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# Economic Growth Theory … Bah Humbug!

## Economic Growth Theory … Bah Humbug!

February 2, 2021

Originally published on Economics from the Top Down

Blair Fix

I’ve written a lot on this blog about the absurdity of marginal productivity theory. But I haven’t said much about the other pillar of mainstream economics: neoclassical growth theory. Today I’ll break that silence.

Neoclassical growth theory is a textbook example of Murphy’s law. Everything that could go wrong with the theory did go wrong. From conceptual foundation to application to interpretation, neoclassical growth theory is a lesson in how not to do science.

I’ve compiled here a list of neoclassical growth theory’s failures. Each one of them is, in my opinion, fatal. Yet none of them has stopped neoclassical economists from applying their theory to the real world.

### Failure 1: A flawed foundation

Neoclassical economics assumes that we can separate society into different ‘factors of production’. This thinking dates back to the origin of political economy itself. The field arose in the 18th century to explain the distribution of income between classes. Here’s how David Ricardo put it:

The produce of the earth … is divided among three classes of the community; namely, the proprietor of the land, the owner of the stock or capital necessary for its cultivation, and the labourers by whose industry it is cultivated.

… To determine the laws which regulate this distribution, is the principal problem in Political Economy.

In attempting to solve Ricardo’s ‘principal problem’, political economists made three foundational mistakes. They assumed that:

1. Wealth is ‘produced’. (It’s not.)

2. Social classes represent different ‘factors of production’. (They don’t.)

3. Wealth production can be attributed to one or more of these ‘factors of production’. (It can’t.)

Once these (false) conceptual foundations were set, the fight became about which factors of production were ‘productive’. The physiocrats (most famously, François Quesnay) thought that land was the source of all value. Classical political economists like Adam Smith, David Ricardo and Karl Marx argued that labor was the source of value. Neoclassical economists like John Bates Clark and Philip Henry Wicksteed added capital to the list of value producers.

The debate about the ‘source’ of value is long, bitter, and largely fruitless. The problem is simple: value has no source. It’s a cultural construct. So it’s impossible to tie value to ‘production’.

That’s the short summary of the problem. The longer summary is that political economists conflated two distinct processes: (1) resource exploitation; and (2) resource distribution. Looking at resource exploitation, political economists saw ‘wealth production’. That was a mistake. Resources aren’t produced. They’re transformed. And this transformation has nothing to do with ‘value’, as humans understand it. Yes, different factions of society are able to demand prices for various resources. But the prices are invented, not ‘produced’. They are a way of organizing the distribution of resources. Prices have nothing to say about the quantity of resource flows (and vice versa).

The easiest way to see the problem is to look outside our own myopia. Humans do the same thing as every other living organism. We transform matter and energy into forms that are useful to us. Attributing this process to a ‘factor of production’ is like attributing an animal’s metabolism to one or more of its organs.

Ask yourself these questions: What portion of a cow’s metabolism is ‘produced’ by the digestive tract? What portion is ‘produced’ by the heart? By the brain? By the hooves?

These questions have no answer because they are conceptually flawed. Organisms function collectively to harvest resources. So it’s meaningless to attribute metabolism to any one organ. The same goes for human societies.

Let’s take the metaphor a step further. Within organisms, we might think of hormones as the equivalent of prices. They convey information that then influences the behavior of different components of the organism. Releasing adrenaline, for instance, takes resources away from the digestive tract and sends them to the skeletal muscles.

Now imagine we analyzed an organism’s body the same way that political economists analyze society. We see that adrenaline influences how resources are distributed within the organism. We then trace the synthesis of adrenaline to the adrenal glands. Aha! We surmise that metabolism is ‘produced’ by this gland. We’ve found our ‘factor of production’!

If this operation seems foolish, that’s because it is. It’s foolish when applied to organisms. And it’s foolish when applied to human societies. Value is not ‘produced’. And there are no ‘factors of production’.

### Failure 2: The one-commodity world

Modern humans consume a dizzying variety of commodities. Not so in neoclassical theory. Dating back to the work of John Bates Clark, neoclassical theory assumes that human societies produce a single commodity.

Why adopt such a bizarre assumption? Because it’s required for neoclassical theory to be logically consistent. According to marginal productivity theory, each factor of production earns its (marginal) product. The problem is that to compare the productivity of different individuals (or groups), everyone must produce the same thing.

In the real world, this condition is never met. So when John Bates Clark formulated marginal productivity theory, he should have concluded that his theory had no foundation. It required a single-commodity world — a world that does not exist. So the theory was a dead end.

That’s not, however, what Clark concluded. Instead, he forged ahead with a theory that said nothing about the real world. He assumed that everyone produced and consumed a single commodity. Then he ‘proved’ that in this fictitious world, everyone earned their (marginal) product.

Neoclassical growth theory continues this dubious tradition. It assumes that there is a single commodity that serves as both capital and output. As with marginal productivity, this bizarre assumption is a requirement for neoclassical growth theory to be logically consistent. If there is more than one commodity, the quantity of both capital and output becomes ambiguous, and the theory falls apart [1]. The consequence is that neoclassical growth theory says nothing about the real world.

### Failure 3: Unmeasurable components

When economists move from the imaginary world of a single commodity to the real world of many commodities, they encounter a problem. The components of their theory — the quantity of capital and the quantity of output — become ambiguous.

This problem was pointed out in the 1950s in a debate now called the ‘Cambridge capital controversy’. Economists Paul Samuelson and Robert Solow, hailing from Cambridge, Massachusetts, forged ahead with neoclassical growth theory. Continuing in the tradition of John Bates Clark, they developed their theory by assuming a single-commodity world. Then they applied this theory to the many-commodity world. The trick to doing so, they argued, was to aggregate all commodities into a single stock. Then you could treat this stock (they claimed) as consisting of a single commodity.

Economists Joan Robinson and Piero Sraffa, hailing from Cambridge, UK, saw through the trick. They pointed out that the capital stock couldn’t be aggregated objectively. (Curiously, they didn’t say the same thing about economic output, even though identical reasoning applies). The capital stock, they noted, was composed of different commodities that had no common ‘natural’ unit. This fact made aggregation ambiguous — it depended on the particular choice of dimension. Worse still, neoclassical economists used prices as their dimension of aggregation. This made the whole operation circular. Neoclassical theory was supposed to explain income as an outcome of productivity. But economists tested their theory by using income to measure productivity.

The critique raised by Robinson and Sraffa is, in my opinion, unassailable. The aggregation of many commodities is always subjective, depending on the particular choices made by the analyst. (I explore this issue in detail here.)

How did neoclassical economists respond to the Cambridge (UK) critique? Mostly by ignoring it. As an example, take Gregory Mankiw’s textbook Principles of Macroeconomics. Used to train a generation of economists, the textbook simply ignores the Cambridge capital controversy — as if it never happened.

### Failure 4: A failed prediction becomes a ‘discovery’

Neoclassical theory proposes that capital and labor are the two ‘factors of production’. Logically, then, it follows that the growth of economic output should stem from the growth of capital and labor inputs.

Let’s put aside, for the moment, the (insurmountable) problems with measuring the growth of capital and the growth of output. Even though the foundations are dubious, economists forged ahead with growth accounting. And still they ran into a problem. The growth of capital and labor, they found, could not explain the growth of ‘output’ (as it was officially measured). Output grew far more than predicted by the growth of capital and labor alone.

The response to this failure was instructive. Instead of seeing this result for what it was — a failed prediction — economists claimed it as a discovery. This trick was first proposed by Robert Solow. He postulated that the growth residual (the portion of economic growth that was unexplained by neoclassical theory) was actually a measurement of ‘technical progress’. With this postulate, Solow invented the field of growth accounting. Economists could now ‘measure’ the growth of technical progress! And so they did. Today, if you google ‘total factor productivity’ (economists’ term for their measurement of technological progress), you’ll get some 3.5 million results.

Despite the ubiquity of growth accounting, economists seem to forget that there’s trouble underneath the hood. Their ‘measurement’ of technological progress is nothing of the sort. It’s actually a failed prediction that is turned into a ‘discovery’. The reality is that neoclassical economists have never measured technical progress. They have measured their own ignorance. [2]

### Failure 5: The accounting identity

Despite the failures of neoclassical growth theory, economists cling to their model. One reason they do so is because they claim that the model ‘works’. By this, economists mean that neoclassical growth theory gives results that are consistent with marginal productivity theory.

Let’s unpack this claim. According to neoclassical growth theory, the growth of output is caused by the growth of capital and labor inputs. Economists postulate that there exists a ‘production function’ that relates output to these inputs. Importantly, this production function must be consistent with marginal productivity theory. That is, it must show that both workers and capitalists earn their marginal product.

The convention, in neoclassical economics, is to assume a ‘Cobb-Douglas’ production function. This is an equation named after the economists who first proposed it (Charles Cobb and Paul Douglas). The equation looks like this:

$Y = A L^{\beta} K ^{\alpha}$

Here Y is economic output, L is labor input, and K is the capital stock. The parameter A was initially assumed to be a constant. Later, however, it came to represent the growth of technological progress.

Let’s talk about the exponents α and β. In the Cobb-Douglas equation, these exponents are free parameters, determined by regressions on real-world data. But if neoclassical theory is correct, these exponents have a special meaning. They should represent each factor’s share of income: α is capital’s share of income; β is labor’s share.

When economists started using the Cobb-Douglas equation, they found that their results supported marginal productivity theory. When they regressed this equation onto empirical data, the fitted values for α and β almost always summed to one — just as they should if they represented each factor’s share of income. Moreover, the fitted value for β was close to labor’s share of income. This seemed to confirm marginal productivity theory.

Writing in 1963, A. A. Walters celebrated this success. The Cobb-Douglas equation seemed to ‘work’, he noted, in two ways:

First, the sum of the coefficients usually approximates closely to unity. The linearity of the production function seems to be a remarkably consistent finding between one country and another. The second important result is the agreement between the labour exponent and the share of wages in the value of output. These two findings have been interpreted as confirmation that the aggregate production function has constant returns to scale and that the marginal productivity of labour is equal to the wage rate.

Despite the many problems in their theory, neoclassical economists clung to this success. Their theory ‘worked’.

The trouble was, the Cobb-Douglas function ‘worked’ even when it shouldn’t. This startling fact was discovered in 1971 by Franklin Fisher. He had created a simulation in which the Cobb-Douglas production function shouldn’t work. The simulation consisted of many different firms, each of which had its own production function. Fisher purposefully designed the model so that the conditions for an aggregate production function were not met. (These conditions, Fisher noted, were “far too stringent to be believable”.)

Now comes the interesting part. Fisher found that he could still fit his simulated economy with an aggregate Cobb-Douglas function. All that was needed for a successful fit, Fisher discovered, was that labor’s share of income be roughly constant over time. This suggested that something strange was going on. Neoclassical growth theory seemed to ‘work’ even when it shouldn’t. How could this be?

In 1974, Anwar Shaikh found the answer. He showed that when labor’s share of income was constant, the Cobb-Douglas function ‘worked’ because it was a tautology. It was a rearrangement of a national accounting identity. To show this fact, Shaikh noted that in the national accounts, national income (Y) is defined as the sum of labor and capitalist income:

$Y = \text{labor income} + \text{capitalist income}$

Labor income, in turn, equals the average wage w times labor hours L. And capital income equals the return on capital r times the value of the capitalist stock K. So national income is defined as:

$Y = wL + rK$

Shaikh showed that with a little algebra, you could rearrange this identity into the Cobb-Douglas equation. The term β in the Cobb-Douglas equation, Shaikh found, was algebraically equivalent to wL/Y — labor’s share of national income.

Shaikh’s discovery was an academic mic drop. The Cobb-Douglas equation ‘worked’ because it had to. It was a mathematical rearrangement of a national accounting identity — an equation that was true by definition. To illustrate the point, Shaikh successfully fit the Cobb-Douglas function to an economy who’s production traced the word ‘HUMBUG’. (See the cover image.)

How did neoclassical economists respond to this devastating discovery? Mostly by ignoring it. Again, take Gregory Mankiw’s macroeconomics textbook as an example. It contains 19 references to ‘accounting identities’. None of them is Shaikh’s identity.

### The winners write history

What’s most frustrating to heterodox economists is not that neoclassical growth theory is irredeemably flawed. It’s okay for a theory to be wrong — that’s part of doing science. What’s infuriating is that neoclassical growth continues to be taught as if it had no problems. Search standard economics textbooks for a discussion of the problems reviewed here, and you’ll likely find nothing.

This blistering silence is a classic example of the winner writing history. Neoclassical economists dominate economics departments in universities, and so they can promulgate their vision without challenge. Neoclassical textbooks are massive lies — the equivalent of a European history book that omits colonialism.

Will the repeated failures of economic growth theory eventually make it into mainstream textbooks? Time will tell. But I’m not holding my breath.

### Notes

[1] In deriving the neoclassical model of economic growth, Robert Solow tucked in a subtle admission. He needed a world with only one commodity so he could “speak unambiguously of the community’s real income”.

[2] I wish I had invented the term ‘measure of ignorance’. But Moses Abramovitz beat me to it. Writing in 1956 about the residual between the measured and predicted growth of output, he wrote: “this element may be taken to be some sort of measure of our ignorance about the causes of economic growth in the United States.”

[Cover image is from Anwar Shaikh’s paper Laws of Production and Laws of Algebra: The Humbug Production Function]

Cohen, A. J., & Harcourt, G. C. (2003). Retrospectives: Whatever happened to the Cambridge capital theory controversies? Journal of Economic Perspectives, 17(1), 199–214.

Felipe, J., & Holz, C. A. (2001). Why do aggregate production functions work? Fisher’s simulations, Shaikh’s identity and some new results. International Review of Applied Economics, 15(3), 261–285.

Fisher, F. M. (1971). Aggregate production functions and the explanation of wages: A simulation experiment. The Review of Economics and Statistics, 53(4), 305–325.

Fix, B. (2015). Rethinking economic growth theory from a biophysical perspective. New York: Springer.

Fix, B. (2019). The aggregation problem: Implications for ecological and biophysical economics. BioPhysical Economics and Resource Quality, 4(1), 1.

Hodgson, G. M. (2005). The fate of the Cambridge capital controversy. In Capital controversy, post keynesian economics and the history of economic thought (pp. 112–125). Routledge.

McCombie, J. S. L. (2001). What does the aggregate production function show? Further thoughts on Solow’s “second thoughts on growth theory”. Journal of Post Keynesian Economics, 23(4), 589–615.

Robinson, J. (1953). The production function and the theory of capital. The Review of Economic Studies, 21(2), 81–106.

Shaikh, A. (1974). Laws of production and laws of algebra: The humbug production function. The Review of Economics and Statistics, 56(1), 115–120.

Solow, R. M. (1956). A contribution to the theory of economic growth. The Quarterly Journal of Economics, 70(1), 65–94.

Solow, R. M. (1957). Technical change and the aggregate production function. The Review of Economics and Statistics, 39(3), 312–320.

Sraffa, P. (1960). Production of commodities by means of commodities: Prelude to a critique of economic theory. London: Cambridge University Press.


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### Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

### Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

### Quartics

Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

# Without Calculus

##### Stage: 5 Challenge Level:

Given that $u > 0$ and $v > 0$ what is the smallest possible value of $1/u + 1/v$ given that $u + v = 5$?

Here is a solution from Danny of Milliken Mills High School, Canada.

Let $S$ be the mininum value,
$S = 1/u + 1/v = (v + u)/(uv)$.
Since $u + v = 5$ then $S = 5/uv$.

The maximum value of $uv$ gives a mininum value of $S$. Given $u + v = 5$ then $v = 5 - u$.

Let $f(u)$ give the value of $uv$ as $u$ and $v$ change. Then $f(u) = u(5 - u) = -u^2 + 5u$. This is a quadratic function and the vertex $(u, M)$ of the graph of $f(u)$ is at $(2.5, 6.25)$. It means the maximum value occurs at $M = 6.25$ when $u = 2.5$ and $v = 5 - u = 2.5$. So the mininum value $S = 1/2.5 + 1/2.5 = 0.8$.

Vassil from Lawnswood High School, Leeds used the same method, and drew two graphs to illustrate it, one to show the possible values of $u$ and $v$ where $u + v =5$ and the other, shown below, the graph of $f(u) = u(5 - u) = -u^2 + 5u$ for the corresponding range of values of $u$ and $v$.

Peter and Koopa of Boston College used the Arithmetic Mean - Geometric Mean inequality (AM-GM) noting as above that minimising $1/u + 1/v$ is the same as maximising $uv$. Here is Peter's solution:

From the AM-GM inequality:

arg = $uv \leq [(u+v)/2]^2 = (5/2)^2 = 25/4.$

The maximum of $uv$ is therefore at equality, where $u = v = 5/2$, yielding $\max(uv) = 25/4$

Hence the minimum value of $1/u + 1/v$ subject to $u+v = 5$, $u$, $v > 0$, is $4/5$.

format: plain


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# Difference between revisions of "2015 AMC 8 Problems/Problem 21"

In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?

$\textbf{(A) }6\sqrt{2}\quad\textbf{(B) }9\quad\textbf{(C) }12\quad\textbf{(D) }9\sqrt{2}\quad\textbf{(E) }32$. $[asy] draw((-4,6*sqrt(2))--(4,6*sqrt(2))); draw((-4,-6*sqrt(2))--(4,-6*sqrt(2))); draw((-8,0)--(-4,6*sqrt(2))); draw((-8,0)--(-4,-6*sqrt(2))); draw((4,6*sqrt(2))--(8,0)); draw((8,0)--(4,-6*sqrt(2))); draw((-4,6*sqrt(2))--(4,6*sqrt(2))--(4,8+6*sqrt(2))--(-4,8+6*sqrt(2))--cycle); draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle); label("I",(-4,8+6*sqrt(2)),dir(100)); label("J",(4,8+6*sqrt(2)),dir(80)); label("A",(-4,6*sqrt(2)),dir(280)); label("B",(4,6*sqrt(2)),dir(250)); label("C",(8,0),W); label("D",(4,-6*sqrt(2)),NW); label("E",(-4,-6*sqrt(2)),NE); label("F",(-8,0),E); draw((4,8+6*sqrt(2))--(4,6*sqrt(2))--(4+4*sqrt(3),4+6*sqrt(2))--cycle); label("K",(4+4*sqrt(3),4+6*sqrt(2)),E); draw((4+4*sqrt(3),4+6*sqrt(2))--(8,0),dashed); label("H",(-4-6*sqrt(2),-4-6*sqrt(2)),S); label("G",(-8-6*sqrt(2),-4),W); label("32",(-10,-8),N); label("18",(0,6*sqrt(2)+2),N); [/asy]$

## Solution

Clearly, since $\overline{FE}$ is a side of a square with area $32$, $\overline{FE} = \sqrt{32} = 4 \sqrt{2}$. Now, since $\overline{FE} = \overline{BC}$, we have $\overline{BC} = 4 \sqrt{2}$.

Now, $\overline{JB}$ is a side of a square with area $18$, so $\overline{JB} = \sqrt{18} = 3 \sqrt{2}$. Since $\Delta JBK$ is equilateral, $\overline{BK} = 3 \sqrt{2}$.

Lastly, $\Delta KBC$ is a right triangle. We see that $\angle JBA + \angle ABC + \angle CBK + \angle KBJ = 360^{\circ} \rightarrow 90^{\circ} + 120^{\circ} + \angle CBK + 60^{\circ} = 360^{\circ} \rightarrow \angle CBK = 90^{\circ}$, so $\Delta KBC$ is a right triangle with legs $3 \sqrt{2}$ and $4 \sqrt{2}$. Now, its area is $\dfrac{3 \sqrt{2} \cdot 4 \sqrt{2}}{2} = \dfrac{24}{2} = \boxed{\textbf{(C)}~12}$.

~savannahsolver


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Q

# Help me solve this In the circuit shown in the figure, each capacitor has a capacity of 3mF. The equivalent capacity between A and B is

In the circuit shown in the figure, each capacitor has a capacity of 3mF. The equivalent capacity between A and B is

• Option 1)

$\frac{3}{4}\mu F$

• Option 2)

$3\mu F$.

• Option 3)

$6\mu F$

• Option 4)

$5\mu F$

152 Views

As we have learned

Parallel Grouping -

$\dpi{100} C_{eq}=C_{1}+C_{2}+\cdots$

- wherein

Hence equivalent capacitance  $C_{eq}= \frac{3*6}{3+6}= 5\mu F$

Option 1)

$\frac{3}{4}\mu F$

Option 2)

$3\mu F$.

Option 3)

$6\mu F$

Option 4)

$5\mu F$

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NOTICE OF MEETING

BOARD OF TRUSTEES
AUDIT, COMPLIANCE, AND MANAGEMENT REVIEW COMMITTEE
TEXAS SCHOOL FOR THE BLIND AND VISUALLY IMPAIRED

NOTICE IS HEREBY GIVEN that a regular meeting of the Audit, Compliance, and Management Review Committee of the Board of Trustees of the Texas School for the Blind and Visually Impaired shall be held on June 2, 2017, beginning at 8:00 a.m. in Room 92 of the Main Building 600, 1100 West 45th Street, Austin, Texas.

If, during the course of the meeting, any discussion of any item on the agenda should be held in closed session, the Audit, Compliance, and Management Review Committee shall convene in such closed session in accordance with Texas Government code, Sec. 551.001, et.seq. Before any such session is convened, the presiding officer shall publicly identify the section or sections of the Code authorizing the closed session. All final votes, actions, or decisions shall be taken in open session.

The subjects to be discussed or considered, or upon which any formal action may be taken are as follows (items do not have to be taken in the same order as shown on this meeting notice):

1. Call to Order

2. Consideration of Approval of Minutes of the January 27, 2017 Meeting

3. Report from the Administrator for Business, Operations and Technology (Pam Darden)
a. Preparations for the Contract Management Audit

4. Report from the Internal Auditor (Jaye Stepp)
a. Consideration of approval of the TAC 202 Information Security Audit
b. Audit Recommendations Tracking Schedule

## TSBVI Teen Competing In Hungary Athletic Competition

Read the story about Davieon Perez, 17, headed to World Games in Hungary this summer to compete in a sport called goalball. You can also watch the News Report on KVUE.

## TSBVI loses treasured individual

Patrick van Geem, a dear friend to many of us, and one of the most knowledgeable professionals in our field, passed away December 30, 2016, after a year-long struggle with cancer. Pat had a long career at Texas School for the Blind and Visually Impaired including work as a classroom teacher and most recently as a consultant with the Outreach Programs.  Patrick is known statewide as the “Braille Bootcamp” fellow because of his work developing materials and training for Braille transcribers and VI teachers.

Beyond his talents as a VI professional, Pat is most often thought of as a kind, intelligent, gentle and funny man.  Pat was generous with both his resources and his energies, especially when it could help a child or teacher.

Pat was helped throughout his illness with the love and support of his partner, Alan Coy, his family and the many close friends who were privileged to be a part of his life. There will never be another person like Pat. Wij houden u Pat en zullen u missen.

A memorial will be held for Pat on January 21, 2017 in the Auditorium at TSBVI beginning at 9:30 AM.  Pat's many friends are invited to join this celebration of his life.

Patrick van Geem, a dear friend to many of us, and one of the most knowledgeable professionals in our field passed away December 30, 2016, after a year-long struggle with cancer. Pat had a long career at Texas School for the Blind and Visually Impaired including work as a classroom teacher and most recently as a consultant with the Outreach Programs.  Patrick is known statewide as the “Braille Bootcamp” fellow because of his work developing materials and training for Braille transcribers and VI teachers.

Beyond his talents as a VI professional, Pat is most often thought of as a kind, intelligent, gentle and funny man.  Pat was generous with both his resources and his energies, especially when it could help a child or teacher.

Pat was helped throughout his illness with the love and support of his partner, Alan Coy, his family and the many close friends who were privileged to be a part of his life. There will never be another person like Pat. Wij houden u Pat en zullen u missen.

Below are some other websites with information that relates to Active Learning and learning styles of individuals in the earliest developmental stages.

Narbethong State Special School

The Narbethong School has a variety of articles, presentations, and information about Active Learning.  Definitely worth exploring their site.

LilliWorks

In addition to being the United States vendor of authentic Active Learning perceptualizing equipment, LilliWorks has some nice footage about Active Learning, especially from a parent perspective.

This website has some great information about how babies (or any sensorimotor or pre-operational level learners) learn and how their brains develop.  These also give you some idea about how long it takes for typically developing children to gain foundational skills and concepts.

BOARD PROGRAM COMMITTEE MEETING

April 8, 2016

1.   Call to Order

Caroline Daley called to order the meeting of the TSBVI Board Committee on Programs at 8:01 a.m. on April 8, 2016 in Room 104 of the Main Administration Building 600, 1100 W. 45th Street, Austin, Texas.

Program Committee members Tobie Wortham and Dr. Anne Corn were also present.

TSBVI staff members participating in the meeting were: Sara Merritt, Short-Term Programs Principal; Susan Hauser, Center for School Resources Director; Ken Miller, Planning, Policy and Procedure Support; Trevor Simmons, LBB Analyst; Chrissy Cowan, Mentor Coordinator; Andy Weir, Media Technician; Steve Landry, Instructional Designer; Kate Hurst, Statewide Staff Development Coordinator; Tad Doezema, Comprehensive Programs Assistant Principal; and Leonard Schwartz, General Counsel.

2.  Consideration of Approval of Minutes of the January 29, 2016 Meeting

Dr. Corn moved that the minutes of the January 29, 2016 meeting be approved and Ms. Wortham seconded the motion.  Committee members were in favor of the motion.

3. Consideration of Approval of Policies

Ken Miller descibed proposed amendments to three Policies.

A.  Policy DAA - Employment Objectives: Equal Opportunity Employment

A minor amendment to the policy regarding discrimination is proposed.  Bankruptcy is a new category that is protected from discrimination.  Reference to this is added to this policy.  The committee discussed the meaning of the word, “prong” and decided to leave the word as it is in the policy.

B.  Policy FFG - Student Welfare: Child Abuse, Neglect, Exploitation, and Improper Child Care Practice

This policy must be reviewed by the Board every year.  A few minor amendments in language are being proposed at this time.  Mr. Miller explained that TSBVI is unique as a school, since staff must report possible abuse, neglect, exploitation, and improper care by TSBVI staff members directly to designated TSBVI officials.  In addition, staff may also make a report of abuse, neglect or exploitation to appropriate outside agencies. Board members had several questions, which were discussed and answered by Mr. Miller.

C.  Policy FL-E - Student Records

This policy has been reorganized to make it simpler, and several amendments are recommended.  Mr. Miller described these proposed amendments:

Mr. Miller and Ms. Hauser described the role of the Texas State Library and Archives Commission in advising TSBVI and in assisting TSBVI with the storage of paper documents.

Mr. Miller recommended that these three policies be approved as corrected, including the elimination of some redundant language.

Dr. Corn made a motion to approve the recommended amendments to these three policies.  Ms. Wortham seconded the motion.  Committee members were in favor of the motion.

4.  Consideration of Approval of the School’s Calendar for 2016-2017

Ken Miller presented the proposed school calendar for the 2016-2017 school year, pointing out various kinds of information contained in the calendar.  Texas law has changed from requiring a certain number of instructional days to requiring a certain number of instructional minutes.  Miles Fain and Susan Houghtling calculated the total number of instructional minutes available in the school day schedule used by TSBVI.  Based on this calculation, the calendar was developed to ensure that the required number of instructional minutes was included in the proposed calendar.

Ms. Daley asked whether the calendar closely follows the Austin Independent School District (AISD) calendar.  Mr. Miller said that the TSBVI schedule follows the AISD schedule as much as possible, but not in every case.

Ms. Wortham made a motion to approve the proposed 2016-2017 school calendar.  Dr. Corn seconded the motion.  Committee members were in favor of the motion.

5.  Report from Short-Term Programs Principal, Sara Merritt:  Summer Program applications and classes

Sara Merritt gave a status report on summer 2016 programs.  This report is given to the Board every year at this time.  The report indicates the number of students to whom each summer program was offered in 2014 and 2015.  The report shows the total number of students who actually attended summer programs in 2014 and 2015.

The report also indicates the number of students accepted for summer 2016 as of 4/1/16.  Summer 2016 numbers will change over the next few months.  So far, 13 students have dropped (for various reasons), and they are being replaced by students from the wait list as they drop out.  This process will continue into the summer.  Short-Term Programs staff constantly monitor students who will come to each program and replace drop-outs with other students, to maximize the number of students who are able to attend each summer program.

Each year, TSBVI Short-Term Programs has increased the number of students offered a summer program.  Partially due to these efforts, the number of students actually attending summer programs has increased slightly.  Four more students attended in 2015 than in 2014, despite the extraordinary efforts that were made to increase attendance.

There has been a decrease in numbers of students applying to TSBVI summer vocational programs, although total numbers of students actually attending TSBVI summer work programs has increased.  Ms. Merritt believes that the decrease in applications is because other vocational programs have been developed across the state, so students are now more able to access work programs closer to their homes.  All departments at TSBVI work across the state to help local schools and agencies better develop local programs, so this is an indication of success in these efforts.

There are a number of possible reasons why students have difficulty attending summer programs.

Families must provide transportation, and many families have limited access to reliable transportation.  Families have competing demands of several children and scheduling conflicts.Some students have difficulty adjusting to change, and a family may decide that a short program will not allow the student adequate time to settle in and adjust to the change.

Ms. Merritt has worked with DARS in supporting families to access services, including Short-Term Programs.  DARS has been able to provide assistance with transportation to summer programs for some families.  Ms. Daley pointed out that DARS is reaching out to families more proactively to ensure that families are able to access Short-Term Programs and other services.

Ms. Merritt said that Short-Term Programs is trying to serve more deafblind students in summer programs, but there is often a challenge in finding qualified interveners for these students.  Short-Term Programs is also working hard to serve more students in popular programs to which students have had to be turned away in the past, for example, Life Skills Camp.  Many students who would like to attend summer programs have a number of intensive medical or health concerns, and considerable effort is required to prepare for those students.

6.  Report from Outreach Department:  Distance Learning

Several Outreach staff gave a presentation on distance learning at TSBVI. Staff participating in the presentation were Kate Hurst, Chrissy Cowan, Steve Landry, and Andy Weir.

A.  Kate Hurst shared information about the TSBVI website.  She demonstrated several elements that can be accessed on the TSBVI website.

a. Delayed viewing of recorded webinars.  There is a Delayed Viewing Request Form that allows people to arrange to view recorded TSBVI webinars.  Some webinars allow a person to receive continuing education credits, and this has been popular for O&M staff especially.  Outreach O&M specialist Chris Tab has marketed this system through Facebook and Twitter, and there has been a big jump in use by O&M staff.  The system includes a list of archived webinar topics, with an indication of whether each webinar provides continuing education credit.  People may view archived webinars online or they can choose to download an MP3 version of the webinar.  Staff around the state love this feature of the TSBVI website.  Dr. Corn said that it is a great resource, and it would have been great to have it many years ago.  Ms. Hurst said that TSBVI staff have been working on it as fast as they can, and that now university teacher training programs are assigning their students to access the system.

b. Advertising of events.  Ms. Hurst showed several examples of advertisements or announcements for events that are posted on the TSBVI website, including:

c. Websites that TSBVI manages collaboratively with other organizations.  Ms. Hurst gave an update on the shared penrickton website, which is devoted to Active Learning approaches to teaching students with visual impairments.  Ms. Hurst described some of the content that is on this website.  The website is a very important resource for learning about Active Learning.  There is a plan to make this site more interactive.  TSBVI staff are working with experts across the world to include more information on this site.  Dr. Corn suggested that the website have sections for additional references to past and ongoing research, in case physicians or parents would like to review the information.  Ms. Hurst said that there is almost no research on Active Learning educational approaches.  Children often show very slow progress, so it is difficult to do a shorter-term reseach study such as a college student might be able to complete.  Ms. Hurst plans to look at assessment tools that can show smaller learning gains, so researchers could use these tools to show student progress over a shorter time period. Ms. Hurst said that the penrickton group has done some anecdotal studies, and they are looking into doing more of these.

B.  Steve Landry made a presentation regarding new eLearning products that are being developed at TSBVI.

Mr. Landry is relatively new to TSBVI.  Previously he has managed online employee training at the Attorney General’s office.  The Attorney General’s office paid \$80,000 per year for the products that they used to create the agency’s online training products, but at TSBVI, he has been working with free programs.  During his employment at TSBVI, Mr. Landry has been surprised at the amount of expertise and detail that is needed to train employees and people new to the field of education of visually impaired students.  He has come to realize the depth of knowledge that is needed for staff to do the jobs at TSBVI.  Mr. Landry has had to figure out how to train staff effectively.  Mr. Landry has developed one product that he shared with the group.  For it, he used Noodle, which is an open source (free) learning platform, used to deliver an interactive powerpoint presentation. The noodle-generated course can mirror a course developed at a college.  In the presentation, the user can answer questions and receive multimedia content.

Mr. Landry has a counterpart in Short-Term Programs, Carrie Farrage, Instructional Developer for Distance Learning.  Ms. Farrage has been working with Mr. Landry to create staff development products.  Ms. Merritt described summer staff training modules developed by Ms. Farrage, who has been learning the accessibility skills that she needs in order to develop TSBVI products.  All training programs developed at TSBVI must be accessible.

Some of the eLearning products in development include:

Mentor Certification for VI Professionals.  Chrissy Cowen explained that new VI teachers in Texas must have a mentor.  Mentors are trained in person on the TSBVI campus, but if they are unable to attend the training in person, this online product can be used instead.

Foundations of Visual Impairment.

TSBVI is working with Dr. Jane Erin at the University of Arizona to create small packages of information intended to give an overview of the field of education of visual impairment that can be presented through eLearning.  Foundations of Visual Impairment is a requirement for all new non-VI-certified direct care staff at TSBVI.  In the past, content has been developed and presented by TSBVI subject-matter experts.  New  TSBVI employees must complete the training during their first year of employment, and promotions are dependent on the staff person completing this course.

Mr. Landry pointed out that eLearning is important for several reasons.  People are very busy, such as traveling or working with students, and eLearning allows people to access  information at their own pace and at convenient times.  ELearning has the potential to save money that might have to be spent on in-person training.

C.  Sara Merritt presented information about in-person distance learning used in Short-Term Programs.

Ms. Merritt pointed out that the Short-Term Programs audience is students.  Ms. Farrage is developing eLearning modules that students can use to learn specific topics, such as a specific piece of technology like JAWS.  The student from a local district participates with the local Teacher of the Visually Impaired and with a Short-Term Programs staff to learn about the topic that the student needs.

7.  Consideration of Approval of Recommendations of the Instructional Materials Committee

Ken Miller reported for the Instructional Materials Committee.  Each year, TSBVI must certify to the state that students have access to instructional materials that cover all the Texas Essential Knowledge and Skills (TEKS).  No new instructional materials were adopted this school year.  It is necessary for the superintendent to sign a certification form and for the Board to approve this submission (p. 86).

Dr. Corn made a motion that the committee accept the report to the State Board of Education as presented.  Ms. Wortham seconded the motion.  Committee members were in favor of the motion.

8.  Report from Comprehensive Programs Assistant Principal, Tad Doezema:  Transition to Adult Life Update

Tad Doezema gave an update on transition to adult life programming at TSBVI.  Mr. Doezema is one of three Assistant Principals at TSBVI.  He supervises the EXIT and the Post-Secondary programs.  He described new things that are happening in Comprehensive Programs in the area of transition to adult life.

A. The EXIT program now includes students with multiple impairments and deafblindness.  These students are served on the EXIT team for the first time.  Those students and their teachers have benefited from this more intentional focus on adult transition.

B.  More transition assessment tools are being identified for use in Comprehensive Programs.  Transition assessments can help students learn about their interests and skills and can help students and their parents think through the student’s postsecondary goals.  Lynne McAlister is the campus-wide transition coordinator, and she has helped staff explore new transition assessments.  For example, the authors of the Transition Planning Inventory (TPI 2) are working with TSBVI staff to see how TSBVI can use the tool.

C.  The Region 13 Service Center recently started an 18+ cohort.  This is a 2-year group organized to develop ideas about what 18+ programming should look like across the state.  TSBVI and TSD are two of six school selected for this project.  The cohort is led by Elizabeth Danner from Region 13 and Dr. Vicki Mitchell, a leader in the field of adult transition for students with disabilities.  The leaders recently came to spend the day at TSBVI doing a “tech assist.”  Two EXIT students were selected (one Basic Skills student and one student who is possibly college bound), and the focus was on these students and their staff and families.  The goal of the time was for Dr. Mitchell to demonstrate how to do a “life plan.”  TSBVI staff observed Dr. Mitchell in order to learn how to help parents and adult students think about their future adult life in detail, and to mirror the future life in school.  This is especially important for students who need consistency and routine.  The plan is done with the idea that the student’s last day of school should mirror the first day of adult life, so the transition is seamless.  The life plan included details of how the student’s days will be spent as an adult, and planning for resources that will be needed.

Mr. Schwartz asked how far out into adult life the planning is done.  Mr. Doezema said that the focus is primarily on what will happen immediately after school ends.  Longer-range future plans are also part of it, especially planning future supports, living environments, and getting on waiting lists for future agencies.

Ms. Daley asked if TSBVI checks in with students who have graduated.  Mr. Doezema noted that the Superintendent’s Office does a graduate survey for statistical data.  Up until now, TSBVI has not provided support and suggestions for former graduates.

Dr. Corn said that learning from former students can help TSBVI better develop its current school programs.  Mr. Doezema is thinking about staff doing follow up with former students as a staff development activity.

Ms. Daley suggested having former students be strongly encouraged to check in the TSBVI at certain intervals after graduation.  This would be an important support for former student who may need some advice or support.

Mr. Doezema said that, in the past, students have stayed at TSBVI until their school eligibiltiy ends and they graduate.  However, most TSBVI students do not live in Austin as adults, and we have recognized that TSBVI needs to collaborate more closely with the student’s local school district to prepare the student to live in his or her local community after graduation.  It is important to consider the advantage of a student returning to the local district and community while still eligible for school, so local supports can be put in place before the student graduates.  Local districts are beginning to appreciate the benefit of having the student return to the local school before graduation.  Even if the district says it is a small community or the school doesn’t have a “program,” it still can be important for the student to return and get supports in place before graduation.  At TSBVI recently, several creative programs have been planned for certain students.  Often, just one person at the local school can help develop a creative program when the student returns.  TSBVI has been actively collaborating with local districts to implement creative, and sometimes shared, programs.  Ms. Daley said that eLearning strategies can help this effort.

Ms. Daley observed that TSBVI’s eLearning efforts could also help students and families after graduation.  Topics about how to handle specific situations after graduation, such as how to deal with the loss of a job or of a supportive employer, would be helpful.

Mr. Doezema said that special education services should be provided before the students turns age 22.  This is a good time for students to learn and practice the activities they will do after graduation. If a students graduates before age 22, the student could come back and re-enroll in school if it becomes clear that additional school services are needed before the student become age 22.   Mr. Doezema said that EXIT is implementing more regular transition meetings with outside agency staff, parents, and students so they can get a good working relationships while the student is in school.

Ms. Merritt pointed out that the school’s funding, role and responsibility ends when the student is no longer eligible for public education (age 22).  The personal bond that a teacher has with a student can continue after the student leaves school, and teachers may keep in touch personally with a student or family after the student graduates.

Mr. Miller asked Mr. Swartz if the school’s enabling legislation needs to be changed.  Mr. Swartz pointed out that students beyond age 22 continue to need services and supports, and Ms. Daley agreed.  Other agencies are available to support adults after graduation.  TSBVI’s role is to provide education, while adult agencies are more focused on providing services.  However, many young adults continue to need education, even after they graduate.

Dr. Corn said that the school should have a role in communicating with students who have graduated, for the purpose of understanding how to improve future school programming.  She would not want TSBVI resources to be diverted to educating adults.  Adult agencies should provide this.  Dr. Corn feels that it is important for TSBVI to communicate and interact with adult agencies.

Ms. Wortham asked if TSBVI provides guardianship information to families and students.  Mr. Doezema said that TSBVI social workers sometimes help with this.  Ms. Merritt said that TSBVI relies on local districts to provide this information.  Mr. Miller said that community organizations, such as ARC in Austin, help with this.  Ms. Merritt and Ms. Daley and Mr. Schwartz discussed why it is important for a parent to consider guardianship for an adult child.

Ms. Daley adjourned the meeting at 10:05 a.m.

## Oops!

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ERSTE SCHRITTE MIT BRACKETS $$\def\N{\mathbb{N}}$$

## Mengen und deren Verknüpfungen

Schwarzschrift Prosa LaTeX LaTeX-Abkürzung
$$\{ 1, 2, 3, 4 \}$$ Mengenklammer auf, 1, 2, 3, 4, Mengenklammer zu \{ 1, 2, 3, 4 \}
$$P = \{ x | x \ ist \ Primzahl \}$$ groß P ist Menge der Elemente klein x, für die gilt: x ist Primzahl P = \{ x | x ist Primzahl \}
$$3 \in P$$ 3 ist Element der Menge P 3 \in P
$$4 \notin P$$ 4 ist nicht Element von P 4 \notin P \nin
$$A \subset B$$ Menge A ist echt in Menge B enthalten  A \subset B \sbs
$$A \subseteq B$$ Menge A ist in Menge B enthalten oder ist gleich der Menge B  A \subseteq B \sbse
$$A \cup B$$ Vereinigung der Mengen A und B  A \cup B
$$A \cap B$$ Durchschnitt der Mengen A und B  A \cap B
$$\O$$ Durchschnitt  \O
$$A \backslash B$$ Menge A ohne die Menge B  A \backslash B \bs
$$\{ \}$$ bzw. $$\emptyset$$ leere Menge als leere Mengenklammern bzw. als Symbol \{ \} bzw. \emptyset  \es
$$\overline A$$ Menge A quer \overline{A} \ol
$$u \circ v$$ Verkettung u \circ v
$$a \times b$$ Kreuzprodukt a \times b

## Spezielle Zahlenmengen

Schwarzschrift Prosa LaTeX LaTeX-Abkürzung
$$\N$$ Menge der natürlichen Zahlen \N  1)
$$\mathbb{Z}$$ Menge der ganzen Zahlen \Z
$$\mathbb{Z}^-_0$$ Menge der negativen ganzen Zahlen einschließlich der Zahl 0 \Z^-_0
Menge der rationalen Zahlen \Q
$$\mathbb{R}$$ Menge der reellen Zahlen \R
$$\mathcal P$$ Potenzmenge \mathcal P

## Verknüpfungen von Zahlen

Schwarzschrift Prosa LaTeX LaTeX-Abkürzung
$$2 +4 = 7$$ 3 plus 4 ist gleich 7 2 +4 =7
$$9 -3 \not= 5$$ 9 minus 3 ist ungleich 5 9 -3 \not= 5 \n=
$$x \pm 3$$ x plus minus drei x \pm 3
$$2 *8 >15$$ 2 mal 8 ist echt größer als 15 2 *8 >15
$$8 :4 <5$$ 8 geteilt durch 4 ist echt kleiner als 5 8 :4 <5
$$x \le 10$$ x ist kleiner oder gleich 10  x \le 10  <=
$$a \ge b$$ a ist größer oder gleich b  a \ge b >=
$$>>$$ viel größer als  \gg
$$<<$$ viel kleiner als  \ll
$$\pi \approx 3,14$$ Die Zahl pi ist ungefähr gleich 3,14 \pi \approx 3,14 \apx
$$(a +b)^2$$ runde Klammer auf, a plus b, runde Klammer zu, hoch 2 (a +b)^2
$$[x -y]^3$$ eckige Klammer auf, x minus y, eckige Klammer zu, hoch 3 [x -y]^3
$$s \sim t$$ s ist proportional zu t (das Zeichen bedeutet auch "ähnlich" (similar))  s \sim t
$$a \hat{=} b$$ a entspricht b  a \hat{=} b
$$7|28$$ 7 teilt die Zahl 28  7|28
$$\pm 7$$ plus minus 7  \pm 7

## Verknüpfungen von Aussagen

Schwarzschrift Prosa LaTeX LaTeX-Abkürzung
$$x \in \N \wedge x < 3$$ x ist Element von N und x ist echt kleiner 3  x \in \N \wedge x < 3
$$\Rightarrow$$ daraus folgt \Rightarrow \Ra
$$x \to \infty$$ x geht gegen unendlich x \to \infty x \to \8
$$x =1 \vee x =2$$ x = 1 oder x = 2  x =1 \vee x =2
$$3x =12 \Leftrightarrow x =4$$ 3x = 12 ist äuivalent zu x = 4 3x =12 \Leftrightarrow x =4  \Lra
$$3x =12 \leftrightarrow x =4$$ 3x = 12 ist äuivalent zu x = 4 3x =12 \leftrightarrow x =4  \lra

## Brüche und Dezimalzahlen

Schwarzschrift Prosa LaTeX LaTeX-Abkürzung
2/3 bzw. $$\frac{2}{3}$$ zwei Drittel bzw. Bruchanfang, 2 durch 3, Bruchende 2/3 bzw. \frac{2}{3}  \f{2}{3}
2/10 bzw. $$\frac{2}{10}$$ zwei Zehntel bzw. Bruchanfang, 2 durch 10, Bruchende NUR \frac{2}{10}  \f{2}{10}
4 3/5 bzw. $$4\frac{3}{5}$$ vier Dreifünftel bzw. 4 Bruchanfang, 3 durch 5, Bruchende 4 3/5 bzw. 4\frac{3}{5}  2)
1/x bzw. $$\frac{1}{x}$$ 1 durch x bzw. Bruchanfang, 1 durch x, Bruchende 1/x bzw. \frac{1}{x}  \f{1}{x}
$$\frac{1}{x +2} \not= \frac{1}{x} +2$$ Bruchanfang 1 durch x plus 2 Bruchende ist ungleich Bruchanfang 1 durch x Bruchende plus 2 \frac{1}{x +2} \not= \frac{1}{x} +2  \f{1}{x}
$$\frac{ \frac{a+b}{2}}{ \frac{x}{a-b}} =1$$ Bruchanfang Bruchanfang a plus b durch 2 Bruchende durch Bruchanfang x durch a minus b Bruchende Bruchende ist gleich 1 \frac{\frac{a +b}{2}} {\frac{x}{a -b}} =1  \f
0,25 = 1/4 0 Komma 25 ist gleich ein Viertel 0,25 = 1/4
$$0,1\overline{6} = 1/6$$ 0 Komma 1 Periode 6 ist gleich ein Sechstel 0,1\overline{6} = 1/6  \ol{6}
$$75\% = 3/4$$ 75 Prozent sind gleich 3 Viertel 75\% = 3/4
2,5 ‰ 2,5 Promille 2,5 \permil 3) \%_0

## Potenzen, Wurzeln, Indizes

Schwarzschrift Prosa LaTeX LaTeX-Abkürzung
$$a^2$$ a zum Quadrat  a^2
$$a^{12}$$ a hoch 12  a^{12}
$$2^{-3} =1/8$$ 2 hoch minus 3 ist gleich ein Achtel 2^{-3} =1/8
$$a^{n+1} \not= a^n +1$$ a hoch Exponentanfang n + 1 Exponentende ist ungleich a hoch Exponentanfang n Exponentende + 1  a^{n+1} \not= a^n +1
$$\sqrt{25} = 5$$ Die Quadratwurzel aus 25 ist gleich 5 \sqrt{25} = 5  \s{25}=5
$$\sqrt{x^2 +y^2} \not= x +y$$ Die Wurzel aus x hoch 2 plus y hoch 2 Wurzelende ist ungleich x plus y \sqrt{x^2 +y^2} \not= x +y \s{x^2 +y^2} \not= x +y
$$\sqrt[3]{8} = 2$$ Die dritte Wurzel aus 8 ist gleich 2 \sqrt[3]{8} = 2  \s[3]{8}=2
$$\sqrt[3]{a^2} =a^{2/3}$$ Die dritte Wurzel aus a hoch 2 Wurzelende ist gleich a hoch zwei Drittel \sqrt[3]{a^2} =a^{2/3}  \s[3]{a^2} =a^{2/3}
$$a_1 + a_n$$ a Index 1 plus a Index n  a_1 + a_n
$$a_{n -1}$$ a Index n minus 1 Indexende  a_{n -1}
$${}^{238}_{95}\mathrm{U}$$ Index und Exponent vor dem Zeichen  ^{238}_{95}U

## Weitere Rechenoperationen, Funktionen

Schwarzschrift Prosa LaTeX LaTeX-Abkürzung
$$f(x) =2x +1$$ f von x ist gleich 2x +1  f(x) =2x +1
$$f(3) =7$$ f von 3 ist gleich 7  f(3) =7
$$f:y=2x+1$$ Die Zuordnungsvorschrift der Funktion f lautet: y =2x +1  f: y =2x +1
$$f: x \mapsto 2x +1$$ Die Zuordnungsvorschrift der Funktion f lautet: x, Pfeil nach rechts, 2x +1  f: x \mapsto 2x +1  \mt
$$(3 ; 7)$$ runde Klammer auf, 3 Semikolon 7, runde Klammer zu (3 ;7)
$$|a|$$ Betrag von a |a|
$$\log_a x$$ Logarithmus von x zur Basis a \log_a x
$$\ln x$$ natürlicher Logarithmus (Logarithmus von x zur Basis e) \ln x
$$\sin \alpha$$ Sinus Alpha \sin \alpha \sin ~a
$$\cos ^2 \beta$$ Kosinus Quadrat Beta \cos^2 \beta \cos ^2 ~b
$$\tan \gamma$$ Tangens Gamma \tan \gamma ~g
$$\cot 45$$° Kotangens 45 Grad \cot 45°
$$\sin (\pi /6)$$ Sinus von Klammer auf Pi Sechstel Klammer zu \sin (\pi /6)

## Geometrie

Schwarzschrift Prosa LaTeX LaTeX-Abkürzung
$$\overline{AB}$$ Strecke AB \overline{AB}  \ol{AB}
$$\triangle ABC$$ Dreieck ABC \triangle ABC \tri ABC
$$\angle BAC$$ Winkel BAC \angle BAC
$$\alpha, \beta, \gamma, \delta, \epsilon$$ Alpha, Beta, Gamma, Delta, Epsilon \alpha, \beta, \gamma, \delta, \epsilon ~a, ~b, ~g, ~d, ~e
$$g \parallel h$$ g parallel zu h  g \parallel h g \| h
$$g \nparallel h$$ g nicht parallel zu h  g \nparallel h
$$g \perp h$$ g senkrecht zu h  g \perp h
$$F \cong F'$$ F kongruent zu F Strich  F \cong F'

## Anmerkungen

Anmerkung 1) Hintergrundinformation für die Übersetzung: Diese Darstellung stammt aus dem Paket amssymb, das in der Vorspanndatei vorspann.tex schon automatisch eingebunden wird. Dort wird auch die Abkürzung \N für {\mathbb N} definiert. Anmerkung 2) Bei der ersten Darstellungsvariante muss zwischen der ganzen Zahl und dem Bruch zwingend ein Leerschritt stehen, um beide voneinander zu trennen. Bei der Übersetzung kann ein solcher Leerschritt in der mathematischen Umgebung nicht mit einem Leerzeichen, wohl aber z.B. mit \; erreicht werden. Anmerkung 3) Der Befehl \permil wird analog zu einem Vorschlag auf einer Dante-FAQ-Seite in der Datei vorspann.tex definiert.

## Analysis

Schwarzschrift Prosa LaTeX LaTeX-Abkürzung
$$n \to \infty$$ n geht gegen unendlich  n \to \infty
$$\lim_{h \to 0}$$ Limes h gegen 0 \lim_{h \to 0}
$$\lim_{x \to x_0}$$ Limes x gegen x Index 0 \lim_{x \to x_0}
$$f\ ' (x), f\ ''(x)$$ f Strich von x, f zwei Strich von x  f'(x), f''(x)
$$\sum_{i=0}^n A_n$$ Summe von i gleich 0 bis n über A Index n \sum_{i =0}^n A_n
$$\int_a^b f(x) dx$$ Integral von a bis b über f von x dx \int_a^b f(x) dx

## Stochastik

Schwarzschrift Prosa LaTeX LaTeX-Abkürzung
$$n!$$ n Fakultät  n!
$${n \choose k}$$ Binomialkoeffizient n über k {n \choose k} aktueller: \binom{n}{k}
$$\sigma \qquad \Omega$$ klein Sigma groß Omega \sigma \Omega

## Analytische Geometrie

Schwarzschrift Prosa LaTeX LaTeX-Abkürzung
$$\vec{x} = (x \; y \; z)$$ Vektor x ist gleich Zeilenvektor x y z Vektorende \vec{x} = (x \; y \; z)  \vec{x} = (x & y & z)  oder
\vec{x} =(x ; y ; z)
$$\vec{y} = \matrix{1 \\ 2 \\ 3}$$ Vektor y ist gleich Spaltenvektor 1 2 3 Vektorende \vec{y} = \mat{c}{1 \\ 2 \\ 3}
[[LaTeX-Manual-Sekundarstufe2#Anmerkung|Anmerkung]]
\vec{y} = \mat{1 \\ 2 \\ 3}
$$A=\begin{bmatrix}1 \; 2 \; 3 \\ 4 \; 5 \; 6 \end{bmatrix}$$ Matrix groß A ist gleich Zeilenanfang 1 2 3 Zeilenende Zeilenanfang 4 5 6 Zeilenende Matrixende  A = \mat{ccc}{ 1 \; 2 \; 3 \\ 4 \; 5 \; 6 }
[[LaTeX-Manual-Sekundarstufe2#Anmerkung|Anmerkung]]
 A = \mat{ 1 & 2 & 3 \\ 4 & 5 & 6 }

## Anmerkung

Diese Kurzschreibweise \mat stammt aus der "Dresdener Abkürzungsliste" mathlib.tex von U. Nitsch. Beim Übersetzen wird diese mit der Datei vorspann.tex automatisch eingebunden. Spaltenvektoren werden hier als Matrizen mit einer Spalte betrachtet. In der ursprünglichen Fassung von \mat musste für jede Spalte der Matrize ein c in geschweiften Klammern angegeben werden. Dies kann man vereinfachen, indem man in der Datei mathlib.tex die Zeile \def\mat#1#2{\left|\begin{array}{#1}#2\end{array}\right|}  ersetzt durch: \def\mat#1{\left(\begin{array}{cccccc}#1\end{array}\right)}  Hier werden vorsorglich sechs Spalten angegeben. Wenn das zu viel ist, stören die überzähligen c aber nicht.

NOTICE OF MEETING - BOARD OF TRUSTEES

FINANCE COMMITTEE

TEXAS SCHOOL FOR THE BLIND AND VISUALLY IMPAIRED

NOTICE IS HEREBY GIVEN that a regular meeting of the Finance Committee of the Board of Trustees of the Texas School for the Blind and Visually Impaired shall be held on November 19, 2015, beginning at 2:00 p.m. in Room 102 of Building 600, 1100 W. 45th Street, Austin, Texas.

If, during the course of the meeting, any discussion of any item on the agenda should be held in closed session, the Finance Committee of the Board shall convene in such closed session in accordance with Texas Government Code, Sec. 551.001, et. seq.  Before any such session is convened, the presiding officer shall publicly identify the section or sections of the Code authorizing the closed session.  All final votes, actions, or decisions shall be taken in open session.

The subject to be discussed, or considered, or upon which any formal action may be taken are as follows (items do not have to be taken in the same order as shown on this meeting notice):

Finance Agenda:

1.   Report from Wells Fargo on the Bert E. Broday Trust Fund (Tom Cagil)

2.   Discussion on developing a framework and policies in preparation for the future transfer of assets upon termination of the Broday Trust

NOTICE OF MEETING

BOARD OF TRUSTEES

AUDIT, COMPLIANCE, AND MANAGEMENT REVIEW COMMITTEE

TEXAS SCHOOL FOR THE BLIND AND VISUALLY IMPAIRED

NOTICE IS HEREBY GIVEN that a regular meeting of the Audit, Compliance, and Management Review Committee of the Board of Trustees of the Texas School for the Blind and Visually Impaired shall be held on November 20, 2015, beginning at 8:00 a.m. in Room 92 of the Main Building 600, 1100 West 45th Street, Austin, Texas.

If, during the course of the meeting, any discussion of any item on the agenda should be held in closed session, the Audit, Compliance, and Management Review Committee shall convene in such closed session in accordance with Texas Government code, Sec. 551.001, et.seq.  Before any such session is convened, the presiding officer shall publicly identify the section or sections of the Code authorizing the closed session.  All final votes, actions, or decisions shall be taken in open session.

The subjects to be discussed or considered, or upon which any formal action may be taken are as follows (items do not have to be taken in the same order as shown on this meeting notice):

1.  Call to Order

2.  Consideration of Approval of Minutes of the September 25, 2015 Meeting

3.   Overview of the Center for School Resources (Susan Hauser)

4.   Report from the Internal Auditor (Jaye Stepp)

a.   Consideration of approval of Public Funds Investment Report

5.   Consideration of Approval of Board Policies:

CFCA       Internal Audit Charter (amend)


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PARALLEL AND PERPENDICULAR LINES

If you know the slopes of two non-vertical lines,
can you easily determine if they're parallel? Perpendicular?
YES!

Parallel lines have the same slope.

Perpendicular lines have slopes that are opposite reciprocals.

This section explores these concepts.

## Slopes of Parallel Lines

Two lines in a plane are parallel if they never intersect; they have the same ‘slant’.
(For a more rigorous study of parallel lines and related concepts, visit my Geometry course: Parallel Lines.)
Slope (or lack of it!) can be used to decide if lines are parallel:

CHARACTERIZATION OF PARALLEL LINES INVOLVING SLOPE
Suppose two lines lie in the same coordinate plane.
Then,
the two lines are parallel
if and only if
(they are both vertical)     or     (they have the same slope)

This statement offers a good opportunity to review the mathematical words ‘if and only if’ and ‘or’ .
The statement has been visually presented to help you see the mathematical sentence structure more clearly.

• The basic set-up is two lines lying in the same coordinate plane.
This condition lurks in the background (is assumed to be true) for all that follows.
• Two mathematical sentences are being compared with the words if and only if:
• first sentence: the two lines are parallel
• second sentence: (they are both vertical) OR (they have the same slope)
• If and only if is a variation of ‘is equivalent to’.
Sentences that are equivalent always have the same truth values—they're true at the same time, and false at the same time.
If one is true, so is the other.
If one is false, so is the other.
• An ‘or’ sentence is true when at least one of the subsentences is true.
The only time an ‘or’ is false is when both subsentences are false.
• The nice, compact statement above is actually saying A LOT.
Take a look, if you want, to increase your appreciation of the brevity of the mathematical language:

## Slopes of Perpendicular Lines

Now, let's move on to perpendicular lines.
Two lines are perpendicular if they intersect at a $\,90^{\circ}$ angle.
For example, the $\,x$-axis and $\,y$-axis are perpendicular.

CHARACTERIZATION OF PERPENDICULAR LINES
INVOLVING SLOPE
Suppose two non-vertical lines lie in the same coordinate plane.
Let $\,m_1\,$ and $\,m_2\,$ denote the slopes of these lines.

The two lines are perpendicular if and only if their slopes are opposite reciprocals: $$\,m_1 = -\frac{1}{m_2}\,$$
Equivalently, the two lines are perpendicular if and only if their slopes multiply to $\,-1\,$: $$\,m_1m_2 = -1\,$$

Make sure you understand the opposite reciprocal equation:

$$m_1 = -\frac{1}{m_2}$$
 one of the slopes is the opposite of the reciprocal of the other slope $\displaystyle m_1$ $\displaystyle =$ $\displaystyle -$ $\displaystyle\frac{1}{m_2}$

For example, what's the reciprocal of $\,2\,$? Answer:   $\frac12$
What's the opposite [of the] reciprocal of $\,2\,$? Answer:   $-\frac12$
So, the numbers $\,2\,$ and $\,-\frac 12\,$ are opposite reciprocals.
Also notice that $\,(2)(-\frac 12) = -1\,$.
So, lines with slopes $\,2\,$ and $\,-\frac12\,$ are perpendicular.

It's easy to see that this is the correct characterization for perpendicular lines, by studying the sketch below:

The yellow triangle, with base of length $\,1\,$ and right side of length $\,m\,$,
shows that the slope of the first line is $\,\frac{\text{rise}}{\text{run}} = \frac{m}1 = m\,$.

Now, imagine that this yellow triangle is a block of wood that is glued to the line.
Rotate this block of wood counter-clockwise by $\,90^{\circ}\,$ (so that the original base is now vertical).
Using the rotated triangle to compute the slope of the new, rotated, line gives:   $\,\frac{\text{rise}}{\text{run}} = \frac{1}{-m} = -\frac1m\,$
Easy! Voila!

Master the ideas from this section


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# Expectation value of an operator

Suppose we have: $$\hat{Q}|\psi_1\rangle=q_1|\psi_1\rangle \\ \hat{Q}|\psi_2\rangle=q_2|\psi_2\rangle$$ with $q_1 \neq q_2$. Then consider the state: $$|\Psi\rangle=\frac{1}{\sqrt{2}}(|\psi_1\rangle-i|\psi_2\rangle)$$ I want to calculate the uncertainty in $\hat{Q}$. Then first we must compute $\langle\hat{Q^2}\rangle$ and $\langle\hat{Q}\rangle^2$. I started with $\langle\hat{Q^2}\rangle$. First, $$\hat{Q}|\Psi\rangle=\frac{1}{\sqrt{2}}(q_1|\psi_1\rangle-iq_2|\psi_2\rangle)$$ Then: $$\langle\Psi|\hat{Q}=\frac{1}{\sqrt{2}}(q_1^{*}\langle\psi_1|+iq_2^{*}\langle\psi_2|)$$ It follows that: $$\langle\Psi|\hat{Q^2}|\Psi\rangle = (\langle\Psi|\hat{Q})(\hat{Q}|\Psi\rangle) =\frac{1}{2}(q_1^{*}\langle \psi_1|+iq_2^{*}\langle \psi_2|)(q_1|\psi_1\rangle -iq_2|\psi_2\rangle )$$ Using the fact that: $$\langle A|bB+cC\rangle = b\langle A|B\rangle + c\langle A|C\rangle \text{ and}\\ \langle bB+cC|A\rangle = b^{*}\langle B|A\rangle + c^{*}\langle C|A\rangle$$ I found: $$\langle\Psi|\hat{Q^2}|\Psi\rangle = \frac{1}{2}(q_1^2-q_2 q_2^{*})$$ Note that $q_1^2 \neq |q_1|^2$. On the other hand, if I act on the ket vector twice to compute $\hat{Q^2}|\Psi\rangle$ I find: $$\hat{Q^2}|\Psi\rangle = \frac{1}{\sqrt{2}}(q_1^2|\psi_1\rangle -iq_2^2|\psi_2\rangle)$$ Then using this expression to compute $\langle\Psi|\hat{Q^2}|\Psi\rangle$ yields: $$\langle\Psi|\hat{Q^2}|\Psi\rangle = \frac{1}{2}(q_1^2 - q_2^2)$$ Now using the representation in the basis: $$\hat{Q}|\Psi\rangle = \frac{1}{\sqrt{2}} \left[\begin{matrix}q_1 \\ -iq_2\end{matrix}\right] \\ \langle\Psi|\hat{Q} = \frac{1}{\sqrt{2}}[q_1^{*} \text{ }iq_2^{*}]$$ Which gives: $$\langle\Psi|\hat{Q^2}|\Psi\rangle = q_1^{*}q_1 + q_2^{*}q_2$$ Why am I getting different results?

• Notice that $\left \langle a \psi + b \phi \right| \neq a\left\langle\psi\right| + b\left\langle\phi\right|$, since $\left\langle a\psi\right| = \left| a \psi \right\rangle^\dagger = (a\left|\psi\right\rangle)^\dagger = a^*\left\langle\psi\right|$. Jun 5 '14 at 18:17
• Is $\hat{Q}$ Hermitian? Jun 5 '14 at 20:25

Assuming $|\psi_1\rangle$ and $|\psi_2\rangle$ are orthonormal, your equation after you write "I found" is incorrect, it should be \begin{align} (\langle \Psi|\hat Q)(\hat Q|\Psi\rangle) &= \frac{1}{2}\Big(q_1^*q_1\langle\psi_1|\psi_1\rangle - i q_1^*q_2 \langle\psi_1|\psi_2\rangle +iq_2^*q_1\langle\psi_2|\psi_1\rangle -i^2q_2^*q_2\langle\psi_2|\psi_2\rangle\Big) \\ &= \frac{1}{2}(|q_1|^2+|q_2|^2) \end{align}
• Does this follow from: $\frac{1}{2}(\langle q_1^{*}\psi_1+i q_2^{*}\psi_2|q_1\psi_1-iq_2\psi_2\rangle)$? Jun 5 '14 at 18:22
• @Ali Yes; just use the properties of the inner product that you wrote down, and you'll get precisely that. The cross terms cancel due to orthogonality of $|\psi_1\rangle$ and $|\psi_2\rangle$. Jun 5 '14 at 18:34
• Did you assume that $\langle A+B|C+D\rangle=\langle A|B \rangle+\langle A|D \rangle +\langle B |C \rangle+\langle B|D \rangle$? Because I can get to what you posted if I use this. But I can't see why this is true. Jun 5 '14 at 18:36
• @Ali Yes (but the first term on the right should be $\langle A|C\rangle$). You can prove that by applying the inner product properties you wrote down twice, once for the first "slot" and once for the second "slot." Jun 5 '14 at 18:38
• @Ali It seems your confusion stems from mixing legitimate bra-ket notation with inner product notation used more commonly in mathematics. In legit bra-ket notation, $|\psi\rangle$ is a vector, the $\psi$ inside is just a label, and $\langle\psi|$ is a corresponding dual vector. In mathy notation, $\psi$ is a vector, the symbol $|\psi\rangle$ has no independent meaning, and $\langle \psi|\phi\rangle$ is the inner product of two vectors $\psi$ and $\phi$. I'd recommending reading up on this stuff; this might help: physics.stackexchange.com/a/57793/19976 Jun 5 '14 at 19:04