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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:

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

KIRK: He won’t break off now. 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: All stop.

SULU: All stop, sir.

KIRK: Z-minus ten thousand meters. Stand by photon torpedoes.

Like Khan, our students are intelligent but have limited experience. I wonder, though, how often we reinforce misunderstandings instead of correcting them?

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:

And if you turn this diamond, it will become a square.
The shape was always a square; the direction it faces doesn’t make any difference.

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
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.
```