Art and Math

  I want to thank O'Reilly for today's entry.  They featured an article which was fantastic where the author admitted to wondering why a person needed to learn mathematics when they have so many more fun things to do.

It wasn't until she started college when she discovered you needed math to create beautiful interactive art she stumbled upon in college.

Through out the article she shows how learning trig opened a whole new world of creation using circles and moving objects.  Even learning about simple triangles sparked multiple layered creations filled with beauty.  The author shows us how triangles can be combined for art.

The most beautiful statement for me as a math teacher was the one in which she connects triangles to polygons.  Made my heart explode in joy.  I admit she describes all of the math learning in association with certain specific graphic programs but she needed the math to create each and every piece of art.

I love the way she connects her art work with various aspects of geometry and trig.  In addition, she ends the article by sharing that math is also used in the development of apps.  I plan to share this article with my students tomorrow at the beginning of each period.

So many of my students are artists who would rather draw than do math so this is perhaps a way of igniting their enthusiasm to continue in art while finding an enjoyable way for them to want to learn math. 

Check out Jinju Jang's article. Share it with your friends and with your students when they ask "When am I ever going to use this?"  I'd love to hear your thoughts.

Linear Interpolation

I am teaching a new math class this semester called the Mathematics of Animation.  It is based off of and uses much of the material from Pixar in a Box at Khan Academy.  Normally I would use most of the material but our bandwidth sucks and I cannot get all the students online to do everything online.  To make it count as a math credit, I choose one or two topics per unit and go into more detail.

So in this case, the topic is linear interpolation which is a way to fill in values you might not have otherwise.  It involves finding the slope and using that to help find your next number.  I would use it to find the value between the 2nd and 3rd dots.  Assuming the x value of the 2nd dot is 9.5 and the 3rd dot is 13.90, I can use it to find the value for x = 11.

 So far, the linear interpolation is being used in the context of the frame and time associated with the animation of a ball.   I began by teaching it in a general sense of the steps needed in the process.  I used the chart which connected the viscosity of water to its temperature.  I wanted to give it to them in several different contexts so they might see how it applied to different situations.

The second context was population growth between two different years.  Same process but a different context.  I was able to show the slope as the population increase each year.  This context worked out much better because they could relate to population growth much better than the viscosity of water. 

For the third context, I took them back to their original context.  The final context was the easiest as it is what they saw when creating animation but they still struggled with the process.  Since I'm a bit frustrated, I looked on the web for ideas on ways to teach this topic but I didn't find anything other than what I was doing. 

I found lots of worksheets, a few videos, and other materials but no articles on ways to teach it effectively.  I did find a super mathematical article on it but nothing I could use with my students.

So I am wondering is there a better way to teach this topic or is it have them find the slope between the two points, find the distance between the original x value and the new x value to find the amount of increase before adding it to the original y value. 

Does anyone have any suggestions?  I'd love to hear from anyone on this topic.  Thank you.

Warm-up

Happy Birthday, Lewis Carroll

I Hate This

I would like to know why elementary teachers insist on teaching students the inequality signs using that stupid alligator eating the fish analogy.  It drives me crazy.  I was working with a college student on piecewise functions and she had to stop, hold up her hand and make eating motions.

Honestly, I never learned it that way and when a student explained it to me, it made no sense what so ever. 

It works if the inequality is written in a standard way such as x<4 but it doesn't work quite as well if you write it as 4 > x. 

I honestly don't know the best way to teach this concept or the best way to present it to high school students.  What I can say is I work on having students connect directly to the signs without using the alligator.

I try to teach inequalities in one variable by relating it to a number line so they have a visual.  For instance x < 4 tells me I am looking for a value smaller than 4.  If you look at the sign, it is like the head of an arrow and points towards the numbers which meet the criteria.  If its 4 > x, I teach students that this states that 4 is going to be larger than any value for x and the sign is telling you that 4 is always bigger than the value you choose.

I think this is more important than using alligators because it helps students relate to the directions rather than trying to remember the eating part.  This tells me they have not developed a real understanding of inequalities and their relationship to numbers.

Fortunately, they drop the alligator when we start graphing systems of linear inequalities but they still have trouble with the concept it is an area which can be the answer.  On a line, its a series of numbers they deal with.  Its like traveling on a road, you just follow it and go in a certain direction but when they graduate to linear inequalities, they suddenly are dealing with a boundary acting like a type of fence marking the end of a region.

So they have to learn to think in terms of areas and where does this region lie.  I sometimes equate it with a fenced ranch.  This is easier for them to relate to.  I'd love suggestions from others on how they teach this topic.  I'd also love to hear your feelings regarding the alligator story.

Depth of Knowledge

I recently ran across the term "Depth of Knowledge" in regard to learning.  Not having heard it used before, I had to investigate.  Yes, I'm a bit like a cat when I run into new things.  I have to find out more or it will bother me until I do.

Depth of knowledge is defined as a way of determining or classifying the level of thinking required by a task to complete.  In other words, the more complex thinking required the more depth of knowledge the task has.

The levels of knowledge are as follows:
1. Recall and reproduction - the lowest level because it requires nothing more than recalling memorized facts.  It does not require any real thinking.

2. Skills and concepts - requires a bit more thinking because the student needs to make a few decisions on how to complete the task but its still rather minimal.

3. Strategic thinking - or really planning how to complete the task by working out the best way to do it, applying knowledge, and some justification.  Such a task could have multiple correct answers which the person has to justify the one they found.

4. Extended thinking which is the highest level because it requires students to synthesize material from multiple sources or transfer knowledge from one area to another. 

The next question becomes how do we apply this to our classes.  I know that unless I see actual examples I have trouble coming up with ideas.  Once I've seen suggestions, I'm fine and can develop my own.

First off, check out this DOK wheel which has a list of verbs for each level so you know which ones to use when you create a wheel.   Words such as define, state, or tell are all level one while develop a logical argument, formulate, hypothesis take it up to level three.  If you start using design, analyze or create, you've designed level four activities.

From this site, you can download a great question stems sheet to help write those level three and four questions.  Its sort of a fill in the blank but it helps me get started and on the right path.  This is a paper which gives ideas for using the depth of knowledge in your math classroom.  It has great definitions for each level with a few ideas so you build a great foundation.

Finally this 13 page resource actually gives ideas at each level for activities, teacher and student based activities along with a list of suggestions which could be applied to various subjects. 

Tomorrow, I'll look more indepth at this topic for its application to math.  Let me know what you think.

Levels of Convincing

  Its great that the election is over and the new president has been installed. We can get back to normal with our daily lives.  Now is the time to look at the idea that their are different levels of convincing.  I'd never thought of that until I read something by Robert Kaplinsky on the subject.

There are three levels of convincing according to his article.   Each level has a different level of convincing needed.  It makes sense.


1.  The first and easiest level is convincing yourself of something.  If this were a trial, you would be the defendant because you begin by convincing yourself you are innocent.  If you believe it, you are more likely to have the ability to convince others.

2.  The next level is to convince your friends of that same thing.  In the trial, you would be convincing your defense lawyer of your innocence so he will take your case and work to convince others. 

3.  The third level is when your friends work on convincing others who might doubt you like the jury. The jury is undecided and are their to listen to your lawyer convince them you are innocent.  While the hardest person to convince is the doubter who is not ready to be convinced.  Someone who needs a lot of convincing to change their opinion much like the prosecuting attorney.

So what would this look like in the math classroom? Well the first level is when a student successfully finds the answer and convinces their partner their answer is correct.  The other student listens to the explanation and agrees with it.  The solution is shared with the rest of the class who have yet to solve the problem and take time to listen.  They are like the jury, ready to be convinced.  The hardest people to share this with are those who already have a solution and know their answer is correct so they are harder to convince.

In reality, convincing in the math classroom means students have to construct an argument which supports their answer such as using a pizza to determine which is bigger, 1/6 or 1/8 rather than saying our teacher last year told us it was so.

In addition, the levels do appear in the classroom because a student has to believe in their argument before they can share it.  The level it takes to convince ones self is different than trying to convince someone who already holds a belief.  Furthermore, it helps develop mathematical thinking.

Remember "Convince me" promotes inquiry and discourse without being judgemental.

If you use it in your  classroom, let me know.

Great Idea

  I am on Robert Kaplinsky's email list.  Recently, I received a lovely email from him on how he gets his students to explain their thinking and he does it without using the words "Explain your thinking."

I sometimes believe when we use those three words cause our students to freeze.  It is like the time I was attempting to learn Inupiaq, the language they speak in parts of Northern Alaska, I was bad at it.  At one point she asked me something. I had no idea what she said so I answered "yes" complete with a stark look of fright on my face.

He chooses problems like XX + XX and use the digits 1 through 9 only once to find the largest number.  He discovered when he asked them how they got their answer or explain their thinking, he'd usually get something like I added 27 + 98 to get 125.  They did not explain their thought process, only what they came up with.

One day he asked them to convince him as to why their answer is right or wrong.  This directive requires students to provide more explanation than just what they got.  They might say their neighbor added a couple of numbers which was lower than theirs.  This is one step more towards a good explanation.

I wonder if we ever take time to teach our students what is expected when they explain their thinking.  Do they know what makes up an explanation.  Its possible the reason we get such a simple answer is because we have not taken the time to help students develop a method for explaining their answers.

I like the idea of asking a student to convince the teacher their answer is right.  It means they have to provide a reason to support their answer.  Even comparing their answer to another is a good start because comparing items is a tactic used in convincing people.

We might want to take a few minutes to discuss what is involved in convincing people.  After all, politicians are always trying to convince people to vote for them, movie trailers are used to convince people to pay money to see the movie, ads are used to convince people to buy a product.  We see this happening all around us.  So how is it applied to mathematics?

That we can discuss with our students to help them learn the art of persuasion. 

Which One Does Not Belong

Which one doesn't belong is a great warm-up activity for math.  It allows students to think about similar characteristics while deciding what might make one different.

Any activity in this group you make should have at least two possible answers.  In addition to selecting the answer, students need to be able to articulate their reasoning.

For instance, in the example I made there are two possible answers.  You could say the triangle because it only has three sides while the other four are quadrilaterals. The other choice is the rounded edged quadrilateral because it does not have the angular vertices.

When you make your own "Which one doesn't belong." think about how students are going to interpret the choices.  I've had students justify their answers with "That one has no R's in it."  I ask them to see if there is a better reason. 

Shapes are the easiest thing to use and you could put all four shapes in different colors so they have to look beyond the appearance.  I've used shapes but I've also used expressions such as x^2y^2, xy, x^2y, xy^2.  The possible answers might be xy because the exponent is one or x^2y^2 because its the only one with both variables squared.

I've even used things like beets, tomatoes, peppers, oranges just to make a change.  This one could be oranges because the others are red or beet because its the only real vegetable.  Of course this has lead to some really great discussions on the fact that beets, tomatoes,  and peppers come in several colors while oranges are pretty much orange.

If I wanted to have them think mathematically, I might show an apple, an egg, a lemon, and a lime and ask them which one does not belong mathematically.  In this case, I might say an apple because its a sphere and not an ellipse.  

Sometimes I've listed terms, or lines, or graphs, in this activity but I've made sure that at least two could be chosen.  I love this as a way to develop mathematical conversation in class and to work on looking past the surface a bit deeper to create reasons with a real basis. 

When introducing the activity, its fine to start with easy examples such as the one with fruits and vegetables.  After they get used to it, throw in the mathematical element which helps them expand their ability to think in deeper ways. 

I love using these in class during my warm-ups.  Let me know what you think.

Warm-up

Warm-up

Algebra on Nudge- The App.

I recently stumbled across a really nice app called Algebra on Nudge.  It focuses on teaching students to solve linear equations right now but shows they hope to do quadratic equations in the future.

Students start with a problem and they can slide terms over to the other side as needed.  The work area is on the left while the computer shows each step completed on the right.

Notice the picture on the right has the problem and highlighted the term needing to be moved.  The student then puts their finger on the number and moves it to the other side.



Once on the other side, a key pad comes up and asks the student to put in the correct operation.

Once the operation is put in, the app puts the sign in so the student sees the complete step.

The problem goes translucent asking the student to complete the math so you are left with the answer to that step.

Once they finish the problem, they get a congrats and stars.

If a student makes a mistake,the program tells them immediately and circles the error.

The linear equations can be one step or multiple steps but it always shows the completed steps and shows errors.  I like this app and the best part of it, its free.

Transfer Learning in Math

  Today I read a description about a class being offered through the state.  It is on visual learning.  The four topics covered in the class include laying the foundation, surface learning, deep learning, and transfer learning.  I've heard of the surface and deep learning but I've never seen transfer learning listed separately.

Transfer learning refers to taking the learning from one situation and applying it to another.

There are four key concepts associated with transfer learning.
1. The initial learning is important to transfer learning.  It is important students master the material during initial learning otherwise they will find it difficult to transfer their knowledge.  In addition, students need to understand the material rather than just memorizing it.  Finally, students need enough time to master and understand the topic being taught.

2. Students who understand abstract representations of the topic find it easier to transfer knowledge.  One way to help develop understanding of abstract representations is to have students solve a problem for one situation, then provide additional examples for students to solve so they work on creating and understanding abstract knowledge. 

Another way is to teach the material in a specific context and then pose several what if scenarios. The third way is to have students create a solution to a class of problems such as figuring out the best travel times for boat trips in a specific part of the country rather than for one trip. It helps to introduce material in multiple contexts rather than a single context so students develop a sense of its wider applications. This includes providing examples for each context.  

3. Transferring knowledge is an active dynamic process and should be viewed as such.  Transferring is the path between what is learned and what is being tested. Part of the process involves selecting and using strategies, knowing available resources, and feedback.  Sometimes, transference of knowledge requires prompting which can help increase the amount being transferred.  One way to improve transference is to help students become more self-aware and actively monitor their own learning.

4.  All new learning needs to be based on prior knowledge.  It is important to activate prior knowledge that many students may already have.  Second, it is possible or students to misinterpret the new knowledge by not correctly using previous knowledge incorrectly.  Finally, some of the new information may seem to be in opposition to what they already know which makes it harder to transfer knowledge. 

Greenl Screen Word Problem

Over the weekend, I created a green screen word problem using my Smart board to create my recording talking about things.  Then I found maps and an actual shot of the Eiffel Tower on google street view.  I put them together on iMovie.

Today, I tried it out in my first period and they really enjoyed it. They requested more of these.  I had the students watch the video one time through before watching it a second time to write down the important information.

They were required to use the Pythagorean Theorem to find the distance from their toes to the top of the tower and then they used trig to find the angle of the hypotenuse. It was cool because it gave them a reference so they could see exactly what they were doing.

It went well and I had fun with it. I'm sharing the video with you so you can see what they saw.

I've had requests for Dubai, Canada, New Zealand, Staten Island, and Los Angeles so far. They are really enjoying it.

Let me know what you think.

Reengaging.

  Yesterday, I looked at reteaching.  While looking for ideas, I came across something from the San Francisco Unified School District Mathematics Department on reengaging students rather than reteaching.

Reengaging gives the feeling of having the students more involved in their learning while reteaching gives the impression the teacher is doing most of the work.

Re-engagement pushes the student to look at their own conceptual understanding of the topic so they can eliminate their misconceptions.  It is said to be more effective because it engages students in a meta-cognitive way.

This method should be employed only after the student has had a chance to learn about the topic.  At this point many students still have misconceptions and its important to clear them up.  It is suggested the teacher give a short assessment to determine common misconceptions.  The teacher should look at the misconceptions to determine what the student was thinking so as to help eliminate it.

Many of my students combine constants and coefficients so 3 + 4x = 7x.  I realized the other day, they're previous teachers never took time to eliminate this particular misconception.  They are used to adding numbers so they added the 3 and 4 without paying attention to the x.

Suggestions for helping students eliminate misconceptions are:

1.  Math Talk - where students explain their thinking on a problem which contains a common error.  Those who made the mistake are encouraged to explain their thinking while those who are correct are also encouraged to explain their thinking.  This activity does need to have rules established so there is no name calling and it has to be done in a safe atmosphere so students want to talk.

2. Revision - Allow students time to revise their work if they have errors.  When I was in school, we called it making corrections but revision sounds gentler and less intimidating.   This goes a bit further than just correcting.  If a student makes a mistake, they identify the mistake, explain why its a mistake and how they need to correct it before correcting it.

One way to prepare and use re-engagement is to have students answer a problem to determine their understanding and figure out where any common errors occur in the process.  Next have students work in small groups to develop a strategy to help themselves find the errors in the problem.  Students will share their strategies with the classroom.

Then students will analyze the problem looking for misconceptions and errors.  During this process they will clarify their thinking by explaining where the mistake is and how to correct it. Finally, they revise their work before trying a similar problem.

For my students  I think I have to use both reteaching and re-engagement because they need to learn the material and learn to determine their own misconceptions.  I like some of the ideas I've read over the past two days.

Ways to Reteach a Topic.

  There are times when we have to reteach material to certain students.  We know we should reteach it using the same methods but what can we do to make reteaching material more effective?

When reteaching content, it is suggested the teacher break the material down into small chunks so students are able to process and learn the material better.  Once they show understanding and mastery of the first chunks, you can introduce more chunks until all of the topic has been learned. If at any point, students show they have not learned the material, try using alternate explanations or problems.

One way to assess understanding is to use a short quiz or questions to determine student misconceptions.  This give a starting point so the teacher can focus on eliminating the misconceptions before moving on.

When reteaching it should consist of controlled, coached, and independent practices.  Controlled practice is when the teacher models and guides the students, usually the first step.  Coached practices are when the students are working more independently and the teacher moves around coaching students while providing feedback, and suggestions.  Finally, independent practice refers to being able to work independently with little coaching or feedback.

It is suggested the independent practice should not be graded but used to analyze strengths and remaining weaknesses. It is the weaknesses which provide the area of reteaching to help students become proficient.

Some days, there is simply not enough time to analyze student understanding to create a proper lesson plan.  Fortunately, there are three easily used strategies for reteaching which can easily be inserted into the lesson.

1. Error Analysis where the teacher posts questions with errors and the students work in small groups to find the errors.  Once they identify the common error, they become more aware of it and can often find it in their own work.  I like this because students often have to be trained to find errors in their work.

2. Create short activities which help students clear up small areas of misunderstanding.  These activities should not be graded, they are just for eliminating misunderstanding.

3. Think aloud where the teacher models their own thinking aloud while mentioning common misconceptions along with explaining why you do what you do. 

Tomorrow, I'm looking at something new to me.  Re-engagement seems to be more student centered while reteaching which seems to be more teacher oriented.  More on that tomorrow.

Warm-up

Warm-up

Practice Makes Perfect or Does It?

How many times have you hear the phrase "Practice makes perfect."?  I've heard it from my grandmother, my parents, my teachers, just about everyone.  I even heard it the other day from another teacher.  He commented the new math program did not have enough practice problems.

I realized one thing about practice and that is you have to learn to do it correctly the first few times or you will impress the wrong moves in your muscles or in your brain. If a student learns the process incorrectly, it takes about 21 days to relearn it correctly.  Imagine 21 days.

Have you taken time to actually analyze the mistakes of your students?  I haven't but I have noticed that certain students make the exact same mistake every time they do the problem.  For instance, when rewriting a two variable equation into slope intercept form, they always combine the x variable and the constant into one term.  Every time.  This means they have the incorrect process imprinted in their brain and it becomes harder to reteach it.

This means you have to reteach the concept which is almost like having to break them of one bad habit and replace it with a better habit. This means you have to reteach the material and have students practice the correct process over a 21 day period so hey have a chance of learning to do it correctly.

Sometimes, they develop the incorrect process because they were absent while other times  its because they didn't learn it properly the first time or perhaps they got careless and just started doing it wrong.  It doesn't matter because we have to make sure students learn and practice it correctly.

I'm figuring out I have to break it down so they practice say the last step on their own, then I have them do another step plus the last step independently until they can work the whole problem on their own.  In addition I try to spread the lesson over several days while practicing previously taught material.  I use homework as another place to practice previously taught material so they have a better chance of learning it properly.

The current thought on this topic is not "Practice makes perfect"  but "Perfect practice makes perfect." Next week I'll provide some ideas for helping students learn the material correctly the first time and looking at effective ways of reteaching.

Feel free to comment and let me know what you think about this idea.

Sum of Interior Angles Hands On.

The first two days of this week, I had students work on an activity designed to help them arrive at the formula for the sum of interior angles.  The first day, I passed out this great worksheet from Great Math Teaching Ideas which has all the regular polygons from triangles to decagons already done and ready to go.

I printed off the three page activity put only passed out pages two and three with the polygons.

The first step is to have students use a protractor to measure one angle and use that to find the sum of the interior angles.  This step has them learning to use and reinforce the use of a protractor.

For the next step, I had students divide the shapes into smaller triangles with the lines starting from the same vertex.  This provides students a chance to create the 180 degree part of the formula.  When everyone had this completed we discussed using the knowledge that triangles have 180 degrees with the number of triangles to calculate the sum of the interior angles.

I asked students to compare their answers which lead to a discussion on why there might be differences between their answers with the protractor and the calculated answer.  It was a great discussion.

I passed out the first page with the chart for names of the polygon, number of sides, and sum of interior angles.  I added two more columns, one for number of triangles and one for each individual interior angle.  One column was added in front on the left side  and one at the end on the right.

Once the chart was completed, I asked questions which lead to the (n-2)180 formula.  They added it to their notes and finished class by answering three of the four questions at the bottom.

Yesterday, they had a chance to apply the actual formula with a worksheet.  Let me know how you teach this.  I prefer activities to introduce topics.  Thanks you all.

iPlugmate

The other day, a friend showed me this awesome thumb drive with a plug for ipads and iphones and a usb one.  It is called iPlugmate and its made by a company called HooToo.  I think he got it off of Amazon during one of their lightening deals.

I borrowed it to transfer some photos off of an iTouch to my computer and it was wonderful.  I didn't have to plug it in, bring up iPhoto, connect them and wait. 

I had to download an app for the free app and I was ready to go.

 As you can see it comes with a plug on each end, the i device on the wider end and the USB plug on the narrower end.  The caps are attached to the thumb drive so they do not get lost.

I simply plugged in the drive to my itouch and instructed the app so save certain photos on it so I could use them on my computer.  The first time I used it, it took me a bit to realize I had successfully transferred the material because the light indicating success is in the middle, right next to transfer interrupted.

Once the photos were transferred, I unplugged it and plugged it into my computer.  It was easy to transfer the photos into iphoto.  I ejected it just like any other thumb drive.

Here is a closer look at the plugs on this thumb drive.  It comes with 32 GB of storage which is enough for what I need.  I can see using it in class.

I often have students do things on the iPad and I have had students mail the presentation to me via e-mail but now, I can just pop it on here and directly transfer it. 

There are times our internet goes down so I do not have to wait till it comes up.  I can use it at home to transfer material from my i devices to my computer and back without hooking it up with the cord.

To say the least, I think it was a great investment for me.  If you've been looking for something like this, check it out on Amazon.  Just look for iPlugmate.

Writing Open Ended Questions

  Yesterday, I talked about visible thinking.  One of the suggestions was to use questions that had no one right answer so students had to explain their thinking.  Unfortunately, it is not always easy to find questions that meet the criteria so its important to know how to change questions from having only one answer to a more open ended one or create your own questions.

Lets first look at taking specific questions found in the textbook and turning them into more open ended questions.

  Take a process question such as "Calculate 47 x 25" and rewrite it to read "Calculate 47 x 25 in two different ways.". Or take a question like "I have a quarter, a dime, and three nickles in my pocket, how much do I have?" and change it to "I have 5 coins in my pocket, how much might they be worth?"."

Instead of finding the volume of a rectangular prism measuring 2.1 by 8.2 by 3.4 and rewrite it so the student is asked to create a word problem where you need to find the volume of a rectangular prism in order to solve it.  You could also rewrite it to require students to find all the possible dimensions of a rectangular prism with a volume of 120 m^3.


Next lets look at techniques for creating open ended questions.

1.  Think of Jeopardy where the answer is given and the contestant provides the question but in this case there is more than one answer.  An example might be "Area of 30 square feet" which gives four possible questions such as "What is a rectangle that is 3 by 10, or 2 by 15, or 1 by 30 or 4 by 5".  This will be a student's first thought but it also allows for questions like "What is a triangle with a height of 10 feet and a base of 6 feet."

2. Use examples that have wrong answers and have the students decide where the mistake is and how to correct it.  It might be a question like "George thinks 24 + 37 equals 51 but Jill says its 511.  Who is correct?  Explain your answer."  In this case both answer are wrong so they have to explain that.  It could also be a question involving one person seeing five rectangles in a design while the other person sees three rectangles and two squares.  The student has to help settle the disagreement.

3. Get menu's or price lists from real world places and have students calculate things like "How much would a school lunch cost if it were bought at this restaurant?" Get a news paper and ask students to speculate on the relationship between the space articles and ads take up on a page.

4. Use the Tell Me All technique which simply asks students to write down what they know on a topic such as fractions, roots, factoring, etc.

Tomorrow we'll look at creating good "Which one does not belong?" questions.  Let me know what you think.

Visible Thinking

  I've been reading up on the topic of visible thinking.  Its something I'm not always as good at using as I should be.

For those of you who are not familiar with visible thinking, it is framework designed to help shape intellectual development.

The goals of visible thinking include developing deeper understanding of the material, increase motivation for learning, improve attitudes towards learning and move the culture towards one of wanting to learn.

One thing is to establish simple rules for exploring ideas student thinking becomes visible through their discussions.Most of my students have trouble expressing their thinking.  Sometimes its because they do not understand the concept, sometimes its due to a lack of  vocabulary.

Visible thinking should involve teachers thinking aloud, students verbally expressing their thinking, students listening to others who are expressing their thoughts, students use discussion to help form understanding, and recording understanding via journals, solving problems, and completing projects.

Some of the recommended ideas for helping students develop visible thinking are:
1. Justifying your thinking through questions such as "What's going on?" or what do you see that makes you think that?"  Questions such as these help students learn to describe things and then support this with evidence.  This type of questioning can be used with everyone from whole class to individuals.

2.  Asking what they think, what still puzzles them, or what are some ways to explore the topic further.  These questions can help connect the current topic with prior knowledge.  This is good to use at the beginning of a topic because it helps students recognize what they already know.  It can also be used to figure out what they still need to learn.

3. Another one is the "I used to think.......but now I think.........." which can be used to show how student understanding has changed.  It might be as simple as I used to think I should add numbers and variables together but now, I think you add numbers to numbers and variables to variables.

4. Use the See, Think, Wonder when introducing new material.  Assign them to preview the section while they consider what they are seeing, what do they think its about, and is there anything they wonder about as they look through the material.

5.  Think about Chalk Talk which is where you list each essential vocabulary word on a piece of chart paper. Each student has a marker and they add their definition, thought, or comment on the word.  They rotate completing this for every word.  This activity can also be used for a mathematical reflection where you place a series of questions, one on each piece of chart paper, and students write down their thoughts, questions, or examples.

6. One way to increase discussion is to present problems to students that have no one right answer.  This could be done via questions such as "Which one does not belong?" or "I have 5 coins in my pocket, what might their total be?"  Have students justify or explain their answer.

These are some simple, easy to implement steps which help increase visible thinking in your classroom.  Let me know what you think. 

Radius Warm-up

Warm-up

Ways to Encourage Persistence

  One of the things we have to work with students is perseverance.  As  you know many of my students are ELL and they find it difficult to continue working on problems and are more likely to give up. 

What are some ways to help students perseverance, so they are less likely to give up and more willing to work their way through the problems.  Unfortunately, research indicates if a student cannot solve a problem within a few minutes, they are more likely to give up.


1.  It is suggested teachers model their own thinking as they solve problems.  This means they talk out loud what is going through their minds as they solve a problem.  This can include discussing being perplexed at one point and sharing what makes it perplexing.  As students see this, they see how to verbalize their own questions.

2. Teachers should also encourage think time.  In other words, students should be allowed time to think about how they are going to solve the problems.  They need time to consider ways to solve the problem so you cannot expect a quick answer.

3.  Encourage students to solving problems using a variety of methods.  By exposing them to a variety of ways, they can choose the method that makes the most sense to them. This also allows them the tools to try different methods if one does not work.

4. Teachers can plan to build persistence.  One way fo doing this is to ask questions such as "Tell me more about it." or "Can you say that in a different way so other understand?" or "Can you put that in your own words."

5. Communicate with students and parents about attitudes on math such as thinking you are not good at math is a learned attitude.   Many parents and students believe the ability to do mathematics is based on talent rather than effort. 

6.  Expect students to share their ideas with others verbally while commenting on others ideas.  They are asked to go to the board to share their ideas and solutions.  They can include representations to show how they see it.

7. Help students realize that it is OK to make mistakes. They should think of mistakes as problems ot be solved or as honing their skills.

I like some of these suggestions and I plan to implement some of them.  I already show students different methods for solving problems but there is more I can do such as verbalize my thinking.  Let me know what you think.

Have a good day.

Ways To Encourage Higher Order Thinking.

  I often wonder how to encourage students to use higher order skills rather than just trying to guess the answer.  I often wonder what techniques I can use to help them.  The textbook has some higher order thinking questions but my students need to have a model on how to do answer them.

I've been working on them with question that may have more than one answer and the justification for their answer but I need more.

Higher order thinking skills are above simple memorization or rote.  It requires them to take the memorized facts to a higher level such as inferring, connecting, manipulating or other such skill.  Suggestions for increasing higher order thinking skills include:

1. Teach the strategies so they know them.

2.  Make sure students understand the important features of one concept that distinguishes it from another.

3. Name the key concept and help students determine the type of concept it is.  Is it concrete, abstract, verbal, nonverbal, or process.

4. Help poorly performing students learn to express themselves by working on verbalizing the steps involved.

5.  Move from concrete to abstract and back so there is a understanding of both.

6. When teaching concepts include false features to help show what its not along with examples of what it is and what its not.

7. Always make sure students have mastered the basic concepts before moving on to the more sophisticated concepts.

8.  Take time to connect concepts beginning with connecting the smaller ones to larger ones.

This two page pdf has some great questions geared to help develop higher order thinking. The questions fall into promoting problem solving, what to do if a student gets stuck, making connections among ideas and applications, encouraging reflections, checking student progress, making sense of mathematics, and encouraging prediction. 

I especially like the questions under helping students when they get stuck. The questions are geared to have students think about the next step.  The questions are such as "How did you tackle a similar problem?" or "Have you compared your work with others in the group?" 

I plan to download these questions to use in class.  I like the questions guide rather than give direct answers.  Check them out and let me know what you think.

Have a good day.

Math Castle

 I downloaded another app designed to help students practice their algebraic skills.  Math Castle has a free version which allows you to access the first level problems for all operations and mixed.  If you like it, you can upgrade to the full version for $1.99.

Once you decide which operation you want to practice, you will be given a page with options for the numbers involved in the problems.
You have a choice of the numbers involved in the problems.  It means the student can start out with easy problems and work up to the problems with larger numbers.

Once the game starts, problems begin to cross the sky in sun like carriers.  the problems are presented in several different ways such as b = 17 -15 or 17 + b = 27 so the student is not always answering questions with the same format.




The way to score gold is to solve the equation, touch the sun so a ball of energy flies down and destroys it if it is the correct answer.  If its wrong, the sun keeps coming but you can try again.

As the student proves their ability, the number of suns increases and begins to arrive faster and faster.  The early ones allow time to use a paper and pencil but as the attack speeds up, it requires the use of mental math to solve the problem.

I found it fun but what I find fun, students do not always find fun.  I am going to have a few students play with it to let me know how they like it.  If they like it, I can add it to the I-pads.  I do like the way it allows students to experience problems in different ways so they do not get used to it in only one format.

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

Algebra Attack

 I downloaded a new math app titled Algebra Attack. It is a free game designed by Tarek Karam to help students improve their algebraic skills by thinking outside the box.

So I downloaded it to check it out.  Each problem solved, means the player moves up one level to a slightly harder problem for solving both equations and inequalities.

Where this app disappears is that it provides both the problem and the numbers you may use to solve it.  So there are restrictions before you begin and the choices do not follow the standard rules of solving two step equations.

This may be good because it requires some higher thought processes to solve the equations.  In addition, at the bottom there is a comment on how many steps it should take you to solve.

If you look to the screen at the left, you have the first problem to solve and it does follow the standard rules.  You subtract three and solve the answer.  You have the opportunity to retry if you get stumped.

The counter at the bottom does keep track of the moves you've made.  If you run out of moves, it will tell you and stop the problem so you have to start again.

As you progress up the levels, the problems and choices get much harder so it takes more thought the higher you go.  I have yet to make it past level 6.

This one shows you a good example of a problem which you have to choose whether to add 21 to first or subtract 7.  I added 21 first, then subtracted 7 till it was at minus 21 before adding 21 again.

If you start at level one again or you redo a level, you get different problems so you are not doing the exact same problems over and over.  They problems change.

There is no mechanism that I found to ask for hints to help you get started or even show the solution so you have to persevere until you get it solved. 

I'm not fully sure how I like this but I think it does help students develop number sense so they are able to solve problems using higher thinking skills.

I'd love to hear what you think about the premise behind this app.  Have a great day.

New Movie

I've been traveling and staying in a hotel the last few days because of weather issues and with luck I'll be getting home today.  Due to this, I've been able to watch television and  see all the ads for movies.  I stumbled across one that I hope to see once its out on DVD.

The movie about three African American females who worked for NASA back in the early 60's, helping with the space program.  Imagine, three women helped in their own way to get John Glenn and others to the moon in 1969.

One is Katherine Johnson who was a trained mathematician.  At this time in history, she had to fight hard to get anywhere due to the societal beliefs.  She was born in West Virginia back in 1918 and obtained a degree in Mathematics in 1937 but it wasn't until 1953, she was hired by NASA.  Her first job was to help analyze data in the Flight Research Division.

She worked on other projects such as the Friendship 7 mission so when John Glenn was due to go up on the Apollo mission, he demanded she check all the numbers because he didn't trust the computers.  When she said ok, he went.  She was known for her mathematical skills. She retired from NASA in 1986.

Mary Jackson is the second one who was born in 1921, received a degree in Mathematics in 1942 and was hired by NASA in 1951.  She ended up working with an engineer in the wind tunnels section.  He encouraged her to become an engineer.  She managed it even though she needed permission to attend the all white classes but when she graduated she became the first African American Engineer at NASA.  She retired in 1985.

The third woman was Dorthy Vaughn who was born in 1910.  She obtained her degree in Mathematics in 1929.  She joined the Langley Memorial Aeronautical Laboratory in 1943, during the war.  She became part of the computing pool, the people who performed the mathematical calculations needed to process data.  Unfortunately, due to her ethnicity, she was assigned to the West Computing Area due to laws of segregation.

In 1949, she was promoted to head the group and was the first African American woman in that position.  She made many contributions and when the company became NASA, integration occurred and she joined the computation division.  She retired in 1971.

After seeing the ads for this movie, I realized there were very few female mathematicians I'd heard about and most of them were scattered through history.  This opened my eyes to the fact there are more women out there hidden in the mists who made huge contributions but were virtually unrecognized.

I think when the new semester starts, I am going to find and create posters of female mathematicians so my girls can see others who made huge contributions.  Let me know what you think.  I also plan to order the

New Years Day