Tuesday, December 23, 2008

The Number Warrior and the Mysterious Minds of Students!

Jason over at Number Warrior, an excellent blog for math teachers, has a short but fascinating post on trying to analyze why students make careless errors when it comes to negative fractional exponents.

I hope he doesn't mind if I repeat my comment over here - I think it raises some important issues for all of us who are trying to help students overcome these apparently 'careless' errors. I also recommend you visit his blog - fascinating stuff...

Jason's post:

So why would a student incorrectly evaluate 16^{-\frac{1}{2}} to be -4 but manage to correctly get on the very next problem that 5^{\frac{1}{4}}\cdot5^{-\frac{9}{4}} is \frac{1}{25}?

I believe this is a case that the knowledge of negative exponents was stored somewhere back there, but because the first problem looked “easy” my students just went for the impulse answer. (Nearly everyone — even students who scored very high overall — got it wrong.) I wonder how I can get students to reach back there more often, because neither gentle admonishments nor fierce reminders seem to work.

My response:

We can speculate about why students make errors, but I’ve learned there are usually several reasons. I found it helpful to simply ask them to explain how they got that result (if they can!).

Some thoughts:
Your 2nd example procedurally involved fractional exponents, but ended up raising the base to a negative integer, not a negative fraction. This is a minor distinction, one extra step, but you never know. Also, I found it helpful to encourage them to write the extra step or two rather than do it mentally. Thus, 16^(-1/2) = 1/(16^(1/2)) might help. in other words, when they have to cope with both the negative and the fraction, make them always do the negative first. Some individuals are simply not detail-oriented and have trouble with precise procedures. I believe left-brained people have fewer of these issues because they are wired to do step-by-step procedures!

Finally, although none will admit to this, some youngsters know how to study for a math test and some simply don’t practice sufficiently. The “I think I know the material” students who didn’t review enough usually get burned on these procedural problems that have that one extra step. Ok, I’m probably over-analyzing all of this - it’s just a darn common error! Happy Holidays!
Dave Marain

My gut feeling is that these kinds of issues which math teachers have to confront daily, beg for considerable dialog. I know I benefited from asking more experienced teachers for advice when so many of my students struggled with certain types of questions. Asking students themselves to analyze their own errors is rarely a waste of time in my opinion. We always want to encourage self-reflection and it's usually good practice to have students correct their errors after receiving their tests back. And, of course, this kind of dialog also serves as a window into their 'mysterious' minds!

I hope this generates some further discussion about 'careless errors' and what we can do to help students cope!

Happy Holidays Everyone!


Robin Schwartz said...

Negative exponents, fractional exponents, the zero exponent, exponent rules, logs – these should turn on the ALERT indicator – be careful -- danger ahead!! Even armed with a calculator, students will get tricked by these types of problems…Jason, did your students use calculators for this exam?

Studying multiple choice questions can help refine thinking skills (and analyze potential errors) through the strength of the comparative.
An example from “Using ‘Good Wrong Answers’ To Achieve Math Confidence and Success” is:

What’s the value of 3^-2?

A) -2/3
B) -9
C) 1/9
D) -6

If they are comfortable with the zero power, they can write 3^-2 = (3^0)/(3^2). Since exponents (and logs) are not logical, I work with learners on extra alertness with this type of content. Unlike some content where they know they are guessing, here they feel sure they have done it right.

Another exercise is shown below:
16^1 =
16^.5 =
16^0 =
16^-.5 =
16^-1 =

By increasing awareness of their thinking process, students can build confidence and enjoyment of Math and even improve their scores.

Robin A. Schwartz
Founder, www.mathconfidence.com

jd2718 said...

We often think we are doing kids a favor by giving examples with natural numbers... In fact, I heavily favor simple variable expressions.

Can a kid rewrite a^(-1/2) without negative or fractional exponents?

I keep flipping back and forth, but that's the side where I keep the emphasis...

I am assuming that absolutely NOTHING in this topic is easy, but that sticking numbers in tends to lull kids, and relax them, and that, as the previous commenter points out, is not what we need to happen.


Dave Marain said...

Some excellent points, Robin and Jonathan...

Robin, I checked out your website and your blog -- interesting. Good luck with your endeavors. The url for your blog needs to be revised slightly.
You may also want to look here to see a development of exponents I published last year. Also, your "Good Wrong Answers" makes a great deal of sense. Students definitely learn from a discussion and analysis of carefully constructed distractors, perhaps even better than from their own errors! They need to be able to explain the error in each!

Jonathan, I definitely see the advantage of working with variables as you suggest. It shifts the focus where it belongs -- the meaning of the exponent itself, not just a computation. Pursuing this further, it gives students the opportunity to practice their rules of exponents which should apply equally well to rationals.

For example: a^(-1/3) can be expressed as (a^(1/3))^(-1) which separates the negative exponent from the fraction. Then we can interchange the order of the operations and compare the results.

This all comes back to the issue of pedagogy and an understanding of differences in cognitive processes of our students. Negative fractional exponents really involves THREE layers of exponents (the negative and the meaning of the numerator vs. the denominator) and many students simply cannot process all 3 simultaneously. Whether we use concrete or variable bases we cannot excape this need to break the exponents down.

Then there is the issue of helping students recall these details in a performance situation. THIS IS SEPARATE FROM UNDERSTANDING, yet I believe it is part of our role as educators! Any mnemonic or device that helps is acceptable, provided we have established meaning first for such 'tricks.'

Thus, in the exponent p/r, if we tell them to think of r as the "root" and remember that the root of a tree is "at the bottom", this may help them remember it. It does not have to do with understanding but that isn't the point of a mnemonic! Understanding is separate.

Jonathan, while I agree that working with variables is essential, I recommend doing both. Ultimately, students need to check the theory with actual calculation. I have found it very helpful to have students evaluate expressions like 16^(3/4) in several ways using the calculator as needed. The calculator should give the correct result of course IF the student can properly enter the expression!

Here's what I mean:
(1) "Jen, I want you to evaluate this using the algebraic form ('law of exponents'):
b^(m/n) = (b^m)^(1/n)."
I want you to compute this in two stages. First 16^3, give us the result pls on your calculator. Ok, now raise that result to the one-fourth. What do you get? Tell us again, Jen, what does it mean to raise a number to the one-fourth? Good..."
(2) "Alright, Mark, now I want you to evaluate the same expression using the algebraic form:
b^(m/n) = (b^(1/n))^m in two steps. First evaluate 16^(1/4) without your calculator! What does the one-fourth mean, Mark? Right! Ok, now raise that result to the what?? Good!"
"How do the final results compare? What rule of exponents justifies that we can interchange the root and the power? Which method is preferable if a calculator is not available? If this were a multiple choice problem, what would it look like? Could you write one?"

Now, I avoided the negative in the above scenario for demonstration purposes, but we can't expect students to process THREE layers until they can handle two!

In the end, how many reading this still believe that the best instruction, if such exists, does not replace the need for students to put pencil to paper and practice many of these over time until it becomes automatic and mechanical. Hey, folks, we can assign it, but it doesn't mean they will do it! Problem is we are NOT assigning as many of these as in the past because we have been told to deemphasize drill. Uh oh, maybe I shouldn't go there on a holiday!

Sorry for the long-winded response! HAPPY HOLIDAYS!

Jason Dyer said...

Wow, that was a lot more feedback than I expected on my offhand post. Thanks!

Robin, this was a no-calculator exam. (I also have part-calculator and part-no-calculator exams, all-calculator exams, and one where I give them data they have to model using their graphing calculator.)

Keep in mind my main question was why the multi-step problem was solved by more students than the single-step problem (where the last step was the same thing). There *were* students able to process all the layers, just they didn't do so when the problem was "easier".

I see this sort of thing occur in other contexts, say where you solve for x with

2x = x - 3


2x + 5 = 7x - 3

and more students will get the latter correct than the former.

Jonathan, I do use a lot of variables, but in that case I wanted to check if the students remembered ^(1/2) was a square root. I guess I could've instructed them to turn x^(1/2) into root form but I'm not sure how to phrase that without giving away what to do.

Dave Marain said...

You have no idea what you started!
I will probably go far afield from your original point, but there are themes here I feel compelled to address.

Seriously, it's educators like you who reflect thoughtfully on their practice who will make the greatest impact on the learning of their students. You are asking, IMO, the central question of teaching:
What do I have to do to maximize the learning of my the greatest number of my students?

Certainly, when a large number of students make a similar error, it gives us pause and makes us want to ask WHY. The question is "HOW" to respond to such phenomena."

Talk about giving them what appears to be a 'simpler' equation to solve, try to have them show you the proper procedure for solving
3x = 2x. Here, of course, we'd like them to think conceptually: Three times what number equals two times what number! This 'apparently' simple equation is not simple at all -- it gets at the heart of the meanings of variables and equations.

I think the issue of error analysis is fascinating and extremely complex. What part is related to the WAY we teach it and what part is HOW they process it. Actually that's an oversimplification, since there are many other variables that go into student performance on a test.

To the original problem, I do believe that it is critical to use the Rule of Four (multiple representations) to help embed the conceptual ideas of negative fractional exponents. It comes back to teaching both conceptually and procedurally. Trying to do both at the same time is very difficult and may not always be the best approach.

Here's what I mean. First, you want students to understand that the function f(x) = 16^x ONLY has positive values, i.e., the range is y > 0. This can be done initially using a table approach to exponents as Robin suggested and as I developed it in my post on motivating exponents. Look here .
Juxtaposing both the numerical xy-table and graph is a powerful construct that you can return to often. When a student gives a negative answer to a negative exponent problem, you can remind them that, for b>0, b^x will NEVER produce negative values - "Remember that graph!"

Ok, that was pedagogy. Teaching procedurally means that we show them the steps to follow, motivating them as much as possible as we go along, i.e., asking students which 'Laws of Exponents" are applied.

I really believe this simple question you raised is at the heart of our practice and why i started this blog: To have meaningful substantive dialog on effective methods of teaching and a better understanding of the many ways in which students make sense of concepts and procedures.

WHY are some errors in math classes so common? I devoted an entire article to this and yet we only scratched the surface. I still believe that there is no perfect lesson! We PRACTICE our profession, refining and improving our techniques daily. We hopefully improve over time, asking better questions, using better techniques. BUT, to state the obvious, no lesson will reach all students all the time. In the end, students must practice many procedural-type problems, make errors, learn from their mistakes and, most of the time, those who perform the best will be those who made the most effort to learn (yes, there will aways be the smart-lazy student who seems to process everything while unconscious!).

Ok, so how does all of this relate to your "simple" original question. We use a variety of methods to help students learn conceptually and we use a variety of methods to help students PERFORM better on tests and quizzes. Robin's idea of using multiple choice format is an excellent one. This drives home the reasoning behind the so-called 'common' mistakes and gives them the opportunity to analyze the reasons behind each. Much better to do this BEFORE the next test, so why not give them a review sheet with a few of these ahead of time. I've done this and generally the results are better, with fewer careless errors. I usually made this 'practice test' more challenging to raise their level of concern. Besides, the greatest learning often takes place the day or two before the exam! They were told to redo it twice as well as redo homework problems and previous quizzes. Note that I said "REDO" not review. "VIEWING" math has nothing to do with performing well! One must "REDO" problems to give oneself the best chance of good performance under timed pressure.

Ok, I've gone light years beyond your original post, but everything is connected to everything else when we start to discuss our practice, yes? Please keep posting your wonderful ideas and experiences. You're giving me lots of material!!
Happy New Year, Jason!

Anonymous said...

Maybe students make careless errors because they couldn´t care less about the problems. I mean this seriously, and I don´t have a solution.
Thanks for the blog, I read it quite regularly
- Ingo

Dave Marain said...

Absolutely, Ingo, some just don't care enough. That's not being cynical, it's calling it like it is.

However, there are others who do care and get confused. But on the whole I believe the most important factor is the quality and quantity of their preparation. Too many students try to to cram at the 11th hour and it simply doesn't work for math. After all the discussion of pedagogy and what WE can do to hewlp students learn, well, you can lead a horse...

Thanks for the support and Happy New Year!