Reddit reviews Classroom Instruction That Works: Research-Based Strategies for Increasing Student Achievement

As to the first part:

It's not that you ask me to back up my claim that rustles my jimmies. I'm taken to task for claims all the time. I majored in Math and will be starting up a masters program soon. I'm used to having my statements challenged.

And while it's true you can't read tone that just implies you need to be even more careful with how you word things. I asked a few people about that message, and from context the three others I asked agreed that it came off as derision. And I'm sure you realize there is a difference between legitimately asking a question, and asking a question derisively.

Probably best we let the misunderstanding go though. So I'll just move on to the second part:


Since you were legitimately asking let me explain a few things about myself. I teach secondary mathematics, in particular I teach several blocks of algebra II. It's honestly pretty heavenly compared to some places where a teacher might have 2-4 preps.

The problem for most of my students is one of transfer. They can understand routines, but not see the connections to other content they have learned. Hell, they may even be able to parrot the quadratic formula, but not actually understand any of it. They can perform the calculations, but don't actually understand anything to apply the reasoning of the problem elsewhere.

I can show you a specific example of a problem I've given recently where I noticed this, but it's up to you. Formatting would make it... kinda screwy, but I can draw it up somewhere and link a video or post a picture on imgur. Just let me know.


So how do we fix problems of transfer? Research by Marzano (who literally wrote the book on instructional techniques,), by Perkins, and by Newmann and Associates (though Newmann's work is getting on in years,) suggests that problems with transfer (and motivation, and general problem solving ability, etc etc) can be solved through authentic, inquiry based coursework with an emphasis on understanding of content over rote memorization.

I don't disagree that rote memorization is necessary. You simply must do quite a bit of it. But with a rote understanding you only get so far. Especially when problems become more abstract, and when which technique to apply isn't spelled out for you.

Its like being able to read Brave New World, but not actually being able to analyze it. In the worst cases it's like being able to analyze Brave New World, but not being able to apply those same analytic techniques when reading 1984.

There's mountains of research out there to support a more constructivist pedagogy.

This outlines a great deal of what I believe, and has links to papers which support the general philosophy. Most of the site is concerned with science education however. Still a pretty good read.


With that we'll head back to the original question.

> What can a person learn about the value and properties of numbers that is not native to rote mathematics but is enlightened in new math?

All sorts of things. But let's be careful and define 'new math.' When I talk about it I mean an inquiry based problem, which relies on a method that isn't mechanically the same as the rote method.

So in the problem at the top of the thread (which is a terrible problem mind you,) the student is asked to play 'spot-the-error' with another students work. There are several issues that make this a shitty problem, but not one without potential. Let's see what those issues are, and why they impede the understanding we want to generate.

What they want the student to see is that you can break subtraction down into parts, essentially taking a huge problem and cutting it into managable pieces.

Instead of just setting up the regular old method, we instead first subtract three 100's, then (and this is what's wrong in the question if memory serves,) subtract a ten, and then subtract 7 1's. Students are basically looking at the decimal expansion.

Then they connect it back to material they learned. They know about number lines. They are forced to connect the routine to their prior knowledge. This is how you build up the ability to transfer content knowledge.

So now students see "Ah, even though I've got a more efficient method of doing this, this lets me see things I've learned with before."

Moreover this re-enforces the idea of subtraction as simply moving backwards along a number line. This makes for a significantly more solid foundation for being introduced to negative numbers.

And once you've got simple counting down you can look ahead to multiplication. Maybe we can find a way to explain that using a number line. Drawing a connection to what a child knows, and showing how our new idea is just a natural extension.

Those skills, visualizing and breaking down problems, analyzing what the components mean, and using your prior knowledge to extend what you already know in order to find a solution, those are the skills you want. Again, I'm not suggesting we supplant routine and mechanical understanding with conceptual understanding. That would be just as problematic (read: dumb.) Instead I'm suggesting that conceptual problems (the new math,) enhance students abilities to tackle questions rooted in the higher order parts of Blooms. And that isn't some random claim pulled out of a hat, it's a claim backed up by a significant portion of the research in modern pedagogical theory.