Year: 2013

Categories
Statistics

A.P. Co-Teaching: Stats Meets Psych

For seven years before becoming an instructional coach, I taught Advanced Placement statistics.  I loved this course, as every day brought a new applied situation,  a new set of data, and a new, rich classroom discussion.  While many of my math colleagues have an aversion to teaching statistics (one friend from another school said to me “you’ll become the loneliest person in your department”), I think teaching the course gives an appreciateion for how we should be approaching data analysis in ALL math courses.  But that’s a post for another day.

This week’s stats twitter chat (#statschat, 9PM on Tuesdays) started with a discussion of the recently released AP Stats items, but later moved to post-exam activities.  As part of this discussion, cross-curricular options came up, and I mentioned a co-taught lesson I have developed with my AP Psychology colleague.  For a number of years, this teacher and I had discussed co-teaching a unit on experimental design, as the AP Psych course outline actually includes a nice chunk of material AP Stats students come to understand.  One section of the description, Research Methods, is right in the AP Stats wheelhouse:

  • Describe how research design drives the reasonable conclusions that can be drawn (e.g., experiments are useful for determining cause and effect; the use of experimental controls reduces alternative explanations).
  • Identify independent, dependent, confounding, and control variables in experimental designs.
  • Distinguish between random assignment of participants to conditions in experiments and random selection of participants, primarily in correlational studies and surveys.
  • Apply basic descriptive statistical concepts, including interpreting and constructing graphs and calculating simple descriptive statistics (e.g., measures of central tendency, standard deviation).

Yey for math featured in non-math subjects!  What a natural fit for a handsome, fun math teacher and a respected social studies teacher to join forces in a class lesson!  Some students comments that it seemed so out of context to have the two of us in the same class together.  Worlds colliding!  Dogs and cats shaking hands!

For two days, we led a discussion on correlation and causation, based on a curriculum module provided by the College Board.  In the “Teaching Statistics and Research Methodology” module, the section “A Lesson on Correlation” by Amy Fineburg was used as a framework for discussion.  Students were provided with an article to read beforehand and was used to generate discussion regarding student ideas of correlation, causation and experimental design.  Our pesentation to the students is given below, and was completed over 2 days.

Ap psych stats methods revised 12 13 from bobloch
Looking forward to sharing this experience at the AP Stats reading next month, along with a similar co-taught experience developed with my school’s AP Chemistry teacher.
Categories
Algebra

Why is “Simplify” So Damn Complicated?

Making my classroom rounds this week, I came across a class reviewing concepts for the upcoming Pennsylvania Keystone Exams in Alegebra 1.  The PA Department of Education provides an eligible contect document with sample items on its website, and the class was working on the following question:

Item

Pretty standard problem.  Factor the numerator and denominator, cancel common factors, and you’re home.  But this class was struggling with the factoring review, so I stepped in with a different approach.  How about taking the given expression, and using a graphing calculator to evaluate it?  Sadly, the class was not familiar with the Table on their TI-84’s, but understood what it did right away:

Calc 1

Some nice discussions emerge here.  What’s with that “error”?  Is our calculator broken?  And some evidence over this function’s behavior emerges.  Note the slowly increasing values of y.

But how does this help us with the question at hand?  A number of students recognized that the correct answer would be the expression which had the same Y-values.  In essence, simplfying produces a different-looking expression with the same outputs as the original.  So, let’s try the answer choices.  Here’s A:

Calc 2

No dice.  Values are much different.  And a fantastic opportunity to discuss the difference between an output of zero, and an undefined output.  But eventually we get to D, and can check the tables:

Calc 3

Looks pretty good, butttt……..what’s with the errors?  And they seem different for some inputs.  But now we can review and discuss domain, and look at those pesky domain restrictions in a new light.

So, am I a bad person for bypassing the factoring review, and encouraging calculator use?  After the discussion, I reminded the class that factoring is a skill they need to have in their toolbox, but the alternate discussion of equivalent forms and assessing values was also worthwhile.  I feel good.

This classroom visit got me thinking about the nature of the word “simplify” in math class.  How often do we ask students to “simplify” in math class, and in what contexts?

Sometimes we want to simplify an expression:

Or maybe we want to simplify a rational expression:

Or perhaps we want so simplify a radical expression:

And make sure you simplify when there is a radical in the denominator (unless you are taking AP Calc, in which case we don’t care about such silliness)

For different situations, we have subtle differences in what it means to simplify, but is there a common goal of simplifying?  Is it just to make things look pretty?    And is a simplified expression always the most useful?  When is it not?

I’m curious if anyone has a short and snappy answer to “what does it mean to simplify an expression?”.  I invite you to participate and contribute your response on Todays Meet (click to participate).  If you have never used Today’s Meet, it is a nice, free way to gather responses.  Simply provide the link and start a conversation!  Feel free to share the link with your students a “bell ringer” activity.  If we get some responses, I’ll make a later blog post about them.

TodaysMeet

Categories
Uncategorized

Last Night at Global Math….

Thanks to Megan Hayes-Golding for hosting last night’s session at the Global Math Department, where I shared some of my Tall Tales for Probability.  The recording is now available.  This was my second time presenting, and it is a unique experience.  I recognize that I talk WAY too fast at times, mostly because I am so excited to share my ideas.  But sitting alone on the couch talking to my laptop, and trying to assess reactions in the chat room make online speaking a wild ride.  I was the 2nd speaker of the evening, and enjoyed Chris Harrow‘s sharing the 4’s Game, and Chris Hunter‘s ideas for cooperative learning.  Always an uplifting experience to be around such excellent educators!

I had a few requests for the video of the hay bales.  I have put it on dropbox, but it is a BIG file.  Let me know if I need to zip it.