Pascal's Triangles and Distribution

Distribution of a perfect square often confuses students. When given (x+3)^2 their first impulse is to square x and square 3. It take a little work for them to see that it is instead (x+3)(x+3). After they get to this point they are typically pretty good at distribution using the FOIL method or the area model. My favorite is obviously the area model because it can be extended to other problems. I've posted about this method earlier.

Here's an extension for students to play with: Pascal's Triangle
Have students identify as many patterns as they can. Ask them to add a new row. Can they make their own triangle with a different rule?  Every time I do this, I learn something more myself about this cool triangle.

          1             (x+y)^0    1
        1  1           (x+y)^1   x+y
      1  2  1         (x+y)^2   x^2 + 2xy + y^2
    1  3  3  1       (x+y)^3   x^3 + 3x^2y + 3xy^2 + y^3
  1  4  6  4  1     (x+y)^4   x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4
1 5 10 10  5 1   (x+y)^5  . . .

Math is Beautiful

My senior honor's thesis in college was on the relationship between math and art. I find the parallels between the two fields fascinating. Just think of the various math classes you might take in college: Abstract Algebra, Modern Algebra one can't help but think of art movements. Now I might have a difficult time relating Abstract Algebra to Abstract Art.  I would have a pretty easy time relating the progress math was making in the Renaissance to the progress seen in paintings.

For instance, thanks to Filippo Brunelleschi in the 1400's we have linear perspective! This is basically what happens when you take the last postulate in geometry (parallel lines never intersect) and allow parallel lines to intersect at infinity. Then go about building a new geometry based on those assumptions. It's both mathematically valid and an amazing tool to depict distance.

Hopefully, you can picture a set of railroad tracks and how someone with no knowledge of perspective would draw them. They would most-likely draw them with parallel lines never intersecting and with evenly spaced ties. Which makes perfect sense because the don't intersect and they are evenly spaced. But in the real world they don't look that way.

Once we apply this new postulate (parallel lines intersect) we get a perspective drawing. We can see in the first picture that the two parallel lines intersect at the horizon (infinity or vanishing point). Using another set of parallel lines (in red) that connect diagonals you can approximate accurate spacing.  Usually art teachers neglect to incorporate a how to measure spacing when teaching perspective drawing. In art class it's usually good enough that there is a vanishing point and that stuff of the same height is in line. But in true perspective as things recede into the distance they get smaller and they also get closer together.


We can do one better. Instead of the diagonals following the old rules of parallel lines (intersection bad). We can have those lines also intersect at infinity. To do this we need our horizon line (Our line at infinity.)

If we continue to extend this idea, we can get a accurate tiling of a floor.  See below.  Look at all of those lovely parallel lines intersecting! The red lines are only in there as diagonals, used to measure spacing. In art they should be erased. In math, we would call that showing your work.
 

Graphing Calculators

I began using a graphing calculator my freshman year of high school. It was a TI-82. I remember being impressed with its ability to quickly graph equations. I also liked how I could see my previous work on the screen and edit typos.

Prior to working with a graphing calculator I had to do my graphs by hand and calculations one step at a time. I would often make a mistake and have to start from the beginning. This slow step-by-step process is a fine and important step in developing a solid foundation in math but it does take a lot of time and is not necessary after those skills are well developed.

Now with a graphing calculator, I could investigate quickly the structure of an equation. Getting to understand not only how to graph but learning all that a graph can tell you.  What happens when you take your base equation and add 3 to it? Will it shift left, right, up, down, stretch, compress, flip...?

I'm a visual thinker. So seeing equations as graphs and seeing situations presented graphically made math make so much more sense. I remember that while sometimes I would get flustered by an equation, I could always figure out how to solve it using technology. Even if I didn't yet have the algebraic skills, my familiarity with technology and graphing made it so I could still figure out a path forward or verify my work. If I had forgotten a rule, I could do a quick investigation by running through a few easier examples until I figured it out again. This made math seem more like a fun challenge in which I had multiple tools for and made the experience a whole lot less stressful.

Throughout high school I continued playing around with my calculator. It wasn't long before I realized it could be programed and those programs could be shared with other students. That opened up both the world of calculator games and my interest in learning how to program. My teacher's always allowed us to have graphing calculator for our tests. So my friends and I would create and share physics programs, games and math programs. A few of them continued on into computer science. I bet all of them still have a graphing calculator at home.

The Area Model for Multiplication

The algorithm for multiplication, I'm sure most adults today can remember learning. And I'm sure a few even understand and appreciate why it works. I personally like it and find it kind of cool. However, there are other methods that are more intuitive and explain how multi-digit multiplication works.

The Traditional Method
It begins with you stacking your two numbers on top of each other. You then proceed to multiple the first number on the bottom to each of the numbers on the top. You then put down a zero and proceeded to the second digit putting that product below the first one. Once you did this to each number you then found the sums of each of those products.


Common misunderstandings/confusions with the traditional method:

• Why do you "carry" when the product is more than 9?
• Why do you add what you "carry"
• Why do you add the products at the end
• Why do you add a zero each time you move onto the next number?

The Area Model 
With this method you begin by breaking the numbers into their hundreds, tens and units parts. You then label each side of a rectangle with those parts, creating smaller rectangles. You then find the product for each smaller rectangle and add them together. Below is an example of 286*43.


Benefits of this Model:

• Relates to the study of Area and Perimeter
• Reinforces the base-ten system
• Easily extended to Algebra concepts like Distribution and Factoring
• Students understand why they are adding each of the products together

How I Teach

I am a "constructivist" teacher because I am a "constructivist" learner. I always needed to know why something worked. If I didn't understand, chances are I would soon forget how to do it. Memorizing is a temporary ineffective solution when the true goal is learning. It might get me through a spelling test but it is no way to learn math. Tricks and over reliance on memorization is my main critique of traditional math education. You might see quick gains but those gains are temporary and not gains that can be built upon.

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When you instead construct your understanding from previous knowledge that new understanding is lasting. It also is easily transferable to other topics. Math is beautifully setup to allow for this construction. It is also far more enjoyable to learn and teach in this way.

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I also naturally look for applications to what I'm learning or teaching. Being able to apply new information helps with authentic incentives. Students want to know why they are learning something for a good reason. "When am I ever going to use this?"  is a valid question.

The Problem with Algorithms

What is an Algorithm Jess? 
It's basically a set of rules to be followed to perform a calculation. They are great for computers. They are easily performed and consistently give the correct answer.

What's Wrong with Teaching Algorithms?
Basically, students are not computers. Any parent can confirm that children are not designed to take inputs and generate consistent outputs. People make mistakes, they have deeper thoughts, they interpret findings, they think "what if this was different," they forget and they get bored.

By teaching algorithms we are teaching only how to get the answer. We are not teaching how math works and the structure/logic supporting the process. When we treat student like computers, we forget that interpretation is hugely important. We also do not allow for the student to understand how the answer came into being. At best, a student has a surface understanding of what just occurred. They view math as a series of boring rules to be memorized and not as the beautiful logical structure that it is. When you approach math as being a series of interconnected topics student can fill in missing pieces with a little problem solving. They can construct there own understanding and continue to build on that understanding. We can given them those tools that allow for independence in math but not if we keep teaching tricks and have them memorize rules.

Examples of Algorithmic Teaching

• Cross Multiplication: every time students see two fractions, whether there's a equals sign in between or not they want to cross multiply. Instead we should be teaching inverse operations and solving equations.
• The Triangle Trick: This is a new one for me that I was introduced to by a science teacher. Apparently, we are unable to teach how to solve one step equations in science classes. So F=MA, is just too difficult to work with if I asked a student to solve for the mass.  So they teach students how to put this equation into a triangle that somehow makes it easier to understand. Here I thought dividing on both sides was straightforward enough. 
• The Distance Formula: Why memorize yet another formula when the Pythagorean Theorem will suffice? 
• Long Division: There's a few other methods that reinforce what your actually doing when you divide.
• Almost any Volume Formula: If you know what volume is and you can find the area of a triangle and circle your pretty well set.
• The Quadratic Formula: I taught my student how to complete the square and then proved the quadratic formula using that method. Then we timed students doing both methods. Completing the square usually won. I do really enjoy the Quadratics Formula song, so I will continue to teach it. 

I could go on and on. When I look back at when I've enjoyed learning or teaching I was not memorizing or practicing a process. When I've enjoyed math the most I've been figuring something out, asking a question I don't know the answer to, extending a topic and going down the occasional wrong path.

Why I Teach

Growing up I wanted to be a lot of things. When I was very young I wanted to be a monkey that worked in a doughnut shop. By high school I was convinced that I should be a artist. In between it was a teacher, a botanist and a physicist. Even after I graduated college with a teaching license in math, I wasn't convinced that teaching was my route. I even tried my hand at bike mechanics for a bit.

At first I tried to avoid teaching and was given sound advice not to go into the profession by teachers and mentors who I looked up to. There are a few reasons why I questioned the profession:

• The first reason is based on how teachers are viewed and treated by society. They are not viewed as a professional and they are not treated as one. For example, there is an expectation that they volunteer their time. Summers are described as vacations instead of unpaid time. They are described and treated as babysitters. They sit through hours of professional development determined by some outside body. The saying "those that can do, those who can't teach" encapsulates society's view of this profession.
  
• Their professional degree is also not valued equally. The average teacher makes roughly $54,000. Whereas, the average professional with a masters degree is making about $90,000. People will say that's because teachers get summers off. So here's the math: $54,000/190 is $284/day, on average teachers are working 50hr/week so that makes $28.4/hr.  If teachers were not salary but instead worked 40 hours per week at that pay rate for 261 days (number of workdays in a year) they should make $59,300. Which is still far less than what their professional peers. 

This view and treatment of teachers has the effect of pushing effective, professional, intelligent, loving teachers away from a field that desperately needs them.

Why do I teach? I teach because I can't help myself. Many of my aunts, uncles, grandparents have shared the same affliction. Teaching is fun. Working with a student who wants to learn is fulfilling. Working with a student who doesn't want to learn is a challenge, and without that challenge I might get bored or complacent.