The Pythagorean what?

“Cognitive learning environments focus on helping students encode information meaningfully to long-term memory so that it can be easily retrieved.” Getting students to “retrieve” previously learned material has always been something that frustrates me. More so in the lower level classes, students can’t seem to remember concepts they have learned in previous years, semesters, and weeks! Math is a very cumulative topic, previous knowledge is almost always necessary to understanding new concepts that are taught. So, I am all for thinking of methods that I can use to help students store some math in their long-term memory so that next time I ask “what is the Pythagorean theorem?”, I will get something other than blank stares in return.
The first thing that came to mind is something I do constantly and that is Organize information so that learners can connect new information to existing knowledge. I usually start each class with a “Do Now” that is a question they have seen before and then build on it in the lesson to explain the new material. An idea I have always liked but never really made it a regular part of my routine is students can journal or blog about a new topic. They would have to summarize it or pretend they are writing a letter to a friend who was not in class the day of the lesson. Maybe encourage them to think about where/when they would use this new concept and maybe come up with a mnemonic (if possible) or method to remember a formula, definition, etc. Students can then read each others blogs and maybe get another view of the topic that might help them to remember it.
I think the idea of groupwork would also benefit students. If students had to explain how to solve a problem to another and talk about the process together I feel that they will better understand it. During a workshop last summer I learned about an ongoing project where everyday a student is selected from the class to type up the notes from that period. They would have to add to the notes explicit directions and explanations of everything that was discussed (applications, questions, steps, etc) and submit this for a grade. I thought this was a great idea but could only see it working well in an honors class. I think it might be a good idea for partner work in a regular or weaker class. Through student discussion and writing I think they might be more apt to listen and remember.

Constructivism

"Constructivist Learning environments focus on giving students authentic, real-world, ill-structured problems to solve." I think constructivist learning environments are very effective ways for students to learn and really understand material. I teach mathematics in high school, and it is always exciting when I find a discovery type lesson for students to actively engage in. I wish I could do these types of lessons more often because I see that students are more prone to remember and understand the concepts when they are working to see where they come from. Students actually come to life during these lessons. Unfortunately time and creativity are factors that prevent me from having these lessons. In the past I have put students into groups and given them an assignment so that they can work together and then as a class we would discuss the findings and answer questions. As a facilitator I guide them through the process through questioning and provide hands on materials. I agree with the readings saying that this constructivist approach does not work well unless students know prerequisites for the material. This is especially true in math, where students are constantly recalling previously learned material. Since time is a constraint in the classroom, I have found that homework projects also work well to get students actively involved and seeking information. I think as teachers it is important for us to encourage this type of learning by providing students with motivating lessons, activities, and projects where they can take some ownership. At the very least I think questioning techniques are an important skill to master to help lead students and have them make connections. Facilitating large and small group discussions on real world problems and connecting these to curriculum is a great way for students to learn. I always feel that if students can talk to each other and explain the mathematics that is being used, then they truly understand the material.