Make Learning Harder, not Easier

I was in a class this week where students were reviewing addition of fractions  They had recently learned multiplication of fractions and several students confused those solutions by adding across numerators and denominators (like you would for multiplication) instead of finding the common denominator and adding the numerators.

This common error underscores a learning principle outlined in Make it Stick, the Science of Successful Learning by Brown, Roediger III, and McDaniel.   They note that many learning environments are designed to produce the familiarity trap.

Beware of the familiarity trap: the feeling that you know something and no longer need to practice it.

The students who knew how to add fractions months ago are now tripped up by that very skill because they have now been taught to multiply fractions and they are confusing the two solutions.

Many teachers tell me that students need to stay focused on one thing until they master it.  The familiarity trap tells us this is not the case.  When we stay with one thing until mastery (a luxury not afforded often in the real world) we will gain a false sense of knowing when we receive a similar problem that requires a different solution.

The solution is interleaving, which is practicing two or more subjects interchangeably so that they not only see the underlying differences between the problems, but will be able to discriminate between the problems and solutions when they face them outside of the sanitized and carefully organized classroom setting.

Let's go back to that fraction conundrum.  Teachers and learners would be better off practicing addition and subtraction of fractions side by side (which is exactly what was happening in those classrooms I saw this week) so they can see the different solutions that are required and be able to choose the proper path.

This  learning structure slows down and seems to frustrate the learner --- and the teacher --- however, it turns out, that it has one overarching, hidden quality - it produces deeper learning.

#thestruggleisreal

Smart In Math

Photo Credit

Came across this excellent list on what it takes to be "Smart in Math" at Math Mama Writes.

'Smart in Math' Before students work in groups, it's important to help them understand that we typically have many misconceptions about what it means to be 'smart'. Typically, people think that someone who is 'smart in math' ...

• answers questions quickly

• always gets the right answer

• doesn't have to work at it

But, really, people who are good at math ...

• are persistent

• wonder about relationships between numbers, shapes, functions, ...

• check their answers for reasonableness

• make connections

• are willing to try things out, experiment, take risks

• are resilient

• want to know why

• contribute to group intelligence by asking good questions

• notice and learn from their mistakes

• try to extend and generalize their results

Students may also need to know how synapses (the connections between neurons that are created each time you learn something new) are strengthened by repeated use. A new connection isn't strong until it's been used:

• multiple times

• in multiple ways

• after a time away

Those first three bullets are not only way off the mark they are detrimental and harmful to the "smart math student". When those students who seem to always get the right answer come up against content that stops them in their tracks, they don't realize how much hard work is needed to keep the engine of their "smartness" running.  Of course, this reminds me of Carol Dweck's work aroud the fixed and growth mindset.  I also appreciate the sense of wonder and discovery that is embedded in the second list.  When I come across the intricacies of math in nature it just boggles my mind - just don't ask me to come up with a proof using all the proper postulates.  That task still gives me nightmares.