Calculus and Kobe Bryant

I used this video in my calculus class to show how math can either support or disprove a movie.

I showed this movie to my students and asked if this looked possible.  One jumped up and said "No way!" while others thought it could happen.  What was interesting was when I asked "Do you have proof?", as the class went silent. 

The math in this movie is incredible. 

Where I thought the students would go was completely the opposite of what happened.  My students timed how long Kobe was in the air, we measured his height in the movie and compared it to his real height to create a scale.  We used integration using gravitational pull to be -9.81m/s^2, to create a velocity and distance function.  The calculus was amazing. 

Whether we proved or disproved the reality is a secret I keep with me as I challenge you to give this to your students and see what they do.  Just watch the movie then ask "Any questions?"  I bet you will get lots.  The secret is then to let them "play" with the math and the movie.

Learning is road that they must travel down themselves and we should only be guiding them not pulling them along by the hand.

Awesome Education Talk in Red Deer

Yong Zhao is coming to Red Deer Wednesday February 8.
Tickets are $10 and are donated to the Red Deer Women's Shelter.

Amazing speaker for parents, future and current educators and as well as anyone interested in Education.

More information, and how to buy a ticket can be found here

Students create the problem to solve

Math 31 Assessment

Option 1: The head of NASA has approached you asking for your assistance plotting a course for the International Space Station.  The tracking device can be found at http://spaceflight.nasa.gov/realdata/tracking/index.html .  NASA needs an equation of the path, as well as the 1^st and 2^nd derivative.  The director is also asking for all relevant information about the path to be explained.  Create a potential problem the satellite might encounter and provide the solution to the problem.

Option 2: Ethan Hunt, an IMF point man, is securing top secret files at the bottom a cylindrical tube.  He is being lowered by rope into the room at a constant speed.  Unfortunately, while being lowered, another man walks, at a constant rate, towards the room.  Ethan is then raised back up out of the view of the approaching man.  While suspended in air, sweat is building at a constant rate until it reaches a critical value and drops onto the floor.  Using the video (a clip can be found here http://www.youtube.com/watch?v=k-oVuQpjG3s) use calculus to help out Ethan and determine various velocities, critical values, and timing to assist him in securing the evidence.

Option 3: Taking what you’ve learned this semester you must create a problem to solve. It must be based on a real world example (or movie world) and must have at least one solution. Be sure to submit your proposed solution in a sealed envelope. Five bonus points will be awarded if you are able to stump your teacher.

No matter which option you choose you must create the question and problem in the given scenario

	Calculus Student	Calculus Student Teacher	Calculus Teacher	Calculus Master
Real Life Application: Is the problem worthwhile solving? What are the implications of the result of the problem? Who would benefit with the knowledge of the answer to your problem?	Only students of this course would see the relevance of this problem. The purpose is built on strictly recalling facts. The solution is only needed to complete this assignment.	Problem is created from a plausible issue with major changes. Purpose is unclear and does not go beyond the needs of the course. Students in this course would only understand the consequences of the solution to the problem.	The problem is created from a plausible issue or problem with minor changes. The purpose is clear and slightly exceeds the needs of this course. Few, outside this course, would benefit from the solution to the problem.	The problem is created from a real life issue or problem. The purpose is meaningful beyond the needs of the course. The solution to this problem adds to the experience of the students’ real world knowledge.
Explanation of math. Are you using basic math knowledge? Are you demonstrating most of the knowledge you learned in this course?	Focuses strictly on basic recall and basic knowledge of the mathematical skills.	Requires few sections to apply higher level thinking to solve the problem. Math is still largely focused on recall of knowledge.	All levels of understanding, from basic to higher level thinking are implemented throughout the problem.	Focuses on higher level comprehension, the use of the combination of multiple skills is evident throughout the entire solution.
Communication of your answer: Does your work follow sequentially throughout your solution? Are there gaps in your communication?	Minimal response is given with multiple gaps in the explanation process.	The use of similar explanation techniques is used throughout the solution.  Communication is limited with various gaps in the problem solving.	Most of the project is easily understood, and organization is mostly logical.	The use of different means of demonstration is illustrated throughout the solution.  The mathematics is clearly communicated as well as the meaning of the solution(s)

Discussion with a first year teacher around marks

Recently, I had a discussion with a first year teacher after 4 months of teaching in her first year, here is how it went:

Her: I’m scared I not going to have enough summative assessments at the end of this year.

Me: How many is enough?

Her: I went around to other teachers and discussed and the general consensus is around 25-30 if I averaged it out.  Some teachers had even given 50.

Me: Do these other teachers teach the same exact students you do?

Her: No, they teach different courses and different subject

Me:  How do the students of other classes and teachers of different subjects have anything to do with your students?

Her: Then who should I be asking?

Me: your students, ask them how important is having 30 assessments, would you prefer less assessment of higher quality or more of lower quality?  We must understand that the more grades we give students, the more we lower the value of everything we have graded to far.

her: Dave, don’t we give more summative assessments to allow for students to have more chances in achieving the best grade possible.

Me: So it sounds as if this strategy has nothing to do with learning.  It sounds as if we are giving more assessments to allow students to get the best score in the “Game” of school.

We then engaged in a discussion around the meaning of assessment and marks.  She started from a belief that there is an actual number of summative assessments which she is required to have for each student.  I hope I showed how wrong this belief was!  Teachers never should have a goal of the number of summative assessments to give students, and in fact we should be aspiring to have zero summative assessments throughout the course and start moving to entirely formative assessments.  The only summative assessment which should occur is at the end of the course, only under the assumption that no more learning can occur.

Facts and Myths of traditional learning in schools

It is little short of a miracle that modern methods of instructions have not completely strangled the holy curiosity of inquiry – Albert Einstein

From “Breaking Free from myths about teaching and learning” by Allison Zmuda, we learn about various myths and facts about learning in the traditional classroom, and my synopsis:

Myth: The rules of this classroom and subject area are determined by each teacher:

This is false as the push for meeting provincial (and state) standards increase, the autonomy of the individual teacher decreases.  Collaboration is being used to enforce compliance as well as standardizing the rules of each classroom.  For learning to be meaningful it must stem from personal experiences, current issues from students, as well as address the personal attributes of each student.  This cannot be achieved by blanket policies which affect all.

Fact: What the teacher wants me to say is more important than what I want to say:

This is truly a sad fact about our education system.  It is entirely summed up by a student who said:

It’s easy to take what the teacher says and regurgitate it without even thinking about what was said, and it’s how we’ve been taught to learn.  When I set out to write a paragraph, I actually thought I should ask my teacher to spell out what he wanted me to write… If I tried to challenge my teacher, all it would take is a little bit of him pushing back to make me drop my argument and look like a deer in the headlights, even if I had a decent argument.  Now that I know how passive I’ve been, I’m ready to make some changes in my learning style.

 Students need to have opinions in classes, and the teachers need to be cognisant of these ideas.

Fact: The point of an assignment is to get it done so that it’s off the to-do list:

In our schools, too many students are feeling overwhelmed to get the assigned readings complete, answer the repetitive math questions, study for the Biology exam and still have time to pursue to their own interests outside of school.  One student has even said

Most students just do the assignment because there is not time to really study it.  We don’t really get a chance to go further into the parts of the topic we are studying that aren’t part of the curriculum because we have already moved to a whole new topic.

We, as educators, must be aware that for students to complete all their “homework” they must take shortcuts and thus truly never understand the material at a deeper level.

Myth: I feel proud of myself only if I receive a good grade:

Students are using grades to truly sort themselves among their friends and classmates.  I believe all teachers have heard comments such as:

I got an F on this exam, but that is ok because I am not good at it.

I got a B like I always do, so I am doing fine.

No one is getting an A, so makes sense that I am not getting one either.

Grades and other extrinsic rewards are actually limiting the potential success of students.  Students are actually seeing the grade as an indicator as to how well they are playing the game of school.  As we push for improving learning we must move away from using grades to motivate students.  Students are proud of the product of their education not the mark they receive on how well they have manipulated their education

Myth: Speed is synonymous with intelligence:

Too many times we are pushing students to complete tasks at a speed which is unnatural to their own learning.  We are stripping education of passion and interest and replacing it with efficiency.  This can be seen in math classes when we teach “math tricks” and justify it by “this is the fastest way to get to the answer”.  Other educators validate this idea of rushing to complete the course due to the amount of material that is needed to be covered in a short amount of time.  The pressure to prepare students for standardized exams and complete the overwhelming curriculum is making it quite difficult for teachers to accept alternate views of learning, instruction using discovery methods, and taking time to allow for each student to deeply understand a topic before moving on.

More myths and facts exist, and I encourage all to read the book…WOW!

In the teaching to which I aspire to

I would truly allow each student to be an individual. To allow students to guide, not only their learning, but as well as their assessment. Students will no longer be "passengers on a fast bus down a lone highway" but instead "participants on a field trip to a field of their choosing" .  Top down mandated policies focused on increasing test scores would be replaced with guidelines, created by and with teachers, to allow for deeper learning. Teacher PD will always be encouraged as the work teachers do outside of school is just as important as the work they do inside the school.  

Outcomes will no longer be delivered in the hundreds, where each is extremely specific, but come in few general outcomes. This will allow for teacher autonomy in classes and allow for learning to be focused around the needs, wants, aspirations, and goals of each student instead of the goals of the system.

Technology, with unfilitered access, will be used in classrooms to create engagement around learning instead of being implemented to trick students in completing mundane tasks.  The goals of each class will be around meaningful questions instead of repeptive answers.

Teaching will be around one single absolute goal; lighting fires of interest and curiosity in each student of the school.

Food Chemistry - Enhancing Learning

Jeff Lerouge is a fellow colleague at my school, one damn good foods teacher, and another teacher who is challenging the idea of using worksheets to teach material.    His twitter is is here @jlerouge, his blog is found here Jeff's blog.

Before, he used to teach different ideas through worksheets, and here is how he is educating students now:  GOOD JOB!

Jeff start with:

What's the big idea? Why did we do this? Did you learn something today?

We've been learning about the pigments found in vegetables and how other ingredients can affect the color of vegetables. We have also been learning about how those same ingredients can have an impact on the texture of vegetables. To gain a better understanding about what we've been talking about, we decided we needed to experiment with the actual pigments and ingredients. What we've really done is do a science experiment and made some observations that helps us make conclusions.

We started with four different vegetables that represent the four colors of pigments found in fruits and vegetables:

• carrots (orange)
• cauliflower (white)
• broccoli (green)
• red cabbage (red/blue)

We then subjected each vegetable to four "treatments" against a control (boiled in plain water):

• salt
• baking soda (alkali)
• lemon juice (acid)
• sugar

After the experiment we came to some conclusions which supported what we learned in theory. It helped so see, touch, taste and smell - enhanced understanding. Here's what we learned:

• the orange pigments in carrots are very stable and are not greatly affected by acid or alkali
• adding sugar made our vegetables firm
• adding acid made the cauliflower stay white and kept the cabbage purple - good result
• adding acid turned the broccoli and unappealing olive green color - bad result
• baking soda made all of the vegetables mushy - bad result
• baking soda turned the red cabbage an unnatural blue green color - bad result
• baking soda turned the cauliflower an unappealing yellow color

On the same day, we also learned about different kinds of rice not by reading about them, but by cooking 10 different varieties and tasting them all. We cooked a number of whole grain rices, something many students who usually eat minute rice or white rice were not used to, especially the different in texture. Students learned that this difference in texture is from the bran layer being present on the grains.

Student guiding his own learning

Below is an example of how a parent allowed his child to guide the learning.  Good job Mr. Hansen!!

My 3rd grade son is deep into multiplication right now and our discussion about groups of things and multiplication eventually led to factors. And while he was pondering how many numbers can be multiplied together to get 12 or 16, he asked a very good question. He asked me "How come there isn't a division table?" and he went on to say "We could make a division table and sell it!"

I explained to him that division tables do not exist because once you put the numbers down the sides, the majority of the inside is empty because most numbers are not (evenly) divisible by the other numbers. I told him that when we do division mentally we actually use the multiplication facts in reverse, as we did with addition. But I could see in his eyes that my explanation didn't phase him much and he was still dreaming of selling millions of division tables to other 3rd graders. So I said "Let's make a division table!" Big smile.

So I opened a spreadsheet and we started making our table. I said to him "let's just go up to 20 for now and we can expand it later", otherwise we would be there for hours if we went all the way to 144 on both edges. So I numbered the rows and columns from 1 to 20 and then like a game of battleship I started calling off the rows and columns and waiting on him for the quotient, or as is the case in the majority of cells, his response "it doesn't divide".

It took less time than I imagined before he saw the light as to why it is difficult to make a good division table. Maybe it was all the dead ends with him responding "it doesn't divide", but he realized that most of the cells are empty and when you are talking about "whole" division you are actually talking about knowing which combinations of numbers are divisible in the first place which is essentially the multiplication table. But something neat happened. He noticed the diagonal of 1's (you can't really miss that) but then he spotted the 2's, so we studied that for a bit and I started coloring the 1's, then the 2's and then the 3's and so forth so that he could see how they each repeat and form a line. And then I pointed out that there were some numbers with no quotients at all, except for 1 and themselves. I colored those red and pointed out that every now and then those patterns of 2's and 3's and 4's line up in such a fashion that they skip a number entirely, we call those primes. They aren't divisible by any number, except 1 and themselves.

In the end (actually there is no end to this) I told him that what we are doing is turning the multiplication table inside out and if we go in and count all of the filled in cells there will be 144 of them (assuming we did the whole table), one for every entry in the multiplication table. You will find all of the 2's and 3's and 4's and so forth but spread out in a 144x144 table, instead of a 12x12 table.

Our division table is here...

http://dl.dropbox.com/u/39455389/DivisionTable.pdf

It was a very productive evening to say the least

More on Differentiated Assessment

A couple of weeks ago, I was watching a dad teach his son how to ride a bike.  The son had a helmet on, elbow pads, and training wheels on the bike.  As the dad put his son on the bike, he walked behind his son as the son rode the bike in circles in the parking lot.

Just recently, I witnessed the same father and son in the school’s parking lot and this time the training wheels were off.  The dad continued to walk behind the son and the son completed the same circles.  At one point the son fell over and the dad quickly picked him up and put him back on the bike immediately.  After about 10 minutes, where the son did not fall once, the dad stopped walking behind the child and the child started to do more complex paths on his bike.

This is how assessment should be!

It would be ridiculous to mandate that all fathers must spend exactly 10 minutes of time with their child until they stop walking behind them; as each child will require a different amount of time to learn the skill. 

It would be ludicrous to allow a son to write a multiple choice test where, if he scored over 50% (even though he wrote it is ok to play in traffic on a bicycle), the father would let him ride alone as the son “passed the test”, since the son doesn’t understand all the safety issues of riding a bike.

It would be unfortunate if all fathers were required to purchase the same bicycle since not every child is the same height, or has the lower body lengths.

Yet all of these ideas are allowed in schools, why is that?

I wonder what school would look like if instead of holding teachers accountable with mandated common assessment we instead allowed teachers to teach students “how to ride a bike”?

I believe, students would learn the skills at a deeper level before moving on, they could learn at their own pace, and each “test” would be different for each student.

Still not convinced?  Reflect on this picture as it represents the traditional testing model of students.

Why we should not have set deadlines in school.

I wonder what school would look like if we didn’t have set timelines or completion dates for the assessments of students.

This is the thought I wanted to address this year in one of my classes.  Instead of having set dates for exams, and a set timeline for project dates, I created a learning environment that is conducive for the needs of every single student in my class.

Let’s first look at the problem of having a timeline for when students must demonstrate their knowledge.

Usually, a teacher makes their year plan around the goal of covering all the outcomes of the course.  This teacher must make predictions on how long it will take to cover each individual outcome, which is usually based upon previous years and other students.  Test dates are then inserted strategically throughout the year to determine when it is best for the class to demonstrate their knowledge.  The problem….the teacher is worrying about the class not the individual students.

I have heard teachers say they teach at a pace such that the “average students” can follow, and my assessment dates are around when the “average student” should be able to demonstrate knowledge.  By definition then we are actually pleasing no one!  Half of the students will feel this day comes too late as they have already learned the material and could demonstrate it classes ago, while the other half believes that the pace is too quick, and they will need more classes until they are comfortable demonstrating the material.  Once again, it is very unlikely that we are meeting the needs of any student by trying to meet the needs of the “average student”.

How have I changed this?

I teach on the same timeline and give students an assessment similar to this.  DA with Derivatives , but instead of taking 3 days for the test (1-2 days for review then the 3^rd to administer the exam) I provide the student with 1-2 days to complete the assessment.  Students who understand the material quickly are able to work on the assessment ahead of time and complete it immediately, while students who need more time can use as much time as possible.  There is no set date for completion. 

What if a student gets behind?

My first comment would be “Behind what?”  Some teachers have this notion that the pace of the class is the pace every student should be learning at, but does this make sense?  Remember these unit plans are created before even meeting our students, so how can we make a plan that addresses the individual student?  Saying that, if a student is not demonstrating the material at an acceptable standard at a time which you feel is detrimental to learning other outcomes, then instead of giving a bad mark and moving on I sit down with this student at lunch, or after school, and ensure this student learns the material.  Is it not our job to educate students?  By giving a test, and saying “sorry you haven’t learned everything, but I am moving on anyways” is actually not completing our job. 

As teachers we must remember, our class sizes may be large and diverse but this is due to the fact that many individual students are making up this group and our assessment style should not be created to meet the needs of the “average student” but the “individual student”.

One day without mandated outcomes

I was recently involved in a session about creativity in the classroom.  The presenter brought forth the fallacies with common and traditional assessment.  After lots of discussion, case studies, videos, and examples of alternate assessment and instruction the session ended.

First off, I have to say that the staff that participated was amazing!  The open minds, deep conversations and already exemplary teaching styles were quite evident.  However, nothing made me more amazed than the comment the principal said at the end of the session.  After the applause to the presenter, the thank yous and handshakes the principal stood up, looked at his staff and said:

“Imagine if you were driving down the highway in a brand new 2011 Mustang.  The cops have all been called away from the road, the deer have been contained, and all other drivers are giving you the full road to explore and try out this new car”

“Ummm…ok” said a teacher in the crowd

The principal paused then started again

“Now let’s take this to a classroom.  One day next week, let’s pretend there are no final exams, no unit tests, no marks to update, no mandated curriculum, and no one from Alberta Education to tell you what to teach.  Take one day and let your students explore their own learning.  Using the taught strategies in this session, open the “learning” doors for students and just let them discover something that as meaning to them”.

If hugging was socially acceptable I would have given this administrator the biggest one ever.  Driving home, I thought of “What would schools look like if all administrators gave this chance to their teachers?  What if every day was this special day?  What if the needs of the students overpowered the needs to teach the curriculum?

I hope all his staff follow through with his directive and truly allow for authentic and autonomous learning to occur, if only for one day.

Awesome job Mr. Principal, I have found another individual who I would feel honored to call my boss!

Halloween Challenge to Educators

This week, I was inspired by NBC's "The Office" to take up a challenge in the mindset around Halloween.  Near the end of the episode title "Spooked", the boss makes the following comment:

Fear places an interesting role in our lives.  How dare we let it motivate us?  How dare we let it into our decision-making, our livelihood, into our relationships?  It's funny isn't it?  We take a day a year to dress up in costume to celebrate fear?

I found myself thinking about this cartoon.

Too many times, I have heard of educators thinking of an innovative way to change instruction, assessment, engagement, classroom management, and so on, but are too scared of the outcome.  This idea never gets implemented and the success (or failure) of it is never experienced.  Educators are left thinking, "What if....?" 

I challenge you, and myself, in Novemeber to take a "new idea", that may have been sitting on the shelf collecting dust, and try it out!  The outcome could be very suprising...of course you could end up with a day of failure, but then again you could experience one day of great success.  Remember, as illustrated by this 4 minute video below creativity starts with a belief.

Simplifying Radicals with Go-Fish

Here is how I am going to attempt to allow my students to discover why and how to rewrite radicals.

Thank you  to https://pumas.gsfc.nasa.gov/files/12_03_06_1.doc for the inspiration.

Simplifying Radicals With Go-Fish

1)      In groups of 3 or 4, take a deck of cards and remove all the jokers and face cards (Jacks, Queens, and Kings) from the deck.  For this activity ignore the suits of the cards.

2)      Deal out all the cards, one at time, to each person.

3)      Fill out the chart as follows:

a.       Write down the square root of the product of all your cards, as an exact number.  (Aces = 1)

b.      Write the decimal approximation, to 2 decimal places

4)      Put your pairs together, and keep the single cards separate.

5)      Fill out the chart as follows:

a.       The product of each pair. (IE if you have 2 3s and 2 4s, you would multiply 3 by 4)

b.      The square root of the product of the single cards, as an exact number.

6)      Multiply the product of the pairs, to the square root of the remaining cards, and round to 2 decimal places.

7)      Repeat for 10 hands.

8)      Answer the questions on the back page.

9)  Explain, why both ways are giving you the same decimal values.

Example:

Jason was dealt a four card hand which consisted of an 8, 3, 5, and another 8.  He filled in the first row and shown.

Hand Number	Square root of the Product	Decimal, to 2 decimal places	Product of the pairs.	Square root of the remaining cards	Product of the pairs and root of the remaining cards
Example		30.98	8		30.98
1
2
3
4
5
6
7
8
9
10