Thursday, September 9, 2010

Is Math Creative?

22 out of 23 preservice K-8 teachers agree:

Math is Creative

They were able to tap into the arguments that people have contrarily, though.  Math is about number-crunching, plug and chug, is boring, about right and wrong.  I asked them to discuss at there tables how to counter those arguments, and they said:
  • Everything can be mathematical - there are numbers everywhere.  I think in numbers!  Any lesson can tie in.
  • Numbers can be manipulated in many different and unique ways.
  • The basic skills are concrete, but the application is creative.  Like physics: need math to describe creatively.
  • The reasoning is creative.  People discovered mathematics.
  • The communication of math is creative.  Explaining how things work.
  • Creative how mathematicians come up with the formulas.  Might have just discovered it accidentally.
  • The different ways to represent: graphs, tables, etc.
  • Math is the universal language.  Everyone can communicate in it.
Creativity is important to me (see these other posts) as a goal for my students, and I think opportunities for creativity add a lot to the likelihood of engagement.

It was very insightful to me how they focused on the communicative aspect of mathematics.  If math is a language, it may or may not be creative, much as uses of language may or may not be creative.  As we share our own genuine thinking, and the way we perceive the world (or a problem),  we create opportunities for creative expression.

The  class moved on to look at how they communicated their work on a problem of making quadrilaterals by folding a square (an extension of this nice 3rd grade problem filmed by Annenberg - Teaching Math, Lesson 20).  And immediately focused on what the answer was and were they right.  After our years of school mathematics, we have definitely been well trained.

In the second workshop, we considered the quadrilateral types, and in particular the idea of nested categories or hierarchical sorting.  I show this weird little travelogue:

(I want to update it and maybe make that into an animoto, but the ppt file is corrupted, so it will be more work than I have time for right now.)

The students then made posters of some of the quadrilateral types, striving for a variety of examples, and to be creative in making the posters. Critique and discussion of the posters brought out the discussion points I was hoping for, like which properties are necessary, considering symmetry as an important characteristic, and whether trapezoids should have exactly or at least one pair of parallel sides.

Trying to make space for creativity is not going to be a one lesson effort, but hopefully a theme for the whole semester. I can't wait to see what happens.

EDIT: updated the slideshow to have more visual cues and a couple extension slides. I wanted there to be more to notice.

EDIT2: added student posters for the quadrilaterals.  We worked on generating a variety of examples.  Some made only the specific types, but some made a variety of types that fit the required properties.  I like having both kinds of posters!

Preassessment, Part II

So I asked my teacher assistants about their dispositions towards growth, because that makes an even bigger difference now that they're trying to help others learn.  My geometry students I asked about... geometry.  It has a benefit of starting to communicate what is in K-8 geometry and it familiarized them with Michigan's standards. (Which were all important up until the common core.  They're still close since we were an Achieve state.)

But instead of quizzing them on content that they've mostly had at some point, I want to know more about how they experience the problems.  (I do peek at their answers, of course.  Data is good.)  In particular, if the answer was available by recall, by one or two steps,  required more thought than that, or if they do not know how to begin the problem, or cannot answer.  I feel like that moves it away from feeling like a quiz, and the results have felt more honest since I started polling for this kind of information.

Here's the assessment and results.  The questions are mostly mild modifications of released items and sample practice items for the big state assessment.

322 Pre Assessment

The lack of comfort with unit conversion and the metric system is quite typical, as is the challenge of recalling and applying formulas.  All of these students have had our course for all elementary teachers, and are typically quite sound on most of the content.  It's quite striking for me that even math majors have these issues.  What chance does a typical student have?  A challenged student?

At the end of the content, I ask them what questions this raised for them about teaching.  Bold and italics are my categorization of their responses.

Questions on teaching K-8 Geometry
Big Teaching Questions:
  • What should the teacher be doing?  How do we find ways of teaching that creative, engaging and instructive? Where do we find the best ideas for lesson plans? How can we use the standards to help plan a fun lesson?
  • What should students be learning? What exercises lead to deeper understanding?  How can we be sure our students understand vs performing procedures?
  • How much time do you spend on a topic? How do you teach everything in a year? What do I do when one or two students don’t understand – do I continue or stop for them? How do you teach so everyone is on the same level?
  • How much work should be shown on each problem by students? (Different at different grade levels?) Should it be shown or mandatory?
  • How to introduce a brand new topic?
  • Are students allowed to use calculators? What tools can they use on assessments?

  • How do teachers organize material to make it easier for students?
  • What is their vocabulary? How do we take that into consideration? How do we teach the language?
  • How will I teach to students with different learning styles?  How do we explain well enough for all students to understand?
  • What do students find most engaging?
  • What variety of solutions do students come up with?

Content Specific:
  • How can I apply this stuff to life?
  • How does it [all this content?] all fit together?
  • What geometry concept is the most difficult for students?  What strategies do we teach to solve geometry problems? How do children do these when I used later knowledge?
  • How much geometry is there in the younger grades?
  • Is measurement hard for students?
  • How to teach formulas without just memorizing? Are there other techniques besides formulas for volume, etc? How do you break formulas down? How do you help kids memorize them?
  • How do you teach conversions so students can remember?
  • Quite Specific:  How do you teach to estimate? How do you teach congruence?  Do students use pi=3.14 or do they need to know more?  How do you find areas of arced shapes?
I'd be interested in ways you can think of making the assessment better, other data I could collect, or what you noticed about the results.

Friday, September 3, 2010

Preassessment, Part I

This is the longest I've gone without blogging in a while, maybe since I've started.  Feels very weird.  Definitely like I've been shirking.

Classes have started this week at Grand Valley, and I wanted to share from my two preassessments.  This first part is from giving the growth mindset survey to our preservice student teacher assistants.  The field experience for the professional program here is one semester in schools in the morning, where the TA teaches at least one unit for one class (though most do much more), and then a semester of traditional student teaching.  After our getting to know you time (Piece of Me plus student interviews of each other), I showed a Prezi on what do we mean by professional development (emphasis on student TEACHER rather than STUDENT teacher).

From the always funny Speed Bump.

Then we did the growth mindset survey.  (Developed with Sue Van Hattum.)  We discussed a few of the questions and made a human bargraph so we could see where people are.  A lot of agreement this semester, and a pretty reform minded bunch, I'd say.


So they're pretty countercultural about their beliefs about mathematics, and at least open to a growth mindset on the surface.  (There's lots of good research contrasting surface beliefs with the beliefs that cause actions.)  And they seem to be open to differentiation practices with how students learn math, strongly endorsing visuals, communication and questioning.  The question for Rebecca and I becomes how do we support and encourage them in this direction.  They are mostly placed in pairs (except for the odd man out), which we think will help, and did in the pilot pair placements.  I'm wondering if the hedging (mostly vs. strongly) is a sign of underlying belief vs. what we've been telling them in the university, or their developing belief vs. cultural truism.

Their comments about 10/15 and 18 were also interesting.

Pick one statement you agree with and explain.

8.  You can greatly change your ability to do math.
  • The biggest thing you can change about math ability is your mindset.  If you believe you can’t, then you won’t be able.  If you believe you can do it, then you will be able to do it.  Might take some extra work.
  • All things take help, and some take extra practice.
17.  Drawing pictures helps me to learn and do math.

  • Being able to see the situation helps me organize. 
  • Drawing pictures help me learn and do math.  They help me to see what is known and what  needs to be solved.
18. Explaining the idea to someone else helps me to learn math.
  • Explaining forces one to view math in different ways.
  • I have experienced this several times myself.  I want my students to have the chance to teach each other.
  • It wasn’t till I started tutoring calculus that I started understanding the underlying complexities and the power of it all.
  • You can say your answer and get it correct, but it doesn’t mean you know math.  You might know 2x3=6 from memorization, but if you can’t say why then you do not know math.
  • Explaining helps me check and see it in a different way.
  • After trying a problem it helps to clarify a question or reinforces my confidence.
  • Forces you to put your thoughts in a logical order and may force you to see the problem in a new light – how the other person sees it.  
  • Helps you learn because it is a good measurement of how well you know something.  It’s easy to explain to yourself, but explaining it to someone else helps a lot.
  • To explain it is to know it so well that you can approach the idea from any angle.
Pick one statement you disagree with and explain.
2.  How intelligent you are mostly determines how well you can do math.
  • There are many subjects and you can be intelligent without math.
3. How well you can memorize mostly determines how well you can do math.
  • Memorization gets a correct answer not understanding.
5.  Boys/men are better at math than girls/women.
  • Girls can do anything as well as boys. (Look around the room.)
9.  How fast you can get a correct answer is a good measure of math ability.
  • You might be the slowest person at getting an answer, but make less mistakes or just need time.
10.  The percent of correct answers on a test is a good measure of math ability.
  • Could be chance.
  • There are so many factors that go into taking a test.
11.  The is one right way to do a math problem.
  • Never is there one way.
15.  You need to focus on getting the right answer to do well in math.
  • You can understand each step and make a simple mistake and get the wrong answer. 
  • Students can learn a lot on the way of solving a probem.   
  • Math is about learning the process.  That is the most important thing. 
  • The right answer may be wrong from rounding error, etc…
Rate your mathematics ability from 0 (none) to 10 (could be a mathematician):

Let me state unequivocally: I have every confidence in each of these teachers' mathematical ability.  This is a measure of their self-perception, not their competence.

6 6
7 7 7 7
8 8.5
9 9 9

This makes me wonder about our schooling.  Why don't they all consider themselves mathematicians?  What damage have we done?  If a college math major with a 2.8 GPA minimum is a 5, what is a struggling high school student?

So, that's my data.  It has me very hopeful for the semester.  I'll be working to support them in their fledgling positive beliefs (hoping their data supports it, too), and in helping them transfer those solid math beliefs to teaching.

What do you see in this data?  I'd love to hear in the comments, if you have the inclination.