I’m sure many of you have been annoyed by the story problems in math classes. They are often boring, worded funny, and require a large suspension of disbelief. You may ask yourself what kind of idiot would write questions like those.
Well, I would.
In my Probability class, one of the projects assigned was to come up with unique questions that would need to be solved using the methods we were learning in class. We didn’t do that well. Now, I don’t mean to insult my classmates. Our questions were clever and fun to think about. However, they fell short when we were asked to solve each others. They became as mundane and silly as the work we had done our of textbooks in previous math classes.
In Jo Boaler’s class, she mentioned a different approach to assigning larger problems. In most American classroom, Mathematics is taught as a subject, rather than a tool. Teachers search for a circumstance applicable to their current curriculum, rather than providing problems and giving the students tools and resources to solve it how they see fit. The questions my class and I came up with were synthesized with answers and a process already chosen. We had worked backward to get the problem, which made them straightforward and fake.
Jo spoke of a school that taught Mathematics as a tool. Instead of giving their students a procedure out of thin air and asking cookie cutter question that didn’t really apply to life outside the classroom, teachers would assign a project, then make a number of tools available for their class. These tools would include what the teachers would think would be of use, such as protractors and straight edges, along with random ones that were unlikely to be necessary. Students would have to think critically about how to go about solving the problem. Some students would even use the ‘unnecessary’ items to problem solve in a new, unconventional way. The students had to choose the procedure, rather than being force-fed one to plug-and-chug throughout the class period. When groups came to the teachers with questions, they would be led in the right direction, or given a concept or procedure useful for how the group decided to overcome their current hurdle. Students would understand why and how the procedure they had just learned worked and how it could be applied to real-world situations. The groups would even share what they learned with others. (I mentioned in an earlier entry how explaining to peers solidifies understanding of the material.) Mathematics was no longer a thing to be memorized and regurgitated, but something to experience first-hand and apply.
Inventing problems with specific information in mind does have a place in the classroom, though. By creating our own, personal questions in my stats class, we were deepening our understanding of how the procedures worked as well as where we could use them in our own lives. It gave us a chance to be creative and invent the most complicated (yet solvable) problem we could.
In both of the EdX courses I took from Stanford, the professors often mentioned how important working with others and discussing justifications can be. I believed it worked– for others. It was one of my “I am awesome and independent” moments. Sure, I would encourage my students to pair up and talk about the questions and answers that came up in class, but it was never something that I had intended to practice myself. This changed Monday, when I swore during my Intro to Probability class.
We were asked how many permutations there were in the stringing of a necklace, consisting of three different-colored beads. We easily came up with six. Next, we were told to discuss within our table groups how many possibilities there were without repeats (Since the necklace would connect, rotations would fall into the same permutation). I came up with two. My partner claimed that there was only one. We argued a bit, then I finally raised my voice and said, “No. See, there are three rotations of this one [where I gestured to my model], and three reversed rotations of that one, and unless you flip the necklace– Shit!” My partner and professor grinned and some of my classmates snickered. My prof then waited as I explained to the rest of the doubters how there was only one unique way to string the necklace.
By justifying my reasoning that there were two options aloud, I was able to hear the flaw in my logic that my partner had seen. I was also able to solidify my understanding of the problem by tailoring my justification to make sense to other students who I had previously agreed with.
Communication is something my prof frequently encourages during class. There is a very small portion of class time spent watching him write on the board or explaining concepts without us interrupting with questions and comments. When we bring something up, he immediately asks us what our interpretation of the issue is and to try to come up with an answer in our own way and at our own pace. Class discussions continue to play an important part in my learning. When I believe I understand a concept, I can turn and explain my thought process to my partner, who then peppers me with questions about every step I’ve made. This helps me find and understand my mistakes and gives me a chance to practice my own teaching skills for the future. When he grasps something first, he’ll explain step-by-step, then ask me to re-explain it to him in my own way, to make sure I ‘get it’. These activities have deepened my level of understanding for the subject and helped me to double-check my work and fix mistakes early on.
While class discussion may seem overrated to introverts and independents (like me), it is a key element in the learning process– particularly due to the requirement of putting thoughts into words and listening to what comes out. I highly encourage classrooms to become less listen-and-repeat and more discussion-based. I was very surprised at how much of a difference it has already made in my education.
One of the themes I learned in my online class (discussed more in my About Page) was having open-ended questions in class. While traditional teaching methods assigned values to be plugged into a formula, open ended questions would give a goal and let the students loose to achieve it with the skills they have acquired– as well as a resource to discover more.
The other day, in my Intro to Math Software class, my professor assigned a project with this theme in mind. We had learned the basics of the program LaTeX, and were given a project to duplicate the code needed to come up with a PDF similar to the one he created. Now, there were multiple codes that would yield to the same tables, graphics, and text formats. There were multiple ways to go about this, and multiple orders to code in to get the same result. He could have asked us to re-type his document, then checked for errors via the PDF. Rather, he asked us to alter a graphic so that it was the same size and rotation as his, and let us play around with the coding until we were satisfied enough to submit it. He gave us a wiki book to look into any alterations we wanted to make to a table, and let us use the method that we were most comfortable with to achieve the same look as he.
So today, I wanted to see how much of a difference open-ended questions made. I took one of his templates for a beamer presentation (similar to a powerpoint) and re-typed what he had. I didn’t understand much. I didn’t remember much of what I typed. Then I went back to the wiki page he linked us to and started to read some of the basics. Once I knew what each code represented, it got easier. If I had gone through his sample presentation and attempted to create one, but with my own twist, I guarantee I would have taken even more from it.
I got a chance to see first-hand what open-ended assignments can do for students’ understanding and memory. I definitely had an easier time creating documents when I had a grasp of what each line of code represented and when I had to experiment a bit to find what I was looking for. And I saw my class grasp most of it, as well. Even with something as complicated as creating a document from pure code, I found that students can assimilate the necessary material better than they would if they were just spoon-fed information and told to spit it back out.