Group Tests - A summary

In my last post (http://robebymathdude.blogspot.com/2014/02/v-behaviorurldefaultvmlo.html), I mentioned at the end that I would be trying group tests this semester in some of my classes.  I want to write a little bit about that here.  First a few related details: 

•  I am scheduled to present on this topic at the AMATYC national conference in November in Nashville, TN.  (http://www.amatyc.org/?page=2014ConfHome)
•  I was motivated to do this by a post at Casting Out Nines by Robert Talbert.  (http://chronicle.com/blognetwork/castingoutnines/2012/05/14/turn-to-your-neighbor-and-take-a-test/) though in his case it was for a summer class and it was for Calculus I students, so a slightly different set of students.  
•   At my school, we have a departmental final exam (with versions) that each student has to take, so I also have to ‘train’ my students toward that also.

Since I was doing this over the long semester, and as I mentioned each student must take an individual final exam at the end of the course, I chose to set up my exams as two parts.  There would be a take home part that the group would work on, and then an in class part that was individual.  However, each student was warned that some of the in class part would include some ‘extension’ type questions and some ‘traditional’ exam questions.

For example, for the “Math for Liberal Arts” class I had these two question as part of the take home part of exam one: (both are based upon questions in the text A Survey of Mathematics by Angel, Abbott, Runde).  The students were given the take home part a week before the in class exam, and the take home part was due at the end of class 2 days before the exam.  The class day it was due was a “review day” where I went around the room to answer any questions the students had over the material, including anything from the take home part.  After the take home parts were turned in, I would post the take home part with answers for the students to review for the in class portion of the exam. Anyway, here are two of the questions and what I did on the in class exam. 

TAKE HOME QUESTION 1:      A yardstick measures 1/4 by 3 by 36 inches. How many yard sticks will fit in a box: 

•  3 inches wide and 36 inches high, if the girth of the box is 30 inches? 
•  6 inches wide and 36 inches high, if the girth of the box is 30 inches?
•  3 inches wide and 72 inches high, if the girth of the box is 30 inches?
•  3 inches wide and 36 inches high, if the girth of the box is 60 inches?
•   6 inches wide and 72 inches high, if the girth of the box is 30 inches?
•  6 inches wide and 36 inches high, if the girth of the box is 60 inches? 
•  3 inches wide and 72 inches high, if the girth of the box is 60 inches?

TAKE HOME QUESTION 2:     An average newspaper contains at least 16 pages and at most 87 pages. How many newspapers must be collected to be certain: that at least two newspapers have the same number of pages?

I.            That at least two newspapers have the same number of pages?

II.            That at least three newspapers have the same number of pages?

III.            That at least four newspapers have the same number of pages?

The students had a few days to write up their solutions with explanation as a group.  Then the individual exam in class had these two follow up questions as part of the exam:

EXAM QUESTION 1:   On the take home part you were given the following problem:

*** A yardstick measures 1/4 by 3 by 36 inches. How many yard sticks will fit in a box:

• 3 inches wide and 36 inches high, if the girth of the box is 30 inches?
• 6 inches wide and 36 inches high, if the girth of the box is 30 inches?
• 3 inches wide and 72 inches high, if the girth of the box is 30 inches?
• 3 inches wide and 36 inches high, if the girth of the box is 60 inches?
• 6 inches wide and 72 inches high, if the girth of the box is 30 inches?
• 6 inches wide and 36 inches high, if the girth of the box is 60 inches?
• 3 inches wide and 72 inches high, if the girth of the box is 60 inches?

***

Your answers should have been 120, 240, 240, 240, 480, 480, and 480 respectively.

So how many of these yardsticks will fit into a box that is 6 inches wide and 72 inches high, if the girth of the box is 60 inches?

How many will fit into a box that is 3*R inches wide and 36*R inches high, if the girth of the box is 30*R inches, where R is some positive number greater than one?  Explain how you know!

EXAM QUESTION 2:   On the take home part you were given the following problem:

***        An average newspaper contains at least 16 pages and at most 87 pages. How many newspapers must be collected to be certain: that at least two newspapers have the same number of pages?

•  That at least two newspapers have the same number of pages?
•  That at least three newspapers have the same number of pages?
•  That at least four newspapers have the same number of pages?

***

Your answers should have been 73, 145, and 217 respectively. 

So how many should we take to make sure that at least 42 papers have the same number of pages? 

What is the formula to use to make sure that there are at least R papers with the same number of pages, where R could be any positive integer?  (You should write the formula in both mathematical symbols AND then in words)

Overall the students seemed to enjoy the format.  I informally interviewed most of the students, and all of the ones I interviewed liked the format of the exams, and the idea of the extension questions.  Several of them commented that it really helped them to understand the idea of mathematics as the process of finding patterns.  It worked so well that my Calculus I students get to experience these this semester.

Thanks for reading, and as always, comments are welcome.

What are tests good for?

Second post!  What are tests good for?

I recently came across a blog post asking  What does a test “test”?  Given that the last comment was from 2011 I am clearly late to the game on this post, but it sparked some thinking on my part.  To recap the post, the major thrust is that tests do not measure what it is generally assumed they measure.  To quote:

….  Tests are usually thought to be reliable indicators (if not measures) of how well students  engage with the material being taught to them, and is often believed to be related in some correlational way with their mathematical ability.

Where a student ranks – their class standing – on a test may well be an indicator of both those things.

I want to argue, however, that what a test really tests is the teacher. A test, in my view and experience, tests a teacher’s ability to set a test at the “right” level…..

Franky [sic] I feel we put too much emphasis on tests and examinations. My own preference is for more collaborative project work in which students can exercise their ability to think, to reason, to plan, and to work toward a goal, utilizing their skills and talents in conjunction with others.

First things first: my department has common post tests for each class.  The exam is roughly 20 questions that cover the basic learning objectives for the course.  We actually have created a test bank for the questions to help with test security, but basically everyone is giving the same exam.  So that means in some sense I have to train the students toward that even if I personally wanted to never give a traditional exam. 

In that vein, I first and foremost see tests as a deadline.  So in that sense a test exams how well you can learn when there is a deadline on learning the material.  I see this as the difference between me saying “You know, at some point I would like to learn about Project Based Learning and use it in my classes,” and me saying “”I just put in a proposal to discuss how I did Project Based Learning this coming semester.”  The second forces me to plan my time accordingly rather than keep putting it off. 

(2)        This relates to my second idea about tests, they allow me as a teacher to know what the students know when we launch into a new set of ideas.  So I have a test in my calculus classes over basic derivative rules before we discuss chain rule.  Then I have some assurance that the students know the basic rules before we launch into chain rule.  Likewise, I have a test over derivative rules before we get into relative extrema for many of the same reasons.  As every teacher knows, having a test does not mean the students will all learn it, but having a deadline seems to help.

(3)          Third, I think tests can be used as a measure of understanding.  For example, after discussing the definition of derivative and the idea of rates of change, the derivative as the slope of the tangent line, and the basic derivative rules we have a test.  On that test I typically include a question like the following:  (Very similar to a question from the Hughes-Hallett Applied Calculus)

The growth graph in the following figure shows the height in inches of a bean plant during 30 days.  On the 15^th day, the plant was growing about _____ inches/day.  Round to 2 decimal places.

In my mind, this question is asking a student to identify the concept in a new area.    In a real sense, this is also helping to differentiate the A, B, and C students.  Almost all of the A students get this, a majority of the B students get this, while a large number of the C students will answer 11/15 instead of 14/15.

(4)          Tests can also be a summative assessment.  Basically, I give a similar problem at the start, in the middle, and at the end of the unit of material.  Then I measure how much the student progressed on the material.  As I said above, at some point I have to distinguish what this student knows about the material in order to assign a grade.  I may not like it, but that is the way my job is set up right now.

(5)          Finally, (because I am tired of writing this, not because I have exhausted all of the ideas behind testing!) there is a test as an indicator of limitations.  You might think of this as showing off by the instructor.  As I say to my students, I can write a test that would take me most of the class time, but that would kill all of you.  But what does that prove?  That I am better at this material than you right now?  We knew that at already.  I think this last point is some of what the blog post is getting at.

To that end, I am trying out group tests in both my statistics classes and my liberal arts classes this semester.  I plan on posting about this in about a month once the first couple of rounds are done.

A Response to Kieth Devlin: 

First blog post!  We shall see how this goes...

The basic rules for this blog:

http://en.wiktionary.org/wiki/round_tuit

1. This will be updated whenever I get around to it.  
2. I hope for once a month, but who knows.
3. All of my posts will be related to mathematics, most likely mathematics education, as that is my profession.
4. While I work at a two year college, I also have children in the K-12 system, so sometimes the posts will overlap.
5. Everything I blog about is my own opinion, and in no way shape or form should be taken as representative of my employer and/or any of the professional organizations I belong to. 

One of the mathematics related blogs I read is Keith Devlin's devlins angle on the Mathematical Association of America (MAA) website.   His post on November 4th discusses the topic of teacher verses curriculum.  His point is that a good teacher can make a bad curriculum into a good learning experience, but even a great curriculum will not help an encounter with a bad teacher become a good learning experience.  ** I strongly suggest you go read it **

I agree, and in fact I would even go further.  It is not enough to be a good teacher with a good curriculum.  You must , It really helps, no You must also be a mathematician.  My friend Kate Owens commented on the difference between "math teacher" and "mathematician" on her blog here.  To quote "Meanwhile, 'mathematician' has something more to do with educational background, training, and hobbies...."  I would like to focus on the hobbies part.  In my mind, if you are involved in helping students learn about mathematics, you need to have as one of your hobbies mathematics.

** Warning, I am about to use myself as an example, which is always a little (or a lot) arrogant.  Also, I really want to avoid any "us vs. them" stuff here. **
If I look around my department, those who I consider really good "teachers" are for the most part also the ones interested in discussing mathematics.  How we are teaching what we teach, good problems, and above all else, reading about more mathematics.  Part of this is the self improvement all people should want to do, but I think a bigger part of this is an awareness that mathematics is at the base a search for patterns and a way of thinking.  The more you search, the more patterns you discover, and the more you expand your mathematical thinking.

For a little example of this, I just thought of another way to help my students understand the  Birthday Problem.  I have always calculated in class, and I have a couple of intuitive ways of explaining it, but just this past week while working on a problem set (just for fun, because solving mathematics problems IS fun!) I thought of another way.

Comments, thoughts, suggestions?!?!  Let me know.