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• 1.  Teacher preparation in mathematics - second course

Posted 07-22-2021 22:10:00
Posting the statistics problem we use to probe student understanding of mean, median, and mode with second-semester students. The college has three four-credit courses in teacher prep math:

1. Nine people decide to share some candy. The most anyone has is nine pieces, and one person does have nine pieces. At least one person ends up with no candy, by choice. The average (mean) number of pieces is four. Four is also the median. That means at least one person has four pieces. (Why?) More people have two pieces of candy than any other number.
2. Find all the possible solutions to the problem. (This is questioning level two, which consists of sorting and contrasting additional information, in this case, additional solutions.)
3. List all of your solutions and take photos of, or draw and scan, the cube representations of your solutions.
4. Ask yourself if you have found all of the solutions and why. (This is statistical reasoning and a type of mathematical proof or conclusive argument.)

And a few student solutions

Enjoy !!!  See you at the AMATYC conference !!

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Ruth Collins
Professor of mathematics education
Walden Univ (retired from a two year school)
Minneapolis MN
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• 2.  RE: Teacher preparation in mathematics - second course

Posted 07-23-2021 20:23:00
Ruth,
Some teachers struggle with asking students to find more than a single solution.  What advice could you give classroom teachers who have never used open-ended tasks that may have more than one solution?

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Nancy Sattler
Dean Emerita & Adjunct Faculty
Terra State CC (emerita)
Fremont OH
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• 3.  RE: Teacher preparation in mathematics - second course

Posted 07-23-2021 21:12:00
I've experimented with the grading rubric accounting for multiple solutions, but I didn't find that particularly successful, and it was particularly unpopular with students on exams, where they struggled to even find a single solution.  I don't think that losing a point on groupwork really makes students think "oh, we should try to find more solutions"; they were just down about not getting full credit on the assignment.

What I think IS helpful is to be consistent about asking students to find multiple solutions, and to especially make sure the problems have multiple solutions in the first place before assigning it.

One classmate of mine in my Masters cohort had a saying that if a problem has step-by-step directions, try cutting those out and letting the students fill in the gap with their own thoughts and inventions and ideas -- and save the step-by-step directions for hints you give on an as-needed basis.

Another technique I've heard my fellow teachers say has worked well is to have students visit other groups that are working on a different solution strategy and take back ideas to their group (the so-called "Spy" technique).  I haven't tried it yet, but I hear many good things about it, and we've been experimenting with it in faculty professional development (peeking at each other's answers) and it's been very generative and helpful.

James Sheldon, Faculty, Mathematics, West Campus & PimaOnline
Pronouns: he/they (see https://www.mypronouns.org for more information on personal pronouns)

I acknowledge my presences on the ancestral lands of the Tohono O'odham. For information on why such land acknowledgments are important, see: https://sites.google.com/view/tosawildcats/land-acknowledgement

• 4.  RE: Teacher preparation in mathematics - second course

Posted 07-24-2021 10:41:00
James
Enjoyed your posting... one comment on the idea of "hints".  If you look at the TIMSS videos, you often see teachers giving hints and generously, and to everyone.  In one section the teacher (who knows the TIMSS cameras are filming him AND his words to a student) were:  Let me give you a hint, the answer is 3.7.  Most teachers do not know how to wait long enough for an answer, or send a student in a different direction in terms of their thinking, without TELLING them what to do.  I guess they think they are helping???

They also do not know how to construct problems, or as you mentioned, cut out directions and let students work their own magic in groups (I LOVED that idea).  SUNY colleges published a book of problems that students worked on together to solve within bigger groups, and THEN the teacher did mini-lectures and cleaned up finer points.  I am pretty sure that the book did not do well... teachers said there was too much work on their part.  Bassarear wrote a book for teacher prep and high school students that had a lot of "answers will vary", and several schools in our area abandoned it after a year or two because their teaching assistants insisted they must teach from a book with real answers. Emphasis mine :(

Admittedly this entire of student-generated learning is a work in progress.  ONE small thing I did with one of the SUNY problems, stacking styrofoam cups (which had a lip).  I gave each group cups to start collecting their data (height of various stacks) but did not give them enough cups to finish the project.  They had to go to other groups and make arrangements to borrow the second group's cups and then vice versa, let that second group "borrow" their cups back.  These are the small skills that make working with colleagues later a smooth thing.  Surprisingly, at least to me,  most of the students were not comfortable with this small change in procedure. They spent time discussing WHO was going to go over there and ask to borrow cups.
Appreciate your posting, thank you

Ruth

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Ruth Collins
Professor of mathematics education
Walden Univ (retired from a two year school)
Minneapolis MN
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• 5.  RE: Teacher preparation in mathematics - second course

Posted 07-24-2021 15:16:00
By the way: I have an article about task construction coming out soon in the MATHAMATYC Educator, so keep an eye out.--
James Sheldon, Faculty, Mathematics, West Campus & PimaOnline
Pronouns: he/they (see https://www.mypronouns.org for more information on personal pronouns)

I acknowledge my presences on the ancestral lands of the Tohono O'odham. For information on why such land acknowledgments are important, see: https://sites.google.com/view/tosawildcats/land-acknowledgement

• 6.  RE: Teacher preparation in mathematics - second course

Posted 07-24-2021 11:06:00

Nancy Sattler
Great question.  The solution in large part would be to change the teachers and their approach to discussions before they actually start teaching.  This is a very long-term solution, and probably not practical.  Spending on in-service is down nationally and think that the other areas of instruction (science, history, and others) are well ahead of mathematics departments in opening up to student talk, instructor listening, and the idea of many answers.

When a student offers an answer that is not the one I was expecting, I usually have them start by telling the class the assumptions they started with. Often the difference is in the assumptions, and their answer is usually another fine and acceptable solution.

We have to change the teachers though, from first grade on up through college, and that in turn will generate classes where the search for more than one answer to a problem is the norm.   If I had a dollar for each student who told me that there is always one answer, no more... I could have retired earlier :)

I posted the candy problem which has multiple answers and many false paths.  :)  You may have remembered this from a former AMATYC presentation way back when :)

Ruth

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Ruth Collins
Professor of mathematics education
Minneapolis MN
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