Friday, August 18, 2017

Scale Models for Houses

Ford, B700, Union Pacific, Usa, America  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.

Thursday, August 17, 2017

Identifying Mistakes.

Error, Bug, Background, Polygon, Pattern  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.


Wednesday, August 16, 2017

Performance Tasks

Ballet, Dancers, Woman, Silhouettes  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.

Tuesday, August 15, 2017

School Started.

Classroom, School, Education, Learning  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.

Friday, August 11, 2017

Gaining Student Attention.

Book, Bored, College, Education, Female 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.

Thursday, August 10, 2017

Teaching Mathematics as Storytelling.

Children, Newspaper, Story Telling  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.

Wednesday, August 9, 2017

Do Students Really Need To Know Full Proofs?

Child, Study, Boy, Proof, Learning  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.



Tuesday, August 8, 2017

Creating Your Own Math Tour

Man, Map, Smile, World, Travel, Vacation 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.

Monday, August 7, 2017

The Flipped Math Classroom.

Classroom, The Engine Room, Computer  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.

Friday, August 4, 2017

Homework - Good or Bad?

Homework, School, Problem, Number, PaperSeveral 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.

Thursday, August 3, 2017

Competition - Good or Bad

Bench Press, Fight, Arm Wrestling 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.

Wednesday, August 2, 2017

Can Babies Count?

Paternity, Baby, Child Care  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.

Tuesday, August 1, 2017

Clothing and Geometry.

Bulgarian, Folk Art, Folk, Art   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. 
Plaid, Yellow, Black, Pattern, Fabric
 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.


Fabric, Textile, Geometric, Blocks, PaleIf 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.
Vintage, Ladies, Romantic, 1920'S
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.