Monday, August 29, 2011

The Nitty Gritty: A Quantway-type lesson in action

One question I have received often about this course is, "what does a class look like?"  So today's blog has the sole purpose of showing you the class atmosphere and a typical large scale lesson.  I say large scale because lessons vary in length.  All lessons are 25, 50, 75, or 100 minutes long.  The one described below was 100 minutes.  I've shown bits and pieces (not all 100 minutes) to give the flavor of the class period.  We've already seen that having lessons of various lengths where the activities change regularly goes a long way to maintain focus and productivity.

The clips I show here show more of Heather so that you can see what the instructor does.  However, most of the period involved students working and the instructor circulating.  It's a back and forth motion, whole group - small group - whole group and so forth.

Warning:  Video quality is moderate and the videographer needs to work on her technique.  I tried to avoid a Blair Witch effect but there are a few shaky moments.

The lesson filmed was called "Higher or Lower."  Its premise problem is that a bargaining unit is negotiating a contract with two pay structure possibilities.  One is a 5% raise; the other is a 3% raise with an additional $1000 added on.  We begin the lesson by describing the scenario.  Students have a few questions to work on related to this such as "which would you pick?", "if you made $20,000, what would your new salary be under each option?", "does the order matter when adding the $1000 and applying the 3%?"  Below, Heather is doing the initial problem description and putting groups to work on these beginning questions.

In the next clip, you get a sense of what the class atmosphere is like.  Small groups, lots of discussion, chairs grouped together.  Heather circulates and helps groups progress without answering everything for them.  Our motto is "answer questions with questions."

Part of this lesson involves taking the percent of a number.  The goal of the lesson is larger than this skill but it's still a vital part.  In the next clip, Heather conducts a mini-lecture explaining how to take the percent of a number.  Many students know a rule but don't understand what they're doing.  Heather works off of a picture, moves to a scaling technique, and then generalizes the process with a rule.  Our goal is to develop agility with skills.  This is hard for students but worthwhile.  It's not just can they do something.  It's do they know when to do the skill and can they perform it in various contexts.  Heather integrates multiple representations, estimation, and reasonableness to work on one of the course goals:  numeracy. 

Not all lessons have a specific skill taught but many do.  MyMathLab is outstanding for skill development.  However, students abuse help aids and often forget to write down legible work.  One goal we have for the materials we're writing is to help students develop good study and work habits so that they're successful in a college level math class.  So we have made a designated page for every MML skill homework.  In the next clip, Heather explains this sheet and what we expect them to do to make the most of the program.

An important note:  the MML assignments that we're building embed the skill in a context to get as much practice as possible with realistic situations.  We do not want students to be in "mimic mode."  The goal is learn the skill and transfer it.

After the mini-lecture, students were asked, "if you wanted to know who each salary structure is best for, what would you do?"  We want them to think about solving large problems before diving in and showing them everything.  They brainstormed ways and we did have students who said, "pick a bunch of salaries and find the outcomes under each."  So students worked on completing a table doing just that and then answering questions with their group about the table.  Questions like, "who benefits the most under each option?" and "can you generalize the calculations?"  Note:  questions are more explicitly defined on paper than my brief descriptions here.  Generalizing a calculation is challenging but begins the process of bringing in variables.  Our method is intentional.  We want them to see how generalizing the calculations allows the use of spreadsheets like Excel to accelerate calculations.  Below, Heather debriefs the class after they've worked on these tasks.

Lastly, Heather shows students Excel and how this problem can take advantage of spreadsheets to gain more insight.  Because there are so many clips, I cut off the end of class.  At the end, we asked students if there was a salary for which the pay structures would give the same amount.  They wanted to try guess and check using Excel (as did we) so we did that until we found the exact amount, $50,000.  Then we helped students write that problem mathematically which led to this equation:  1.05D = 1.03D + 1000 where D is the salary in dollars.  We can't solve this yet but it was interesting for students to see variables used and for equations to come into play.  They're used to being given an equation to solve so it was surprising to them to see one develop organically.  We're building to the point of being able to solve one.  That comes in a later unit.

An additional homework assignment on paper accompanies the lesson and MML assignment.  Paper homework is conceptual and applied, looking similar to test type questions.  They're not long assignments but they're deep and challenge students to explore concepts further.

It's a lengthy blog, for sure, but I wanted to give a real feel for what we're doing in the classroom.  More blogs and videos to come.

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