Math Hombre

Our standard (non-thesis) capstone is a course called The Nature of Modern Mathematics. For me, this is a math history course. 

Our essential questions:

• what is math?
• what is its nature? (Is it invented or discovered? Is it completable? Is it beautiful?)
• what are the important ideas of math?
• how do I do math?
• what is the history of math?
• who made/discovered math?
• what are the important milestones?

• what do mathematicians do now?
• who are they?
• what are the big open questions?

I love teaching this course. 

The first assignment is a pre-assessment of sorts, asking them to start blogging with a short post on what math is and what are the milestones they know about.  Given their responses, I think we can see that this is going to be a good semester. What have college majors learned about math? We have about a third future elementary teachers, a third secondary teachers, and a third going on for graduate school or the corporate world. You might be able to see a stong influence of calculus courses, geometry and discrete mathematics. 

The amazing Ben Orlin

This blogpost is in case you would find what they think about math interesting, or if it might start you thinking about what your students think about math. I sorted their responses by my own weird classifications.

Here is the list of all their blogs. If you read just one, try Brandon's.

• patterns
• about trying to find universal patterns that we can apply to infinite situations or problems.
• a way of thinking about patterns throughout the universe. Math is interpreting and studying these patterns to find more patterns.

• about pattern recognition

• the study of patterns in the world and in our minds and how they connect to each other.

• a tool
• all the computational things we learn throughout life, but it is also a tool and language humans use to make sense of the world around us.
• a collection of tools that we use to quantify and describe the world around us. We use mathematics very similarly to how we use language. Using language, we can identify objects, convey ideas, and argue. Math can be used in the exact same way when communicating scientific ideas, defining mathematical objects, and proving theorems. The most interesting relationship between language and mathematics is that both can be utilized to describe events and objects that do not exist in the physical universe.

• logical science
• a framework we use to understand, and like science, it is not reality itself
• the study of everything around us. It is how we quantify structures. It's a science that deals with logic. It is a measurement of the physical space around us. It is so much more then just a simple discipline or school subject.
• a logical way of explaining everything in the world and you can find math everywhere you go
• a quantifiable way to explain physical phenomenon but also includes ways to predict imaginary situations.
• a numeric and logical explanation of the world around us.
• our human desire to give order and regularity to the world.

• a language
• a language used to study and discuss patterns found in nature.

• using logical and analytical thinking to derive solutions to the problems we see from all directions
• the use of objects that have been given accepted values and meanings to help us to quantify the world around us.

Things We Forget

• context.  Math gives us a common ground from which to clearly and accurately communicate with the world.  Math transcends language.
• much more than just numbers, it can be used theoretically to answer some of worlds most unexplainable phenomenon. We are in the age of information where researchers and engineers are making breakthroughs everyday using advanced computes powered by mathematical formulas and theories.
• a way of explaining what happens around us in a logical and numerical way, but there is also so much more to math than just numbers and logic.  New discoveries in mathematics are occurring all the time to describe anything and everything about the world, and with these the definition of math is growing as well.  So for me, the best way I could define math is by likening it to an infinite series, how mathy of me.  Just like with the next term in the series, each new discovery broadens the scope of mathematics and as a result the definition becomes that much different than before.
• literally everything

The brilliant as usual Grant Snider

• x 3 Number
• x2 counting
• Egyptian numeration
• zero as a number
• the acceptance of i as a number
• the acceptance of irrationals as numbers
• x2 e
• x2 pi
• x3 Measurement
• Quantifying time and number systems in Egyptian times
• a definite monetary system
• x4 number operations (+, –, x, ÷)
• proportional reasoning
• functions
• The coordinate plane
• x2 the discovery of infinity

• x2 Proof
• when mathematical concepts could be argued and verified through what we all now recognize as a proof.
• the first math proofs for example the geometry proofs by the Greek mathematicians
• x2 the power of communication
• symbols
• how to communicate what we know to others outside the math world
• The movement into abstraction.

• x7 geometry
• x2 pyramids
• x3 non-Euclidean
• x3 algebra
• x2 to predict, plan, and control the environment
• ballistics
• x2 trigonometry
• x5 calculus
• the computer age of statistics

Usually he says "practice"! (Sydney Harris)

• Pythagoras and his theorem
• x7 Euclid
• x4 Elements

• way to prove concepts and communicate mathematically

• Al Khwarizmi
• Galileo
• Descartes
• Newton and his Laws
• Leibniz
• Blaise Pascal's invention of the mechanical calculator

• x4 The Pythagorean theorem
• the realization that the Earth was round and not flat
• x3 Euler’s Identity
• (I swear this is the closest thing the real world has to magic.)
• The Nine Point Circle
• The Seven Bridges of Konigsberg
• Euler’s Method

If you want to answer those questions in the comments, I'd be fascinated. Or if you want to share what you notice about their responses.

I shared before about the math learning inventory I sometimes give. The greater the diversity of learners, the more inclined I am to give it. It surveys how students think they learn mathematics.

A basic break down of the learning preferences assessed follows from a Silver, Strong, Perrini and Tuculescu article, The Thoughtful Classroom:Making Students as Important as Standards.

• Mastery learners want to learn practical information and procedures (what)
• Interpersonal learners want to dialogue and collaborate (how)
• Self-expressive learners want to use their imagination to explore (how)
• Understanding learners want to learn why things work (what)

I do not think learning styles are determined for a student, but finding out what students think they prefer seems like a worthwhile preassessment to me. If I want to be a culture changer, I need to understand from what I wish to progress. The form I use is at Scribd, but I'm happy to email the Word document, too.

I thought I'd share how this semester's group of preservice elementary teachers look, and how that affects my planning. Mostly it doesn't change what I want them to experience, but it has a reasonably big impact on how I implement it. Also, with an education class, it's a chance to point out the tension between surface and core beliefs.




Especially worrisome is when you compare my inventory with their composite. At the conference where I first saw this inventory I had a number of students.  They divided us up by table based on the results and I was alone! So I'm aware of my distance from students on this and how it can affect my persective.


Whoa. All over the place. This is a caution to me to not require group work always, and to look for ways to make space for some individual problem solving. Since relationship with teacher is part of this, I'm trying to be more personal to supplement for the people at the top end of this scale.


Very encouraging. Many times when learners have had negative math experiences this category can be quite low. I have to think about how to support those four students, though. So much of our class revolves around this kind of why thinking. And I feel that it is crucial for future teachers. I hope to have experiences that justify this approach to math. Maybe the way conceptual understanding furthers skill is the entry point?
 


Post Script: went through the survey with my 8th grader - very interesting discussion.  Discussing why he was answering what he did on the individual questions was enlightening. It makes me wonder about using this as an interview tool. Even for a student I know pretty well, we got to some new ground.

Post Post Script:  forgot to add the other questions' data. Even if it's mostly for fun, some illumination is to be had.

Motto

• 16 - Show me how and let me practice.
• 2 - I want to know why.
• 1 - Let me play with it. (My choice)
• 0 - Let’s talk about it and hear everyone’s ideas.

Assignment

• 10 - p. 101: 1-39 odd
• 6 - Draw a picture that shows the ideas. (My choice)
• 3 - Work with a group to make a math skit.
• 0 - Report on a math controversy.

I have to work to understand the mindset of 1-39 odd. Other than comfort with the situation you know.

Boy, it's been hard to find time to write lately.  Hopefully I can get a teaching and a more mathy post up today.

I was first exposed to the Mathematical Learning Styles Inventory last year at Math in Action, our local math education conference.  (It's the end of registration season for that - if you're close to Grand Rapids, MI give it a thought if you could come present!  Fillable pdf speaker form.)  The teaching center folks from Central Michigan University presented it, and had us take it and discuss.  An interesting bit for me was having a group of my students there.  When they had us move to tables based on our strongest style, all my students were seated at my lowest style!  Hmmm - was I providing them with appropriate work and activity.
I know learning styles are a controversial topic in some arenas.  But I think of them as preferences and predilections.  I do not see them as exclusive or predetermined or limiting.  As with most assessments, they provide data about what your students prefer or think they prefer.  They might indicate areas where I need to provide more or more explicit support for my students.  Hopefully they indicate areas where students can become stronger with more experience and application.

Strong, Silver and Perrini do a good job at laying out their inventory in the 2001 ED Leadership article, Understanding Student Differences.  One of the things they describe are five great suggestions from their research:

• Have simple, deep standards 
• Differentiate
• Increase the role of assessment and feedback
• Start writing curriculum that appeals to students
• Collaborate with colleague

They have examples and exposition of all five points in their article.  They introduce their learning styles in the section on differentiation.  A basic break down follows from a Silver, Strong, Perrini and Tuculescu article, The Thoughtful Classroom:Making Students as Important as Standards.

• Mastery learners want to learn practical information and procedures (what)
• Interpersonal learners want to dialogue and collaborate (how)
• Self-expressive learners want to use their imagination to explore (how)
• Understanding learners want to learn why things work (what)

I dislike their names for the styles, but what can you do.  Notice that two are really about what objectives you're shooting for, and two are about how you reach them, although all four styles have elements of both.  Their inventory is 25-ish multiple choice questions with an option for each style, that you give 5, 3, 1 or 0 points.  (How do they calibrate these?)  Their subsequent research has shown some strong validity to the types, and correllated academic struggle with instruction mismatch with learner type.  In particular, how learners that struggle in mastery classrooms are often remanded to remediation with an even stronger mastery-style.  Turns out that is ineffective.

I reworked the inventory for an inservice with middle school teachers because I thought that most secondary students would see it as repetitive and onerous.  Especially if they have to score their own inventory.  Also, people often complained about hard choices and having to choose between equally weighted (to them) options.  I thought about just giving nine points to distribute among the choices, but that would take even longer for students. (Although I like the elegance.)  So ultimately I decided to break up the questions and have students indicate strong, partial or dis- agreement by giving items 2,1 or 0 points.  That turns this into an informal assessment vs. a research-based inventory.  Sorry!  But I can share:



Math Learning Inventory (Adapted)

I gave it to my preservice K-8 teachers, and they had interesting results.  The different structure seemed to lead to more balanced styles numbers, though still indicating a preference.  It sounds like some of the middle school teachers are going to give it to their students, and I'll share their feedback on the assessment if they do.

Of course, if you try it with your students, I would love to hear how it goes!

I also made an easy data recording sheet to get a glimpse of your students' results quickly. (Plus bonus line plot lesson!)  Out of my cold-depleted class, I had strongest traits as follows: 7 mastery, 9 Inter-personal, 0 Understanding and 10 Self-expressive.  (Ties I counted in both categories.) I expected more mastery learners, as that's what most math teachers tend to be.  I personally have high scores in everything but mastery, so you can see why I worry that my preferences keep me from seeing my students.  But even more interesting (and relevant for planning) to me were the distributions.

The relatively high interpersonal scores fit with the engagement level during discussions, and the relatively low understanding scores fit with the lack of engagement when more formal reasoning is the topic.  The split on self-expressive seems to fit my crazier assignments, where some dive in and some ... do not.  I tend to struggle with what to do for mastery learners, as that doesn't fit my view of mathematics well.

I do want to reiterate what I put on the assessment: This shows the ways you are comfortable to learn math now. People can learn new ways to learn math and try ways that other people like. Everyone can learn to do math better.

In other words, remember the importance of a growth mindset.

PS> I've added a new post describing my use of this inventory and the results from a group of preservice teachers.

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?

Students:

• 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.

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.

Teacher, Teach Thyself on Prezi

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.

5
6 6
7 7 7 7
8 8.5
9 9 9
10

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.