Working Around Student Addiction

  I had hall way duty the other day just after lunch and I noticed how  many kids were plugged in to their mobile devices listening to music.  I often hear students tell me, they can't study without the music playing but if you watch them carefully, they spend more time changing songs and texting then they do actually studying.

An unfortunate side effect of this is that student attention span is dropping and its getting harder to hold their attention.  So what can we do as teachers to create a lesson and a classroom that works with a short attention span and a desire to move to something else.

This Power Point has some wonderful information on creating a pacing that will keep students interested in class.  I like the first real comment in the presentation where the author states that "good pacing creates an illusion of speed."  I've had good lessons like that.  So with good pacing, students do not have time to watch the clock.

One suggestion to create good pacing is to have several different activities designed to teach a single objective.  It is the flow from one activity to another that creates the pacing.  In addition to changing activities, one might consider changing the presentation, or the way students are grouped to help change the pace.

When you hold student attention, you help the student learn.  Unfortunately, there is no single answer for the question "How long is a student's attention span?"  There are lots of different answers but I've discovered it all depends on the students you have so you have to decide the length based on them.

Another suggestion is to break up the lecture into micro lectures or chunks. In other words, these are very small focused chunks of direct instruction that last mere minutes and are followed by a student activity.  This method of delivery makes the pace feel brisk rather than slow. 

In addition, the multiple activities allow for multiple starts and finishes which capitalizes on the idea that students learn best at the beginning of the lesson, followed by the end.  Its in the middle they don't learn as much.  It is also suggested that breaks between activities are made crystal clear so students use these as markers to help keep tract of the material.  These breaks are like bookends because the segments are clearly marked.

Check this out for more information and details.  Although it is a general presentation, it is easily applied to math.

Edutopia has a nice short article on pacing which adds to the above.
1.  Create a sense of urgency so they know it is important.  The author recommends a timer to help students know they must get it done within this segment of time.
2.  Make the goals clear so students know exactly what they are supposed to be learning.
3. Create smooth transitions so the lesson has few interruptions and everything is planned a head.  Have the next activity set up and ready to go so there is little or no dead time.
4. Be sure materials are prepped for the next activity.  Ask yourself if the next activity requires individual paper or could it be done on the screen?
5. Present instructions visually so they are written down on the board or screen.
6. Check for understanding and adjust accordingly.
7. Select the most effective method of presenting the material

I like the information I found in both articles.  I know I need to do it but I always seem to run out of time.  I think I'm going to have to really sit down and do some serious planning this summer so I have better lessons prepared for the fall. I plan to write a bit more on this tomorrow.

Are Spreadsheets in Math Really Effective

As a math teacher, I wondered if there was a "correct" way to use spreadsheets that deepens understanding rather than being just a time filler.  Since most places have adapted the Common Core, teachers are required to improve the student's depth of understanding.  As you know, I've found all sorts of spreadsheet activities but what is the best way to use them.  So today I'm exploring the best ways to do use spreadsheets.
According to an article in Education World, spreadsheets can be effectively used but it depends on how, when, and why they are used.

There is an idea that a blank speadsheet is like the student of a student .  In other words, the student has to enter the correct information into the spreadsheet so the spreadsheet "learns" to do the math.  The formula's have to be right for the spreadsheet to produce the correct answers.

The article indicates that spreadsheets are a good way to:
1. Extend patterns.
2. Explore algebra through formulas and functions in a variety of ways.  Check the article out for the details.

Just know that spreadsheets can be used to improve understanding and communicate mathematically.  Spreadsheets allow us to speak math with formulas and functions.  We can extend generalizations, and we can develop meaningful real life uses.  The two pieces of advice they give are stay focused on math and keep it simple.

According to an article in ERIC digest, spreadsheets are the ideal vehicle to connect arithmetic with algebra and allows a person to move back and forth quickly.  Spreadsheets help students develop an understanding of mathematical concepts, seeing relationships among different types of representations, and discover how a change in one place effects the other parts.  It also allows them to develop mathematical reasoning because they are freed from worrying about calculations and algebraic manipulation.

SERC has a short page on Why teachers should use spreadsheets in math class.  One of the first things stated is that most students have spreadsheets on their computers or have access to it via google docs.  In addition, most students will need to use it when they graduate and head off to work.  Because they are so easy to set up, spreadsheets increase both the breadth and depth of the class.  It has been suggested that spreadsheets can increase critical thinking by its very nature.  It also improves qualitative fluency and a student has to apply the process when they use one because they are modeling the data rather than just reading it.

Finally for tonight, Education has a lovely write-up that summarizes the whole topic. It can be summarized in this way.  Spreadsheets can be used to:
1. Save time.
2. Organize displays of information.
3. Allow "What if" questions to be displayed.
4. Increase motivation to work with mathematics.

So all in all, it looks like spreadsheets are great if used properly.  That is good to know and helps me prepare my lessons for next year.

I Did It Again!!!!!!!!

Today I had a couple problems on the warm-up dealing with things on sale.  As we went over the problems, I realized I was doing this the same way I had learned. In fact, every time I teach this or do any problems, I always teach it the same way.

 This is one of those topics that I always teach in one way.  I do not show a second way of doing it. By this I mean, I have the students the cost by the percent discount and then subtract the product from the cost.

Example:  You buy a $500 coat for 20 percent off.  So 500 x .20 = $100 and $500 - $100 = $400.  You paid $400 for the coat. This is the way I've always done it and its the way the books usually teach it.

As I was writing on the board, I realized I could easily teach it in at least two other ways.
First, I could have showed the students that if the store takes 20% off, you are actually paying 80% for the coat so its just as easy to multiply the original cost by 80% so that you don't have to subtract anything.  I know this intellectually but I forget it when I'm teaching it.  The nice thing about doing it this way is that it is extremely easy to create a visual representation.  I often make a rectangle, divide it into 5 parts where each part represents 20%.  I color in the 20% off part, leaving 80% uncolored so students can see where the 80% comes from.

Second, I could have shown them they could easily have divided the 500 by 5 so they know that each 20% represents $100.  So if they take 20% off, they take $100 off the cost or 80% would be 4 - 100's or $400.

If we are teaching visualization of the concept as a way of improving understanding, why do we revert to the multiply and subtract method?  Probably because its the way we learned and its our default way of doing things.  I check out suggested ways of teaching percentages using visualization and the method I saw most often, used a 10 by 10 grid to represent 100%. 

I am not sure how well that would work because the percentages used in high school are more often looking at either a mark-down such as when something is on sale or a mark-up for when you have your own business.  So I don't think that a 10 by 10 grid would work unless there is a way to remove or add to it to represent discounts, sales tax, mark-ups etc. 

So when I realized I was reverting to my usual way of teaching percentages, I made a conscientious decision to try teaching it more than one way.  I need a poster that states "There is more than one way to do things!" that I can hang in my room as a reminder to me.

Fractions and Spreadsheets and More

  Have you ever wondered how to use spreadsheets to teach fractions?  I haven't but I've been thinking about it for a while because most of my lower performing math students have no visualization of fractions.  The can perform the mechanical mathematical processes but if asked to draw or illustrate the problems, they can't do it.

I've had students draw pictures but the pictures are not always carefully drawn so the fractions are not even.  So I did some research on fractions and found a site called What if Spread Sheet Math.

They have a page of download able spreadsheets dealing with fractions, decimals, and common denominators.  The fraction spread sheet uses histograms to compare fractions using Excel.  It allows students to input fractions, guess which one is larger and which one is smaller before checking the answer.  The second downloadable spreadsheet is designed to help students learn decimal place value which is another topic many of my students have trouble with.  The third sheet helps students find common denominators using a spreadsheet.

I looked through the site and found much much more that could be used in a math classroom in different ways.  Under Arithmatic, there are activities that show the distributive property, factoring (which is used in factoring polynomials etc), area, shapes, square numbers, ratio tables that introduce slope, decimals and percents, multiplying integers, decimal division, etc.

Under Algebra they have growth spreadsheets, linear functions, solving equations, quadratic functions, power functions, polynomial functions, sine functions, the inverse of of a function, composition of functions, systems of functions, Interest, projectile motion and more.

Geometry has spreadsheets on building a house, shapes, parenthesis and pi, triangles, etc.  In addition, there is a section on financial reasoning and probability and statistics.  I was amazed at the number of spreadsheet activities contained in this website. 

I love the wide choice of activities that i can pick and choose to integrate into my classroom.  The beautiful thing is they even include an introduction to spreadsheet section to ease students into using spreadsheets in a way that is different than in a computer class.  This provides a cross curricular connection and reinforces their learning.

Check it out and see if there is something you can use in class.

Got One Working

Just this year, I learned about paper circuitry and I've been trying to find ways to use it in my math classes.  Before I do it in class, I'm going to make sure I know what I'm doing.  

For my first project, I decided to try creating a circuit to illustrate several points on the line y = 3/4X - 5.  I ended up with a parallel circuit with 5 major points on the line.  I used small LED's to light the points such as (0, -5) or (4, -2).

Once I got the circuit set up on construction paper, I placed the graph paper over the circuit and it lit nicely.  I did poke holes in the graph paper, so the lights fit through the paper and shown better.

 Tonight, I tried setting up a blinky slider series circuit for a two simultaneous linear equations.  I got the circuit built but nothing is happening.  I'm not sure what is wrong but I may try it again using parallel circuits to see if that would work better.

Do I know what I'm doing?  No, that is why I'm experimenting to see what works and what doesn't.  I may not be able to get the blinky slider working with this many LED's on it.  I don't know if I got it all set up properly but I'm not giving up.  I want to create a whole selection of these activities to integrate into my class next year to show students that Math does not always have to be figuring everything out on paper and that is it.

Before I started this linear equation, I had to choose  the equation of the line I wanted to create.  I needed a table of points for reference and then I had to figure out where everything went on the graph paper and on the construction paper.  I got it to work the first time.  But when I moved to trying to set up a systems of equations so the lights for the points for an X value would blink together before the next points and so on, I ran into trouble. 

I tried a series which is just not working so I think I'm going to retry it using a parallel to see if I can get it to work showing all the points at once.  Then I can try it again with the moving lights.  First step is to redo part of the circuit to see if I can get it working as is.  If not then I'll go back to the drawing board.

The other thought behind using paper circuits is to see if I can help build perseverance in the students who like to give up as soon as they think its too hard.  I admit that when it comes to electronics, I often put things off to the side while my mind works things out and the same applies to math.  I have to let my mind percolate.  Perhaps, I need to help my students learn to let things simmer in their minds until they figure out a method of attack.

I'm off to work on the circuit to see if I can get it working. As soon as I get it working, I will let you know and share the final product with you.  Have a good day.

Jigsaw Puzzles + Math

  I love working jigsaw puzzles.  I even have a jigsaw puzzle app on my iPad so I can work a puzzle any time.  So I wondered if there is a way to use jigsaw puzzles in Math.  It turns out there is.  It just doesn't use jagged pieces, it uses square, triangle, rectangular, or other sharp edged shape.

I created a small sample jigsaw puzzle using trinomials and their factors to fill out the rectangle.  I chose trinomials with a leading coefficient of 1 so as to factor faster.  The idea is to cut this apart, then reassemble it.

Other topics that could be used this way:
1. Trinomials with leading coefficents other than 1.
2  Slope
3.  Equations of a line between two points.
4.  Adding, subtracting, multiplying, or dividing fractions.
5.  Adding, subtracting, multiplying, or dividing decimals.
6. Calculating percent.

Simple easy to make but what about a more complex puzzle that would be more recognizable as a jigsaw puzzle.  Why not have students work out problems such as completing the square and use that as the base for the puzzle. The student can cut it up after laminating it and its ready to be reconstructed.

On the other hand to add a technological twist to puzzle making,  use a puzzle making app or web based app that will take any picture and turn it into a puzzle.  Take the problem the student worked, snap a photo of it, turn it into a puzzle and let them put it together.

I think the type of puzzle I have a picture of is actually the best to use in math because it requires the student to work out the problems in order to put it together but I'm no expert.

Video Note Taking

I regularly show short videos in class but I haven't done anything to help students learn to pick out what they need to know in the video.  In the past, when I've had them take notes, they just scribble anything down regardless of relevance.  In college, when I took notes, I'd take almost everything down since I didn't know how to take proper notes.

I have a goal of working with students next year on note taking in general and video note taking in specific.  One site suggests students  watch the video and then write down notes afterwards but I don't think that is as effective with ELL students.  Another way is to take notes while watching the video.  I know from past experience, they will write down unimportant things like "a girl rode a bike down the street" which has nothing to do with the rate times time equals distance.

I found this site which focuses on the best way to take notes for online and off line courses. The author makes several great suggestions starting with taking the notes by hand because most of us get distracted when we try to take notes on the computer.  Even if you have an app to take notes on, you often have to switch between the app and the video.  I found mention of a web tool that allows a person to watch the video while taking notes on a split screen but I cannot find any information on it since the last review of the tool in 2014.  The only URL seems to lead to an expired domain.  If I can find it, I'll provide information in a future

The second thing suggested is to only take down the "meat" and not every single thing the professor says.  This can easily be applied to watching short videos because they can write down the equations or short phrases.  The best suggestion I saw is for students to review their notes.  The author did comment that students should do notes by hand but they can type in the notes on the computer or note taking app.

Since I'm off Monday, I've left a Numb3rs episode to watch.  I found a nice graphic organizer for students to use to take notes as they watch the show.  Its a Key Facts Graphic Organizer from Freeology which asks them to find events, people, key terms, and facts.  It gives them a guide since I won't be there so they have a way of organizing the material.

Next fall, I am planning on giving students lessons on take notes off of math videos as that does require a different skill than say watching a video in English or Social Studies and they may not be able to transfer those particular skills.  It is said that you loose 40% of what you saw in a video within 20 minutes after its over.  So I'm adding another skill to my list of things for next year.

I Need Tape!

  As I read about ways to improve my teaching, I hit research that suggests we incorporate movement into our daily lessons to help students learn better.  Currently, I have students get up to get papers or iPads so I have movement but I'd like to expand the movement to including it for learning. 

I've seen several ideas that have movement built in to the activity but most of the activities require me to create a number line or grid using tape.  I have a room that is mostly carpet except for a small area next to the door which is tile.

I can do number lines on the tiled part but I end up making the units much smaller and its hard to do a good number line students can stand on. So I need to find out what type of tape would work well on carpeting.  It needs to come off easily and not leave any residue.  I'm thinking maybe painters tape but I don't know.  I've got a trip planned to the hardware store in May to get their advice.

I can use the tape to set up a grid system around my desks so I can use movement to teach:
1. Slope
2. Linear equations - finding a line from the two points, finding the equation from the line, etc.
3. Solving systems of equations.
4. Learning the coordinate system.
5. A number line for positive and negative numbers, adding and subtracting signed numbers.
6. Visual representations of percents, decimals, fractions, etc.
7. Matrices.
8. Polygons - area, perimeter, internal angles, external angles.
9. Similar and congruent Triangles.
10.  Solving certain types of equations.

So I need a tape I can put down and take up as needed that will not hurt my carpet.  I want to do more of this type of activity in my class so I keep students interested and focused.

More on Descartes Dots and Graphing

I reviewed Descartes Dots earlier in this blog but I've discovered that some students have difficulty using it if they are classified as ELL.  I've been working with my Pre-Algebra class which is 75% designated ELL students.  I can't always go as fast as I would like.

If you haven't run into it, it is a program designed to help students practice graphing on a coordinate plane by creating pictures using the provided coordinates.  You have to set point 1, set point 2, set the line, set the first point again at the end of the line.  We spent about 15 minutes on it but they will need more practice.

We ended up sitting on the floor with my students sitting around me.  I brought up the app and showed them how to do it step by step.  When I paused, I had them do the move before I demonstrated the next step.  A couple of the students got it but most of them are going to need additional practice.

I started the students on the Coordinate Plane app which relies on students finding the point by taping on the coordinate. At the end the app connected all the dots but it only uses the first quadrant.  So once they were comfortable, I moved them to Descartes Dots which uses all four quadrants.  Once they are comfortable using all four quadrants, I'll have them do an extention.

I am going to have students create a drawing on graph paper and write down the coordinates of the drawing  so other people can recreate the pictures.  Once, I've collected the drawings with coordinates, I'm going to make copies and then pass them out to other students to "test" how well they did.

Back to Descartes Dots.  There is both a free version and paid version but I use the free version and I really love it.  I find it fun and I love the pictures that are created and you never know what it will be.  Have a great time and enjoy.

Exponents

  Every time I work with exponents in any math class, I notice many of the students still do not have a good understanding of exponents.  I always review exponents in every math class because I end up using them in some way.  So I've been trying to find a visual way to help students understand the concept.

At this point, you are probably wondering why I have a picture of three dogs.  Well, they are a way of expressing exponents in a safe nonnumerical way.  I came across this great page showing how to teach exponents using art.

I might have the above picture to represent dog^2 = dog times dog.  or I might say dog squared = dog time dog.  In this visual representation activity you can use anything as the main object that you are creating exponents with.  It might be ice cream cones, cars, snow machines, anything.

I love it because it shows the base beautifully while letting students utilize the whole concept of exponents.  You could easily create a picture going the other way.  The dog picture would be excellent for dog times dog = dog^2.

Another visual representation is through the use of fractals which is something I'd never even considered. PBS has a great lesson which includes a 3 minute video on using fractals to show positive exponents.  It was great.  It uses the Sierpinski triangle in the video to help show the process.  The activity comes with the whole lesson plan.  Its really nice. 

PBS also had a lesson using the same Sierpinski triangle only for negative exponents.  This one comes with both the lesson plan and the worksheet.  I love the way the short video introduces negative exponents.  The positive exponent video looked at triangles while the negative exponent video looked at lengths of the triangle edges.  I plan to use this the next time I review exponents.

Now I just have to figure out how to teach the laws of exponents with a visualization other than writing it all out.  An example might be (x^2)^3 = (x^2)(x^2)(x^2).  They can see its x^6 but I'm wondering if there is another way of visualizing it.  I'm going to think about it and maybe play around at home.  I hope to put something up on it soon.

What Do You Do?

I think everyone has that one or two students who are always finished first, even when you give them slightly harder work.  They understand what they are doing and rip through the assignment.  You check the work and there are few  if any mistakes. What do you do at this point?

I finally found a solution which is working out but it means I have to be ready with the "next" assignment.  I let them work on the next assignment so when the rest of the class is ready to do it, they are my "assistants". 

They help me by going around the room answering questions so I don't feel like a gerbil running around in circles. This helps these students retain the material better and it implements peer tutoring which has been found to be beneficial.  As my students sometimes comment "The explain it in a way I understand better." 

I am starting to have students actually ask me for work so they can help out in class. Last night at study hall, I had two students who completed Friday's work so they can help me in the room.  They think its cool being totally caught up and ahead.  I like it because it keeps me so I"m thinking further than just the lesson I'm on.

I admit that with four weeks of school left, I"m to the point of looking at my day by day teaching rather than planning ahead.  This also gives me something to add to my planning for next year.  Yes I"m one of those teachers who plan ahead, create lesson plans, and decide how to teach things.  I don't rely on "last years" plans because every year I have a different group of students with different needs.

Another  reason I like doing this is simply that it is moving it from being teacher run to being more student run.  When students take ownership of their learning, they are more likely to do it and learn better.  I'm happy with this development.

Introducing Radians

I finally found a great video to use as an introduction to radians.  It clearly defines  a radian after giving a little bit of history on degrees.  I loved it because it explained everything in an easy manner so students can see it.

Check out Khan Acadamy for this video on the introduction to radians.  He shows how a radian is when the arc length of the circle equals the radian.  I've never seen such a clear description.  He goes on to show the relationship between degrees and radians.

The last part of the video explains why pi/180 or 180/pi is used as a conversion factor setting up to show the next video on converting between the radians and degrees.  This video is a perfect set up for the topic.

This leads into having students practice converting from degrees into radians and back.  Once they are comfortable, I'm going to reteach arc length using radians and again Khan Academy has a great video on it.  Its only three minutes long but it is a great introduction to the topic.

So when I have students redo finding the arc length in radians, they should find it much easier to do since they've already done it in degrees and they've explored the relationship between degrees and radians.

I think this will help my ELL students tremendously. They often need these additional steps.  I look forward to seeing how well they learn this.  I have not taught radians in Geometry before but with the change in standards I have to teach basic trig earlier in the sequence.  It used to be I didn't touch it before pre-Calculus but due to the demands of Common Core, I have to do this.

I am going to admit something to the rest of the world.  I never learned what a radian was when I was in college.  It was one of those things we accepted but never needed to know where it came from.  I am happy to learn more about it.  I love that I can still learn something new.  Yeah!

Circumference and Area, How I Love Thee, Let Me Count The Ways.

It started off as a normal Monday in Geometry.  My students could not remember the work they did Friday.  They could not remember how to find the circumference or the area which they need for the arc length or area of a sector.  As part of the warm-up, they had to find the arc length of a circle, the area of a sector, and the volume of a cone.

So I started out the lesson telling them that the circumference formula and area formulas are their best friend.  If they know them, they know how to find arc length and area of sectors. One of the best way's I've found is for the students to use a fraction of 360 degrees time either the formula for the circumference or the area.

Most of the material I've seen for teaching these topics use radians.  I realize that radians are the usual choice but for my ELL students I'm finding they have too much trouble relating to it at the moment.  When I use radians, they often get confused so I have them use the fraction of 360 which is much easier for them to use because they "see" where it comes from.  In addition, most of the problems in the math book are in degrees an not radians.

Once I have students comfortable using degrees, I plan to introduce the concept of radians.  This is actually the perfect spot to introduce radians as it gives them something to relate the new concept to.  They have established prior knowledge I can build upon.

My first step will be to have them draw circles that can be divided in to 4, 6 or 8 parts.  This provides a visualization for radians and the circle being 2 pi. They can create their own unit circle for degrees and radians.  Once they see that the radians are a fractional part of the circle, I think they might find it easier to calculate arc length using radians.

I like using a hands on activity to help introduce the material.  CPalms has a nice introductory activity for radians using pipe cleaners. This lesson comes with the plans, the worksheets, the resources, everything you need to conduct the activity.  Better Lesson also has a complete lesson ready to go. This lesson is part one of a two part lesson on radians and includes a bit of history which is nice.  I like having a couple activities to choose from when I introduce radians to the students.  I'm set for tomorrow.

Bar Modeling

  Recently, I've seen bar or strip modeling showing up more and more in several different aspects of math.  Bar modeling is often associated with Singapore math which uses it heavily in their program.

If you are not sure what it is, there is a lovely presentation that explains so much of it.   This 52 slide presentation is mostly geared at elementary but it has some excellent videos embedded and provides some great examples.

The Pensive Sloth has a great example for using it with algebra.  I wish I'd seen this earlier in the year with I introduced solving one step equations with  my students but I do see a way I can use it with them and linear equations.  I think it would be a wonderful way to reinforce solving equations while teaching about linear equations.

NCTM (National Council of Teachers of Mathamatics) has a great 6 page article on using visual tools in middle school.  It has some great information and illustrations to help the teacher learn to use these modeling tools in class.  The article is from the Oct 2015 issue of the middle school teaching magazine.  I've got the issue at home so I'll go read it in detail to do some planning.  I like that the author even shows ways of using visualization in several different ways.

Teacher Tube has a lovely video on dividing decimals which shows using the strip methods, fractions, basic math facts, and what to do if you don't remember the basic math facts.  Its nice because of the first example where he shows how the drawing relates to the actual division problem.

This site defines what a tape model is, how it is used and provides suggestions for its use in upper elementary, middle school, and high school.    It includes suggestions for implementing it, videos, and reminds teachers to make sure that the topic can be done via bar modeling.  This article suggests it as method or differentiation with students who lack conceptual understanding.

There are several youtube videos that show how to use strip modeling to show how to solve simple equations etc but I have not been able to review them because my home internet is down and the one at work has blocked you tube.

I just wish there was more information available using it in middle school and high school because we all have students who need something like this to help them learn better.

Virtual vs Concrete Manipulatives

  Recently, I've been wondering how effective virtual manipulatives are because I know that typing uses different muscles and does not provide the same kinesthetic experience that handwriting does.

After quite a bit of reading it appears that both have merits but it appears the bottom line in any manipulator use is that the teacher must be comfortable with the use of them.

First of all, the problem with concrete or physical manipulative is the individual pieces that can get lost.  On the other hand some students need to feel the physical piece in their hands.  Its like some of us love the feel and smell of paper in our hands.  It helps our memories.  On the other hand, virtual manipulatives can take a while to download. If the internet is messed up, it can make use of the virtual manipulatives difficult to use and they become frustrating.  I know from personal experience, students often see either one as a toy to play with.

One issue that come up is simply making sure you don't over use manipulatives including calculators to the point that students loose track of the math.  It was noted that sometimes the lessons focus more on the manipulatives than the math.  I've had my students in Algebra I using an app to help them learn to factor basic trinomials.  I realized my students have not figured out that the c and b go in specific places in the diamond.  They had the connection on paper but not on the app.  This gives me something to work on with them on Monday.

The interesting thing about virtual manipulatives is that most papers were not looking only at virtual representations of the physical manipulative such as the base 10 boards etc but as including visualizations of the concept which may be why it is often suggested both types be used in the classroom.  

A commonality among everything I read comes down to three or four main points.
1. Manipulatives either virtual or concrete should be used in all grades from K to 12 to help students develop a greater understanding of the mathematical concepts.
2.  We need to monitor the use manipulatives so students do not focus only on manipulatives but on the concept.
3. In general, manipulatives are not used enough in the upper grades/middle school/high school.
4.  Many teachers do not have proper training in using manipulatives effectively in the classroom.

I fit the last one.  When I went through teacher training, there was nothing on using manipulatives in high school.  Only elementary teachers receive that training.  I've had to teach myself about them.  I suspect there are others out there in the same boat.  Have a nice day.

Bummer

I just found out that I may not have my classroom set of ipads next year in my classroom full-time.  It turns out the last Tech Director did not keep track of equipment the way he should have and much of the inventory went missing.  Right now, there is no money in the budget for replacements and my iPads might have to be put in the general use supply which will be checked out.

That being the case, I have to explore two different options for next year.  I need to weigh the pros and cons of each option to see which is most viable.

The first is doing BYOB where I have students put three or four apps on their devices so we can still use some of the technology in class.  The problem with this option is not every student has a device and many of the devices are android based so I'll have to see if the apps I want are available on both. I know I"ll want a calculator, a note taking app, a mind mapping app and perhaps a graphing app such as Desmos.  I will have to make plans over the summer to see which I want to use.

The second is to arrange to check out ipads to use once or twice a week.  I don't like this option as much because there is no guarantee that the iPads will have the apps I want.  So for this option, I will have composition books available to write all the notes, etc in.

There is one other factor thrown into this mess is that I work at a very small high school with no more than 100 students.  Many of the students blow off the year and flunk various math classes.  We've reached a point where we cannot separate students the way we used to so they are going to be moved on.  This means, I'm going to have to do more differentiation in the hopes that I can help students who are missing certain skills get caught up so they do not fall farther behind.

It is going to be interesting to teach without iPads in my room.  I have gotten used to them and I really enjoy having them there for when things suddenly change and I have to readjust what I"m doing.  Sudden changes include fire drills, floods, power outages that cause the plumbing to go down.

I am still hoping I get to keep the iPads but only time will tell.  In the meantime, I will create plans that could easily incorporate the mobile devices while including more visual math and hands on to make learning better.

Percentages

  Its amazing how many of my students have trouble with the idea of percent being something per 100.  My father years ago told me that students do much better if you put it in terms of money rather than teach it as a decimal. 

So many times I remind them to think of 100 pennies and the percent is talking about how many pennies you have out of the 100.  I realize that its not precise but it helps my ELL students understand it better.

When I taught percentages at a community college, I learned to use the is/of = #/100 to solve those "What is 20% of 18" type problems.  It made it easier for students to solve that type of problem because they knew where to put the various numbers.

Better Lessons has a nice visual lesson on percents. This is a complete lesson plan that relies on a video, worksheets with percent model and a smartboard presentation. The video is more of a reflection but you do get enough information to present the lesson.

The one thing I love saving sales flyers from the Sunday paper for an assignment.  I have students "shop" for things and they have to calculate their savings from the shopping trip.  This makes it more realistic, especially if you are having them grocery shop.  Other times, I might have students plan to remodel a room and calculate the savings they received on the materials.

Another time, I had students decide to be a business.  They had to research the wholesale cost of the item and calculate the mark-up so they knew what they would have to sell it for so they could meet expenses. 

I just found a lovely article from George Mason University on using Art to help teach fractions, decimals and percentages. The idea is students use three colors of construction paper to create art on 10 by 10 grids much like the American artist Ellsworth Kelly from the 1950's.  Once they've created the art, they have to figure out the fraction, decimal and percentage of each color.   For more advanced students have them use an 8 by 8 grid. This makes it so they are not using a base of 100 for their percentage.

So now I have two new ideas for teaching percentages. The biggest issue my students have living in such a remote area is the lack of stores with sales.  The sales flyers I have to use are from the Bush Sales Flyer or the flyer from the supermarket in the hub two about 135 miles away.  The Bush Flyer is great because it includes sales info from Loews and Home Depot.  Its the best I can come up with so we make do with it.

Have fun trying the art activity.

Visual Math

  After reading the paper about visual math, I wanted to know more about it because it explains one reason I should not always get on students when they use their fingers to help them remember things.  It wasn't until I was an adult that someone showed me the trick with your fingers for nines.  I wish I'd known it when I was a kid in school.  It might have saved some frustrations on the 7 x 8 and the 9 x 6 calculations.


I discovered a website dedicated to visual math.  The site has lessons, exercises, and games for mostly Pre-algebra topics.  I checked out the integer chapter which has material spread out over several pages in small chunks. In addition, the pages are set up so a narrator reads each page.  This means someone with a print disability can still learn the material and allows for more independence.  Each page has lovely moving graphics that illustrate the concept presented on the page.  The exercises and games are geared more for elementary students but there are a couple that are worth using because they require the students to use higher order thinking skills. For instance, one game has students choosing the shortest route they can take from start to finish while crossing a square grid with assigned values for each segment.  That game takes a lot of thought.  I don't know if it requires flash because I used the site on my computer.

This site is dedicated to teaching fractions visually.   This site has a ton of material starting with lessons in pdf or power point form showing division, multiplication, addition, subtraction, comparing, renaming, etc of fractions.  The material is all visual and well done.  There are also games that use the help students practice using fractions and come in both a flash and non-flash version.  Each topic comes with worksheets to practice fractions and matching answer sheets. The worksheets use either circles, bars or both so students are able to see everything in more than one way. 

There is a short lesson plan for each topic that starts with an online pretest, the lesson, three online practice sessions, certain worksheets, and a post test so that the teacher does not need to do much planning.  There are additional worksheets available should a student need it to get ready for the post test.

This is a unit that could be used from elementary all the way up to high school age.  I wish I'd known about this at the beginning of the year because my Pre-Algebra class because all of them were weak in fractions and this would have allowed them to work at their own pace through the material. I'm going to keep this in mind for next year.  I'm also going to recommend this to other teachers in middle school and upper elementary as  scaffolding for their students.

Free This Week

Jo Boaler of You Cubed over at Standford University is offering the materials for the new visual math network without a login for the week.

The material consists of a 17 page paper in which she discusses how important visual representations are to people learning math.  This means all levels from very young to high school and beyond.

The paper discusses what the brain science research has to say about this particular topic.  One interesting thing I found in this part of the brain has to do with how the brain uses fingers, even past the normal expected use in school.  They talk about embodied cognition or how we use our body such as gesturing, pointing, etc to help convey mathematical ideas. If we don't have the words, we often will draw something in the air to convey it.  Apparently, it is much better for individual students to create their own gestures, rather than using ones supplied by the teacher because it helps them remember better.

In fact the paper gives two or three examples of visual math that were used in the classroom.  The paper concluded three important things.

1. Replace the idea with  strong mathematical learners memorize and calculate well with the idea that visual representations actually help students learn better.

2.  Successful mathematicians use finger representations in their minds. 

3. Mathematical instruction needs to include more visual representations.

Please down load the paper itself and read it so you can get the full picture. It was so interesting to see I'm doing some things right.  Today, I had my Algebra II class draw representations for completing the square.   A couple of my students after the example I put on the board had light bulbs go off and were off and running.  A few more could see it better and when we start doing the problems on Thursday, they feel they won't have that much trouble doing it.

The second item to download is a 26 page paper that has details on the visual math activities used in research for the paper.  This pdf has everything needed to complete the various activities.  Some activities are designed for young children to develop their finger sense and discrimination while others work for Algebra and other classes. 

The final item is a check list for the teacher to use to see if they use enough visual math now.  Its like a self reflection on your current teaching.

I plan to print off the paper, highlight and read it to see how I can improve my teaching.  I love trying to find visual representations for the topics I teach.  I admit, some topics such as dividing fractions can be very difficult to figure out but I've done it.  So go check it out, read it and enjoy it.

Arcs and Sector Areas

Currently in Geometry, the students are working on area, circumference, arcs, area of sectors, etc.  They are struggling to relate the topics and try to look at each topic as a separate topic.  I want to find ways that might help them learn to flow from one topic to the next.

I found this 39 page Pdf which contains a whole unit on arcs and area of sectors. It has everything from the pre-assessment to the lesson plan to the activities and the post assessment.  This is great in that it introduces radians so it would be perfect between circles and the trigonometric ratios.  I wish I'd found this about two weeks ago when I was starting this.  That being said, I could easily use parts of this to help my students learn the material better through the use of activities rather than relying on worksheets.  It is labeled as a concept development packet.

This site has a great pdf of cards with real life problems for finding circumference, area, radius, and other interesting things.  For instance one card has you calculate a sheet of gold to cut coins from but the sheet has to be of the smallest size so there is less waste.  I like the variety of problems since the problems are not all circumference, or area, or diameter.

This slide share shows where circles are found in real life from theater to car tires, to the earth, to manhole covers.  It has some good information and would be a great introduction to the card activity.

The Department of Education in Virginia has a great 5 page lesson plan and activities for arc length.  There are two activities included.  One finds the length between two ends which are actually arcs and requires use of the Pythagorean Theorem while the other is an activity that requires students to cut a 10 inch cake into 12 pieces.  The questions require thought so it will take them a while to complete.

This activity has students working to figure out the area of a large cookie that is at least 8 inches in diameter.  They end up calculating area and circumference of the cookie, the area of their slice and the length of the edge (arc) of the cookie.  When done they can eat it.

Finally, this power point presentation has some great practice problems for students that are done in such as way as to have a friendlier feel than most worksheets.  There are instructional slides scattered through out the problems.  I like the way its organized.

Have fun checking it out.

Mathmatical Modeling part 2

Yesterday, I wrote about a great pdf that would make a great introduction to the topic. Today, I"m looking at several places that offer lesson plans on mathematical modeling.

One of them is the University of Indiana in Bloomington.  The page offers 40 modeling lesson plans tested by teachers and staff for students in grades 7 to 12.  The lessons actually include a senario and ask that the students create a proposal that includes the mathematical model to show why their suggestion is the best one.  Others are either experiments that require them to come up with rules for certain things while others are performance tasks based on real life situations such as needing to protect art in a museum.

The teachers college at Columbia University has a wonderful 25 page sample of the  Mathematical Modeling Handbook with the introduction and the first project with everything needed to conduct the lesson.  The introduction gives some everyday situations and discusses how it modeling should be taught in schools. 

Annenberg learner has a great lesson on mathematical modeling using circular movement, and transmission ratios followed by one on mosquito population and exponential growth.   The lessons come with everything you need and utilize hands on activities to help students see connections.

Back to the Plus Maths site for a complete package on mathematical modeling. This package shows some of the ways mathematical modeling is used in medicine and nature, economics, politics, and human interaction, games, sports, and art.  They offer three levels - explicit has students using the math, middle ground that introduces mathematical modeling and gives a glimpse of equations, the bigger picture that looks beyond what can be done in the classroom, and the try it yourself are problems from NRICH to do in the classroom.

The University of Arizona has two semesters worth of posters showing student work from a class on mathematical modeling.  The posters are one page summaries of the work they did in the class and contains everything from the problem to results, references and potential applications of the results.  Each poster includes a listing of scientific challenges for the problem.  Some of these are beyond my ELL student's understanding but a few, such as the one which explores why the same store might charge different prices for an item is something they can relate to.

Finally from the University of Texas at Austin is a nice page of mathematical modeling projects with units on science and engineering, art, music, and entrepreneurship. Each unit comes with everything needed from teacher instructions to the worksheets to samples showing how the results can be put together.  Each topic has several subtopics such as under fine arts - geometry, there are two different lesson plans, one on symmetry of design while the other looks at kaleidoscopes.

I think I'm going to see where I can work this in my math classes so students get a chance to experience mathematical modeling.  I know I can use one on population growth anytime I teach exponential functions.  So I have a start.  Have a good day.

Mathematics Modeling

  Yesterday while searching for information on the math used by the CDC, I came across a great 72 page pdf on math modeling .  It's eight sections cover it all from the general introduction to putting everything together.  Every chapter has a summery and an activity at the end.  The activities are set up so by the end the student has completed their own mathematical model.

Each chapter focuses on a specific aspect of mathematical modeling.  Chapter two is solely on defining the problem statement including the types of problem statements and possible interpretations of those questions.   The authors use quite a few mind maps to illustrate the process used to help define the problem.

Chapter three looks at the process of making assumptions for the topic.  This chapter puts the material from the previous chapter into focus and helps create a clearer view of the topic.  It gives some great information and shows step by step how the assumptions fit together.

Chapter four focuses on what aspects are the variables in the problem, both independent and dependent.  The authors provide three different examples so students can see how independent and dependent variables are designated.  The authors explain that the problem determines which is the independent and dependent variables.  By this point, students have created their initial mathematical model.

Chapter five concentrates on approaches to using model to create initial data. The authors give a set of questions to help the student proceed with this step. Again, there are three examples and each example is shown with two or three different approaches so they can see how this works. The examples include any math and graphs to show results using these specific approaches.

Chapter six examines analyzing the data in two ways.  The first is does the answer make sense and the second looks at ways to look at a more detailed analysis of the data.  Again using examples the authors show how to analysis the data in both ways and provide a list of things to check for.  They even state that students might want to refine the model and run it again if there is time.

Chapter seven ties everything thing together to complete the process including how to write the results up.  It includes do's and don'ts for writing the results, and the recommended structure.  At the very end, there is a final write-up of a model done on recycling so students can see what the end results should be.

This is a well done guide put out by the Society of Industrial and Applied Math.  Check it out, download it, use it.  Tomorrow, I'll talk about sites with suggested lesson plans on mathematical modeling.

Center for Disease Control

Yesterday a lunch, we ended up discussing how the influenza of 1918 caused native populations in the area to decline rapidly. One of the guys brought up how a Japanese balloon during WW II landed and brought disease with it that killed quite a few locals. 

Every time there is a break out of something, the CDC or Center for Disease Control gets involved.  It doesn't even have to be a disease. Years ago when they put that additive M something in the fuel, Alaska had health issues because a byproduct was formaldehyde.  Due to the cold winters, the formaldehyde stayed at ground level and people developed problems.  The CDC came up an investigated.  So this asks the question - "What math does the CDC use?".

The Washington Post ran a great article that discusses the spread of Ebola.  The article includes some great infographics and discusses the math involved. It would be easy to create a worksheet or guide to accompany the article so students can write down information as they read.

This article discusses what can happen when a wrong value is used in calculating risk. A business sold flooring made in China that may not meet U.S. air quality standards.  In addition, this article explains more about the error.  Its quite good in that it shows how a small error can cause underestimation of the danger. This article showed the underestimation and how it changed when the error was corrected.

This pdf discusses the ways statistics is used within infectious disease epidemiology.  It is a 17 page document that discusses frequency distribution, ratios, proportions, and rates, morbidity frequency rates, point vs period prevalence and other topics.  There are lots of examples because this is used to train health officials.

From We Use Math comes an interview with an epidemiologist who explains the types of math they do and how she uses it in her job.  It is a short article that could be used as an introduction.

I found a great pdf for math modeling I am going to share tomorrow because its perfect for a entry on that topic.

Firefighter Math

Have you ever wondered what mathematics firefighters use in their job?  How much is actually what we teach in the classroom?  It turns out we teach the foundation in several math classes but we don't supply the application of the topics.

After checking into the math that police officers use, I wondered what math firefighters used.  Low and behold, I found an online self-paced course for firefighter math.  It covers math that a fire fighter is going to need in the general sense such as ratios and proportion and then provides a fire fighting application with several practice problems.

Each topic is set up with a general introduction, examples, followed by practice problems. These practice problems are online multiple choice problems with immediate feedback.  I like the clear explanations you are given if you are correct.  I tried a couple and when I was wrong, the program told me I was incorrect, try again.  I did learn a fair bit.

This is set up nicely so I could use it in my geometry class to show how volume of cylinders has a real life application by having students work their way through the hose section.  It looks easy to assign the sections to students as needed.

It appears this is the only real site that discusses firefighting math but I found a great article on Fire and Math that talks about the use of natural logs and Fourier's law of heat transfer in arson investigation. This 6 page article is great because it explains natural logs and Fourier's law of heat transfer before providing specific examples.

This site has a 400 page document that talks about all the factors on fires.  It has detailed information on the factors and talks about mathematical modeling and computer modeling for Fire Dynamics.  There are so many more factors than I ever realized. This would make a good basis for a project.

Have fun exploring these sites.

SLO's

My district finally got around to doing Student Learning Objectives or SLO's.  Until the professional learning opportunity where they taught us about SLO's, I was under the impression that an SLO was like an IEP for each student. 

It turns out I was wrong.  Its simply a plan to teach a specific standard or part of standard.  I choose the what, how, when, and do it.  You set a goal for improvement and it gives you some real data.

I chose my Algebra II class for my first SLO. I looked at simple factoring from the standards and created my plan of action.   I found a nice pretest I gave my students and the majority of them freaked out and didn't do well at all.  Most had a zero percent but that means the only place they can go is up. 

My plan started students off with a sheet to learn the diamond factoring.  I taught it, passed out worksheets and when I moved on to simple factoring using that, I had them use the Diamond factoring app to practice throughout the unit.  At each step, they saw a video to introduce each step, followed by "I do, We do, You do."  They practiced using IXL.  Finally I gave them a set of practice problems. 

The post test is this week.  I told them they could take it any day this week.  So far half the class has taken it and the results ranged from 50 to 100% which shows real growth.  I'm happy with having to use the SLO.

Before this, I didn't know how to write a unit that would help me focus on the best way to instruct my students.  Usually, I've had administrators who had us make pacing guides for the whole year.  This never really gave me the focus I needed.  I also had someone who decided we should look at the previous year's state test, analyze that and use it to teach but that didn't always give the students a good mathematical background. 

I really feel as thought I hit a milestone in my teaching and I know how to prepare properly.  I love it when I made a breakthrough in my teaching.

Math Journals

  I am rethinking the idea of journals after reading about a teacher who has students write about all sorts of things from warm-ups to homework problems.

It used to be a math journal was only used for students to journal their thoughts and to summarize their learning.  In the last few years, the use of journals has changed.


Now journals can be used to:
1.  Take interactive notes.

2. Work their guided practice problems through.

3. Self-reflection.

4. Answer open ended questions.

5. Explain their thinking.

6. Share their answers on  homework problems.

According to  one document , journals serve several purposes.  They help
1. Increase student awareness of how they learn and remember.

2. Provide a record of student thinking.

3. To help students see that writing is a way of learning.

4. Provide a context for recalling previous learning and summarize current learning.

5. Provide a record of the challenges students face when learning new material.

Journal entries can be student or teacher directed.  Student directed journaling can involve their explaining how they feel, what they think, what they need to practice.  Teacher directed journaling covers material where the teacher asks the students to explain how a strategy works,  how to solve a problem, explain how to do something, explain an error in a problem, construct and model an answer for a problem, and support a point of view on why a certain way of doing something might be the best way.

One source recommends the teacher create an example journal filled with examples of what the students are expected to put in their journals.  This example journal is a quick way for students to check to see if they are doing things correctly.  Even though this suggestion is geared for elementary students, I can see using it in the middle school and high school for ELL learners.

I like the idea for creating an example journal for my classes because it becomes part of the I do, We do, You do method of teaching and it would help me plan what I'm doing in my class ahead of time.  I bet I could have students write down their answers to any scavenger hunt and comment on what parts they had trouble with.  It would provide another assessment.

Difference of Squares

Today I introduced factoring the difference of squares using the same drawing I use for multiplying binomials.  This was actually one of the best ways I've ever used to provide visually why it works the way it does.

I started by drawing a representation of 4X^2 -16 so they could see the two parts that existed.  I asked what is this drawing missing. The students noticed there was no x's in the drawing.  I asked them why would there be nothing showing.  After a bit of thinking and guessing they finally came up with the idea that the missing elements were opposites of each other.

So in a different color, I added in the missing x's.  One set of four is positive, one set is negative.  We drew in the missing parts to this square.  It was great because this is the first time students could connect the drawing with the equation and add in the missing parts. 
The final step in the process of drawing was to add the lengths of the side.  The 4x^2 was easy because it is 2x due to 2 x's.  The numbers were  4 and -4

It was so easy for them to see all.  I just realized, I should have cut the x's in half so I had 8 positive and 8 negative values.  I'll change that the next time I do it.

In Algebra I, I introduced multiplying binomials whose result is the difference of squares.  I drew the boxes, filled in the values and showed how the x values cancel each other out by erasing those area.  I loved the way the students kind of went "Oh."  When they did their guided practice, most of them just whizzed through the whole set and if they made a mistake, they had no trouble understanding why.

This method is going to show up in my box of teaching tools.  Tomorrow, I'm going to be using the same idea to help my Algebra I learn the pattern for binomials squared.  Yes!

Completing the Square

May I start off with the fact that I hate teaching the topic of completing the square.  I always have and I always will.  If all else fails, I run the equation through the quadratic formula, find the half way point, run that number through the formula and I have the three points I need for a rough graph.  Now, I carry a graphing app on my phone and I graph the equation, read it, and write the equation from the graph.  Unfortunately, I have to teach it as part of the curriculum.

So before I can teach completing the square I need to make sure that students at least know there are perfect squares as that is part of the form.  So then the question becomes, what is the best way to introduce the topic to high schoolers.  The Math = Love blog provides a great introductory activity to help students visualize the process.

This is the first time I've seen an activity that uses manipulatives to introduce the topic .  I have a whole set of Algebra tiles in my classroom for my students to use.  I see how this activity could be used to teach perfect squares and difference of squares.  Two uses out of one activity.

This file has a complete 5 page lesson plan that uses the idea from the hands on activity used to introduce the topic.  It comes with warm-up, the lesson, a think-pair-share, and practice problems.  It does a good job of connecting the visual with the process.  The directions are clear and it is ready to use.

 This final link is perfect because it uses the box method to show students how to complete the square.  This is great because the box method is one of the methods I use to teach students how to multiply binomials.   I love how its used rather than relying on the standard formula of taking half of the middle term, square it, and add to both sides. 

Due to these sites, I have a new way of teaching completing the square that I believe will be more effective than the way I've taught it in the past. 

Literal Equations

My students have such a problem with literal equations, especially when they have to rewrite them.  It might be because I do not give them enough practice in rewriting the equations themselves.

I am guilty of teaching formulas and literal equations in only one way and I don't teach them to find different things.  In Geometry, I had them find the area only.  I did not ask them to find the base given the area.

CPalms has a nice lesson plans with everything needed to introduce the topic.  I like the way it includes the necessary materials to accompany the lesson.

One way I've taught this type of thing is with sticky notes on the board that I could physically move around.  I write the variable on the sticky note, put it on the board as an equation, then move it around step by step.  I've passed out a ton of sticky notes to the kids to use on their white boards to go through the same process.

Hands on High School Math recommends using cards such as 3 x 5 cards cut in half with variables and operations written on them.  So you could create R x T = D using one card for each term.  Then they can rearrange the cards to form the new equations such as D/R = T.  I like the idea of using cards better because I can laminate them for future use.  This is much better than using sticky notes.

The Math = Love site suggests a scavenger hunt as a way to practice rewriting literal equations.  I enjoy the way she has it set up although I've been known to set one up using QR codes.  The only thing I missed in the examples was the statement on rewriting it to find x or y.  This is important.  I think I'll do one with the standard equations such as I = PRT or A=LxW.

I would add one thing to this and that is give students some numbers to use in the various equations to get answers.  My students are not good at going from the literal formulas to substituting values in for real answers.  I'm hoping by adding this in, my students might be able to transfer their knowledge.

Better Lesson has a nice fully developed lesson from a brainstorming introduction, to a guided notes and practice, a partner activity and a closing activity that includes an exit ticket.  I like that this has the power point presentation for the guided notes, the worksheets for the practice activity and a do now or warm-up at the beginning.  I like the way the lesson is fully developed and ready to go.  I plan to try this.

These resources when integrated will provide a nice unit for literal equations.  I will be teaching it in about two weeks.

Math Identity

  The March issue of the National Council of Teachers of Mathematics for Middle School Teachers magazine has a really nice article on developing math identity.  I wasn't sure what math identity is in this context because it was not something they talked about when I was in teachers training.

Apparently math identity is a the frame around knowledge, skills, habits, attitudes, beliefs, and relationships students need to successfully learn math.

It is suggested that their mathematical identity is connected with all their other identities.  Consequently, teachers have the ability to shape a student's mathematical identity.  One way to do that is to support a flexible mindset. 

Have students work in groups so there each member has a particular role, give them group type activities so that no students dominate the interactions.  Next, have students keep a math journal which could include warm-ups or bell ringers, classwork, open ended reflections and problems, and possibly homework.  In addition, the teacher needs to communicate expectations that every student will learn mathematics and contribute to the mathematical learning of others.

It is important to focus on their abilities rather than their deficits. It is also important to help students learn to believe in themselves as thinkers and problem solvers.  This means focusing on their skills and talents rather than learning the algorithm.  As part of this, students need to know it is possible to have more than one correct way to do any problem. 

There are four major things that they say we as teachers must do.

1. Give all the students a chance to write or talk to a partner before anyone answers a questions publicly.  This would be a great time to use silent conversation where two people share one piece of paper and carry on the conversation by writing it out.

2. Set routines so students can develop ideas privately and share to a small group before presenting to everyone else.  Think, pair, share would be a great starting point to have this happen.  So have them work in pairs, then in groups of four before sharing to the class.

3. Have the students either individually or as a group give a rating to indicate with hands or fingers, where they stand in terms of moving on.

4. Have the students set goals for the class.

Four easy things to help students develop a flexible mindset and a strong mathematical identity.  This article has helped me see where I need to go in my teaching.  I'm happy I found it.