Z-minus 10,000 Meters, Mr. Spock

Stu­dents come to our class­rooms with many assump­tions and mis­con­cep­tions, and it is the teacher’s job to antic­i­pate them, rec­og­nize them, and cor­rect them. Here are a few that I have seen or heard about:

• When you add or sub­tract, always line up the num­bers on the right
• When you mul­ti­ply, the answer is always bigger
• Rock­ets work because the exhaust pushes against the Earth
• Mag­nets stick to any­thing made of metal
• Christo­pher Colum­bus was try­ing to prove the world was round
• The Amer­i­can Rev­o­lu­tion was fought over high taxes

Many stu­dent mis­un­der­stand­ings are sim­ply a lack of expe­ri­ence. There is a scene in the 1982 movie, Star Trek II: The Wrath of Khan, where Khan, the vil­lain, is try­ing to hunt down our heroes. Kirk flies the Enter­prise into a neb­ula in order to obscure the ship from Khan’s scan­ners. After a few min­utes, Spock makes an obser­va­tion about Khan:

Often in the name of mak­ing our lessons acces­si­ble or under­stand­able we sim­plify con­cepts and use stereo­typ­i­cal exam­ples. Con­sider geom­e­try, for instance. When we draw shapes, they always look essen­tially the same:

Stan­dard pat­tern block shapes

Tri­an­gles are always equi­lat­eral and point up. Rec­tan­gles are always wider than they are long and are par­al­lel to the ground. At the extreme, we even refer to shapes by dif­fer­ent names depend­ing on their ori­en­ta­tion. I actu­ally heard this state­ment dur­ing a math les­son once:

Try these sug­ges­tions to avoid rein­forc­ing the mis­con­cep­tions of your students:

• Know your own mis­con­cep­tions. Begin with the assump­tion that you may have picked up your own wrong ideas in school or from pop­u­lar media. Review the mate­r­ial ahead of time and look for places where you your­self didn’t quite get it right. (Inci­den­tally, if you read any of the items in my orig­i­nal list and thought, “What’s wrong with that?” you may want to do a lit­tle research and find the sub­tle prob­lems with them.)
• Plan ahead for stu­dent mis­un­der­stand­ing. Learn the places where your stu­dents are likely to get con­fused or have pre­con­ceived ideas about a topic. Many mis­con­cep­tions are com­mon and repeated, so it’s easy to pre­pare for them.
• Use a wide vari­ety of exam­ples. Delib­er­ately choose exam­ples that stretch stu­dents’ think­ing. Use coun­terex­am­ples to help them bet­ter define con­cepts in their minds.
• Let stu­dents con­struct their own def­i­n­i­tions. By let­ting stu­dents build def­i­n­i­tions and expla­na­tions around exam­ples you use, you are encour­ag­ing them to ana­lyze the exam­ples and under­stand the con­cept deeply instead of just mem­o­riz­ing a sen­tence some­one else has pro­vided them. After they attempt to build a student-​​friendly expla­na­tion, you can come in and pro­vide more pre­cise vocab­u­lary where nec­es­sary to give them a more con­cise way to express it.
• Expect stu­dents to explain and jus­tify their rea­son­ing. Some­times stu­dents are able to apply a rote algo­rithm accu­rately and get a cor­rect answer to a prob­lem with­out really under­stand­ing what they are doing. Ask­ing them to explain, even when their process seems obvi­ous to you, will give you insight into whether their think­ing is accu­rate or has flaws that need to be corrected.

Soon after Kirk changed his tac­tics to account for Khan’s mis­con­cep­tion, he was able to sneak up behind Khan’s ship, ulti­mately win­ning the bat­tle. While it is unlikely that the mis­con­cep­tions our stu­dents carry through school will result in such life or death cir­cum­stances, we can make our own jobs eas­ier by pre­vent­ing them in the first place.

SPOCK
                             Sporadic energy readings port side,
                             aft. Could be an impulse turn.

                                           KIRK
                             He won't break off now. If he
                             followed me this far he'll be back.
                             But from where...?

                                           SPOCK
                             He's intelligent, but not experienced.
                             His pattern indicates two dimensional
                             thinking...

                   Kirk looks at him, smiles.

                                           KIRK
                             Mr. Saavik, all stop.

                                           SAAVIK
                             All stop, sir.

                                           KIRK
                             Descend ten thousand meters. Stand
                             by photon torpedoes.