The Smallest Things Can Cause Huge Problems for Students

preemptive

Pre-Emptive Explanation

It is often the case, for the mathematically-insecure, that the slightest point of confusion can completely undermine their determination. Consider a beginning Algebra student that is learning how to evaluate functions like:

f( x )=3x x 2 +1 f( 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb WaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaaG4maiaadIha cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa qaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaaaaaa@43D9@

A confident student is likely to make the same error as the insecure student, but their reactions will be totally different. Below would be a typical incorrect answer that students will make:

f( 2 )=3( 2 ) 2 2 +1 f( 2 )=6+4+1 f( 2 )=11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb WaaeWaaeaacaaIYaaacaGLOaGaayzkaaGaeyypa0JaaG4mamaabmaa baGaaGOmaaGaayjkaiaawMcaaiabgkHiTiaaikdadaahaaWcbeqaai aaikdaaaGccqGHRaWkcaaIXaaabaGaamOzamaabmaabaGaaGOmaaGa ayjkaiaawMcaaiabg2da9iaaiAdacqGHRaWkcaaI0aGaey4kaSIaaG ymaaqaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaacqGH9aqp caaIXaGaaGymaaaaaa@4F4E@

The correct answer is 3, and the mistake is that -22 = -4, because it is really subtract two-squared. And when students make this mistake it provides a great chance to help them learn to read math, especially how exponents are written and what they mean.

Here’s what the students actually read:

f( x )=3x x 2 +1 f( 2 )=3( 2 )+ ( 2 ) 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb WaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaaG4maiaadIha cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa qaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaacqGH9aqpcaaI ZaWaaeWaaeaacaaIYaaacaGLOaGaayzkaaGaey4kaSYaaeWaaeaacq GHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa ey4kaSIaaGymaaaaaa@4E85@

A confident student will be receptive to this without much encouragement from you. However, the insecure student will completely shut down, having found validation of their worst fears about their future in mathematics.

There are times when leaving traps for students is a great way to expose a misconception, and in those cases, preemptively trying to prevent them from making the mistake would actually, in the long run, be counter-productive. Students would likely be mimicking what’s being taught, but would never uncover their misconception through correct answer getting. Mistakes are a huge part of learning and good math teaching is not about getting kids to avoid wrong answers, but instead to learn from them.

But there are times when explaining a common mistake, rooted in some prerequisite knowledge, is worth uncovering ahead of time. This -22 squared is one of those things, in my opinion, that is appropriately explained before the mistakes are made.

 

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