Date: 2/04/08 I. Assessment of Stuent Learning Outomes 1. Name an Contat Information of Person(s) Responsile for Program s Assessment Terry Kiser, Chair Department of Mathematis & Statistis, zip 0525 898-6111 tkiser@suhio.eu Dr. LaDawn Haws Assessment Coorinator Department of Mathematis & Statistis, zip 0525 898-5271 lhaws@suhio.eu 2. Goal Statements & Stuent Learning Outomes [General Content] Grauates are profiient in performing asi operations on funamental mathematial ojets an have a working knowlege of the mathematial ieas an theories ehin these operations. GC1 Demonstrate asi skills an oneptual unerstaning of ifferential, integral, an multivariale alulus. GC2 Demonstrate asi skills an oneptual unerstaning as relating to funamental mathematial ojets introue in our egree ore, suh as, sets, funtions, equations, vetors, an matries. GC3 Demonstrate asi unerstaning of proaility an statistis, relevant to their option in the major. GC4 Demonstrate more tehnial skills an more in-epth an roaer oneptual unerstaning in ore mathematial areas (suh as, analysis, geometry/topology, algera, applie math, statistis), relevant to their option in the major. [Critial Thinking/Prolem Solving] Grauates use ritial thinking an prolem solving skills to analyze an solve mathematial & Statistial prolems. CT/PS1 Interpret an translate prolems into appropriate mathematial language. CT/PS2 Solve prolems y applying appropriate strategies an interpreting the results. 1
[Communiation] Grauates ommuniate mathematis effetively in a manner appropriate to areer goals an the mathematial maturity of the auiene. Com1 Demonstrate the aility to effetively an aurately write on mathematial topis relevant to their mathematis option an appropriate to their auiene. Com2 Demonstrate the aility to effetively an aurately speak on mathematial topis relevant to their mathematis option an appropriate to their auiene. [Proofs Profiieny] Grauates have a asi profiieny in the omprehension an appliation of proofs. PP1 Stuents an rea mathematial proofs, extrat the key ieas use in the proof, an onvey the logi ehin the proof. PP2 Stuents emonstrate the aility to write their own rigorous an logially orret proofs. [Tehnology] Grauates know how to use tehnology tools (e.g., graphing alulators, omputer algera systems) appropriate to the ontext of the prolem. Teh1 Stuents use tehnology to manipulate mathematial ojets (e.g., funtions, equations, ata sets, et.). Teh2 Stuents use tehnology to onut mathematial explorations, to moel prolem senarios, an to analyze mathematial ojets. [Life-long Learner] Grauates are aware of the important role of mathematis an have the interest an aility to e inepenent learners an pratitioners. LL1 Stuents emonstrate the aility to apply mathematis an statistis to new ontexts (e.g. in other lasses, the workplae, grauate shool, lasses they teah). LL2 Stuents reognize an appreiate the role that mathematis an play in their future an in soiety in general. 3. Course Alignment Matrix: See the attahe Exel spreasheet 4. Learning Outome(s) Assesse to Date an Planne for 2007-08 2
Stuent Learning Outome GC1 1 GC1 PP1 GC4 PP2 Sample an Sample Size Math 121 Common Final Sp 06: 104 Math 121 Common Final Sp 07: 10 Math 330 Inlass assignment Sp 07: 10 Math 330 Final Exam Sp 07: 17 Math 330 Final Exam Sp 07: 17 GC1/GC2 Math 120 Common Final Sp 08 GC1/GC2 Math 121 Common Final Exam Sp 08 GC4 PP1/PP2 Math 346, Math 361, Math 449, Math 450 Sp 08 Measure Ahieving was set at Aeptale or Exemplary ase on the Ruri given in the Appenix Profiient was set at Aeptale or Exemplary ase on the Ruri given in the Appenix Profiient was set at Aeptale or Exemplary ase on the Ruri given in the Appenix Profiient was set at Aeptale or Exemplary ase on the Ruri given in the Appenix Profiient was set at Aeptale or Exemplary ase on the Ruri given in the Appenix TBC We plan to o a omplete assessment of the General Content goals in Math 120 y assoiating groups of prolems on the ommon final to all the funamental topis relate to these outomes. TBC Similar to Math 120 aove, we plan to o a omplete assessment an, where appropriate with overlapping ontent from Math 120, we will analyze this assessment as oming at a Pratie level. TBC We plan to assess this outome in all options in the major y onuting emee assessment in 4 upper ivision ourses in eah option. It will e a thorough assessment involving multiple prolems on all exams allowing us to trak stuent progress through the ourse. Math 420 Sp 08 TBC We in t omplete the assessment in Math 420 planne for Spring 2007 that was to e in oorination with the assessment from Math 330. So, this will e a Spring 2008 task. Perent of Stuents Ahieving 54.8% 60.0% 70.0% 35.3% 70.6% 1 This was iential to the assessment in Spring 2006 in the row aove; it was repeate to extrat math majors only. 3
Com2 for the Math E option Math 342 Inlass presentations Sp 08 Teh1/Teh2 Math 230 Projets Sp 08 TBC This ourse is only require for the Math E ut the epartment has plans for a new senior projet/apstone ourse, whih will serve all other options TBC Math 230 fulfills our Computer Literay requirement. 5. Analysis / Interpretation of Results Stuent Learning Outome GC1 Sample an Sample Size Math 121 Common Final: 10 Measure Profiient was set at Aeptale or Exemplary ase on the Ruri given in the Appenix Perent of Stuents Profiient 60.0% The ommon final for Math 121 was group grae y all of the instrutors. Sine one of the instrutors was a memer of the epartment s Assessment ommittee, that person grae this prolem for all setions an, at the same time, pike out our math majors to apply the ruri. The results of this assessment were isusse at a Fall 2007 ourse oorination meeting. This type of meeting is hel at the eginning of eah semester in all multisetion ourses an in this ase it involve all urrent alulus instrutors. Given the asi nature of this prolem, there was onsierale surprise in the outome an isussion followe as to why the % of stuents at a profiient level was so low. However, the most prominent response was onern over how we are implementing assessment eause there seems to e very little that one oul garner from a single emee prolem like this. There was onsensus that if this is going to e meaningful an helpful in shaping the urriulum in the future, it has to e far more sustantial. If we want to gain insight on the level of profiieny for a ertain stuent-learning outome, then we nee to over that outome with more in epth assessment, with multiple prolems proviing a variety of perspetives. Further oorination is neee to e onsistent with a pakage of prolems aross setions ut this seeme oale espeially in this ourse an in Math 120 where we alreay have a Common Final in plae. Stuent Learning Outome PP1 Sample an Sample Size Math 330 In lass assignment: 10 Measure Profiient was set at Aeptale or Exemplary ase on the Ruri given in the Appenix Perent of Stuents Profiient 70.0% 4
As esrie in the prolem statement, stuents saw the instrutor provie a proof for a very similar prolem. Then, a few lass perios later, stuents were given this prolem as an in-lass assignment. The intent was to hek their aility to unerstan a given proof, that is, to follow the logi ehin the proof an to e ale to extrat the key ieas, whether it is rea on their own or presente to them. This oul e assesse in a variety of ways; one way is to follow up the presentation of the original proof with a written assignment where stuents are aske to explain in their own wors the logi ehin the proof or the key ieas an another way is to see if they an apply the ieas from the original proof in writing their own proofs. In this assessment, in aition to eing irete at a speifi stuent-learning outome, we are hoping to gain experiene an a etter unerstaning of the effetiveness of the ifferent ways of onuting assessment for this outome. Two faulty, one eing the epartment s new Assessment Coorinator, sore the prolem. They are the only ones to have isusse these results at this time. During Fall 2007, it was planne to get feeak from the instrutor who taught the ourse when the assessment took plae, as well as from other faulty who often teah this ourse, ut that has een postpone to Spring 2008. However, it seems lear that for this speifi prolem, the level of profiieny is aeptale, espeially, sine this is the first exposure to proofs for most stuents. The main question that nees to e aresse an requires feeak from Math 330 instrutors, is the effetiveness of this approah to assessing PP1 an the valiity of this speifi prolem in terms of how losely aligne it is to the original proof presente to the stuents. Stuent Learning Outome GC4 Sample an Sample Size Math 330 Final Exam: 17 Measure Profiient was set at Aeptale or Exemplary ase on the Ruri given in the Appenix Perent of Stuents Profiient 35.3% Two faulty, one eing the epartment s new Assessment Coorinator, sore the prolem. They are the only ones to have isusse these results at this time. During Fall 2007, it was planne to get feeak from the instrutor who taught the ourse when the assessment took plae, as well as from other faulty who often teah this ourse, ut that has een postpone to Spring 2008. In riefly isussing the low level of profiieny, one oservation ame up. Continuity isn t a ommon topi in Math 330 an may e a little avane for this introutory proofs ourse. It is more typial to stik to very asi ontent areas that are suitale for introuing the logi ehin oing proofs ut not as relevant to future upper ivision math ourses. There has een some isussion aout the effetiveness of this approah in preparing stuents for more avane ourses. So, for Spring 2007, the instrutor wante to experiment with other ontent areas, whih, at the same time, woul allow us to o a follow up assessment in one of our apstone ourses, Math 420: Avane Calulus, sine ontinuity is a ore onept in Avane Calulus. We suspet that most instrutors woul expet mastery of a efinition suh as this in Avane Calulus ut it may e expeting too muh for the typial stuent in Math 330. 5
For Spring 2008, we are planning to o a follow up assessment with this exat same prolem as well as a follow up to anther emee assessment on one of our proofs profiieny outomes that took plae on this final exam an to hek for Mastery of this outome. Stuent Learning Outome PP2 Sample an Sample Size Math 330 Final Exam: 17 Measure Profiient was set at Aeptale or Exemplary ase on the Ruri given in the Appenix Perent of Stuents Profiient 70.6% The prolem was sore y two faulty, one eing the epartment s new Assessment Coorinator. They are the only ones to have isusse these results at this time. During Fall 2007, it was planne to get feeak from the instrutor who taught the ourse when the assessment took plae, as well as from other faulty who often teah this ourse, ut that has een postpone to Spring 2008. However, it seems lear that for this speifi prolem, the level of profiieny is aeptale, espeially, sine this ourse provies the first exposure to proofs for most stuents. When ouple with the emee assessment for GC4, whih also took plae on this exam, this level of profiieny seems surprising an will require some thought an investigation to explain. Stuents in t perform well on the prolem asking them to state the efinition of ontinuity ut unerstaning this efinition is ritial to e ale to o a proof of ontinuity! It will e interesting to get feeak from Math 330 instrutors ut one initial thought is that the formal efinition reveals the stuent s lak of profiieny with using mathematial notation preisely ut, eviently, not a lak of unerstaning. This prolem, although, seemingly more hallenging in requiring them to o a proof of ontinuity, involves a speifi algerai example, whih may make it oneptually easier. Our epartment is ehin in getting a long-term assessment plan in plae an in isussing the results of availale assessment in terms of what programmati ations an revisions in assessment o these results suggest. However, starting Spring 2008, we will have an Assessment Coorinator on oar an we have a plan in plae for this semester, whih will result in signifiant new assessment. Also, we starte the semester with a series of Course Coorination meetings to get instrutors on oar to help with the emee assessment an to help plan follow up meetings so we an have a more involve epartment in losing the loop. Last year, assessment was primarily ommittee riven an our goal for this semester is get more epartmental involvement. This has not een ompletely laking. Last spring, we ha a series of 5 meetings all irete towar assessment. This was valuale in shaping our perspetive on onuting assessment in a way that will e more meaningful an enefiial to the epartment. For the first time, we saw the full yle of assessment take plae as the epartment ame to the onlusion that signifiant hanges to our urriulum were neee if wante to keep an value 6
our stuent-learning outomes for Communiation (whih everyone inee felt was important). This report will e upate soon with further etails. 6. Planne Program Improvement Ations Resulting from Outomes (if appliale) 7. Planne Revision of Measures or Metris (if appliale) 8. Planne Revisions to Program Ojetives or Learning Outomes (if appliale) 9. Changes to Assessment Sheule (if appliale) 10. Information for Next Semester II. Appenies (please inlue any of the following that are appliale to your program) A. Assessment Data Summaries B. Measurement Stanars (Ruris, et.) C. Survey Instruments Math 121 Common Final Spring 2007 GC 1 Emee Assessment Prolem & Ruri Prolem Statement: Write the integral(s) that alulates the shae area shown. This prolem was esigne to help us assess a general ontent SLO onerning oneptual unerstaning in seon semester alulus. Speifially, this prolem heks a stuents unerstaning of the onnetion etween the integral an area trappe etween two urves. f(x) This prolem purposefully goes eyon the asi onept, whih, a in general, involves the integral of the ifferene of the two funtions, in requiring areful treatment of the ouns for the integral to eal with left out areas an positive/negative area. g(x) 7
Exemplary Aeptale Marginal Unaeptale Integral/Area Conept Ruri Eviene emonstrates that the stuent has mastere this one aspet of the outome as it relates to the appliation of the integral to area an at a high level of sophistiation in ealing with the more iffiult ouns. Eviene emonstrates that the stuent has a general unerstaning of the onnetion of the integral to area ut is laking the sophistiation to eal with the exlue area eyon x = or the negative area elow the x-axis. Eviene that the stuent has some unerstaning of the onnetion to area is provie ut it is too inomplete to efinitively say it is at an aeptale level. In aition to missing how to eal with oth of the exlue areas, their answer is weak in showing an unerstaning of the asi priniple that the area is the integral of the ifferene of the two funtions. Eviene that the stuent has mastere this outome is very inomplete. Answers might involve the integral of the ifferene of the two funtions ut inappropriately an in a manner that suggests it is a memorize formula with little or no unerstaning ehin it. Sample answers for eah ategory Exemplary f (x)x + ( f (x) g(x)) x f (x) x g(x) x Aeptale Corret answers as aove ut inorret notation, suh as, missing x or parentheses. Marginal ( f (x) g(x)) x ( f (x) g(x)) x Unaeptale 8
( f (x) g(x)) x (or variations of this showing no attention to the atual shae area.) a f (x) x Math 330 Spring 2007 Emee Assessment PP1 PP1is a stuent-learning outome assoiate with our goal on Proofs Profiieny. This outome is irete towar a stuent s aility to rea a mathematial proof an show their unerstaning y eing ale to extrat the key ieas an to onvey the logi ehin the proof. We ve hosen to assess this outome y asking stuents to write a proof for a prolem that is very losely aligne to one they ha seen fairly reently eforehan. Prolem Statement Stuents were shown in lass the proof that if f(x) an g(x) are ontinuous at, then (f + g)(x) is also ontinuous at. On this prolem, they were aske to prove that if f(x) an g(x) are ontinuous at, then so is (f - g)(x). Base on the efinition of ontinuity, stuents must show that given ε > 0, there exists a δ > 0 suh that whenever x < δ, (f g)(x) - (f g)() < ε. (1) Proof: Given ε > 0, sine f is ontinuous at, there exists δ 1 > 0 suh that whenever x < δ 1, f (x) f () < ε 2. Similarly, sine g is ontinuous at, there exists δ 2 > 0 suh that whenever x < δ 2, g(x) g() < ε 2. Let δ = min{ δ 1, δ 2 }. (2) Then, whenever x < δ, (f g)(x) - (f g)() = f (x) g(x) - f() + g() = f (x) f() + g() - g(x) (3) < f (x) f() + g() - g(x) (y the triangle inequality) (4) = f (x) f() + g(x) - g() (5) < ε /2 + ε /2 = ε (from ontinuity of f an g) (6) Ruri/Comments: Steps (1) (this isn t neessarily a step that they must put in their proof ut the iea must e evient in their proof), (2), an (6) are iential to the proof given in lass. The main ifferene with the proof given in lass are that in steps (3) (5) stuents must eal with the new minus sign an realize that g() g(x) = g(x) g() so they an orretly ring in the assumption of ontinuity of g. Note: a goo stuent oul go from step (3) to step (5); unfortunately, a weak 9
stuent might o that, too, ut not neessarily knowing why perhaps, they just knew from the in-lass prolem that this is where they were suppose to en up. Step (1) shows an unerstaning of the efinition, whih rives the proof, an step (2) is the set up for the proof orret work through these two steps is neessary for a Marginal rating. To e at an Aeptale level, stuents also nee to know how the proof must en an show step (6) even if their work in the intermeiate steps is somewhat inorret. To have mastere this type of proof an e at an Exemplary level, stuents must eal with the new feature the minus sign orretly an this means not skipping steps so the eviene is learly shown. Math 330 Spring 2007 Emee Assessment GC4 GC4 is a General Content stuent-learning outome irete towars more tehnial skills an more in epth oneptual unerstaning. Math 330, in proviing an introution to oing mathematial proofs, is a ourse where stuents first get expose to the important role of efinitions. Given this fous, we hose to o an emee assessment on the Final Exam given in Spring 2007 where stuents were aske to state the efinition of ontinuity, whih is one of the main onepts in the ourse. Prolem Statement: Given a funtion f : D an D, state what it means for f to e ontinuous at. Definition: (Part 1) ε > 0, δ > 0 suh that if x D with (Part 2) x < δ, then f (x) f () < ε. Ruri/Comments: The asi iea is that if x is lose to then f(x) will e lose to f(). This is apture in Part 2 of the efinition an is require to e at a Marginal level. It is ritial that stuents use the orret quantifiers for ε an δ as iniate in Part 1 - so, to rise to an Aeptale level, this must e evient in their efinition, although, perhaps not in the logially orret orer or plae. By the en of the semester, stuents shoul e ale to give preise efinitions - so, to e Exemplary, the efinition essentially has to e entirely orret. Math 330 Spring 2007 Emee Assessment PP2 PP2 is a Proofs Profiieny stuent-learning outome irete at the stuent s aility to write logially orret mathematial proofs. Math 330 is the ieal ourse for assessing this at the Introue or Pratie level as the main goal of the ourse is not ontent riven as muh as to provie an introution to oing mathematial proofs. In this single ourse, we expet stuents to evolve in profiieny from an Introue level of expetation to a more Pratie level y the en of the en of the ourse. We hose to assess their proof writing aility at the Pratie level through an emee assessment on the Final Exam given in Spring 2007. One of the topis 10
overe on the Final was Continuity an a founational type of prolem at this level in this topi is to show ontinuity of a speifi an algeraially simple funtion. This prolem also was hosen eause this is a ore topi in Avane Calulus, whih will allow us to o a follow up assessment at the Mastery level using an iential prolem type. Prolem Statement: Let f (x) = 3x 2. Show that f is ontinuous at = 4. Stuents are expete to know how the efinition rives the proof an that the key step is to etermine how to hose δ in terms of an aritrary ε that will ensure that f (x) f () < ε. They are often taught to o the srath work to fin δ efore starting to write their proofs. Proof Srath work: f (x) f (4) = (3x 2) 10 = 3x 12 = 3 x 4. This reveals that, sine we want f (x) f (4) < ε, we nee to keep x 4 < ε 3 [Start of the proof] Let ε > 0. Take δ = ε 3 - here s δ (1) (2) Then, if x 4 < δ, f (x) f (4) = L (repeat srath work) = 3 x 4 < 3( ε 3 ) = ε (3) Ruri/Comments: As state aove, the key step is showing their unerstaning of the efinition, that is, to show how to make f (x) f (4) < ε an unerstaning that the ontrol omes from x 4. Getting through the srath work either outsie or in the proof - is require to e at a Marginal level. To e Aeptale, the proof must expliitly show that they know how to hoose δ from this srath work an they unerstan the orret quantifiers for ε an δ even if there is some slight logial error in the orer or plaement in setting them up. To e Exemplary, the proof must have the srath work inserte orretly showing they also have the orret unerstaning of how the proof ens. 11
Math 121 Common Final Emee Assessment Prolem & Ruri Prolem Statement: Write the integral(s) that alulates the shae area shown. This prolem was esigne to help us assess a general ontent SLO onerning oneptual unerstaning in seon semester alulus. Speifially, this prolem heks a stuents unerstaning of the onnetion etween the integral an area trappe etween two urves. This prolem purposefully goes eyon the asi onept, whih, in general, involves the integral of the ifferene of the two funtions, in requiring areful treatment of the ouns for the integral to eal with left out areas an positive/negative area. f(x) a g(x) Exemplary Aeptale Marginal Unaeptale Integral/Area Conept Ruri Eviene emonstrates that the stuent has mastere this one aspet of the outome as it relates to the appliation of the integral to area an at a high level of sophistiation in ealing with the more iffiult ouns. Eviene emonstrates that the stuent has a general unerstaning of the onnetion of the integral to area ut is laking the sophistiation to eal with the exlue area eyon x = or the negative area elow the x-axis. Eviene that the stuent has some unerstaning of the onnetion to area is provie ut it is too inomplete to efinitively say it is at an aeptale level. In aition to missing how to eal with oth of the exlue areas, their answer is weak in showing an unerstaning of the asi priniple that the area is the integral of the ifferene of the two funtions. Eviene that the stuent has mastere this outome is very inomplete. Answers might involve the integral of the ifferene of the two funtions ut inappropriately an in a manner that suggests it is a memorize formula with little or no unerstaning ehin it. 12
Sample answers for eah ategory Exemplary f (x)x + ( f (x) g(x)) x f (x) x g(x) x Aeptale ( f (x) g(x) ) x g(x) x (Missing that the area elow the x-axis is negative.) Marginal ( f (x) g(x)) x ( f (x) g(x)) x f (x) x Unaeptale ( f (x) g(x)) x (Showing no attention to the atual shae area.) a Please sumit omplete reports eletronially to your ollege ean an to the provost s offie (warnok@suhio.eu). 13