Written Assessments - Accurate?

  Last night, I began a book on assessment in which the author noted that many students may have decent reading comprehension but they may have difficulty expressing their thoughts in written form so they tend to score lower on normal assessments.

In addition, if a student reads at well below grade level, they may not understand what is being asked, so they may not be able to express their answer.

This caused my thinking to extend the the possibility that students who do not read on grade level may struggle to understand most standardized tests they take and the results may not accurately reflect their knowledge. So we might assume they learned less than they actually have.  I felt as if I had a big "Ah ha" moment, realizing that most of the sources we use for data driven instruction may not be providing us with accurate assessments.

Fortunately, the author had a suggestion to obtain more accurate results.  The author suggested we need to look at alternative ways of assessing so students have a better way of showing what they know.  The author suggested using a voice to text feature so they can dictate their answer instead of trying to type or write an answer. 

Other suggestions included letting students discuss how they solved a problem using Flipgrid or video instead of providing a written explanation to accompany their solutions.  With Flipgrid or a video, they can show the problem, discuss their work to provide the teacher with something to assessment.

This idea would even work with a class project because the students who need this method of assessment, could make a video to insert into the final product.  Furthermore, it is possible to pair students who write well with those who do not to produce something to show how they did the work.


If you notice the idea is to find a way for the student to show what they know without requiring a ton of writing so the teacher can determine what they know.  Understand, I am not suggesting the teacher throw out all written assessments because most schools use some sort of standardized assessment to show student growth, satisfy state demands to use "their" test, etc. 

I am suggesting teachers provide alternative ways of assessment to let students show their knowledge using methods with or without writing.  Even those that require writing, students can use voice to text answers, print them off, and contribute that way.  Tomorrow I'll be sharing suggestions to assess students using methods that may be used as a more informal assessment.

Let me know what you think, I'd love to hear your thoughts on this topic.

Warm-up

Warm-up

How to Write in Math.

  I have recently, as in this year, begun having students write a sentence for each step explaining what they are doing.  I work with ELL students so this is a first step.  Next, they need to add why they did something because people need to know both what was done and why. 

For the current test, I've already told them they will be able to earn half the points back but they will have to correct the mistake, explain why they made the mistake and what they have to look out for the next time.  I keep getting their complaint that Math is not English.  I'm sure there are other students out there who object for the same reasons.

Aside from being able to communicate ideas in an understandable way, writing good mathematical explanations helps improve your understanding of the material.  The physical writing of the information , requires careful thought and helps the individual learn and retain the information.

Writing mathematical ideas is more than just showing computations with the correct answer at the end.  You might be able to follow a process much like people do in a factory but you may not know why you are doing a particular step.  The explanations expand the process to include the ideas behind the calculations.

When writing in math, students should be encouraged to use correct grammar and complete sentences.  In addition, the writing should be divided into individual paragraphs rather than just one long paragraph.  Furthermore, equations should be separate so they stand out, even if you include information on that equation.  The best way is to give each equation its own line so it is easier to read.

In addition, it is important to define what each variable represents within an equation or formula as a way of providing context.  Context is important so the reader is aware of the situation.  In a linear equation, x may represent the slope or rate of change while in a quadratic it might represent time.  This would be the place you might make a note to yourself or reader of any restricted domains.

Furthermore, include pictures of diagrams to represent the visual element of the mathematics.  My students hate writing anything more than the basics and hate including diagrams because they see it as a waste of space.  I've been working with them on this but they find it hard to change the way they do it because they've done it this way for years.

Do you remember when they said to read your writing out loud so you can hear mistakes, awkward spots, and other issues for your English paper?  Well the same thing applies to mathematics.  When the material is read out loud including equations, to see if everything is written correctly.

Another thing to keep in mind is to use symbols and words properly.  An example of this is using the equal sign "=" when you mean next or implies.  If the symbol is used incorrectly, the writer is conveying improper information and changing the meaning of the text.

Again when showing the sequence of mathematical steps, place each step on its own line.  If you are explaining each step, it is best to write the explanation on one line, then the equation on the next, followed by the next explanation on another line, etc.  This makes it so much easier to read and understand. 

It is strongly recommended that people do not start sentences with a formula.  It is better to begin with a transitional word such as since, however, etc. It is also recommended when writing something mathematical the writer determine who the audience is because that will determine how it is written.  Even when writing for yourself, pretend you are writing for another student.  When writing for yourself, you might decide not to give as complete an explanation as you might.

Even when taking notes, these suggestions apply because notes are there to help us learn and refresh our memories.

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

Frustrations upon Frustrations.

There are days I get so frustrated with teaching.  I had to cut way back on the use of technology this semester for three reasons.  The first is because we've done so much testing that the computer based tests take up most of the band width.  The second has to do with the iced over relay up on the mountain so internet isn't great at best and finally is the attitude of the principal, the superintendent, and a couple other teachers keep insisting the only way for our students to learn well is to use the textbook and all the worksheets associated with the text book.

I think the only reason student do well with worksheets and the textbook is because that is the only way they've been taught up to now.  They are so used to learning this way that they resist any other form of learning.  In addition, they do not use their mobile devices for anything other than for playing games, taking pictures, and texting people.  They text more than they call.

When I use technology in the classroom, I have to work with the students to teach them how to use the application because they do not like exploring on their own.  I honestly do not know why.  Another teacher I work with noted our students will watch the same video time and time again.  They will also draw the same pictures repeatedly.  One of the middle school teachers commented in general people out here find the old physical action movies like the 3 stooges funnier than the ones relying on the use of puns and multiple word meaning.

I loved it when I could use the internet more last semester.  I spoke with another middle school teacher and we both want our students to learn using technology.  I love Google Classroom because I can post everything from warm-ups to assignments to links all in one place.  I am not questioned by students who miss a day or two because its all right there.  In addition, those who have to travel on school trips do not get behind.

I want my students to learn to use spreadsheets for math.  I want them to take the results from those spreadsheet activities add them to a write up in a word document before turning in.  I'd like them to use Flipgrid or some other application so they can create a video presentation on a mathematical topic.

I want my students to learn to use their mobile devices as productivity tools rather than play toys.  I sometimes feel like a fish swimming against the tide because I feel as if I fight the admin and the students themselves.  I want to make their learning more relevant.  Maybe next year I can do it all since we will have a brand new admin and several new teachers.

I'll keep you posted.  Let me know what you think.  Have a good day.

Real Life Inverse Functions.

  I don't know about you but I always teach inverse functions after I teach composite functions.  It has always been easier since you use composite functions to show equations are inverses of each other.

The easiest inverse functions are powers and their equivalent roots such as squared and square roots.  We especially use those two when taking the area of a square room and finding its length by taking the square root.

Another example is when you call someone and their caller id shows your number. What about when you go traveling and you have leftover Peso's?  When you convert it back into American, you are performing an inverse function.  Those currency exchange places you find at airports do the conversions all day long.  Years ago, my sister worked for a company that had dealings with other companies in Canada.  She called me one day to find out how much they needed in Canadian to pay for something so I had to do the calculations. 

On the other hand, if you have convert from metric to standard, you've performed an inverse function because living in the United States, we had to use a formula to convert from standard to metric so when we convert back, we've just performed the inverse function.  Last summer I traveled on a European carrier and their weight limits were all in kilograms so I had to make sure my suitcase did not go over the limits.  I'm going to Germany this summer via a different carrier and I have to deal with kilograms so sometimes we have to use inverse functions regardless of our desires.

Other inverses include things like you know my speed but I need to find how far I traveled to get somewhere so you use the inverse to find how far you went and vice versa.  Its also derivatives and integrals in calculus. On a side note, there is this great book by Elizabeth Moon called Mathamagics.  In it, spells are derivatives and counter spells are antiderivatives.  The heroine keeps messing up when trying to counter spells but her man, a mathematician, is great at casting and countering spells.  Even the chapter numbers are mathematically correct.

One other way to look at inverses is that information is converted in a way that it is not lost in the conversion or it allows us to solve problems much easier.  For instance it is much easier to use ln than e when solving e^x = 50.  Although it is not done as much, people used to use logs to do math when they had certain functions to calculate results from.

Sorry about yesterday but the internet at work went down so I couldn't do anything. Its been down at home most of the time so I haven't even been able to work there.  I went two days with no landline, no cell service, no internet and so I apologize if I miss a day here and there.

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

Neural Coupling in Math

The other night, while reading one of my new books, I came across something about neuro coupling. Apparently, it is what happens when someone tells a story, one that captures your attention type of story, your brain fires in the same way as the storyteller's at the same time.

Cohesion between the storyteller and the listeners brains is created by mirror neurons.  In addition, good stories pull in listeners through narrative transport.

Research indicates when people are presented with facts, only two areas in the brain are active but if you get the same facts presented in a story more areas of the brain light up.

Furthermore, under normal circumstances the brain tends to stray by engaging in daydreams but when there is an interesting story, the brain experiences no daydreaming.  None!

So how does one create Neural coupling in math when most of the time its taught as facts?  That was my first thought when I read about neural coupling but realized we could possibly create lessons to follow the parts of a story so we might build neural coupling with our students.

First is the introduction or exposition where we are introduced to the hero and the situation.  This is the make or break point for any book.  If the author does not grab your attention, you won't finish the book, so the introduction to a lesson should grab the student's attention.  Next is the main body where we introduce and guide students through the material.   The last part is the climax and ending where students show they have mastered the material.

The lesson has to be taught so we gain the attention of the student and keep it through the whole lesson.  We need to create the neural coupling at the beginning of the lesson and keep student involvement till the end.  This can be hard because most upper level math teachers have been trained to create lessons depending on who the current best practices guru is.  Oftentimes the lessons are basically the same as the previous guru but with new names.

It seems to me that if a neural coupling occurs through storytelling, it should also occur while watching a movie because the viewer often becomes involved in the same way we would while listening to a story.  This means we have a few ways to get students involved from the beginning.


This means when we plan a lesson we should begin with some sort of hook to engage students.  Teaching Like a Pirate by Dave Burgess has a wonderful collection of types of hooks a teacher can use to grab their attention.  I have used the movie trailer one to build interest in an upcoming topic.  He has enough ideas, a teacher can create a wide selection of hooks.

To keep them hooked, there are the 3 act tasks by various authors, Estimation and Splats by Steve Wyborney and other great activities.  In addition, there are some great videos out there that are designed to keep student interested.

The hard part is the practice but once they've done some work, they can show their understanding through Kahoot, Jeopardy, or other activity that engages them.  Once we have them engaged, they are more likely to be involved and learn the material.

Have a great day and let me know what you think.  I'd love to hear.

Warm-up

If a snow machine can go 85 mph how long will it take to go 1087 miles?

Ice Covered Relay

I cannot publish a picture of the relay because I do not have rights to it so I am including the link to the site with the picture.  Please check it out and know why I sometimes skip a day.  I do not know when it will be up.  I do know I have to do things from school because the school has something to keep us up and running through state testing.

The Three Act Math Problem

If you have taught math for any length, you are familiar with three act math lessons. The idea behind them to introduce the problem in act one using say a video or  photo which introduces the conflict or problem with as few words as possible. The visual should be clear and easy to understand.  The second act helps students look for solutions using tools, etc while the third act helps resolve or show the solution.

This type of problem actually helps promote a growth mindset.  Most 3 act math lessons are set up so any math student can attempt a solution, even lower performing students feel capable of trying to solve the problem. Another reason these are good is that there are always more than way to find a solution.

The second act is designed to allow students to explore multiple avenues to find the same solution.  Furthermore, the second act is where the discussion happens on what information is still needed  before students work in groups trying to find a solution.  Last, is act three where the answer is revealed to the students.  There are times when their solutions will not match the actual answer because there other factors in play but it opens up the floor to a discussion on the reality of mathematical models. In addition, the reveal also allows students to discuss errors they made along with correcting those errors.

The nice thing about these three act math problems is they allow students the freedom to explore solutions in their own way.  They also promote conversation which helps improve communications and collaboration.  For many students, these 3 act math problems based on real life situations make more sense than the standard problems provided in a textbook.  These types of problems allow students to shine through and show they are math people. 

In addition, three act math problems  are wonderful for helping students develop the skills they need to solve real life problems because they have to understand the situation or problem, determine which information is needed and usable, what information they still need, what questions they need to ask in order to find the needed information, where to go to find the information they might need, and the problems help build perseverance.

Sometimes the 3 act math problem can be based on something ridiculous like the number of post it notes needed to cover a  whiteboard or the amount of money one can fit in a classroom.  Sometimes its not as "real life" as people would like but if it engages students fully does it matter?  The problems can sometimes be the hook in and of themselves.

If you don't want to make them yourself, check out those by Dan Meyer who started the 3 act math problems, Robert Kaplinsky, Andrew Gael, John Orr, and several others.  Just do a search and so many will show up that you'll never have to make them. 

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

Internet on the Fritz

  If you've read my blog for the past few weeks, you know the school and the village are having internet problems due to relay tower covered in ice a couple feet thick.  Just the other night, the internet company sent a couple techs out to set up a military grade satellite dish they are loaning us but the internet is still limited.

Since it is the state testing time of year, the first priority is to get students tested.  The whole schedule got thrown out due to the internet going in and out.  When testing is going on, teachers can only use the internet to take roll.

If I want a video from Youtube, Khan Academy, Edpuzzle or other website, I have to come in before  or stay after to download it so it can be used in my class.  I cannot stream it during school or the tests get kicked off. 

I have been wanting to have my students work problems out on Google slides but that won't happen right now.  I love posting assignments on Google Classroom but again, I can't as long as our bandwidth is so limited we even have to limit the number of students tested.

My students ask when will be able to play Kahoot again but that will be another week or two because we have MAP testing to see how much growth our students have made since Christmas.  They love playing Jeopardy but that is not possible for the moment unless I find a Smart version.

I realize there are better pieces of equipment to use than Smart boards and presentations but I can use the Smart board equipment already installed in my classroom without using bandwidth except to download predone activity.  I can use it to show the video's I'd normally have students watch on their own devices. 

What I do have are several math apps for students to use which allow them to practice certain skills since they work well off line but I have a few apps that will not work without the internet.  I almost feel crippled without access to all my usual internet based materials.

 The routers and such are about 8 years old so they are not very efficient.  The school is hoping to put in new routers and other equipment to make the internet more efficient and easier to use. We are hoping this will allow everyone to use the internet in class for education.  We may not be able to stream videos but we should be able to have students work together on Google slides to produce a PDF explaining the similarities between regular fractions and algebraic fractions or a slide show on exponential functions, common logs, natural logs and e.

I hope next year is better so I can do more with technology to help my students develop a deeper understanding and learn skills they can use in the future.  Let me know what you think. 

Mistakes Cause The Brain To Grow

The brain has an interesting way of reacting when a mistake is made.  When a mistake is made, electrical impulses travel through the brain causing one of two things to happen.

First, there is increased electrical activity due to the belief the brain itself is conflicted between knowing the right answer and making a mistake. This happens even if the person does not know they made an error.  The second occurs when the person is aware they made a mistake and pays attention to that mistake.

Mistakes cause our brain to grow, even when we are not aware we made a mistake because the brain is aware and struggles with it. Just because we make mistakes does not mean we are not math people.  It means we are leaning and growing.  The ability to grow is referred to as brain plasticity which is the ability to change, grow, and rewire as needed. 

Researchers have taken scans of the brain which shows the brain does grow when we make mistakes because there are more synapses firing than before.  Unfortunately, the fear people have of failure can prevent them from trying more challenging things which is opposite to the truth.  Failure helps our brains develop.  As your brain grows, you acquire more mastery of the topic until you have learned it.

Furthermore, we need to make sure our classrooms are mistake friendly rather than unfriendly.  When the classroom is perceived as mistake friendly, students are willing to work harder.  A mistake friendly classroom means that mistakes are valued and so is all student thinking rather than just a few.  Students should not feel ashamed of any mistakes they make, especially if they can connect mistakes with improved brain function. It is important we help students understand that making mistakes is helping us learn. 

Unfortunately, people and students often feel as if making mistakes means they are not good at math.  This is not true. They just need practice.  It has been suggested that we praise their perseverance on solving a problem, their use of different ideas to solve a problem rather than telling them they are smart can change their self image.

Often the attitude that effort is the same thing as innate ability can influence their decision to try a problem.  In other words, if they thing a problem is too hard, they won't even try because they think they will never be good at math.  In addition, some teachers spend too much time on getting the right answer rather than understanding the process so they can work any problem.

So take time to celebrate the mistake, focus on the process, so students want to learn.  I read a suggestion from my current book where the author suggests the teacher use the words not yet rather than a failing grade and take time during class to grade a paper with the student to discuss mistakes rather than grading and returning it.  I want to try this to in the hopes to make it more acceptable to make mistakes.

Let me know what you think.  I may or may not be on everyday because the relay tower is covered in ice and my internet is down more than up.  In fact, the internet is sporadic at school and may be slightly more reliable since they set up a special dish.  Just giving you a warming.

Correcting Test Mistakes.

  Up until now, I've required students to correct all their errors on tests before they could retake the test but most of my students will not make corrections unless they get points for it.  They do not want to retake the test, especially if they have to do the retake it after school.

They enter high school with the idea that everything turned in is worth a grade and they are allowed to make corrections for half the points. In addition, they do not need to explain why they made a mistake.

Furthermore, once they get any work back, its done with and they can throw it out.  They don't need it.  So I need to make changes to my teaching to help them learn to correct mistakes.  I read one blog where the person does not allow retakes.  What she does is she has the students make corrections and explain what they did wrong using complete sentences.  For this, they get half the points back.

I think this idea would work better in my class than what I've been doing because it meets the student need to earn back points but it makes them identify the error they made.  They have to determine if they dropped or ignored negative signs, or if they multiplied wrong.  I have a sheet they can fill out to turn in with the original test but I can add a small table on the bottom to summarize the type of mistake.  This small table will provide me with data I can use to see the type of mistakes they are making.

This method does three things. First it helps the student learn to spot their own errors, thus they learn to do this on tests when they check their work.  It also helps them learn to check their work on tests.  Most of my students when asked if they checked their work, glace at the problems before handing it in.

Second, it provides me with data I can use to plan my lessons because if they are all making the same type of error, it tells me I need to reteach it or provide additional scaffolding.  In addition, it also means they can see the type of mistake they make and learn to check their problems.

The third thing it does is to help students communicate their weaknesses with me in written form.  It is important for students to formulate their mistakes while using language because it helps make them aware of the problem.

So beginning next year, this will become a regular requirement for tests in my classroom. That is if they want the extra points. Don't worry, they will only earn half the points they missed but for some students it might help them obtain a passing grade.

Geometry Nets

I finally got around to teaching three dimensional shapes in Geometry.  Much of that has to do with between one half and two thirds of my students being off for some school trip.  Its hard to move forward when there is always an excuse for why they couldn't do their homework. 

I love using nets when teaching three dimensional shapes because students can use them to find the formula's for surface area and it allows them to see volume.   Many times I print out the nets and have them cut the patterns out before doing the work but this time I changed it a bit because I had the flu and felt absolutely cruddy.

This time I put a list on the board of a cube, rectangular prism, pyramid, cone, and cylinder with no clues.  I just said draw one shape on a piece of paper.  I refused to tell them how to do it.  I refused to help them in any way except when they wanted help with the rectangular prism.  Then I handed them a tissue box to explore.

I even refused to give them a minimum size the finished shape had to be.  One of the girls asked if one inch was too small for each side in the cube and I said it might be too small to easily use for the next step so I'd make it a bit bigger.  She redid it using a two inch side.

 Most students got the first three figures done easily.  Two students got to the cone and decided a isosceles triangle was perfect and one young man stayed after school working on his cylinder pattern.  One young man kept muttering that I wasn't teaching them because I wouldn't tell them how to do each shape.

Later this week, I will have them cut the nets out and fold them into the appropriate shape to see if they were correct.  I already know the cone patterns are not quite right but I want them to figure that out themselves.  This is the perfect opening for a discussion about what shape is at the top of a cone.  I plan to show a picture so they can see the edge to note its really a circle.

Once the nets are correct, they are going to try to figure out the formula for the surface area of each three dimensional shape.  I've found in the past that it is easier for them to "see" the connection between the shapes and the formulas if they have a net in front of them. 

The volume can be a bit harder to see for shapes such a pyramids and cones but the nets can open a wonderful discussion on why its 1/3rd the volume of a cylinder or a cube.  In addition, its going to be a challenge for them to figure out the height because it will be different than the height of the triangles in the pyramid so I'll introduce the Pythagorean theorem to this part.  I want them to see it applies in other circumstances than the length of a ladder propped against the side of the building.

When a student is able to derive the formulas and make the connections to the "why"s they are more likely to remember the material.  I hope they come out with a better understanding of surface area versus volume and the difference between lateral height versus regular height.

Please let me know what you think, I'd love to hear.  Its possible other entries might be a bit late since I am still having issues with my internet.  The relay that serves this village and a couple others has been covered with snow and ice since March 7th, making both cell and internet service intermittent. It has even gone down at work for days.

Warm-up

Warm-up

Order of Operation Issues

I mentioned this earlier this week but decided I wanted to look into the topic of order of operations due to the way it is traditionally taught.  All of the posters hanging in the hallway made by elementary students were all based off of PEMDAS - "Please Excuse My Dear Aunt Sally".

Each operation was on a separate line beginning with parenthesis, exponents, multiplication, division, addition, with subtraction at the bottom.  Each poster was identical.

One big problem with teaching it this way is that when students reach high school, they believe the multiplication is done before division and addition before subtraction. They don't understand that multiplication and division are done in order as you read the problem left to right. The same applies to addition and subtraction.

Too many elementary students never get the part about addition or subtraction depending on which one you get to first when reading the equation from left to right or multiplication or division  under the same circumstances.  When they arrive in high school, they apply the order of operations incorrectly due to this misconception.

In addition its rather difficult to undo the incorrectly learned methods when they arrive in my classes.  I know at least one 6th grade teacher who works to undo the damage but often times it is so ingrained in them that they cannot unlearn PEMDAS to do the order correctly

I have actually had students argue with me that they did the math right after doing all the multiplication followed by the division, then the addition, and finally the subtraction.  I've tried reteaching it but honestly it is so ingrained in them that it is very difficult to undo the learning.

Furthermore, when you start adding a variety of different grouping symbols and multiple exponents, the student use of this mnemonic breaks down because it does not state what to do with multiple grouping signs. I have to remind my students to work from the inner most pair of parenthesis, brackets, or other grouping symbols because PEMDAS only talks about parenthesis. 

In high school we tend to use more complex equations where we might have 2 + 4^2 all divided by 3^2 and you have to discuss implied groupings due to the way things are written.  My students do not understand implied parenthesis because they've spent their elementary years seeing only cases that follow PEMDAS.

I believe the first step is to make sure elementary teachers understand the actual way the order of operation works so they are not just teaching students the same "Please Excuse My Dear Aunt Sally" they learned.  They need to include the left to right statement to make it more complete.

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

Inquiry Maths

  Do you want to have your students do more exploration of their thinking in regard to mathematical topics but you don't know how to create the prompt to do that?  Do you look at many of the prompts you have know and think "Hmmmm?"

I found a web site with prompts that allow students to explore a certain topic.  It gives them the chance to check out different paths while doing more learning independently. 

The site has prompts for Numbers, Geometry, Algebra, and Statistics.  Each prompt has information on how to set up situations for introducing the prompt and there is information on what ideas or concepts it has students looking for.  These probes come before the lesson.

The site includes detailed information for creating your own prompts by explaining what it should and shouldn't have along with an example of how a prompt was created.  In addition, it has links on further guidance for making new prompts. 

In addition, the site provides a variety of cards such as practice a procedure, reflect on the inquiry so far, and others so students have some guidance in learning to work the prompts independently.  Most of us cannot throw students into this type of prompt without helping them learn to do it.  You have the choice of several different sets or you can learn to make your own.

Furthermore, the site explains how to assess while students are doing the inquiry, at the end of the inquiry, or after the inquiry.  There are questions, suggestions, and forms to help do this.  The idea with all this is for the student to take over directing the way the class goes but I know for my students they only want to work with worksheets and notes.  They really hate trying to learn on their own.

This is another site with inquiry based activities but the information is shared via video's which show how the teacher implemented it in the classroom.  It has examples for primary, middle, and high school students along with information for a math fair and possible problems.  I actually held a math fair many many years ago because i had a class filled with extremely low performing students and they had a blast along with those who came.  We geared it for elementary students and they all came.

Another site, the math giraffe, has entries on this topic from good questions to use, to structuring, to the benefits, to its use in geometry.  Each entry has some great information and ideas to get you started.

Honestly, I do not think my students are ready to go down this road fully but I would like to use more inquiry in my class to help them develop a deeper understanding of the material.  It is one step at a time and maybe I'll get them there. It is all a matter of doing it.

Visual Literacy and Learning.

  Visual literacy is defined as being about to interpret information from graphs, images, charts, pictures, and scenes.  This is an important skill because you find all of these in some form in the newspaper, on the internet and it is important to be able to read and interpret them.

Visual information is becoming more commonplace because technology provides us with so many graphs, images, charts, pictures and scenes that students must acquire visual literacy.

Visual language in mathematics is especially important because it makes it much easier to communicate ideas or items without the use of written language.  I've had my student write down the instructions to draw a square and they get frustrated because they know what a square is but they cannot describe how to draw one yet if I let them use a picture, they could communicate it in a much easier manner.

Visual learning consists of five different skills.

1. Observation - to really see the item and be able to answer questions like "What is it?" or "How does it work?" or "How does it differ from a square?"

2. Recognition or the ability of a student to recall the meaning of the visual sign such as a red octagon means stop in many countries or its a rectangle because it has 4 90 degree angles but two sets of  parallel sides of different lengths.

3. Interpretation of the visual that leads us to understanding what we see.  It involves questions such as "What does this tell me?" or "How does this model work?"

4. Perception involves answering the question "What comes next?" so we are able to analyze or propose a conjecture.

5. Self-expression allows us to convey the information to others.

In mathematics, information should be conveyed in visual, verbal, and numerical form  so they are able to understand the complexities.  When ideas are conveyed in these three forms, they aid in true communications.  If you look at sketch note taking, it uses all three to provide information.

In order to understand abstract concepts in mathematics, students need to "see" how they work by using a series of images to solve the more complex problems.  For instance, in a word problem, you break the parts down and visualize each part, perhaps even draw something showing the relationships or when looking for the surface area of a three dimensional shape such as a pyramid, you create a net so you can calculate the area for each surface.

Each type of visual provides us with information such as graphs are designed to convey information we capture via our eyes while charts show us relationships.  Students need to develop the ability to create mental images or images to make the jump from concrete to abstract.  Even accurate estimation requires this ability.

It has been found that sketching and modeling are extremely important in note taking.  My students are always asking "Do I need to write that down?"  Usually, I say "Its up to you but this is important towards your learning."  Now I am going to tell them yes, you need to take all the notes.

When they sketch notes about math, it helps students figure it out.  They have the capacity to internalize the information, find relevancy and make connections. Technology is making things more and more visual so our students have to learn visual literacy to keep up with the changing landscape of information. 

So what are some strategies to help students become visually literate? Here are some:

1. Display materials in visual form so students become used to seeing them this way.

2. Encourage students to draw images to explain or solve a problem.

3. Have students work together to model new concepts. 

4.  Encourage students to make sketching a daily part of their note taking.

5. To check understanding of a concept, have students create a visual without numbers or words to convey it.

6. Use the visuals to show relationships such as between linear equations and their graphs, or ratios and their relationship between proportion and scale.

7.  Create concept maps to show relationships between mathematical concepts.

I hope you found something in this you can use.  Let me know what you think, I'd love to hear from you.  Have a great day.

Why Is It Important To Analyze Mistakes?

I work with students who come to high school with the idea they do not need to show their work and once it is turned it, its all over.  When I return the work, all they want to do is throw it out.

Consequently, they really do not know how to examine their work to find the errors nor do they use the mistakes to help them learn the material better.

I admit, I'm partially at fault because I do not require them to analyze then their mistakes nor do I have them verbalize why they made the mistake.  Thus they continue to make the same mistake as they progress through the course.

Its a bit late this year to start this since school ends in about a month but I can make it part of the course requirements next year.  If students learn to identify the types of mistakes they are making, it means they can learn to identify it when they check their work.  In fact, most of my students have never learned to check their work before turning it in. 

I read somewhere recently, where the teacher required her students to go through all returned papers, identify the type of error they made and record it in their journals as a way of keeping track of this.  Other teachers only use this for tests where they have students write down the problem, correct it, explain what the mistake was before they can retake the test.

It is said that understanding where you go wrong and how you go wrong can lead to better understanding and improved grades.  In addition, it  is a habit good math students have.  Furthermore, as a teacher, you can write your next test to that it contains problems based on student errors.  We have to teach students to analyze their errors so they can improve.

Most errors fall into five categories:

1. Mechanical errors such as transposing numbers, dropping a sign, being too much in a hurry, or making a simple mathematical mistake such as 2 x 3 = 5.

2. Application errors such as misapplying a step in the process.

3. Knowledge based error where they do not understand the concept or do not understand the term.

4. Order of operation error where they do not understand when they can do things slightly out of the traditional order.  The elementary school has Order of Operation posters made by students hanging on the wall and they all have addition, subtraction, multiplication, and division in that order without the left to right based on what comes first.  This means students arrive in high school thinking they have to add before they can subtract, multiply before they can divide.

5. They have incomplete knowledge so they need to practice to get it.

It is recommended students keep a journal for analyzing all mistakes, patterns of errors, review concepts that give a student problems, review all previous assignments and tests, ask for help as soon as possible, and always diagnose problems immediately.

Using Technology to Teach Reading and Writing.

I love having students write in my math class.  The common complaint is "This is not English!" and my reply is that you have to learn to communicate mathematical ideas so you need to practice. 

 We know that writing math is much different than say for English or Social Studies because it is multimodal in nature with language, numbers, formulas, mathematical representations, and visual representations. 

In addition, there are words which have different meanings in math than they do in English, such as product, or less than. Its important students understand the multiple means both general and mathematically.  so its good to provide opportunities for students to use that language.

One way is to create videos with one or five scenes depending on the topic.  Set it up so they need to create a story board, the script, and illustrations or visuals and have a practice before they film it.  If they work in groups, you usually have someone  who is an artist to create the visuals, someone else who loves to write and can produce the script, the person who loves being on screen, and the one who would rather be behind the camera.

Another choice if they do not want to do a video, might be to create a 30 second radio or internet spot that is voice only.  Again they have to create a story board, prepare the script, before they can record the commercial.  If they prefer, the group could produce a book using pages, word, or book creator which must have illustrations, the information in their own words, etc.

The topics could be based on vocabulary such as words with both a general and mathematical meaning, real world applications of mathematical formula such as the exponential growth of Uber and AirBnB, or perhaps how stores set the markup of product.  There are possible topics in the newspapers or online including reading graphs, determining which graph is more accurate, etc.

The activity could require them to defend a position and justify it.  Many students have trouble with that because they are not used to defending anything in math.  Furthermore, common core wants them to justify or explain their choices and these activities allow that to be incorporated.

This type of activity requires research, planning, writing understanding the math itself, and putting it all together into a final package.  Its important to have students use the language of mathematics and learn how to communicate their ideas.  This is more of a authentic application rather than many of the questions on tests. 

Furthermore, many students such as ELL students need to practice the reading and writing to develop their language skills and other students have never learned to express their ideas in verbal form because many classes focus only on solving mathematical problems.  This is a way to make it more fun and relevant while allowing time to practice.

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

Warm-up

Warm-up

Knitting, Crocheting, and Mathematics.

  In most geometry classes, students spend time learning about angles, shapes, and relationships but we don't usually do much with them once the unit or units are over.

The other day, I scrolled through knitting and crochet books, looking for those on sale and found a book showing people how to crochet circles, squares, triangles, hexagons, pentagons and spirals. 

I remember seeing a baby blanket made up of a ton of squares sewn together.  After checking things out, it is possible to make a bunch of triangles, hexagons, or pentagons that are sewn together to create a blanket.  In addition, you can make spheres, cones, and cylinders to create heads, arms and legs, or tails.

If you prefer doing something a bit more mathematically esoteric, how about crocheting a hyperbolic plane that resembles the edges of a lettuce leaf.  Its too hard to make one out of paper that stays together so a math professor and her husband figured out a pattern based on an algorithm.  I've actually seen someone making one out of yarn.  I did put the link to the instructions.

You could also crochet the twisted ribbon structure of Lorenz manifold. If you are not up to date on the Lorenz manifold, it comes from a paper written back in 1963 about chaotic weather systems and is believed to be the start of chaos theory.  Further more, if you'd like to represent cyclic systems you could knit a tube which provides a great visual of following a pattern back to where it started.

If you wanted to move into fashion, you could knit Mobius cowls or a swan tea cozy.  Perhaps you could make a curve of pursuit pillow out of an equiangular spiral or perhaps a dragon?  Now if you'd like to explore a really cool site with all sorts of things like Klein bottle hats or hyperbolic baby pants, fractals, etc, go ahead.  Unfortunately, not all of the links are live.  Some are broken and some lead to patterns which do cost but some of the products are neat.

If you prefer to stay with the bare bones of math in regard to knitting or crocheting, there the pattern which uses repetitions of stitches, gauge based on a specific yarn so if you use a different yarn, you have to adjust.  When you increase the number of stitches, you are adding but when you remove or decrease stitches, you subtract.  If you need to calculate the cost of the project you use multiplication and if you are knitting a sock, you have to divide the number of stitches equally among the needles.

I think its awesome the way knitting and crocheting have expanded to create visual representations of all sorts of mathematics.  Check out Sarah-Marie Belcastro's webpage, the home of Mathematical Knitting with loads of links, examples, and articles on mathematical knitting.  She created the hyperbolic baby pants.

I'd love to hear what you think.  Drop me a line.

Using Paper Airplanes in Math

  This is the topic for those kids who are always making paper airplanes when your back is turned and try to get them airborne before you notice anything.  Why not take advantage of that and make a lesson based on how far a paper airplane can fly.

So lets look at a few activities you can do with your students to promote mathematics while allowing them to have a great time and be engaged at the same time.

No matter which mathematical concept you are trying to teach, you have to start with the airplane.  Why not start with a general discussion on favorite plane patterns and why in this type of activity you might want to use the same plane, rather than a variety.  Once this is done, it is time to let the students fold the plane using a template. This way they all have the same plane to start with.

So you are ready for the activity where students test their planes but make sure there is a throwing line with a judge so no one steps over and gets an unfair advantage.  Each student throws their plane 3 times and each time the distance should be measured and recorded in feet or meters. If you want to add another layer, have students make the distance negative if the plane flies backwards.

Once all the planes have been flown, the data recorded, and you are back in the classroom.  Its time to explore a variety of mathematical topics.  Some of these activities could be done for the whole class while others you might want for each plane.

1.  Calculate the average or mean for each set of flights.
2.  See where the mode is if any for the class
3.  Is there a median?
4.  Find the range of distances for the whole class.
5. Create a line plot of all the flights for the whole class as a way of visualizing mean, median, or mode.
6. Use the line plot to make a box plot so students can categorize their data.
7. The same data can be used in a histogram.
8.  Use a spreadsheet to produce bar graphs and histograms.
9.  The same data can be organized into a stem and leaf plot.
10. Discuss and/or calculate statistical deviations.

If you have students use a stopwatch to record the time it took each plane to fly the distance, they can calculate the rate or speed of the plane using the D = RT formula.

All of these possibilities from the simple act of making and throwing a paper airplane.  I have one week just before the end of the year where I'll do this because its awkward to try to teach after one week of doing cultural activities and just before the end of school.  This is something that they will love.

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

The Art of Mathematics

  The other day, I came across the "Art of Mathematics" website which I found absolutely fascinating.   The site is devoted to helping people discover the math in liberal arts.  Although, it is actually designed for college level courses, its resources can be used in high school or even younger with some adjustment.

They have created eleven ebooks filled with ideas for teaching the art of mathematics through one liberal art lens.  All of their books begin with The Art of Mathematics: Topic

1.  Art and Sculpture is designed to help the student discover the connection between visual art and mathematics.  The book has students explore a variety of topics from origami, to perspective, projections, and the interplay between dimensions, to the mathematics of 3D panoramic photographs and many other topics.

2. The idea of Calculus takes the student on a journey through the idea and concepts of Calculus by finding the area and length with fractals, string art to explore the relationship between curves, tangents, and derivatives, architecture, and a few other topics.

3. Dance which allows students to see the connections between mathematical ideas and concepts and dance ideas and concepts. This is done through Contra dancing, Salsa dancing, and using the maypole among other activities.

4. Games and Puzzles takes students on a journey connecting math with the structures and patterns found in games and puzzles through the use of a Rubik cube, Suduko, and Hex.   The exploration of each of these games introduces students to understanding how to create their own games or learning how computers think.

5. Geometry which uses activities to see the connection between math and the world around us.  Some of the activities include Flatlands to explore how we perceive what we are looking at, building a 3 dimensional piece of art to understand space, and other concepts.

That is just the first five books.  The other books focus on connecting the ideas and concepts of one type of art to the ideas and concepts of math using Knot Theory, Music, Number Theory, Patterns, the Infinite, and finally Truth, Reasoning, Certainty, and Proof.  Each book provides several inquiry based activities which allow students to explore the topic themselves rather than having the teachers stand in front of the class to "teach" the same material.

In addition, if you check out "classroom" resources, the site has provided information on inquiry based classrooms and using the materials provided along with grouping students, mathematical discussion etc.

Oh and the best thing of all is that all of these ebooks are free.  Just go to the site and download the books, one or all and have fun.  I realize some of these are a bit late for this year but you can always plan to use them next year in your class.

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

Dance, Transformations and Geometry.

  In Geometry, students are studying reflection, rotation, and translation, so the other day, I had them get up out of their seats to move.  We held our hands in a position so we had an invisible partner. 

We moved four steps to the right while saying "Translation, two, three, four"  Then we turned in place with "Rotation, two, three, four, followed by a half turn, pause, second half turn while saying "Reflection".  We repeated this several time but by the end of it, my students knew those three types of transformations and could explain them in their own words.  

In college, I took modern dance for my P.E. credit because  I was not and still am not the athletic person who plays hours and hours of basketball, volleyball or other sport.  I love to dance, so I was able to get my credit.  As I created the dance on the spot, I realized that most choreographed dances involve transformations to move the dancer around the stage using one of those three transformations or perhaps even dilation.  In addition, the choreographer looks at angles, parallel and perpendicular movement, shapes, patterns, or symmetry. 

Lets look at how each of the above connects between mathematical ideas and dance. 

1. Translations occur when the dancer changes position in a vertical, horizontal, or combination.  This is one way they are able to move around the stage.

2. Rotations are when the dancer does a pirouette in place or across the stage in which case it would be a translation with several rotations.  Usually rotations are 360 degrees but they don't have to be.

3.  Reflections happen with two different groups are dancing the same moves in opposite directions as if there is an line of reflection running between the dancers. 

5. Angles - Dancers use angles in different ways from the arrangement of the whole body so a leg might be at a 45 degree angle to the torso or the dancer might be in a deep plie so the legs for 90 degree angles when bent at the knee.  Angles could appear in the movement of the dance itself.

6. Parallel and perpendicular movement often occurs as the dancers move around the stage in lines or they head towards each other such as when Baby ran to Patrick Swayze in Dirty Dancing so he could lift her over his head.  She ran straight at him so if you had one line through Patrick Swayze and another running from her to Patrick, the lines are perpendicular.

7. As for shapes, many choreographers have the dancers move so at the end of a set of steps, the dancer has moved in the shape of a rectangle, or a circle, or even a spiral. 

8.  Patterns are applied to dance in the form of AABA or ABAB where A represents one set of moves while B represents another set of moves so the patterns are based on the combination ad repetition of the sets.

9.  Symmetry often occurs in the actual.  First you move 4 steps to the right and move the same number back so they are exactly the same but in the opposite direction.  The moves are symmetrical.  Sometimes, symmetry appears when you have an equal number of dancers on the left side of the stage and the right side.  The audience loves symmetry because it looks right. When its not symmetrical, the audience feels the dance is jarring.

So these are some places you find transformations and other geometrical concepts in dance.  To introduce the topic, show a dance video and have students look for certain things that represent transformations, or angles and identify these as they watch the video.

If you have any dancers in your class, you could let them create their own dance using geometric concepts.  Or get your filmmakers to create a short film or animated film to show where these concepts appear in dance.

I hope you found this interesting today.  Let me know what you think, I'd love to hear.  Have a great day. Tomorrow I'll share information on e-books designed to create a more inquiry based classroom using art and sculpture, dance, and other "liberal arts" topics.

Cryptography and Matrices

  Cryptography or the ability to code and distribute data or messages so only the party receiving the information is able to read it.  One of the most famous cryptography machines is the Enigma machine from World War II. What is interesting is that Enigma is not a machine, it is the name of a manufacturer who produced a series of machines before and during World War II.  In fact, the Enigma machines inspired other companies to produce their own cipher machines.

This situation provides math teachers with a perfect activity to introduce students of all ages to cipher machines and matrices as used in real life.  Students relate better when given an example of when they might use it.  Yes, I know they won't be working for the government but they could share messages so no one else would understand them.

Now the process involved in encrypting a machine is rather complex.  It involves several steps described as follows.
1.  Write out the message.

2. Create an alpha- numeric substitution so each letter has a numerical substitution.  Save 0 for spaces and use 1 to 26 for the letters but make sure its not sequential i.e. A=4, B = 5, C = 6 and so on.  Mix them up so its more like A=4, B = 9, C = 5, D = 21, etc. 

3. Choose an invertible square n x n matrix to help encode the message.  Then take your message and translate it into numerical form using the substitution chosen in step 2.

4.  Convert the message into an m x n matrix.  n representing the number of columns comes from step 2 and m represents the number of rows which changes depending on how many letters are in the message.

5. Now you multiply the matrix from step 3 and the one from step 4 to create the encoded message you are trying to keep secret.  You take the resulting matrix, write the letters out in a string and send it.

6.  The person receiving the message then takes the numerical values, places them in the same size matrix as step 4 and multiplies by the inverse of the n x n matrix from step 3 to get the m x n matrix in step 4. 

7.  Using the original substitution key, you translate the numerical values from the m x n matrix to find the original message.

If that gets a bit confusing, check this example to see how it works.  Engage NY has a nice little lesson on this topic that is well organized and set up to run stations so students rotate from one to another as they work through the activities.  There is lots of detailed examples of the math in using matrix along with the answers.

You do not have to wait until students reach advanced class to use this activity.  There are now matrix calculators available for both online and mobile devices so no more having to sit there with fingers as you multiply each term by each term and then spend hours trying to figure out where you messed up.  These apps allow you to integrate activities like this sooner so you can give students a taste of real life.

This topic is really appropriate for a cross curricular unit on World War II because you have this for the math component, social studies can discuss the enigma machine and other ways they communicated or the various intelligence agencies,  Language arts can focus on reading a fictional tale and science could build their own enigma machine.

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

Happy Easter.