I’ll be honest, until just recently I always assumed when students made mistakes with negative numbers all I needed to do was to remind them that a negative times a negative equals a positive, or a negative times a positive is negative, and that once they could spit that information back to me if they did continue to make mistakes it was due to carelessness.
Then, I looked at an algebra test my middle school son had taken. The teacher assured me there was no patterns indicating a lack of understanding, just a lot of careless mistakes. However, when EVERY mistake involved negative numbers, as a statistician, I couldn’t accept this as being due to chance.
So I asked Dr. Livie Carducci, a “guest sage,” could this be mere chance or is my son, who can spit back the rules of negative numbers, be missing an important concept? Livie explained to me the history of negative numbers and that as concepts go, it’s a relatively new one providing an indication that the concept of negative numbers is really a complex topic (e.g., http://nrich.maths.org/5961). This got me thinking … as at least some of my students every semester struggle with locating a z-score on a normal distribution (or sampling distribution of the means). This is, as such, something I have them practice two or three times before I even continue with concept of z-scores. Students who cannot locate the z-score often are fine with the positive numbers, but it’s the negative numbers that cause them grief.
I typically just require them to visit me during office hours, as I cannot take up valuable class time teaching about negative numbers where I proceed to take them through a pictorial exercise that requires them to draw a host of things on a number line. Without realizing it, I have been teaching these particularly weak students about negative numbers. However, as so many students still have trouble in the classroom, I am now revisiting the idea of “valuable class time” with regard to this assignment. In truth, it only takes about 10 minutes for the weakest of my students to master this material.
What I do first? Instead of having an image of the normal distribution, I first have them draw a number line with the integers – 4 to +4. I often find even having them draw out the number line takes a great deal of their attention when they hit the negative side. Then I have them find positive numbers, first whole numbers, then 1/2 numbers, then 1/10 numbers. Then they have to find the negative version of the same value. Reteaching at this point is often required, so I find it critical to carefully watch where the students are placing their negative z-scores. FYI — each value is written on its own number line. After practicing this for a while, I start having them shade in the values above, below, or between to values on the number lines.
Only after they can do this on a number line, do I have them draw the entire normal distribution. Again, they first just find the location of the z-score. Then I have them shade in the area of interest (above the z-score, below the z-score, or between two z-scores).
If you are working with a population who really is lacking in the concepts of negative numbers, here is a link to three really great activities to help students to truly comprehend negative numbers http://www.resourceroom.net/math/integers.asp.
The rest of the activity I have completed during class time with all students. I have students draw 8 normal distributions, labeling mu, -4 sigma to + 4 sigma, then on the next line they write the corresponding z-scores. For one (or two) of the distributions, I start off giving them the values of mu and sigma, then they have to find the X values that correspond to the z-scores. Now, we calculate a z-score and they locate it on the normal distribution, they then must shade in the area of interest (e.g., above the z-score). Next, I introduce the standard normal table (found in the back of statistics books), and the students find the proportion for the area shaded. Driving home the point to not memorize the steps involved in this process, but to figure out what you want, and visualize it with a picture helps all students in understanding what these values mean. Yes, this activity takes up most of the 50 minute class period, and now if I add the previous activity where students practice on a number line to develop a richer understanding of negative numbers, it will most certainly take up an entire class period, but think about how central this concept is for students’ future success.
If a student doesn’t understand negative numbers, they will not understand z-scores, which will put them at a HUGE disadvantage in understanding standard error and the sampling distribution of the means, making it virtually impossible for the to understand the z-test as a hypothesis test statistic. Is it even possible to understand the t-test, F-test, or any form of hypothesis testing if you cannot understand standard error and the z-test? And it all starts with the negative number … without it, none of the rest of this is possible.
It’s funny … I am about to start my 25th year teaching, and I have always prided myself on being able to identify the underlying cause of a students’ error. Yet, had my son not so consistently make the same “careless” error on an important test, I probably would have started this year attributing a very common mistake I see my students make with negative numbers as careless, and not conceptual. Meanwhile, this conceptual gap could hold them back throughout the course. I wonder what other conceptual gap that I have attributed to carelessness?
As always, comments are welcomed.
*** For those of you who may not be able devote class time to work on this concept, I encourage you to provide your students with the free educational video by Khan Academy on negative numbers http://www.khanacademy.org/video/negative-numbers-introduction?playlist=Pre-algebra . ***