6 replies on “Exponents – Positive Integer Powers”

I strongly suspect when they see this again the student will realize his/her mistake immediately. If not try working through the (correct) expansion term by term. If they still don’t get it, which I doubt, there’s a fundamental confusion of addition and multiplication. Make a point of saying that 27 = 3*9, redirecting their incorrect thought into a correct statement.

But really, as much as I try not to write these off to “autopilot”, I can’t think of any other explanation.

While I can understand your supposition that there may possibly be a lack of attention to precision, I see this type of mistake regularly with my lower-level 7th grade students. This is my student, and I know factually that this is common for him/her (there’s some FERPA action for you).

Taking Michael’s “deeper” approach, I have formed a hypothesis about these types of mistakes. I am of the firm belief that this is a result of a student’s lack of number sense. Two of the characteristics of strong number sense are number magnitude and the effect of operations at a mental computation level. The students I have witnessed in my career that struggle with math also struggle with “visualizing” quantities. They have been subjected to trying to memorize procedures without regard to developing the ability to first visualize what a reasonable answer would look like. Hence, this type of mistake.

I also have the suspicion that a lot of the math curriculum currently on the market and a lot of the instructional practices in math classrooms are not doing a good enough job to develop and nurture number sense in students. This is, of course, my opinion and I don’t have a lot of evidence/research to back up that claim, but I definitely think it is something worth looking into. For what it’s worth, I know I will be sure to examine this closely while writing curriculum and evaluating textbook series in the future.

I agree that number sense is an issue, but I think there’s another prong to this. This student has *some* understanding of exponents. He/she knew that an exponent can be written as a repeated multiplication expression. Hence all the multiplication symbols between all the 3s. However, despite what he/she wrote, it appears that the student was really thinking about *adding* nine 3s.

Students have an uncanny ability to ignore symbols and perform the operations they *think* they should be doing. I’ve seen it on this site with number sentences, and I’ve also seen it in my own classroom with word problems as well. Students will completely ignore what a question is asking, choosing instead to answer the question they *think* they were asked.

Maybe it has to do with engagement? If a student is not fully engaged in the problem, then they are working on automatic pilot, which leads them to use procedures they’re familiar with without realizing they don’t apply in a given situation. Often, if you can get a student to slow down and be more present in a problem, they can avoid mistakes they would be prone to make otherwise. We need ways to differentiate true misunderstandings from these sorts of automatic pilot errors.

I’ve been thinking about this since I wrote my reply yesterday. I wonder if it’s an issue of dissonance. I know that cognitive dissonance is a state required for a person to be able to learn. Maybe the problem is that most assignments do not put students into a state of dissonance. As a result, they aren’t even aware that they are making such huge mistakes. When you don’t have dissonance, you assimilate what you see into your current worldview. If the student’s current worldview is primarily filled with thoughts of elementary school math, then no wonder they apply those skills without even realizing the mistakes that are being made.

As an adult looking at this site, we are put into dissonance because we can easily see that the student did something wrong, and immediately we start trying to figure out why. The more baffling the mistake, the more dissonance it might cause us. How can classroom assignments create that same level of engagement in students?

The student posted two equivalent solutions (3^9 and 3*3*3…*3), but neither are equivalent to 27.

I think the student can see that 3*9=27, but factoring farther and correctly writing the solution seems to be where the problem is. Showing an intermediate example might help to illustrate the technique.

[…] really, as much as I try not to write these off to “autopilot”, I can’t think of any other […]

## 6 replies on “Exponents – Positive Integer Powers”

I strongly suspect when they see this again the student will realize his/her mistake immediately. If not try working through the (correct) expansion term by term. If they still don’t get it, which I doubt, there’s a fundamental confusion of addition and multiplication. Make a point of saying that 27 = 3*9, redirecting their incorrect thought into a correct statement.

But really, as much as I try not to write these off to “autopilot”, I can’t think of any other explanation.

While I can understand your supposition that there may possibly be a lack of attention to precision, I see this type of mistake regularly with my lower-level 7th grade students. This is my student, and I know factually that this is common for him/her (there’s some FERPA action for you).

Taking Michael’s “deeper” approach, I have formed a hypothesis about these types of mistakes. I am of the firm belief that this is a result of a student’s lack of number sense. Two of the characteristics of strong number sense are number magnitude and the effect of operations at a mental computation level. The students I have witnessed in my career that struggle with math also struggle with “visualizing” quantities. They have been subjected to trying to memorize procedures without regard to developing the ability to first visualize what a reasonable answer would look like. Hence, this type of mistake.

I also have the suspicion that a lot of the math curriculum currently on the market and a lot of the instructional practices in math classrooms are not doing a good enough job to develop and nurture number sense in students. This is, of course, my opinion and I don’t have a lot of evidence/research to back up that claim, but I definitely think it is something worth looking into. For what it’s worth, I know I will be sure to examine this closely while writing curriculum and evaluating textbook series in the future.

I agree that number sense is an issue, but I think there’s another prong to this. This student has *some* understanding of exponents. He/she knew that an exponent can be written as a repeated multiplication expression. Hence all the multiplication symbols between all the 3s. However, despite what he/she wrote, it appears that the student was really thinking about *adding* nine 3s.

Students have an uncanny ability to ignore symbols and perform the operations they *think* they should be doing. I’ve seen it on this site with number sentences, and I’ve also seen it in my own classroom with word problems as well. Students will completely ignore what a question is asking, choosing instead to answer the question they *think* they were asked.

Maybe it has to do with engagement? If a student is not fully engaged in the problem, then they are working on automatic pilot, which leads them to use procedures they’re familiar with without realizing they don’t apply in a given situation. Often, if you can get a student to slow down and be more present in a problem, they can avoid mistakes they would be prone to make otherwise. We need ways to differentiate true misunderstandings from these sorts of automatic pilot errors.

I’ve been thinking about this since I wrote my reply yesterday. I wonder if it’s an issue of dissonance. I know that cognitive dissonance is a state required for a person to be able to learn. Maybe the problem is that most assignments do not put students into a state of dissonance. As a result, they aren’t even aware that they are making such huge mistakes. When you don’t have dissonance, you assimilate what you see into your current worldview. If the student’s current worldview is primarily filled with thoughts of elementary school math, then no wonder they apply those skills without even realizing the mistakes that are being made.

As an adult looking at this site, we are put into dissonance because we can easily see that the student did something wrong, and immediately we start trying to figure out why. The more baffling the mistake, the more dissonance it might cause us. How can classroom assignments create that same level of engagement in students?

The student posted two equivalent solutions (3^9 and 3*3*3…*3), but neither are equivalent to 27.

I think the student can see that 3*9=27, but factoring farther and correctly writing the solution seems to be where the problem is. Showing an intermediate example might help to illustrate the technique.

[…] really, as much as I try not to write these off to “autopilot”, I can’t think of any other […]