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

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.

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.

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);
  }

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.

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.


http://realteachingmeansreallearning.blogspot.ca/2014/01/math-in-new-setting.html

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.  

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.

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
subtracting
• estimating sums and differences
• solving problems involving addition and
subtraction.

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.

http://realteachingmeansreallearning.blogspot.ca/2014/01/math-in-new-setting.html

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.

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

82-19


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.

My new Final Exam

Over the past years I have been trying to make the end of my courses a celebration not a date students dread.  I say that because, I have seen students cry, stress, and create all sorts of excuses, whenever I brought up their "30% final Exam".

This year, I have taken an entirely different approach.  Instead of a 30% exam, which consists of one part multiple choice, one part numerical response, and one part written, I have changed it to the following:

1 part entirely written worth 15%, and a presentation worth 15%.  In our school, there are certain criteria for which a student can exempt a final exam.  In my course I have made the presentation mandatory, and a student can only exempt the written portion.

How does the presentation work?

Students get the following BIG ROCK outcomes of my course:

Students are then required to create a presentation around these outcomes.

What does it have look like?  How long should it be? How many questions should they have? Does it have to have videos? 

The answers to all these are "Up to the student!"

Now, since I have over 40 calculus students, I do not have time to watch 40+ presentations, so I do allow students to collaborate in groups of up to 3.  HOWEVER THEY DO NOT GET GROUP MARKS.

During the presentations I ask questions and depending on the individual answers I differientate the assessment mark for each individual student. 

The presentations usually take 40-60 min in length with disucssions, questions, and even some learning occuring.

My students no longer dread the final and one even told me

"Mr. Martin I spent more time creating my presentation then I did preparing for my diploma in my other course, which is worth 50% of my mark"

What's wrong with Math Education?

"You can't teach that now, because then what am I going to teach next year!"

Ever heard this?  Ever said it?  Ever been a part of a conversation with this used?

I know I answered "Yes" to all three.  When curriculum is designed linearly, then we encounter such problems.  You have to know Y before I teach you X.  However, what if curriculum was designed around the questions student asked in class?

In two of my classes this year, I have eliminated "Units".  I can now demonstrate how math flows from one concept to another instead of teaching through 5 disjoint units, and no longer do I have to answer "You will learn that later"

Check out the clip below

Repetition in math class

Practice makes perfect! or Perfect Practice makes Perfect!

These, and others, are comments I hear as to why we need practice in math classes.  This "practice" can be seen by worksheets, flash cards, and multiplication tables.  I, however, disagree with this notion.

The problem occurs when we look at what the students are supposedly practicing when they complete the above tasks. 

These tasks promote the idea of efficiency over understanding.  I remember back to when I used to administer "Mad Minutes" (basic questions which have to be completed in a minute or less, and their mark was based on only the ones which were answered correctly.  If a student did not answer it, or did so and was incorrect, this student would lose marks).  On multiple occasions, I saw students writing down numbers which made no sense, simply due to the pressure put on them during this timed exam. 

Of course, how can students complete deep math questions if they don't understand their basics?  Well I have a great story against this...and the main character in this story is me!  I still to this day, 30 years old with a bachelor degree in Mathematics, and 3 courses left in my Masters of Mathematics, cannot recite the multiplication tables.  I struggle deeply with my 7 and 8 times tables.  Does this make me a weak math student?  Does this imply I will not be able to answer deep questions?  I would hope not one person would answer yes to either of these questions.  However, the way I used to assess math, through repetition, would never allow myself to succeed in my own courses. 

If it was not for my own mother, who strongly refused to use flash cards at home, I would probably have grown up hating mathematics.

As a math teacher I needed to understand that the beauty of mathematics does not come from memorization of basic facts, but instead the use of basic facts to solve problems which a person may encounter on a daily basis.  Does understanding basic facts allow for students to solve problems quicker? Of course, but should we judge the quality of answer solely based on the time given?

I have given tasks to my students, some of which are upcoming blog posts, where students have chosen to complete multiple questions, of similar types, to come to a conclusion.  The difference in these tasks, however, is the word "choice".  If we allow students to decide how many problems he/she needs to solve, to demonstrate higher level thinking, then I guarantee your students will start to see the beauty of mathematics as a wonderful, sometimes chaotic, subject which is not limited to solving petty details.

Prime numbers and Cicada

Lost on how to find meaningful ways to introduce prime numbers?

Why don't we look at Cicadas 

These are winged insects that evolved around 1.8 million years ago.  What is interesting is that their life cycles follow prime numbers.  They emerge, mate and die quickly during the spring of either the 13th or 17th year.    On these such years, they build an exit tunnel where millions quickly exit and overcome any predators, such as birds, by using their vast numbers to their advantage.

Why every 13 and 17 years?

This way, it will be impossible for any other life cycle to line up with the life cycle of the Cicada.

For example, lets suppose they emerged every 12th year.  This would have animals with life cycles of 2, 3, 4, and 6 years to line up perfectly every time they emerged.

The life cycle being a prime number is, what scientists predict, what has been their greatest strength in their survival for over 1 million years.

Science and Religion

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I used to believe there was a dichotomy between science and religion. It seemed in the past, that when science could not explain why something was true, we turned to religion and simply attributed to God. There have also been arguments against the existence of a God,
from The problem of Evil to Michael Martins proof, A Disproof of Gods Existence. Even after growing up in a Catholic home, I believed that one person was either religious or logical and scientific, but not both; a Venn-diagram with no overlap. 8 years ago, this all changed when I started teaching at Notre Dame; a Catholic high school. When I make references to Church and Religion, I will be referring to the Catholic Church.

Once I started working at Notre Dame, I met many Catholic Science teachers and was shown that you can be a Catholic logical thinker. My eyes were opened to the reality that this Venn-Diagram does have an overlap. I believe, like others in the past, I did not understand the implications that Religion has on Science and also how Science impacts Religion. I will show, through my own stories how Science and Religion can, and do, coexist.

When I first started teaching Science, I was worried I would go against the teachings of the Church when I started to address the age of our Planet. Before, I took the literal sense of the Word in the book of Genesis which stated that the Earth was formed in six days, some 6000 years ago. Science has shown that the Universe is around 15 Billion years old. An obvious contradiction! After some research, on how I will address this in my Science Class, I read Frank Sheed (1982) say

"one shouldnt be forced to choose between evolution and creation."

and he continues on to say that
"Creation answers the question why does everything exist, why there isnt nothing? While evolution, is a theory, as to how come the Universe did develop once it existed." (Pg. 58)

As well, when Genesis was written, humanity did not fully understand the workings of the universe and these six days just corresponded with the Babylonian creation myth Enuma Elish, and does not really mean God created the Earth in six day at all. Religion now fully accepts the fact the Earth is not 6000 years old and that uses science to determine when God created the Universe
One discussion which arose in my class was around the idea of evolution for Apes versus the creation of humans by God. Again, before looking deeper I believed these were two contradictory ideas, but instead they are not. The Church has said it is not against the idea of evolution, and in fact evolution exists in our current era (Some people will not grow molars), and that the evolution of the body is an almost certain fact. However, religion teaches us that the soul, in which is inside our body, was not the result of any certain evolution but instead infused by God Himself; again not contradictory, but two theories from the same thread of truth.

Due to the limit of words I will not go further into detail, but science and religion also agree on when life starts, how pre maritial sex will lead to increased divorce rates, and many other theories. Science and religion are both logical, deal with science and fact, and truthfully it is
because of Science that my faith is strong.

Permutations and Combinations Lesson 1

Here is my lesson plan for my first lesson in Math 30-1 on Permutations and Combinations. Which covers the outcome: Apply the fundamental counting principle to solve problems.

(My students sit in groups of 4 and 5)

First show the following funny video,

Next, show this video:

After which, ask the question "How many different possible pin numbers could there be?"

Give the students about 2-3 minutes to discuss and then re-ask the last question
"Is there any more information you need?"

This is where you can go in any direction you please.  In my lesson, the gentleman in the movie has either a 4 or 5 digit PIN number (we don't know).

Now allow students to work for approximately 10 minutes.

After this is done, ask the class:

How did you arrive at your number?  Here I would actually have students come up and solve on the board

What assumptions did you make?

Is there a way we could arrive at the answer more efficiently?

Take this time to discuss that you should multiply the different possibilities of having a 5 digit pin, and the possibilities of having a 4 digit pin, and then you should ADD these answers together.

Next show the students the following picture of a hand knit mitten, and explain the following:

Jennifer makes this mitten out of four different parts, the fabric of the entire mitten, the middle "tree or leaf" part, the bead, and the strings which tie them together, and has multiple different colours for each part.

Then ask: How many different mittens can Jennifer make?

I would let students talk as long as they needed until they realized they are missing a lot of vital information.  Ask for any questions or information they might need (just like the previous question) and provide them with the following: (You can change as you see fit)

5 different colours of yarn for the mitten
3 different tree/leaf colours
2 different beads
5 different colours for the string.

Give time to solve and then ask

How did you arrive at your number?  Here I would actually have students come up and solve on the board

What assumptions did you make?

Is there a way we could arrive at the answer more efficiently?

Next, you can go the link https://order.bostonpizza.com/EN#content=/Menu/ViewMenu/&CategoryItemsContainer=/Menu/CategoryGroup/dfa5509b-935b-4776-b157-bfefef2ab654

Which shows that Boston Pizza currently has 4 different types of wings with 21 different flavours of each type of wing.

The problem:  Red Deer Rebels (or whichever local hockey team you want) is having dinner and orders 8 different double orders of chicken wings, how many different combinations could there be?

Again, using the same process students will need to know if you can have more than 1 flavour, and you can have up to 2 flavours PER double order, or they could be the same flavour as well.

After, ask the three crucial questions again, with some leading if needed.

If you have more time I would ask the following question:

Should Alberta, currently, be concerned with the number of phone numbers in the province and truly needed to add the 3rd area code (587)?

Following the same procedure of asking if they require more information and then the three crucial questions of debrief.

How my Dog taught me about Math

Over the summer, my wife and I adopted a puppy found in a local dumpster.  After reading about 3 or 4 books I decided I was going to teach my dog how to sit, stay, roll over, bark and……mathematics.  

How do you teach a dog math?

Very easy, but first you have to teach your dog how to bark.  Once this is done, I have trained my dog to bark twice every time I say "One plus one".  She barks three times when I say "Two plus one".  Lastly, I have trained her to remain silent when I say "Four times zero".

She understands math correct?

Before an argument is started, I do not believe she truly understand math, but only has memorized mathematical commands.  I wonder how many students go through math class with the knowledge similar to my dog; memorized facts, but has very little understanding.  

Years ago, my class was set up in a way that I was training dogs, not teaching students.  I would give students questions out of context, assign redundant homework, and lastly reward speed and repetition with marks.  My dog has taught me a more valuable lesson than I could ever provide to her; there is a large difference between memorization and knowledge.

Math and Super Mario Brothers

Usually I would hand out a worksheet on calculating when a function is closest to a point and have students complete 10-12 questions.  This year, I took the a different approach by bringing Super Mario Brothers in my class.

Using this picture,

I informed my class about Mario Brothers; when you jump, with Yoshi, you can jump again.  I then posed the question, "When would be the best time to jump off Yoshi if you want to get to the top level?"

Using Calculus, and geogebra you can calculate the path of Yoshi and the co-ordinate of the top level to get:

From here we calculated the equation of the parabola, and a distance function based on any point (x, f(x)) on the function.  Ultimately, we calculated the closest distance Yoshi comes to the point, and when to double jump to get the coin.
Students enjoyed this more than completing the 10 questions on the worksheet.  Feel free to use and fix as you see fit.