Empowering Students and Teachers - Freedom Within Limits (Part 2)

Task-oriented classrooms can provide an appropriate balance between freedom and limits.  When students are given a task, they are given the freedom to learn.  If that task is too easy, they won't need a teacher!  If the teacher tries to insert themselves into the conversation, students will feel that their progress is impeded by the intrusion.  Tasks that are just a little beyond their reach require students to rely on the instructor while at the same time conveying the responsibility that learning is their responsibility.  Students who are not accustomed to this type of learning may rely too much on the instructor out of habit.  However, with a little encouragement, students can quickly assume the role of responsible learner.

Traditional Math Teaching vs. Effective Math Teaching

I hate to be so blunt, but the title explains it all and how I feel about traditional math teaching.  What's the difference?  To me, the key difference would be in the foundation of each philosophy.  It is there that we part ways.

Traditional teaching, in a nutshell, believes that all knowledge can be ordered sequentially and split up into topics.  Then, in order to get that knowledge from teacher to student, the teacher simply tells the student what the concept or skill is and demands exorbitant amounts of practice.  Proceed to the next concept.  Supposedly at the end of this exercise, the student is now competent.  One problem is that the student is simply doing mindless tasks over and over again.

It's not that telling students things cannot produce understanding.  It can.  But in order for that to happen, the student has to have fired neural circuits that relate to that information before it is given to them.  Even then, I believe that they need as reasonable effort as possible before they are given conceptual information.  I'm not talking about conventional knowledge.  Yes, just tell them that.  How would they ever figure that out.  What I am talking about is conceptual breakthroughs that took mathematicians decades, centuries, or millennia to discover.  So proceed with what is reasonable.

Effective math teaching requires students to do the work of discovery and conceptual construction.  The learning environment will be carefully constructed so that students feel that they did all of the work.  When students feel like they did all of the work of discovery, mindsets change.  Confidence grows.  Little do they know the teacher spent ridiculous amounts of time designing the learning setting for that occur.

Those familiar and comfortable with traditional teaching will ironically say that the teacher isn't teaching.  It takes time for students, parents, and administrators to shift their paradigm to see that the status quo is all for show.  When students begin to excel with effective math teaching methods, they will resent stepping foot into another traditional classroom.

Exploring Task Design

Wondering what will come of picking a few random secondary math topics.

1. Area of a Circle - The area of a circle is pi(r^2):  How many 1 x 1 units fit inside of a circle?
2. Pythagorean Theorem - Given a right triangle with sides a, b and hypotenuse c, a^2 + b^2 = c^2:  
1. The Egyptians used the 3,4,5 fact to make buildings square and many other peoples knew about the relationship over the past few thousand years.
2. I think this theorem really captures the essence of mathematics and what mathematicians do.  How in the world did anyone come up with that relationship?  It seems the best answer to this would be an interest in the study of triangles and relationships between side lengths.  Yet, SO many applications!  What can we discover about the properties of triangles?  are there any relationships between side lengths? angles?  As a mathematician embarks on this quest, this surprising result emerges.  Then they seek to prove the relationship they observe generally.
3. Exponent Product Property - a^n * a ^m = a ^(n+m):  
1. This operation also seems to get at some of the core ideas behind the work of mathematicians.  So much work just to get to this result.  What kind of foundational work led up to this discovery?  How can we simplify relationships between varied operations or functions?  What kinds of properties do operations have?
2. Using this operation with Natural Numbers, this almost seems definition-like.  

What can be learned from this exercise?  Students can't experience what it's like to be a mathematician without building theorems or properties from scratch.  The Pythagorean Theorem just can't be taught in a couple of days.  It has to be part of some larger picture and exploration.  The question(s) asked need to be a natural outgrowth of previous explorations.

This is a tall order.  In order to provide such an environment, the teacher needs to have a strong background in the development and progression of mathematics, or at least the designer does.   

Designing Curriculum to Align with Current Research

I came to the realization today of why I have struggled so much to create quality math instruction.  In my previous attempts to design tasks, I have reasoned through the lens of traditional teaching and learning.  This has made it almost impossible to design tasks according to current educational research.  I have been trying to reconcile the two.  It can't be done.

I discovered this as I reasoned through what I would call a great task for the 21st century math classroom (the open math task below):

• Traditional:  What is the area of a 6 x 8 rectangle?
• "Real" math:  Why is the area of a 6 x 8 rectangle 48?
• Open math: How many rectangles can you find with an area of 48?

I would categorize the open math task as a skill development task.  A conceptual development task might be: How much space is contained inside of a rectangle?  How many 1 x 1 squares fit inside these rectangles?  Any rectangle?  Why will the are of a rectangle always be length x width? 

Both types of tasks are necessary for students to grapple with for the construction of knowledge.  Concept development tasks come first, as they do not require desired know-how to complete.  Students may or may not find something.  It is an exploration.  Skill development tasks require repeated application of know-how to complete.

Traditional math education compiles all of the theorems, definitions, tools, and properties that have been discovered and arranges them in a supposed sequential order of difficulty.  The teacher then tells the pupil what was discovered and prepares countless exercises for acquisition of that knowledge.  It doesn't work.  Why?  Because what the student is doing is not even remotely similar to what actual mathematicians do or have done.

Why all those pesky definitions to memorize?  What's the point in knowing that a^2 + b^2 = c^2?  How could anyone have discovered something so seemingly random?  Why did mathematicians even ask the questions they asked that led to discovery?  What do mathematicians do?

I believe that every theorem and property answers a question or multiple questions.  These are the questions that give meaning and life to mathematics.  It seems an impossible task to teach others mathematics without actually engaging in the work of mathematicians.  

Once the question(s) that could or should be asked is/are discovered, how then does the teacher use this to improve mathematics instruction?  Students should experience some of the real angst that mathematicians have felt through the ages.  Careful scaffolding can reveal some of the questions that could be asked as they learn how to think like mathematicians. 

In subsequent posts I will explore the creation of some secondary math tasks.