aspects in any of the attainment targets. This position was supported by arguing that that those proposing such an approach with a view to raising the self-esteem of ethnic minority pupils and to improving mutual understanding and tolerance between races, were afflicted with an attitude that was ‘misconceived and patronising’.

y to succeed. The forthcoming revision is a much more limited exercise, but the principle still applies. The exercise will proceed collaboratively, with full consultation, and on the basis of firm evidence that it works. (Tate, 1998)

In relation to the latter there is the INTRODUCTION

A high-quality mathematics program is essential for all students and provides every student with the opportunity to choose among the full range of future career paths. Mathematics, when taught well, is a subject of beauty and elegance, exciting in its logic and coherence. It trains the mind to be analytic – providing the foundation for intelligent and precise thinking.

To compete successfully in the worldwide economy, today’s students must have a high degree of comprehension in mathematics. For too long schools have suffered from the notion that success in mathematics is the province of a talented few. Instead, a new expectation is needed: all students will attain California’s mathematics academic content standards, and many will be inspired to achieve far beyond the minimum standards.

These content standards establish what every student in California can and needs to learn in mathematics. They are comparable to the standards of the most academically demanding nations, including Japan and Singapore – two high-performing countries in the Third International Mathematics and Science Study (TIMSS). Mathematics is critical for all students, not only those who will have careers that demand advanced mathematical preparation but all citizens who will be living in the twenty-first century. These standards are based on the premise that all students are capable of learning rigorous mathematics and learning it well, and all are capable of learning far more than is currently expected. Proficiency in most of mathematics is not an innate characteristic; it is achieved through persistence, effort, and practice on the part of students and rigorous and effective instruction on the part of teachers. Parents and teachers must provide support and encouragement.

The standards focus on essential content for all students and prepare students for the study of advanced mathematics, science and technical careers, and postsecondary study in all content areas. All students are required to grapple with solving problems; develop abstract, analytic thinking skills; learn to deal effectively and comfortably with variables and equations; and use mathematical notation effectively to model situations. The goal in mathematics education is for students to:

De