Teaching math through Coding

I recently started teaching Computer Science 10 and 20 and I use the program Processing.  It is a free program and entirely based in a geometric space.  The cross curricular links in this program are amazing!  I want to share how my Grade 10 students were introduced to higher level math concepts while working with this program.

First, here is a program Sean wrote:

int[] numb = new int[5];
void setup() {
  size(800, 800);   background(255);   numb[0]=0;  numb[1]=200;  numb[2]=400;   numb[3]=600;  numb[4]=800;
}
void draw() {
  line(numb[int( random(0, 5))], numb[int( random(0, 5))], numb[int( random(0, 5))], numb[int( random(0, 5))]);
}

The picture it creates is:

Now in case you don't understand processing what is drawing does is takes the numbers 0, 200, 400, 600, and 800 and creates a line from all possible co-ordinates created from these numbers to all other possible co-oridinates.  For example a line from (200, 600) to (800, 800).  It does it in a random pattern, but after running for some time all possible lines are drawn. 

After Sean drew this I asked him "How many lines have been drawn?"  This is a typical Math 30-1 question, a course in which Sean has never been in yet.

After some thought he asked if it would be "5 times 5 times 5 times 5 times 5?"  or 3125.  This is of course, a great way to start the problem but is too high as you can't have a line from (0,0) to (0,0).  Also he didn't account that the line from (0, 400) to (600, 800) is the same as the line from (600, 800) to (0, 400).  At this point the bell rang and we will finish the conversation tomorrow.  However in Grade 10 Computer Science he was introduced to a Gr 12 Math concept called "Fundamental Counting Principle" and "Permutations and Combinations".

Next was Ex who wanted to create a scene where a sun rises and sets. His original project had the sun follow a straight line to the top of the screen and then follow a straight line back down to the horizon.  Following a "^" shape in the sky.  This of course is not how the sun moves, as it would move more in a parabolic shape. 

Unfortunatly, Ex has only taken Math 10 and not have heard of a "Parabola".  Consequently, I sat with him and we played with his code.   Instead of it following "y=-x+10" I asked him to put in "x^2" and to watch what will happen.  Instantly he was surprised to see his sun move in a different fashion than before.

He asked how do we move the sun right in the sky, as he wanted the sun to be at the highest point in the middle of the screen.  What he was asking was "How do we horizontally move the parabola?".  Again this is Math 30 concept.  Through some guided discovery, Ex realized that by replacing x with x-h we move the parabola left and right.  

 Here is his final code.

int xPos=0; float xPos2=260; int positionX =50; int positionY = 100; int Switch = 0;
void setup() {
  size(500, 500);  smooth();
}
void draw() {
  background(130, 200, 255);  fill(255, 238, 21);  ellipse(xPos, xPos2, 100, 100);  xPos=xPos+1;
  xPos2=0.005*(xPos-260)*(xPos-260);
  if(xPos<=0){
    background (0);
     }
     
  noStroke();  fill(15, 80, 0);  rect(0, 300, 500, 400);  fill(40, 40, 40);  rect(200, 230, 100, 70);
  fill(65, 65, 65);  rect(235, 250, 30, 50);  triangle(300, 190, 300, 230, 202, 230);  ellipse(240, 280, 5, 5);
  fill(53, 43, 32);  rect(140, 230, 20, 70);  fill(6, 62, 0);  ellipse(150, 220, 60, 60);  fill(112, 112, 112);
  rect(0, 355, 500, 100);  fill(191, 191, 9);  rect(0, 400, 50, 10);  rect(75, 400, 50, 10);  rect(150, 400, 50, 10);
  rect(225, 400, 50, 10);  rect(300, 400, 50, 10);  rect(375, 400, 50, 10);  rect(450, 400, 50, 10);
  }

Math in Google Docs

How to use LaTex and Google Docs together.

Cross Competencies in Alberta

In 2016, the Alberta Government is going to remove the "silos of learning" occurring in our schools.  No longer will only the English teachers teach literacy, and the Math teachers teach numeracy.  Here is an example of a problem that a student could face, with examples of how different grades could react to this problem.

Your community is planning to build a new recreation centre and is looking for residents of the area to share ideas.  You have the opportunity to offer your suggestions to the planning committee.  Think about the activities you would like to do at the centre.  Research what other communities offer at their recreation centres.  Considering the needs and interests of your community, select a format that will best communication your ideas to the planning committee.  Use your research to support your ideas.

Examples of how certain grades could address this task.

Grade 1- "We looked at pictures of really cool rec centres.  Then we drew pictures of things we want in our new rec centre like indoor soccer fields and rinks for learning to skate."

Grade 8- "I worked collaboratively with my skateboarding friends to create a PowerPoint presentation for the committee.  We would like a skateboarding park because we need a safe place to ride and learn new tricks."

Grade 12-"I wrote a speech advocating for a public library in the centre, recorded it as a podcast, and submitted a copy to the planning committee"

As you can see that the passions and interests of the students are just as important as the learning outcomes and content of the lesson.  Also you can see how many subject areas could be involved in this problem.  In 2016, cross-curricular competencies will be implemented.  Here is a diagram of the competencies.

How will these fit in with "Learning Outcomes" and the "Literacy and Numeracy" Benchmarks?  Here is another diagram showing the connection.

After another day, with over 100 other educators, I have learned more the 2016 Alberta Curriculum redesign.  As always, I have to say I am excited.  First, here is the implementation timeline

Another focus of this drastic change will be to address that the following are lacking in our current curriculum.

• Growth of the whole child.
• A student centred approach.
• Personalized learning.
• Supporting teacher decision making.
• Providing consistency across subject/discipline areas and in assessing and reporting.

The plan is to address these problems by ensuring that all students K-12 achieve the following cross-curricular competencies:Now, of course, not everything is going to change and here a list of what will remain the same.

• Alberta Education will still determine what students need to learn.
• There will be grades and subjects.
• The basics will remain.
• Teachers will continue to teach.
• There will be some form of provincial assessment.

The key here is that students will now face problems where they must use strategies taught in Math, Science, English, and Social together.  Currently, we have "silos" of learning occurring, since what is taught in Math has very little relevance to the lesson taught in Social Studies only one hour later.  

Lowering Standards or increasing classes?

I think it is time to go back to traditional assessments.  Why? I am tired of large class sizes.... See, larger class sizes are the result of changing assessment.  

First, I abolished grades in my class, then instead of standardizing assessments, I actually personalized my assessment after some time I removed deadlines for assignments.  Why did I do all this?

I did this because my failure/drop rate in my calculus classes was extremely high.  The first 2 years, of teaching Calculus, I had a failure/drop rate of 40-50% of the class.  I would start with classes around 38 and end with classes around 18.  In one class, students had bets on what would be the final number of students.  This had to stop!  

Over the course of 2 years, I realized that my teaching was not the problem; it was how I assessed students.  I made all students know the material by Friday, assessed with a Multiple Choice, Written Response exam, and never let a child have a chance to be reassessed. In addition I would assign over an hour of homework each day.

Below is the result.  If there is one line you look at, it should be the orange one.  The orange line is the percentage of students who have failed/dropped my calculus class in each year.  The time is over 4 years.

I currently have a drop/failure rate of 4-5%.  I do believe I can get this to 0!  Was I joking about going back to traditional assessment? Yes!! Was I joking about classes being larger? No!  However, this is not a bad thing!  Here are what the other lines are

Light blue-The percentage of "traditional assessments" I use in my class.
Green-The class average on my Final Exam (This has been the constant over the 4 years)
Purple- The final class average.

Results:

• The number of traditional assessment is directly related to my drop/failure rate. 

What is also pretty cool is you can see, by the green and blue lines, that the "standard" or "average" of my class has not dropped significantly!!  In fact, my class average has increased.  More kids completing the course and even a higher average....Remember these kids are not doing homework, prepping for exams, or completing worksheets.

There was one year, in which the Final Exam marked dropped, as it was due to the fact that I was perfecting my open ended projects.

Conclusion: If you want small class sizes, please use traditional assessments.  If you want a low drop/failure rate, please click on the links at the top and learn more.

Dealing with Change

The following is from http://www.enablingchange.com.au/Summary_Diffusion_Theory.pdf

When something is changing you need to realize the following occurs.

Innovators: The adoption process begins with a tiny number of visionary, imaginative innovators. They often lavish great time, energy and creativity on developing new ideas and gadgets. And they love to talk about them. Right now, they’re the ones busily building stills to convert cooking oil into diesel fuel and making websites to tell the world about it. Unfortunately their oneeyed fixation on a new behaviour or gadget can make them seem dangerously idealistic to the pragmatic majority. Yet no change program can thrive without their energy and commitment.

How to work with innovators:
• Track them down and become their “first followers”, providing support and publicity for their ideas.
• Invite keen innovators to be partners in designing your project.

Early adopters: Once the benefits start to become apparent, early adopters leap in. They are on the lookout for a strategic leap forward in their lives or businesses and are quick to make connections between clever innovations and their personal needs.

How to work with early adopters:
• Offer strong face-to-face support for a limited number of early adopters to trial the new idea.
• Study the trials carefully to discover how to make the idea more convenient, low cost and marketable.
• Reward their egos e.g. with media coverage.
• Promote them as fashion leaders (beginning with the cultish end of the media market).
• Recruit and train some as peer educators.
• Maintain relationships with regular feedback.

Early majority: Assuming the product or behaviour leaps the chasm, it may eventually reach majority audiences. Early majorities are pragmatists, comfortable with moderately progressive ideas, but won’t act without solid proof of benefits. They are followers who are influenced by mainstream fashions and
wary of fads. They want to hear “industry standard” and “endorsed by normal, respectable folks”.

How to work with the early majority:
• Offer give-aways or competitions to stimulate buzz.
• Use mainstream advertising and media stories featuring endorsements from credible, respected, similar folks.
• Lower the entry cost and guarantee performance.
• Redesign to maximise ease and simplicity.
• Cut the red tape: simplify application forms and instructions.
• Provide strong customer service and support.

Late majority: They are conservative pragmatists who hate risk and are uncomfortable your new idea. Practically their only driver is the fear of not fitting in, hence they will follow mainstream fashions
and established standards. They are often influenced by the fears and opinions of laggards.

How to work with the late majority:
• Focus on promoting social norms rather than just product benefits: they’ll want to hear that plenty of other conservative folks like themselves think it’s normal or indispensable.
• Keep refining the product to increase convenience and reduce costs.
• Emphasise the risks of being left behind.
• Respond to criticisms from laggards.

Laggards: Meanwhile laggards hold out to the bitter end. They are people who see a high risk in adopting a particular product or behaviour. Some of them are so worried they stay awake all night, tossing and turning,
thinking up arguments against it. And don’t forget they might be right! It’s possible they are not really not laggards at all, but innovators of ideas that are so new they challenge your paradigms! In the early stages,
where you are focusing on early adopters, you can probably ignore the views of laggards, but when you come to work with late majorities you’ll need to address their criticisms, because late majorities share many of their fears.

A great video to follow this up: http://www.ted.com/talks/simon_sinek_how_great_leaders_inspire_action