Ecuaci´on de Korteweg-de Vries modificada por disipaci´on

As part of their development of the FCI, Hestenes et al. attempted to create questions that covered six conceptual dimensions of force: kinematics, New- ton’s first, second, and third laws, the superposition principle, and kinds of force [8]. Huffman and Heller claimed that if the FCI was indeed constructed along these six dimensions, then factor analysis could be used to test the va-

in this report are displayed in Fig. 1. More such data un- doubtedly exists but goes unreported because the gains are so embarrassingly minimal.

For survey classification and analysis purposes I define: ~a! ‘‘Interactive Engagement’’ ~IE! methods as those de-

signed at least in part to promote conceptual under- standing through interactive engagement of students in heads-on (always) and hands-on (usually) activities which yield immediate feedback through discussion with peers and/or instructors, all as judged by their

literature descriptions;

~b! ‘‘Traditional’’ (T) courses as those reported by in- structors to make little or no use of IE methods, relying

primarily on passive-student lectures, recipe labs, and algorithmic-problem exams;

~c! Interactive Engagement ~IE! courses as those reported by instructors to make substantial use of IE methods; ~d! average normalized gain^g&for a course as the ratio of

the actual average gain^G& to the maximum possible average gain, i.e.,

^g&[%^G&/%^G&max5~%^Sf&2%^Si&!/~1002%^Si&!,

~1! where^Sf&and^Si&are the final~post! and initial ~pre!

class averages;

~e! ‘‘High-g’’ courses as those with (^g&)>0.7;

~f! ‘‘Medium-g’’ courses as those with 0.7.(^g&)>0.3; ~g! ‘‘Low-g’’ courses as those with (^g&),0.3.

The present survey covers 62 introductory courses enroll- ing a total of 6542 students using the conceptual MD or FCI exams, and ~where available! the problem-solving Mechan- ics Baseline~MB! test. Survey results for the conceptual and problem-solving exams are presented below in the form of

graphs. In a companion paper,17~a!intended to assist instruc- tors in selecting and implementing proven IE methods, I tabulate, discuss, and reference the particular methods and materials that were employed in each of the 62 survey courses. Also tabulated in Ref. 17~a! are data for each course: instructor’s name and institution, number of students enrolled, pre-/post-test scores, standard deviations where available, and normalized gains. Survey information was ob- tained from published accounts or private communications. The latter usually included instructor responses to a survey questionnaire15~c! which asked for information on the pre-/ post-testing method; statistical results; institution; type of students; activities of the students; and the instructor’s edu- cational experience, outlook, beliefs, orientation, resources, and teaching methods.

As in any scientific investigation, bias in the detector can be put to good advantage if appropriate research objectives are established. We do not attempt to access the average effectiveness of introductory mechanics courses. Instead we seek to answer a question of considerable practical interest to physics teachers: Can the classroom use of IE methods in-

crease the effectiveness of introductory mechanics courses well beyond that attained by traditional methods?

III. CONCEPTUAL TEST RESULTS A. Gain versus pre-test graph—all data

To increase the statistical reliability ~Sec. V! of averages

over courses, only those with enrollments N>20 are plotted

in Fig. 1, although in some cases of fairly homogeneous instruction and student population ~AZ-AP, AZ-Reg, PL92-C, TO, TO-C! courses or sections with less than 20 students were included in a number-of-student-weighted av- erage. Course codes such as ‘‘AZ-AP’’ with corresponding enrollments and scores are tabulated and referenced in Ref. 17~a!. In assessing the FCI, MD, and MB scores it should be kept in mind that the random guessing score for each of these five-alternative multiple-choice tests is 20%. However, com- pletely non-Newtonian thinkers~if they can at the same time read and comprehend the questions! may tend to score below the random guessing level because of the very powerful interview-generated distractors.1~a!,12~a!

It should be noted that for any particular course point

(^G8&,^S_i8&) on the^G& vs^Si& plot of Fig. 1, the absolute

value of the slope of a line connecting (^G8&,^S_i8&) with the point~^G&50,^Si&5100! is just the gain parameter^g8&for

that particular course. The regularities for courses with a wide range of average pretest scores @18<(^Si&)<71# and

with diverse student populations in high schools, colleges, and universities are noteworthy:

~a! All points for the 14 T courses (N52084) fall in the Low-g region. The data17~a!yield

^^g&&14T50.2360.04sd. ~2a!

Here and below, double carets ‘‘^^X&&N P’’ indicate an

average of averages, i.e., an average of ^X& over N courses of type P, and sd[standard deviation @not to be confused with random or systematic experimental error~Sec. V!#.

~b! Eighty-five percent ~41 courses, N53741! of the 48 IE courses fall in the Medium-g region and 15% ~7 courses, N5717! in the Low-g region. Overall, the

Fig. 1. %^Gain& vs %^Pre-test& score on the conceptual Mechanics Diagnos- tic~MD! or Force Concept Inventory ~FCI! tests for 62 courses enrolling a total N56542 students: 14 traditional (T) courses (N52084) which made little or no use of interactive engagement~IE! methods, and 48 IE courses (N54458) which made considerable use of IE methods. Slope lines for the average of the 14T courses^^g&&14Tand 48 IE courses^^g&&48IEare shown,

as explained in the text.

Figure 2.1: Average normalized gain of scores on the FCI from administrations at 62 courses

lidity of these categories and the FCI as a whole [50]. They specifically asked, does the FCI measure a unified concept of force, multiple factors of a “force concept,” or does it not measure a force concept at all?

Huffman and Heller used factor analysis on student CI scores to deter- mine whether performance on different FCI questions could be attributed to a single factor in their knowledge (a unified concept of force) or if the FCI questions mapped to six factors that might correspond to the six concep- tual dimensions outlined by Hestenes et al. According to their analysis, high school student responses on the FCI grouped into two statistically signifi- cant factors: four (of twenty-nine) questions grouped on one factor and three questions grouped on a different factor. University student responses on the FCI grouped into only one significant factor composed of five questions [50]. They found that student responses seemed to cluster around the contexts the problems described rather than the concepts the problems evaluated.

Because so few questions on the FCI grouped to significant factors, Huff- man and Heller concluded that the FCI measures neither a single unified concept of force nor the six conceptual dimensions of force [8, 50]. Huff- man and Heller recommended that the FCI could still be used cautiously for diagnostic purposes and more freely to evaluate instruction, but the FCI should not be used as a placement exam. They also concluded that instruc- tors should not evaluate student understanding of the different components

of force separately using the FCI (i.e., instructors should not use the FCI to see if students understand acceleration but not velocity) [50].

Huffman and Heller claim the FCI revealed that “to a physics instruc- tor, the six conceptual dimensions originally proposed by the authors of the inventory seem like logical categories. Students, however, apparently view these items quite differently than do instructors [50].” They concluded that students have a fragmented understanding of force making their knowledge context dependent. Huffman and Heller explained that

When a student answers a test question, it is difficult to deter- mine the extent to which the test question is measuring students’ understanding of the concept and the extent to which the test question is measuring students’ familiarity with the context [50]. To summarize their argument, students do not provide consistent answers to FCI questions that test the same concept, so it is debatable whether the FCI is a valid examination.