Success in Mathematics

Transcription

1 Success in Mathematics Tips n hw t study mathematics, hw t apprach prblem-slving, hw t study fr and take tests, and when and hw t get help. Math Study Skills Be actively invlved in managing the learning prcess, the mathematics and yur study time: Take respnsibility fr studying, recgnizing what yu d and dn t knw, and knwing hw t get yur Instructr t help yu with what yu dn t knw. Attend class every day and take cmplete ntes. Instructrs frmulate test questins based n material and examples cvered in class as well as n thse in the text. Be an active participant in the classrm. Get ahead in the bk; try t wrk sme f the prblems befre they are cvered in class. Anticipate what the Instructr s next step will be. Ask questins in class! There are usually ther students wanting t knw the answers t the same questins yu have. G t ffice hurs and ask questins. The Instructr will be pleased t see that yu are interested, and yu will be actively helping yurself. Gd study habits thrughut the semester make it easier t study fr tests. Studying Math is Different frm Studying Other Subjects Math is learned by ding prblems. D the hmewrk. The prblems help yu learn the frmulas and techniques yu d need t knw, as well as imprve yur prblem-slving prwess. A wrd f warning: each class builds n the previus nes, all semester lng. Yu must keep up with the Instructr: attend class, read the text and d hmewrk every day. Falling a day behind puts yu at a disadvantage. Falling a week behind puts yu in deep truble. A wrd f encuragement: each class builds n the previus nes, all semester lng. Yu re always reviewing previus material as yu d new material. Many f the ideas hang tgether. Identifying and learning the key cncepts means yu dn t have t memrize as much. Cllege Math is Different frm High Schl Math A cllege math class meets less ften and cvers material at abut twice the pace that a High Schl curse des. Yu are expected t absrb new material much mre quickly. Tests are prbably spaced farther apart and s cver mre material than befre. The Instructr may nt even check yur hmewrk. Take respnsibility fr keeping up with the hmewrk. Make sure yu find ut hw t d it. Yu prbably need t spend mre time studying per week. Yu d mre f the learning utside f class than in High Schl. Tests may seem harder just because they cver mre material. 1

2 Study Time Yu may knw a rule f thumb abut math (and ther) classes: at least 2 hurs f study time per class hur. But this may nt be enugh! Take as much time as yu need t d all the hmewrk and t get cmplete understanding f the material. Frm a study grup. Meet nce r twice a week (als use the phne). G ver prblems yu ve had truble with. Either smene else in the grup will help yu, r yu will discver yu re all stuck n the same prblems. Then it s time t get help frm yur Instructr. The mre challenging the material, the mre time yu shuld spend n it. Prblem Slving (Hmewrk and Tests) The higher the math class, the mre types f prblems. In earlier classes, prblems ften required just ne step t find a slutin. Increasingly, yu will tackle prblems which require several steps t slve them. Break these prblems dwn int smaller pieces and slve each piece divide and cnquer! Prblem types Prblems testing memrizatin ( drill ), Prblems testing skills ( drill ), Prblems requiring applicatin f skills t familiar situatins ( template prblems), Prblems requiring applicatin f skills t unfamiliar situatins (yu develp a strategy fr a new prblem type), Prblems requiring that yu extend the skills r thery yu knw befre applying them t an unfamiliar situatin. In early curses, yu slved prblems f types 1, 2 and 3. By Cllege Algebra yu expect t d mstly prblems f types 2 and 3 and smetimes f type 4. Later curses expect yu t tackle mre and mre prblems f types 3 and 4, and (eventually) f type 5. Each prblem f types 4 r 5 usually requires yu t use a multi-step apprach, and may invlve several different math skills and techniques. When yu wrk prblems n hmewrk, write ut cmplete slutins, as if yu were taking a test. Dn t just scratch ut a few lines and check the answer in the back f the bk. If yur answer is nt right, rewrk the prblem; dn t just d sme mental gymnastics t cnvince yurself that yu culd get the crrect answer. If yu can t get the answer, get help. The practice yu get ding hmewrk and reviewing will make test prblems easier t tackle. Tips n Prblem Slving Apply Pólya s fur-step prcess: The first and mst imprtant step in slving a prblem is t understand the prblem, that is, identify exactly which quantity the prblem is asking yu t find r slve fr (make sure yu read the whle prblem). Next yu need t devise a plan, that is, identify which skills and techniques yu have learned can be applied t slve the prblem at hand. Carry ut the plan. Lk back. Des the answer yu fund seem reasnable? Als review the prblem and methd f slutin s that yu will be able t mre easily recgnize and slve a similar prblem. 2

3 Sme prblem-slving strategies: use ne r mre variables, cmplete a table, cnsider a special case, lk fr a pattern, guess and test, draw a picture r diagram, make a list, slve a simpler related prblem, use reasning, wrk backward, slve an equatin, lk fr a frmula, use crdinates. Wrd Prblems are Really Applied Prblems The term wrd prblem has nly negative cnntatins. It s better t think f them as applied prblems. These prblems shuld be the mst interesting nes t slve. Smetimes the applied prblems dn t appear very realistic, but that s usually because the crrespnding real applied prblems are t hard r cmplicated t slve at yur current level. But at least yu get an idea f hw the math yu are learning can help slve actual real-wrld prblems. Slving an Applied Prblem First cnvert the prblem int mathematics. This step is (usually) the mst challenging part f an applied prblem. If pssible, start by drawing a picture. Label it with all the quantities mentined in the prblem. If a quantity in the prblem is nt a fixed number, name it by a variable. Identify the gal f the prblem. Then cmplete the cnversin f the prblem int math, i.e., find equatins which describe relatinships amng the variables, and describe the gal f the prblem mathematically. Slve the math prblem yu have generated, using whatever skills and techniques yu need (refer t the fur-step prcess abve). As a final step, yu shuld cnvert the answer f yur math prblem back int wrds, s that yu have nw slved the riginal applied prblem. Studying fr a Math Test Everyday study is a big part f test preparatin. Gd study habits thrughut the semester make it easier t study fr tests. D the hmewrk when it is assigned. Yu cannt hpe t cram 3 r 4 weeks wrth f learning int a cuple f days f study. On tests yu have t slve prblems; hmewrk prblems are the nly way t get practice. As yu d hmewrk, make lists f frmulas and techniques t use later when yu study fr tests. Ask yur Instructr questins as they arise; dn t wait until the day r tw befre a test. The questins yu ask right befre a test shuld be t clear up minr details. Studying fr a Test Start by ging ver each sectin, reviewing yur ntes and checking that yu can still d the hmewrk prblems (actually wrk the prblems again). Use the wrked examples in the text and ntes - cver up the slutins and wrk the prblems yurself. Check yur wrk against the slutins given. Yu re nt ready yet! In the bk each prblem appears at the end f the sectin in which yu learned hw d t that prblem; n a test the prblems frm different sectins are all tgether. Step back and ask yurself what kind f prblems yu have learned hw t slve, what techniques f slutin yu have learned, and hw t tell which techniques g with which prblems. Try t explain ut lud, in yur wn wrds, hw each slutin strategy is used (e.g. hw t slve a quadratic equatin). If yu get cnfused during a test, yu can mentally return t yur verbal 3

4 Als capsule instructins. Check yur verbal explanatins with a friend during a study sessin (it s mre fun than talking t yurself!). Put yurself in a test-like situatin: wrk prblems frm review sectins at the end f chapters, and wrk ld tests if yu can find sme. It s imprtant t keep wrking prblems the whle time yu re studying. Start studying early. Several days t a week befre the test (lnger fr the final), begin t allt time in yur schedule t reviewing fr the test. Get lts f sleep the night befre the test. Math tests are easier when yu are mentally sharp. Taking a Math Test Test-Taking Strategy Matters Just as it is imprtant t think abut hw yu spend yur study time (in additin t actually ding the studying), it is imprtant t think abut what strategies yu will use when yu take a test (in additin t actually ding the prblems n the test). Gd test-taking strategy can make a big difference t yur grade! Taking a Test First lk ver the entire test. Yu ll get a sense f its length. Try t identify thse prblems yu definitely knw hw t d right away, and thse yu expect t have t think abut. D the prblems in the rder that suits yu! Start with the prblems that yu knw fr sure yu can d. This builds cnfidence and means yu dn t miss any sure pints just because yu run ut f time. Then try the prblems yu think yu can figure ut; then finally try the nes yu are least sure abut. Time is f the essence; wrk as quickly and cntinuusly as yu can while still writing legibly and shwing all yur wrk. If yu get stuck n a prblem, mve n t anther ne. Yu can cme back later. Wrk by the clck. On a 50-minute, 100-pint test, yu have abut 5 minutes fr a 10-pint questin. Starting with the easy questins will prbably put yu ahead f the clck. When yu wrk n a harder prblem, spend the alltted time (e.g., 5 minutes) n that questin, and if yu have nt almst finished it, g n t anther prblem. D nt spend 20 minutes n a prblem which will yield few r n pints when there are ther prblems still t try. Shw all yur wrk. Make it as easy as pssible fr the Instructr t see hw much yu d knw. Try t write a well-reasned slutin. If yur answer is incrrect, the Instructr may assign partial credit based n the wrk yu shw. Never waste time erasing! Just draw a line thrugh the wrk yu want ignred and mve n. Nt nly des erasing waste precius time, but yu may discver later that yu erased smething useful (and/r maybe wrth partial credit if yu cannt cmplete the prblem). Yu are (usually) nt required t fit yur answer in the space prvided - yu can put yur answer n anther sheet t avid needing t erase. In a multiple-step prblem, utline the steps befre actually wrking the prblem. Dn t give up n a several-part prblem just because yu can t d the first part. Attempt the ther part(s). If the actual slutin depends n the first part, at least explain hw yu wuld d it. Make sure yu read the questins carefully, and d all parts f each prblem. Verify yur answers. Des each answer make sense given the cntext f the prblem? If yu finish early, check every prblem (that means rewrk everything frm scratch). 4

5 Getting Assistance Get help as sn as yu need it. Dn t wait until a test is near. The new material builds n the previus sectins, s anything yu dn t understand nw will make future material difficult t understand. Use the Resurces Yu Have Available Ask questins in class. Yu get help and stay actively invlved in the class. Visit the Instructr s Office Hurs. Instructrs like t see students wh want t help themselves. Ask friends, members f yur study grup, r anyne else wh can help. The classmate wh explains smething t yu learns just as much as yu d, fr he/she must think carefully abut hw t explain the particular cncept r slutin in a clear way. S dn t be reluctant t ask a classmate. G t Math Help Sessins r ther tutring sessins n campus. Find a private tutr if yu can t get enugh help frm ther surces. All students need help at sme pint, s be sure t get the help yu need. Asking Questins Dn t be afraid t ask questins. Any questin is better than n questin at all (at least yur Instructr/tutr will knw yu are cnfused). But a gd questin will allw yur helper t quickly identify exactly what yu dn t understand. Nt t helpful cmment: I dn t understand this sectin. The best yu can expect in reply t such a remark is a brief review f the sectin, and this will likely verlk the particular thing(s) which yu dn t understand. Gd cmment: I dn t understand why f(x + h) desn t equal f(x) + f(h). This is a very specific remark that will get a very specific respnse and hpefully clear up yur difficulty. Gd questin: Hw can yu tell the difference between the equatin f a circle and the equatin f a line? Okay questin: Hw d yu d #17? Better questin: Can yu shw me hw t set up #17? (The Instructr can let yu try t finish the prblem n yur wn), r This is hw I tried t d #17. What went wrng? The fcus f attentin is n yur thught prcess. Right after yu get help with a prblem, wrk anther similar prblem by yurself. Yu Cntrl the Help Yu Get Helpers shuld be caches, nt crutches. They shuld encurage yu, give yu hints as yu need them, and smetimes shw yu hw t d prblems. But they shuld nt, nr be expected t, actually d the wrk yu need t d. They are there t help yu figure ut hw t learn math fr yurself. When yu g t ffice hurs, yur study grup r a tutr, have a specific list f questins prepared in advance. Yu shuld run the sessin as much as pssible. D nt allw yurself t becme dependent n a tutr. The tutr cannt take the exams fr yu. Yu must take care t be the ne in cntrl f tutring sessins. Yu must recgnize that smetimes yu d need sme caching t help yu thrugh, and it is up t yu t seek ut that caching. Department f Mathematics and Cmputer Science, SAINT LOUIS UNIVERSITY 5