classroom structure investigation pattern sequences and series tasks

In the Footsteps of Gauss

I like to introduce each topic with a task or activity. These do not necessarily have to be long, but should activate mathematical thinking. The idea has slowly evolved for me throughout my short career. They are the amalgamation of the ideas of a “motivational set” and discovery learning. I felt that both components are positive things to include in a math class, but both had severe implementation problems.

The motivational set is far too passive. In my college, a picture, story, or conversation could serve as a motivational set. It was essentially a transition tool that was completely void of any mathematics. Every lesson begins with the same routine whether it be a national anthem, attendance, or a short time of homework recap, but each learning experience needs to begin with an active brain. I found that the purpose of the motivational set was important, but needed a stronger method to get brains engaged in the day’s learning. 

Discovery learning has become the hottest issue in mathematics education today. (Barring, possibly, the Khan Academy’s claims of educational reform). I was introduced to several activities where the mathematical process is reversed and students play with the math in order to create meaning through manipulation. I noticed a gap widening between two types of high school teachers: there were those that wanted to implement inquiry, but didn’t have the class time, and those who didn’t want to implement, but gave time as their limiting factor. Although one of these groups consists of liars, the problem is the same. There needed to be a time efficient way to implement discovery learning into the classroom. Prompts and tasks keep a very strong curricular tie, and only take a fraction of the class time. Many of them are well known questions.

This past week, I introduced the topic of Arithmetic Series to my grade 11s by asking them to sum the numbers from 1 to 100. This activity was paired with the story of C.F. Gauss. In elementary school, Gauss was asked the same question and found an ingenious pattern to solve it. My students were told that he found a pattern, but were not given a hint on what it was. They were to get into pairs and try to find a shortcut to generalization.

This is where the criticism comes in from traditional teachers. Why, if I was so concerned with lack of class time, would I send students away for 15 minutes to discover a pattern that may be far over their heads? I have many reasons for this, and all of them were apparent in various students as they attempted to follow in the footsteps of Gauss. 

1) It puts the students at the center of the learning
Immediately, the students took the initiative. They formed their groups and began to converse. They began bouncing ideas off each other, while I circulate and sit in on various group discussions. When I visit, it is not to re-direct thought, but to force them to recap their thoughts. These recaps reveal holes in their logic (if they exist), and serve to trace their progress. 

2) It hooks into prior knowledge
Students bring different math experiences into different tasks. Some have seen a similar problem, and begin to take the role of instructor in their group. One student knew that the numbers needed to be lined up in order to create pairs. Although her memories were foggy, she used this starting point to lead her group into valuable learning. Enough of the previous problem was forgotten, that the group still needed to justify their process. Even if they have not seen a problem before, many numeracy skills are involved. The ideas of T-charts and Odd/Even sums were tossed around by several groups.

3) It builds strong peer discourse in the classroom
Students will gravitate to those that are like-minded, so after the groups have the opportunity to solidify their answers, I call the whole class back and recap all solutions. This begins powerful peer discourse. Students will ask for explanation, and, as the teacher, I never provide it. It it the groups’ responsibility to defend their work. I simply act as a moderator if the conversation stagnates or becomes exhaustingly cyclical. 

4) The struggle makes the solution more meaningful
The naysayers are correct; not every student will “discover” the math exactly as the book or curriculum intends. Not every student figured out that the numbers from 1-100 line up nicely into 50 pairs of 101. Many tried a different route. When the group work is discussed as a class, and an elegant solution revealed, everyone shares in the success. When the method of pairs was discussed there was an audible sigh and gasp throughout the class. The solution meant more because they were invested in the problem. One girl even said, “wow, that’s a cool way to think about it!”.

5) Students may stumble upon an alternate learning
Just like some may not get any answer, some students may come up with powerful answers that you may not expect. I love these occurrences more than any other. One student called me to his desk during the task and declared that he had found “the” pattern. I quickly corrected him:

“You have found ‘a’ pattern”
He went on to show an interesting result:
Sum (0-9) = 45
Sum (10-19 = 145
Sum (20-29) = 245
he concluded that he could answer the problem by summing the first 10, and then adding 100 for each additional group of 10. I had a little bit of time to probe deeper. What if you only wanted to sum every other number? What if we started at 15 and summed to 100? How can you generalize your pattern using sequence notation? This discovery was a highlight of the class for everyone involved. The fact that I was surprised meant the world to the student.
Prompts provide a unique hybrid to discovery learning and a motivational set. Finding or designing an interesting problem allows students to begin a topic in control. They also provide an invaluable reference tool for the teacher. Asking them to remember the Gauss Activity reverts them to the meaning of a formula or idea. There is no coincidence why these activities remain forefront in their memories–students will remember the concepts that they had a hand in forming. 

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