Within questions on truth tables it is not possible to distinguish general assumed and general tested skills across the full range of possible truth tables. They vary in their degree of complexity and therefore more complicated propositions will require more knowledge. Therefore, the skills tested on easy examples will become assumed skills for the more complicated examples.
Regardless of the difficulty of the proposition, students should be able to read a truth
table, i.e. to identify the statements involved, indicate the compound statement for which
the truth table is built, understand the size of it (i.e. the number of rows containing all possible combinations of truth values the statements can take) and to understand what the truth values under the compound proposition mean.
Going one step further, students are expected to know how to produce the truth table themselves. There are four sections in a truth table (see Table 5.1.1a) and students are expected to know how to fill them in.
Table 5.1.1a: Format of the truth tables.
Section 1 Section 2
Section 3 Section 4
Firstly, the compound statement for which truth values have to be found should be written down in section 2. Then, its propositions (p, q, r etc.) should be identified and listed in a single row of section 1. After that students should show ability to lay out a truth table in a logical way in section 3. What is meant by this is that they should use a pattern when filling in rows with truth values below the statements listed in section 1. The standard method is as follows: the columns under the letter representing the first statement should be divided in half (i.e. 21 parts for the first statement), the top part should contain T’s
whereas the bottom F’s; the area under the next letter should be divided into 22 parts,
each part should be filled in alternately with T’s and F’s starting with T’s. Depending on the number of statements the area under the nth statement would be divided into 2n parts
and each part should have T’s and F’s listed alternately giving 2n rows. Lastly, it is
expected from students to know that section 4 should be filled in with truth values for the proposition written in section 2. This is the only section that is actually tested by the question developed here. However, students are expected to learn how to fill in sections 1-3 from the many examples they are exposed to – see Chapter 10 for further discussion. Asking students to find truth values of propositions with one operator allows clear specification of the tested skill. Three elementary operations can be distinguished: negation, conjunction, disjunction, as well as few more complicated operations that can be built up from these three (e.g. implication,
p q
p q
or equivalent). All of them form a set of eight connectives that are used within the questions and are as follows:1. Negation – if p is a statement, the negation of p is ‘not p’ denoted ~p. This means that if p is true, ~p is false; if p is false, ~p is true. It is a unary operator since it acts on just one statement (in contrast to all the others listed below).
2. Conjunction – if p and q are statements, the conjunction of p and q is ‘p and q’, denoted p q . It is true only when both are true and false in other cases.
3. Disjunction – if p and q are statements, the disjunction of p and q is ‘p or q’, denoted
p . It is false only when both are false and true in other cases. q
4. Implication (conditional statement) – if p and q are statements, the conditional of p and q is ‘if p then q’, denoted p . It is false only when first statement is true and q
the second is false, true in other cases. It is best thought of as a ‘broken premise’. 5. Equality (biconditional statement) – if p and q are statements, the biconditional of p
and q is ‘p if and only if q’, denoted p . It is true only when both statements have q
the same truth value and false in other cases.
6. Exclusive disjunction (XOR) – if p and q are statements, the exclusive disjunction of p and q is denotedpq. It is true both statements have different truth values, false in
other cases.
7. Logical NAND – denoted p │ q . It is false when both p and q are true, true in other cases.
8. Logical NOR – denoted
pq
. It is true when both p and q are false, false in other cases.All of the above operations have truth values assigned. They are defined in words above, but all the possible combinations of truth values can also be presented in the table called truth table (Table 5.1.1b).
Table 5.1.1b: Truth table containing logical operators.
~ ~ → ↔ ⊕ | ↓
T T F F T T T T F F F
T F F T F T F F T T F
F T T F F T T F T T F
Generally, one could be asked for the equivalent representation of statements with operations 4-8 above using
,
and.
However, all the statements can also be constructed using only NAND or only NOR (Epp, 2010, pp. 74-75). For example, the NOT, AND, and OR connectives can be written using NAND alone in the following way:~ p ≡ p │ p,
p q ≡ ( p │ q ) │ ( p │ q ), p q ≡ ( p │ p ) │ ( q │ q ), or using NOR alone:
~ p ≡ p ↓ p,
p q ≡ ( p ↓ p ) ↓ ( q ↓ q ), p q ≡ ( p ↓ q ) ↓ ( p ↓ q ).
The three remaining operations could be expressed using NAND or NOR only by using their equivalent forms, e.g.
p q
p q
,
p q pq pq ,
p q pq pq ,
and the results for
,
and.
These relations are not tested here, but are recommended in Section 11.5 for future work.For more complicated propositions, knowledge of connectives’ truth values is now an assumed skill. As for mathematical operations, logical operations must also be performed according to precedence rules. Therefore, this is another assumed skill. However, different sources give different order of precedence. According to Epp (2010), ~ should be performed first, then operations ˄ and ˅ as they are considered to be coequal, as are → and ↔ that should be evaluated third. On the other hand, some authors, including Rosen (2007), state that operations should be executed in the following order: ~, ˄, ˅, →, ↔. However, Rosen also suggests using parentheses for clarity. To avoid ambiguity, parentheses are used throughout when conjunction and disjunction, or conditional and biconditional statements are both present. Consequently all the questions have been formulated in the way that students that were taught either of two rules should be able to solve them.
5.1.2. Objective questions
Word input type questions
Because the possibility of constructing varying questions is limited (see Section 5.1), all of them are of the word input type. Figure 5.1 shows an example of students being asked to type their answers in the provided input box. The submitted answer should be in an appropriate format: a string of the mixture of three characters: ‘T’, ‘F’ and comma. Letter T indicates true value for operation, F stands for false value and commas are used to separate T’s and F’s. Participants are also instructed that there should be no spaces between and no full stop at the end of the answer. Hence, answers should be of T,T,F,T,F format. If they input spaces the marking function removes them; this also removes trailing spaces which are not even visible to the students.
Different styles
The database of the truth tables’ questions contains questions that are clones of each other. Every question is of different style as compound propositions were obtained by changing the number of statements and operators.
The number of statements used for each problem varies between two and three. The number of propositions specifies the size of the truth table. Number of rows in the table depends on the number of combinations of values that propositions can have and is equal to 2 , where n is the number of propositions. Therefore, tables with two variables will have 4 rows, with three variables 8 rows, whereas with four variables 16 rows. 16 rows makes the table big and difficult to read (especially for CAA where one would need to scroll up and down), so, although quite possible, no questions have been created that will result in this size. It could also be a problem when displaying the table on the computer screen when a big font size is being used. In any case, it is arguable that such questions would not test any new mathematical skill that could not be tested with 3 propositions.
When creating questions on truth tables, eight operators have been used and pre- allocated to each problem. Questions of different styles were created by changing operators and the location of parentheses or their omission. In the situation when questions contain one type of operator, it is clear what knowledge has been tested, whether the student assimilated the tested knowledge, or if more time should be spent to understand specific areas. Formulating problems with mixture of connectives makes questions more interesting and harder to solve. On the other hand, inferring which skills are missing when student gets such a question wrong is more problematic. For instance,
getting wrong truth values of statements like a(ba) (see Figure 5.1.2a or Figure
5.1.2b) may be caused by not knowing one or both of the following: truth values for one or more operations used in the question, i.e. negation, implication, conjunction, disjunction; or precedence rules together with importance of brackets.
In this instance, questions were grouped with respect to the number and type of operators involved in the problem. This allows the test setter to decide what skills will be examined. Questions were classified in the following way:
Two or three operators of the same type (˄, ˅, → or ↔, coupled with negation), with parentheses to avoid ambiguity (Figure 5.1). These questions test students understanding of truth values for specific connectives;
Two operators of different types (˄, ˅, → or ↔, coupled with negation). Symbols ˄ and ˅ as well as → and ↔ do not appear together. No parentheses are used (Figure 5.1.3). These questions test knowledge about rules of precedence and of course about truth values;
Two, three or four operators of different type (˄, ˅, → or ↔, coupled with negation), with parentheses (Figure 5.1.2a and Figure 5.1.2b). These questions are more interesting and harder to solve, so require more thinking, more time and more knowledge (e.g. importance of brackets). The more complex the expression is, the more important it is to establish the order in which the table should be built up; Three to five operators of different type, including XOR, NAND and NOR, with
parentheses. These questions test students’ knowledge of ⊕, | and ↓.
Random parameters
Within questions on truth tables a few parameters are set randomly. Firstly, names (e.g. Martin, Claudia, Tania) with which the question starts and proposition names (e.g. , , ) are selected at random from the set of all possibilities. These trivial changes are known as ‘surface effects’ since they do not alter the nature of the questions in any mathematical way. Secondly, missing values from 1 to 5 (or any other number specified by the question designer) are randomly placed in the table. At this point however, possible entries are restricted (see ‘Restrictions on possible rows and columns for missing values’ in Section 5.1.4 for coding) to make sure that there will be no situation when student is asked for values of direct statements (i.e. , , ) or their negations (i.e. ~ , ~ , ~ ). Therefore, one can be certain that knowledge is tested rather than just the ability to rewrite. Furthermore,
the number of missing values can be easily changed (see ‘Number of truth values to be found’ in Section 5.1.4 for coding), and different restrictions can be set up.
An illustration of parameters that change every time the question is run is presented in Figure 5.1.2a and Figure 5.1.2b below.
Figure 5.1.2a: Realisation 1 of the truth table question (two operators of different type, with parentheses).
Figure 5.1.2b: Realisation 2 of the truth table question (two operators of different type, with parentheses).
5.1.3. Feedback
Questions in the database are formative and summative (see Section 1.1) at the same time. They test the knowledge, but also provide students with detailed and automatic feedback if the submitted answer is wrong. An example of such a feedback screen with description is presented in Figure 5.1.3 below.
Figure 5.1.3: Feedback of the questions on truth tables (two operators of different types, without parentheses).
As in all the questions, here also students are presented with the question itself, the submitted response and correct answer, as well as the related material button at the bottom of the screen. The part that is specific to this question is the completed truth table. Together with the truth values for the given proposition, students are provided with the order that the table must be built up indicated in red colour. Combining all this information together should allow students to identify their mistakes and understand what the right answer is. The effect of this information on students’ learning will be presented in Chapter 10.
5.1.4. Technical content
Technical content on truth tables will be presented in this section for the selection of questions shown earlier.
All the coding will be shown on the example presented in Figure 5.1 and this is as follows: Number of truth values to be found
Number of entries participant is asked to find is specified in a direct way but can be easily changed as this decision depends on the question designer.
Number of statements and attributed at random letters
terms = new Array;
i = Math.ceil(7*Math.random()); terms[1] = varname_logic(i,0); terms[2] = varname_logic(i+1,0); Nvar = terms.length-1;
varname_logic is a function in brunel_logic template (see Appendix E) that assigns
consecutive letters from the string of eleven characters ‘pqrstuvwxyz’ as propositions. For example, if ‘i’ gives value ‘q’ in our example then second proposition is ‘r’.
Nvar within coding stands for a number of propositional variables (Nvar = 2 for the
example in Figure 5.1). Since every array starts with the 0 index, and it has been left empty, 1 has to be subtracted in order to get the correct number of statements. Here,
terms[1] represents the first statement and terms[2] the second one.
Size of the table
Nrow = Math.pow(2,Nvar); Ncolumn = Nvar+10;
To find the number of needed rows (in addition to headers’ row) Math.pow(a,b) JavaScript function has been used. This method returns the value of the first parameter (2 in our example) to the power of the second parameter (Nvar above).
The number of columns is specified by the person creating the question and depends on the compound statement of which truth table has to be found (Nvar was 2, after adding 10 the table has 12 columns).
Restrictions on possible rows and columns for missing values
allowed_rows = new Array(1,2,3,4); allowed_columns = new Array(4,7,10);
Missing values of the table are located in every row, but only in selected columns. Columns that are allowed for questioning are those with operators rather than with single
or negated statements. This is easily done by direct listing numbers representing rows or columns.
Assigning truth values of the proposition in the table
elements= gettruthtable(MathML_terms,Nvar,Ncolumn); display_elements= gettruthtable(MathML_terms,Nvar,Ncolumn); elements[i][Nvar+1] = elements[i][1]; elements[i][Nvar+4] = NOT(elements[i][2]); elements[i][Nvar+7] = elements[i][1]; elements[i][Nvar+9] = elements[i][2]; elements[i][Nvar+8] = IF(elements[i][Nvar+7],elements[i][Nvar+9]); elements[i][Nvar+5] = IF(elements[i][Nvar+4],elements[i][Nvar+8]); elements[i][Nvar+2] = IF(elements[i][Nvar+1],elements[i][Nvar+5]);
Function gettruthtable(MathML_terms,Nvar,Ncolumn) is stored in the brunel_logic template and is given in Appendix E. It returns matrix elements for truth tables that are 0’s and 1’s. The function sets up first Nvar columns in a standard way for every question.
MathML_terms is an array containing the table headers. Functions ‘NOT’ and ‘IF’ are also
defined in the brunel_logic template and are responsible for assigning the truth values for operators. This allows for coding in a naturalistic and understandable way.
Referring to Figure 5.1, the entry for the missing value indicated by ‘2’ is in Nvar+5=7th
column and is obtained by IF operation on the statements from columns 6th (Nvar+4) and
10th (Nvar+8).
Conversion of 0’s and 1’s to F’s and T’s respectively
for (i = 1; i <= Nrow; i++ ) { for (j = 1; j <= Ncolumn; j++ ) {
elements[i][j]=TF(elements[i][j],undecided_string); }}
This is a loop that starts at the 1st row, increases by 1 each time and continue to run as
long as the Nrowth row. The same happens with columns and therefore all the elements of
the table are converted into T’s and F’s because of the TF (a,undecided_string) function from the brunel_logic template.
5.2. Translations
The database comprises of eight questions on translation of symbolic logic into English statements and vice versa. Questions within this subtopic are multiple choice, but as always of different styles. They differ from each other by the instructions, the number of statements and the type of operators (different connectives and many English translations of symbols).
English statements are sentences of subject matter from the ‘Shrek’ movie (Shrek, 2001), in order to make questions more interesting and engaging.
5.2.1. Assumed and tested skills
Within questions on translations it is assumed that students know basic English
vocabulary that enables them to understand the meaning of the sentence. It is further
assumed that students understand the rules of English grammar. In particular, it is important to be able to distinguish tenses and conditional statements. For example, the sentence ‘If I am carrying an umbrella or it will not rain, I will not get wet’ consists of three simple sentences, namely, ‘I am carrying an umbrella’ contains a subject, a verb in the present continuous tense and an object; whereas both ‘it will not rain’ and ‘I will not get wet’ contain the subject ‘I’, verbs in the future tense and no objects.
Translating an English language sentence into a logical statement is a process of abstraction and the use of replacement rules. Students are tested on three parts of this process. Firstly, students have to be able to understand the meaning of punctuation,
especially the comma. Inserting commas into the sentence makes the meaning of the
sentence clear. For instance, the sentence ‘If I do not revise today’s lecture then I will not be able to understand the future topics and I will fail the exam’ has different meaning depending on whether the comma is either placed before ‘then’ or before ‘and’. Symbolically we have r
~u f
versus
~ r u
f . Despite the common-sense meaning of the sentence, in the second version the exam is failed regardless of whether revision takes place or not. When the compound sentence contains three or more simple sentences, commas help to identify where the brackets should be inserted in the symbolic translation of the English statement. The importance of parenthesis has been explained in Section 5.1.1 so connection between commas and brackets is visible.
Secondly, students are tested on their ability to understand the logical meaning of
English words such as: not, and, or, unless, if/then, since, etc. and their symbolic representatives. Some of these are presented in Section 5.1.1, but the wider selection of most common translations as given in Epp (2010), Rosen (2007), Ikenaga (2009) and Halpin (2006) is shown in Table 5.2.1.
Table 5.2.1: Possible translations of logical operators into English. Symbolic Form Translation into English
~ Not p
It is not the case that p
q p and q p but q p as well as q p moreover q p however q p although q p even though q q p or q → If p, then q If p, q q follows from p p implies q q if p Whenever p, q p is sufficient for q
p is a sufficient condition for q a sufficient condition for q is p p only if q
q whenever p q when p
q is necessary for p
q is a necessary condition for p a necessary condition for p is q provided (that) p, q
q provided (that) p in case p, q q in case p
↔ p if and only if q p iff q
p is necessary and sufficient for q if p then q, and conversely
p is equivalent to q
pq
Unless p, then qUnless p, q If not p, then q
The language here conveys additional meaning beyond logic. For example, ‘p but q’ implies ‘p and q’ are both true, but given the truth of p one would not expect q.
Bearing in mind the first two skills, the third tested skill can be defined, i.e., the ability to
parse English sentences into symbolic propositions. Analysing the above sentence
and setting r = ‘it will rain’, u = ‘I am carrying an umbrella’, w = ‘I will get wet’, translating English conjunctions into symbolic connectives, as well as remembering the link between commas and parenthesis, the symbolic form of the English statement ‘If I am carrying an umbrella or it will not rain, I will not get wet’ would be