Maths Teaching Tip #1-Remembering Whats Its Like Not to Know

Math Teaching ??? #1 – Remembering Wh?t It’s L?k? ??t t? Know

I h?d b??n t? London ? f?w times b?f?r?, s? ? knew m? w?? ?r?und pretty well. ??v?rth?l?ss, ? ?lw??s carried ? map. ?? ? felt sur? th?t ? w?uld n?t h?v? ?n? trouble finding m? w?? t? m? appointment w?th ? local education official-especially s?n?? h? h?d g?v?n su?h good directions: “??k? th? Northern L?n?; g?t ?ff ?t th? Elephant ?nd Castle; g? straight ?ut th? door ?nd cross t? th? ?th?r side ?f th? road; g? u? th? f?rst street ? couple hundr?d meters; ?ur office ?s ?n th? left, ?ust b?f?r? th? park. Y?u ??n’t m?ss it.”

That sounded pretty easy. ? h?d ridden th? Northern L?n? ?f th? Underground dozens ?f times, ?lth?ugh ? h?d n?v?r g?tt?n ?ff ?t th? Elephant ?nd Castle. ?? ? g?t ?ff th? Tube ?t th? correct st?? ?nd w?nt u? th? escalator, thinking t? head straight ?ut th? door. That’s wh?n m? troubles began. Wh?n ? g?t t? th? top ?f th? escalator, th?r? w?s n?t ? door t? g? straight ?ut from-there w?r? f?v? doors, ?ll distributed ?r?und th? circumference ?f ? circular-shaped exit/entry area! ?h? official hadn’t mentioned th?t. ? h?d n? idea wh??h direction t? exit. ?? mu?h f?r “g? straight ?ut th? door!”

But ?ll w?s n?t lost. ? h?d m? trusty map, ?nd ? knew th? n?m? ?f th? street ? w?s headed f?r, s? ? ?ust headed ?ut th? nearest door t? l??k f?r th? street sign. ?s ? emerged, ? discovered th?t th? tube st?? w?s ? round island surrounded b? s?v?r?l wide lanes ?f swirling traffic, w?th streets radiating ?ut ?n s?v?r?l directions. ?h? street signs ?n London ?r? embedded ?n th? walls ?f th? buildings, ?nd n?n? ?f th?m ??uld b? s??n fr?m wh?r? ? stood. (Wh?t ?s th? practicality, ? wondered, ?f street signs th?t ?r? ?nl? visible ?n?? ??u’v? ??tu?ll? turned ?nt? th? street? D? th?? serve t? offer reassurance t? people wh? ?lr??d? kn?w wh?r? they’re going?!)

It t??k ? long time f?r m? t? wander ?r?und th?t circus (well, th?t ?s wh?t th?? call ?t) unt?l ? finally f?und th? r?ght street. ? finally arrived ?t m? appointment s?m?wh?t late ?nd r?th?r perturbed. ?ut th? experience w?s n?t lost ?n m?. ?h? man h?d g?v?n m? directions th?t described exactly wh?t h? d?d ?v?r? day. ?ut h? failed t? t?k? ?nt? account th?t ? h?d n?v?r b??n th?r? b?f?r?. ?nd th? fact th?t h? d?d n?t remember wh?t ?t w?s l?k? t? b? th?r? f?r th? f?rst time caused h?m t? omit ?m??rt?nt ?nf?rm?t??n, wh??h rendered h?s directions meaningless t? m?. ?h?? w?uld ?nl? m?k? sense t? ? person wh? h?d ?lr??d? b??n th?r?! “Y?u ??n’t m?ss ?t,” indeed.

It struck m? ?s ? left m? appointment th?t th?s w?s ? perfect metaphor f?r wh?t ?ft?n g??s wrong w?th math education. ? ?n?? heard ? teacher introduce fractions t? h?s class b? pronouncing “numerator,” ?nd “denominator,” writing th?m ?n th? board, quizzing h?s pupils ?n th? correct spelling ?f th? w?rds, ?nd th?n verbally defining th??r meaning. Wh?l? h?s presentation w?s technically correct, ?nd w?s ?n accurate description ?f h?w h? thought ?f fractions ?v?r? day, th? lesson w?s meaningless t? m?n? ?f h?s students b???us? ?t provided n? connection t? physical ?r visual experience. ?h? instructor h?d forgotten wh?t ?t w?s l?k? n?v?r t? h?v? s??n ?r considered ? picture ?f ? fraction b?f?r?, ?r t? h?v? divided ?n object ?r groups ?f objects ?nt? fractional parts. ?? h?d forgotten wh?t ?t w?s l?k? t? n?t kn?w ?b?ut fractions. ?s ? consequence, h?s instructions w?uld m?k? sense m??nl? t? students wh? ?lr??d? knew ?b?ut fractions; but th? lesson w?uld g? r?ght ?v?r th? heads ?f ?th?r students, ?v?n wh?n they’re diligently paying attention.

Fortunately, m?st teachers n?w kn?w better th?n t? ?r?s?nt ? fraction lesson l?k? that-although th?t style ?f presentation ?s st?ll pretty mu?h th? norm ?n algebra classes! ?? introduce fractions, ?t ?s m?r? typical f?r th? teacher t? b?g?n b? drawing ? circle ?n th? blackboard, drawing vertical ?nd horizontal diameters thr?ugh ?t, shading three ?f th? f?ur r?sult?ng parts-and th?n proceed t? t?ll th? students th?t s?n?? th?r? ?r? f?ur parts altogether, ?nd three ?f th?m ?r? shaded, w? call th?s “three fourths.” ? f?w teachers m?ght consider th?s ?n? illustration sufficient t? define ?ll fractions. ?ut m?st teachers w?uld provide s?v?r?l pictures ?f d?ff?r?nt fractions, ?nd th?n ?sk volunteer students t? n?m? th?m properly. ?h?? th?n consider th??r introduction complete.

This type ?f presentation s??ms t? m?n? teachers t? cover ?ll th? bases, s? th?? ?r? surprised ?nd dismayed t? discover l?t?r th?t ? couple ?f th??r students st?ll h?v? n? understanding ?f basic fractions whatsoever. Naturally, teachers feel ? n??d t? account f?r th?s “? taught it-but th?? d?dn’t learn ?t” situation. ?n days g?n? b?, teachers w?uld simply label th?s? students ?s stupid, lazy, ?nd incompetent; th?? w?r?n’t paying attention, th?? w?r?n’t f?ll?w?ng directions, th?? w?r?n’t tr??ng hard ?n?ugh, th?? w?r?n’t focused, th?? d?dn’t care. Nowadays, ? d?ff?r?nt label ?s invoked: th? students d?dn’t learn th? lesson b???us? th?? h?v? learning disabilities.

But th?r? ?r? ?th?r reasons wh? th?s seemingly effective presentation ?s v?r? mu?h l?k? telling ? first-time visitor t? London t? g?t ?ff ?t th? Elephant ?nd Castle ?nd g? “straight ?ut th? door.” ?f th? teacher ?s d??ng ?ll th? drawing ?n th? board, th? teacher owns th? drawings, n?t th? students. ??m? pupils m?k? better sense ?f th? teacher’s drawings wh?n th?? copy th?m ?nt? th??r ?wn paper. F?r th?m, feeling th? ?nf?rm?t??n thr?ugh th??r ?wn fingers ?s m?r? effective th?n m?r?l? l??k?ng ?t s?m??n? else’s pictured thought. ?ut ?v?n wh?n th? lesson requires students t? copy th? teacher’s drawings, s?m? students copy th? drawings incorrectly, b???us? th?? fail t? notice ?m??rt?nt details, ?r fall b?h?nd ?nd b???m? confused ?r flustered. ?? th?? st?ll d?n’t learn th? lesson th?t ?s supposedly b??ng taught.

Even ?f th??r drawings ?r? perfect, pupils ??n st?ll fail t? connect th? pictures t? th? fraction nomenclature voiced b? th? teacher. Wh?l? th? teacher ?s proclaiming “…?nd that’s wh? w? call ?t three fourths…” s?m? students ?r? busy studying th? picture, noticing th?t three sections ?r? shaded ?nd ?n? ?s n?t. Wh?l? th??r minds ?r? completely occupied w?th t?k?ng ?n th?s visual ?nf?rm?t??n, th?? m?? n?t ?v?n hear th? teacher’s voice ?t ?ll. ?t ?s easy f?r teachers t? assume th?t b???us? th?? s??d s?m?th?ng, ?v?r??n? heard ?nd understood wh?t w?s said-forgetting h?w m?n? times ? day th??r students fail t? respond t? th? sound ?f th??r voice telling th?m t? ?ut th??r books ?w??, ?r t? ?ut th??r pencils d?wn, ?r t? b? quiet. ?v?n ?f th? students d? hear wh?t ?s s??d, th? teacher’s w?rds ??n s?m?t?m?s provoke n?th?ng but confusion: “Wh? ?s h? calling ?t three fourths, wh?n ?n? ??rt ?s white ?nd three parts ?r? shaded? ?h?t d??sn’t m?k? sense!”

And th?r? ?s st?ll m?r? th?t ??n g? wrong, ?v?n wh?n th? students understand th?t th?? sh?uld count h?w m?n? parts th?r? ?r? altogether, ?nd h?w m?n? ?f th?t total ?r? shaded. Wh?n writing th? fraction, th? learners m?? write th? total number ?f parts ?n top, ?nd th? number ?f shaded parts ?n th? bottom. ?r th?? m?? write th? fraction correctly, but read ?t fr?m th? bottom u?, ?nst??d ?f fr?m th? top d?wn. ?r th?? m?? us? th? ordinal number terminology incorrectly: “third fourth,” “three fours,” “thirds f?ur,” ?t?. ?h?r? r??ll? ?r? f?v? doors ??u ??n g? ?ut ?t th? Elephant ?nd Castle-and ?v?n m?r? ways t? misconstrue ? simple introductory lesson ?n basic fraction identification.

One imprtant key t? avoiding th?s? instruction land-mines ?s f?r th? teacher t? remember wh?t it’s l?k? n?t t? kn?w. Wh?t ?s ??t?nt??ll? confusing ?b?ut th? subject? Wh?t ??n g? wrong? Wh?t steps ?f learning ?r? prerequisite t? ?th?r steps? ?t ?s helpful f?r th? teacher t? adopt th? attitude ?f ?n actress ?n ? stage play. ??f?r? th? f?rst performance, th? actress rehearses h?r ??rt thoroughly-and naturally, sh? kn?ws h?w th? play ends. ?ut wh?n ?t ??m?s time t? perform ??t ?, Scene ?, sh? acts ?s ?f sh? d?dn’t ?lr??d? kn?w th? outcome ?f th? play. ?h? acts ?n ? w?? th?t ?s appropriate f?r th? b?g?nn?ng ?f th? play.

So th? math teacher sh?uld guide h?r students ?t th? b?g?nn?ng ?f th? lesson w?th th? attitude ?f s?m??n? wh? d??sn’t ?lr??d? kn?w wh?t ?t ?ll m??ns. ?n guiding h?r students’ exploration ?f th? subject, th? teacher’s w?rds sh?uld g?v? voice t? th? questions th?t ?r? emerging ?n th? students’ mind-or th?t ?ught t? b?. ?h? students’ attention must b? skillfully directed w?th simple commands ?nd questions. ??r? ?s ?n example ?f h?w t? d? th?s w?th ? lesson th?t introduces fractions.

The teacher hands ?v?r? student ? copy ?f ? ??g? th?t h?s m?n? pictures ?f fractions (th?r? ?r? m?n? ways t? d? th?s, but pictures ?f “pizzas” will d? f?r n?w). ???h pizza h?s ?nl? ?n? shaded slice, n? matter h?w m?n? slices th?r? ?r? altogether. ?h? f?rst pizza ?s ? picture ?f “?n? fourth.” ?h? teacher s??s, “?v?r?b?d? touch th? f?rst pizza ?n ??ur ??g?. Count ?ll th? slices. Y?s, count th? shaded slice, t??. ??w m?n? slices ?r? th?r? altogether? Write th?t number ?n ? piece ?f scratch paper.” ?h? teacher writes th? number ?n th? board ?nd l??ks t? m?k? sur? th?t ?v?r??n? h?s f?ll?w?d th? directions precisely. “??w draw ? l?ttl? l?n? ?v?r th? f?ur.” ?h? teacher models h?s instruction ?n th? board, ?nd qu??kl? inspects th? students’ work, offering guidance t? students wh? h?v? s?m?h?w managed t? draw th??r l?n? und?r th? f?ur ?nst??d ?f ?v?r ?t. “??w count h?w m?n? slices ?r? shaded… Y?s, ?ust ?n?. ??w write th?t number ?b?v? th? l?n? ??u drew. ?v?r?b?d? touch th? top number ?nd s?? ‘one.’ ??w touch th? bottom number ?nd s?? ‘fourth.’ Wh?t d? w? call th?s fraction? That’s r?ght: ‘one fourth.’ Good. ??w let’s l??k ?t th? n??t pizza.”

[By h?v?ng th? students count ?ll th? parts f?rst ?nd th?n th? shaded ??rt, th? teacher h?s sh?wn h?w t? determine th? denominator ?nd th? numerator-even th?ugh th? specific nomenclature h?s n?t ??t b??n introduced. ?f th? students h?d counted th? non-shaded ??rt f?rst, s?m? ?f them-in spite ?f verbal instructions-would h?v? automatically counted th? shaded ?n?s n??t, r?th?r th?n th? total amount. Task order ?s ?m??rt?nt ?n shaping th? direction th?t th? students’ thinking takes.]

Continuing th? lesson, th? teacher g?v?s exactly th? s?m? directions f?r th? n??t f?ur ?r f?v? pizzas. ?h?n h? tells th? students, “??w turn ??ur pencil ?r?und s? ?t l??ks l?k? you’re going t? write w?th ??ur eraser. Count ?ll th? slices ?n th? n??t pizza. Pretend t? write th?t number ?n ??ur scratch paper. ??w draw ?n imaginary l?n? ?v?r th? number. ??w m?n? slices ?r? shaded? ?h?n write ?n imaginary ‘one’ ?v?r th? l?n?. Wh?t ?s th?s fraction called?” ?w? ?r three s?m?l?r examples follow.

“Now ?ut ??ur pencils d?wn. Count h?w m?n? slices th?r? ?r? altogether ?n th? n??t pizza. Pretend t? write th?t number w?th ??ur finger, ?nd draw ? l?n? ?v?r ?t. ??w m?n? ?r? shaded? Pretend t? write th?t number ?b?v?. Wh?t ?s th? n?m? ?f th?s fraction?”

“Now ? h?v? ? challenge f?r ??u. Wh? ??n n?m? th? f?rst f?v? fractions?” ?h? teacher calls ?n ? volunteer. ?h?n ?n?th?r volunteer names th? n??t f?v? fractions. “??w ? n??d tw? volunteers wh? will ??t ?s partners.” ?h? teacher hands ?n answer key t? ?n? ?f th? partners ?nd s??s t? th? ?th?r partner, “??m? ???h fraction. Y?ur partner will check ??ur accuracy w?th th? answer key. Wh?n ??u answer correctly, sh? will s?? ‘Yes.’ Wh?n ??u ?r? wrong sh? will s??, ‘Try again,’ ?nd ??u will h?v? t? figure ?ut th? r?ght answer.” ?ft?r th? partners model th? n?w activity, th? teacher g?v?s ?n answer key t? ???h pair ?f students, ?nd t?g?th?r th?? practice proving th??r mastery ?f th? n?w lesson.

A lesson su?h ?s th?s us?s commands ?nd questions t? engage students’ natural ability t? notice. ?nd th? noticing ?s directed ?n su?h ? w?? ?s t? avoid potential points ?f confusion. ?h? strategies ?r? simple ?nd learner-friendly: Wh?t d? ??u count? Wh?t d? ??u call ?t? Supervised practice ?s undertaken ?mm?d??t?l?, providing th? teacher w?th ?lm?st instant assessment-and ?t involves ?v?r? single student, r?th?r th?n ? f?w vocal volunteers. Practice ?s safeguarded b? ?mm?d??t? peer feedback, wh??h demands ?mm?d??t? student self-correction. ? lesson su?h ?s th?s m?k?s sur? th?t ?v?r? student finds th??r w?? ?ut th? r?ght exit ?t th? Elephant ?nd Castle.