I’m writing problems to seed Opus, a free and open-submission problem bank for K-12 teachers. My friend and I started Opus in May, and we are hard-at-work inputting problems, trying to push it to the point where it is worthy of a community.
As I write more and more problems, I am running up against a definite hierarchy of problem-writing difficulty. It goes: procedural, conceptual, applied, from easiest to hardest.
As I write more and more problems, I am running up against a definite hierarchy of problem-writing difficulty. It goes: procedural, conceptual, applied, from easiest to hardest.
Here are my latest attempts (posted to Opus):
Procedural
For each of the following: identify the base, identify the power, write the expression in words, write the expression in expanded form, and simplify the expression.
For each of the following: identify the base, identify the power, write the expression in words, write the expression in expanded form, and simplify the expression.
To write a good procedural problem, I have to make sure that the problem demands the appropriate skill. In this case, we’re practicing whole number and fractional powers. I try to provide a good mix of introductory problems, and to avoid unnecessarily laborious computations. Minimal thought.
Conceptual
Expand and simplify each of the powers below.
Conceptual
Expand and simplify each of the powers below.
Which power is different from the rest? Explain why it is different.
To write a good conceptual problem, I have to present examples that reveal a pattern and ask good questions to help the student understand that pattern. In this case, I provide examples of powers with a negative sign in front of the base, prompt the student to use expanded form to simplify, call attention to the discrepancy, then ask the student to explain it. More thought.
Applied
To write a good conceptual problem, I have to present examples that reveal a pattern and ask good questions to help the student understand that pattern. In this case, I provide examples of powers with a negative sign in front of the base, prompt the student to use expanded form to simplify, call attention to the discrepancy, then ask the student to explain it. More thought.
Applied
What is the pattern? How could you express the number of people in each generation as a power? Why does using a power make sense? How many people will there be in the sixth generation?
To write a good applied problem, I have to come up with a relevant and, ideally, interesting situation, at just the right level of complexity. Coming up with a good application is difficult, because K-12 math tends to be simple, and interesting applications tend to be complex. My pool is further limited by the fact that most students have a relatively short list of life experiences, and a relatively short list of topics that they consider inherently interesting.*
*What they eventually consider interesting is, of course, subject to influence, but it takes a powerful force to overcome inertia.
In this case, we are looking at the growth of a family – not a particularly thrilling example, but probably more interesting than, say, a petri dish (people tend to like people). I doubt that students lie in bed at night wondering how many descendants they will have - in-laws included - if each generation observes a constant rate of reproduction, but perhaps it is enough to justify the effort. The small numbers and multiple questions are supposed to make it manageable for a novice. This problem took the most thought, and I’m not really satisfied.
I am guessing that the hierarchy of problem-writing difficulty affects all problem-writers, and that this hierarchy has led to a shortage of conceptual and, especially, applied problems in the classroom. I would also guess that this shortage is impacting student outcomes. If practice makes improvement, and students are getting more procedural practice than conceptual or applied practice, then it makes sense that our students are better at executing procedures than they are at explaining concepts or applying their understanding to novel situations.
To write a good applied problem, I have to come up with a relevant and, ideally, interesting situation, at just the right level of complexity. Coming up with a good application is difficult, because K-12 math tends to be simple, and interesting applications tend to be complex. My pool is further limited by the fact that most students have a relatively short list of life experiences, and a relatively short list of topics that they consider inherently interesting.*
*What they eventually consider interesting is, of course, subject to influence, but it takes a powerful force to overcome inertia.
In this case, we are looking at the growth of a family – not a particularly thrilling example, but probably more interesting than, say, a petri dish (people tend to like people). I doubt that students lie in bed at night wondering how many descendants they will have - in-laws included - if each generation observes a constant rate of reproduction, but perhaps it is enough to justify the effort. The small numbers and multiple questions are supposed to make it manageable for a novice. This problem took the most thought, and I’m not really satisfied.
I am guessing that the hierarchy of problem-writing difficulty affects all problem-writers, and that this hierarchy has led to a shortage of conceptual and, especially, applied problems in the classroom. I would also guess that this shortage is impacting student outcomes. If practice makes improvement, and students are getting more procedural practice than conceptual or applied practice, then it makes sense that our students are better at executing procedures than they are at explaining concepts or applying their understanding to novel situations.
But the shortage of good conceptual and applied math problems is not just impacting students’ K-12 math achievement. It is also impacting their excitement about math, their confidence in their mathematical abilities, and, by extension, their decisions about whether to pursue math in their collegiate and professional careers. The real power of conceptual and applied problems is that these problems help students understand why the procedures work and why they matter, and understanding “why” makes them want to learn more math.
As a student and as a tutor, I felt this shortage quite keenly. As a K-12 student, I benefited from really solid procedural and conceptual instruction, and - relative to the available assessments - I was achieving. But I was never convinced that math mattered – until, of course, many years later, when I graduated into the world. Then, last year, as a tutor, I found myself facing the same situation that my teachers faced– a surplus of educational ambition, but a shortage of applied problems.
Of course, it’s not really accurate to say that there is a shortage of conceptual and applied problems “in the world.” Right now, hundreds of thousands of math teachers are custom-crafting excellent applied problems for their students. In toto, there are tens of millions of masterfully crafted problems. But there is a local shortage in that no single teacher can access all of them.
We started Opus so that master problem-crafters can share their problems with one another and decide, as a community, which problems are best.
As a student and as a tutor, I felt this shortage quite keenly. As a K-12 student, I benefited from really solid procedural and conceptual instruction, and - relative to the available assessments - I was achieving. But I was never convinced that math mattered – until, of course, many years later, when I graduated into the world. Then, last year, as a tutor, I found myself facing the same situation that my teachers faced– a surplus of educational ambition, but a shortage of applied problems.
Of course, it’s not really accurate to say that there is a shortage of conceptual and applied problems “in the world.” Right now, hundreds of thousands of math teachers are custom-crafting excellent applied problems for their students. In toto, there are tens of millions of masterfully crafted problems. But there is a local shortage in that no single teacher can access all of them.
We started Opus so that master problem-crafters can share their problems with one another and decide, as a community, which problems are best.
