CHAPTER 5 RESULTS SHOWING SCOPE AND QUALITY OF RESEARCH ON
8.4 DIFFICULTIES WITH SYMBOLIC AND MATHEMATICAL
8.4 DIFFICULTIES WITH SYMBOLIC AND MATHEMATICAL
(2006) elaborates that when Grade 10 students described pH as a measure of the ‘strength’ of an acid or base, they used ‘strength’ to mean: “how powerful or reactive a substance was” which suggest they associated pH with the tendency of an acid or base to be a proton donor or acceptor. Following a discussion of some of this research, Demerouti et al. (2004) and Oversby (2000b) clarify the difference between a weak acid (characterized by partial dissociation or ionization) and a weakly acidic solution (as measured by a pH of 4, 5 or 6). (Oversby’s suggestion of using the word ‘potent’ to distinguish a strong acid has yet to receive general acceptance.) As a result of this argument it is a misconception to assume that a particular pH characterizes acid-base strength. Students who do so may be unaware that pH will vary according to concentration, so a strong acid may have a pH close to 7 if it is in a dilute solution.
Qualitative aspects of a concept such as pH should be taught in a way that forms a sound base for later studies of quantitative aspects (Hawkes, 1994). In this matter, Oversby (2000b) gives a qualitative interpretation of the pH scale, at an appropriate level for junior secondary students who have not yet formally encountered acid-base strength or ionic concentration, [H+], or pH calculations as summarised in the propositional knowledge below.
• Solutions with pH 1 to 3 are described as strongly acidic. (9.3.1.1)
• Solutions with pH 4 to 6 are described as weakly acidic. (9.3.1.2)
• Solutions with pH 7 are neutral. (5.1.1)
• Solutions with pH 8 to 10 are described as weakly alkaline. (9.3.2.1)
• Solutions with pH 11 to 13 are described as strongly alkaline. (9.3.2.2)
• pH of a solution depends on the concentrations [H+] and [OH–]. (9.4.2)
The research described above shows the existence of the conception in four contexts but there are some problems. While Sheppard (2006) reports on a comprehensive study of the difficulty, only limited confirmation (that is single instances of the conception) comes from Ross and Munby (1991) and Smith and Metz (1996). Furthermore, only Botton (1995) reports investigating the student conception of strong bases, but in a model concept map he includes the difficulty described above as an acceptable proposition. Consequently, the difficulty description needs to be separated to show different classifications for difficulty descriptions concerning acids and bases, as follows:
Difficulty R12.1 pH is a measure of acid strength. (Level 3) Difficulty R12.2 pH is a measure of base strength. (Level 2)
The implications of the difficulty for student conceptual development in the quantitative aspects of pH are important and so further research would be useful. Such research needs to find out whether students make a direct link between strength and pH, or if the conception follows from
the idea that strength indicates concentration (Difficulty R1 in Section 8.2.1.1). Appropriate remedial strategies will depend on the source of the difficulty.
8.4.2 Difficulties concerning limits to pH
8.4.2.1 Difficulty R13: The function pH = – log10[H+] has upper and lower limits
The idea that there are upper and lower limits to the pH function has been reported by three researchers (see Table 8.4 below). Furthermore, Oversby (2002b) observed a student’s confusion when the range was given inappropriately as 1 to 14, with neutral as 7 in the middle, because the student had calculated that 7.5 was midway between 1 and 14.
Table 8.4 Student conceptions of limits for pH
Lower limits Upper limits Educational level of students
Author
1 14 Tertiary Zoller (1996)
0 or 0.01 or 1 9, 13 or 14 Pre-service teachers Dhindsa (2002)
1 14 Teacher Oversby (2000b)
When introducing the concept of pH, Sörenson (1909) noted that it would usually be a positive number, but in exceptional cases where [H+] was greater than 1 mol.dm-3 it would have a negative value. In this regard, Oversby (2000b) clarifies that the practical limits are from –2 to 16. Consequently, the limits that students put on pH values are inappropriate, both theoretically and in practice. They indicate little understanding of the mathematical relationship shown by the symbols defining pH. This leads to the following propositional statements for the difficulty, as suggested by Dhindsa (2002):
• pH usually applies to dilute solutions. (9.4.3.4)
• When [H+] = 1.0 mol.dm-3 pH is 0. (9.4.3.2.1)
• When [OH–] = 1.0 mol.dm-3 pH is 14. (9.4.3.2.2)
Based on the common aspects across the reported research, I can describe the difficulty as: the pH scale has upper and lower limits. Having been identified in several contexts, I can classify the description at Level 3++. Further research may show whether students conceive the limits as theoretical or practical, what specific limits they tend to use, and perhaps where they see the midpoint of the scale.
8.4.2.2 Difficulty R14: pH has discrete integer values
As reported by Dhindsa (2002), “Students have difficulty in viewing continuity between numbers of pH”. These students gave the highest value of pH for an acid as 6 (2% prevalence) and the lowest value for a base as 8 (19% prevalence). He gives typical student reasoning as:
“pH 7 is neutral, therefore, an acid has to have pH less than or equal to 6”. From this research, which used a written questionnaire and individual interviews with two cohorts of pre-service teachers in Brunei, I can classify the student conception as partially established in more than one context, i.e. Level 3+.
Further insight into the difficulty comes from research by Demerouti et al. (2004) which shows student responses (with justification of their choices), to pairs of complementary multiple- choice items. When asked about the pH of a 10-8 mol.dm-3 solution of HCl, a few students responded with “7” instead of the authors’ preferred response: “just under 7” However, I cannot interpret this statement as evidence for the belief that pH is not continuous between numbers because I do not know the student frame of reference, and the multiple-choice format precluded other responses that were less than 7. Moreover, the incidence was small (2%). I have, however, used this research to rephrase the description of the difficulty more explicitly than that given by Dhindsa; it becomes: pH has discrete integer values. It remains to be confirmed whether this description will be stable in further contexts.
Concerning integer values, Oversby (2000a) notes that ‘weakly acidic’ is often taught as applying to pH values of 4, 5 or 6 rather than a range of values. The modern operational definition gives pH in terms of electrolytic measurements, such as with a pH meter (Hawkes, 1994; McNaught & Wilkinson, 1997) in which case it is a continuous variable. Although pH calculated by –log [H+] may differ in the first decimal place from the measurement, this is near enough for most approximations, especially for dilute solution (Hawkes, 1994). Calculations using activity, instead of concentration are more accurate – having an uncertainty of ± 0.02 (McNaught & Wilkinson, 1997). This propositional knowledge could be presented for students as follows:
• pH measured with a pH meter gives continuous values. (9.2.3)
• pH calculations with concentrations are accurate to ± 0.1. (9.4.3.3.1.1)
• pH calculations with activities are accurate to ± 0.02. (9.4.3.3.1.2)
8.4.3 Difficulties concerning the effect of equilibrium systems on pH
Difficulties with chemical equilibrium have been extensively researched in general chemistry (see Gabel & Bunce, 1994). In this section, three difficulties specific to acid-base equilibria are discussed.
8.4.3.1 Difficulty R15: pH = –log [H+] means using [H+] only due to acid or base.
Three independent triangulated research studies have shown that students use the formula:
pH = –log [H+] in a simplistic algorithmic fashion, using only the acid concentration to calculate the pH of very dilute acidic solutions. Pinarbasi (2007) reports the following interview quotations, which show a student ignoring the relationship between acidity and pH in favour of an algorithm:
Student A: ...according to pH = -log [H+], the pH [of 10-8 M HCl] will be 8.
Interviewer: But, this is an acid solution, isn’t it?
Student A Yeah...but the equation says that its pH is 8
Student B: If we added a large amount of water into this [10-5 M HCl] solution, we can make the pH of 8.
Similar reports to this quotation are summarised in Table 8.5 below. In all these studies students are shown to be applying the formula to calculate pH for both acidic (pH 8 and 10) and basic solutions (where students appear to use the pOH). In the case of very dilute solutions such as these, calculations should also take into account the self ionization of water (Skoog et al., 1996). The pH values calculated by Skoog et al.’s method are given in the fifth column of the table. These are all extremely close to 7, and moreover are slightly below 7, for acidic solutions or just above 7 for basic solutions.
Table 8.5 Summary of research into student conceptions of pH for very dilute solutions
Concentration of acid/base in
aqueous solution Student conception of pH Incidence #Acceptable answer *pH calculated Education level of students
Authors
10-8 mol.dm-3 HCl 8 12% Just below 7 6.98 Grade 12 Demerouti et al. (2004) Kousathana et al.
(2005) 10-8 mol.dm-3 HCl 8 70% Not given 6.98 Pre-service
teachers
Pinarbasi (2007) 10-10 mol.dm-3 HCl 10 Not given 7 7.00 Tertiary Watters & Watters
(2006) 10–8 mol.dm-3
NaOH.
8 6% Just above 7 7.02 Grade 12 Demerouti et al. (2004) Kousathana et al.
(2005)
# from authors
*By the present author using the method of Skoog et al. (1996)