Dear Professor Steinbring

Dear Professor Steinbring,

CONCEPT IMAGE REVISITED

Erhan Bingolbali and John Monaghan

In July of this year you wrote, “Based on the advice received, the Editors feel that your manuscript could be reconsidered for publication should you be prepared to incorporate major revisions. When preparing your revised manuscript, you are asked to carefully consider the reviewer comments which are attached, and submit a list of responses to the comments.”

John and I have rewritten our article, taking into account your and the reviewers’ comments. We have resubmitted the article on the Eidtorial Manager and this letter is your requested “responses to the comments”. We make three points (reminders) before addressing the comments:

1. You agreed that the final resubmission date was the end of October.
2. John wrote a letter to you listing aspects of reviewer #2’s comments that were not feasible to take into account in a resubmission, e.g. “I strongly encourage them to share the data in Tables 1 and 5 with the MM and ME chairman and say: this is the situation, now how are we going to fix it?” and you replied agreeing with this.
3. John informed you that a strange thing happened with a sister article (submitted to another journal) to the ESM submitted article – it was initially rejected and then published without revision (the editor informed us that the ‘reject’ letter was a clerical mistake). This caught us unaware as we submitted to ESM in the period when we thought this other article was rejected. There were certain areas of overlap in the two articles (only in terms of the outline of the research and some of the data presented – the focus is very different). Anyway, overlap between the articles has been minimised to ‘irrelevant’ in the resubmission and we have noted this other article in the acknowledgements.

We hope that the revised article satisfactorily addresses your and the reviewers’ criticisms.

Yours sincerely,

Erhan Bingolbali and John Monaghan

Response to reviewers’ comments

NB For brevity we only respond to questions or recommendations other than those covered in note 2 above. Reviewers’ comments are presented in boxes.
Reviewer #1

The authors' claim that they do no care of concept definition. We ask them to justify this strong choice.

Due to changes made in the revision we now do deal with concept definition (new section 4). We have thus amended the sentence “We are not really interested in concept definition …” and have stated why this is so. The amended sentence reads “We are less interested in concept definition than we are in concept image in this article because we are not purely cognitivist theorists, i.e. from our standpoint factors such as students’ departmental affiliation enter into the mix that results in students’ concept images.”

At page 14-15 (full text) there is the explanation of the coding of answers that does not seems useful, being this coding rather obvious. At p. 16 the explanation of the coding is useful.

Given that p.16 comment is considered useful we have reduced the student quotes but have not cut the explanation of the coding.

We recommend to present the whole questionnaire so that the reader may have a holistic idea of the full work proposed to students. In the present version of the paper the questions are presented separately.

We have not attended to this because we are aware that ESM likes articles to be as short as possible. We feel this is an editorial decision and will do this if the editor directs us so.

There is some English mistakes, e.g. the sentence "we have our collapsed original categories".

This has been attended to. Indeed Table 1 has been represented to make it shorter and possibly clearer.

Reviewer #2:

The meat of this paper lies in the questionnaire … And so the question became: why? Why do the two groups have such different concept images of the notion of a derivative? And the answer is in Table 5. The ME's received 133 minutes of instruction on the rate of change notion of the derivative; with scant time (about 10 minutes) and NOT A SINGLE EXAMPLE given to its slope-of-the-tangent-line interpretation! The MM's were given a similar amount of instruction on the slope-of-the-tangent-line interpretation, with scant time and NOT A SINGLE EXAMPLE given to its rate-of-change interpretation!

…………………..

As I understand it, this paper is trying to show that a dominant concept image emerges in our psyche because of social factors. We have several concept images of a derivative and we can move equally well between them, but if given a choice of where more than one version can be applied, we gravitate toward a personally dominant concept image-and this gravitation is caused by social factors. In this study, the MM's will move to the slope interpretation and the ME's will move toward the change with respect to the independent variable interpretation. But it seems to me that this hypothesis can not be tested if we have only one concept image in our repertoire. And it makes no sense whatsoever to say the ME's used X while the MM's used Y, if the ME's knew only X and the MM's knew only Y. They didn't know the other interpretation because is wasn't taught in a serious way.

Aha, this is an excellent point – but it took us a while to appreciate it because we knew more than we presented but it was not until we tried to make sense of reviewer #2’s comments that we realised that we needed to provide the reader with additional information. We will explain. We attempted to present relevant results as fully but as succinctly as possible but we were too succinct and relied too much on Table 5. The resubmission has a new section 4 (and no Table 5) with three subsections in which we describe each calculus course and make some general comparative comments on them. We think the course descriptions show the distinctiveness of the courses but also show some commonality and we make the point in the third subsection that “it is not possible to completely divorce tangent and rate of change approaches”. Are we ‘changing our tune’? No, we are simply providing a fuller picture of the students’ course experiences.

Now how does the resubmission, with a new section 4, address reviewer #2’s comments above? Well, his/her “NOT A SINGLE EXAMPLE” comments still hold but with regard to “if the M_’s knew only Z” comments – (i) we hope that it can now be seen that the ME students were presented with tangent descriptions even though they did not work on tangent examples (ii) although the M students did not work on rate of change examples does encapsulate rate of change ideas and could be applied to (Appendix) questions 3 and 4 (as, indeed, it was by some M students).

How can one reasonably expect that the students, through their own musings, develop the theory for the approach not taught to them? How can the instructors completely ignore one of the major interpretations of the derivative (be it the slope or rate-of-change approach)? And worse, why wasn't the definition of the derivative even mentioned in either group? Isn't that also part of one's concept image of a notion? Something really seems peculiar here, and it has nothing to do with the students being MM or ME majors and the various cultures in which they are studying. But things get even worse.

Another good point but we think our response to the previous point covers this point.

Let's look at the exam itself. To me, the questions seem to be trivial, and certainly not something one would ask university level students who were exposed to these notions in high school. Yet if we look at the numbers in Table 1, we see proof that the students seem to know NOTHING, and this is AFTER instruction! After instruction, only 13 out of 32 mathematics majors could correctly find the slope of the tangent line from the graph which is presented in question 1a). And recall, this was the interpretation that was stressed in their classrooms. This is scandalous. The students can see that the tangent line at the point (5, 3) also goes through the point (-1, 0), and yet at the end of the course they couldn't compute f′(5)? Is this a joke? And with ME students, only 16 students out of the 50 tested could answer this correctly? This is very hard for me to swallow-particularly when I assume (because it is not stated), that most of these students passed the course. They received credit for having taken a calculus course, and yet they couldn't answer a question that is often encountered in the ninth grade (to find the slope of a line given the coordinates of two points on it).

Well, we think we’re allowed to disagree with reviewers. We agree that the questions are not difficult but disagree that they are “trivial”. Indeed, reviewer #2 makes a mathematical mistake him/herself (it goes through (-2.5, 0), not (-1, 0)). Question 1 has been used in other studies on non-Turkish students and, to our recollection, the results there are sometimes worse than our results. “know NOTHING” is an exaggeration.

OK, I could go on and on about the students' obvious ignorance, but something really seems to be askew here. Had I constructed such a test, I would have forced the students into a particular mode of thought in a more subtle way. E.g., I would have given them the graph of f′(x) without algebraic description, and asked them to sketch the graph of f(x) and also that of f′′(x). This would have told me who was thinking along the line of a derivative as being the slope of a tangent line. I would have also asked if all continuous curves have a derivative at all points in its domain, and its converse, if all differentiable functions are continuous at all points in its domain. I would have done this analytically and also graphically. But all of this is moot-the students in this study couldn't even answer the questions on this test-and that is embarrassing.

One of us teaches calculus and does much of what the reviewer does. But this, as stated in the article, was a ‘naturalistic’ study, i.e. we observed what was done, it was not our teaching.

My point in all of this is that the authors set us up to see which concept image is dominant
in the students-but this implies that they have more than one interpretation of the derivative. Yet the data clearly shows that the students in this study didn't even have a single interpretation of the derivative, let alone multiple interpretations of it! Look at the responses to question 2b) (find the equation of the tangent line L at the point (1,1) which is on the curve f(x) = -2x3 + x2 + x + 1). AFTER instruction only 17 out 50 ME students correctly answered this question; and only 18/32 of the MM students correctly answered it. This question is trivial, and the responses of the students again point to the fact that something is really wrong at their university.

Apart from disagreeing, as noted above, that the question is “trivial” we feel this is another good point but we also feel we have addressed this in our response beginning “Aha, this is an excellent point …” above.

Reviewer #3

From my point of view, there are some repetitions in the different sections such that on the one hand the article could be shortened without losses in information. On the other hand, it might be helpful to present (and to discuss) some more consequences, as, e.g., in with way the students' departmental affiliations should be taken into consideration when calculus is taught, or which concept images might be regarded as more appropriate or worthy from the one or the other department's point of view, that means, which (and how) changes in concept images should be supported in calculus courses.

If the repetitions are pointed out to us, then we will be happy to attempt to shorten the article. Regarding “more consequences”, there are lots of things to say regarding this research. We have an article in another journal in which we consider consequences for the mathematical education of engineers and a chapter in a forthcoming Springer book on situated cognition in which we consider ‘institutionalisation of knowledge’. In this article we focus on concept image. All the ‘messages’ we receive about ESM amount to ‘keep it short and snappy’. So we hope you will excuse our not taking on board further issues.