Friday, November 28, 2014

Teachers; more than "teach"ers

Julie changes one student every day as he is mentally unable to change himself.  She has to wipe, clean, undress and dress this student every single day.

Jason, after school, coaches 15 boys basketball.  He has to plan drills, prepare his students both mentally and physically for the next game, book hotels and busses, and ensure each player is caught up in all classes.

Lindsey listened in silence as her student confessed that one night she drank too much and danced naked in front of other students.  After calling a counselor and organizing a third party psychologist to come in, Lindsey watched for days as this student dealt with turmoil, bullying, and public ridicule.  Through various supports, which Lindsey organized, this student over came this incidence and regained her mental health.

Chris wakes up every morning at 5 am to ensure he is at school to drive the cross-country running team on the bus.  Being one of the few teachers who have a bus license, he is called to drive the team to and from the track every morning.  While they train, Chris sits, enjoys a book, listens, and waits for the coach, whom is another teacher, to finish the daily training so he can drive the students back.

Brenda spent her Saturday on the phone with local RCMP discussing the mental health of a suicidal student.  She then drove to the hospital to sit with the student, as his family abandoned him and Brenda was the closest person he had left.  Only knowing the student for 3 months, she had very little information to give to the RCMP however the time she spent with him on Saturday may be the only reason he is alive today.

Carol has students who arrive to school starving.  Knowing that it is hard to focus when your stomach is louder than the people around you, she cooks a simple breakfast in her classroom every morning.  Usually coming out of her own pocket, she shops once a week to buy bagels, bread, eggs, fruit, and other simple breakfast items. 

The root word of teacher may be “teach” however this word represents only a part of the day of a teacher.   Teachers do teach, as well as act as parents, friends, shoulders, counselors, emergency contacts, cooks, bus drivers, coaches, and most importantly non-judgmental people.

Wednesday, November 5, 2014

TV around a corner

What is the largest TV that could fit around a corner in your house?

This was the question we answered in my calculus class.

First I showed this applet and had students play with it

Next I asked what do we need to know?

Students asked for the width of the hallways which are 0.8 m and 0.9 m wide.

Next we realized that actually to determine the largest TV we actually need to MINIMIZE the length of the line.  As the smallest line will be the line that can fit around the entire corner.

Calling the, angle between the TV and the 0.8 m wall, theta you get the equation of the TV length at any angle to be

Next taking the derivative, and solving for when it equals 0, gives us

Substituting this back into the equation gives us a TV (or any rigid object) with a length of 2.4 m or 94.45 in across.  

We then did have a discussion around what assumptions are we making?  Some are...
  • The TV has no depth at all
  • The TV will scrap across the wall
  • The TV is out of the box

Wednesday, October 22, 2014

Bowling and Math

Recently, I joined a weekly bingo league and realized that simple addition and daily physical activity could be integrated together.  First, give each student a bingo sheet which has all the numbers 0-99 (inclusive) on it.  In Red Deer, Heritage Lanes has these already made up.  An example of one box might look like

Essentially, you have a total of 4 boxes, each with 25 squares and therefore all numbers will appear once.

How to play:

Take the last two digits of your TOTAL score on each frame of bowling and cross off the respective number on your bingo sheet.  First to a line, X, blackout, etc..wins!  Students would play at least 3 games with the same sheet for all games.

Where is the math?

Students, most likely, would play the first game not caring what score they receive and simply crossing off the scores.  Starting the second game, students will probably start becoming strategic towards the scores they want.  This is where the math will come out, and you will want to do some teaching on how bowling scores work.

Crucial knowledge includes:

  • Pins are worth 2,3,5,3,2 from left to right
  • Strikes are worth 15 points plus the score of the next two balls thrown. (Frame ends)
  • Spares are worth 15 points plus the score of the next ball thrown. (Frame ends)
  • The 10th (final) frame, you throw 3 balls no matter what you knock down on each ball.

Here is what recently happened on my team:
One of the bowlers needed a score of 68 to complete a line and was currently at a score of 17.  He threw a strike and therefore the machine doesn't update your score until his next 2 balls are scored and I saw him doing some math on the back of his bingo sheet.

He realized that essentially he has 32, and the next two balls are worth double points, as they count towards the next frame as well as the previous strike.  Quickly, him and I talked about how he needed 36 points.

There are many options to get this, but one essential question he asked is "Can I get another strike, or will this put me over?"  The answer to this will determine how he throws the first ball in the next frame.

If he throws another strike then, the first strike is now worth 30 points plus the next ball thrown, and the second strike will be worth 15 points plus the next 2 balls thrown, and therefore he essentially would have a score of 62 and the next ball would be worth triple points.  Which means if he throws another strike, then a 2 pin and gutters the 2 balls after this (to complete the third frame) he would be at 68.

What I realized is that the 3 adults on my team (all over age 25) had to think about this problem and it wasn't easily solved.  I wonder if this could help students learn simple addition and multiplication in a context and for a purpose.

If you teach younger grades and want to embed movement into your math classes, I suggest a field trip to a local bowling place. If you are in Red Deer, then I advise you to go to Heritage Lanes, as these sheets are already made.

Friday, September 26, 2014

Coding and the equation of a circle

A student was creating a tower defence game in my computer class, doing so he learned what the equation of a circle is.  This idea is a Gr. 12 math idea, and he did this in Grade 10.  Here is what happened...

He was coding a certain tower in his game and he asked me "How do I code the tower to only attack units which are within 200 pixels?"  I first asked if he could draw me a picture of what he wanted, and below is computer graphic of what he drew..

I then said, "What do you have?" He then showed me that he created variables:

t_x = x value of the turret
t_y= y value of the turret
u_x= x value of the unit
u_y= y value of the unit
He had currently coded that if the following two inequalities were true the tower would attack.
At a quick glance we realize that this creates a square around the turret not a circle.  This he had already realized.  He then said, "How do I test if the straight line distance is less than 200?".  We then drew a picture as follows:

 He then said "Well I know that once the line from the turret to the unit is less than or equal to 200, the turret will attack but what inequality do I create?"  A student, next to him, said "Would pythagorean theorem work?".  The problem we had was to label the other two sides.  Minutes passed while I let him think, and finally he asked if this would work
 I said.. "lets try it".. sadly the turret would attack the unit if the unit was within 200 units of the origin not the turret.  Once again, I refused to simply give him the answer and I asked him, "what could we do to change from the origin to the turret?"  He replied with "Well the turret isn't always the origin, so we would have to test the distance.. and so can we do.."
I then asked, "Why did you use the absolute value before?" Which is responded "because the code needs to take the positive value, and if the unit was to the left or below the turret I need it to become positive....but....wait....squaring is positive, so can I just remove the absolute value?"  We tried and here was his final test

When tested, this worked perfectly.  Keep in mind this child is in Grade 10, and completed an outcome from Gr. 12 mathematics.

Discovering a Variable

I wanted to see if I could get students to "create" or "discover" the idea of a variable.  To try this, I completed the following in my ESL (English as a Second Language) math class.

We first started with a discussion around language, and how math is the "Universal Language".  Next, we talked about "What is the best way to learn a language?"  The students agreed that we should learn how to translate from our language into math would be a great start.  I then told them how I once ordered 2 pepperoni pizzas and 3 Hawaiian pizzas and it cost $70.00, and I asked the class if there is a way we could translate this into math?  One student came up and wrote,
2 Pepperoni + 3 Hawaiian = $70.00 
We then had a discussion how, currently, we would not be able to deduce how much each pizza cost, however this would count as a translation.  I then asked how would you translate "4 Pineapple Pizzas, 3 bottles of Coca-Cola, and 1 Meat lovers, costing $92.00"? Another student came to the board and wrote
4 Pineapple + 3 Coca Cola + 1 Meat Lovers = 92
The class again agreed this was sufficient.  At this time, a student in the back was getting irritated at how easy and time consuming this one.  I asked him to go to the front and in front of everyone translate "3 super size fries, 2 Extra large Coca-cola, and 1 double, extra bacon, cheeseburger costs $21".  He let out a big "UGH!", and asked me to repeat.  As I repeated he wrote...

3 F + 2 C + 1 CB = 21

He looked at me with a smile, and some of his classmates started to laugh.  I then told him "I said supersize fries, not Fs", which he responded with "Yeah this F is supersize fries".  We then had a dialogue around what CB could mean.  After some time, a student asked "Could that be Cheese times burgers?",  and almost immediately a student yelled "but C is coca-cola, so coca-cola times burger?". The student, at the board then changed his answer to   

3 F + 2 C + 1 B = 21

I then wrote on the board

3D +2C = 13

and asked "What does that mean?".  The answers ranged from "3 Dogs and 2 cats cost 13" to "3 bags of dill pickles and 2 bags of cheetos is $13".  We then decided, as a class, that it is important to create a legend at the top.  Therefore we went back and wrote legends such as "F is Super size fries, C is extra large Coca-Cola..."

This was my attempt at students creating their own knowledge of variables.

Tuesday, June 10, 2014

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;
    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);

Thursday, May 8, 2014

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.

Thursday, April 17, 2014

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.


  • 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.

Tuesday, April 15, 2014

Dealing with Change

The following is from

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:

Monday, April 14, 2014

Curriculum Redesign in Alberta

Before you read on, I ask that you stop and think for a couple of minutes about "What should school look like?"

WHY do we need to change?

My reason comes from "WHY do I teach?":  I teach because I believe in a classroom that is structured different for each student in the class.  My "perfect class" would be focused on meeting the needs of the students not the system.  Students in my class should be thinking critically, not only learning what to learn but also how to learn.  Innovation and creativity will be at the core of all my lessons, as the focus will be creating opportunities for my students for the future and for the world outside the walls of my classroom.  Students can work at different paces, and implement various learning strategies to achieve the goals of my class.  I believe in a classroom that allows me to dig deep in various interests of students without worrying about losing time.  Personalized learning will be allowed to flourish in my classroom as the passion and interests of my students will be just as important as pencils and paper.

I don't believe that any teacher or any educational stakeholder, in Alberta, is 100% content around the current education system.  If this is true, then isn't it about time we change?

Our education system needs to prepare students to be successful in a future world that will be defined by global interaction, competition, engagement and networks. It needs to ensure Alberta’s young people will have the knowledge, skills and attitudes to be prepared for jobs that do not exist yet and in industries that are emerging or evolving.

HOW will curriculum change?

Inspiring Education involved parents, teachers, students, business, and many other educational stakeholders and listened to them around "What should change in Education?".  The comments called for more student centred, personalized, authentic learning experiences that will result in youth becoming engaged thinkers and ethical citizens, with an entrepreneurial spirit.  The vision was for an education system which is significantly different from that of yesterday and today.

The Alberta Government listened and is creating a curriculum with a different focus.  We need to invest in our students and empower them to bring out their potential.  We are emphasizing the development of key competencies in our students, cultivating engaged thinkers, ethical citizens and entrepreneurial spirits.  We’re recognizing that not all students learn the same way, and that textbooks and classrooms are just one way for them to experience education.

Lastly, so WHAT will change?

All classes will focus on core competencies, which will be integrated into the curricular outcomes.  The competencies are:
  • Know how to learn
  • Think Critically
  • Identify and solve complex problems
  • Manage information
  • Innovate
  • Create opportunities
  • Apply multiple literacies
  • Demonstrate good communication skills and the ability to work cooperatively with others
  • Demonstrate global and cultural understanding
  • Identify and apply career and life skills

This change will allow any classroom to become the teacher's "dream classroom".  An example of how my class has changed.


Friday, April 11, 2014

11 Reasons why we need the new math

Answer to Top 11 Reasons against New Math.  Recently, I have read an article titled "Top 11 Reasons to Return to old math".   Below is each point, and my rebuttal.  

1) Johnson has failed to admit any mistakes and adequately correct them
There are approved textbooks, but no Mandated textbooks.  Teachers have the choice and freedom to determine what is best for their own classes.  Some teachers follow textbooks closely (the book they choose) and others don't even use textbooks at all.

2. Twice the number of math illiterate kids
Has other things in Alberta changed over the last 4 years? If you they have then how can we ignore everything that has changed and point the blame entirely on the math curriculum?

3. Kids need direct teaching and practice to attain mastery
 Direct Instruction is in the new math curriculum. The curriculum tells us WHAT to teach not how to teach.  If you feel that there is a teacher which only does discovery then I advise you to phone the teacher.  If it is a problem that with what the teacher is teaching then you call the government.

4. Major support for conventional math, little for discovery/inquiry math
If you support that some students should be taught with memorization and some students learn best through inquiry then you support the new math, if you believe that we should ignore individual student differences and force memorization on all then you support the old.  Which do you want?

5. Johnson not open to input
Johnson has offered many symposiums and sessions, which are open to parents and community, to attend and ask questions.   Johnson has gone to many school boards and discussed concerns, and has shown how most just don't "get" the new math as it allows students to learn under conditions which are best for the individual student not the best for the group.

6. New math is harming kids, making them hate math
People enjoyed math before the change? Really?  When I tell people I am a math teacher, with a masters in mathematics, I get looks of disgust and often asked "WHY?".  Forcing students to all memorize caused many to hate math.

7. New math has created inequitable, two-tier education in Alberta
This is more opinion than fact.  I teach in a school with various income levels and all of these kids are representative in my calculus class, and all are doing quite well.

8. Educators say we need to get back to conventional math teaching
Many educators are wanting the ability and autonomy to teach how they want. The irony is that going back is actually putting more restrictions on the teacher, while keeping this way is actually providing options for teachers.

9. New math was brought in without any credible classroom studies showing it’s better or even effective
Lots of research was done.  Again, the words "discovery, inquiry, 21st Century" are not found ANYWHERE is the curriculum.  The curriculum dictates to the teacher WHAT to teach, and what you will find are the words "personal strategies".  Meaning that a student can determine how to answer a question based on what makes the most sense for them.

10. Direct instruction and memorization can lead to creativity, deeper understanding, critical thinking
This might be true...for some, while for others inquiry can lead to creativity.  Both strategies are embraced in today's math classrooms, while only one group can be "creative" in the old style.

11. Conventional math advocates are open to discovery-inquiry techniques
 All strategies can be used in Today's Classrooms.

Lastly, if you aren't convinced here is what an actual "new math classroom" looks like.

Thursday, April 3, 2014

Zombies meet Mathematics

Below is how I brought in "28 days later", "World War Z", or "The walking dead" into my calculus class to introduce points of inflection.  At this point my students have been taught derivative rules, relative maximums and minimums, but not yet application of second derivatives.

First ask the class
What would the graph of "Zombie population" vs "time" look like?
Have them explain their answers and why.  Instead of telling them, play the following game.
1) Number the students from 1- X
2) Put a table on the board with Days, and Number of Zombies as the headings.
3) Draw a random number (I used a simple random number generator)-The number becomes the Zombie.
4) Each following day draw N numbers where N= the number of zombies on the previous days.  If the number of a student is drawn they become a zombie and will attack the next day.

Your chart should probably start like:

On Day 2 you would have drawn 1 number, on day 3 you would have drawn 2 numbers, etc.
Of course, it will slowly stop doubling due to some numbers being drawn more than once.  For example if number "10" was drawn on day 2, and again on day 3, then it represents a case where a zombie attacked another zombie (stupid zombies!).

You can then graph the data and it should look like a horizontally stretched out "S". 

Now lets integrate calculus. I used the following equation, (however if you find, or create, a better equation please let me know) Also you could also use base 2, as it looks very similar.  My world ends roughly after 28 days (since I love the show 28 days later).
Where Z(t) is the population of zombies, in billions, at year t.  The graph should be

From here you can answer the following questions:

Now you want them to lead you towards points of inflections, so here is how I suggest you do it:
At this point we have discussed relative maximums and minimums are there any of these on the graph? No.  Alright, is there anything "special" going on?
Then let them talk, explain, discuss.  You want them pointing towards the middle, and determining that the derivative here is a maximum.  You can then relate how you determine relative maximums of functions to determine that you simply make the second derivative equal 0.  This is what we call a Point of Inflection!

Please change, tweak, use, etc as you see fit.  

Friday, March 28, 2014

Why multiples strategies makes sense.

I received this story from a teacher who prefers not to be named:

I have been a teacher for the past five years. Although I am not a math teacher now, I did have the opportunity to teach math when I completed my APT a few years ago.

Now before I proceed, I should inform you that I struggled with math when I went to school. I barely passed Math 30 Pure and quite frankly, despised the class. So when I found out that my placement was teaching primarily grades 4 and 5 math I was concerned.

I was given the task of teaching addition to the students. Simple right? Wrong! The new math had me very confused as there were multiple ways to teach students how to add. Growing up, I had learned one way: start on the right and work your way to the left, carry the one, and so on. The math that I was required to teach had me doing things that I had never learned to do. It took time for me to understand this new way of thinking.

Now, at this point you're probably thinking that I'm bashing the new math curriculum. However, I am doing quite the opposite. When I actually sat down with the material and tried to teach myself how I would teach these youngsters this different way of adding, I began to understand! All of the sudden the old algorithmic way that I had learned didn't matter anymore. This was a new way that

I could understand because it was teaching to how I learned! Teaching my students was also a success! My struggling students were able to see different ways of learning, and although they still had their challenges at times, I was able to explain to them why we add this way.

Many times people have shown me that other current math teachers have signed a petition around bringing back the old curriculum and this experience has shown me why.  I assume that these teachers simply don't "get it".  They are trying to show our current students the poor strategies they were shown.  Do this and you will get the right answer, without any explanation.

I hated how math was taught to me because I was forced to solve one very specific way and now that I have learned multiple ways, math has become more enjoyable.

Thursday, March 27, 2014

Petition around Math

I am fearful that people who have signed the recent "Back to Basics" Math petition truly don't understand the devil they are asking for.  In our current math curriculum you will see objectives such as

Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:
• using personal strategies for adding and
• estimating sums and differences
• solving problems involving addition and
Also similar objectives around multiplication and division.  The petition, and other critics, are upset that "memorization" is not needed and "discovery learning" is forced.  This is completely, and utterly, incorrect.  I have searched the Alberta's Math Curriculum for the word "discovery" and not one incident of the word exists.  What does exist is: Personal strategies.  Meaning the strategy of one child could be vastly different than the next child.  This curriculum is simply not forcing discovery in the classroom.

The curriculum simply tells all Alberta Educators what they must teach, but it does not, and hopefully never will, tell teachers how to teach these outcomes.  The petition, on the other hand, wants to do just that.  It wants to invade our classrooms and mandate to the professional teacher that every child must memorize.

Imagine if we did this for all outcomes.  Students cannot move a grade forward until they memorize the following facts.....  We would have most graduates who are simply "Siri" clones, and also students who truly hate the idea of learning.  Of course some, the ones who strive on memorization tasks, would get a great education.

I am not claiming that no student should memorize basic math facts at a young age, nor should every child be forced to discover the facts.  All I want is to keep the autonomy to the professional; the teacher.  I trust our Alberta teachers to know which students should use manipulatives, flash cards, centers, collaborative, or independent learning tasks.  I trust that some students will memorize, some will discover, and some will complete activities which are a hybrid of both.

What I do not want is to force all students to memorize.  Are there some people who loved mad minutes?  Sure.  Are there some students who learn best through discovery? Also yes.  To force every student to learn the same way is alienating some.

This is why the petition is causing alarm to me.  They want a culture where the individuality of the student, the teacher, and the lesson is abolished.

There are lots of pictures out there around how horrible the new math worksheets are, or how horrible the lessons are, but I want to remind you that the "how" part is up to the teacher not the government. Also, we should be aware that the context, in which the photo is taken, most likely is lost in the photo.

The problem is that not everyone completes addition, subtraction, multiplication, and division the same.  There are algorithmic ways, and many mental strategies.  It is ludicrous that a child should be told "Don't do it the way you understand, you must memorize another way".  People can solve "82-19" multiple ways. Does this mean that one way should be norm? How you solved that problem should be the exact way the next person does?

At the root of the new math curriculum is simply "Differentiated Instruction".  Each student is taught using more than just pencil and paper, but also tying into their passions and interests.  Which do you want for your child?  To be formed into a clone, or to be allowed to blossom into their own character?

Lastly, here is a link to what the new math curriculum looks like in my class.  Let me know if you have a problem with me allowing students to solve the same problems, in different fashions.

Saturday, March 15, 2014

Madden 14 and Calculus

Recently I was playing Madden 14 with a friend on my XBOX 360 when he asked me

Dave, does the wind make that much of a difference when I punt the ball?  If so, should I change my angle a lot, or a little?

I replied with "It would change your angle, but by how much I don't know".  This question also sparked my most recent calculus optimization problem.

I asked the students in my class
If a football punter is kicking into the wind, should he/she worry about the angle they kick at?
Also showed this video

What came of this was a great 10 minute  discussion around questions such as

  • How windy is it?
  • How much should the angle change by before we worry about it?
  • How fast does the ball move?
After doing some "googling" we decided to work on the following problem
If a punter can kick a ball at  40 kph, and there is a head wind of 20kph, what angle should the punter kick it at so the ball travels the farthest distance?
Here is a possible solution: (Yes I do realize we ignored air resistance, as I said I assumed Madden 14 did the same)

When you kick a ball there are vertical and horizontal components and we determined the following, starting at acceleration and integrating, with respect to time.(Using h=horizontal, v=vertical)

We then discussed physics, that if I kick a ball with velocity 40 at an angle of theta, then the vertical velocity is 40sin(theta), and horizontal velocity is 40cos(theta), as well as initial distances were 0, and subtracting 20 off the horizontal velocity due to the wind, and so the formulas become
Now, we realized we have two variables in the equation we are optimizing (d_h(t)) and so we needed a way to relate the angle and time.  Knowing at the top of the kick the vertical velocity would be 0, and that the total time in the air would simply be double this value we get the total time to be: (setting v_h(t)=0, then multiplying by 2)
Now we substituted this into the horizontal distance equation, and took derivative (knowing that the derivative equals 0 at a max), and solved for theta
I allowed my students to solve this last equation graphically to determine when it equaled 0.  

We got an angle of 32 degrees.
We then discussed how easy it would be to simply make the wind speed a letter, the kicker speed another letter and ultimately create a program which could do this for any experience.

I am in the process of making an app and selling it to all football teams and retiring in the next year..jokes!!

Feel free to use, tweak, comment, etc.

Thursday, March 13, 2014

Larger shoes makes your child smarter!

Yes that is right, the larger the shoe of the student the better the child is at math.

Of course you are asking, where is your proof?

Well, I tested students in a K-12 School on basic math skills.  I then ranked them according to shoe size, and I noticed that the larger the shoe, the better the score...on average of course.

If you buy this research, then I have to inform you it is bogus, however if you knew right away there were some critical flaws then I invite you to read on.

See there are more variables at play then simply shoe size.  Age, years in school, gender, socio-economic status, language, etc are just some of the other variables.  However critics of new math are using this same logic above to make claims that the new curriculum is making our students less smart.

See, PISA is a test administered every 4 years and since the last test marks have dropped by 6%.  First, there is a problem with using PISA  and next a lot has changed in our Country, schools, and communities then simply math instruction.

Should we ignore every other variable and pick one out of a hat and attribute this change to it?  If so, then we can also prove that shoe size is linked to math scores.  However we need to realize that in the last 4 years,

  • Class sizes have increased
  • Immigration population has increased
  • English Language learners population has increased
  • Education Funding has been reduced.
  • Special education projects, such as AISI in Alberta, has gone to 0
  • Our culture has changed
This is to name a few.  Lastly, if they want to blame the new math curriculum, because it doesn't "teach the basics", you might want to know that the students who wrote the last PISA test.......were taught the basics under the old math curriculum.

Wednesday, March 12, 2014

Case for the new math curriculum

I have seen many pictures, articles, and petitions on why the new math needs to leave our schools.  Here is a quick explanation, and "Myths around the new math curriculum".

First, I want to ask you to determine what is


Take a moment and complete it.  Don't worry there is no test, just please don't use a calculator.

The answer is 63.  Now did you:

1) Borrow one from the 8 to get 7(12)-19, then say the ones are 3 and then 7-1 is 6, so the answer is 63?

2) Did you add 1 to 19 to get 20, then added 60 to get 80, then added 2.  Finally, added 1+60+2 to get 63?

3) Did subtract 20 from 82 to get 62, then added 1 to get 63?

4)Did you do it a different way?

Finally, which way is the best way?  Which way should your child learn?

If you answered "NUMBER 1 MUST BE THE WAY TO DO IT" you are in favor of the old math curriculum.

If you answered "Number 1,2,3, or 4 is a way to do it" then you are in favor of the new math curriculum.

Myth: Discovery Math is a mandatory strategy to be used in K-12 curriculum.
Fact: "Discovery" or any synonym, cannot be found anywhere in the curriculum at all.  The government tells the teachers WHAT to teach, but not HOW to teach.

Myth: The teachers are simply no longer teaching children.
Fact: The teachers are not teaching each child the same.  Differentiated instruction is now part of the classroom. Students A, and B may be taught differently; one with manipulatives, one with without, based on the needs of the child.

Myth:Children don't need basic math facts.
Fact: Basic math facts are part of the curriculum.

Myth: Students are becoming dumber.
Fact: While the PISA score has dropped 2%, the students who wrote the last PISA test were taught in the old curriculum.  Therefore, if you think this is a problem you should be advocating for the new math.

Myth: Teachers hate the new math.
Fact: Teachers have the choice and autonomy to teach however they want, and therefore some extremely creative and innovative things are occurring in classrooms.

Lastly, if your child comes home with a different way of completing math than you were taught, then ask them to explain how they are solving the problem.  Lets not forget that there is not only ONE way to solve a problem.

Thursday, February 27, 2014

Curriculum Redesign

I currently have been to many curriculum redesign talks, symposiums, and discussions and I have never been more excited in my life around educating children in Alberta.  I am truly inspired!! Now of course not all feel this way and so I bring to you "why you should be excited."

First, David Staples from the Edmonton Journal says,

It has had a profound influence, so much so that if you have a child in school after 2016, they will get a fundamentally different education than you got.
This makes me happy.  I am not claiming I received a bad education, but I do feel that I want something better for my own daughter.  Currently we have a model where tests, which are focused on the low level of bloom's taxonomy, are being used rampant throughout our culture.  If you think this should be the norm I bring you a quote from Seth Godin,

As soon as we associate reading a book with taking a test, we've missed the point of literacy
If you are still not convinced that our curriculum needs a change, here is what Jeff Johnson, Alberta's Education Minister, says,

Our [current] packed curriculum stifles creativity in the classroom. It’s too packed. There’s too much stuff to try to get through and it doesn’t allow enough flexibility to individualize learning, which is going to be really key in the future. 

How will the new curriculum work?  Again, David Staples sums it up fairly well when he says,

Teachers will do far less direct instruction so that students can increase their knowledge base. Instead, students will focus on more group and project work. Teachers are to act as guides who assist kids as they explore areas of interest.

Finally, the goal of education will be creativity and innovation and to truly inspire students to learn more.  I think back to my math classes, which I usually finished with 95-100%, was I inspired to become a mathematician, an engineer, a scientist, or even to attend the next level of math?  From some classes yes, but this was due to the excellent teacher I had in front of me, not because of the excellent curriculum. Why did this happen? Well...just to name a few

  •  Gr. 3 teachers are having a hard time inspiring students while they teach their 1352 mandated outcomes
  • High School Biology teachers are having a hard time bringing in creativity when their curriculum focuses on memorization of terminology.
  • Grade 12 teachers are worried around the 50% diploma and finding it hard to bring in innovation to their classrooms.

Since when we can't measure what is important we start making what we measure important.  This is what happened over the last centuries, more and more low level skills became important because that is what we can measure.  Critics, of this change, will use standardized test score data to justify that this change is not excellent for the students in our classes, and my quick response is "how do multiple choice, standardized tests, like PISA show creativity, innovation, interests, and passions of the students?".  See they can't, but they do show how well students can memorize and regurgitate; which is why these critics say these skills are important. 

What does it look like?  Well I have been changing, my own classes, already to meet the needs of the future curriculum and here is a photo which sums it up. 

You can't change the input or the process of education and not expect a different output.  Since the output is different, the way we measure this "new output" must also change.  

Lastly, I ask you...."Were you inspired while you were in school because of what (not who taught) you learned?  Did you want to attend the next level of schooling because of the number of outcomes the teacher covered?  Did you feel that your creativity and innovation was welcomed and could flourish while studying for a 50% diploma?"

If you answered yes to all of these, then maybe I have it wrong and I am simply an anomaly, however if you answered no then I ask you to join me and become inspired with the new curriculum.


Stop Stealing Dreams, Seth Godin

Monday, January 27, 2014

Problems with PISA argument

First I don't believe we should put our faith in PISA scores due to the following:

Why my Child won't write optional Standardized Testing
As well as my other posts around why standardized testing is flawed.

However, lets for a moment assume that PISA scores are accurate.  We will also negate the fact that Alberta Class sizes have increased, ignore the fact that AISI funding is at 0, and lastly forget that our ESL population is at an all time high.

We shall enter the "Wonderland".

Well, first the argument is that we need to go "BACK TO BASICS" with elementary math classes.  Students should be "Drilled and killed" with mathematical facts and flashcards should be the norm.  Since elementary classrooms have removed these activities the PISA scores have dropped.

Well this is great, however this argument is FLAWED!

See the 2012 PISA test is completed by 15 year olds or Grade 10.  Which means in 2011 they were in Grade 9, 2010 they were in Grade 8..., 2003 in Grade 1.  The Alberta curriculum changed , and had the following implementation years...

Which means...drumroll...these students experienced elementary classrooms teaching the "old" math. Which is, of course, the exact math critics want to bring back.  These students are not the ones we should be looking at, but instead the next data set to come, as these students will be full time "new" math. (Keep in mind I don't think we should look at any PISA scores, but we are in Wonderland)

Next, someone might say that the PISA drop is due to poor retention levels.  As the new curriculum is not allowing students to retain mathematical information in the long term.  This poor retention level is causing the frustration when students try to complete higher level math.

Well..again this is flawed.

Most people have heard of PISA, but have you heard of PIAAC testing? See now this is an exam, also done on the international level, but tests 16-65 year olds.  Keep in mind, all these people experienced 100% "old" math.

Since we are assuming that Standardized test scores are accurate, lets take a look at how our country scored on this...

As you can clearly see Canada was under the OECD average in every age group in Numeracy levels.  Since all these participants took the "old" math, I guess retention was an issue then as well.

What does this mean?

I believe people are following the wonderful quote of Henry Ford
When I asked people what they wanted, they replied with faster horses

What we forget is that sometimes the "Better" system is one which we have never seen or done before, and this is another reason why I support the new math curriculum.