Teachers in Saskatchewan, Canada have had a lot to deal with lately in the classroom. The ongoing political battle has effected hours of direct instruction in a very real way. I quickly noticed my classes becoming disjointed with large amounts of time between each encounter with the mathematics. Needless to say, I entered today’s lesson in Math 9 with a little apprehension. A Friday morning after 2 days of job action and a long weekend didn’t sound like the most nurturing of environments. I decided that the time was ripe to attempt a lesson that has been in my mind for a couple of months; the following account is the story of the task, presentation, student reaction, and important learnings.
The students were introduced to the concepts of Mean, Median, and Mode earlier this week. It had been 3 days, so I quickly refreshed their memories with a standard problem. I gave them a list of data, and had them compute (in pairs) the three measures of central tendency. After some painful re-hashing and peer tutoring, the class was then alerted that we were going to take a major shift. With a quick swipe of the eraser, I eliminated the data set and left only 4 facts on the board:
n = 19
mean = 4.47
med = 4
mode = 2
I asked them if they could re-construct the data set using these facts. It should be noted that I unfairly took advantage of my students’ laziness. When asked if they should write down the set of numbers to solve the original problem, I told them not to bother. Do I feel guilty? Meh.
I expected the class to complain about this crazy task. Some began to rack their working memories for the last remaining traces of the numbers–but with little luck. After these efforts fizzled out (2 mins max), I posed the problem I actually intended them to solve:
Find the data set with the following attributes:
n = 1
mean = 3
med = 3
mode = 3
Soon the fear of wrong answers wore off and partners were conversing with other groups. After about 3 minutes, we decided that the only data set that fit all four was :
{3}
I asked very quickly for an explanation and verification of the facts, and then erased the “n=1” from the board. I turned toward the class (with a dramatic pause) and repeated the exact same question:
Find the data set with the following attributes:
mean = 3
med = 3
mode = 3
Group work began immediately. An electric hush fell over the room as they worked for 10 solid minutes discovering numerous data sets that fit the description. As I circulated, I overheard phrases like, “what about the median” and “won’t that change the mode”. At this point, I began to ask questions that required students to check their process. “How do you know you need to add a large number?” and “Why did you decide to start your set with 1?”. In due course, we generated a class list of sets, including one that could go on “forever and ever”.
{1,2,3,3,3,4,5,5,6} *
{1,2,3,3,3,6}
{3,3}
{3,3,3,3,3,3,3,3…}
{1,3,3,5}
{2,3,3,4}
{1,1,1,3,3,3,3,4,5,6}
The list itself contains predicted responses and revealed a lot about how each group thinks about numbers. Certain groups have been trained to think that lists start with one. This seems natural; we start school in grade 1; we begin counting with 1–why not begin listing with 1? Other students took the ingenious route of understanding what a list of 3’s does. When we had the list, I asked for observations about it. I often do this with my class to begin the processes of pattern recognition, problem solving, and problem posing. The list of observations was excellent!
“Number 1 is wrong”
“None of the patterns include a zero”
“All use only whole numbers”
“There are no negatives”
“They are all in ascending order”
Although I would love to detail the conversation along each of these points, I would rather use them to illustrate the growing nature of such a problem. These student inquiries can be used to explore the concept of Mean, Median, and Mode in a much deeper way. What began as mathematical play, quickly turned into serious mathematics. I would have loved to set up think-tanks to explore how negatives can be used in sets of numbers. Maybe a particular ambitious student would take on the trouble of finding an integer mean with fractions as data points. (I imagine this would lead quickly to pairs of fractions that sum to 1, but I can never be sure where student thought leads). I chose to go into an interesting conversation about zero. I touched quickly on the history of zero, what adding zero to a data set would do to each of the three measures of central tendency, and constructing sets with a mean of 0. The last topic ran naturally into that of negative numbers.
I assigned the class to “fix” the first entry on our list, and describe, in words, their thought process. The topic of writing in mathematics is one for another day.
My point is, changing the focus of a very routine mathematical exercise changed the way the students saw the topic. It began to grow right before them. It is quite artificial for me to separate the extensions; it has already been shown that the topics intertwine. Posing questions in a playful atmosphere unlocks student drive. For an approach often coined, “Fuzzy Math”, it sure led me and my students into very serious (and curriculum supported) mathematics.
NatBanting