EDPuzzles

I started using EdPuzzle in my classroom this week.  In case you are unaware of them, this site allows you to choose already created videos or to create your own. 

For my Algebra II class, I created several videos at EdPuzzle, all dealing with solving systems of equations. 

I signed into the site, selected a video, added questions at various points, and cut the excess off the length so it covered only what I wanted the students to see.

It was wonderful. I asked the exact questions I wanted answered. They could answer the questions correctly if they listened to the video.  I used short answer questions but I could have used a multiple choice or true or false instead.  If I'd chosen to, I could have created a new sound track for the video, or even inserted a comment to all the students.

I created a document with links and instructions for the whole topic of solving systems of equations either by graphing, substitution, or elimination.  For the first video, they watched, and answered questions but it was obvious they were not paying close attention.  Come the second video, they started repeating segments to find the correct answer.  They even asked questions of material so they understood it better.

It freed me to monitor student learning.  I could answer individual questions, clarify misunderstandings and all the sudden I felt like I was reaching each student rather than trying to lecture up at the front and hoping for the best.

My Foundations of Math has four groups, each working at a different level.  Since I cannot teach one lesson, I'm starting to look for already created videos I can assign to students in each group.  I just started and so far, it looks like its going to work well.  With questions embedded into the videos, they must think about the answers.

The best thing, is that I can export the video to and assign the video to any class in google classroom.  That is fantastic because I can personalize the instruction making them more independent learners.  I just love using this site so much.  It is actually making my job easier in the long run.  Check them out.

Let me know what you think.  Have a great day.

Elementary Math

  Yesterday, I was speaking to the 6th grade teacher.  He is good at math and spends 1.5 hours everyday working on math with his students.  He wants them prepared for high school. 

Unfortunately, he stated that the 5th grade teachers from last year did not do a good job with the students.  Too many of them arrived not even knowing how to do fractions.

As you know, I work in a one school district with grades pre-k to 12th grade all in one building with three wings and a gym.  Unfortunately, we often have years where teachers are there one year before moving on. 

Last year was one of those where almost everyone in the elementary school turned in their resignation.  We often get new teachers who are just out of training and are scared of math.  If something is not in the math book, they won't do it.  Many do not know how to supplement their teaching with activities to liven things up.

The thing about textbooks, is they are only as good as the teacher.  If the teacher does not have a full understanding of math, they will stick to the book and refuse to venture out into helping students learn more than the process for completing problems.

 It wasn't until I began looking into the why do we do things this way that I discovered the reason we change fractions in a division problem into a multiplication problem. I've learned cool ways to show primes in pictorial form. I've even learned to explain why anything to the zero power is one.  If I hadn't taken time to research how these topics and more, I would be still teaching processes without including the background to provide understanding of the concepts.

I wouldn't even know how important it is to have students include written explanations for each step.  I've just started doing it and my students already hate it but it forces them to really think about what they are doing.

I wish more elementary teachers showed an interest in learning the why's in math rather than just teaching according to the textbook and using only supplemental materials.  The only suggestion I have on this topic is to provide short weekly classes on one topic to provide the background needed to fill out their understanding.

If you have any ideas let me know.  I would love to hear from people.  Have a good day.

Graphic Novels In Math.

After reading Solution Squad, I wondered if there were other graphic novels I could use in my Math classes to create interest.  I found a few more, some I've seen, some I haven't but I'll share several to give you a better idea.

Imagine using graphic novels instead of relying on standard textbooks.  It might be easier to get students to read and pay attention to the material.

It turns out there are several graphic novels out there to learn math from.  One series is done in Manga style.  The three books in this series are the Manga Guide to Calculus, the Manga Guide to Linear Algebra and the Manga Guide to Statistics, and the Manga Guide to Regression Analysis.

All four books are rated at 4 stars or higher on Amazon.  These types of graphic novels are referred to as EduManga which combines Manga with Educational content and is all the rage in Japan.  All of these books have been translated from Japanese to English.  These books take the material and encase it into stories so they become an intricate part of the story.

In addition, Larry Gonik has a series of the Cartoon Guide to Calculus, the Cartoon Guide to Statistics, and the Cartoon Guide to Algebra. Larry Gonik may be better known for his Cartoon Guides to various science subjects.  I became familiar with him when I read his Cartoon Guide to Genetics.  His three books rate 4 stars or higher at Amazon.  His books use humor to convey mathematical topics. 

The last set of books for today are the ones titled Manga Math Mysteries.  This series of eight books was written for eight to ten year old students but could easily be used with older students who are English Language Learners or who have print disabilities.

Each book covers a different topic such as whole numbers, time and temperature, probability, geometry, fractions, money, measurement and distance.  The main characters are youngsters who take martial arts and who solve the mysteries together.  Each books is about 45 pages long but done in standard comic strip format.

Let me know what you think about the idea of using Manga in the Math class.  Have a good day and enjoy.

Solution Squad

  Over the weekend, I became acquainted with a graphic novel of super heroes who solve mathematical problems.

These super heroes are better known as the Solution Squad. Each member has a superpower that are actually math based.

One member can perform complex calculations in her head, while another is able to create electromagnetic triangles in the air. 

The third member can represent absolute value, the fourth member is able to copy the powers of anyone around her just by a touch, the fifth is one of a pair of twins who can speed across the land at hyper speed while her twin can fly up or down and withstand huge changes in pressure.

This graphic novel was created by Jim McClain as a way to teach math.  He sells a hard copy of the graphic novel but you can get a digital copy of the first adventure.  There are links for all of these.

I purchased the Solution Squad in Primer, the digital copy and the lesson plans separately.  I love the super heroes have a boat named with a great mnemonic from trig.  They have a super plane again possessing a cool mathematical name. We get to read about their adventure with

The teachers guide provides lesson to accompany the graphic novel complete with the activity sheets.  I love the way he includes the Fundamental Theorem or Arithmetic as one of the activities which shows how one can fully factor any composite number using only prime numbers. 

In addition, students are exposed to Goldbach's Conjecture which states any number greater than 2 can be expressed as the sum of two primes.  Students are given the chance to practice the conjecture.  Another fun math concept is the idea of Happy Primes vs Unhappy Primes.  In a Happy Prime, you add the squares of all the digits in the number to get a second number.  Repeat the process with the each additional number until you get 1.  This is a lesson plan I could leave for a sub to do.

Check out the blog and the additional Web Comics.  I have another tool for my toolbox.  Thanks to Twitter for this lead.

I could easily use the Solution Squad in my Pre-algebra and Foundations of Math classes.  I think it would keep my students interested. 

Warm-up

Warmup

Desperation and Inspiration.

  Yesterday, in geometry, I had students work with points, lines, and rays so they could develop a better foundation for future topics.

The activity was to look at various scenarios such 2 points, 2 lines, 3 lines, 2 rays, etc and they had to suggest an arrangement that produced one, two, or three points.

The two points was easy since all they had to do was stack the points to create one point but they could not arrange them in any way to create two points other than leaving them separate which violates the rules.

With so many blank looks on faces, I know they really couldn't picture solutions to the activity so I pulled out a large number of sticks and passed them out so students could play with them to find the answer.  They gave me crazy looks but I convinced them to arrange and rearrange the sticks until they found solutions.

It was great watching them working together, sharing sticks and ideas.  I had to convince one group of girls that the vertex of the angle did not count as a point.  Instead the angle sides had to cross to be counted.  I loved the way students would think about the various arrangements before deciding if they matched the requirements.

I've used the activity before but this is the first time I've used manipulatives of any type.  It made a tremendous difference in that students had lower frustration levels and enjoyed the activity more.  From now on, I am going to use manipulatives for this exercise.

I was desperate so I found the sticks in my closet because I didn't have anything else that might work for this activity.  I save things "Just in case".  This met the description of "Just in case."  It is amazing how desperation gives us that bit of motivation to get really creative.  I've used sticky notes when teaching function notation, substitution, and any other topic requiring a number replace a variable.

I've created number lines on the floor to help students learn to use positive and negative numbers.  They loved walking back and forth to find the answer various problems.  I've used desks for explaining the angles associated with traversals and for the coordinate plane.  Each idea came out of desperation.

Let me know what you think.  I love hearing from people.

3 Dimensional Shapes.

When I teach the volume of three dimensional shapes in geometry, I use a very specific order.  I begin by giving a list of shapes to students so they can research nets on the internet.  They must draw their own nets on paper using a ruler and pencil.  I realize this is old fashioned but they see the number of squares, rectangles, triangles, or circles used to create each 3 dimensional shape.

This is important because this information helps students derive the formulas for volume and surface area.  They use put their nets together so they have the shapes in hand when they derive the formulas.  It makes visualization much easier.

Next I have them work their way through the app Geometry: Volume of Solids Lite.  The app uses videos to explain how to find volume on sphere, cone, cube, cylinder, and pyramid. This reinforces the formula's they derived from the models.

Of course, I give problems to work but I just stumbled across one more thing I can integrate into the topic to add another dimension.  In order to make it work, the students first must know how many edges, faces, and vertex each shape has so they can answer riddles for each shape. I found a slide share site with several riddles or they could make a set of their own.

This would be a great topic to have groups of students create a slide show at the end of the unit.  Each group should include the shape, its net, equations with examples for volume and surface area, real life uses of the shape complete with pictures and a write up, and finish with a riddle.

Its nice for students to find out they need to calculate the surface area of the inside of a rectangular prism (box) in order to buy the correct amount of paint, or find the area of one surface so they can purchase enough flooring for a room. Most of my students are unaware this is the most common use for surface area.

By finding real life uses of volume and surface area, they find a connection between class and the world around them.  Let me know what you think.  I'd love to hear.  Have a great day.

I am trying to get away from using only worksheets so students have a chance

Revelation

  Today, I started teaching function notation to my Algebra I classes.  They got as far as learning to take the value given and substituting it into the equation to get an answer.

After class, I realized I need to take this one step further by having students connect the input and output as giving a location on a line.

I fear I am guilty of teaching functions, function notation and linear equations as separate entities rather than showing their relationships.

I found a wonderful chart, I wish I had to introduce functions. It gives examples such as selecting K on a snack machine after putting money into it (input), the machine processes the information, and a Twix candy bar comes out. (output) or A zero on the phone is pushed (input), the phone sends a signal to a receiver (process), and the operator answers (output).  Practical examples of input, process, and output.

By introducing the topic this way, it identifies prior knowledge that can be built up.  It is easy to relate this to the value for x being the input, substituting the value for x and doing the math in the equation, with the output being the answer.  Once students learn function notation, the next step would be to relate this to a graph. 

Since functions and graphs are related it is important students be taught the two together so they get a chance to understand the connection.  In addition, its nice to relate function notation with an actual value representing a point on the graph to that point. 

When I went through my teacher training program, the world still treated this topics as separate topics.  In fact, most of the math I taught was taught in isolation from other mathematical topics.  So as I read, think, and reflect, I find more and more connections I try to share with my students.

Once I get students used to thinking in terms of function notations, functions, and graphs, I want to take them to the next step showing the four ways points can be represented but I hope to do it so they see the representations are related. 

I'll let you know how this goes.  I'd love to hear your thoughts.  I'm off.  Have a great day.

HIstory in Math and Books

  The first week of school is one of the hardest because students are changing schedules, figuring out they have to be up earlier than during the summer, or returning from their summer trips.

Our high school allows one week for students to make any changes to their schedules.  It is also the week where students who are transferring to boarding schools disappear from the rolls. 

Some of my students know school has started but are still on their summer sleep schedule.

This is the perfect time to sneak a history project into class before starting the real work.  I would have done it but my iPads are not available yet.  It might be another two or three weeks before I get them but I'm going to keep this in mind for next year. The idea is to create either an ebook or a slide presentation as a class.

Yesterday, I provided a brief history of numbers but you could use famous mathematicians, culturally based math systems, mathematical tools, measuring systems or any topic you desire.  The first two days could be used to have students research topics individually or in small groups.  They need to research detailed information.  I might assign the a student to find out more information on the bone, another to research the Sumerian system for commerce.  In other words, break the topics down so students know what to look for.

This is a great time to remind students about plagiarism, copyrights on pictures, etc to remind them they have to follow the rules any time even outside of the English classroom.  I've been working with my students to provide a URL for where they got the information.

Once everyone has had a chance to research their topic, they need to put together the information. There are three ways to do this.  The class can create a book using iBook Author, Book Creator, or Google slides.

Each chapter can cover one topic or more than one topic depending on how you want it divided.  I might have the three Greek mathematicians in one chapter while Fibonacci might be another chapter if the book is on the history of numbers but I'd have them do individual chapters if I was making a book on mathematicians.

Its all up to you.  Let me know what you think.  Have a great day.

History of Numbers

One of these days, I want to take time to have students explore the topic of numbers.  Where did they come from?  How did we get the current ones when previous ones were based on pictures or combinations of lines?  How did it end up as base ten when the Babylonians used base 20?

The numbers we use today are referred to as Hindu - Arabic numbers by the Italian Mathematician Leonardo Pisano also known as Fibonacci. 

He studied in North Africa, where he learned about these numbers and brought them back with him to Italy.  The beauty of the system is that it uses just 10 digits in combination to create the numbers from 0 to infinity. 

Historically, Archeologists believe the first evidence of counting is shown on the Ishango Bone found in the Congo.  This bone has lines done in such a precise manor, they conclude this is possibly the first evidence of tally marks used to keep track of something. 

However, numbers and counting did not come into being until the rise of cities when a method was needed to keep track of commerce.  Sumeria is attributed to have the first system of counting with tokens.  People were given a token to represent items such as pigs.  If a farmer had 5 pigs, he received 5 tokens but if he slaughtered one pig, he removed a token so he had 4 tokens to show ownership of 4 pigs.

Originally, they kept the tokens in a bag which was marked also but someone got the idea that as long as the bags were marked, you didn't need the tokens inside.  Thus a transition was made to using clay tablets to keep records and people were hired to mark the tablets so no one could give himself more than he had.

Later on, the Egyptians added using the number one to indicate measurement in addition to counting when they created the cubit.  The cubit was their standard measurement like most countries use the meter as their base.  They also created a symbol for each number.

The Greeks added the concept of odd and even numbers to the number system.  Greece is noted for several mathematicians such as Euclid, Pythagoras, or Archimedes, who contributed significantly to the development of mathematics and whose ideas are still used.

The Romans took control of Greece.  With them they brought their system of numerals which could only be used for addition and subtraction. Even them, the problems had to be worked on a board which is the predecessor to the abacus.

Then in about 500 AD someone in India came up with the concept of zero.  The mathematicians in India also developed the idea of 10 digits so they could express extremely large numbers easily.  Their system is referred to as Arabic numbers.

About 250 years later, Persians invented fractions so items less than a whole could be expressed.  About the same time, the concept of zero arrived which allowed Muslims to advance mathematically. By 1200 AD, the numbers arrived in North Africa where Fibonacci studied. Eventually, he brought the system back to Europe but it took a while to replace Roman numerals.

Once the Catholic reformation occurred, merchants were quick to adopt the new system because they were now allowed to charge interest where before they could not because it was classified as a sin.  The new system allowed them to calculate interest out to the 12th place.

Tomorrow, I'll discuss things that can be done with this information but in the meantime, I hope you enjoyed this brief history of the number system.

Warmup

Warm-up

Scale Models for Houses

  Students often find ratios difficult to manage without having a real context to relate to.  Years ago, I saw the results of a class which was quite impressive.

The students started by designing a house.  They had to draw plans for their dream house complete with bathrooms, bedroom, kitchens,  etc.  The plans had to be done correctly with a scale, doors, windows and everything normally expected.

Once the plans were created, they had to build a scale model out of wood so they could see it, just like those scale models you see on television shows.  The models give a better sense of proportion for the rooms when we cannot visualize it.

When they build a model, they see that a 2 foot by 3 foot bathroom might be a bit small, or the 30 by 60 foot bedroom might overwhelm the rest of the house.  There is always an esthetic element one has to keep in mind when designing a house.  I know I hate houses with two sides heading back at a diagonal.  To me, the main room acquires a squashed feel.

In addition, I prefer open floor plans with a higher ceiling so the rooms feel bigger.  If the rooms are divided up so you have a separate living room from the kitchen, and the dining room, it can feel more cramped. By building a scale model, they see if the feel is as they desire.

 I like some of the new apps because they allow a person to populate the rooms with furniture and drapes for a more complete feel.  A 10 by 10 foot bedroom may seem large enough empty but when you add a bed, dresser, desk, chair, and night stand, it may suddenly seem too small.  These apps supplies furniture in the correct size.

So with a couple of apps, the students has designed the house and populated it with furniture.  Then building the model, gives a better feel so you can move around it and check it out.  It also adds the hands on element for those who need a physical component.

Please let me know what you think.  I'd love to hear from you.

Identifying Mistakes.

  I've decided to start the year with error analysis in my two lowest performing classes.  These are the classes with students who do not have a solid foundation in Math.  These are the students who throw out a paper once its been returned.  They do not know how to learn from their mistakes.

I found this lovely sheet of addition and subtraction problems, some correct and some with mistakes.  The sheet requires them to check the math and make corrections as needed.  I took this one step further and requested they identify the mistake made by writing it down.

One of my students asked if I wanted to turn them all into teachers.  I laughed and explained they need to know how to identify the type of error they made so they can get better in math.  Many students actually worked together trying to determine what was done incorrectly.  Most of the mistakes were things like forgetting to add the carried number, forgetting to borrow, adding instead of subtracting.

The reason for this exercise is that I do not plan to record grades until a student has made all the corrections and included the reason for the mistake.  I hope this exercise will help students determine where they still do not fully understand the topic.

Will the idea work?  I don't know yet.  Most of my students have the mindset, even in English, of I've done it once, why work on it more?  In English, they think the first draft should be the final draft and hate rewriting to improve it. In Math, they did the problems so that's it.  Its important to have them learn the material correctly the first time but if not, they need a tool to learn self correction.

I suspect this attitude develops in math because many of the elementary teachers do not stress correcting the work.  I admit, I've not done this except on tests in the past but since reading that students need to know how to explain steps, etc, I am changing my focus to include correcting even daily work.

I'll report back in a few weeks to let you know how it goes.  Let me know what you think.  I would love to hear.

Performance Tasks

  Today is Wednesday.  That means its a short day so I started students on performance tasks.  If you are not aware, performance tasks allow students a chance to determine an answer based on information provided.

I found some lovely performance tasks at Inside Mathematics.   All the tasks are divided into grade level complete with answers and examples for grading.

Although I teach High School, many of my students are English Language Learners who need a bit more scaffolding.  I chose a task from the 7th grade level in which they had to determine which of the cereals had a higher protein level.  I enjoyed the task because it had them using ratios in context.

I sort of walked them through the task but I required them to be a bit more independent than last year.  I began the class by asking them "What is a ratio?"  The answers to this question indicated they were not sure what a ratio is.  After some discussion, I asked "What are some examples of ratios used in real life?"  This stopped everyone cold because they connect ratios with the math classroom and not with life outside of school.

I guided them to snow machines and ATV's because these engines use an oil to gas ratio.  I have no idea what it is but I've heard the ratio has to be right.  These machines also have miles per gallon and miles per hour ratios.  They started getting the idea because someone suggested certain stats in basketball.

Unfortunately, they struggled with setting up a proportion to determine the amount of cereal required for 9 grams of protein.  They already knew that a person got 12 grams of protein from 100 grams of cereal.  It took a bit but they managed to find the answer.

Disaster struck when they had to compare two ratios to determine which cereal had the higher ratio of protein.  Several students based their answer on the denominator of the ratio written in fraction form.  They did not bother looking at the numerator.  Because one denominator was 9 and the other 25, they assumed the one with 9 was bigger. 

This lead to a discussion on comparing fractions and needing a common denominator.  One student suggested finding decimal values instead which was fine but several students set up the division problem incorrectly. 

The great thing about this exercise was the way it exposed weaknesses in student knowledge.  This will make it easier for me to start the year and work on strengthening these areas.  This helps me plan future topics.

Let me know what you think.  Have a good day.

School Started.

  Sorry about not publishing anything yesterday but I had to get my room ready for school.  School started today.  I have one class this year, "Fundamentals of Math" which is going to be my try it out class.

This class is for upper level high school students who have not done well in other math classes.  My idea is to take the last set of MAP results, group them according to their scores, assigning work based on what they need.

I plan to do this using Google Classroom because I can place all the work there for each group.  Just to let you know, I will be adjusting groups based on the strand.  Google classroom will make it easy to create collaborative assignments using Google slides or docs.

In addition, I can create Hyperdocs so each group can work on their own while I provide some one on one time with students who are lacking basics.  I discovered Hyperdocs this summer and fell in love with them because I can create a document with all sorts of interactive links. Anytime, I have them watch a video, I hope to make it so they have to answer questions at various times so they really pay attention to what is being said.  Too many of my students will just enjoy the video because they do not know how to watch it to learn.

This is the class I am going to try things out on to make it more student centered so they are guiding their learning rather than having me teach it so much.  I could teach it the traditional way but I have skill levels from low to on level, making it harder to do the old one methods fits all.

Many of these students have failed earlier math classes either because they do not have certain skills or they do not learn in a traditional classroom.  I hope by changing the way conduct the class, students might find success they have not attained before.

I will let you know how it goes.  Have a good day and let me know what you think.

Warmup

Warmup

Gaining Student Attention.

Recently, I've been reading several books, looking for ways to improve my methods so I keep student interest and create hooks to improve their desire to pay attention.

The first suggestion, I ran across was actually via a short video where the teacher recommended you wait till all the students are quiet.  If you start while some of the students are talking, they are going to ask you to repeat the information.  By waiting till they are quiet, they are actually going to pay attention and hear instead of being involved in their own conversations.

The next four suggestions come from Pow Toons blog and the third suggestion is seconded in the book "Teach Like a Pirate"
1. Change your focus from teaching a topic to teaching for the student.  Make it so they feel as if they benefit from the content.  Creating benefit creates desire so its important to create desire. Instead of telling them what they are going to learn, create headlines to tease them with upcoming topic.   Headlines make a promise designed to create desire so students want to learn.

2.  Convince students they are going to miss out on the benefits if they do not pay attention. As part of this, let students know what might happen if they miss the information.  They need to know the pain of lacking information.  You provide the motivating reason for learning the material.

3.  Create a movie trailer designed to capture student attention.  I have done with using imovie on my Mac.  I created a spy trailer teasing students with a preview of the next unit.  It caught their attention because when it was done, they wanted to see it a second time.    The suggestion is based on the fact Hollywood always teases audiences with upcoming movies before showing the current movie.  They build desire which is what a teacher does by creating trailers for the next topic.  Don't tell, show.

4. Be willing to use animated videos which have both an auditory and visual component to help meet student needs.  If you have students create their own animated videos, you have a kinesthetic component. With all the web sites and apps, its easy to create animated videos.

I'd like to thank Pow Toons for these ideas.  I plan to try the three that I've not used before.  I can hardly wait to try.  Please let me know what you think.

Teaching Mathematics as Storytelling.

  After reading that short article in Medium, I decided to explore the idea of teaching mathematics as story telling.  A more generalized topic.

Having gone through a traditional teacher training program, I was never exposed to the idea of teaching mathematics as story telling.  To help me understand how to teach this way, I found a book by Rina Zazkiz and Peter Liljedahl on this very topic.The link goes to a 33 page sample for the book you can buy on

The authors spend the first two chapters discussing the different types of stories and elements needed to make up a story before discussing storytelling in a variety of contexts.

Beginning with chapter five, the authors discuss different types of stories and provide examples of each type.  I do well when presented with examples so I can pinpoint the elements of the story.

In chapter 20, the authors take time to explain how to create a story from scratch and provide examples before addressing using existing stories.  It comes in a pdf file, that can be downloaded.  I downloaded so I can read up on it in more detail.


One article I read proposed Dan Meyer with his Three Act problems and Karim Ani of Mathalicious 
are modern math storytellers with the activities they have created.  Activities which  hook the students, provide the math within the context of a story, and allow them time to figure out answers.  I'd always seen them as performance tasks for students to work on.  I'd never actually considered them as teaching mathematics as storytelling but they are, aren't they?  They tell a story filled with information and ask a question to be solved. 

I'm impressed with the idea of using storytelling to teach math because it is human nature to enjoy listening to stories and as shown, most people remember the key ideas of any story they read or hear.  Since reading this information, I've wondered if it would work having high school students read elementary level mathematical based picture books and then writing a book report discussing the math in the book. 

I'm thinking of books like those in the Sir Circumference series or Counting on Frank, or One Grain of Rice.  Since I work with English Language Learners, this activity might improve their comprehension of the written word.  They are great at decoding but their ability to comprehend is way behind.

Let me know what you think.  I'd love to hear.  Have a great evening.  As you read this, I am in my hotel room in Helsinki waiting for a conference to start. If I learn anything, I'll share it with everyone, next week when I get home.

Do Students Really Need To Know Full Proofs?

  I read a wonderful article in Medium by Junaid Mubeen where he questions the need for students to remember entire proofs in Math.

He speaks of a friend who can reconstruct proofs from a math class he took 7 years previously yet had not been practicing mathematics in at least 5 years.

So how did he remember the proofs.  It turns out, he remembered a couple of key idea, not everything.  He was able to fill in the rest of the proof because he understood the ideas and their relationships.   It is compared to remembering information about a novel.  People remember the important parts, not every single detail.  Its like weaving the important ideas together into a story so as to remember more.

The author observes that he memorized every single step of a proof without understanding the main ideas and how they related to each other. So he struggles to remember the steps of the proof.  It has been suggested people take the material and create story lines out of the material because our memories remember key elements of stories better and longer.

Stories also show the interplay between memory and thinking.  So people's ability to recall information is predicated upon their ability to think.  Unfortunately in math, facts are often presented in a disjointed way, making it harder for people to understand the material. 

According to the author,"mathematics is an act of storytelling that supports the dual goals of memory and understanding".  He believes a good proof will tell a story filled with turns and twists, and integrates the key elements into the story. 

After reading this, I fear I am guilty of teaching proofs for students to learn as written, just the way I learned.  This article provided a new perspective on teaching proofs.  Perhaps, I should have students identify the key ideas in a proof so they understand and remember the material.

I'd love to hear your opinion on this idea.  Please feel free to comment.  I love the idea.  Have a good day.

Creating Your Own Math Tour

I spent three days in Iceland, including one day I spent on a tour bus checking out some of the most famous sites of Iceland.  On the bus, they gave us tablets with the route using google maps.  The pins on the map provided background information automatically based on the GPS information.

In addition, I wrote about two tours in England which used google maps to create tours of mathematically inspired buildings.  I thought about having my students create one of mathematically inspiring buildings but there are not buildings that are truly inspiring.

So instead, I decided to have students research buildings such as the Leaning Tower of Piza, or the Roman Colosseum and use those for a tour of the mathematics of ancient buildings.  Google Earth allows peoples to place pins on the map and the pins can have pictures or information attached to them.  

Or students can research mathematically interesting buildings world wide and use that information as the basis of a second tour.  There are articles out there on this topic.  Some buildings recommended by the articles include the Great Pyramid of Giza in Egypt, The Taj Mahal in India, the Parthenon in Greece, The Parabola House in Japan, The United Nations Headquarters in New York City, The Mobius Strip Temple in China, and The Tetrahedral Shaped Church in Colorado.


Imagine assigning each student a specific building they have to research its location, size, mathematical information, and find pictures they can post on a class google earth map.  They are responsible for the interactive element of the tour.

For instance, if I chose to create the pin for the Roman Colosseum, I might talk about the elliptical shape of the building, find the length and width so I could show the approximate formula of the building.  I might figure out its height, research the number of people who could sit in it.  In addition, I might do a search for mathematical information on it. 

Once I have all this information I can put it together, create the interactive pin on the map as my part of a world wide tour of mathematically interesting buildings.  If someone wanted to take this a step further, they could take the same information, put it on google slides and turn it into an ebook. 

Kasey Bell posted these great instructions on her blog so you can turn google slides into an ebook.  One assignment, two results.  Let me know what you think.  I'd love to hear from you.

The Flipped Math Classroom.

  School starts in the next week or so and I've spent the summer looking for materials to use in my classroom.  I plan to use things I find to make the classroom more student centered so I spend more time with the students and less time actually teaching. 

I decided to check out the idea of a flipped classroom again when I came across this site with materials from videos to work already posted and ready to go for Algebra I, Algebra II, Geometry, and Pre-Calculus.

Each section is set up so a student watches the video while taking notes before trying the practice and check section.  These are followed by an application set before ending with a mastery check.  Each beginning video is set up to be watched directly on line or can be downloaded to be used offline.

 In addition, students can download a packet of material including a note guide so students have guidance in what notes to take. The pack includes practice problems, and applications. Furthermore the site includes answers to the problems and a corrective assignment complete with answers.

The whole course is divided into semester one and semester two.  Each unit has a review of the material covered and there is a calendar available to put down the pace needed to complete each assignment.  You just need to fill it out.

I love the classes are set up already but I'd add a bit more to make it a bit more interactive but the basics are there.  This is set up as a flipped classroom but if you live in a place where there are issues with the bandwidth, one could always download the videos and show them in class.

In addition, the material here could be used for students who are traveling or sick and are missing class.  The biggest use I see for this, is with students who need a higher level math than may be offered or providing alternative explanations or work.

Check it out and let me know what you think?  Have a good day.  As you read this, I am enjoying myself in Finland.  I'm hoping to speak with some Finnish teachers to learn more about their schools. From what I understand, they teach less and students do better.  If I find out anything, I'll let you know.

Warm-up

Warm-up

Homework - Good or Bad?

Several educators I follow on Twitter are arguing against the use of homework, especially assigning 25 math problems ever night.

I took one class where they recommended assigning homework so students practice what they've already learned.

This is not an easy question to answer but what is clear is homework in early elementary school is not advised because there is no real evidence it helps the brain.
By middle school homework should take no more than an hour and a half and in high school, no more than 2 hours. 

It appears teachers are assigning more homework in early elementary because of the pressure students are under to read by grade three.  In high school homework has increased so students have a better chance of being accepted into the college of their choice.

What I can say with absolute certainty is there is very little concrete evidence regarding homework and what little there is?  Well, it doesn't say is where the evidence comes from.  Many sites listed the reasons homework is good but the reasons are for learning life skills such as time management and seldom was learning the material given.  The reasons for not assigning homework include allowing more time for students to relax at night and go to bed earlier so they are not as tired in the morning.  

Two things continually came up in regard to homework.  If you give it, it should be designed with the development of the brain and how it works.  Second, one should get a bit more creative when designing it such as assigning an online game to help develop a foundation of knowledge for the new topic.  It has also been suggested students be given a choice of several things they can do for homework.
 
In addition, the homework task, even as a game, should not take more than 15 minutes to complete or it may no longer be effective.  If the assignment is too long, it can interfere with home life.  One clear facet of well designed homework is students need to know the reason for the assignments and it has to be authentic.

Homework should never be busy work or students may never get around to completing it.  Still, I see no clear cut answer on homework - yes or no?  But I've found a few things that make an assignment more effective.  In the meantime, I have a book to read on the topic and I will get back to you on that.

Let me know what you think.  Have a great day.

Competition - Good or Bad

I subscribe to a daily email which gives information on things happening in the world in regard to teaching mathematics.

Today, I got one that kind of surprised me where they say competition is a good way for students to improve their math skills.

I know, the first thing popping into your mind are those timed math fact tests where everyone tries to get the maximum number of problems correct in the least amount of time.

This is not what the author is talking about. The author is talking about those math competitions with unique problems designed to test a student's knowledge to solve problems.

Problems found in the American Mathematics Competition, Math Counts, or Math Olympiad can change a student's life.  It provides a way for students to discover their talents because these problems are not run of the mill.  They require thought and the ability to discover a way to solve problems.

Most of the problems presented require more than plug numbers into a formula.  They require students to take apart the information, change it into a more usable form so the problem could be solved.  These problems require higher level thinking to solve.

These problems require students to "read" the information in great detail so they know exactly what they have, what they have to find, and approach its solution from a variety of angles before settling on the correct one.  It also requires they check their work for calculation errors and reasonableness.

Although, students can win and eventually head off to international locals to compete with others, who says we cannot include problems from these competitions in our daily or weekly instruction?  I know a teacher who used to post weekly problems for students to solve.  If they successfully solved the problem, they earned additional points.  She'd then post the solution and a new problem. 

One way to find these problems is to search for previous competitions.  Many include the answer so students could copy the answer down but if you ask for them to explain the reasoning behind each step, they have to think about it and it shows their understanding.

I'm in Iceland, so I'll be posting in the evenings here which means you'll see this around mid day.  Let me know what you think about this.  Have a good day.

Can Babies Count?

  Yes, you read the title correctly.  The question "Can babies count?" is based on something new, researchers are discovering. 

I will say the answer is no, they cannot count at least count in the traditional way of starting with one.  However, it appears babies recognize certain concepts dealing with more or less.  More general ideas.

According to research, babies are sensitive to the concept of more than or less than relationships when applied to number, size, and duration of exposure.  It is noted if a baby is exposed to one concept, they are able to guess what the other looks like.

The author of a study at Emory University has concluded babies use information to organize their experiences during their first few months of life.  Further research indicates children by the age of 1.5 years old are learning to count to six using the fundamental one by one routine.

Both articles indicated children tend to stare longer at items they are processing so if an adult stands in front of six items and points to each one while counting out loud, children pay more attention than if an adult just points at one example while repeating the number.  However, children are not able to really begin to  develop the process of counting till at least the age of two.

Even NPR published something on this topic.  NPR looks at a different study that concluded we are born with the ability to know math due to evolution.  Furthermore, it states animals are born with the ability to do certain types of math which help them navigate through their environment. This particular study mentions the better high school students are at numeracy when they graduate, the more likely they are do well in life.

The thought behind these studies is to develop a set of indicators so we know when students enter kindergarten if they have the skills to succeed in math.  After following preschool students for two years, checking them for 12 different skills and following them into kindergarten and first grade, researchers discovered estimation is extremely important.

In addition, it is important for young children to have a solid grasp of cardinal numbers. Early results indicate preschool teachers should focus on core skills rather than trying to cover it all. The authors have designed additional studies to explore it in more detail.

This supports something I've known for a while and that is students who are not up to grade level in math by the end of 3rd grade, they may not graduate from high school because they do not have the skills needed to succeed.

Let me know what you think. I am writing this while sitting in the Sea Tac Airport awaiting my flight to Washington D.C. where I catch a plane to Iceland for a few day before heading off to Finland.  I will be doing my best to stay on schedule.

Have a good day.

Clothing and Geometry.

   Geometry makes an appearance in two different ways in regard to clothing.  First, it is often used as a pattern on the material or used to shape dresses in the 20's and 30's.

So let's look at the use of geometry in patterns.  The picture to the left shows geometric patterns often found embroidered on ethnic clothing.  These designs are essentially edgings and provide decoration.  I can tell you from personal experience, it is easy to create these patterns using a cross stitch.

One of the most common geometric patterns found on materials are simple stripes.  Stripes are nothing more than parallel lines of different thicknesses.  Simple and easy.  Take it one step further by having perpendicular stripes and you end up with plaids. 

 Look carefully and you'll see squares of yellow, white, and black, along with rectangles in those colors plus gray.  Look carefully and you'll see narrow stripes of color.  Easy to do with perpendicular lines.


If you've ever gone through the fabric store, you may have seen checkered fabric made out of alternating square of material.

I do not wear checks but I love to use this type of material for smocking because it makes it so much easier.  In smoking you have to gather points the same distance apart if you want it to look right.

I've also used checkered fabric as a table cloth.  When I headed off to college, my mother packed one to use.  She thought it might cheer up my dorm room.  I hated to tell her, I didn't have a table to put it on.

In the time period of 1920 to 1930, Art Deco became famous with both material and style.  The predominant form was geometric even if the patterns were not. Dresses were designed to hang loosely from the body in a flowy manner.

Part of this was accomplished through the use of Art Deco materials which had strong colors and geometric shapes providing a modernistic feel.

In addition a couple of designers created outfits using squares, rectangles and triangles.  One designer, Madeline Vionnet created a dress where she used four rectangles of white silk, two for the front and two for the back. These four pieces were hung from two points to create a diamond shaped dress which allowed a lot of movement.

Madeline Vionnet is also known for her Chilton influenced dress which is made a long rectangles sewn to a plain top with the last three inches or so on each side hanging loose to create a flowing look.  You can see a picture of it here. 

The other designer, Paul Poiret, advocated dresses be cut along straight lines and constructed of rectangles.  His dresses hung from the shoulders, offering more freedom and turning three dimensional shapes into two.

Real uses of geometry in real life.  Let me know what you think, I'd love to hear from you.  I hope you all have a great day.