6.4: Black Holes (Part 1)
Singularities
A provocative feature of the Schwarzschild metric is that it has elements that blow up at \(r = 0\) and at \(r = 2m\). If this is a description of the sun, for example, then these singularities are of no physical significance, since we only solved the Einstein field equation for the vacuum region outside the sun, whereas \(r = 2m\) would lie about 3 km from the sun’s center. Furthermore, it is possible that one or both of these singularities is nothing more than a spot where our coordinate system misbehaves. This would be known as a coordinate singularity . For example, the metric of ordinary polar coordinates in a Euclidean plane has \(g^{\theta \theta} → \infty\) as \(r → 0\).
One way to test whether a singularity is a coordinate singularity is to calculate a scalar measure of curvature, whose value is independent of the coordinate system. We can take the trace of the Ricci tensor, \(R^a_a\), known as the scalar curvature or Ricci scalar , but since the Ricci tensor is zero, it’s not surprising that that is zero. A different scalar we can construct is the product \(R^{abcd}R_{abcd}\) of the Riemann tensor with itself. This is known as the Kretschmann invariant . The Maxima command lriemann(true) displays the nonvanishing components of R abcd The component that misbehaves the most severely at r = 0 is R trrt = \(\frac{2m}{r^{3}}\). Because of this, the Kretschmann invariant blows up like \(r^{−6}\) as \(r → 0\). This shows that the singularity at r = 0 is a real, physical singularity.
The singularity at \(r = 2m\), on the other hand, turns out to be only a coordinate singularity. To prove this, we have to use some technique other than constructing scalar measures of curvature. Even if every such scalar we construct is finite at \(r = 2m\), that doesn’t prove that every such scalar we could construct is also well behaved. We can instead search for some other coordinate system in which to express the solution to the field equations, one in which no such singularity appears. A partially successful change of coordinates for the Schwarzschild metric, found by Eddington in 1924, is (see problem 8):
\[t → t' = t − 2m \ln(r −2m).\]
This makes the covariant metric finite at \(r = 2m\), although the contravariant metric still blows up there. A more complicated change of coordinates that completely eliminates the singularity at \(r = 2m\) was found by Eddington and Finkelstein in 1958, establishing that the singularity was only a coordinate singularity. Thus, if an observer is so unlucky as to fall into a black hole, he will not be subjected to infinite tidal stresses — or infinite anything — at \(r = 2m\). He may not notice anything special at all about his local environment. (Or he may already be dead because the tidal stresses at \(r > 2m\), although finite, were nevertheless great enough to kill him.)
Event Horizon
Even though \(r = 2m\) isn’t a real singularity, interesting things do happen there. For \(r < 2m\), the sign of \(g_{tt}\) becomes negative, while \(g_{rr}\) is positive. In our + − −− signature, this has the following interpretation. For the world-line of a material particle, \(ds^2\) is supposed to be the square of the particle’s proper time, and it must always be positive. If a particle had a constant value of \(r\), for \(r < 2m\), it would have \(ds^2 < 0\), which is impossible.
The timelike and spacelike characters of the \(r\) and \(t\) coordinates have been swapped, so \(r\) acts like a time coordinate.
Thus for an object compact enough that \(r = 2m\) is exterior, \(r = 2m\) is an event horizon: future light cones tip over so far that they do not allow causal relationships to connect with the spacetime outside. In relativity, event horizons do not occur only in the context of black holes; their properties, and some of the implications for black holes, have already been discussed in section 6.1 .
The gravitational time dilation in the Schwarzschild field, relative to a clock at infinity, is given by the square root of the \(g_{tt}\) component of the metric. This goes to zero at the event horizon, meaning that, for example, a photon emitted from the event horizon will be infinitely redshifted when it reaches an observer at infinity. This makes sense, because the photon is then undetectable, just as it would be if it had been emitted from inside the event horizon.
Infalling Matter
If matter is falling into a black hole, then due to time dilation an observer at infinity “sees” that matter as slowing down more and more as it approaches the horizon. This has some counterintuitive effects. A radially infalling particle has
\[\frac{d^{2} r}{dt^{2}} > 0\]
once it falls past a certain point, which could be interpreted as a gravitational repulsion. The observer at infinity may also be led to describe the black hole as consisting of an empty, spherical shell of matter that never quite made it through the horizon. If asked what holds the shell up, the observer could say that it is held up by gravitational repulsion.
There is actually nothing wrong with any of this, but one should realize that it is only one possible description in one possible coordinate system. An observer hovering just outside the event horizon sees a completely different picture, with matter falling past at velocities that approach the speed of light as it comes to the event horizon. If an atom emits a photon from the event horizon, the hovering observer sees it as being infinitely red-shifted, but explains the red-shift as a kinematic one rather than a gravitational one.
We can imagine yet a third observer, one who free-falls along with the infalling matter. According to this observer, the gravitational field is always zero, and it takes only a finite time to pass through the event horizon.
If a black hole has formed from the gravitational collapse of a cloud of matter, then some of our observers can say that “right now” the matter is located in a spherical shell at the event horizon, while others can say that it is concentrated at an infinitely dense singularity at the center. Since simultaneity isn’t well defined in relativity, it’s not surprising that they disagree about what’s happening “right now.” Regardless of where they say the matter is, they all agree on the spacetime curvature. In fact, Birkhoff’s theorem tells us that any spherically symmetric vacuum spacetime is Schwarzschild in form, so it doesn’t matter where we say the matter is, as long as it’s distributed in a spherically symmetric way and surrounded by vacuum.
A particularly nice way of summarizing and understanding these issues is with the use of a Penrose diagram, as discussed in section 7.3 .
Expected Formation
Einstein and Schwarzschild did not believe, however, that any of these features of the Schwarzschild metric were more than a mathematical curiosity, and the term “black hole” was not invented until the 1967, by John Wheeler. There is quite a bit of evidence these days that our universe does contain objects that have undergone complete gravitational collapse, in the sense that their mass M is contained within a radius \(r \lesssim M\) (in geometrized units). These objects are probably black holes, although doubts have been raised recently as to whether they are in fact other objects such as naked singularities. 10 Supposing that black holes do exist, there is also the question of what sizes they come in.
We might expect naively that since gravity is an attractive force, there would be a tendency for any primordial cloud of gas or dust to spontaneously collapse into a black hole. But clouds of less than about 0.1M \(\odot\) (0.1 solar masses) form planets, which achieve a permanent equilibrium between gravity and internal pressure. Heavier objects initiate nuclear fusion , but those with masses above about 100M \(\odot\) are immediately torn apart by their own solar winds. In the range from 0.1 to 100M \(\odot\) , stars form. As discussed in section 4.4 , those with masses greater than about a few M \(\odot\) are expected to form black holes when they die. We therefore expect, on theoretical grounds, that the universe should contain black holes with masses ranging from a few solar masses to a few tens of solar masses.
Observational Evidence
A black hole is expected to be a very compact object, with a strong gravitational field, that does not emit any of its own light. A bare, isolated black hole would be difficult to detect, except perhaps via its lensing of light rays that happen to pass by it. But if a black hole occurs in a binary star system, it is possible for mass to be transferred onto the black hole from its companion, if the companion’s evolution causes it to expand into a giant and intrude upon the black hole’s gravity well. The infalling gas would then get hot and emit radiation before disappearing behind the event horizon (Figure \(\PageIndex{1}\)). The object known as Cygnus X-1 is the best-studied example. This X-ray-emitting object was discovered by a rocket-based experiment in 1964. It is part of a double-star system, the other member being a blue supergiant. They orbit their common center of mass with a period of 5.6 days. The orbit is nearly circular, and has a semimajor axis of about 0.2 times the distance from the earth to the sun. Applying Kepler’s law of periods to these data constrains the sum of the masses, and knowledge of stellar structure fixes the mass of the supergiant. The result is that the mass of Cygnus X-1 is greater than about 10 solar masses, and this is confirmed by multiple methods. Since this is far above the Tolman-Oppenheimer-Volkoff limit, Cygnus X-1 is believed to be a black hole, and its X-ray emissions are interpreted as the radiation from the disk of superheated material accreting onto it from its companion. It is believed to have more than 90% of the maximum possible spin for a black hole of its mass. 11
Around the turn of the 21st century, new evidence was found for the prevalence of supermassive black holes near the centers of nearly all galaxies, including our own. Near our galaxy’s center is an object called Sagittarius A*, detected because nearby stars orbit around it. The orbital data show that Sagittarius A* has a mass of about four million solar masses, confined within a sphere with a radius less than 2.2 × 10 7 km. There is no known astrophysical model that could prevent the collapse of such a compact object into a black hole, nor is there any plausible model that would allow this much mass to exist in equilibrium in such a small space, without emitting enough light to be observable.
The existence of supermassive black holes is surprising. Gas clouds with masses greater than about 100 solar masses cannot normally form stable stars, so supermassive black holes cannot be the end-point of the evolution of heavy stars. Mergers of multiple stars to form more massive objects are generally statistically unlikely, since a star is such a small target in relation to the distance between the stars. Once astronomers were confronted with the empirical fact of their existence, a variety of mechanisms was proposed for their formation. Little is known about which of these mechanisms is correct, although the existence of quasars in the early universe is interpreted as evidence that mass accreted rapidly onto supermassive black holes in the early stages of the evolution of the galaxies. As of 2016, an explanation getting a lot of attention is that in the early universe, there was a brief period in which the ambient conditions allowed the creation of supermassive black holes by direct collapse. 12
A skeptic could object that although Cygnus X-1 and Sagittarius A* are more compact than is believed possible for a neutron star, this does not necessarily prove that they are black holes. Indeed, speculative theories have been proposed in which exotic objects could exist that are intermediate in compactness between black holes and neutron stars.  Although there is no evidence that these theories are right or that these objects exist, we are faced with the question of how to determine whether a given object is really a black hole or one of these other species. The defining characteristic of a black hole is that it has an event horizon rather than a physical surface. We currently have two ways of probing the structure of these stars at the radii where general relativity predicts the existence of an event horizon.
If an object is not a black hole, then by conservation of energy any matter that falls onto it must release its gravitational potential energy when it hits that surface. Cygnus X-1 has a copious supply of matter falling onto it from its supergiant companion, and Sagittarius A* likewise accretes a huge amount of gas from the stellar wind of nearby stars. By analyzing millimeter and infrared very-long-baseline-interferometry observations, Broderick, Loeb, and Narayan 13 have shown that if Sagittarius A* had a surface, then the luminosity of this surface must be less than 0.3% of the luminosity of the accretion disk. But this is not physically possible, because there are fundamental limits on the efficiency with which the gas can radiate away its energy before hitting the surface. We can therefore conclude that Sagittarius A* must have an event horizon. Its event horizon may be imaged directly in the near future. 14
A second approach is through the observation of gravitational waves. As discussed in more detail in ch. 9 , 2016 saw the first direct observation of gravitational waves. The waveform that was detected ( Figure 9.2.2 ) fits very well with the predictions of general relativity for the merger of two black holes. It seems very unlikely that a waveform with this time-scale and characteristic shape could have been produced unless general relativity’s description of black holes is correct in detail.
Singularities and Cosmic Censorship
Informal Ideas
Since we observe that black holes really do exist, maybe we should take the singularity at r = 0 seriously. Physically, it says that the mass density and tidal forces blow up to infinity there.
Generally when a physical theory says that observable quantities blow up to infinity at a particular point, it means that the theory has reached the point at which it can no longer make physical predictions. For instance, Maxwell’s theory of electromagnetism predicts that the electric field blows up like r −2 near a point charge, and this implies that infinite energy is stored in the field within a finite radius around the charge. Physically, this can’t be right, because we know it only takes 511 keV of energy to create an electron out of nothing, e.g., in nuclear beta decay. The paradox is resolved by quantum electrodynamics, which modifies the description of the vacuum around the electron to include a sea of virtual particles popping into and out of existence.
In the case of a black hole singularity, it is possible that quantum mechanical effects at the Planck scale prevent the formation of a singularity. Unfortunately, we are unlikely to find any empirical evidence about this, since black holes always seem to come clothed in event horizons, so we outside observers cannot extract any data about the singularity inside. Even if we take a suicidal trip into a black hole, we get no data about the singularity, because the singularity in the Schwarzschild metric is spacelike, not timelike, and therefore it always lies in our future light cone, never in our past.
In a way, the inaccessibility of singularities is a good thing. If a singularity exists, it is a point at which all the known laws of physics break down, and physicists therefore have no way of predicting anything about its behavior. There is likewise no great crisis for physics due to the Big Bang singularity or the Big Crunch singularity that occurs in some cosmologies in which the universe recollapses; we have no reasonable expectation of being able to make and test predictions or retrodictions that extend beyond the beginning or end of the universe.
What would be a crushing blow to the enterprise of physics would be a singularity that could sit on someone’s desk. As John Earman of the University of Pittsburgh puts it, anything could pop out of such a “naked” singularity (defined formally later ), including green slime or your lost socks.
Penrose’s cosmic censorship conjecture states that the laws of physics prevent the formation of naked singularities from nonsingular and generic initial conditions. “Generic” is a necessary addition to Penrose’s original 1969 formulation, since Choptuik showed in 1993 that certain perfectly fine-tuned initial conditions allowed collapse to a naked singularity. 15 As of 2017, evidence is accumulating that cosmic censorship is false. This is discussed at greater length later .
Formal Definitions
The remainder of this subsection provides a more formal exposition of the definitions relating to singularities. It can be skipped without loss of continuity.
The reason we care about singularities is that they indicate an incompleteness of the theory, and the theory’s inability to make predictions. One of the simplest things we could ask any theory to do would be to predict the trajectories of test particles. For example, Maxwell’s equations correctly predict the motion of an electron in a uniform magnetic field, but they fail to predict the motion of an electron that collides head-on with a positron. It might have been natural for someone in Maxwell’s era (assuming they were informed about the existence of positrons and told to assume that both particles were pointlike) to guess that the two particles would scatter through one another at \(\theta\) = 0, their velocities momentarily becoming infinite. But it would have been equally natural for this person to refuse to make a prediction.
Similarly, if a particle hits a black hole singularity, we should not expect general relativity to make a definite prediction. It doesn’t, because the geodesic equation breaks down.
We would therefore like to define a singularity as a situation in which the geodesics of test particles can’t be extended indefinitely. But what does “indefinitely” mean? If the test particle is a photon, then the metric length of its world-line is zero. We get around this by defining length in terms of an affine parameter.
Definition
A spacetime is said to be geodesically incomplete if there exist timelike or lightlike geodesics that cannot be extended beyond some finite affine parameter into the past or future.
This is also a pretty good working definition of what we mean when we say that a spacetime contains a singularity, although it may not be optimal for all purposes. 16 The Schwarzschild spacetime has a singularity at r = 0, but not at the event horizon, since geodesics continue smoothly past the event horizon. Cosmological spacetimes contain a Big Bang singularity which prevents geodesics from being extended beyond a certain point in the past .
Actual singularities involving geodesic incompleteness are to be distinguished from coordinate singularities, which are not really singularities at all. In the Schwarzschild spacetime, as described in Schwarzschild’s original coordinates, some components of the metric blow up at the event horizon, but this is not an actual singularity. This coordinate system can be replaced with a different one in which the metric is well behaved.
Example 1: A harmless blow-up
Let’s define coordinates (t, y) in the region of spacetime where you’re sitting and reading this book. Let (0, 0) be your current time and position, and for convenience let this be an inertial frame (so that your motion is not geodesic). The Riemann tensor, expressed in these coordinates, has a component R tyyt = \(\frac{2Gm}{r^{3}}\), where m and r are the mass and radius of the earth. This has the finite value of 1.5 × 10 −6 s −2 , which expresses the strength of a tidal effect near the earth’s surface.
Now define a new coordinate u = y 3 . Applying the tensor transformation law, we have
\[R_{tuut} = R_{tyyt} \left(\dfrac{\partial y}{\partial u}\right)^{2},\]
which is infinite at y = 0. This example demonstrates that we cannot test for a singularity by looking for a blow-up of the components of a curvature tensor at certain coordinates or as the coordinates approach some limit.
There are two types of singularities: curvature singularities and non-curvature singularities.
 These indicate that tidal forces blow up to infinity, and would destroy any observer.
The reason curvature scalars are useful as tests for a curvature singularity is that since they’re scalars, they can’t diverge in one coordinate system but stay finite in another (cf. example 1). A sufficient condition for a singularity to be a curvature singularity is if timelike or lightlike geodesics can only be extended to some finite affine parameter, and some curvature scalar (not necessarily every such scalar) approaches infinity as we approach this value of the affine parameter.
But we should not expect this to be a necessary condition for a curvature singularity. Example 2 below shows that the most commonly occurring curvature scalars may not be enough to catch the presence of a singularity. This is not too surprising, since curvature scalars do not suffice to tell us everything there is to know about the curvature of a spacetime (example 3).
Example 2: Incompleteness with finite curvature scalars
Consider the 1+1-dimensional spacetime described by the metric
\[\begin{split} ds^{2} &= A(dt^{2} - dx^{2}) \\ A &= \frac{1}{1 + e^{t}}, \end{split}\]
with \(− \infty < x < \infty\) and \(− \infty < t < \infty\). For large negative t it is indistinguishable from Minkowski space. The following Maxima code computes its Riemann tensor and the scalar curvature R and the Kretchmann invariant K.
The results for the two curvature scalars are
\[R = (1 + e^{-t})^{-1}\]
and
\[K = (1 + e^{-t})^{-2},\]
both of which are finite everywhere; they go from 0 at large negative times to 1 at large positive times.
From these results we would not imagine that there was any singularity present, but blow-ups of curvature scalars are only a sufficient condition for geodesic incompleteness, not a necessary one. Consider the timelike curve x = 0, which by symmetry is a geodesic. If we integrate the proper time along this geodesic, we get a finite limit as t → \(\infty\). Since proper time qualifies as an affine parameter, this geodesic is incomplete.
But it is not so obvious that this spacetime is “really” singular. It is possible that we could smoothly extend it beyond t = \(+ \infty\). If so, then the singularity at t = \(+ \infty\) would be a kind of fake singularity, of the type that we could obtain simply by chopping off the part of Minkowski space with t ≥ 0.
Example 3: Vanishing curvature scalars
We remarked above that curvature scalars do not in general suffice to tell us everything about the curvature of a spacetime. In fact, there is an entire class of curved spacetimes such that every curvature invariant vanishes everywhere. Schmidt 17 gives the example
\[ds^{2} = du dv - a^{2} (u) dw^{2},\]
where a is an arbitrary nonlinear function. The eigenvalues of this metric are 1, −1, and −a 2 , so its signature is + − −, i.e., this is general relativity in 2 + 1 dimensions. A computation shows that the space is not flat, since, e.g., \(R_{uu} = − \frac{a''}{a}\). The u and v directions are lightlike, so this metric represents a wavelike disturbance traveling at the speed of light. (Since the Ricci tensor doesn’t vanish, this isn’t a vacuum solution, and we don’t have a gravitational wave in vacuum. Such waves, as described in ch. 9 , are transverse and can only exist in 3 + 1 or more dimensions.)
The lightlike character of u and v motivates us to consider coordinate transformations of the form (u, v) → (uD, \(\frac{v}{D}\)), because in the case a = 0, which is flat, this would be a Lorentz boost with a Doppler shift factor D. In the case where D approaches zero, we are chasing the wave at a velocity approaching c, so the wave Doppler-shifts to undetectability. All components of the Riemann tensor, as well as their derivatives, approach zero.
Now consider any curvature scalar I that is expressible as a continuous function of the Riemann tensor and its derivatives. By continuity, I approaches zero as D → 0. But curvature scalars are scalars, so they are invariant under coordinate transformations. It therefore follows that I = 0 identically, regardless of the value of D. Thus we have a spacetime that, although curved, has no nonvanishing curvature scalars anywhere.
17 “Why do all the curvature invariants of a gravitational wave vanish?,” arxiv.org/abs/gr-qc/9404037
References
10 See sec. 6.3 , and, e.g., Joshi et al., arxiv.org/abs/1304.7331.
11 Gou et al., “The Extreme Spin of the Black Hole in Cygnus X-1,” http://arxiv.org/abs/1106.3690
12 See, e.g., http://arxiv.org/abs/1402.5675
13 arxiv.org/abs/0903.1105
14 arxiv.org/abs/0906.4040
15 Phys. Rev. Lett. 70, p. 9
16 Geroch, ”What is a singularity in general relativity?,” Ann Phys 48 (1968) 526.


<IP_ADDRESS>: Hymn to Beauty
O Beauty! dost thou generate from Heaven or from Hell?
Within thy glance, so diabolic and divine,
Confusedly both wickedness and goodness dwell,
And hence one might compare thee unto sparkling wine.

Yea, art thou from the planets, or the fiery womb?
The demon follows in thy train, with magic fraught,
Thou scatter'st seeds haphazardly of joy and doom,
Thou govern'st everything, but answer'st unto nought.
O Loveliness! thou spurnest corpses with delight,
Among thy jewels, Horror hath such charms for thee,
And Murder 'mid thy mostly cherished trinklets bright,
Upon thy massive bosom dances amorously.
The blinded, fluttering moth towards the candle flies,
Then frizzles, falls, and falters—"Blessings unto thee"—
The panting swain that o'er his beauteous mistress sighs,
Seems like the Sick, that stroke their gravestones lovingly.
What matter, if thou comest from the Heavens or Hell,
O Beauty, frightful ghoul, ingenuous and obscure!
So long thine eyes, thy smile, to me the way can tell
Towards that Infinite I love, but never saw.
From God or Satan? Angel, Mermaid, Proserpine?
What matter if thou makest—blithe, voluptuous sprite—
With rhythms, perfumes, visions—O mine only queen!—
The universe less hideous and the hours less trite.


<IP_ADDRESS>: New Thoughts
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The integration of information technology has influenced the structure of organizations. The increased ability to communicate and share information has led to a “flattening” of the organizational structure due to the removal of one or more layers of management.
Another organizational change enabled by information systems is the network-based organizational structure. In a networked-based organizational structure, groups of employees can work somewhat independently to accomplish a project. In a networked organization, people with the right skills are brought together for a project and then released to work on other projects when that project is over. These groups are somewhat informal and allow for all members of the group to maximize their effectiveness.


13.2: Examples
Here’s another problem:
Here is a large square made up of 100 smaller unit squares. The unit squares along the border of the large square are colored red. Without counting one-by-one, can you figure out how many red squares there are in the picture?
Clearly describe how you figured out the number of red squares, and how you know your answer is correct.
Justin calculated the number of squares as \((10 \times 4) - 4\). He justified his answer this way:
Since the dimensions of the big square are \(10 \times 10\), there are 10 squares along each of the four sides. So that gives me 40 red squares. But then each corner is part of two different sides. I’ve counted each of the corners twice. So I need to make up for that by subtracting 4 at the end.
Justin showed this picture to justify his work:
- What do you think about Justin’s solution? Are you convinced? Could he have explained it more clearly?
- Was Justin’s solution different from your solution or the same?
- Notice the color coding in Justin’s picture. What do the colors represent? Why did he use the colors the way he did?
There are lots of different ways to calculate the number of colored squares along the border of a \(10 \times 10\) square. Below are the calculations several other students did. For each calculation, write a justification and draw a picture to show why it calculates the number of squares correctly. Think about using color in your picture to make your work more clear.
- Valerie calculated \(10 + 10 + 8 + 8\).
- Kayla calculated \(4 \times 9\).
- Linda calculated \((10 \times 10) - (8 \times 8)\).
- Mark calculated \((4 \times 8) + 4\).
- Allan calculated \(10 + 9 + 9 + 8\).
Now suppose that you have a large \(6 \times 6\) square with the unit squares along the border colored red. Adapt two of the techniques above to calculate the number of red unit squares.
For each technique you used, write an explanation and include a picture. Think about how to use colors or other methods to make your picture and explanation more clear.
Now suppose that you have a large \(25 \times 25\) square with the unit squares along the border colored red. Adapt two of the techniques above to calculate the number of red unit squares.
For each technique you used, write an explanation and include a picture. Think about how to use colors or other methods to make your picture and explanation more clear.
- Suppose that you have 64 red squares. Can you use all of those squares to make the border of a larger square in a picture like the one above? If yes, what are the dimensions of the larger square? If no, why not?
- What if you have 30 red squares? Same questions.
- What if you have 256 red squares? Same questions.
Describe some general rules:
- If you have a large \(n \times n\) square with the border squares colored red, how many red squares will there be? Justify your answer with words and a picture.
- If you have red squares, is there a quick test you can do to decide if you can use all of those squares to make the border of a large square? Can you tell how big the square will be?


10.9: Solution Stoichiometry
Learning Objectives
- Determine amounts of reactants or products in aqueous solutions.
As we learned previously, double replacement reactions involve the reaction between ionic compounds in solution and, in the course of the reaction, the ions in the two reacting compounds are “switched” (they replace each other). Because these reactions occur in aqueous solution, we can use the concept of molarity to directly calculate the number of moles of reactants or products that will be formed, and therefore their amounts (i.e. volume of solutions or mass of precipitates).
As an example, lead (II) nitrate and sodium chloride react to form sodium nitrate and the insoluble compound, lead (II) chloride.
\[\ce{Pb(NO_3)_2 (aq) + 2 NaCl (aq) → PbCl_2 (s) + 2 NaNO3 (aq)} \label{EQ1}\]
In the reaction shown above, if we mixed 0.123 L of a 1.00 M solution of \(\ce{NaCl}\) with 1.50 M solution of \(\ce{Pb(NO3)2}\), we could calculate the volume of \(\ce{Pb(NO3)2}\) solution needed to completely precipitate the \(\ce{Pb^{2+}}\) ions.
The molar concentration can also be expressed as the following:
\[1.00\,M \,\ce{NaCl} = \dfrac{1.00 \; mol \; \ce{NaCl}}{1 \; L \; \ce{NaCl} \; \text{solution}} \nonumber\]
and
\[1.50\,M\, \ce{Pb(NO3)2} = \dfrac{1.50 \; mol \; \ce{Pb(NO_3)_2}}{1 \; L \; \ce{Pb(NO_3)_2} \text{solution}} \nonumber\]
First, we must examine the reaction stoichiometry in the balanced reaction (Equation \ref{EQ1}). In this reaction, one mole of \(\ce{Pb(NO3)2}\) reacts with two moles of \(\ce{NaCl}\) to give one mole of \(\ce{PbCl2}\) precipitate. Thus, the concept map utilizing the stoichiometric ratios is:
so the volume of lead (II) nitrate that reacted is calculated as:
\[0.123 \; L \; \ce{NaCl} \, \text{solution} \times \frac{1.00 \; mol \; \ce{NaCl}}{1 \; L \; \ce{NaCl} \; \text{solution}} \times \frac{1 \; mol \; \ce{Pb(NO_3)_2}}{2 \; mol \; \ce{NaCl}} \times \frac{1 \; L \; \ce{Pb(NO_3)_2} \; \text{solution}}{1.5 \; mol \; \ce{Pb(NO_3)_2}} = 0.041 \; \ce{Pb(NO_3)_2}\,L \; \text{solution} \nonumber\]
This volume makes intuitive sense for two reasons: (1) the number of moles of \(\ce{Pb(NO3)2}\) required is half of the number of moles of \(\ce{NaCl}\), based off of the stoichiometry in the balanced reaction (Equation \ref{EQ1}); (2) the concentration of \(\ce{Pb(NO3)2}\) solution is 50% greater than the \(\ce{NaCl}\) solution, so less volume is needed.
Example \(\PageIndex{1}\)
What volume (in L) of 0.500 M sodium sulfate will react with 275 mL of 0.250 M barium chloride to completely precipitate all \(\ce{Ba^{2+}}\) in the solution?
Solution
| Steps for Problem Solving | Example \(\PageIndex{1}\) |
|---|---|
| Identify the "given" information and what the problem is asking you to "find." |
Given: 275 mL BaCl 2 0.250 M \(\ce{BaCl2}\) or \(\displaystyle \frac{0.250\; mol BaCl_2}{1\; L\; BaCl_2\; solution}\) 0.500 M \(\ce{Na2SO4}\) or \(\displaystyle \frac{0.500\; mol Na_2SO_4}{1\; L\; Na_2SO_4\; solution}\) Find: Volume \(\ce{Na2SO4}\) solution. |
| Set up and balance the chemical equation. |
\(\ce{Na2SO4(aq) + BaCl2(aq) -> BaSO4(s)} + \underline{2} \ce{NaCl (aq)}\) An insoluble product is formed after the reaction. |
| List other known quantities. |
1 mol of Na 2 SO 4 to 1 mol BaCl 2 1000 mL = 1 L |
| Prepare a concept map and use the proper conversion factor. | |
| Cancel units and calculate. |
\(\displaystyle 275\cancel{mL \; BaCl_2 \; solution}\times \frac{1\cancel{L}}{1000\cancel{mL}}\times \frac{0.250 \cancel{mol \; BaCl_2}}{1 \cancel{L\;BaCl_2 \; solution}}\times \frac{1 \cancel{mol \; Na_2SO_4}}{1 \cancel{mol \; BaCl_2}}\times \frac{1\; L \; Na_2SO_4 \; solution}{0.500 \cancel{mol Na_2SO_4}}\) = 0.1375 L sodium sulfate |
| Think about your result. | The lesser amount (almost half) of sodium sulfate is to be expected as it is more concentrated than barium chloride. Also, the units are correct. |
Exercise \(\PageIndex{1}\)
What volume of 0.250 M lithium hydroxide will completely react with 0.500 L of 0.250 M of sulfuric acid solution?
- Answer
-
0.250 L \(\ce{LiOH}\) solution
Contributions & Attributions
This page was constructed from content via the following contributor(s) and edited (topically or extensively) by the LibreTexts development team to meet platform style, presentation, and quality:
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Paul R. Young , Professor of Chemistry, University of Illinois at Chicago, Wiki: AskTheNerd; PRY askthenerd.com - pyoung uic.edu; ChemistryOnline.com
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Henry Agnew (UC Davis)


1.7E: Exercises for Cylindrical and Spherical Coordinates
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Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems.
For exercises 1 - 4, the cylindrical coordinates \( (r,θ,z)\) of a point are given. Find the rectangular coordinates \( (x,y,z)\) of the point.
1) \( \left(4,\frac{π}{6},3\right)\)
- Answer
- \( (2\sqrt{3},2,3)\)
2) \( \left(3,\frac{π}{3},5\right)\)
3) \( \left(4,\frac{7π}{6},3\right)\)
- Answer
- \( (−2\sqrt{3},−2,3)\)
4) \( (2,π,−4)\)
For exercises 5 - 8, the rectangular coordinates \( (x,y,z)\) of a point are given. Find the cylindrical coordinates \( (r,θ,z)\) of the point.
5) \( (1,\sqrt{3},2)\)
- Answer
- \( \left(2,\frac{π}{3},2\right)\)
6) \( (1,1,5)\)
7) \( (3,−3,7)\)
- Answer
- \( \left(3\sqrt{2},−\frac{π}{4},7\right)\)
8) \( (−2\sqrt{2},2\sqrt{2},4)\)
For exercises 9 - 16, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface.
9) [T] \( r=4\)
- Answer
-
A cylinder of equation \( x^2+y^2=16,\) with its center at the origin and rulings parallel to the \(z\)-axis,
10) [T] \( z=r^2\cos^2θ\)
11) [T] \( r^2\cos(2θ)+z^2+1=0\)
- Answer
-
Hyperboloid of two sheets of equation \( −x^2+y^2−z^2=1,\) with the \(y\)-axis as the axis of symmetry,
12) [T] \( r=3\sin θ\)
13) [T] \( r=2\cos θ\)
- Answer
-
Cylinder of equation \( x^2−2x+y^2=0,\) with a center at \( (1,0,0)\) and radius \( 1\), with rulings parallel to the \(z\)-axis,
14) [T] \( r^2+z^2=5\)
15) [T] \( r=2\sec θ\)
- Answer
-
Plane of equation \( x=2,\)
16) [T] \( r=3\csc θ\)
For exercises 17 - 22, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates.
17) \( z=3\)
- Answer
- \( z=3\)
18) \( x=6\)
19) \( x^2+y^2+z^2=9\)
- Answer
- \( r^2+z^2=9\)
20) \( y=2x^2\)
21) \( x^2+y^2−16x=0\)
- Answer
- \( r=16\cos θ,\quad r=0\)
22) \( x^2+y^2−3\sqrt{x^2+y^2}+2=0\)
For exercises 23 - 26, the spherical coordinates \( (ρ,θ,φ)\) of a point are given. Find the rectangular coordinates \( (x,y,z)\) of the point.
23) \( (3,0,π)\)
- Answer
- \( (0,0,−3)\)
24) \( \left(1,\frac{π}{6},\frac{π}{6}\right)\)
25) \( \left(12,−\frac{π}{4},\frac{π}{4}\right)\)
- Answer
- \( (6,−6,\sqrt{2})\)
26) \( \left(3,\frac{π}{4},\frac{π}{6}\right)\)
For exercises 27 - 30, the rectangular coordinates \( (x,y,z)\) of a point are given. Find the spherical coordinates \( (ρ,θ,φ)\) of the point. Express the measure of the angles in degrees rounded to the nearest integer.
27) \( (4,0,0)\)
- Answer
- \( (4,0,90°)\)
28) \( (−1,2,1)\)
29) \( (0,3,0)\)
- Answer
- \( (3,90°,90°)\)
30) \( (−2,2\sqrt{3},4)\)
For exercises 31 - 36, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface.
31) [T] \( ρ=3\)
- Answer
-
Sphere of equation \( x^2+y^2+z^2=9\) centered at the origin with radius \( 3\),
32) [T] \( φ=\frac{π}{3}\)
33) [T] \( ρ=2\cos φ\)
- Answer
-
Sphere of equation \( x^2+y^2+(z−1)^2=1\) centered at \( (0,0,1)\) with radius \( 1\),
34) [T] \( ρ=4\csc φ\)
35) [T] \( φ=\frac{π}{2}\)
- Answer
-
The \(xy\)-plane of equation \( z=0,\)
36) [T] \( ρ=6\csc φ\sec θ\)
For exercises 37 - 40, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in spherical coordinates. Identify the surface.
37) \( x^2+y^2−3z^2=0, \quad z≠0\)
- Answer
- \( φ=\frac{π}{3}\) or \( φ=\frac{2π}{3};\) Elliptic cone
38) \( x^2+y^2+z^2−4z=0\)
39) \( z=6\)
- Answer
- \( ρ\cos φ=6;\) Plane at \( z=6\)
40) \( x^2+y^2=9\)
For exercises 41 - 44, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle φ in radians rounded to four decimal places.
41) [T] \( \left(1,\frac{π}{4},3\right)\)
- Answer
- \( \left(\sqrt{10},\frac{π}{4},0.3218\right)\)
42) [T] \( (5,π,12)\)
43) \( \left(3,\frac{π}{2},3\right)\)
- Answer
- \( (3\sqrt{2},\frac{π}{2},\frac{π}{4})\)
44) \( \left(3,−\frac{π}{6},3\right)\)
For exercises 45 - 48, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.
45) \( \left(2,−\frac{π}{4},\frac{π}{2}\right)\)
- Answer
- \( \left(2,−\frac{π}{4},0\right)\)
46) \( \left(4,\frac{π}{4},\frac{π}{6}\right)\)
47) \( \left(8,\frac{π}{3},\frac{π}{2}\right)\)
- Answer
- \( \left(8,\frac{π}{3},0\right)\)
48) \( \left(9,−\frac{π}{6},\frac{π}{3}\right)\)
For exercises 49 - 52, find the most suitable system of coordinates to describe the solids.
49) The solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length \( a\), where \( a>0\)
- Answer
- Cartesian system, \( \big\{(x,y,z)\,|\,0≤x≤a,\;0≤y≤a,\;0≤z≤a\big\}\)
50) A spherical shell determined by the region between two concentric spheres centered at the origin, of radii of \( a\) and \( b\), respectively, where \( b>a>0\)
51) A solid inside sphere \( x^2+y^2+z^2=9\) and outside cylinder \( \left(x−\frac{3}{2}\right)^2+y^2=\frac{9}{4}\)
- Answer
- Cylindrical system, \( \big\{(r,θ,z)\,|\,r^2+z^2≤9,\;r≥3\cos θ,\;0≤θ≤2π\big\}\)
52) A cylindrical shell of height \( 10\) determined by the region between two cylinders with the same center, parallel rulings, and radii of \( 2\) and \( 5\), respectively
53) [T] Use a CAS or CalcPlot3D to graph in cylindrical coordinates the region between elliptic paraboloid \( z=x^2+y^2\) and cone \( x^2+y^2−z^2=0.\)
- Answer
-
The region is described by the set of points \( \big\{(r,θ,z)\,|\,0≤r≤1,\;0≤θ≤2π,\;r^2≤z≤r\big\}.\)
54) [T] Use a CAS or CalcPlot3D to graph in spherical coordinates the “ice cream-cone region” situated above the xy-plane between sphere \( x^2+y^2+z^2=4\) and elliptical cone \( x^2+y^2−z^2=0.\)
55) Washington, DC, is located at \( 39°\) N and \( 77°\) W (see the following figure). Assume the radius of Earth is \( 4000\) mi. Express the location of Washington, DC, in spherical coordinates.
- Answer
- \( (4000,−77°,51°)\)
56) San Francisco is located at \( 37.78°N\) and \( 122.42°W.\) Assume the radius of Earth is \( 4000\)mi. Express the location of San Francisco in spherical coordinates.
57) Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are \( (4000,−43.17°,102.91°).\)
- Answer
- \( 43.17°W, 22.91°S\)
58) Find the latitude and longitude of Berlin if its spherical coordinates are \( (4000,13.38°,37.48°).\)
59) [T] Consider the torus of equation \( \big(x^2+y^2+z^2+R^2−r^2\big)^2=4R^2(x^2+y^2),\) where \( R≥r>0.\)
a. Write the equation of the torus in spherical coordinates.
b. If \( R=r,\) the surface is called a horn torus. Show that the equation of a horn torus in spherical coordinates is \( ρ=2R\sin φ.\)
c. Use a CAS or CalcPlot3D to graph the horn torus with \( R=r=2\) in spherical coordinates.
- Answer
-
a. \(ρ=0, \quad ρ+R^2−r^2−2R\sin φ=0\)
c.
60) [T] The “bumpy sphere” with an equation in spherical coordinates is \( ρ=a+b\cos(mθ)\sin(nφ)\), with \( θ∈[0,2π]\) and \( φ∈[0,π]\), where \( a\) and \( b\) are positive numbers and \( m\) and \( n\) are positive integers, may be used in applied mathematics to model tumor growth.
a. Show that the “bumpy sphere” is contained inside a sphere of equation \( ρ=a+b.\) Find the values of \( θ\) and \( φ\) at which the two surfaces intersect.
b. Use a CAS or CalcPlot3D to graph the surface for \( a=14, b=2, m=4,\) and \( n=6\) along with sphere \( ρ=a+b.\)
c. Find the equation of the intersection curve of the surface at b. with the cone \( φ=\frac{π}{12}\). Graph the intersection curve in the plane of intersection.
Contributors
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org .
Links to CalcPlot3D added by Paul Seeburger (Monroe Community College).


3.2.1G: Nonmaterial Culture
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Non-material culture includes the behaviors, ideas, norms, values, and beliefs that contribute to a society’s overall culture.
Learning Objectives
- Analyze the different ways norms, values and beliefs interact to form non-material culture
Key Points
- In contrast to material culture, non-material culture does not include physical objects or artifacts.
- It includes things that have no existence in the physical world but exist entirely in the symbolic realm.
- Examples are concepts such as good and evil, mythical inventions such as gods and underworlds, and social constructs such as promises and football games.
- The concept of symbolic culture draws from semiotics and emphasizes the way in which distinctively human culture is mediated through signs and concepts.
- The symbolic aspect of distinctively human culture has been emphasized in anthropology by Emile Durkheim, Claude Lévi-Strauss, Clifford Geertz, and many others.
- Semiotics emphasises the way in which distinctively human culture is mediated through signs and concepts.
Key Terms
- social construct : Social constructs are generally understood to be the by-products of countless human choices rather than laws resulting from divine will or nature.
Culture as a general concept consists of both material and non-material culture. Material culture is a term developed in the late 19 th and early 20 th centuries, that refers to the relationship between artifacts and social relations. In contrast, non-material culture does not include physical objects or artifacts. Examples include any ideas, beliefs, values, or norms that shape a society.
When sociologists talk about norms, they are talking about what’s considered normal, appropriate, or ordinary for a particular group of people. Social norms are group-held beliefs about how members should behave in a given context. Sociologists describe norms as laws that govern society’s behaviors. Values are related to the norms of a culture, but they are more global and abstract than norms. Norms are rules for behavior in specific situations, while values identify what should be judged as good or evil. Flying the national flag on a holiday is a norm , but it exhibits patriotism, which is a value . Wearing dark clothing and appearing solemn are normative behaviors at a funeral. In certain cultures they reflect the values of respect and support of friends and family. Different cultures honor different values. Finally, beliefs are the way people think the universe operates. Beliefs can be religious or secular, and they can refer to any aspect of life. For instance, many people in the U.S. believe that hard work is the key to success.
Members take part in a culture even if each member’s personal values do not entirely agree with some of the normative values sanctioned in the culture. This reflects an individual’s ability to synthesize and extract aspects valuable to them from the multiple subcultures they belong to.
Norms, values, and beliefs are all deeply interconnected. Together, they provide a way to understand culture.
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Unit 2: Realism
Unit 2: Realism
The dates for Realism as a movement vary, from as early as 1820 to as late as 1920. Although Realism is, in many ways, a rejection of Romanticism, it does address some of the same concerns about the industrial revolution that Wordsworth had expressed earlier. The passage of years increased the number of authors who noted the failures of industrialization, especially where pollution and quality of life were concerned. The British period of Victorianism (1837-1901) saw a gradual shifting from Romanticism to Realism. Poets such as Tennyson and Robert Browning are more properly transitional poets: products of Romanticism, but who express themselves in more realistic terms. It is important to remember not only that literary movements are not set in stone, but also that they are not always identified the same way by their own time period. When Charles Baudelaire wrote his seminal work,
The Flowers of Evil
(1857), he was praised as a poet of Romanticism by Gustave Flaubert, even though most modern scholars locate Baudelaire in Realism, and later poets of Modernism cite him as an early example of their own movement.
Romanticism was slowly but surely replaced with an attempt to see the world as it is. As later generations would note, it is difficult to represent reality in its entirety in one poem, play, short story, or novel. Early Realists tended to include more portrayals of middle class and/or lower class characters, who previously were not the main subjects of literature. In Europe, writers such as Ibsen wrote about the middle class specifically, using ordinary occurrences (at least, ordinary for the middle class experience) as the stuff of drama. Authors such as Henry James occasionally were criticized for novels in which very little seems to happen, since ordinary events are rarely as dramatic as the situations regularly found in Romanticism. In some cases, the attempt to be more realistic led to many works that focused on the negative aspects of humans, leaving out the positive aspects to avoid Romantic overtones.
Instead of Romanticism's belief in the power of emotion and intuition to achieve insight, Realism addressed more concrete issues, usually with direct and straightforward language. In American Realism, Mark Twain's use of dialect falls into this category. Twain's
Adventures of Huckleberry Finn
(1884) is realistic in its use of common, everyday speech, which includes dialect and slang. The novel also offers a realistic portrayal of the complicated friendship that slowly develops between Huck, a poor white boy, and Jim, a runaway slave.
Drawing by E.W. Kemble of Huckleberry Finn with a rabbit and a gun, from the original 1884 edition of the book. License: Public Domain.
Another way that Realism distinguished itself from the previous movement was in its examination of human psychology. Robert Browning looked at the psychology of narcissists, murderers, and psychopaths in many of his dramatic monologues. Fyodor Dostoyevsky's novels take place as much in the minds of the characters as they do in the interactions among characters. In
Notes from Underground
(1864), Dostoyevsky uses Realism to explore the allure (and dangers) of Romanticism, including its continuing hold on audiences. Dostoyevsky's "Underground Man" attempts to explain not only why he does what he does, but claims that the reader is no different—a tactic used by Baudelaire in his poem "To The Reader" at the beginning of The Flowers of Evil.
The Russian writer Fyodor Dostoevsky in 1863. Black and white photograph. License: Public Domain
As a general guideline, Realism tended to point out society's problems (and the problems with the Romantic view), but offered observations, rather than suggested changes. Naturalism, a subset of Realism often treated as a separate movement, was regularly motivated by a desire to improve the world. Naturalism concerned itself with the poorest members of society in particular, and social change was the goal. Naturalism was criticized for being even more focused on the negative aspects of life than regular Realism. Emile Zola's novel
Germinal
(1885) is perhaps the most famous example of Naturalism. In it, Zola depicts the lead-up to and aftermath of a coal miners' strike with a stark realism that shocked readers. His unsentimental portrayal (in almost journalistic fashion) of events angered both conservatives (reluctant to admit the brutal working and living conditions of the poor) and socialists (unhappy that the workers were not Romantic heroes). Eventually, Modernism began in literature as Realism and Naturalism were ending, overlapping for a brief period of time. Perhaps not surprisingly, Modernism would claim to be more real than Realism—or, as the artist Georgia O'Keeffe said, "Nothing is less real than realism" (Haber), preferring abstract art as a way to arrive at a more complete image of (one type of) truth.
Written by Laura Getty


Note
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Comprehend examples of Secondary English assessments
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Understand the assignments used in each type of assessment described
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Understand different ways to assess a student in Secondary English
Introduction
Secondary English is definitely not one of the easier subjects to assess. There could be several different answers for one assignment and it is up to the teacher to decipher those answers according to his or her rubric. History, science and math are easier in the sense that there is always one answer. English, especially secondary, allows for multiple answers, all differing. Although memorization and comprehension is used in all subjects, it lends itself more to the sciences and history compared to secondary English. Reading comprehension, interpretation and writing are somewhat harder to assess because students create the pieces of work and that work is graded via a general rubric the teacher has created.
What are assessments and how are they utilized?
There are three common assessments used by educators in the classroom; summative, formative and performance. Summative assessments are usually given periodically throughout a school year to determine at a particular point in time what students know and do not know. A lot of students and educators reference standardized testing, end-of-unit tests and end-of-chapter tests with summative assessments. The teacher is informed of the students' progress and learning at the end of a lesson plan with the summative assessment. They help evaluate the effectiveness of programs, alignment of curriculum, or student placement in specific programs. They are too far down the road to be used in classroom learning adjustment. Summative assessments are typically used after the classroom has taken a number of formative assessments.
Formative assessments inform both teachers and students about student understanding at a point when timely adjustments can be made, unlike summative assessments. They are typically associated with practice tests, quizzes and group work. They require a lot of student-teacher communication and interaction to ensure the student is learning and understanding the curriculum before they take the summative assessment. According to Robert Stake, "When the cook tastes the soup, that’s formative; when the guests taste the soup, that's summative," (Earl, Lorna, 2003).
The third most common assessment used in education today is called the performance assessment. Teachers use this type of assessment to discover how a student, "understands and applies knowledge," (Coyle, G., 2009). "They can be used to evaluate reasoning, products, and skills that can be observed and judged using specific criteria. Tasks that have more than one acceptable solution often lend themselves well to a performance based assessment," (Coyle, G., 2009). Performance based assessments are being utilized more and more with students who show ineffective progression on standardized testing. A student who doesn't perform well on the SAT and ACT, isn't necessarily behind in education; they just might not test well. "Many states are incorporating performance-based assessments into their standardized tests or adding assessment vehicles such as student portfolios and presentations as additional measures of student understanding. These rigorous, multiple forms of assessment require students to apply what they're learning to real world tasks," (How Should We Measure Student Learning?: The Many Forms of Assessment, 2008).
Listed below are other well-known assessments used in education today; the objective assessment, subjective assessment, self assessment and interactive assessment.
 These are good for testing recall of facts and can be automated.
-
Subjective assessments:
The teacher's judgment determines the grade; essay tests. Essay tests take longer to answer, longer to grade and therefore only include a small number of questions, focusing on complex concepts.
-
Self assessments:
Assignments that provide for quick student feedback. They help the learner check if they have mastered a topic, provide opportunity to measure learning progress, are usually voluntary and may allow multiple attempts, inform the learner, but not the teacher.
-
Interactive assessments:
Designed to allow learner's to perform a task. Can be graded or not. Examples: students could conduct a virtual experiment rather than an experiment in a physical laboratory, language software might have sophisticated speech recognition software to provide feedback about pronunciation, creation of an online tool, such as a virtual instrument. Students could perform a song, and the software provides feedback about accuracy and timing."
-
(MSU, Virtual University Design and Technology, 2007).
Note
•
Formative Assessment examples:
• Anecdotal records
• Quizzes & essays
• Diagnostic tests
• Lab reports
•
Summative Assessment examples:
• Final Exams
• Statewide exams (SOL)
• National exams
• Entrance Exams (SAT, ACT, GRE)
•
Performance Assessment examples:
• Conducting and creating a science experiment
• Giving a speech
• Writing a Research paper
• Operating machinery or equipment, i.e., photography or a computer program
(Classroom Assessment: Basic Concepts, 2009)
Ways to assess Secondary English
English teachers are advised to always give encouraging feedback along with their suggestions. Handing back a student's work with red marks scribbled all over their assignment isn't encouraging for future assignments. "Informative feedback should be provided rather than just a mark or grade. In addition to identifying areas where improvement should be made, teachers can write constructive and encouraging comments, as these motivate students to do well," (Curriculum Development Council and the Hong Kong Examinations and Assessment Authority, 2007). There is no one "universal" way to grade an English assignment, each assignment is different.
Different works equals different grading:
-
-
-
-
-
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"When assessing an oral presentation, the emphasis can be put primarily on content and fluency.
-
When going through book reports, the focus can be upon learners’ ideas and personal responses.
-
When marking compositions, it is advisable to provide learners with comprehensive feedback on content, accuracy, appropriateness, presentation and organization."
-
(Curriculum Development Council and the Hong Kong Examinations and Assessment Authority, 2007).
Teaching secondary English is an everyday challenge. Educators are constantly creating new ways to assess the children in their classes. Why? Because they want to involve their students and ensure that everyone is learning. Students in high school today are children of the Digital Age; they expect everything fast and are very hands-on learners. In the article, "Beyond Reading Check Quizzes," by Diane Vanaskie Mulligan, a veteran high school English teacher in Massachusetts, we are given several examples of creatively assessing a high school English class. Mulligan says, "By shifting the focus from making students prove they've done their homework to helping them engage with texts, teachers can set students up for success," (2008). Participation in structured discussions, one paragraph responses and "the old-fashioned" reading journal are just a couple of ways to heighten student interaction and grades without quizzes and tests, (Mulligan, 2008). Not only does this aid the teacher in the amount of work they are required to grade, it also gives the students the ability to learn creatively.
Another great way to assess a student in secondary English is through student portfolios and their development. A portfolio can be started at a young age; either in middle school and carried through high school or as a freshmen in high school. A portfolio can show a teacher and a student the progression and learning curve over a number of years. It's good for either short or long term use in the classroom and should include all written work, written assessments and published work, (Streich, 2008). "Properly maintained portfolios contain student work that assists curricular evaluation, demonstrates individual student progress, and can be used in student placement," (Streich, 2008). Students can also use their portfolios to gain admittance into advanced placement classes in college along with entrance into that university.
Check this site out! It lists and describes the Stands of Learning requirements according to the Department of Education for Secondary English, grades 9-12. [1]
Secondary English Assessment vs Math & Science Assessment
Assessing a student in English class is much different than assessing a student in math or science. The biggest difference between English, math and science assessments is the ability to list one or more answers. In English you are expected to use your mind creatively. Students are expected to overly develop the plotline in an essay and are authorized to act dramatic while reading a Shakespeare play outloud in class. In math and science there is almost always one single answer, never any grey area. Unlike math and science, English allows students to cross into a lot of grey area depending on the type of assignment. Another key point that differentiates the two areas is "do-over's." In a typical English class, students are allowed to turn in rough drafts and outlines of projects or assignments before receiving a final grade. In science and math, students are taught the lesson, they may have one or two homework or class work assignments and then they are tested. There are no "do-over" chances.
Secondary English Assignments vs. Math & Science Assignments
Note
Different Types of Assignments
English: Reading & Writing
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-
-
-
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Essays
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Vocabulary work either individual or in pairs
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Creative writing
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Poetry or writing portfolios
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Collaborative research papers
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Grammar worksheets
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Collaborative book reports
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Group discussions about literary topics (4 students to a group)
Math & Science
-
-
-
-
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Data Analysis
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Algebra
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Geometry,
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Chemistry, periodic table of elements
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Word problems
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Factors, sums, multiples
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Linear equations
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Statistics
-
Lab reports
Conclusion
No matter the subject you are assessing, assessment is always hard. Your job as a teacher is to honestly assess each student in your classroom. Teachers must have the ability to differentiate between their students and the types of assessments that work for them and that particular assignment. Assessing secondary English is definitely a challenge, but so is math and science assessment. They differ in the types of assessments used and how a teacher assess the assignment.
Exercise \(\PageIndex{1}\)
1. According to the article, what is an example of a secondary English assessment assignment?
-
-
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a. Oral Presentation
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b. Creating a science experiment
-
c. Research paper
-
d. Both A & B
2. Mrs. Martin, a 12th grade English teacher, has assigned a written response essay to her students. The topic of choice is up to them as long as it discusses a topic regarding William Shakespeare's
Hamlet
, which they just, finished reading. What kind of assessment is this?
-
-
-
a. Objective
-
b. Self Assessment
-
c. Subjective
-
d. Interactive
3. What can a student place in a portfolio?
-
-
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a. Anything, as long as it's their personal work.
-
b. Only graded, approved, written assignments
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c. Items from their senior year of high school
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d. Tests, quizzes, essays and papers
4. Mr. Ray, an 11th grade English teacher, has placed his students in groups and assigned each a number of essays, articles and poems that need to be laid out and arranged for the junior class newspaper. He has assigned his students the task of design, lay-out and organization via the computer program, Adobe InDesign. They have until the end of their two-hour block to complete this. What kind of assessment is this?
-
-
-
a. Formative Assessment
-
b. Self-Assessment
-
c. Summative Assessment
-
d. Interactive Assessment
-
Answer
-
1. d
2. c
3. a
4. d
References
Curriculum Development Council and the Hong Kong Examinations and Assessment Authority, (2007). English Language Education Key Learning Area. pgs 117,118. www.edb.gov.hk/FileManager/EN/Content_5999/eng_lang_final.pdf
MSU, Virtual University Design and Technology, (2007). Teaching and Learning: Types of Assessments. vudat.msu.edu/assess_types/
Classroom Assessment: Basic Concepts.
http://fcit.usf.edu/assessment/basic/basica.html
Earl, Lorna, (2003). Assessment as Learning: Using Classroom Assessment to Maximise Student Learning. "Thousand Oaks, CA, Corwin Press".
http://www.wyoaac.org/Lit/assessment%20for%20learning%20of%20learning%20as%20learning%20-%20Earl.pdf
Coyle, G. (2005). Performance based assessment. B. Hoffman (Ed.), Encyclopedia of Educational Technology. coe.sdsu.edu/eet/articles/pbassess/start.htm
Edutopia Staff, (2008). How Should We Measure Student Learning?: The Many Forms of Assessment. "Edutopia: What works in public education, The George Lucas Educational Foundation".
http://www.edutopia.org/assessment-introduction
Streich, Michael, (2008). Portfolios Gauge Progress & Growth of Students, Short and Long Term Benefits of a Lifetime Portfolio Program in HS.
Mulligan, Diane Vanaskie, (2008). Beyond Reading Check Quizzes, Assessment in the High School English Classroom.


2.8: Summary
This chapter has outlined the different components of health assessment. Different assessments are done in different contexts and are dependent on the type of patient, health care professional, and environment.
The health assessment is an opportunity to develop a therapeutic relationship with the patient, optimize communication, promote health, and provide patient education as necessary. Throughout the assessment, be aware of ensuring patient safety, privacy, and dignity.
The assessment must always be documented according to the agency policy, and any unusual findings must be reported comprehensively, in conjunction with other pertinent findings, to appropriate members of the health care team.
Key Takeaways
- Health assessment refers to a systematic method of collecting and analyzing data for the purpose of planning patient-centred care.
- A pain assessment should be measured at the beginning of the physical assessment, and comfort measures taken as necessary.
- Safety considerations should be followed throughout any physical assessment.
- Components of health assessment include conducting a health history, performing a physical examination, and communicating and documenting the findings according to agency policy.
- The amount of information gained during a health assessment depends on several factors, including the context of care, patient needs, and the health care professional.
- The types of health assessments are head-to-toe, focused, initial, and emergency assessment.
- The data collected during the health assessment is organized and interpreted to initiate or continue a plan of care.
Suggested Online Resources
1. Auscultation Assistant, The. This website provides audio clips of heart murmurs and lung sounds.
2. BC Patient Safety & Quality Council: 48/6 Model of care. This resource offers a model of care for hospitalized seniors (aged 70 and older) in British Columbia. It is an integrated care initiative that addresses six care areas of functioning through patient screening and assessment (assessments are completed only where screening shows areas of concern) within the first 48 hours of hospital admission.
3. Canadian Patient Safety Institute (CPSI): The Canadian framework for teamwork and communication . This framework provides health care providers with techniques to improve teamwork and collaboration.
4. Canadian Patient Safety Institute (CPSI): The safety competencies. Developed by the Safety Competencies Steering Committee of CPSI, this interprofessional patient safety framework identifies the knowledge, skills, and attitudes required by all health care professionals to practise safely.
References
Anderson, B., Nix, N., Norman, B., & McPike, H. D. (2014). An evidence based approach to undergraduate physical assessment practicum course development. Nurse Education in Practice , 14(3), 242-246.
Assessment Skill Checklists. (2014). Checklists from Nursing 1019: Clinical Techniques (BCIT BSN program course).
Jarvis, C., Browne, A. J., MacDonald-Jenkins, J., & Luctkar-Flude, M. (2014). Physical examination and health assessment . Toronto, ON: Elsevier-Saunders.
Perry, A., Potter, P., & Ostendorf, W. (2014). Clinical skills and nursing techniques (8th ed.). St Louis, MO: Elsevier-Mosby.
Rosdahl, C. B., & Kowalski, M. T. (2007). Textbook of basic nursing . Philadelphia, PA: Lippincott Williams & Wilkins.
Stephen, T. C., Skillen, D. L., Day, R. A., & Jensen, S. (2012). Canadian Jensen’s nursing health assessment: A best practice approach . Philadelphia, PA: Wolters Kluwer-Lippincott.
Wilson, S. F., & Giddens, J. F. (2013) Health assessment for nursing practice ( 5th ed.) St Louis, MO: Mosby.


13.6: Disabilities
- Discuss how various physical and mental disabilities might affect communication
Impossible is an opinion—not a fact.
—Paul Hockey
General perception and understanding of those who are different is not unlike ancient cultures’ understanding of the world: flawed (i.e., the belief the world was flat) and with large areas marked “the great unknown.” This is particularly the case when it comes to people with disabilities. This lack of understanding is due to a combination of factors including a lack of exposure to people with disabilities, the amorphous definition of disability, and privacy and discrimination concerns. People with disabilities are under-represented in media and entertainment—a situation that the Alliance for Inclusion in the Arts is seeking to address. The ADA (Americans with Disabilities Act) text defines an individual with a disability as a person who has a physical or mental impairment that substantially limits one or more major life activities, a person who has a history or record of such an impairment, or a person who is perceived by others as having such an impairment. Further, people with disabilities are not required to disclose their disability, and indeed, they are often advised not to—at least in the job search process. [1]
So how do we approach the great unknown? Let’s start by considering our perceptions of people with disabilities. In the award-winning short Different , by Tahneek Rahman , we see two young people navigating a new relationship. The actors’ perceptions and emotions are shaped and colored by what they think they see; then, like a kaleidoscope, reality changes. As you watched the film, how did you feel when you looked through the young girl’s eyes as she peered through the bushes?  How often do we do this with people who are wired differently or have a different range of abilities?
It’s important to note that not all disabilities are visible or accompanied by obvious cognitive or processing challenges. It’s possible to work with someone for a long time before you learn that they have a disability. Some disabilities can be invisible or usually unobtrusive (MS, for example, which can lay dormant and then flare up). Additionally, some mental illnesses (like ADHD or OCD) can impact a person’s work, but are completely invisible.
Empathy goes a long way in bridging knowledge and communications gaps. For a start, watch one or more of the videos in Soul Pancake’s How You See Me series .
Perhaps our perception of ability—and disability—says more about us then it does about others. So before we discuss communication specifics, let’s broaden our perspective of what’s possible. Erik Weihenmayer is one of seven disabled athletes to have successfully climbed Mount Everest, the highest mountain in the world at 29,029 feet above sea level. The only blind person to summit Everest to date, Erik went on to complete the “Seven Summit” challenge, scaling the highest peaks of each of the seven continents. Erik subsequently co-founded No Barriers, a non-profit organization that helps youth, veterans, and people with disabilities achieve transformative challenges. The organization’s motto: Unleash the Human Spirit.
Indian amputee Arunima Sinha is another of the seven disabled Everest summiters. In the aftermath of a train accident that severed her leg, Arunima “pondered on the most impossible dream I could set for myself. I decided to climb Everest.” Erik and Arunima are remarkable athletes and people, but there are thousands of similar stories—people who, by birth or circumstance, found themselves at a relative disadvantage and yet prevailed and indeed, thrived. Clearly, they did not see their disability as a limitation. Tapping this human spirit is critical to business success in a competitive global economy.
Overcoming Communication Challenges
People with disabilities can experience unique communication challenges whether they have sensory impairments (blindness or deafness), cognitive disorders (autism spectrum disorder, post-stroke challenges), or physical disabilities (head trauma or neurological injury). In particular, some communication difficulties in the workplace can include the following:
- Difficulty speaking: speech may be unclear, interrupted by stuttering, or abnormally slow, fast or irregularly paced
- Difficulty with listening for extended periods or listening to multiple people participating in a conversation
- Difficulty reading manuals with dense amounts of text
- Difficulty keeping track of procedural material without the help of notes or hands-on experience
- Difficulty interpreting language that has implied meaning such as indirect requests or offers for help, or certain types of humor
- Difficulty interpreting body language, the emotions of others, or other non-verbal language
- Difficulty communicating with unfamiliar people; this can include eye contact
Whether individuals have disclosed a disability or not, the way you approach a communication breakdown or misunderstanding matters. If you do not understand something a person says, do not pretend that you do. Ask the individual to repeat what he or she said and then repeat it back. Try to ask questions that require only short answers or a nod of the head. Concentrate on what the person is saying and do not rush to a conclusion about what you think they mean. Do not speak for the individual or attempt to finish her or his sentences. If you are having difficulty understanding the individual, consider writing as an alternative means of communicating, but first ask whether this is acceptable.
Other things to consider are:
- If you are in a public area with many distractions, consider moving to a quiet or private location.
- Be prepared to repeat what you say, orally or in writing.
- Offer assistance completing forms or understanding written instructions and provide extra time for decision-making. Wait for the individual to accept the offer of assistance; do not “over-assist” or be patronizing.
- Be patient, flexible and supportive. Take time to understand the individual and make sure the individual understands you.
If you notice a communication breakdown or misunderstanding, it is of utmost importance to treat everyone with dignity, respect, and courtesy. Be patient, be supportive, and take as much time as necessary to listen to the individual because it can make all of the difference.
Developing an Accessible Workplace
The Society for Human Resource Management’s (SHRM) Developing an Accessible Workplace toolkit is an excellent resource, addressing the business case, physical accessibility and information accessibility. Technology accommodations might include use of large display screens, screen readers, and/or voice recognition software. In order to communicate effectively across a range of abilities, businesses also need to design materials—from onboarding and ongoing communications to training and development—with accessibility in mind. Rather than “retrofitting” materials and programs to accommodate a person’s particular disability, a best practice is to use a Universal Design for Learning (UDL) approach. Briefly, UDL is a research-based educational framework that guides the development of flexible learning environments that can accommodate individual learning differences.” [2] The concept and language of UDL was inspired by the universal design movement, proposing that “products and environments be designed to be usable by all people, to the greatest extent possible, without the need for adaptation or specialized design.”
As with many conceptual frameworks, there are different approaches to achieving UDL. The following 7 core principles provide the framework for an online educational certification program at Saddleback College:
- Principle 1: Inclusive & Fair
- Principle 2: Straight-Forward & Clear
- Principle 3: Flexible & Fair
- Principle 4: Explicitly Presented
- Principle 5: Supportive
- Principle 6: Minimize Effort
- Principle 7: Appropriate Learning Space
Delving into UDL is beyond the scope of this section and course. However, even at the summary level, these principles clearly contribute to a fair, inclusive and effective workplace. The upside is that applying the design principles also makes information more accessible to people for whom English is a second language. The clarity that the design principles require also contributes to clear communication across other diversity dimensions—for example, cultures and ethnicities.
There are also simple, practical adjustments we can make in our one-on-one interactions that will facilitate effective communication. The following eight recommendations, adapted from a toolkit for medical practitioners, are equally relevant to communicating with people with disabilities in the workplace (Heath Care for Adults with Intellectual and Developmental Disabilities):
- Allocate additional time to achieve the communication objective.
- Be aware of your tone of voice and nonverbal signals.
- Moderate your speaking pace and give the person with a disability adequate time to process and respond to what you’ve communicated.
- Actively confirm the person’s understanding and your own understanding of what he or she communicated.
- Focus on abilities rather than disabilities.
- Use specific rather than abstract language; for example, “bring a pen and paper” rather than “get ready for the meeting.”
- Stage conversations in areas that are relatively quiet without distracting activity or background noise.
Just as our history is not our destiny, our frame of reference doesn’t need to limit our future possibilities—individually or collectively, as a business or society. Disability rights are not only civil rights, they’re human rights—the right to strive to achieve our full potential, whatever that is. As one of the testimonials on Nike’s Equality Campaign page phrased it: “we all deserve a starting line.”
- The Language of Disability: Do’s & Don’ts – Alliance for Inclusion in the Arts
- A Guide to Disability Rights Laws
- Nike’s Equality Campaign page
- Communicating With and About People with Disabilities
- Isaacs, Kim.  Monster. Accessed August 5, 2019. Disclosing a disability: www.monster.com/career-advice/article/disclose-disability-on-resume
- Rose, David H., Anne Meyer, Nicole Strangman, and Gabrielle Rappolt. Teaching Every Student in the Digital Age: Universal Design for Learning. Association for Supervision & Curriculum Development, 2002. www.ascd.org/Publications/Books/Overview/Teaching-Every-Student-in-the-Digital-Age.aspx
Contributors and Attributions
- Disabilities. Authored by : Nina Burokas. Provided by : Lumen Learning. License : CC BY: Attribution
- Overcoming Communication Challenges. Provided by : Lumen Learning. License : CC BY: Attribution
- accessibility. Authored by : renma . Provided by : Pixabay. Located at : pixabay.com/en/accessibility-disability-1682903/. License : CC0: No Rights Reserved
- Communicating With and About People with Disabilities. Provided by : United State Department of Labor. Located at : www.dol.gov/odep/pubs/fact/communicating.htm. License : Public Domain: No Known Copyright
- Mount Everest as seen from Drukair. Authored by : shrimpo1967; derivative work by Papa Lima Whiskey 2. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/File:Mount_Everest_as_seen_from_Drukair2_PLW_edit.jpg . License : CC BY-SA: Attribution-ShareAlike


1.4: The Role of the Viewer
An artist or craftsperson has an audience in mind when creating a work of art. Sometimes the audience is the artist. Most of the time, however, the audience—the viewer—is someone else. It may be an individual or a group of people the artist personally knows, or people the artist knows will be viewing the work in a specific context or with a certain purpose. The artist may also consider what meaning or impact the work of art will have for people who view it at an unknown time or place in the future, perhaps with little information about the artist or the work itself. Or, the artist may feel the need or desire to express an emotion and have no concern for how the viewer will react to the work, or even if the viewer will understand the work and why it was created.
As the viewer of a work of art, then, we are often aware that we do not have full knowledge of what the artist intended or, at times, even what the artist depicted. Not having that information, however, is not necessarily frustrating nor does it dampen our enjoyment of the piece. Instead, we may find the colors vibrant, or the subject intriguing, or the composition relaxing; in other words, we may simply enjoy looking at the work of art without feeling the need for particulars about it or the artist. But, there are other times when it is helpful to have some information about the artist or artwork for us to better understand and appreciate what we are looking at.
Sites exist around the world where images were painted or inscribed on cave walls during the Upper Paleolithic Period, c. 40,000-12,000 BCE. The majority of the images are of animals, but outlines of hands, human figures, instruments such as bows and arrows, and designs such as spoked wheels or parallel lines can also be found. They possess a number of notable features, including the fact that these images were painted over tens of thousands of years on every continent except Antarctica. Despite significant differences, the types of subjects depicted during all that time and in all those places are remarkably similar. But, as they were made during the pre-historic period, that is, before humans kept written records, all we know about them is what we can interpret by looking at the images themselves and by studying other objects we have found from the same places and time periods.
Scholars have put forth numerous ideas about why the images were made and what they could mean. The animals depicted include horses, bulls, bison, and deer, all of which were hunted during that span of approximately 30,000 years. For that reason, some scholars hypothesize the paintings acted as a form of sympathetic magic , expressing the hope or giving thanks for a successful hunt by depicting the animals hunted. If the images were associated with such activities, crucial for the survival of those who created them, then their makers, as scholars further speculate, were shamans , or spiritual leaders of the group. A shaman is an individual with the power to interact with the physical world and the otherworld of spirits in order to maintain harmony between the two, predict the future, cast spells, and cure the sick. Venturing into a cave, where all light from the outside world quickly disappears, is akin to a journey into another realm of existence. The images painted, seen only by fire, would have flickered and danced on the walls as if they depicted visitors from another world. We do not know who saw the paintings other than those who created them, but in the Panel of Spotted Horses within the Chapel of Bison in the Pech-Merle de Cabrerets Cave, France, the handprints also present are evidence that there were others who viewed them. (Figure 1.15) The prints were made by placing a hand on the wall and blowing paint around it, perhaps through a hollow, reed-like object. Are they meant to identify or document those who were present, to indicate their hoped-for powers as hunters or their inclusion as part of a shamanistic experience? We do not know, but even with the little information we have as viewers today, we can nevertheless enjoy the painting’s beauty and mystery.
Figure 1.15 Replica of the Pech-Merle de Cabrerets Cave painting, (public domain, "HTO")
A labyrinth , or maze, such as the one in the floor of the nave of Chartres Cathedral (1194- 1250), France, is another example of an image or object found in a number of places, but about which we have little information. (Figure 1.16) A labyrinth is similar to a maze but generally has only one intricate and twisting path to the center. (Figure 1.17) There are labyrinths in the floors of numerous medieval Gothic cathedrals in Europe that were built in the twelfth to fifteenth centuries. The labyrinth at Chartres Cathedral was built in the thirteenth century and, at 42.3 feet in diameter, it fills the width of the nave , or central area of a church. While there is documentation that clergy performed dances during Easter celebrations upon labyrinths found in other cathedrals in France, no such records exist regarding Chartres. What it does seem to have in common with other labyrinths, however, is being used as a path to circumambulate , or walk, by visitors to the church who were on a pilgrimage or journey of faith. As was true of many Gothic churches, Chartres Cathedral held a relic , an object thought to have belonged to or been part of a holy person’s body, in this case, a garment believed to be the tunic worn by the Virgin Mary when she gave birth. Pilgrims traveled to Chartres to venerate this relic as a demonstration of their religious devotion. While there, pilgrims and other visitors might follow the stones of the labyrinth while in prayer or a state of meditation; the inevitable outcome of the complex and turning path leading to the center mirrors the certainty that prayer will lead the believer to God. The repetitive and focused movement of walking while absorbed in prayer enhanced the devotional experience for the worshiper—who was also the viewer of the labyrinth—on both a physical and a spiritual level.
Figure 1.16 Labyrinth at Chartres Cathedral, (CC BY-SA 3.0, "Maksim")
Figure 1.17 Diagram of the Labyrinth of Chartres Cathedral, (CC BY-SA 3.0, "Ssolbergj”)
John Haberle (1856-1933, USA) was a painter who was born and spent most of his life in New Haven, Connecticut. He was well known for his trompe l’oeil works such as A Bachelor’s Drawer : paintings that were so realistic they “fooled the eye.” (Figure 1.18) Precisely rendering objects on a two-dimensional surface as if they were in three-dimensional space, he was able to create an illusion of reality that was meant to draw in his viewers, who were briefly unaware of the trick he was playing upon them. Quickly recognizing the painting was in truth an uncannily accurate semblance of actual objects, the viewer then became a participant in the artist’s game of deception.
Figure 1.18 A Bachelor’s Drawer, Artist: John Haberle, Author: Met Museum, (OASC)
The various objects in A Bachelor’s Drawer , including photographs, paper currency, theater ticket stubs, newspaper clippings, a thermometer, and a hair comb, that appear to be haphazardly fixed to a wooden drawer front are visually interesting because they are so life-like. Once the viewer shifts focus to look at these everyday and commonplace items—the sorts of things you take out of your pocket at the end of the day, often intending to throw them away—and think about what they are, we also wonder what they might mean. And, that is exactly what Haberle intended his viewers to do.
The artist even rewards his viewers for their close attention to the many details in his painting by placing some important ones in the center: several fragments of newspaper articles, including one stating, “A New Haven artist has plunged himself into trouble by making too perfect greenbacks in oil.” Viewers who knew Haberle’s work would probably have been aware the statement was true. Haberle frequently depicted paper currency in his paintings, in spite of having been warned to stop doing so by the U.S. Secret Service, which was formed in 1865 to stop the distribution of counterfeit money. Those who appreciated his work knew Haberle took pleasure in making it clear he was ignoring that demand.
A Bachelor’s Drawer , painted 1890-1894, would turn out to be the artist’s last trompe l’oeil painting of currency, though, as the exacting work had strained his eyes to the point that he could no longer paint such fine detail. Some of the other objects Haberle included, and the title of the work itself, seem to be referring to the end of an era. The pamphlet titled “How to Name the Baby,” prominently displayed in the upper right, partially covers the postcard showing a finely-dressed dandy with his dashing moustache that is placed directly above a discreetly covered photograph of a nude woman. They all lead down to a small photograph that appears to be stuck in the bottom (painted) frame, which is a portrait of the artist. Was he the bachelor who once had the freedom to attend the theatre, but is now taking up the life of a young father? This trail of clues is typical of the dry humor in Haberle’s work, here turned on himself, with an open invitation for his viewers to share the joke with him.