The final answer
BOBBY’s question was simply brilliant. It was beautiful like a minimalist painting, just three square matrices sitting in the middle of the page with question marks for the missing numbers.
But despite its simplicity the question was a great test of understanding the process of matrix multiplication… but it also clearly showed that Bobby understood the process sufficiently well to create this question without any prompting.
Writing their own exam questions was the Final for my Algebra 2 classes this summer.
Each student was asked to choose a topic we’d done during the year and create a question. They also needed to list the math concepts and skills needed to answer their question as well as prepare a model answer with notes about likely common errors. Students were also encouraged to add a supplementary question aimed at probing deeper or look for ways to test a number of ideas within one question.
The students then swapped questions and attempted answers. The original authors marked the answers. The authors also explained the answers and ideas if a student got stuck on their question.
So, Cassie gave Bobby a second chance with her question built around the need to understand the Quadratic Formula, without mentioning the formula in the question.
Other questions were pleasingly complex. Becky created a log problem where the base was itself an algebraic expression which was the result of a simplifying task.
Some students also exposed errors in the model answers.
The result of the exercise was some good-humored but thoughtful mathematical discussion that continued the learning process while also revealing something of each student’s knowledge and math thinking skills.
This Final did not produce a series of right and wrong answers and a percentage at the end. But it did avoid all the damage that such traditional finals can inflict on student self-esteem and confidence. Anyone who doubts just how serious this can be should read Testing, Motivation and Learning from the Assessment Reform Group and reviewed earlier on this blog. Teachers, who almost by definition fitted well into the schooling system, frequently utterly fail to comprehend the stress repeated testing causes those not so fortunate, and indeed even many of those who can manage the system.
My own view is that traditional finals tell the teacher nothing that they should not know already. They are more ritual with their roots less in thoughtful pedagogy and more in the no-pain-no-gain vision of education and its emphasis on sorting, ranking and the motivating power of humiliation.
In my algebra 1 classes I tried a compromise… in my school the pressure is mounting to dumb down and get back to those mythical and failed so-called basics to get those damned state test scores up.
Students had the traditional exam, written by a colleague. But instead of sitting in silence and in rows, students were encouraged to try a few questions and then bring them to me. Students got to move around, chat in the small queue and get some immediate feedback plus a one-to-one lesson for a few seconds.
I got to see just what each student could do or not… and keep the teaching going targeted on individual misconceptions.
Sure, by the end of the lesson there had been considerable collaboration among some students as correct answers filtered around the room as if by osmosis. But even that can be a learning process, better than stress and the fear of inevitable failure.
The final answer must be: Do nothing to destroy hope.
Try Bobby’s question for yourself:
Find the missing numbers.
[A] x [B] = [C]
where a11 = 2
a12 = ?
a21 = ?
a22 = 5
and b11 = 2
b12 = 6
b21 = 3
b22 = 17
and c11 = 13
c12 = ?
c21 = 23
c22 = ?