BLOOM S TAXONOMY Topic Following Bloom s Taonomy to Assess Stdents Smmary A handot for stdents to eplain Bloom s taonomy that is sed for item writing and test constrction to test stdents to see if they nderstand the corse material at all levels from knowledge to evalation. Athors Atar Kaw, Jim Eison Date Febrary 20, 2003 Following Bloom s Taonomy to Assess Stdents Eaminations are an integral part of taking most college corses. Only in a few corses sch as Capstone Design or Independent Stdy wold yo not take an eamination. Why do instrctors se eams? One obvios fnction is assigning a corse grades. Other less obvios bt even more important reasons for sing eams inclde Motivating stdents to learn and meet the objectives and goals of a corse Provide feedback to stdents so that they know their weaknesses and strengths It also tells the instrctors whether they are meeting their objectives The prpose of this handot is to help yo better prepare for eams by eploring the different types of qestions that instrctors formlate. To check yor mastery at epected levels in the corse Comptational Methods (Nmerical Methods), I am following a widely sed approach to item-writing and test constrction. This approach is called Bloom s taonomy. So, what is Bloom s taonomy? In 1956, an edcational psychologist Benjamin Bloom was chairing a committee of higher edcation eaminers who were asked to develop a system that wold define whether stdents learned what they were taght. This system came to be known as Bloom s taonomy. In the cognitive domain, Bloom s taonomy provides a gideline to develop test qestions at levels of increasing competence as follows: knowledge comprehension application analysis synthesis evalation. Bt, what do these categories of competence mean? I am going to briefly eplain each category, follow it by an eample from the topic of Nmerical Differentiation from the Comptational Methods corse, and then eplain how the eample fits Following Bloom s Taonomy to Assess Stdents -1
the category. The eamples given are mltiple-choice bt cold be rephrased to be written as short answer, fill in the blanks or problem soltions. 1. Knowledge This level checks for basic knowledge and memorization. It is simply recall of information or knowledge. Eample: The definition of the first derivative of a fnction f() is f ( + ) + f ( ) a) f '( ) = f ( + ) f ( ) b) f '( ) = f ( + ) + f ( ) c) f '( ) = lim 0 f ( + ) f ( ) d) f '( ) = lim 0 Eplanation: In this problem, I am asking yo a qestion from the pre-reqisite corse of Calcls I. This formla is reintrodced in the Comptational methods corse to develop methods for nmerical differentiation. The qestion simply checks whether yo have memorized the definition of a derivative of a fnction. 2. Comprehension This level checks for literal nderstanding and checks if yo can apply the general concepts to a problem. Eample: The eact derivative of f() = 3 at =5 is most nearly a) 25.00 b) 75.00 c) 106.25 d) 125.00 Eplanation: In this problem, I am asking yo to find the eact derivative of a fnction. Yo need to know how to find eact derivatives of simple fnctions to be later able to nderstand nmerical differentiation methods and concept of tre errors to show how well nmerical differentiation works. 3. Application This level checks whether yo are able to se the concepts and apply them. These can be problems where yo are asked to apply a nmerical method to a simple problem. Following Bloom s Taonomy to Assess Stdents -2
Eample: Using forwarded divided difference with a step size of 0.2, the derivative of f()=e at =2 is a) 6.697 b) 7.389 c) 7.438 d) 8.179 Eplanation: In this eample, yo are applying the one of three nmerical methods yo were taght in class to find the first derivative of a fnction. 4. Analysis This level checks whether yo can scrtinize a problem. Yo may be asked to eamine a comple problem. To be able to solve it, yo will have to break it into simpler parts. Yo shold be able to see the connection between the parts. Eample: d A stdent finds the nmerical vale of ( e ) = 20. 219 at =3 sing a step size of 0.2. d Which of the following methods did the stdent se to condct the differentiation? a) Backward divided difference b) Calcls, that is, eact c) Central divided difference d) Forward divided difference Eplanation: In this eample, yo are now asked to find which method has been sed. This involves being able to se the formlas of all the methods and see which one has been sed. I can also assess whether yo know the difference between the three nmerical methods of finding the first derivative. 5. Synthesis This level checks whether yo can pt concepts together to form a whole. One may need to se mltiple pieces of information to be able to solve the problem. Eample: d Using backward divided difference scheme, ( e ) = 4. 3715 at =1.5 for a step size of d d 0.05. How many times wold yo have to halve the step size to the find ( e ) at =1.5 d before two significant digits can be considered to be at least correct in yor answer? Yo cannot se the eact vale to determine the answer. a) 1 Following Bloom s Taonomy to Assess Stdents -3
b) 2 c) 3 d) 4 Eplanation: Here, yo are going to se the backward divided difference scheme several times. Yo need to synthesize the nmerical differentiation reslts with yor mastery of Approimation and Errors to see how many significant digits are at least correct in yor answer. Therefore, this is an eample of bringing several concepts together. 6. Evalation This is the level where yo make a jdgment. This is what yo wold be doing first when yo apply the concepts learned in this corse in another corse or in a practical engineering or science problem. The factors yo wold consider to make a jdgment in Comptational Methods may be to redce error, increase speed of comptation, choose a particlar method, make initial estimates, adopt step sizes, etc. Eample: In a circit with an indctor of indctance L and resistor with resistance R, and a variable voltage sorce E(t), di E ( t) = L + Ri dt The crrent, i, is measred at several vales of time as Time, t (secs) 1.00 1.01 1.03 1.1 Crrent, i 3.10 3.12 3.18 3.24 (amperes) If L= 0.98 Henries and R=0.142 ohms, how wold yo find E(1.00), what wold be yor choice for most accracy. 3.24 3.10 a) E ( 1) = 0.98( ) + (0.142)(3.10) 0.1 b) E ( 1) = 0.142*3. 10 3.12 3.10 c) E ( 1) = 0.98* + 0.142*3. 10 0.01 3.12 3.10 d) E (1) = 0.98* 0.01 Eplanation: Here, the problem is not jst finding the vale of the derivative, bt also how it is going to be sed to eventally evalate the voltage in a real-life problem. Also, yo are asked to pick p the most accrate formla based on yor crrent knowledge of nmerical Following Bloom s Taonomy to Assess Stdents -4
differentiation. Additionally, the data given is discrete and hence yo do not have the lry of having data at any point yo desire, as is the case in derivatives of continos fnctions. Atar K Kaw of Mechanical Engineering and Jim Eison of Teaching Enhancement Center, both at the University of Soth Florida, wrote this handot. We hope that yo are clear abot the levels at which yo will be tested. If yo have any qestions, please call Atar Kaw at 813-974-5626 or e-mail me at kaw@eng.sf.ed. Best of lck! Following Bloom s Taonomy to Assess Stdents -5
Appendi A Qestions for Nonlinear Eqations In this appendi, we give si sample qestions for nonlinear eqations. Find the right answer bt also find ot the motivation behind the incorrect answers as well. 1. Secant method of finding roots of nonlinear eqations falls nder the category of methods. A. bracketing B. graphical C. open D. random 2. The Newton-Rahpson method formla for finding the sqare root of a real nmber R from 2 the eqation R = 0 is, A. i i+ 1 = 2 B. 3i i+ 1 = 2 C. 1 R = i+ 1 i + 2 i D. 1 R = i+ 1 3i 2 i 3. Assming an initial bracket of [ 0,5], the second (after 2 iterations) iterative vale of the root t of te 4 = 0 is A. 0 B. 1.25 C. 2.5 D. 3.75 4. The absolte relative approimate error at the end of an iteration in bisection method can be written in terms of the lower and pper gess, l and, respectively as A. B. + + Following Bloom s Taonomy to Assess Stdents -6
C. D. + + 5. The root of 3 = 4 is fond by sing Newton-Raphson method. The sccessive iterative vales of the root are given in the table below Iteration Vale of Nmber Root 0 2.0000 1 1.6667 2 1.5911 3 1.5874 4 1.5874 At what iteration nmber wold yo trst at least two significant digits in yor answer? A. 1 B. 2 C. 3 D. 4 6. The ideal gas law is given by pv = RT where p is the pressre, v is the specific volme, R is the niversal gas constant, and T is the absolte temperatre. This eqation is only accrate for a limited range of pressre and temperatre. Vander Waals came p with an eqation that was accrate for larger range of pressre and temperatre given by a p + ( v b) = RT v 2 where a and b are empirical constants dependent on a particlar gas. Given the vale of R = 0.08, a = 3.592, b = 0.04267, p = 10 and T = 300 (assme all nits are consistent), one is going to find the specific volme, v, for the above vales. Withot finding the soltion from the Vander Waals eqation, what wold be a good initial gess for v? A. 0 B. 1.2 C. 2.4 D. 3.6 Following Bloom s Taonomy to Assess Stdents -7
This material is based pon work spported by the National Science Fondation nder Grant No. 0126793. Any opinions, findings, and conclsions or recommendations epressed in this material are those of the athor(s) and do not necessarily reflect the views of the National Science Fondation. Following Bloom s Taonomy to Assess Stdents -8