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.

Supporting the new ideas of the Math Curriculum

Anna Stokke, basically ripped apart the new ideology behind the math curriculum in Alberta, and across the Country, in this article http://www.winnipegfreepress.com/opinion/westview/why-our-kids-fall-behind-in-math-129938903.html

The new math curriculum believes, among many things, that:

·         Students learn by attaching meaning to what they do, and they need to construct their own meaning of mathematics.

·         Students need to explore problem-solving situations in order to develop personal strategies and become mathematically literate. They must realize that it is acceptable to solve problems in a variety of ways and that a variety of solutions may be acceptable.

·         Curiosity about mathematics is fostered when children are engaged in, and talking about, such activities as comparing quantities, searching for patterns, sorting objects, ordering objects,  creating designs and building with blocks

Before I critique Anna’s arguments, we must realize that when someone talks about “meeting” or “raising standards”, they are usually referring to applying conventional, or traditional, instructional techniques in the classroom.  Usually this really means, “Everything we are doing is OK, we just need to be harder on students”. 

Anna Stokke, will have us believe that we have lowered standards in our math classes and we need to raise the standards back up.  Her first argument about the new math curriculum is

“… practise and memorization of number facts are no longer a priority in our schools. Children are instructed to use overly complicated "horizontal" methods to work out simple arithmetic problems. Now, simply knowing how to produce an answer to a basic multiplication question, no matter how long it takes, is considered to be a sufficient indicator of fluency.”

The reality is that being told number facts, and forced to regurgitate these facts, will not create an environment which is conducive to deeper learning.  The best way a student can understand, not just recite, mathematics is through discovering the facts on their own.  For some, this discovery process can take seconds, while for others it may take an entire class.  The role the teacher plays in this is by acting as a tour guide.  By keeping the students on the “path” towards the discovery, it will ensure that all students create their own individual, and innovative, techniques in learning mathematics.

Next, she says that 

“Kids are set up for failure if they are not required to memorize basic number facts. Without the memorized facts, they will become hung up on these simple numbers when they are trying to solve more difficult problems.”

I often ask myself, which skills do I want for my own children?  Do I want them to be taught the skills to memorize, and ultimately be able to complete algorithmic tasks, or the ability to create and design which will lead them down a road towards a career with heuristic tasks? 

The reality is that the age old saying that “Practice makes perfect”, does not apply to deep learning in mathematics.  Traditionally, students have been given worksheets which have multiplication facts on them and are given a strict time limit to complete these (named Mad Minutes), but in actuality the mark given on these “Mad” sheets really do indicate to anyone as to knowledge of the student writing them. 

The weak students truly suffer the most from this model of proficiency-driven, because they find these tasks dull, repetitive, and entirely unusable in the world outside the walls of the classroom.  I would agree that sometimes, knowing facts as the days of the week are important, but these facts should be a by-product of use and application not out of necessity. 

Another fact is, by spending time forcing these facts to be memorized is truly interfering with a child’s innovative and creative ability.  Most of these worksheets require low level thinking and usually can lock the thought process of a student into understanding simple algorithms.  Eleanor Duckworth summed it up the best,

“Knowing the right answer requires no decisions, carries no risk, and makes no demands.  It is automatic.  It is thoughtless.” 

Is this the environment in which math students should be learning?

Last she makes the statement,

“…children need to practise arithmetic skills, without calculators, do an adequate amount of pencil-and-paper math, and memorize times tables in order to become proficient in math.  Children need to be given time to do a reasonable amount of math daily at school and this needs to be a priority.”

This closely sounds like a case for homework at younger levels and there is not one shred of evidence that supports the idea of homework in elementary grades.  Assuming this is not her argument, I will critique her argument for repetitive daily work in school. 

When we teach math as “routine skills” students may get the correct answer, at an efficient rate, but they will most likely be clueless about the significance of their of their answer.  The National Research Council calls it “Mindless mimicry mathematics” and can no longer be the norm in our classrooms.  Another math educator, William Brownell, over 70 years ago explained “one needs a fund of meanings, not a myriad of ‘automatic responses..’ Drill does not develop meanings.  Repetition does not lead to understanding. 

There are many studies that support this view of thinking.  If we look at an environment where students are not learning effectively, it is usually due to an overwhelming desire to maintain traditional beliefs and practices.  Once again classrooms need to be aware that

It is important to realize that it is acceptable to solve problems in different ways and that solutions may vary depending upon how the problem is understood.

Rate of change of beaker.

Here is a lesson you can do with your students on related rates of change:

Before this, ensure your students understand at least similar triangles and have seen some related rates questions before.

1) Bring students to a chemistry lab with different beakers, or bring different beakers into your class.  **You will need a water tap**

2) Divide the students into groups of 3 or 4 and provide each group with one of each of the following beakers, shown in the picture below.

 3) Provide the following question to the students:


 Hypothesis:  Will the height change at a constant rate or will it change throughout, if we were fill these beakers up with water? Explain.  If the rate of change of the height of the water changes, when will the rate be the largest, and when will it be lowest.



Procedure: Turn the tap on and record the time it takes to fill the beaker up to the last marking on the glass, using a constant stream of water.  Record this three times and average your times.  What does this represent?



Fill up both beakers, using this stream of water.  Was your hypotheis correct?  If not, what did you notice?



Calculations:  If you were to fill up the cylindrical beaker with the stream of water how fast would the height change when it was half full? When it was entirely full?



If you were to fill up the conical beaker with the stream of water how fast would the height change when it was half full?  When it was entirely full?



I have used this before with my students and watched how they had to determine not only WHAT to measure but HOW to measure it.  Feel free to use, change, tweak as you see fit.

Here is video one student made:

**SOON TO BE UPDATED**

And his Prezi:

Copy of Dante Bencivenga's Unit 3 Assessment on Prezi

Cut the Rope in math class.

I provided my students with a laptop, which had Geogabra on it, and the following picture:

I then showed, on my iPad, how the rockets travels around the point.  When the rope is cut, the rocket will fly tangent to the curve.

I used this in my calculus class and asked "Determine the exact location the rocket should be when the rope is cut"

Students determined the equation of the circle, found the derivative and made it equal to the slope between any point (x,y) on the circle and the co-ordinates of the mouth of the monster (or goblin).
**WARNING, this is a hard question**

After, I was thinking this could be used in lower classes, by changing to one the questions below:

•  allow students to use the tangent button on geogabra, and ask for the reference angle.
• Determine the amount of time which should pass before cutting the rope.
• There are always two tangent lines on a circle through an exterior point, are there two times you could cut the rope in the picture?

There are more I could think of.

My students enjoyed the activity, and feel free to change and adapt as you see necessary. 

Angry Birds and calculus:

First provide the students with a laptop with Geogebra, and the following photo.

Introduce the meaning of Piecewise functions, and how the yellow bird, when clicked, shoots off at a tangent to the curve.

Ask the students to describe what are the two functions that create this curve (Parabola and a line).

Using prior knowledge have students graph the maximum point on the parabola, and use the dot where the yellow bird took off at a tangent (B), to create the equation of the parabola.  Have students graph the parabola in geogabra, overtop the picture, to ensure all calculations have been done correctly.

Next, ask students how to determine the equation of line.

We will need either a) two points or b) a slope and a point.  Both of which is impossible, without the use of the tangent button in Geogabra.  I explained, we can calculate this without that button!

Have students pick another “point on” the parabola (c), and to calculate the slope between B and C.  Ask how do we make this slope more accurate to the slope of the tangent…

Move C closer to B..

Have students move C closer to B, while still staying on the parabola and calculate the new slope.  Eventually move C as close as possible to B.  The slope should be -0.5  Below is a picture of one of the students’ work

Next using the point B and the slope you can create the equation of line.

From here your choice for extensions: I had students graph the piecewise functions on their calculator and got the following image

Lastly, here is a video on how to determine the slope at a point:

Learning can occur on a final exam

Historically, I would say that the least amount of learning occurs during exam week.  This year, however, I have tried to make this statement false through giving my students a new final exam.  When the first group presented, not only did I see, hear and was taught about unique examples of how to use calculus, I can actually say the group still LEARNED through the assessment. 

Below is a video of their PowerPoint presentation, which they used as well as supplemented with dialogue. 

After the presentation they opened the floor to me by giving me 10 minutes of questioning.  Most of the questions they answered swift and correctly, until I asked “Do all functions, on a closed interval, have an absolute maximum?” . Their answer was “Yes”.  Of course this is incorrect.

Now here is how the learning occurred… First I will address the traditional way of assessment:

If this was a traditional final exam, I would have marked this question wrong and moved on to the next question and continued marking.  These students would never have received any feedback, as in the past, I have yet to see many students come back to see WHAT they did incorrectly on a final exam.  These students would have then gone on to university/college with this false knowledge.

How it has changed this year…

I didn’t let this false information continue.  I then asked, “What if I told you, I could draw a function on a closed interval without an absolute maximum?”  The girls looked at each other with confused eyes, and pondered the idea.  After some passing minutes, one replied accusing me of a liar.  I wanted to ensure they didn't continue on with this false information, so I then asked, “Is there any kind of function that continues upward on forever?”.  One quickly answered, “WAIT! A function could have a vertical asymptote and therefore have no absolute maximum.  I guess you weren’t lying”, the other girl smiled and agreed. 

After this, I wanted to ensure they had a true understanding and therefore I asked, “Can a function, without any asymptotes, on a closed interval, not have an absolute maximum?”  The enlightenment has occurred!  Both girls whispered quietly, and then turned and replied “If the curve has an open point at the highest point, then it would not have an absolute maximum”.

Learning had occurred, and yet it was a final exam.

I continued with my questioning, which they answered correctly, and I am happy to say that this experience has been a success with the first group!

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.

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)

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