First International Congress on Tools for Teaching Logic Teaching Logical

8/14/2019 First International Congress on Tools for Teaching Logic Teaching Logical

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First International Congress on

Tools for Teaching Logic

Teaching Logical reasoning in high school

Antonia Huertasmhuert [email protected] ec.es

Teacher of t he Spanish secondary school and l ogician.Barcelona

Spain

Sept ember 6, 2000

A bst ract

T his report deals wit h t wo general feat ures in t he usual curri culum in

high school. On t he one hand there is no explicit inclusion of Logical

Reasoning in secondary studies, on t he other hand it is commonly admit -t ed t hat t he logical reasoning is a crucial part of general educati on in any

of t he areas most closely associat ed wit h Logic: Phil osophy, M athemat ics,

Comput er Science, etc.

Logic courses are present in t he coll ege level, in Phil osophy, Mat hemati cs,Computer Science and Li nguist ics courses; and t here are not doubt t hat

logical reasoning is in t he basis of any subj ect r elated with Logic. But t he

sit uati on i s di erent in t he pre-college curr iculum, whereas other elds

are expli cit in t he secondary curri culum, Logic seems t o have been leftoutside it.

In t his paper I rst explain t he presence of logical reasoning t aught in

t he high school curr iculum ( mainly as a Logical mathemat ical reasoning).

Secondly I claim for t he recognit ion of Logic as a subj ect mat t er of t hesecondary curriculum in its own right.

1 L ogical r easonin g t aught in hi gh school

In this level, students are ages from 12 to 18, usually divided into two dier-ent cycles (12-16 and 16-18, which is t he case of t he new Spanish educationsyst em, t he one I know bet t er) . One of t he most representat ive charact erist icsfrom an educati onal point of view i s that st udents are in t he adolescent phase:at t hat moment import ant physical and psychological changes can aect t heirbehaviour. To mot ivate many of t hese t eenager st udents is somet imes a di cult


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t hing because t hey are i n t he phase of opposit ion t o t he est ablished world :

parents, teachers, learning. I do not want to start a psychological debate, but It hink t hat t his is a very dist inct ive feat ure when t eaching at t hat level.

T he high school curri culum is usually st ructured in at least two dierentst ages, t he rst one deals wit h common and basic areas such as Language, His-t ory, Mat hemat ics, Sciences, et c., and t he higher ones deal wit h more variateand specic subj ects, such as Philosophy, Economics, Physics, Psychology, et c.In t he rst stage t here is not any specic reference t o Logic in t he curriculum,.but it is an accepted t hing t hat st udents somehow learn t o reasoning logicall yin dierent areas of t he curriculum. Mathemat ics seems t o be t he principal areawhere student s have to learn logical reasoning. In t he higher st ages, speciallywhen st udents have a course in Phil osophy containing Logic (mainly propo-sit ional classical calculus at a very introduct ory level) t hey t hink t hey learn

reasoning wit h Philosophy as well as wit h Mathemat ical courses. But froman inst rumental point of vi ew t here is no doubt t hat Mathemat ics is the cur-rent context for logical r easoning in high school. In fact t his is connected wit hlearning / discovering induct ive and deduct ive abili t ies.

2 D iscover ing induct ive and deduct ive abilit ies

in mathematical courses.

When I ask my st udents i f learning Mathemat ics can be an enjoyable act ivi tyor if it is always a hard, painful t hing, most of t hem answer : yes, i t can be

enjoyable, but only if t hey (M athemat ics) are understandable and many oft hem recognize t hat t he t eacher is very import ant t o underst and M at hemat ics.What do st udents mean wit h understandable and thus enjoyable Mat hemat ics?.When t hey are l earning an algori t hm in calculus or algebra or using a newmat hemat ical language, t hey use expressions such as it is easy , it is hard or when do I use it ? , but in t hese cases t hey dont have to underst and or notsomet hing. To understand Mat hemat ics means t o follow t he underlying logicalreasoning t hat allows them t o const ruct a mathemat ical abst ract concept or asolving st rategy. T hus, it seems t hat t o enjoy learning Mat hemat ics is connectedwit h using l ogical (deduct ive and i nduct ive) abil it ies successfull y.

Now, t ake t hese adolescent crit ical and insecure st udents and t ry t hem t olearn how t o develop induct ive and deduct ive abili t ies in t he context of Mathe-mat ics. It is oft en a very di cult t hing. We must t ake int o account t hat t hereis an intr insic di culty in mat hemat ical language and met hods t hat are obst a-cles when exercising t he essay-err or met hod that develops t he logical abil it ies.The abst ract languageof Mat hemat ics, l ow moti vat ion and the insu cient pres-ence of t he mat hs t eacher in t he process of learning r easoning (i n Spain t hereare about 30 st udents per group) make underst anding mat hs a di cult t hing.


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Learning some part s of Mat hemat ics is not possible wit hout using logical

reasoning, but what I also defend here is t hat learning logical reasoning wit houtmat hemat ical language is possible and necessary. I t hink t hat t he fact of teach-ing logical reasoning basically in mathematical courses is a special problem fort he less bril li ant st udents. When st udents have a previous st ructur ed logicalway of t hinking they are able t o bet t er learn Mathemat ics and it improves t heircapacity t o bet t er reasoning generally. But when st udents have t o manage bothwit h Mathemat ics and Logical Deduct ion, t he di cult ies increase and makeMat hemat ics understandable and t hen only a hard t hing t o learn.

Now I shall make a sket ch of t he main part s of t he mat hemat ical curri culumconnected wit h t he logical/ cognit ive domain in secondary school.

Processes of codicat ion and decodicati on using numbers, graphics, func-t ions or specic languages.

Known conceptually natural numbers, integers, fractions, decimal and ir-rational numbers, it s representat ion and operativi ty.

Elaborat e geomet ri cal models by using plane and solid geomet ry.

St udy funct ionsas a t ool t o describe informat ion and to const ruct models.

To use dierent mat hemat ical languages in t he appropriate sit uat ions.

To use t he mat hemat ical way of reasoning.

Discover t he beauty of t he mat hemat ical str uct ures.

When t eaching t hese parts of Mat hemat ics in high school t here are manypart s of Logic t hat are involved:

T he development of language and relat ions: verbal, symbolic, graphic.Syntaxis.

Formali zat ion of information in dierent languages.

Using precision and rigor.

Using connecti ves and i nferencerules (M odus Ponens, No contr adict ion) .

Using rest ri cti ons and condit ions (necessaries, su cient )

Representation of relations wit h diagrams.

Const ruct ion of well dened concept s (pr imi t ive and deri ved concept s)

Proofs and counterexamples.


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Using language and met alanguage.

Paradoxes (as mat hemati cal games).

All t hese logical concept s are also logical-mathemat ical concept s. At t hatlevel t he border is very t hin. But t hey are Logic concepts in t heir own ri ght .From my point of view t hey are independent from t he Mathemat ics concepts,t hey can be taught independently of Mat hemat ics; and t eaching L ogic in highschool would even i mprove t he result s when t eaching M athemat ics.

3 I nclusion of Logic in t he curriculum of high

school

In t his section I present some ideas about how Logic might be brought into highschool level. Before that, I would like to mention t he import ance of promot -ing and facil it ati ng logical reasoning at an early age. If st udents are exposed t oscholar experi ences t hat involve abil it ies such as exploring, conject uring and r ea-soning logically, as well as t he abili ty t o use a variety of mat hemat ical met hodst o solve problems t hey wi ll gain in logical/ mat hemat ical power .

In high school level, an appropriate logical curri culum should consist int eaching t he explicit logical not ions and t echniques t o const ruct proofs andcounterexamples, especiall y in t he higher grades. In t he rst st age (12-16) t he

curriculum in Logic should contain :

Formali zat ion i n di erent languages (also in algebraic languages). Workwith connectives, negation, disjunction.

Work on particularization and generalization.

Work on relat ions wit h dierent representat ions. Dening relations andset s.

St udy elementary concept s on Set Theory.

Elementary Computat ion concept s: programs, languages, algori t hms, et c.

Give st rategies t o solve nonrout ine problems (some of t hem wit h a mat h-emat ical context and some of t hem about everyday sit uat ions). Expresswit h algori t hms t he st rategy and whit precision the solut ion.

Give proofs in very simple cases.


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In higher st ages (16-18) t he st ress must be on pr oof and deduct ion, and a

possible curri culum could contain t he following points:

More Set Theory. Cardinalit y.

Presentation of formal languages. (Elements of a formal l anguage)

Proofs. Elements of a logical proof. Examples wit h dierent languages.Also mat hemat ical proofs as an important case.

Const ruct ion of counterexamples.

Int roduct ion t o T heoret ical Comput at ion.

Met alanguage and paradoxes.

The more general goal of Logic in high school should be to understand thevery nat ure of t hesubject as an ult imately deduct ive discipli ne and in t hehighergrades t o underst and t he role of Logic in Mathemat ics and Comput at ion.

In a report of the Commit tee on Logic and Education of the A.S.L twost rat egies were presented t o work on a general recognit ion of L ogic as a subj ectmat t er i n it s own right in t he pre-college curri culum:

(1) Work for a subst antive addit ion t o the Mathemat ics curriculum.(2) Work t oward t he day when all t eachers wil l have an appreciat ion of

Logic as an import ant discipli ne .

Both t hings are connected wit h a more general t ask: dispel t he negat ive

view of many of t he academic and administ rative coll eagues t hat modern Logicis symbol pushing, and explain t henecessit y of Logic st udies int o t hepreparationof mathematical and computer high school teachers.

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First International Congress on Tools for Teaching Logic Teaching Logical