Week 5 of 525

As this class comes to an end, our final topic of blogging discussion involves Identifying Similarities and Differences, Homework and Practice, and Generating and Testing Hypotheses.  I believe that the purpose of teaching students to identify similarities and differences is to help them restructure their understanding of the given content.  Through the classification (and consequently declassification) of certain concepts, a person deepens their understanding of a topic.  I remember a specific instance when I asked my Pre-Algebra students to identify "like terms" in a given algebraic expression by listing all like terms together in a column.  I think that this worked well because it forced the students to examine what terms were similar (and why) and different (and why).  I also remember an instance where I failed to explain that "x" and "1x" meant the same thing, but looked different.  This ended up causing an assignment to go rather poorly, because I asked the students to simplify expressions that they didn't entirely understand.  Explaining how these two are similar/different would have cleared up their confusion.  As far as the implementation of technology is concerned for identifying similarities and differences; spreadsheet software, Venn diagrams (Inspiration), and online graphic organizers allow for an easy means of compare/contrast of any content.  My question regarding these types of technologies would be how to apply the use of a Venn Diagram in a math class.  I've always been a fan of this type of visual aid, but haven't quite found the best way to implement it into the classroom.

The next topic of conversation was regarding homework and practice.  I believe that the purpose of asking students to do homework and practice their content is to review and apply what they've learned.  Like many other skills, acquiring knowledge demands practice before mastery can be demonstrated.  I remember when I took an extra day on the topic of one step equations (dealing with fraction coefficients), giving my seventh graders an additional worksheet of practice.  When we took a test several days later, I was very pleased with the results as a whole, and I believe the extra practice affected these results.  On the other hand, I also recall a lesson with my Advanced Math students where I made homework optional.  My logic behind the decision was to give them a sense of what some college classes are like.  However, I believe that this experiment failed because the students failed to see the importance of practicing content.  As this was essentially the first time that they had been given the freedom to not to do their homework, they exercised that freedom promptly (along with putting up low test scores).  I think that even though this lesson failed, the students learned the importance of practicing their content, largely due to their poor test scores.  Some technological resources that help promote homework and practice include, but are obviously not limited to BrainPop, IKnowthat, Flashcard Exchange, and CoolMath.com.  My big question with implementing technology as a means of homework practice would be how to find a good balance between online and offline resources?  Surely not all assignments should be done online, but every once in awhile doesn't seem very productive either. 

The final topic of discussion was generating and testing certain hypotheses.  I believe the purpose behind making students generate, and then consequently test a hypothesis is to engage them in complex mental processes, along with enhancing their understanding of the content.  I recall several years ago, while teaching Geometry, covering a unit on inductive/deductive reasoning.  Whereas this is not generating a specific hypothesis, the students where forced to look at real-life word problems that demanded they draw conclusions from a given set of information (the foundation of testing hypotheses).  I feel this section worked well because it allowed the students to take a break from shapes, angles, etc.; focusing on the foundation of mathematics, which is logic.  However, I also recall my first year of teaching, when I more or less let my students attempt solving geometric proofs with little to no initial instruction.  This left several students confused and without a real starting point on the problem.  I feel that if I would have explained to them that by using what they know, systematically gaining a little bit of information at a time, proving their hypothesis would not be overly difficult.  Some different types of technological aids that promote dealing with hypotheses include Smog City, Primary Access, and Darwin Pond.  My question regarding technology in this respect would be how to find a decent proof-writing program online.  I think that students hate proof writing more than most mathematical topics, and having some type of electronic aid could really promote student interest in the area.