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.

Angry Birds and Calculus

During my first unit, instead of assessing with a traditional exam I used an open ended project.  One of my students submitted the following video to answer question 3:

As you can see, he used a timer on his phone, a ruler and the game Angry Birds.  Here is his work

Geometric Sequences Reductions

Here is an activity I created to introduce geometric sequences. 

1) Watch the movie below:

2) Determine the dimensions of the last image if the original image was 210 × 297 mm.


**I had a copy of the original and final image and had students measure the original on their own, and then check their answer by measuring the final image.**

3) Extenstions: If we combined all the images, determine the total area.

If I extended this sequence to infinite, what would be the limit of the sum of the sequence?

Intro to Tangents

Here is an activity I used in my Math Calculus class to introduce tangents at a point.

I gave each student a copy of the video below.

Then, in groups of 2, students had to answer the following questions (which is part of the comments in the full YouTube video)

1)  Watch the video

2)  Determine the average speed for the first 5 minutes.

3)  Determine the average speed for the entire trip.

4)  At the end of the video, the speed of the vehicle is 109  km/h, explain any discrepancies from your calculation in part 3.

5)  Determine the average speed of the vehicle after 15 min of driving.

6)  Determine, with the least amount of error, the speed of the vehicle  after 15 min of driving.

7)  Explain, if any, the difference from part 5 and part 6.

Feel free to use, change as you seem necessary.

For the parts that is hard to see:

The time at each break in the bottom right corner is the total driving time to that point.
The distances at each break are -7.6 km/16.7 km/23.3km/32.1km

I do apologize the video is quite horrible with YouTube.  The original is much better.  If you require a copy of the original, email me and I can get you a copy.

2 movies about math

Here are two YouTube movies which, through edititng, could become an engaging math problem.

If you would like to use either and can't edit them to fit your needs, please let me know and I can edit or "beep" out certain parts of the movie.


If you do use them, please let me know how so I can share with others.

Creativity in Calculus

This year I gave my students an open-ended project.  Below is a video that a student submitted.  The project was entirely open-ended and when I assigned it I informed my students that it will be worth NO MARKS!  Here is what I got from one student.  (I will be uploading more but most are too large for YouTube). 

Three students completed a 17 min video, another student completed an amazing powerpoint video, while a third group integrated calculus into a the world of Harry Potter.  I truly was inspired when I witnessed the level of creativity and engagement completed by the students.

Was the motivation really from marks?  Can't be that, as there were no marks.  Maybe the motivation was from the chance to be autonomous and creative?  I did not force my students to complete the project on a specfic outcome, nor did I force the tool they were to use to illustrate the outcome.  All the task was "Show me something cool you have learned"

Here is the project that my department head created and I tweaked.

 Math 31/35  Multi-Media Project

To date you have covered several concepts within your math units.  You now have the opportunity to use your other talents to share your knowledge with others via a multi-media presentation!

What kind of presentation?

That’s up to you….Powerpoint…video tape an interview with an expert… videotape yourself teaching a concept…videotape a skit showing mathematical instruction… a website.. a math lab…an everyday application of a learned concept…the possibilities are unlimited.

Some questions you might want your presentation to answer:

• “When am I ever going to use this math?” and/or
• “How could I help other students better understand a concept?”

Your presentation could focus on a single concept or on an entire unit

Do I have to do this on my own?

You can if you want, but that’s up to you.  You can form a small group, I would suggest 3 or 4, but if you need more talk to me.  Your group members can be from this class, any Math 31/35 class, or friends outside of class that have a talent that could add to your presentation.

What do I get out of this?

You gain an increased understanding and appreciation for mathematics.  You would also be leaving a legacy behind here at the school, as I intend to use these projects to introduce and supplement unit material to future Math 31/35 students.

How do I get involved?

All you need to do is fill in the attached Project Proposal sheet and have all members of your group sign the release that allows me to use the material.  Then let the fun begin!

What if I start it, but don’t finish, or if there are problems with the group?

Keep me informed on your progress.  Whatever support or assistance you need I will try to provide.(ex.  Video camera, computer access, talking to group members, assisting with an expert to contact etc..)    Whatever you have accomplished by the due date I would like you to submit, even if it doesn’t turn out exactly as you thought it would.  If you need an extension you must speak to me early regarding the reason for the request and then we will discuss it.

When is it due and how long does it have to be?

The Project Proposal sheet is due to me by Thursday, Febuary 11.

The presentation is to be submitted by Tuesday, April 14^th or sooner.  

As for the length of the project, the only direction I can give you is that it needs to be long enough to answer the question that you have set out.

Using post-it notes and a flip cam to answer math questions

 Does the price of engagement and real learning have to be in the $1 000s?

The price of a flip cam - $159. 99

The price of a pack of post it notes - $5. 00

The price of class time – 84 minutes

The price of true engagement – Priceless

The above items are all that was required to have students complete a statistics question with meaning and creativity, which is illustrated by the video.

What was the task?

In class we were investigating the average shots on goal per game in an NHL game.  The class was put into groups of 3 or 4 students and given various questions.

With this project I tried to create meaning to the questions, and not just give students meaningless numbers.

Students worked in groups and then had to create stop videos using post-it notes and illustrating how they solved each question. 

What truly made me smile….

At the end of class, my students asked me what I was going to do with the videos.  One student even asked if we could upload them onto YouTube.  I informed them that I would compile them all, remove the names, and attach some music.  The uproar that occurred was amazing.  Students were upset that I was going to remove the names!  Here are the comments:

“I am proud of this and want to show it off!”

“Can I get a copy of our video to upload on my USB drive?”

“Don’t delete our names, I want my name on in it”

Students were truly proud of their work.  In my 4 years of teaching, I have yet to witness a student be proud of the worksheet he/she completed, or the meaningless task he/she accomplished.  I am not implying this has never occurred, but I have not witnessed it yet in my own classes. 

I still remember back to my first year teaching when a student informed me, “Worksheets suck and I told my teacher that.  He still gives them to me.  I now hate [the subject]”.    What amazed me by this comment was that the student started to hate the course and not the teacher.

The most eye opening experience for me was when a student told me, “The way you have asked me to do math was what made it difficult”.  I had to truly leave my ego aside and embrace this comment.  Maybe I was the one who made the class difficult!   This comment was from 3 years ago, and in a class with the same lesson plan every day:

1)      Review homework

2)      Give students hand-outs to follow while I write on the board questions of increasing difficulty.

3)      Show and complete a word problem

4)      Hand out worksheet

5)      Give Pg. XX questions 1-XX odd for students who finish early

Since then, I have burned this lesson plan alive!

Many people have asked me, “why did you start changing the way you teach?”, and my true answer is “Because I don’t want to hear another comment like that from a student”

 I also have to say thanks to Geoff, or twitter: @emergentmath for recommending I look at the site:

http://emergentmath.wordpress.com/2011/02/11/post-it-notes-animation/

What kind of teacher are you?

Meaning first, videos second.

Usually on the last day before Christmas, a christmas movie or activity is completed or "busywork."  This year I decided, instead of watching a movie, why not let the students create one?  Today, I tried an activity that I read about on a blog.  My students were put in groups of 3-4 and given 11 questions of review.  They were then given the following instructions with a flipcam (mini video camera).

Directions

Figure out the solution to three different problems on paper and check to make sure it is correct. Check your solution with the answer key.

Plan out a 2 minute or less presentation. See Presentation below.

Write out the section/problem number(s) on your whiteboard or desk.

Write out the question with any key information on your whiteboard or desk.

Write out part of the problem’s solution on your whiteboard or desk.

Remember each person will do some explaining in the video.

Make sure that your writing is big enough to see through the video.

Get a “FLIP” and read the instructions about “The FLIP”. See below.

Once you are finished with your presentation, return the “FLIP” and begin working on the rest of the homework assignment. Make sure you put a note-card with your names and section/problem into the black bag of the FLIP.

Presentation: 2 minutes or less (redo your video if it is over 2 minutes)

VIDEO

1. Introduce yourselves: first names only.

2. Read the section/problem number(s) from the writing on your whiteboard or desk.

3. Read the question and any key information from the writing on your whiteboard or desk.

4. Explain the solution that you have written out on your whiteboard or desk so far.

5. Finish the problem by actually writing in front of the “flip” while explaining the solution.

6. Thank the audience for their time and to have a good day.

The students started to collaborate and solve the questions.  Surprisingly, on the last day before Christmas, students were engaged on meaningful tasks.  Below are some of the videos that were created.  I published one of them from my ESL student explaining in English how to do math.

ESL Student talking math

Another student discussion elimination.

I cannot find the original blog were this activity was created, if you do know the author please comment so I can give him/her the recognition he/she deserves.