#### Study 1, Problem 1—

Analysis of the student responses to Question 1 in Study 1 revealed a number of categories that we have labeled A to D. Category A students were pre-conceptual and basically had no understanding (S1 and S9). Students in category B had a naive-conceptual understanding of the meaning of pH (HCl is an acid, and therefore, the solution should be acidic); however, they did not have the mathematical skills necessary to manipulate calculations related to pH. Category C students could calculate the pH from the concentration given but missed the point about the actual concentration of hydrogen ions, and their understanding of logs was still flawed. A sound conceptual understanding (Category D) would have been demonstrated if students could discuss the properties of water, its dissociation, and that the concentration of H^{+} ions in water is significantly higher than any additional provided by HCl at 10^{−10}M. These students would have been competent with the mathematical manipulations or even intuitively recognized that mathematical manipulations in this case were not necessary.

*Category A* responses were given by students 1 and 9. They adopted guesswork and expressed clear indications that they had very little understanding of pH. They displayed a lack of interest or motivation in doing the problem consistent with the adoption of an avoidance strategy.

Typical comments from these students included:

S1: “Forget it, it says pH!”

S9: “When it's acidic it gets rid of electrons.”

*Category B* responses were expressed by four students (S2, S3, S4, and S7). They appeared to have some understanding that pH related to the concentration of hydrogen ions in solution and a qualitative understanding of the behavior of acids and bases. They knew that solutions of low pH are acidic or that water is neutral and therefore, since HCl is a strong acid, a solution of HCl must have a pH below 7. Their understanding reflected a traditional definition, but they could not operationalize their understanding.

During her interview, S2 revealed many of the problems that students were confronting. Firstly, she acknowledged that pH meant “if it is acid or base,” but then with probing, she stated “they, ah, acid, add a hydrogen, no it's having a hydrogen, H^{+} is an acid.” Eventually she remembered that “pH equals log of H^{+} (sic).” At this point, she paused and sought help in using the calculator. Clearly, S2 had no understanding of logs and responded as she struggled with the calculation “all you do is put that in the calculator, that's all you get taught.”

She eventually calculated a value of pH 10 which was queried as follows:

I: Would you expect that to have a pH of 10?

S2: Because I remember doing that in class, and that's one part they just said you just know that.

S2: Oh that's right, you take it away because it must be a strong acid.

I: Hang on, we have 10^{−10}M, what does that mean?

I: No, what does 10^{−10}M of anything mean?

S2: Oh that would be big because you would go 1, 2, 3.

S2: Oh, so you go backwards.

I: So what does that mean?

I: Do you mean weak or … ?

I: No, I am just trying to get your terminology correct, so what does having 10^{−10}M mean?

S2: Not a very strong concentration.

After some scaffolding, she attempted to calculate the result; she took the mean of pH 7 and pH 10 and got pH 8.5.

S4 also relied on a naive understanding of pH and arrived at the answer 7 quickly by remembering that the pH of water was 7. S4 did have a conception of pH as “the amount of hydrogen ions floating around” and that the amount of HCl added was very small and would not make much difference. S4 relied on memory to solve the problem and associated pH with a physiological observation as revealed in this exchange.

I: Yes, so what does the pH mean?

S4: What does the pH mean, I suppose like whether it's an acid or a base.

I: So how do you work it out?

S4: Water is neutral, it is actually neutral because 7 is neutral but isn't it 7.4?

I: No, that's blood plasma.

This same student, however, had no understanding of logarithms and struggled with the mathematics, eventually giving up:

S4: “I don't know what the log actually is, I only know where the button is on my calculator.”

Another student, S3, arrived at the answer 7 but could neither manipulate the mathematics nor proceed further when the interviewer queried the response:

S3: “I was thinking the negative log of the hydrogen ion concentration, then I couldn't get to the concentration of H^{+} ions. Then I thought, OK, what is the pH in a solution of water, so I thought maybe neutral.”

*Category C* responses were provided by four students (S5, S6, S8, S10) who approached the problem by application of the formula and generally got a value of pH 10. They either were satisfied with this response or tried to apply further algorithms to reach a response, particularly if they were concerned with the answer. For example, S8 applied the formula for calculating pH and got the answer 10. He did not really comprehend the number meaning of pH 10, saying “The number looks scary.”

Three other students, after asking for a calculator, arrived at the answer 10 because they remembered that pH is the negative logarithm of the hydrogen ion concentration. After it was pointed out to them by the interviewer that they had neglected the water, they proceeded to either add 10 and 7 and come up with 17 (then realized that 17 was not on the pH scale, S5), subtract 7 from 10, giving the answer 3 (S6 and S10), or even subtract 10 from 14 to arrive at the answer 4 (S6). Thus although these students understood the general concept of pH and were able to do the primary calculation, their mathematics were flawed (poor understanding of logarithms) as they tried to manipulate the numbers to arrive at an answer less than 7 without really understanding what they were doing. Depressingly, there were no students in Category D.

#### Study 1, Problem 2—

Problem 2 provided an opportunity to explore understanding of acids and bases on a deeper level. The problem comprises three components: the first dealing with the effect of pH and p*K*_{a} on the ionization state of the amino acid side chain. This required an understanding of the concept of the acid dissociation constant p*K*_{a}. The second part required an understanding of the correlation between the ionization states of active site amino acids with pH optimum for the enzyme. The third component required an understanding of the effect of microenvironments on p*K*_{a}. Student responses to this problem were categorized in five levels, depending on evidence of their capability to solve each component of the problem.

In *category A* (S1, S2, S3, S7, S9), students did not understand what the problem was asking and were not able to do any part of the problem. Two students already identified as pre-conceptual in their response to Problem 1 had similar responses to Problem 2. For example, S9 did not attempt the question, expressing little understanding of pH or p*K*_{a}, and S1, a mature age student, admitted she had not done pH at school and so did not attempt the problem.

“No there is no way I can do that.”

“I never had that drummed into me, so I don't know how to take the first step.”

S2 was unable to get anywhere with the problem and had no useful understanding of the meaning of p*K*_{a} or equilibrium constants and kept getting confused with optical isomers, isoelectric point, polar *versus* nonpolar, and plus and minus charges. Interview data supported the assertion that this student approached learning in a surface manner, accumulating a lot of information by memory, but was unable to make coherent links between these ideas.

S3, although possessing a naive conceptual understanding of pH, did not understand the relationships among pH, p*K*_{a}, and ionization state and drew incorrect conclusions for the first part of the problem from misguided interpretation.

S3: “Here it says what is the ionization state, protonated or deprotonated, of each residue. So pH 5.2 is acidic, so there are a lot of H+ ions floating around, and I thought that OH^{−} and H+ would probably form H_{2}O, and this here will be negative and that would be protonated there (drew incorrect conclusions).”

I: What about Asp-52? Is that protonated as well?

S3: I thought that would be also protonated as it has the same carboxyl group.

S7 attempted unsuccessfully to recall the relationships among p*K*_{a} and pH and ionization state. As in the previous problem, S7 relied on memory and displayed limited understanding of concepts and the relationships between them. He attempted to explain the pH activity profile in purely descriptive terms as a bell curve and eventually lost interest: “I don't actually know, I am not in a frame of mind to do that.”

“First bit I remembered that the p*K*_{a} value and the pH, they are very similar and kind of I think, I can't actually remember how they correlate in trying to work out whether it's protonated or deprotonated, but from memory it's something like:

If pH is higher than the p*K*_{a} value, then it will be protonated, and if not it will be deprotonated, but I know there is more to it than that, so it's a bit of a stab in the dark really.”

In *category B*, the student (S10) had an intuitive understanding of p*K*_{a} and ionization, although she required coaching to realize it. She could not advance to other parts of the question and did not understand what the rest of the question was asking.

In *category C*, one student (S4) had a basic understanding of the state of ionization of the two amino acids but tried to explain the activity profile in terms of just wanting to get the mid point between the two p*K*_{a} values. She was almost on the verge of making the connection between ionization state and function:

“It's deprotonated, is it not that because it has released its hydrogen, it is now able to bond to other things or do whatever it needs to do?”

In category D, student S8 needed assistance to work out the first part but could not really relate this to the pH activity profile, although he did have a conception of the principles behind the third part of the question.

“But these residues have obviously been linked to something else, I mean linked to other different amino acids, and probably that might contribute to the character.”

In *category E* (S5, S6), the students were successful on the first part of this problem and were able to answer one other part of the question at a basic level. S5 had a conceptual understanding of the effect of p*K*_{a} and pH on the ionization state of the amino acid side chains and drew the correct conclusion. S5 almost reached an explanation for the activity profile but did not manage to get the third part out. With further probing, she gave a rudimentary explanation for the third part.

S6 originally had the ionization states the wrong way around but conceptually knew what the problem was asking, although relying heavily on memory. With probing, S6 was the only one to associate the ionization state of the amino acids with the enzyme reaction mechanism in terms of transfer of electrons. She had a rudimentary understanding of the effect of environment on p*K*_{a}.

The categories of students' responses to Problem 2 are presented in Table I. No student was able to make completely the connections between different aspects of the problem. For the first part of the problem, students relied heavily on their memory of a sentence in the study guide relating pH and p*K*_{a} to determine the ionization state of amino acids. This sentence stated that when the pH is less than the p*K*_{a}, the proton is *on*, and when the pH is greater than the p*K*_{a} the proton is *off*, and therefore, the students assumed that the charge was positive below the p*K*_{a} and negative above the p*K*_{a} value. Only the students in category E understood the Henderson-Hasselbalch equation relating ionization state to pH and p*K*_{a}.

A summary of all data collected is presented in Table II. Only two students performed at a mastery level in their final assessment (S6 and S8). During the interview these, students had displayed a category C on Problem 1 and, respectively, categories E and D levels of understanding on Problem 2. There is a relationship between each student's performances in both problems. Student performing at category 1 level on Problem 1 tended to perform poorly on Problem 2. With the exception of students 4 and 5, performance on these problems predicted their performance on the final pH problems in the examination. The relationship to final grade obtained is more unpredictable, reflecting the wider range of concepts being tested.

The problems presented to students in this study tested their knowledge of pH and their ability to use mathematical procedures associated with the application of this knowledge. Conceptual knowledge about pH appears to be fragmented, with students relying on memory of facts. Nearly all students demonstrated only a rudimentary understanding of mathematics and depended on their calculators to generate numbers that had no real meaning to them. It might be argued that this situation is transient and that as students proceed through the degree program, a deeper understanding of the concept and proficiency in the application of knowledge is developed. To test this assertion, the prevalence of students' naive understandings of pH was subsequently explored among students in the third year of their degree.