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Student: "I see the fractions. Where are the pizzas? My teacher says to draw a picture if I don't understand a word problem, so I should draw a pizza cut into 600 slices? How do I slice the pizza into an hour?"

Student: "I see the fractions. Where are the pizzas? My teacher says to draw a picture if I don't understand a word problem, so I should draw a pizza cut into 600 slices? How do I slice the pizza into plane travel?"

Then the students must think about meaning. They might need a caring to teacher to gradually build them up to this kind of challenge, but there's really no comparison as to which set of exercises will make students think more meaningfully. In addition to being pedagogically difficult, though, this is also motivationally hard as students and parents - and most teachers - tend to care most about what's on the test, and these kinds of exercises are rarely on tests.

Then the students must think about meaning. They might need a caring teacher to gradually build them up to this kind of challenge, but there's really no comparison as to which set of exercises will make students think more meaningfully. In addition to being pedagogically difficult, though, this is also motivationally hard as students and parents - and most teachers - tend to care most about what's on the test, and these kinds of exercises are rarely on tests.

Sometimes teachers say that when given $\frac{26}{6}$, you should type $26\div6$ into the calculator. But I challenge you to find any textbook in North America that explains why this might be a reasonable thing to do and why the fraction bar and the $\div$ are always interchangeable while the fraction bar and, say, $\times$ are not. Even harder: find any textbook or test or teacher or tutor that asks students to compare and contrast $\frac{26}{6}$ vs $\frac{26\div6}{6\div6}$ vs $\frac{26\div2}{6\div2}$ vs $\frac{26}{6}\div6$ vs $\frac{\frac{26}{6}}{6}$ vs $\frac{6}{\frac{26}{6}}$ vs $4.33333...$ vs $4\frac{1}{3}$ etc.

One reason that students are not tested this way: they tend to bomb horrifically, immediately. If a student who types $26\div6$ into his calculator when he sees $\frac{26}{6}$ - only because he was told to yesterday - then a teacher can feel that student understanding is confirmed. The student gets a good report card, moves on to the next topic or grade-level, can apply to a prestigious educational program, etc.

Sometimes teachers say that when given $\frac{26}{6}$, you should type $26\div6$ into the calculator. But I challenge you to find any textbook in North America that explains why this might be a reasonable thing to do and why the fraction bar and the $\div$ are always interchangeable while the fraction bar and, say, $\times$ are not. Even harder: find any textbook or test or teacher or tutor that asks students to compare and contrast $\frac{26}{6}$ vs $\frac{26\div6}{6\div6}$ vs $26\div6$ vs $\frac{26\div2}{6\div2}$ vs $\frac{26}{6}\div6$ vs $\frac{\frac{26}{6}}{6}$ vs $\frac{6}{\frac{26}{6}}$ vs $4.33333...$ vs $4\frac{1}{3}$ etc.

One reason that students are not taught or tested this way: they tend to bomb horrifically, immediately. If a student who types $26\div6$ into his calculator when he sees $\frac{26}{6}$ - only because he was told to yesterday - then a teacher can feel that student understanding is confirmed. The student gets a good report card, moves on to the next topic or grade-level, can apply to a prestigious educational program, etc.