Friday, January 19, 2018

Providing Timely Feedback.

Feedback, Confirming, Board, Blackboard  I will be the first to admit that I struggle with ways to provide timely feedback.  I teach 6 different math classes, one of which is combined Algebra II and College Prep while another is called Fundamentals of Math made up of three groups all at different levels.

I finally realized students who asked "Is this right" provided a perfect opportunity for immediate feedback.  On the other hand, I don't always have the time to provide feedback as soon as I want. 

Feedback is much more than grading assignments and returning them.  A 95% indicates they know the material while a 50% tells you they know less but it doesn't tell you what they don't understand yet unless you analyze the missed problems.

it is important to provide task specific feedback where the teacher comments on things to improve.  I've been commenting that two or three words does not provide enough information when answering a question asking them to explain something.  In addition, it is good to ask students to provide self reflection on their understanding of the activity or material so the teacher can immediately adjust their teaching.

There are five steps in providing effective feedback to students regardless of the subject.

1.  It has to be immediate before they've learned it wrong.  One way is to post the answers so students can check to see if they did the problems correctly.  Another way is to utilize technology that provides that immediate feedback.  The goal here is to make sure students know they have not gotten it right before doing it wrong has become habit.

2. Feedback must target the students' specific need so feedback should be personalized and not applied to the whole class in a generic manner.  There are numerous sites which allow students to work a few problems and the site either provides the student with targeted feedback or gives the teacher a printout with information that can be used to help the student.

3. Feedback should be concrete such as when I noticed one of my students forgot to carry during multiplication.  Once I noticed that, he began to watch for that and began getting the problems right.

4. Do not just point out the errors, give them tools to help do the work.  Help students build upon previous successes so they continue to improve.  Remember a teacher cannot teach the material one day and expect them to take work home to do independently.  They will not have had time to learn.  Spend time helping them understand the concepts before expecting them to work outside of school.

5. Allow some of the feedback to help build their confidence because the more confident they are in math, the more likely they are to persevere and work their way through harder problems.

This is just a start.  I'll revisit the topic in the future.  Let me know what you think.  I'd love to hear.  Have a great day.

Thursday, January 18, 2018

360 degree photos and Math.

Lens, Fish Eye, Macro, Photography  If you follow any crowd funding places, especially the ones with cutting edge technology, you'll have seen several offering fish eye lenses for various phones. I was recently offered a fish eye lens to pop on my iPhone to make 360 degree pictures and videos.  I don't own an iPhone so it makes no sense to buy it.

I've been trying to figure out how to incorporate 360 degree material into my class but its taken me a bit.  However, I  have found a few things. 360 degree videos are often referred to as spherical videos which use Mobius transformations for purposes of editing.

which are used to map a one to one mapping from one domain to another domain. 

Mobius transformations are used to map a one to one from on domain to another.  In addition, these are transformations of the sphere such as making regular rotations of the sphere, zoom like transformations, and other similar effects.  The first transformation turns the pixel coordinates into angles, the second transformation takes it from the equirectangular projection while the third deals with complex numbers.

On the other hand if you look at a 360 degree photo also known as a fish eye projection, it appears distorted but its not. Its actually a three dimensional projection onto a two dimensional plane.  This is what gives it the peculiar look.  There are programs which convert 360 degree photos into landscape shots so they look more "normal."

In simpler terms, the math involved takes an image which is circular in shape and creates a more rectangular shape through the use of "uncurling" the lines. Think of it this way, your source image is  2l by 2l and you want to make it so its destination image is 4l by 1l. Only the pixels inside the inscribed circle of the source make it to the destination image.

The pixels along the top of the destination image come from the circumference of the source image. The formula for the math conversion is (4l-x)/4l * 2pi where x is the Cartesian X axis.  In addition, when iterating from left to right on the destination image is the same as going clockwise on the source image. 

The radius (remember a circle has a radius) is calculated as l - y where y is the Cartesian Y axis.  So iterating from top to bottom on the destination image is the same as going from the edge to the center of the source image.

There are software programs out there with the math already present so you can convert your photos without having to do the math but its nice to know what these programs do when the conversion is carried out.

Let me know what you think.  I'd love to hear.

Wednesday, January 17, 2018

Persistence is Stamina

Dog, Water, Run, Movement, Joy  As I looked up ways to increase student persistence, it came up as increasing student stamina.  I like that idea.  Helping students develop persistence is not just a goal in mathematics, it is needed for all subjects.

The English teacher comments on how students believe that one draft is enough to produce a wonderful award winning epic while the social studies teacher noted that once students get slightly behind, they feel as if they cannot catch up and give up.

There are ways to help students build  persistence or stamina.  Students need  persistence or stamina so they can work through challenges, deal with failures, and meet all goals they set for themselves. Unfortunately, most refuse to believe that learning math is like learning a sport. 

 One of the first things is to help students change their mindset from can't to can.  Too often, they have a little voice in their head which convinces them they cannot do it. That little voice tends to override the teacher saying they can learn.   One way is to praise them when they are focused on meeting specific goals.

Next, give a gentle push when they run into a bump by using encouraging words.  Its ok to let students know they can take a bit of a break before resuming work. Many students just give up rather than going back and many math teachers want students to finish it all in one sitting rather than acknowledging the brain needs breaks to function at peak efficiency.

Furthermore, it helps if the teacher models persistence by sharing an incident where the teacher overcame something.  It could be something as simple as replacing the toilet that broke but explaining the problems they faced.  It might be when they were in college and struggled through a class.

In addition, it is good to give students optional ways to talk to themselves so rather than saying "Its too hard." the could say, "I could ask the teacher for help."  or "It'll get easier with a bit more practice."  Most students tell themselves they can't do it rather than encouraging themselves.

The teacher also needs to hold students to high expectations while letting them know they can meet those expectations and providing them with the tools they need to do that.  Its also important to incorporate technology because it can provide immediate feedback.

One thing teachers tend to avoid is taking the time to explain how the brain learns and how it changes as it is used. Finally, incorporate repetition of these strategies so students become comfortable with them and increase their stamina.

Let me know what you think.  I'd love to hear.

Tuesday, January 16, 2018

Vectors in Real Life

Turn On, Turn Off, Arrow, Success, Trend  I usually teach vectors about once every other year and usually when I have to teach a semester or two of physics because its easy to integrate vectors there but easier than in Math.  Vectors are one of those topics that are much harder to teach without a context.

I had to do a bit of research so I can teach it in my College Prep math class.  I needed to know more about their use in the real world because its important to show relationships and connections.

Many of the ways vectors are used in real life are done so without using the word vector.  This is a good thing to shae with students. So on to the way vectors are used in real life.

1.  Air traffic controllers give pilots a specific heading (direction) with a specific distance (magnitude) along the planned route.Vectors are also used in flight patterns to take into account the wind blowing with or against or cross the plane's flight.

2.Vectors are used to aim cannons and other heavy artillery.

3. In baseball, any player must run in a certain direction going a certain distance in order to intercept a ball.  The player also has to predict where the ball will be to catch it, not aim for where it is now.

4. In basketball, football and golf, the player uses vectors to determine the trajectory of the ball in order to make a basket, get it to the right player to get it to the goal or in the hole.  In football, the quarterback must take his own movement into account, the receiver's movement and the path of ball, all of which can be represented by vector arrows.

5.  When someone asks for directions, they are given them via vectors such as two blocks left, then three blocks right.

6.  Many video games such as Angry Birds use vectors in the shooting of birds at objects.  The vectors include the amount of the pull used to launch the bird and  the angle of launch.  This is actually the most relevant use of vectors for my students.  Vectors are used in video game development to describe the location of an objects or physical simulations of objects.

There you have it, six different situations where vectors are used.  No one thinks of arrows or directions with speed but they use it every day instinctively.  If people want they can create a diagram of a football play off of the television or a shot from a basketball game but we know what we are doing without knowing all the proper mathematical terms.

Let me know what you think.  I'd love to hear.


Monday, January 15, 2018

Writing in Mathematics.

Write, Plan, Business, Startup, Start-Up  We hear more and more that students need to have the ability to explain what their thinking is but writing can play a much more important part in the math classroom.

Most math teachers are not trained in writing.  We are trained in mathematical formulas and solving for unknowns.  Thus when we have to integrate writing, we try or ask the English department but they don't have a reference to writing in mathematics.

It appears there is a minimum of two types of writing.  First is writing to learn which uses short or informal writing tasks designed to help students think about key concepts and ideas.  These activities are sprinkled throughout the lesson and focus on the concept, not on proper writing techniques.  This type of writing is found in journal writing, logs, written responses, etc.

The other type of writing is referred to as writing to demonstrate knowledge in which they show what they've learned and show their understanding of concepts and ideas. This type of writing requires students to write for a specific audience using more formal language and are checked for grammer, punctuation, etc.

To turn writing into a learning experience, it should include more than just copying notes.  It should personalize the writing by asking them to reflect, ask questions, which helps them better understand the concept. 

It is best not to assume students know how to write in mathematics. There are suggestions the teacher can implement to help students learn to write for mathematics. 

1.  Explain the strategy and its purpose.  Include real life examples if possible and who the audience is.  This gives students more understanding they are not only writing because the teacher said so. 

2. Model samples of the type of writing you ask students to do.  it is important to include the type of thinking that goes into the process both before and during. 

3.  Give students a chance to practice the strategy or process in small groups before having them do it alone.

4. Provide timely feedback and have students use the feedback when they rewrite the piece. 

5. Encourage them to become more independent.

As far as strategies go, most math teachers lack knowledge of strategies, especially if they are not trained in writing or work with students who may lack mathematical vocabulary. I found this 33 page write up filled with possible strategies.  Each strategy is described with information on how it helps the student, implementation suggestions, and examples. The pfd includes 14 writing to learn strategies and three writing to demonstrate knowledge. 

Some of these strategies I've seen before but most of them I haven't so I will have several more strategies to incorporate into my teaching, especially with my lowest math group, many of whom read and write poorly.

Let me know what you think.  I'd love to hear.