17.4: Inequalities
Basic Theory
In this section, we will study a number of interesting inequalities associated with martingales and their sub-martingale and super-martingale cousins. These turn out to be very important for both theoretical reasons and for applications. You many need to review infimums and supremums.
Basic Assumptions
As in the Introduction, we start with a stochastic process \( \bs{X} = \{X_t: t \in T\} \) on an underlying probability space \( (\Omega, \mathscr{F}, \P) \), having state space \( \R \), and where the index set \( T \) (representing time) is either \( \N \) (discrete time) or \( [0, \infty) \) (continuous time). Next, we have a filtration \(\mathfrak{F} = \{\mathscr{F}_t: t \in T\} \), and we assume that \( \bs{X} \) is adapted to \( \mathfrak{F} \). So \( \mathfrak{F} \) is an increasing family of sub \( \sigma \)-algebras of \( \mathscr{F} \) and \( X_t \) is measurable with respect to \( \mathscr{F}_t \) for \( t \in T \). We think of \( \mathscr{F}_t \) as the collection of events up to time \( t \in T \). We assume that \( \E\left(\left|X_t\right|\right) \lt \infty \), so that the mean of \( X_t \) exists as a real number, for each \( t \in T \). Finally, in continuous time where \( T = [0, \infty) \), we make the standard assumptions that \( t \mapsto X_t \) is right continuous and has left limits, and that the filtration \( \mathfrak F \) is right continuous and complete.
Maximal Inequalites
For motivation, let's review a modified version of Markov's inequality, named for Andrei Markov.
If \( X \) is a real-valued random variable then \[ \P(X \ge x) \le \frac{1}{x} \E(X ; X \ge x), \quad x \in (0, \infty) \]
Proof
The modified version has essentially the same elegant proof as the original. Clearly \[ x \bs{1}(X \ge x) \le X \bs{1}(X \ge x), \quad x \in (0, \infty) \] Taking expected values through the inequality gives \( x \P(X \ge x) \le \E(X; X \ge x) \). Dividing both sides by \( x \) gives the result (and it is at this point that we need \( x \gt 0. \)).
So Markov's inequality gives an upper bound on the probability that \( X \) exceeds a given positive value \( x \), in terms of a monent of \( X \). Now let's return to our stochastic process \( \bs X = \{X_t: t \in T\} \). To simplify the notation, let \( T_t = \{s \in T: s \le t\} \) for \( t \in T \). Here is the main definition:
For the process \( \bs X \), define the corresponding maximal process \( \bs U = \{U_t: t \in T\} \) by \[ U_t = \sup\{X_s: s \in T_t\}, \quad t \in T \]
Clearly, the maximal process is increasing, so that if \( s, \, t \in T \) with \( s \le t \) then \( U_s \le U_t \). A trivial application of Markov's inequality above would give \[ \P(U_t \ge x) \le \frac{1}{x} \E(U_t; U_t \ge x), \quad x \gt 0 \] But when \( \bs X \) is a sub-martingale, the following theorem gives a much stronger result by replacing the first occurrent of \( U_t \) on the right with \( X_t \). The theorem is known as Doob's sub-martingale maximal inequality (or more simply as Doob's inequaltiy ), named once again for Joseph Doob who did much of the pioneering work on martingales. A sub-martingale has an increasing property of sorts in the sense that if \( s, \, t \in T \) with \( s \le t \) then \( \E(X_t \mid \mathscr{F}_s) \ge X_s \), so it's perhaps not entirely surprising that such a bound is possible.
Suppose that \( \bs X \) is a sub-martingale. For \( t \in T \), let \( U_t = \sup\{X_s: s \in T_t\} \). Then \[ \P(U_t \ge x) \le \frac{1}{x} \E(X_t; U_t \ge x), \quad x \in (0, \infty) \]
Proof in the discrete time
So \( T = \N \) and the maximal process is given by \( U_n = \max\left\{X_k: k \in \N_n\right\} \) for \( n \in \N \). Let \( x \in (0, \infty) \), and define \( \tau_x = \min\{k \in \N: X_k \ge x\} \) where as usual, \( \min(\emptyset) = \infty \). The random time \( \tau_x \) is a stopping time relative to \( \mathfrak F \). Moreover, the processes \( \{U_n: n \in \N\} \) and \( \{\tau_x: x \in (0, \infty)\} \) are inverses in the sense that for \( n \in \N \) and \( x \in (0, \infty) \), \[ U_n \ge x \text{ if and only if } \tau_x \le n \] We have seen this type of duality before—in the Poisson process and more generally in renewal processes. Let \( n \in \N \). First note that \[ \E\left(X_{\tau_x \wedge n}\right) = \E\left(X_{\tau_x \wedge n}; \tau_x \le n\right) + \E\left(X_{\tau_x \wedge n}; \tau_x \gt n\right) \] If \( \tau_x \le n \) then \( X_{\tau_x \wedge n} = X_{\tau_x} \ge x \). On the other hand if \( \tau_x \gt n \) then \( X_{\tau_x \wedge n} = X_n \). So we have \[ \E\left(X_{\tau_x \wedge n}\right) \ge x \P(\tau_x \le n) + \E(X_n; \tau_x \gt n) = x \P(U_t \ge x) + \E(X_n; \tau_x \gt n) \] Similarly, \[ \E(X_n) = \E(X_n; \tau_x \le n) + \E(X_n; \tau_x \gt n) = \E(X_n; U_t \ge x) + \E(X_n; \tau_x \gt n) \] But by the optional stopping theorem, \( \E\left(X_{\tau_x \wedge n}\right) \le \E(X_n) \). Hence we have \[ x \P(U_t \ge x) + \E(X_n; \tau_x \gt n) \le \E(X_n; U_t \ge x) + \E(X_n; \tau_x \gt n)\] Subtracting the common term and then dividing both sides by \( x \) gives the result
Proof in continuous time
For \( k \in \N \), let \( \D^+_k = \{j / 2^k: j \in \N\} \) denote the set of nonnegative dyadic rationals (or binary rationals ) of rank \( k \) or less. For \( t \in [0, \infty) \) let \( T^k_t = (\D^+_k \cap [0, t]) \cup \{t\} \), so that \( T^k_t \) is the finite set of such dyadic rationals that are less than \( t \), with \( t \) added to the set. Note that \( T^k_t \) has an ordered enumeration, so \( \bs{X}^k = \{X_s: s \in T^k_t\} \) is a discrete-time sub-martingale for each \( k \in \N \). Let \( U^k_t = \sup\{X_s: s \in T^k_t\} \) for \( k \in \N \). Note that \( T^j_t \subset T^k_t \subset [0, t] \) for \( t \in [0, \infty) \) and for \( j, \, k \in \N \) with \( j \lt k \) and therefore \( U^j_t \le U^k_t \le U_t \). It follows that for \( x \in (0, \infty) \), \[ \left\{U^j_t \ge x\right\} \subseteq \left\{U^k_t \ge x\right\} \subset \{U_t \ge x\} \] The set \( \D^+ \) of all nonnegative dyadic rationals is dense in \( [0, \infty) \) and so since \( \bs X \) is right continuous and has left limits, it follows that if \( U_t \ge x \) then \( U^k_t \ge x \) for some \( k \in \N \). That is, we have \[ \{U_t \ge x\} = \bigcup_{k=0}^\infty \left\{U^k_t \ge x\right\} \] The maximal inequality applies to the discrete-time sub-martingale \( \bs{X}^k \) and so \[ P(U^k_t \ge x) \le \frac{1}{x} \E(X_t; U^k_t \ge x) \] for each \( k \in \N \). By the monotone convergence theorem, the left side converges to \( \P(U_t \ge x) \) as \( k \to \infty \) and the right side converges to \( \E(X; U_t \ge x) \) as \( k \to \infty \).
There are a number of simple corollaries of the maximal inequality. For the first, recall that the positive part of \( x \in \R \) is \( x^+ = x \vee 0 \), so that \( x^+ = x \) if \( x \gt 0 \) and \( x^+ = 0 \) if \( x \le 0 \).
Suppose that \( \bs X \) is a sub-martingale. For \( t \in T \), let \( V_t = \sup\{X_s^+: s \in T_t\} \). Then \[ \P(V_t \ge x) \le \frac{1}{x} \E(X_t^+; V_t \ge x), \quad x \in (0, \infty) \]
Proof
Recall that since \( \bs X \) is a sub-martingale and \( x \mapsto x^+ \) is increasing and convex, \( \bs X^+ = \{X_t^+: t \in T\} \) is also a sub-martingale. Hence the result follows from the general maximal inequality for sub-martingales.
As a further simple corollary, note that \[ \P(V_t \ge x) \le \frac{1}{x} \E(X_t^+), \quad x \in (0, \infty) \] This is sometimes how the maximal inequality is given in the literature.
Suppose that \( \bs X \) is a martingale. For \( t \in T \), let \( W_t = \sup\{|X_s|: s \in T_t\} \). Then \[ \P(W_t \ge x) \le \frac{1}{x} \E(|X_t|; W_t \ge x), \quad x \in (0, \infty) \]
Proof
Recall that since \( \bs X \) is a martingale, and \( x \mapsto |x| \) is convex, \( |\bs X| = \{|X_t|: t \in T\} \) is a sub-martingale. Hence the result follows from the general maximal inequality for sub-martingales.
Once again, a further simple corollary is \[ \P(W_t \ge x) \le \frac{1}{x} \E(|X_t|), \quad x \in (0, \infty) \] Next recall that for \( k \in (1, \infty) \), the \( k \)-norm of a real-valued random variable \( X \) is \( \|X\|_k = \left[\E(|X|^k)\right]^{1/k} \), and the vector space \( \mathscr{L}_k \) consists of all real-valued random variables for which this norm is finite. The following theorem is the norm version of the Doob's maximal inequality.
Suppose again that \( \bs X \) is a martingale. For \( t \in T \), let \( W_t = \sup\{|X_s|: s \in T_t\} \). Then for \( k \gt 1 \), \[ \|W_t\|_k \le \frac{k}{k - 1} \|X_t\|_k\]
Proof
Fix \( t \in T \). If \( \E(|X_t|^k) = \infty \), the inequality trivial holds, so assume that \( \E(|X_t|^k) \lt \infty \), and thus that \( X_t \in \mathscr{L}_k \). The proof relies fundamentally on Hölder's inequality, and for that inequality to work, we need to truncate the variable \( W_t \) and consider instead the the bounded random variable \( W_t \wedge c \) where \( c \in (0, \infty) \). First we need to show that \[ \P(W_t \wedge c \ge x) \le \frac{1}{x} \E(|X_t|; W_t \wedge c \ge x), \quad x \in (0, \infty) \] If \( c \lt x \), both sides are 0. If \( c \ge x \), \( \{W_t \wedge c \ge x\} = \{W_t \ge x\} \) and so from the maximal inequality above, \[ \P(W_t \wedge c \ge x) = \P(W_t \ge x) \le \frac{1}{x} \E(|X_t|; W_t \ge x) = \E(|X_t|; W_t \wedge c \ge x) \] Next recall that \[ \|W_t \wedge c\|_k^k = \E[(W_t \wedge c)^k] = \int_0^\infty k x^{k-1} \P(W_t \wedge c \ge x) dx \] Applying the inequality gives \[ \E[(W_t \wedge c)^k] \le \int_0^\infty k x^{k-2} \E[|X_t|; W_t \wedge c \ge x] dx \] By Fubini's theorem we can interchange the expected value and the integral which gives \[ \E[(W_t \wedge c)^k] \le \E\left[\int_0^{W_t \wedge c} k x^{k-2} |X_t| dx\right] = \frac{k}{k - 1} \E[|X_t| (W_t \wedge c)^{k-1}] \] But \( X_t \in \mathscr{L}_k \) and \( (W_t \wedge c)^{k-1} \in \mathscr{L}_j \) where \( j = k / (k - 1) \) is the exponent conjugate to \( k \). So an application of Hölder's inequality gives \[ \|W_t \wedge c\|_k^k \le \frac{k}{k - 1}\|X_t\|_k \, \|(W_t \wedge c)^{k-1}\|_j = \frac{k}{k - 1}\|X_t\|_k \|W_t \wedge c\|_k^{k-1} \] where we have used the simple fact that \( \|(W_t \wedge c)^{k-1}\|_j = \|W_t \wedge c\|_k^{k-1} \). Dividing by this factor gives \[ \|W_t \wedge c\|_k \le \frac{k}{k - 1} \|X_t\|_k \] Finally, \( \|W_t \wedge c\|_k \uparrow \|W_t\|_k\) as \( c \to \infty \) by the monotone convergence theorem. So letting \( c \to \infty \) in the last displayed equation gives \[ \|W_t\|_k \le \frac{k}{k - 1} \|X_t\|_k \]
Once again, \( \bs W = \{W_t: t \in T\} \) is the maximal process associated with \( |\bs X| = \{\left|X_t\right|: t \in T\} \). As noted in the proof, \( j = k / (k - 1) \) is the exponent conjugate to \( k \), satisfying \( 1 / j + 1 / k = 1 \). So this version of the maximal inequality states that the \( k \) norm of the maximum of the martingale \( \bs X \) on \( T_t \) is bounded by \( j \) times the \( k \) norm of \( X_t \), where \( j \) and \( k \) are conjugate exponents. Stated just in terms of expected value, rather than norms, the \( \mathscr{L}_k \) maximal inequality is \[ \E\left(\left|W_t\right|^k\right) \le \left(\frac{k}{k - 1}\right)^k \E\left(\left|X_t\right|^k\right) \] Our final result in this discussion is a variation of the maximal inequality for super-martingales.
Suppose that \( \bs X = \{X_t: t \in T\} \) is a nonnegative super-martingale, and let \( U_\infty = \sup\{X_t: t \in T\} \). Then \[ \P(U_\infty \ge x) \le \frac{1}{x} \E(X_0), \quad x \in (0, \infty) \]
Proof
Let \( Y_t = -X_t \) for \( t \in T \). Since \( \bs X \) is a super-martingale, \( \bs Y \) is a sub-martinagle. And since \( \bs X \) is nonnegative, \( Y_t^+ = X_t \) for \( t \in T \). Let \( U_t = \sup\{X_s: s \in T_t\} = \sup\{Y_s^+: s \in T_t\} \) for \( t \in T \). By the maximal inequality for sub-martingales, and since \( \bs X \) is a super-martingale we have for \( t \in T \), \[ \P(U_t \ge x) \le \frac{1}{x} \E(Y_t^+) = \frac{1}{x} \E(X_t) \le \frac{1}{x} \E(X_0), \quad x \in (0, \infty) \] Next note that \( U_t \uparrow U_\infty \) as \( t \to \infty \). Let \( x \in (0, \infty) \) and \( \epsilon \in (0, x) \). If \( U_\infty \ge x \) then \( U_t \ge x - \epsilon \) for sufficiently large \( t \in T \). Hence \[ \{U_\infty \ge x\} \subseteq \bigcup_{k=1}^\infty \{U_k \ge x - \epsilon\} \] Using the continuity theorem for increasing events, and our result above we have \[ \P(U_\infty \ge x) \le \lim_{k \to \infty} \P(U_k \ge x - \epsilon) \le \frac{1}{x - \epsilon} \E(X_0) \] Since this holds for all \( \epsilon \in (0, x) \), it follows that \( \P(U_\infty \ge x) \le \frac{1}{x} \E(X_0) \).
The Up-Crossing Inequality
The up-crossing inequality gives a bound on how much a sub-martingale (or super-martingale) can oscillate, and is the main tool in the martingale convergence theorems that will be studied in the next section. It should come as no surprise by now that the inequality is due to Joseph Doob. We start with the discrete-time case.
Suppose that \( \bs x = (x_n: n \in \N) \) is a sequence of real numbers, and that \( a, \, b \in \R \) with \( a \lt b \). Define \( t_0(\bs x) = 0 \) and then recursively define \begin{align*} s_{k+1}(\bs x) & = \inf\{n \in \N: n \ge t_k(\bs x), x_n \le a\}, \quad k \in \N \\ t_{k+1}(\bs x) & = \inf\{n \in \N: n \ge s_{k+1}(\bs x), x_n \ge b\}, \quad k \in \N \end{align*}
- The number of up-crossings of the interval \( [a, b] \) by the sequence \( \bs x \) up to time \( n \in \N \) is \[ u_n(a, b, \bs x) = \sup\{k \in \N: t_k(\bs x) \le n\} \]
- The total number of up-crossings of the interval \( [a, b] \) by the sequence \( \bs x \) is \[ u_\infty(a, b, \bs x) = \sup\{k \in \N: t_k(\bs x) \lt \infty\} \]
Details
As usual, we define \( \inf(\emptyset) = \infty \). Note that if \( t_k(\bs x) \lt \infty \) for \( k \in \N_+ \), then \( (x_n: n = s_k(\bs x), \ldots t_k(\bs x)) \) is the \( k \)th up-crossing of the interval \( [a, b] \) by the sequence \( \bs x \).
So informally, as the name suggests, \( u_n(a, b, \bs x) \) is the number of times that the sequence \( (x_0, x_1, \ldots, x_n) \) goes from a value below \( a \) to one above \( b \), and \( u(a, b, \bs x) \) is the number of times the entire sequence \( \bs x \) goes from a value below \( a \) to one above \( b \). Here are a few of simple properties:
Suppose again that \( \bs x = (x_n: n \in \N) \) is a sequence of real numbers and that \( a, \, b \in \R \) with \( a \lt b \).
- \( u_n(a, b, \bs x) \) is increasing in \( n \in \N \).
- \( u_n(a, b, \bs x) \to u(a, b, \bs x) \) as \( n \to \infty \).
- If \( c, \, d \in \R \) with \( a \lt c \lt d \lt b \) then \( u_n(c, d, \bs x) \ge u_n(a, b, \bs x) \) for \( n \in \N \), and \( u(c, d, \bs x) \ge u(a, b, \bs x) \).
Proof
- Note that \(\{k \in \N: t_k(\bs x) \le n\} \subseteq \{k \in \N: t_k(\bs x) \le n + 1\}\).
- Note that \( \bigcup_{n=0}^\infty \{k \in \N: t_k(\bs x) \le n\} = \{k \in \N: t_k(\bs x) \le \infty\} \).
- Every up-crossing of \( [a, b] \) is also an up-crossing of \( [c, d] \).
The importance of the definitions is found in the following theorem. Recall that \( \R^* = \R \cup \{-\infty, \infty\} \) is the set of extended real numbers, and \( \Q \) is the set of rational real numbers.
Suppose again that \( \bs x = (x_n: n \in \N) \) is a sequence of real numbers. Then \( \lim_{n \to \infty} x_n \) exists in \( \R^* \) is and only if \( u_\infty(a,b, \bs x) \lt \infty\) for every \( a, \, b \in \Q \) with \( a \lt b \).
Proof
We prove the contrapositive. Note that the following statements are equivalent:
- \( \lim_{n \to \infty} x_n \) does not exist in in \( \R^* \).
- \( \liminf_{n \to \infty} x_n \lt \limsup_{n \to \infty} x_n \).
- There exists \( a, \, b \in \Q \) with \( a \lt b \) and with \( x_n \le a \) for infinitely many \( n \in \N \) and \( x_n \ge b \) for infinitely many \( n \in \N \).
- There exists \( a, \, b \in \Q \) with \( a \lt b \) and \( u_\infty(a, b, \bs x) = \infty \).
Clearly the theorem is true with \( \Q \) replaced with \( \R \), but the countability of \( \Q \) will be important in the martingale convergence theorem. As a simple corollary, if \( \bs x \) is bounded and \( u_\infty(a, b, \bs x) \lt \infty \) for every \( a, \, b \in \Q \) with \( a \lt b \), then \( \bs x \) converges in \( \R \). The up-crossing inequality for a discrete-time martingale \( \bs X \) gives an upper bound on the expected number of up-crossings of \( \bs X \) up to time \( n \in \N \) in terms of a moment of \( X_n \).
Suppose that \( \bs X = \{X_n: n \in \N\} \) satisfies the basic assumptions with respect to the filtration \( \mathfrak F = \{\mathscr{F}_n: n \in \N\} \), and let \( a, \, b \in \R \) with \( a \lt b \). Let \( U_n = u_n(a, b, \bs X) \), the random number of up-crossings of \( [a, b] \) by \( \bs X \) up to time \( n \in \N\).
- If \( \bs X \) is a super-martingale relative to \( \mathfrak F \) then \[ \E(U_n) \le \frac{1}{b - a} \E[(X_n - a)^-] \le \frac{1}{b - a}\left[\E(X_n^-) + |a|\right] \le \frac{1}{b - a} \left[\E(|X_n|) + |a|\right], \quad n \in \N \]
- If \( \bs X \) is a sub-martingale relative to \( \mathfrak F \) then \[ \E(U_n) \le \frac{1}{b - a} \E[(X_n - a)^+] \le \frac{1}{b - a}\left[\E(X_n^+) + |a|\right] \le \frac{1}{b - a}\left[\E(|X_n|) + |a|\right], \quad n \in \N \]
Proof
In the context of the up-crossing definition above , let \( \sigma_k = s_k(\bs X) \) and \( \tau_k = t_k(\bs X) \). These are the random times that define the up-crossings of \( \bs X \). Let \( Y_k = X_{\tau_k \wedge n} - X_{\sigma_k \wedge n} \) and then define \( Z_n = \sum_{k=1}^n Y_k \). To understand the sum, let's take cases for the \( k \)th term \( Y_k \):
- If \( \tau_k \le n \) then \( Y_k = X_{\tau_k} - X_{\sigma_k} \ge b - a \). By definition, the first \( U_n \) terms are of this form.
- If \( \sigma_k \le n \lt \tau_k \) then \( Y_k = X_n - X_{\sigma_k} \ge X_n - a \). There is at most one such term, with index \( k = U_n + 1 \).
- If \( \sigma_k \gt n \) then \( Y_k = X_n - X_n = 0 \).
Hence \( Z_n \ge (b - a)U_n + (X_n - a) \bs{1} \left(\sigma_{U_n + 1} \le n\right) \) and so \( (b - a)U_n \le Z_n - (X_n - a) \bs{1} \left(\sigma_{U_n + 1} \le n\right) \) Next note that \( \sigma_k \wedge n \) and \( \tau_k \wedge n \) are bounded stopping times and of course \( \sigma_k \wedge n \le \tau_k \wedge n \).
- If \( \bs X \) is a super-martingale, it follows from the optional stopping theorem that \[ \E(Y_k) = \E\left(X_{\tau_k \wedge n}\right) - \E\left(X_{\sigma_k \wedge n}\right) \le 0 \] and therefore \( \E(Z_n) \le 0 \). Finally, \(-(X_n - a) \bs{1} \left(\sigma_{U_n + 1} \le n\right) \le (X_n - a)^-\). Taking expected values gives \[(b - a) \E(U_n) \le \E(Z_n) + \E[(X_n - a)^-] \le \E[(X_n - a)^-]\] The remaining parts of the inequality follow since \( (x - a)^- \le x^- + |a| \le |x| + |a| \) for \( x \in \R \).
Additional details
The process \( \bs Z = \{Z_n: n \in \N\} \) in the proof can be viewed as a transform of \( \bs X = \{X_n: n \in \N\} \) by a predictable process. Specifically, for \( n \in \N_+ \), let \( I_n = 1 \) if \( \sigma_k \lt n \le \tau_k \) for some \( k \in \N \), and let \( I_n = 0 \) otherwise. Since \( \sigma_k \) and \( \tau_k \) are stopping times, note that \( \{I_n = 1\} \in \mathscr{F}_{n-1} \) for \( n \in \N_+ \). Hence the process \( \bs I = \{I_n: n \in \N_+\} \) is predictable with respect to \( \mathfrak F \). Moreover, the transform of \( \bs X \) by \( \bs I \) is \[ (\bs I \cdot \bs X)_n = \sum_{j=1}^n I_j (X_j - X_{j-1}) = \sum_{k=1}^n \left(X_{\tau_k \wedge n} - X_{\sigma_k \wedge n}\right) = Z_n, \quad n \in \N \] Since \( \bs I \) is a nonnegative process, if \( \bs X \) is a martingale (sub-martingale, super-martingale), then \( \bs I \cdot \bs X \) is also a martingale (sub-martingale, super-martingale).
Of course if \( \bs X \) is a martingale with respect to \( \mathfrak F \) then both inequalities apply. In continuous time, as usual, the concepts are more complicated and technical.
Suppose that \( \bs x: [0, \infty) \to \R \) and that that \( a, \, b \in \R \) with \( a \lt b \).
- If \( I \subset [0, \infty) \) is finite, define \( t^I_0(\bs x) = 0 \) and then recursively define \begin{align*} s^I_{k+1}(\bs x) & = \inf\left\{t \in I: t \ge t^I_k(\bs x), x_t \le a\right\}, \quad k \in \N \\ t^I_{k+1}(\bs x) & = \inf\left\{t \in I: t \ge s^I_{k+1}(\bs x), x_t \ge b\right\}, \quad k \in \N \end{align*} The number of up-crossings of the interval \( [a, b] \) by the function \( \bs x \) restricted to \( I \) is \[ u_I(a, b, \bs x) = \sup \left\{k \in \N: t^I_k(\bs x) \lt \infty\right\} \]
- If \( I \subseteq [0, \infty) \) is infinte, the number of up-crossings of the interval \( [a, b] \) by \( \bs x \) restricted to \( I \) is \[ u_I(a, b, \bs x) = \sup\{u_J(a, b, \bs x): J \text{ is finite and } J \subset I\} \]
To simplify the notation, we will let \( u_t(a, b, \bs x) = u_{[0, t]}(a, b, \bs x) \), the number of up-crossings of \( [a, b] \) by \( \bs x \) on \( [0, t] \), and \( u_\infty(a, b, \bs x) = u_{[0, \infty)}(a, b, \bs x) \), the total number of up-crossings of \( [a, b] \) by \( \bs x \). In continuous time, the definition of up-crossings is built out of finte subsets of \( [0, \infty) \) for measurability concerns, which arise when we replace the deterministic function \( \bs x \) with a stochastic process \( \bs X \). Here are the simple properties that are analogous to our previous ones.
Suppose again that \( \bs x: [0, \infty) \to \R \) and that \( a, \, b \in \R \) with \( a \lt b \).
- If \( I, \, J \subseteq [0, \infty) \) with \( I \subseteq J \), then \( u_I(a, b, \bs x) \le u_J(a, b, \bs x) \).
- If \( (I_n: n \in \N) \) is an increasing sequence of sets in \( [0, \infty) \) and \( J = \bigcup_{n=0}^\infty I_n \) then \( u_{I_n}(a, b, \bs x) \to u_J(a, b, \bs x) \) as \( n \to \infty \).
- If \( c, \, d \in \R \) with \( a \lt c \lt d \lt b \) and \( I \subset [0, \infty) \) then \( u_I(c, d, \bs x) \ge u_I(a, b, \bs x) \).
Proof
- The result follows easily from the definitions if \( I \) is finite (and \( J \) either finite or infinite). If \( I \) is infinite (and hence so is \( J \)), note that \[ \{u_K(a, b, \bs x): K \text{ is finite and } K \subseteq I\} \subseteq \{u_K(a, b, \bs x): K \text{ is finite and } K \subseteq J\} \]
- Since \( I_n \) is increasing in \( n \in \N \) (in the subset partial order), note that if \( K \subset [0, \infty)\) is finite, then \( K \subseteq J \) if and only if \( K \subseteq I_n \) for some \( n \in \N \).
- Every up-crossing of \( [a, b] \) is an up-crossing of \( [c, d] \).
The following result is the reason for studying up-crossings in the first place. Note that the definition built from finite set is sufficient.
Suppose that \( \bs x: [0, \infty) \to \R \). Then \( \lim_{t \to \infty} x_t \) exists in \( \R^* \) if and only if \( u_\infty(a, b, \bs x) \lt \infty \) for every \( a, \, b \in \Q \) with \( a \lt b \).
Proof
As in the discrete-time case, we prove the contrapositive. The proof is almost the same: The following statements are equivalent:
- \( \lim_{t \to \infty} x_t \) does not exist in in \( \R^* \).
- \( \liminf_{t \to \infty} x_t \lt \limsup_{t \to \infty} x_t \).
- There exists \( a, \, b \in \Q \) with \( a \lt b \) and there exists \( s_n, \, t_n \in [0, \infty) \) with \( x_{s_n} \le a \) for \( n \in \N \) and \( x_{t_n} \ge b \) for \( n \in \N \).
- There exists \( a, \, b \in \Q \) with \( a \lt b \) and \( u_\infty(a, b, \bs x) = \infty \).
Finally, here is the up-crossing inequality for martingales in continuous time. Once again, the inequality gives a bound on the expected number of up-crossings.
Suppose that \( \bs X = \{X_t: t \in [0, \infty)\} \) satisfies the basic assumptions with respect to the filtration \( \mathfrak F = \{\mathscr{F}_t: t \in [0, \infty)\} \), and let \( a, \, b \in \R \) with \( a \lt b \). Let \( U_t = u_t(a, b, \bs X) \), the random number of up-crossings of \( [a, b] \) by \( \bs X \) up to time \( t \in [0, \infty) \).
- If \( \bs X \) is a super-martingale relative to \( \mathfrak F \) then \[ \E(U_t) \le \frac{1}{b - a} \E[(X_t - a)^-] \le \frac{1}{b - a}\left[\E(X_t^-) + |a|\right] \le \frac{1}{b - a}\left[\E(|X_t|) + |a|\right], \quad t \in [0, \infty) \]
- If \( \bs X \) is a sub-martingale relative to \( \mathfrak F \) then \[ \E(U_t) \le \frac{1}{b - a} \E[(X_t - a)^+] \le \frac{1}{b - a} \left[\E(X_t^+) + |a|\right] \le \frac{1}{b - a} \left[\E(|X_t|) + |a|\right], \quad t \in [0, \infty) \]
Proof
Suppose that \( \bs X \) is a sub-martingale; the proof for a super-martingale is analogous. Fix \( t \in [0, \infty) \) and \( a, \, b \in \R \) with \( a \lt b \). For \( I \subseteq [0, \infty) \) let \( U_I = u_I(a, b, \bs X) \), the number of up-crossings of \( [a, b] \) by \( \bs X \) restricted to \( I \). Suppose that \( I \) is finite and that \( t \in I \) is the maximum of \( I \). Since \( \bs X \) restricted to \( I \) is also a sub-martingale, the discrete-time up-crossing theorem applies and so \[ \E(U_I) \le \frac{1}{b - a} \E[(X_t - a)^+] \] Since \( U_t = \sup\{U_I: I \text{ is finite and } I \subset [0, t]\} \), there exists finite \( I_n \) for \( n \in \N \) with \( U_{I_n} \uparrow U_t \) as \( n \to \infty \). In particular, \( U_t \) is measurable. By property (a) in the theorem above , there exists such a sequence with \( I_n \) increasing in \( n \) and \( t \in I_n \) for each \( n \in \N \). By the monotone convergence theorem, \( \E\left(U_{I_n}\right) \to \E(U_t) \) as \( n \to \infty \). So by the displayed equation above, \[ \E(U_t) \le \frac{1}{b - a} \E[(X_t - a)^+] \]
Examples and Applications
Kolmogorov's Inequality
Suppose that \( \bs X = \{X_n: n \in \N_+\} \) is a sequence of independent variables with \( \E(X_n) = 0 \) and \( \var(X_n) = \E(X_n^2) \lt \infty \) for \( n \in \N_+ \). Let \( \bs Y = \{Y_n: n \in \N\} \) be the partial sum process associated with \( \bs X \), so that \[ Y_n = \sum_{i=1}^n X_i, \quad n \in \N \] From the Introduction we know that \( \bs Y \) is a martingale. A simple application of the maximal inequality gives the following result, which is known as Kolmogorov's inequality , named for Andrei Kolmogorov.
For \( n \in \N \), let \( U_n = \max\left\{\left|Y_i\right|: i \in \N_n\right\} \). Then \[ \P(U_n \ge x) \le \frac{1}{x^2} \var(Y_n) = \frac{1}{x^2} \sum_{i=1}^n \E(X_i^2), \quad x \in (0, \infty)\]
Proof
As noted above, \( \bs Y \) is a martingale. Since the function \( x \mapsto x^2 \) on \( \R \) is convex, \( \bs{Y}^2 = \{Y_n^2: n \in \N\} \) is a sub-martingale. Let \( V_n = \max\{Y_i^2: i \in \N_n\} \) for \( n \in \N \), and let \( x \in (0, \infty) \). Applying the maximal inequality for sub-martingales we have \[ \P(U_n \ge x) = \P(V_n \ge x^2) \le \frac{1}{x^2} \E(Y_n^2) = \frac{1}{x^2} \var(Y_n)\] Finally, since \( \bs X \) is an independent sequence, \[ \var(Y_n) = \sum_{i=1}^n \var(X_i) = \sum_{i=1}^n \E(X_i^2) \]
Red and Black
In the game of red and black, a gambler plays a sequence of Bernoulli games with success parameter \( p \in (0, 1) \) at even stakes. The gambler starts with an initial fortune \( x \) and plays until either she is ruined or reaches a specified target fortune \( a \), where \( x, \, a \in (0, \infty) \) with \( x \lt a \). When \( p \le \frac{1}{2} \), so that the games are fair or unfair, an optimal strategy is bold play : on each game, the gambler bets her entire fortune or just what is needed to reach the target, whichever is smaller. In the section on bold play we showed that when \( p = \frac{1}{2} \), so that the games are fair, the probability of winning (that is, reaching the target \( a \) starting with \( x \)) is \( x / a \). We can use the maximal inequality for super-martingales to show that indeed, one cannot do better.
To set up the notation and review various concepts, let \( X_0 \) denote the gambler's initial fortune and let \( X_n \) denote the outcome of game \( n \in \N_+ \), where 1 denotes a win and \( -1 \) a loss. So \( \{X_n: n \in \N\} \) is a sequence of independent variables with \( \P(X_n = 1) = p \) and \( \P(X_n = -1) = 1 - p \) for \( n \in \N_+ \). (The initial fortune \( X_0 \) has an unspecified distribution on \( (0, \infty) \).) The gambler is at a casino after all, so of course \( p \le \frac{1}{2} \). Let \[ Y_n = \sum_{i=0}^n X_i, \quad n \in \N \] so that \( \bs Y = \{Y_n: n \in \N\} \) is the partial sum process associated with \( \bs X = \{X_n: n \in \N\} \). Recall that \( \bs Y \) is also known as the simple random walk with parameter \( p \), and since \( p \le \frac{1}{2} \), is a super-martingale. The process \( \{X_n: n \in \N_+\} \) is the difference sequence associated with \( \bs Y \). Next let \( Z_n \) denote the amount that the gambler bets on game \( n \in \N_+ \). The process \( \bs Z = \{Z_n: n \in \N_+\} \) is predictable with respect to \( \bs X = \{X_n: n \in \N\} \), so that \( Z_n \) is measurable with respect to \( \sigma\{X_0, X_1, \ldots, X_{n-1}\} \) for \( n \in \N_+ \). So the gambler's fortune after \( n \) games is \[ W_n = X_0 + \sum_{i=1}^n Z_i X_i = X_0 + \sum_{i=1}^n Z_i (Y_i - Y_{i-1}) \] Recall that \( \bs W = \{W_n: n \in \N\} \) is the transform of \( \bs Z \) with \( \bs Y \), denoted \( \bs W = \bs Z \cdot \bs Y \). The gambler is not allowed to go into debt and so we must have \( Z_n \le W_{n-1} \) for \( n \in \N_+ \): the gambler's bet on game \( n \) cannot exceed her fortune after game \( n - 1 \). What's the probability that the gambler can ever reach or exceed the target \( a \) starting with fortune \( x \lt a \)?
Let \( U_\infty = \sup\{W_n: n \in \N\} \). Suppose that \( x, \, a \in (0, \infty) \) with \( x \lt a \) and that \( X_0 = x \). Then \[ \P(U_\infty \ge a) \le \frac{x}{a} \]
Proof
Since \( \bs Y \) is a super-martingale and \( \bs Z \) is nonnegative, the transform \( \bs W = \bs Z \cdot \bs Y \) is also a super-martingale. By the inequality for nonnegative super-martingales above : \[ \P(U_\infty \ge a) \le \frac{1}{a} \E(W_0) = \frac{x}{a} \]
Note that the only assumptions made on the gambler's sequence of bets \( \bs Z \) is that the sequence is predictable, so that the gambler cannot see into the future, and that gambler cannot go into debt. Under these basic assumptions, no strategy can do any better than bold play. However, there are strategies that do as well as bold play; these are variations on bold play.
Open the simulation of the red and black game. Select bold play and \( p = \frac{1}{2} \). Play the game with various values of initial and target fortunes.


4.1: Formal vs. Informal Fallacies
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A fallacy is simply a mistake in reasoning. Some fallacies are formal and some are informal. In Chapter 2, we saw that we could define validity formally and thus could determine whether an argument was valid or invalid without even having to know or understand what the argument was about. We saw that we could define certain valid rules of inference, such as modus ponens and modus tollens . These inference patterns are valid in virtue of their form, not their content. That is, any argument that has the same form as modus ponens or modus tollens will automatically be valid. A formal fallacy is simply an argument whose form is be invalid, regardless of the meaning of the sentences. Two formal fallacies that are similar to, but should never be confused with, modus ponens and modus tollens are denying the antecedent and affirming the consequent . Here are the forms of those invalid inferences:
Denying the antecedent
p ⊃ q
~p
∴ ~q
Affirming the consequent
p ⊃ q
q
∴ p
Any argument that has either of these forms is an invalid argument. For example:
1. If Kant was a deontologist, then he was a non-consequentialist.
2. Kant was not a deontologist.
3. Therefore, Kant was a not a non-consequentialist.
The form of this argument is:
- D ⊃ C
- ~D
- ∴ ~C
As you can see, this argument has the form of the fallacy, denying the antecedent. Thus, we know that this argument is invalid even if we don’t know what “Kant” or “deontologist” or “non-consequentialist” means. (“Kant” was a famous German philosopher from the early 1800s, whereas “deontology” and “non consequentialist” are terms that come from ethical theory.) It is mark of a formal fallacy that we can identify it even if we don’t really understand the meanings of the sentences in the argument. Recall our Jabberwocky argument from chapter 2. Here’s an argument which uses silly, made-up words from Lewis Carrol’s “Jabberwocky.” See if you can determine whether the argument’s form is valid or invalid:
- If toves are brillig then toves are slithy.
- Toves are slithy
- Therefore, toves are brillig.
You should be able to see that this argument has the form of affirming the consequent:
- B ⊃ S
- S
- ∴ B
As such, we know that the argument is invalid, even though we haven’t got a clue what “toves” are or what “slithy” or “brillig” means. The point is that we can identify formal fallacies without having to know what they mean.
In contrast, informal fallacies are those which cannot be identified without understanding the concepts involved in the argument. A paradigm example of an informal fallacy is the fallacy of composition. We will consider this fallacy in the next sub-section. In the remaining subsections, we will consider a number of other informal logical fallacies.
Composition Fallacy
Consider the following argument:
Each member on the gymnastics team weighs less than 110 lbs. Therefore, the whole gymnastics team weighs less than 110 lbs.
This arguments commits the composition fallacy . In the composition fallacy one argues that since each part of the whole has a certain feature, it follows that the whole has that same feature. However, you cannot generally identify any argument that moves from statements about parts to statements about wholes as committing the composition fallacy because whether or not there is a fallacy depends on what feature we are attributing to the parts and wholes. Here is an example of an argument that moves from claims about the parts possessing a feature to a claim about the whole possessing that same feature, but doesn’t commit the composition fallacy:
Every part of the car is made of plastic. Therefore, the whole car is made of plastic.
This conclusion does follow from the premises; there is no fallacy here. The difference between this argument and the preceding argument (about the gymnastics team) isn’t their form. In fact both arguments have the same form:
Every part of X has the feature f. Therefore, the whole X has the feature f.
And yet one of the arguments is clearly fallacious, while the other isn’t. The difference between the two arguments is not their form, but their content. That is, the difference is what feature is being attributed to the parts and wholes. Some features (like weighing a certain amount) are such that if they belong to each part, then it does not follow that they belong to the whole. Other features (such as being made of plastic) are such that if they belong to each part, it follows that they belong to the whole.
Here is another example:
Every member of the team has been to Paris. Therefore the team has been to Paris.
The conclusion of this argument does not follow. Just because each member of the team has been to Paris, it doesn’t follow that the whole team has been to Paris, since it may not have been the case that each individual was there at the same time and was there in their capacity as a member of the team. Thus, even though it is plausible to say that the team is composed of every member of the team, it doesn’t follow that since every member of the team has been to Paris, the whole team has been to Paris. Contrast that example with this one:
Every member of the team was on the plane. Therefore, the whole team was on the plane.
This argument, in contrast to the last one, contains no fallacy. It is true that if every member is on the plane then the whole team is on the plane. And yet these two arguments have almost exactly the same form. The only difference is that the first argument is talking about the property, having been to Paris, whereas the second argument is talking about the property, being on the plane. The only reason we are able to identify the first argument as committing the composition fallacy and the second argument as not committing a fallacy is that we understand the relationship between the concepts involved. In the first case, we understand that it is possible that every member could have been to Paris without the team ever having been; in the second case we understand that as long as every member of the team is on the plane, it has to be true that the whole team is on the plane. The take home point here is that in order to identify whether an argument has committed the composition fallacy, one must understand the concepts involved in the argument. This is the mark of an informal fallacy: we have to rely on our understanding of the meanings of the words or concepts involved, rather than simply being able to identify the fallacy from its form.
Division Fallacy
The division fallacy is like the composition fallacy and they are easy to confuse. The difference is that the division fallacy argues that since the whole has some feature, each part must also have that feature. The composition fallacy, as we have just seen, goes in the opposite direction: since each part has some feature, the whole must have that same feature. Here is an example of a division fallacy:
The house costs 1 million dollars. Therefore, each part of the house costs 1 million dollars.
This is clearly a fallacy. Just because the whole house costs 1 million dollars, it doesn’t follow that each part of the house costs 1 million dollars. However, here is an argument that has the same form, but that doesn’t commit the division fallacy:
The whole team died in the plane crash. Therefore each individual on the team died in the plane crash.
In this example, since we seem to be referring to one plane crash in which all the members of the team died (“the” plane crash), it follows that if the whole team died in the crash, then every individual on the team died in the crash. So this argument does not commit the division fallacy. In contrast, the following argument has exactly the same form, but does commit the division fallacy:
The team played its worst game ever tonight. Therefore, each individual on the team played their worst game ever tonight.
It can be true that the whole team played its worst game ever even if it is true that no individual on the team played their worst game ever. Thus, this argument does commit the fallacy of division even though it has the same form as the previous argument, which doesn’t commit the fallacy of division. This shows (again) that in order to identify informal fallacies (like composition and division), we must rely on our understanding of the concepts involved in the argument. Some concepts (like “team” and “dying in a plane crash”) are such that if they apply to the whole, they also apply to all the parts. Other concepts (like “team” and “worst game played”) are such that they can apply to the whole even if they do not apply to all the parts.
Begging the question
Consider the following argument:
Capital punishment is justified for crimes such as rape and murder because it is quite legitimate and appropriate for the state to put to death someone who has committed such heinous and inhuman acts.
The premise indicator, “because” denotes the premise and (derivatively) the conclusion of this argument. In standard form, the argument is this:
1. It is legitimate and appropriate for the state to put to death someone who commits rape or murder.
2. Therefore, capital punishment is justified for crimes such as rape and murder.
You should notice something peculiar about this argument: the premise is essentially the same claim as the conclusion. The only difference is that the premise spells out what capital punishment means (the state putting criminals to death) whereas the conclusion just refers to capital punishment by name, and the premise uses terms like “legitimate” and “appropriate” whereas the conclusion uses the related term, “justified.” But these differences don’t add up to any real differences in meaning. Thus, the premise is essentially saying the same thing as the conclusion. This is a problem: we want our premise to provide a reason for accepting the conclusion. But if the premise is the same claim as the conclusion, then it can’t possibly provide a reason for accepting the conclusion! Begging the question occurs when one (either explicitly or implicitly) assumes the truth of the conclusion in one or more of the premises. Begging the question is thus a kind of circular reasoning.
One interesting feature of this fallacy is that formally there is nothing wrong with arguments of this form. Here is what I mean. Consider an argument that explicitly commits the fallacy of begging the question. For example,
1. Capital punishment is morally permissible
2. Therefore, capital punishment is morally permissible
Now, apply any method of assessing validity to this argument and you will see that it is valid by any method. If we use the informal test (by trying to imagine that the premises are true while the conclusion is false), then the argument passes the test, since any time the premise is true, the conclusion will have to be true as well (since it is the exact same statement). Likewise, the argument is valid by our formal test of validity, truth tables. But while this argument is technically valid, it is still a really bad argument. Why? Because the point of giving an argument in the first place is to provide some reason for thinking the conclusion is true for those who don’t already accept the conclusion. But if one doesn’t already accept the conclusion, then simply restating the conclusion in a different way isn’t going to convince them. Rather, a good argument will provide some reason for accepting the conclusion that is sufficiently independent of that conclusion itself. Begging the question utterly fails to do this and this is why it counts as an informal fallacy. What is interesting about begging the question is that there is absolutely nothing wrong with the argument formally.
Whether or not an argument begs the question is not always an easy matter to sort out. As with all informal fallacies, detecting it requires a careful understanding of the meaning of the statements involved in the argument. Here is an example of an argument where it is not as clear whether there is a fallacy of begging the question:
Christian belief is warranted because according to Christianity there exists a being called “the Holy Spirit” which reliably guides Christians towards the truth regarding the central claims of Christianity. 1
One might think that there is a kind of circularity (or begging the question) involved in this argument since the argument appears to assume the truth of Christianity in justifying the claim that Christianity is true. But whether or not this argument really does beg the question is something on which there is much debate within the sub-field of philosophy called epistemology (“study of knowledge”). The philosopher Alvin Plantinga argues persuasively that the argument does not beg the question, but being able to assess that argument takes patient years of study in the field of epistemology (not to mention a careful engagement with Plantinga’s work). As this example illustrates, the issue of whether an argument begs the question requires us to draw on our general knowledge of the world. This is the mark of an informal, rather than formal, fallacy.
False dichotomy
Suppose I were to argue as follows:
Raising taxes on the wealthy will either hurt the economy or it will help it. But it won’t help the economy. Therefore it will hurt the economy.
The standard form of this argument is:
1. Either raising taxes on the wealthy will hurt the economy or it will help it.
2. Raising taxes on the wealthy won’t help the economy.
3. Therefore, raising taxes on the wealthy will hurt the economy.
This argument contains a fallacy called a “false dichotomy.” A false dichotomy is simply a disjunction that does not exhaust all of the possible options. In this case, the problematic disjunction is the first premise: either raising the taxes on the wealthy will hurt the economy or it will help it. But these aren’t the only options. Another option is that raising taxes on the wealthy will have no effect on the economy. Notice that the argument above has the form of a disjunctive syllogism:
A v B
~A
∴ B
However, since the first premise presents two options as if they were the only two options, when in fact they aren’t, the first premise is false and the argument fails. Notice that the form of the argument is perfectly good—the argument is valid. The problem is that this argument isn’t sound because the first premise of the argument commits the false dichotomy fallacy. False dichotomies are commonly encountered in the context of a disjunctive syllogism or constructive dilemma (see chapter 2).
In a speech made on April 5, 2004, President Bush made the following remarks about the causes of the Iraq war:
Saddam Hussein once again defied the demands of the world. And so I had a choice: Do I take the word of a madman, do I trust a person who had used weapons of mass destruction on his own people, plus people in the neighborhood, or do I take the steps necessary to defend the country? Given that choice, I will defend America every time.
The false dichotomy here is the claim that:
Either I trust the word of a madman or I defend America (by going to war against Saddam Hussein’s regime).
The problem is that these aren’t the only options. Other options include ongoing diplomacy and economic sanctions. Thus, even if it true that Bush shouldn’t have trusted the word of Hussein, it doesn’t follow that the only other option is going to war against Hussein’s regime. (Furthermore, it isn’t clear in what sense this was needed to defend America.) That is a false dichotomy. As with all the previous informal fallacies we’ve considered, the false dichotomy fallacy requires an understanding of the concepts involved. Thus, we have to use our understanding of world in order to assess whether a false dichotomy fallacy is being committed or not.
Equivocation
Consider the following argument:
Children are a headache. Aspirin will make headaches go away. Therefore, aspirin will make children go away.
This is a silly argument, but it illustrates the fallacy of equivocation. The problem is that the word “headache” is used equivocally—that is, in two different senses. In the first premise, “headache” is used figuratively, whereas in the second premise “headache” is used literally. The argument is only successful if the meaning of “headache” is the same in both premises. But it isn’t and this is what makes this argument an instance of the fallacy of equivocation.
Here’s another example:
Taking a logic class helps you learn how to argue. But there is already too much hostility in the world today, and the fewer arguments the better. Therefore, you shouldn’t take a logic class.
In this example, the word “argue” and “argument” are used equivocally. Hopefully, at this point in the text, you recognize the difference. (If not, go back and reread section 1.1.)
The fallacy of equivocation is not always so easy to spot. Here is a trickier example:
The existence of laws depends on the existence of intelligent beings like humans who create the laws. However, some laws existed before there were any humans (e.g., laws of physics). Therefore, there must be some non-human, intelligent being that created these laws of nature.
The term “law” is used equivocally here. In the first premise it is used to refer to societal laws, such as criminal law; in the second premise it is used to refer to laws of nature. Although we use the term “law” to apply to both cases, they are importantly different. Societal laws, such as the criminal law of a society, are enforced by people and there are punishments for breaking the laws. Natural laws, such as laws of physics, cannot be broken and thus there are no punishments for breaking them. (Does it make sense to scold the electron for not doing what the law says it will do?)
As with every informal fallacy we have examined in this section, equivocation can only be identified by understanding the meanings of the words involved. In fact, the definition of the fallacy of equivocation refers to this very fact: the same word is being used in two different senses (i.e., with two different meanings). So, unlike formal fallacies, identifying the fallacy of equivocation requires that we draw on our understanding of the meaning of words and of our understanding of the world, generally.
1 This is a much simplified version of the view defended by Christian philosophers such as Alvin Plantinga. Plantinga defends (something like) this claim in: Plantinga, A. 2000. Warranted Christian Belief. Oxford, UK: Oxford University Press.


29.5B: Exercise-Induced Muscle Damage
Exercise damages muscles due to eccentric and concentric muscle loading and often results in delayed onset muscle soreness (DOMS).
- Explain the process of exercise-induced muscle damage
Key Points
- Delayed onset muscle soreness is a result of damage in muscle sarcomeres during contractions.
- The acute but delayed nature of muscle soreness is indicative of an inflammatory immune response.
- This disruption of the muscle fibers triggers white blood cells to increase following the induced muscle soreness, leading to the inflammatory response observation from the induced muscle soreness.
Key Terms
- eccentric contraction : The elongation of a muscle while under tension due to an opposing force being greater than the force generated by the muscle.
- concentric contraction : A type of muscle contraction in which the muscles shorten while generating force.
Exercise damages muscles due to eccentric and concentric muscle loading. Resistance training, and particularly high loading during eccentric contractions, results in delayed onset muscle soreness (DOMS). Previously attributed to the accumulation of lactic acid during exercise, it is now understood that DOMS is due to structural damage in sarcomeres, particularly to the z-disks and contractile filaments. Z-disks provide the structural support for the contractile filaments of the sliding filament mechanism. Overloading of muscles damages these connections and the orientation of the contractile filaments.
The Sarcomere : Delayed onset muscle soreness is caused by structural damage to the Z disk and myosin and actin filaments.
Causes of Exercise-Induced Muscle Damage
Acute inflammation of the muscle cells, as understood in exercise physiology, can result after induced eccentric and concentric muscle training. Participation in eccentric training and conditioning, including resistance training and activities that emphasize eccentric lengthening of the muscle such as downhill running on a moderate to high incline, can result in considerable soreness within 24 to 48 hours, even though blood lactate levels, previously thought to cause muscle soreness, were much higher with level running. This has been noted especially in marathon runners whose muscle fibers revealed remarkable damage after both training and marathon competition. The onset and timing of this gradient damage to the muscle parallels the degree of muscle soreness experienced by the runners.
This disruption of the muscle fibers triggers white blood cells to increase following the induced muscle soreness, leading to the inflammatory response observation from the induced muscle soreness. Elevations in plasma enzymes, myoglobinemia, and abnormal muscle histology and ultrastructure are concluded to be associated with the inflammatory response. High tension in the contractile-elastic system of muscle results in structural damage to the muscle fiber and plasmalemma and its epimysium, perimysium, and endomysium. The mysium damage disrupts calcium homeostasis in the injured fiber and fiber bundles, resulting in necrosis that peaks about 48 hours after exercise. The products of the macrophage activity and intracellular contents (such as histamines, kinins, and K+) accumulate outside the cells. These substances then stimulate the free nerve endings in the muscle; a process that appears accentuated by eccentric exercise, in which large forces are distributed over a relatively small cross-sectional area of the muscle.
The acute, and delayed nature of muscle soreness is indicative of an inflammatory immune response. Damage to the sarcomeres causes aninflux of white blood cells, leading to inflammation, which is itself associated with increased plasma enzyme concentration, myoglobinemia, and abnormal muscle structure and histology. A further response to sarcomere damage is necrosis following damage to the mysium, which peaks about 48 hours following exercise.
The muscle adapts rapidly to the structural damage caused by exercise, and further soreness and damage in later exercise events is mitigated.


2.2: Prewriting
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Prewriting is an essential activity for most writers. Through robust prewriting, writers generate ideas, explore directions, and find their way into their writing. When students attempt to write an essay without developing their ideas, strategizing their desired structure, and focusing on precision with words and phrases, they can end up with a “premature draft”—one that is more writer-based than reader-based and, thus, might not be received by readers in the way the writer intended.
In addition, a lack of prewriting can cause students to experience writer’s block. Writer’s block is the feeling of being stuck when faced with a writing task. It is often caused by fear, anxiety, or a tendency toward perfectionism, but it can be overcome through prewriting activities that allow writers to relax, catch their breath, gather ideas, and gain momentum.
The following exist as the goals of prewriting:
- Contemplating the many possible ideas worth writing about.
- Developing ideas through brainstorming, freewriting, and focused writing.
- Planning the structure of the essay overall so as to have a solid introduction, meaningful body paragraphs, and a purposeful conclusion.
Discovering and Developing Ideas
Quick Prewriting Activities
Quick strategies for developing ideas include brainstorming, freewriting, and focused writing. These activities are done quickly, with a sense of freedom, while writers silence their inner critic. In her book Wild Mind , teacher and writer Natalie Goldberg describes this freedom as the “creator hand” freely allowing thoughts to flow onto the page while the “editor hand” remains silent. Sometimes, these techniques are done in a timed situation (usually two to ten minutes), which allows writers to get through the shallow thoughts and dive deeper to access the depths of the mind.
Brainstorming begins with writing down or typing a few words and then filling the page with words and ideas that are related or that seem important without allowing the inner critic to tell the writer if these ideas are acceptable or not. Writers do this quickly and without too much contemplation. Students will know when they are succeeding because the lists are made without stopping.
Freewriting is the “most effective way” to improve one’s writing, according to Peter Elbow, the educator and writer who first coined the term “freewriting” in pivotal book Writing Without Teachers , published in 1973. Freewriting is a great technique for loosening up the writing muscle. To freewrite, writers must silence the inner critic and the “editor hand” and allow the “creator hand” a specified amount of time (usually from 10 to 20 minutes) to write nonstop about whatever comes to mind. The goal is to keep the hand moving, the mind contemplating, and the individual writing. If writers feel stuck, they just keep writing “I don’t know what to write” until new ideas form and develop in the mind and flow onto the page.
Focused freewriting entails writing freely—and without stopping, during a limited time—about a specific topic. Once writers are relaxed and exploring freely, they may be surprised about the ideas that emerge.
Researching
Unlike quick prewriting activities, researching is best done slowly and methodically and, depending on the project, can take a considerable amount of time. Researching is exciting, as students activate their curiosity and learn about the topic, developing ideas about the direction of their writing. The goal of researching is to gain background understanding on a topic and to check one’s original ideas against those of experts. However, it is important for the writer to be aware that the process of conducting research can become a trap for procrastinators. Students often feel like researching a topic is the same as doing the assignment, but it’s not.
The two aspects of researching that are often misunderstood are as follows:
- Writers start the research process too late so the information they find never really becomes their own setting themselves up for way more quoting, paraphrasing, and summarizing the words of others than is appropriate for the 70% one’s own words and 30% the words of others ratio necessary for college-level research-based writing.
- Writers become so involved in the research process that they don’t start the actual writing process soon enough so as to meet a due date with a well written, edited, and revised finished composition.
Being thoughtful about limiting one’s research time—and using a planner of some sort to organize one’s schedule—is a way to keep oneself from starting the research process too late to
See Part V of this book for more information about researching.
Audience and Purpose
It’s important that writers identify the audience and the purpose of a piece of writing. To whom is the writer communicating? Why is the writer writing? Students often say they are writing for whomever is grading their work at the end. However, most students will be sharing their writing with peers and reviewers (e.g., writing tutors, peer mentors). The audience of any piece of college writing is, at the very minimum, the class as a whole. As such, it’s important for the writer to consider the expertise of the readers, which includes their peers and professors). There are even broader applications. For example, students could even send their college writing to a newspaper or a legislator, or share it online for the purpose of informing or persuading decision-makers to make changes to improve the community. Good writers know their audience and maintain a purpose to mindfully help and intentionally shape their essays for meaning and impact. Students should think beyond their classroom and about how their writing could have an impact on their campus community, their neighborhood, and the wider world.
Planning the Structure of an Essay
Planning Based on Audience and Purpose
Identifying the target audience and purpose of an essay is a critical part of planning the structure and techniques that are best to use. It’s important to consider the following:
- Is the the purpose of the essay to educate, announce, entertain, or persuade?
- Who might be interested in the topic of the essay?
- Who would be impacted by the essay or the information within it?
- What does the reader know about this topic?
- What does the reader need to know in order to understand the essay’s points?
- What kind of hook is necessary to engage the readers and their interest?
- What level of language is required? Words that are too subject-specific may make the writing difficult to grasp for readers unfamiliar with the topic.
- What is an appropriate tone for the topic? A humorous tone that is suitable for an autobiographical, narrative essay may not work for a more serious, persuasive essay.
Hint: Answers to these questions help the writer to make clear decisions about diction (i.e., the choice of words and phrases), form and organization, and the content of the essay.
Use Audience and Purpose to Plan Language
In many classrooms, students may encounter the concept of language in terms of correct versus incorrect. However, this text approaches language from the perspective of appropriateness. Writers should consider that there are different types of communities, each of which may have different perspectives about what is “appropriate language” and each of which may follow different rules, as John Swales discussed in “The Concept of Discourse Community.” Essentially, Swales defines discourse communities as “groups that have goals or purposes, and use communication to achieve these goals.”
Writers (and readers) may be more familiar with a home community that uses a different language than the language valued by the academic community. For example, many people in Hawai‘i speak Hawai‘i Creole English (HCE colloquially regarded as “Pidgin”), which is different from academic English. This does not mean that one language is better than another or that one community is homogeneous in terms of language use; most people “code-switch” from one “code” (i.e., language or way of speaking) to another. It helps writers to be aware and to use an intersectional lens to understand that while a community may value certain language practices, there are several types of language practices within our community.
What language practices does the academic discourse community value? The goal of first-year-writing courses is to prepare students to write according to the conventions of academia and Standard American English (SAE). Understanding and adhering to the rules of a different discourse community does not mean that students need to replace or drop their own discourse. They may add to their language repertoire as education continues to transform their experiences with language, both spoken and written. In addition to the linguistic abilities they already possess, they should enhance their academic writing skills for personal growth in order to meet the demands of the working world and to enrich the various communities they belong to.
Use Techniques to Plan Structure
Before writing a first draft, writers find it helpful to begin organizing their ideas into chunks so that they (and readers) can efficiently follow the points as organized in an essay.
First, it’s important to decide whether to organize an essay (or even just a paragraph) according to one of the following:
- Chronological order (organized by time)
- Spatial order (organized by physical space from one end to the other)
- Prioritized order (organized by order of importance)
There are many ways to plan an essay’s overall structure, including mapping and outlining.
Mapping (which sometimes includes using a graphic organizer) involves organizing the relationships between the topic and other ideas. The following is example (from ReadWriteThink.org, 2013) of a graphic organizer that could be used to write a basic, persuasive essay:
Persuasion Map. Copyright 2013 IRA/NCTE. All rights reserved. ReadWriteThink materials may be reproduced for educational purposes.
Outlining is also an excellent way to plan how to organize an essay. Formal outlines use levels of notes, with Roman numerals for the top level, followed by capital letters, Arabic numerals, and lowercase letters. Here’s an example:
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Introduction
- Hook/Lead/Opener: According to the Leilani was shocked when a letter from Chicago said her “Aloha Poke” restaurant was infringing on a non-Hawaiian Midwest restaurant that had trademarked the words “aloha” (the Hawaiian word for love, compassion, mercy, and other things besides serving as a greeting) and “poke” (a Hawaiian dish of raw fish and seasonings).
- Background information about trademarks, the idea of language as property, the idea of cultural identity, and the question about who owns language and whether it can be owned.
- Thesis Statement (with the main point and previewing key or supporting points that become the topic sentences of the body paragraphs): While some business people use language and trademarks to turn a profit, the nation should consider that language cannot be owned by any one group or individual and that former (or current) imperialist and colonialist nations must consider the impact of their actions on culture and people groups, and legislators should bar the trademarking of non-English words for the good of internal peace of the country.
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Body Paragraphs
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Main Point (Topic Sentence): Some business practices involve co-opting languages for the purpose of profit.
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Supporting Detail 1: Information from Book 1—Supporting Sentences
- Subpoint
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Supporting Detail 2: Information from Article 1—Supporting Sentences
- Subpoint
- Subpoint
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Supporting Detail 1: Information from Book 1—Supporting Sentences
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Main Point (Topic Sentence): Language cannot be owned by any one group or individual.
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Supporting Detail 1: Information from Book 1—Supporting Sentences
- Subpoint
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Supporting Detail 2: Information from Article 1—Supporting Sentences
- Subpoint
- Subpoint
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Supporting Detail 1: Information from Book 1—Supporting Sentences
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Main Point (Topic Sentence): Once (or currently) imperialist and colonialist nations must consider the impact of their actions on culture and people groups.
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Supporting Detail
- Subpoint
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Supporting Detail
- Legislators should bar the trademarking of non-English words for the good and internal peace of the country.
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Main Point (Topic Sentence): Some business practices involve co-opting languages for the purpose of profit.
- Conclusion ( Revisit the Hook/Lead/Opener, Restate the Thesis, End with a Twist— a strong more globalized statement about why this topic was important to write about)
Note about outlines: Informal outlines can be created using lists with or without bullets. What is important is that main and subpoint ideas are linked and identified.
Activities
- Use 10 minutes to freewrite with the goal to “empty your cup”—writing about whatever is on your mind or blocking your attention on your classes, job, or family. This can be a great way to help you become centered, calm, or focused, especially when dealing with emotional challenges in your life.
- For each writing assignment in class, spend three 10-minute sessions either listing (brainstorming) or focused-writing about the topic before starting to organize and outline key ideas.
- Before each draft or revision of assignments, spend 10 minutes focused-writing an introduction and a thesis statement that lists all the key points that supports the thesis statement.
- Have a discussion in your class about the various language communities that you and your classmates experience in your town or on your island.
- Create a graphic organizer that will help you write various types of essays.
- Create a metacognitive, self-reflective journal: Freewrite continuously (e.g., 5 times a week, for at least 10 minutes, at least half a page) about what you learned in class or during study time. Document how your used your study hours this week, how it felt to write in class and out of class, what you learned about writing and about yourself as a writer, how you saw yourself learning and evolving as a writer, what you learned about specific topics. What goals do you have for the next week?
During the second half of the semester, as you begin to tackle deeper and lengthier assignments, the journal should grow to at least one page per day, at least 20 minutes per day, as you use journal writing to reflect on writing strategies (e.g., structure, organization, rhetorical modes, research, incorporating different sources without plagiarizing, giving and receiving feedback, planning and securing time in your schedule for each task involved in a writing assignment) and your ideas about topics, answering research questions, and reflecting on what you found during research and during discussions with peers, mentors, tutors, and instructors. The journal then becomes a record of your journey as a writer, as well as a source of freewriting on content that you can shape into paragraphs for your various assignments.
Works Cited
Elbow, Peter. Writing Without Teachers . 2nd edition, 1973, Oxford UP, 1998.
Goldberg, Natalie. Wild Mind : Living the Writer’s Life. Bantam Books, 1990.
Swales, John. “The Concept of Discourse Community.” Genre Analysis: English in Academic and Research Settings , 1990, pp. 21-32.
For more about discourse communities, see the online class by Robert Mohrenne “ What is a Discourse Community? ” ENC 1102 13 Fall 0027. University of Central Florida, 2013.


2.7: Unit Heaters
Click play on the following audio player to listen along as you read this section.
Unit heaters are often installed at ceiling height or recessed into a wall, and consist of a heating element, a built in blower fan and louvers to direct the direction of heated airflow. They are small and compact in size, yet have a high capacity for heating a room or area.
If a single unit heater is to be installed, its heated airflow should be directed towards the area of highest ambient heat loss, such as a window or door, however care should be taken to make sure that no surface is heated beyond a safe level.
If multiple unit heaters are installed their air flow should be directed in a circular motion along the exposed outer walls.
Control of unit heaters is similar to that of baseboard heaters, in that there must be a temperature activated switch, either installed directly on the heater unit, or more effectively, in another part of the room. A built-in fan switch is also provided to allow the fan to circulate air during warm weather without energizing the heater elements.
Unit heaters are available anywhere from 2 kW to 60 kW, with units on the lower side, up to approximately 5 kW usually being single-phase AC and controlled by low-voltage thermostats and relays, while units rated above 5 kW are usually three-phase.


18.8: Global Leadership
- Describe the importance of nurturing global leadership
Strong leadership is a cornerstone of all successful companies. Strong global leadership begins with a global mind-set. A global mind-set is “characterized by (1) tolerating, accepting, and understanding diversity with an inclusive mind-set; (2) a broad and universal perspective of business; and (3) thinking openly, free from cognitive cobwebs.” [1] People with a global mindset are able to keep an open mind around cultural nuances and make thoughtful decisions with these differences in mind. They are inclusive and understand the benefits of diversity. Global mindset is the foundation of all strong global leaders, as it lays the groundwork for cultural acceptance and understanding.
Mindfulness
Mindfulness is a key component to a global mindset and global leadership. Mindfulness is, “a state of mind characterized by heightened awareness of self and the surrounding environment, and to be non evaluative and nonjudgmental in experiencing the present.” [2] Strong leaders need to be aware of their surroundings and understanding of those they interact with. Mindfulness is an invaluable skill for leaders to have in their toolkit. Mindfulness and a global mindset work together to help form a strong global leader. Global leaders interact with a large number of people across multiple cultures. Therefore, in order for global leaders to be successful, they need to be mindful of their surroundings and view the world through a global mindset.
Employee Development
In today’s day and age, international expansion of many companies is becoming the norm. Employees who want to expand their horizons, and gain valuable experience and insight into other cultures may consider taking company assignments in foreign countries. So, how do these employees prepare to be strong global leaders? Training is the best way to nurture global leaders and ensure they have all the knowledge and tools necessary to be successful. International companies who are invested in training global leaders will rapidly reap the benefits.
A Shrinking World
Since technology continues to shrink the world around us, global leaders will continue to play a key role in international expansion. Strong global leaders are able to keep track of cultural differences, as well as compliance and legal concerns, that may differ from one country to the next. Equally as important, successful global leaders are able to seamlessly infuse differing cultures, languages, and personalities to form one strong and cohesive team. When an international company has a unified team, they are better able to quickly and effectively expand. A strong international team also contributes to productivity and profit. If a company has strong global leaders, they are better able to grow and unify the company. Therefore, all international companies should strive to nurture global leadership on their teams.
Sources
Mansoor, Fatima. “ The Importance of Developing Global Leaders for Company Growth: Extraordinary Business .” Aepiphanni Business Consulting, October 17, 2017.
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Chandwani, Rajesh, Narendra M. Agrawal, and Ben L. Kedia. “Mindfulness: Nurturing Global Mind-Set and Leadership.”
Thunderbird International Business Review
58, no. 6 (2015): 617–25. ↵
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Ibid. ↵
Contributors and Attributions
- Global Leadership. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution
- Authored by : blackmachinex. Provided by : Pixabay. Located at : pixabay.com/photos/office-people-team-group-woman-4480198/. License : CC0: No Rights Reserved . License Terms : Pixabay License
- Authored by : MichaelGaida. Provided by : Pixabay. Located at : pixabay.com/photos/globe-map-country-borders-old-3383088/. License : CC0: No Rights Reserved . License Terms : Pixabay License


1.7: Whole Number Place Value
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- Janet Stramel
- Fort Hays State University
The focus of chapter 6 was on number sense and a relational understanding of numbers. Number sense is linked to place value and an understanding of the base-ten number system. The progression of place value across grades K-5 is critical for understanding:
- Decomposing numbers in base ten
- Reading and writing numbers
- Rounding numbers
- Comparing numbers and quantities in base ten.
By the end of kindergarten, students are expected to count to 100 and count sets of 20 (KSDE, 2017a). In kindergarten, counting is based on a ones approach – the number 15 means 15 ones. According to Wright, et. al. (2006), there is a progression to understanding ten:
- children understand ten as ten ones,
- children see ten as a unit, and
- children easily work with units of ten.
Consider the phrase, “ten ones make one ten.” And now think like a child. How does this make sense? Students in phase one count a set of items and think of that set as ones. When students move to phase two, they begin to see a group of ten as a unit, such as a group of ten ones.
We use the base-ten system; using ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Every number can be represented using these digits. In the base-ten system, the value of each place is always 10 times the value of the place to the immediate right. When moving one place left, the value of the place is multiplied by 10.
In the base-ten system, the “places” are ones, tens, hundreds, thousands, etc. And the digit in each place represents 0-9 of those units. Students learn that ten ones makes a unit, called a ten. They then learn that two-digit numbers are composed of ones and tens.
Place value is a fundamental concept in the elementary grades, and understanding place value is essential in learning mathematics. Place value is the value of the digit in its position. For example, the number 358 has three columns or “places,” each with a specific value. In 358, the 3 is in the “hundreds” place, the 5 is in the “tens” place, and the 8 is in the “ones” place. But more importantly, the value of each digit is 3 hundreds (or 300), 5 tens (or 50), and 8 ones.
Base-Ten Blocks
Base-ten blocks provide a hands-on model of our base-ten number system. The smallest cubes are called units. The long, narrow blocks are called rods. The flat, square blocks are called flats. The large cubes are called cubes. The size relationships of the base-ten blocks are perfectly designed to help children discover that it takes 10 units to make one rod, or 10 ones to make one ten; and 10 tens to make one hundred.
Base-ten blocks are ideal for students to physically manipulate “the numbers” so they can conceptually understand the concepts of place value. Nothing can replace the physical manipulatives used to help students make the connections from concrete to abstract understanding.
An example showing the relationship among the manipulatives, the place or position, and the place value of the digits in a number is below.
The number 235 is written in standard notation . This shows the “value” of each digit. There are two hundreds, three tens, and five ones. Additionally, in the number 235: 2 is in hundreds place and its place value is 200, 3 is in tens place and its place value is 30, 5 is in ones place and its place value is 5.
Understanding the place value of digits in numbers helps in writing numbers in their expanded form . For instance, the expanded notation of the number 235 is 200 + 30 + 5.
Read more about the Base-Ten Number System by clicking here.
Developing Whole Number Place Value Concepts across the Grades
When thinking about place value and base-ten understanding, first consider the 5 Strands of Mathematical Proficiency from the National Research Council (2001).
Mathematical proficiency is based upon five interwoven components:
- Conceptual understanding – comprehension of mathematical concepts, operations, and relations
- Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
- Strategic competence – ability to formulate, represent, and solve mathematical problems
- Adaptive reasoning – capacity for logical thought, reflection, explanation, and justification
- Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy (NRC, 2001, p. 116).
Read more about the 5 Strands of Mathematical Proficiency by downloading the book, “Helping Children Learn Mathematics.”
According to James Brickwedde (2012), “place value and the base ten system is an early and easy entry point for students to begin to explore this agility. Without this level of flexibility and fluency, students are limited to inefficient strategies or are overly dependent upon tactical procedures they know only through rote application.”
Your language matters when teaching students about place value. Speak in value, not in digits. For example, the value of the 2 in 26 is two tens or 20.
Kindergarten
Kindergarten is the first time many students work with numbers greater than 10 using manipulatives and/or drawings. Kindergarten students separate (decompose) a set of 11-19 objects into a group of 10 and some other ones. Experiences with double ten frames will help students to understand this concept.
Notice the grouping of the ten. It is obvious that there is one group of ten with some others left over. Students will immediately see that 14 is made up of one ten and four ones.
The “teen” numbers are one group of ten and some more ones. Notice the “teen” numbers that do not follow a particular pattern in the counting sequence. First, think of the number names 11-19: eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen.
- Eleven and twelve are special number words that do not have “teen” as a suffix.
- The verbal counting of “teen” numbers is backwards: the ones digit is said before the tens digit. For example, 37 is read as thirty-seven; tens to ones. But the number 14 is read as fourteen; read ones to tens.
When teaching the “teen” numbers, ask students to read the number as well as describe the quantity. For the number 18, students should read “eighteen” and then say, “18 is one group of ten and eight ones.” Additionally, students should record the number sentence 18 = 10 + 8.
By the end of kindergarten, students should be able to compose and decompose numbers between 11 and 19 into tens and some ones. When teaching to compose and decompose numbers, students must use manipulatives and drawings.
Source: KSDE Flip Book Kindergarten (2017b)
Check out the task “ What Makes a Teen Number? ” at the Illustrative Mathematics website. This activity helps students decompose teen numbers using ten-frames and number sentences.
First Grade
In the first grade Domain Number and Operations in Base Ten, the second Cluster is understand place value. Thinking about “10 ones makes 1 ten” is confusing for students. In first grade, students should see 10 ones as 1 ten and that 1 ten has the same value as 10 ones.
Students need many opportunities to practice grouping 10 ones into a bundle of 10. In first grade, students begin to unitize . They see that a group of ten objects is also one ten.
Watch this video “ One Is One, Or Is It? ” narrated by Christopher Danielson for more information on unitizing.
Students must be given numerous experiences using ten-frames, snap cubes, or other groupable models to help develop the concept of place value.
Suggested Activity using Ten-Frames:
Give students 13 counters.
Teacher: “Do you have enough to make a ten?” “Would you have any left over?” “If so, how many left overs do you have?”
Student: “I have filled up one ten-frame and have 3 counters left over. The number 13 has 1 ten and 3 ones.
In first grade, students explore the decade numbers (10, 20, 30, 40, 50, 60, 70, 80, 90) as groups of ten with no ones left over. Students use manipulatives to group, or bundle, groups of ten. As students count any number up to 99, they should group/bundle tens and some more. Furthermore, they should focus on the mathematical language associated with the quantity. For example, the number 62 can be expressed as 62 ones; or 6 groups of ten with 2 left over.
In addition, students should read numbers in standard form AND using place value concepts. Read the number 62 as “sixty-two” as well as 6 tens and 2 ones.
First grade students “compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the relational symbols >, <, =, and ≠”. According to Dougherty, Flores, Louis, & Sophian (2010), students should determine whether quantities are equal or not equal before using “greater than” and “less than.” Once students know the two quantities are not equal, then teachers can have a discussion about “how” they are not equal by asking the following questions. “Which one is greater?” “Which one is less?” These discussions must occur before using the greater than and less than symbols.
Give students extensive experiences to explain their thinking using words and models before using the symbols. Students can use manipulatives to model the two numbers, as well as pictures or number lines. Again, it is critical that students develop the concept of composing and decomposing tens and ones. This is foundational to understanding place value and involves number relationships and promotes a flexibility with mental computation. Furthermore, students must learn through a progression of representations. They begin with concrete models, then move to pictorial models, and then abstract models.
Second Grade
In second grade, students work on decomposing numbers by using place value. Give students extensive experiences with manipulatives as well as pictorial representations of numbers. Students also should be able to decompose numbers into hundred, tens, and ones in several different combinations. For example, 268 could be shown as 2 hundreds, 6 tens, and 8 ones. But is also correct to show 268 as 26 tens and 8 ones, OR 1 hundred, 16 tens, and 8 ones. There are several ways to represent the number 268:
Building on their work in first grade, second grade students need multiple opportunities to count and bundle groups of hundreds. It is critical that students understand that 100 is 10 tens as well as 100 ones.
Students in second grade also learn to read, write, and represent a number of objects in various forms
- Base-ten numerals – 268
- Number names – two hundred sixty-eight
- Expanded notation – 200 + 60 + 8
- Unit form – 2 hundreds, 6 tens, 8 ones
Notice, in the number names, the word “and” is not used between any of the whole number words. Save the word “and” for decimals.
Second grade students build on their work in first grade to compare two numbers and use the relational symbols of <, >, =, and ≠. Give students extensive experiences to explain their thinking using words and models before using the symbols. Students can use manipulatives to model the two numbers, as well as pictures or number lines.
Another way to ensure that students develop a conceptual understanding of the relational symbols (<, >, =, and ≠), use a number line. Be sure to put the greater than and less than symbols above the number line.
Then share the two numbers you want to compare; for example 14 and 7. Explain to students that you want to compare where 14 is in relation to the number 7. So you can move the two relational symbols above the 7.
Ask students where 14 is in relation to 7. Is it to the left, which would be lesser number? Or is it to the right, which would be greater number?
Using relationships truly focuses on the numbers and on tricks.
Additionally, second grade students fluently add using strategies based on place value. For example, using the place value strategy for 56 + 27, a student might say:
- I decomposed 56 and 27 into tens and ones.
- 5 tens plus 2 tens is 7 tens.
- Then I added the ones; 6 ones and 7 ones is 13 ones.
- Then I combined the tens and ones. 7 tens plus 13 ones is 83.
Source: 2017 Kansas Mathematics Standards Flip Book 2nd Grade
Rounding Whole Numbers
The first time students encounter rounding whole numbers is in third grade. Students “use place value understanding to round whole numbers to the nearest 10 or 100.” Additionally, students should have a deep understanding of place value and number sense in order to explain and reason about their rounded answers.
When teaching students about rounding , you are deciding which number is closest. For example, 127 is closer to 100 than it is 200. If students don’t know this, then there is a gap in their place value understanding.
A strong understanding of place value is critical for students to understand rounding of whole numbers. Don’t begin by telling students the rules; use a number line so that students can develop a conceptual understanding of rounding. Students need to learn when and why to round numbers by first identifying possible answers and the halfway point.
Using an open number line to teach students about rounding will help both their rounding skills and strengthen place value understanding. For example, round 167 to the nearest ten.
Source: KSDE Flip Book 3rd Grade (2017c)
Teaching Greater Than and Less Than
Don’t teach greater than and less than using methods such as “Pac Man” or “Alligator” or other aids. When you use these, students don’t grasp the full meaning of the relational symbol. If you must use one of these “learning aids,” be sure to include the mathematical name and symbol with it. But first, explicitly teach that the symbols have names. Second, ask students to read the entire inequality, reading the numbers and symbols left to right, in the same way they would read a sentence.
For example, when students write 5 < 8, they would read this sentence as “five is less than eight.” This is a critical step in learning these relational symbols. Additionally, when students read the inequality aloud, they can recognize errors. For example, 5 < 8, if students say five is greater than eight, that does not make sense and they can correct their mistake.
Calculators in the Elementary Mathematics Classroom
The NCTM (2015) Position Statement, “Calculator Use in Elementary Grades,” states:
Calculators in the elementary grades serve as aids in advancing student understanding without replacing the need for other calculation methods. Calculator use can promote the higher-order thinking and reasoning needed for problem solving in our information- and technology-based society. Their use can also assist teachers and students in increasing student understanding of and fluency with arithmetic operations, algorithms, and numerical relationships and enhancing student motivation. Strategic calculator use can aid students in recognizing and extending numeric, algebraic, and geometric patterns and relationships.
Calculators in the elementary classroom are essential in helping students make sense of mathematics and reason mathematically. Teachers need to plan for the strategic use of calculators that will support student thinking and assist them in making connections to real-world situations.
Read NCTM’s Position Statement “ Calculator Use in Elementary Grades .”


19.6: Emerging and Reemerging Infectious Diseases
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Learning Objectives
- Give examples of emerging and remerging infectious diseases
An emerging infectious disease (EID) is an infectious disease whose incidence has increased in the past 20 years, and could increase in the near future. Emerging infections account for at least 12% of all human pathogens. EIDs are caused by newly identified species or strains (e.g., SARS, AIDS) that may have evolved from a known infection (e.g., influenza), or spread to a new population (e.g., West Nile virus ), or to an area undergoing ecologic transformation (e.g., Lyme disease). They could also be reemerging infections, such as drug resistant tuberculosis. Of growing concern are adverse synergistic interactions between emerging diseases and other infectious and non-infectious conditions leading to the development of novel syndemics.
SARS
Severe acute respiratory syndrome (SARS) is a viral respiratory disease in humans which is caused by the SARS coronavirus (SARS-CoV). Between November 2002 and July 2003, an outbreak of SARS in Hong Kong nearly became a pandemic, with 8,422 cases and 916 deaths worldwide (10.9% fatality), according to the World Health Organization (WHO). Within weeks, SARS spread from Hong Kong to infect individuals in 37 countries. The last infected human case of the outbreak occurred in June 2003, and there was a laboratory-induced infection case in 2004. SARS is not claimed to have been eradicated (unlike smallpox), as it may still be present in its natural host reservoirs (animal populations) and may return to the human population. During the outbreak, the fatality of SARS was less than 1% for people aged 24 or younger, 6% for those 25 to 44, 15% for those 45 to 64, and more than 50% for those over 65. For comparison, the fatality of influenza is usually under 0.03% (primarily among the elderly), but rose to 2% during the most severe pandemic to date.
HIV
Human immunodeficiency virus infection/acquired immunodeficiency syndrome (HIV/AIDS) is a disease of the human immune system caused by the human immunodeficiency virus (HIV). During the initial infection a person may experience a brief period of influenza-like illness. This is typically followed by a prolonged period without symptoms. As the illness progresses it interferes more and more with the immune system, making people much more likely to get infections, including opportunistic infections, and tumors that do not usually affect people with working immune systems.
Influenza
Influenza, commonly known as the flu, is an infectious disease of birds and mammals caused by RNA viruses of the family Orthomyxoviridae, the influenza viruses. The most common symptoms are chills, fever, sore throat, muscle pains, headache (often severe), coughing, weakness/fatigue and general discomfort. Although it is often confused with other influenza-like illnesses, especially the common cold, influenza is a more severe disease caused by a different type of virus. Influenza may produce nausea and vomiting, particularly in children, but these symptoms are more common in the unrelated gastroenteritis, which is sometimes inaccurately referred to as “stomach flu” or “24-hour flu. ”
West Nile Virus
West Nile virus (WNV) is a mosquito-borne zoonotic arbovirus belonging to the genus flavivirus in the family flaviviridae. This flavivirus is found in temperate and tropical regions of the world. It was first identified in the West Nile subregion in the East African nation of Uganda in 1937. Prior to the mid 1990s, WNV disease occurred only sporadically and was considered a minor risk for humans. However, there was an outbreak in Algeria in 1994, with cases of WNV-caused encephalitis, and the first large outbreak in Romania in 1996, with a high number of cases with neuroinvasive disease. WNV has now spread globally, with the first case in the Western Hemisphere being identified in New York City in 1999; over the next 5 years, the virus spread across the continental United States, north into Canada, and southward into the Caribbean Islands and Latin America. WNV also spread to Europe, beyond the Mediterranean Basin. A new strain of the virus was recently (2012) identified in Italy. WNV is now considered to be an endemic pathogen in Africa, Asia, Australia, the Middle East, Europe and in the United States, which in 2012 has experienced one of its worst epidemics.
Tuberculosis
Tuberculosis, MTB, or TB (short for tubercle bacillus) is a common, and in many cases lethal, infectious disease caused by various strains of mycobacteria, usually Mycobacterium tuberculosis. Tuberculosis typically attacks the lungs, but can also affect other parts of the body. It is spread through the air when people who have an active TB infection cough, sneeze, or otherwise transmit their saliva through the air. Most infections are asymptomatic and latent, but about one in ten latent infections eventually progresses to active disease which, if left untreated, kills more than 50% of those so infected.
Key Points
- Emerging infections account for at least 12% of all human pathogens.
- Emerging infections deseases are caused by newly identified species or strains that may have evolved from a known infection or spread to a new population or area undergoing ecologic transformation, or be reemerging infections, such as drug resistant tuberculosis.
- Adverse synergistic interactions between emerging diseases and other infectious and non-infectious conditions leading to the development of novel syndemics are of growing concern.
Key Terms
- emerging infectious disease : An emerging infectious disease (EID) is an infectious disease with an incidence rate that has increased in the past 20 years and could increase in the near future. Emerging infections account for at least 12% of all human pathogens.
- pathogens : A pathogen or infectious agent (colloquially known as a germ) is a microorganism (in the widest sense, such as a virus, bacterium, prion, or fungus) that causes disease in its host. The host may be an animal (including humans), a plant, or even another microorganism.
- species : In biology, a species is one of the basic units of biological classification and a taxonomic rank. A species is often defined as a group of organisms capable of interbreeding and producing fertile offspring.