Atlanta:  Traffic, traffic, and more traffic…

Coming Soon! Peach Pass available in Spring 2011.
I-85 Express Lanes in Atlanta open in Summer 2011.

View the Peach Pass video to see lots of accessible math connected to a real community issue. Learn more about the I-85 Express Lanes.

• What’s the difference between an express lane, a toll road, and a turnpike?
• Are you charged by the mile or by the minute?
• Why is the target speed 45 miles per hour?  What is the target speed for other express lanes?
• How will the Peach Pass know when I should pay (because I have less than 3 people in my vehicle) and when I can ride toll free?
• What is the mathematical model that determines the toll?  We know it is positively correlated.  Will the model be linear, exponential, or some other type of function?
• What will the revenue generated by the Peach Pass be used for and who controls these monies?
• Are the Peach Pass and other E-ZPass-type cards cost effective or just convenient?
• How do the other locations listed in the video charge for the use of their express lanes?  How do other states collect this money?  Utah, for example, uses an ExpressPass.

The Pennsylvania Turnpike is the oldest turnpike in our country.  Beginning in January, 2011 there was a rate increase; cash tolls increased 10% while E-ZPass tolls increased by 3%.  Is there a savings to use the E-ZPass, or is it just for convenience?  Since there is a Pennsylvania Toll/Mileage calculator, we can investigate the cost to drive on the PA turnpike.  Would this help indicate a reasonable rate for driving on any toll road or express lane?

To see if there is a pattern to the cost, I chose to collect data entering the Pennsylvania Turnpike at Interchange 57-Pittsburgh and then vary the exiting interchange for a class 1 vehicle with 2 axels.  I wonder what the toll rate for an 18-wheeler would be compared to my passenger vehicle.  My learners have many choices.  They may choose to start at any entry point on the turnpike and vary their exiting interchanges.  I suppose they could vary both the enter and exit interchanges.  They could also change the type of vehicle to investigate the charges and the rates for different size vehicles.

Is there a pattern to the data?  It the relationship linear, exponential, logistic?

To see the relationship between the data, we graph.

Cash toll charged vs. miles driven on the PA Turnpike:

E-ZPass toll charged vs. miles driven on the PA Turnpike:

To compare the two data sets, graph on the same grid.

More questions:

1. What are the mathematical models that could represent these data sets?
2. What are the meaning of the slopes of these lines?
3. Is it cost efficient to purchase the E-ZPass?
4. Is there a relationship between the E-ZPass toll charged and the Cash toll charged?
5. What is the mathematical model that could represent these data?
6. What is the meaning of the slope of this line?

• How does the rate charged by the PA Turnpike compare to the rates of other turnpikes?
• How does the rate charged by a turnpike compare with the charge on a toll road or express lane?
• From \$0.60 to \$6.00 is a pretty big swing in cost to use the 16 miles of the I-85 express lane in Atlanta.  How will traffic volume be determined since tolls go up when traffic volume increases and the toll is lowered when traffic volume decreases?
• How do the toll roads, turnpikes, and express lanes in other countries compare to our toll roads, turnpikes, and express lanes?  How do they compare in cost, in speed, and in access?

Can our learners aquire the needed content through a problem or project based approach?  Will they find the content more interesting and engaging?

As we learn more about problem-based learning and project-based learning, would this be type of lesson help learners see the application of content? … the blending of content? … the relevance of content?

I think so.  Are we willing to experiment?… to learn by doing?

s=v*t + 0.5a*t^2 ~ Wanneer heeft u vorige bezoek?

If you teach a foreign language, what is the most important or essential for your learners?

• Is it that they are grammatically correct?
• Is it that they can read a book or watch a video in that language and understand the message?
• Is it that they can communicate with others when in a country where this language is the primary language?
• Is it that they can solve problems that arise while in a country where this language is the primary language?

While being grammatically correct and reading are both very important, I’m pretty sure that we would agree that being able to communicate with others and solve problems in another language would be more important.  If I’m sick, I need to find a doctor or the hospital.  Right?

I made several new friends at last week.  Culturally, we are very different.  For example this came across my Twitter timeline from @mvanast:

Laatste ontbijtje hier. Gek dat ze deken dat je ijsthee wilt, als je thee bestelt.”

Roughly translated it says “Last breakfast here.  Crazy that they bring you iced tea when you order tea.”  It is shocking to expect hot tea and be served iced tea.  Michel speaks (and tweets) in English as well as Dutch.  I, on the other hand, speak (and tweet) in English only.  I can use the Live Search translator to read Michel’s messages.  He does not need a translator to read my tweets.

Our colleagues, the teachers of foreign languages, want their learners to be prepared when they visit another country.  We want our learners to understand the culture, the climate, the traditions and the customs as well as the language itself.  Is this true for me, a teacher of mathematics, the language of science? I dare to say that most, if not all of my colleagues teaching foreign language have been to a country where the language they instruct is spoken.  Have we, the teachers of mathematics?

Are we teaching a language when
we have never visited the lands where the language is spoken?

Has it been so long since we have visited these lands that
we have forgotten about the culture and the traditions that are important?

Photo by @fnoshese, 2010 (Cross River, NY)

What are the components of the culture, climate, traditions and customs? What are the conventions and must-knows for the lands our learners use our language to survive, function, and thrive?  What serves as the Live Search translator for our learners when they are immersed in one of these lands?  How can we, teacher-learners, develop opportunities for “foreign exchange programs” and visit these lands to experience the culture and practice our language?

“If we teach today as we taught yesterday,
we rob our children of tomorrow”
~ John Dewey

Many of my tribe gathered and learned together this weekend.  (If you have not read The Element: How Finding Your Passion Changes Everything by Sir Ken Robinson, run don’t walk to your preferred book supplier to read a copy.)  The 2011 T³ International Conference begins today in San Antonio.  As a regular practice, the 300+ T³ Instructors from all over the world gather on Thursday, the day before this conference, to learn in team, to work together for our professional develop.  It is an amazing experience.

It has spurred more thinking about technology and teachnology.  We say over and over again “It’s not about the technology; technology is a tool.  It is about the lesson; it is about LEARNING.”

Is it?

If I don’t understand the technology, will I ever get to the content, pedagogy, and learning?

Ruth Casey and I, along with 6 other members of the TI-Nspire CX Publish View team, spent our Thursday together teaching and learning about Publish View for the latest version of TI-Nspire.  It is SO COOL.  You can drop images, video, hyperlinks, interactive spreadsheets, interactive graphs and so much more onto a sheet to organize resources and opportunities for learning.  If you blog, you can think of it as a math-science dynamic glog (a graphic blog – see glogster.edu for another type of glog) where learners can interact with the science and the math.  The new Publish View allows documents to be embedded; now our learners will be able to interact with the lesson online.  How GREAT is that?

The Publish View team spent the day teaching the “how to” of using Publish View.  We used Stopping Distances as our foundation.   The physics, math, and application of the Stopping Distances lesson requires strong content and skill.  My T³ tribe knows the math or the science or both.  How many know both?  Who could know it all?  I want to know more about both the math and the physics.  I want to know more about the social impact and ramifications of distracted driving.  I want to know what I can do to help our communities experiences less of the tragic consequences of distracted driving.

I want to know more.

…and I am a teacher teaching other teachers.  I want to be known as a learner.  I want to lead learning, to be led to learning, to collaborate in learning.

The technology becomes teachnology for me.  I can now wonder about the physics successfully and safely.  I am motivated to find and interact with resources and to ask my favorite physics teacher to help me learn.  Don’t you think your favorite English teacher will be all over the topic of distracted driving as will your favorite language teachers, art teachers, technology teachers, etc.? Can we find common ground to integrate our disciplines for and with our learners?

The Publish View of TI-Nspire software became teachnology for me because it prompted my learning; it spurs me to new levels of learning, inquiry, and collaboration.  The teachnology offered me the opportunity to think creatively, to integrate ideas from multiple sources, to blend and blur content lines, and to need to work collaboratively with a team that spans multiple disciplines.

If we value problem-finding as much as problem-solving, then we must learn to utilize teachnology to prompt learners to ask questions, to investigate, to discover, and to collaborate.

In our session Thursday, we never got to the math, science, or social lesson
…or did we?

In many cases, it is about the technology.  The technology must become teachnology.  We need our learners to problem-find and problem-solve.  We aspire to facilitate and motivate self-directed learning.  We value creativity, collaboration, and critical-thinking. This compels us to understand that  is not about Mac vs. PC, iPad vs. tablet, iPhone vs. Blackberry, dry erase markers vs. colored chalk.  It is not about being 1:2, 1:1, 2:1.  (I’m well past being 1:1; aren’t you?)

It is about experience, inquiry, discovery; it is about learning – all of which can be accomplished without technology IF we are satisfied and willing to stay in a closed community of learners that we can see and talk with daily.  But, if we want to broaden your community of learning and support, don’t we need technology?  If we want to learn, serve, and lead with others, don’t we want and need to be more globally connected?

Learning something new is difficult.  It takes patience and perseverance. Shouldn’t we model this for our learners?

Now is the time.  The negative self-talk has got to stop.  We can do this.  Our expectations have to rise.  We must believe in the best of ourselves and others.  We should stop making excuses and learn.  We need technology; we need teachnology.  Stop saying “I can’t or they can’t.”  Let’s change our language to “Can you help me?  Can I help you?”  Isn’t that what we want our learners to do?  How many times do we coach our students – complain, actually – about that very phrase?  Stop saying “I can’t or we can’t.”  Change the language to “Can you help me? Will you help us?”  Stop saying what we can’t do.  Focus on what we can do.  We are learners!

It is about technology and teachnology.

It is about learning and Learning!

Helping Students Level Up

Formative assessment takes many forms. I generally put these forms into two categories: formal and informal.  Informal formative assessment happens all the time, planned and unplanned through questioning and observation.  As I float through the room and look at student work, I am assessing struggle and success.  As they work together to calibrate their work and communication, they formatively assess for struggle and success.

Formal formative assessment happens when my learners are challenged to scrimmage with the information we are learning, when they go one-on-one with assessment items.

We’ve been struggling with the traditional descriptors for the Guskey-style 4-point rubric.

Level 1: Beginning
Level 2: Progressing
Level 3: Proficient
Level 4: Exceptional

How do you explain to a 13-year old that they are progressing rather than beginning? How do you explain to a parent that their child is proficient but not exceptional?  Do you have time to sit down one-on-one with every child and counsel them with the level of feedback to help them improve?  Feedback is powerful and necessary for growth.  We have 15 essential learnings with multiple learning targets for the year.  How can we develop an assessment system that helps our learners self-assess and calibrate their understanding?  How many times does a child come back after school and say “I don’t get it.”  They can’t ask a question.  They don’t know what to ask.

We have been saying…

Level 1 is what was learned as 6th graders.
Level 2 is what was learned as 7th graders.
Level 3 is the target; it is where we want you to be as 8th graders.
Level 4 is the challenge for those ready for more.

While not totally accurate, it has helped our young learner understand and gauge how much work needs to be done.  These descriptions worked well as long as we were learning about linear functions.  These descriptions failed me this week.  My descriptions failed us this week.  Modeling learning, we try again.  Here’s the new attempt.

Level 1:  I’m getting my feet wet.
Level 2:  I’m comfortable with support.
Level 3:  I’m confident with the process.
Level 4:  I’m ready for the deep end.

The success we’ve had offers our students the opportunity to level their understanding of each learning target in the progression of an essential learning.

We started our linear functions unit using asking our learners to identify their understanding of each learning target using the Graphing Linear Functions Rubric shown below.

We then gave them a diagnostic assessment to help them calibrate what they thought with what they could produce.  (It was very interesting, and the process prompted many discussions about what we think we can do versus what we can do.)  They immediately asked to complete the Graphing Linear Functions Rubric again.  They asked to chart their own progress!  After each formal formative assessment, students returned to the Graphing Linear Functions Rubric to chart their progress and to seek intervention or enrichment.

As we progressed through the unit, we used leveled formative assessments to continue to self-assess and calibrate.  The components of these formal formative assessments include

• Assessment questions.  Questions are leveled using the language of the essential learnings.
• Answer key (answers only).  Students self-check and then correct in teams.
• Table of Specification.  Students calibrate their work level with the expected level.
• Solutions.  Students can use our work to improve their communication and understanding.
• Differentiated Homework.  Students are assigned (or choose) work at an appropriate level, working to level up.

The table of specifications helps our learners self-assess and calibrate their learning and understanding as we are working through the targets and skills.

The change in response from our students is remarkable.  The improvement in our communication is incredible.  Students now come in after school, sit down with me, and say “Ms. Gough, I can write the equation of a line if you give me a slope and a point, but I’m having trouble when you give me two points. Can you help me?”  Look at the language!  We are developing a common language.  Our learners can articulate what they need.  Regularly in class a child will ask “Is this level 3?”  They are trying to calibrate our expectations.

We are now able to differentiate and intervene for and with our learners.  I have always struggled with what to do for my fastest learners.  I need them and their peers need them to coach and work collaboratively; they need to learn more.  Finding the right way to balance these needs has been a struggle until now.  My favorite story about enrichment happened last week.  MR – very quiet, hardly speaks in class – literally skipped down the hall talking to me from 2 doors down.  “I left class yesterday confused about level 4, but I used your work from the webpage last night and now I’ve got it!”

Self-assessment, self-directed learning, appropriate level of work that is challenging with support, and the opportunity to try again if you struggle are all reasons to offer students formative assessment with levels.  Making the learning clear, communicating expectations, and charting a path for success are all reasons to try this method.

Sending the message “you can do it; we can help” says you are important.  You, not the class.  You.  You can do it; we can help.

In addition to reading the research of Tom Guskey, Doug Reeves, Rick Stiggins, Jan Chappius, Bob Marzano and many others, we’ve been watching and learning from TED talks.  My favorite for thinking about leveling formative assessments is Tom Chatfield: 7 ways games reward the brain.

If you are interested in seeing more formative assessments, you can find them sprinkled throughout our assignments on our webpages.

The TI-Nspire files shared during my T³ International Conference in San Antonio are linked below:

I’d love to know what you think; do you have suggestions or advice?

Stopping Distances

… written in collaboration with Ruth Casey and Sam Gough.

Distracted driving is any non-driving activity a person engages in that has the potential to distract him or her from the primary task of driving and increase the risk of crashing.
~
From D!straction.gov
The Official US Government Website for Distracted Driving

A typical rule for the distance you should follow behind a car is given by the “three second rule.” To determine the right following distance, select a fixed object (a tree, a sign, an overpass ..) on the road ahead. When the vehicle ahead of you passes the object, begin counting “one one thousand, two one thousand, three one thousand.” If you reach the object before you complete the counting, you’re following too closely.

• When you see an object in your path, can you stop your car instantly?
• What happens between the time you realize that something is in your path and when the car actually stops?
• How much distance has been covered before the car has stopped?
• How does your reaction time affect these distances?

As an introduction, watch Vehicle Stopping Distance from teacher’sdomain.org or Think! – Slow Down which is embedded below.  (Warning…it is tough to watch.  A dummy is used, but you should preview before you show it to students.  I like it because you can see the screeching tires and the struggle to stop.)

If you are interested in the physics, check out the Vehicle Stopping Distance Calculator from Computer Support Group and their online division, csgnetwork.com.

For an experiment of calculating your reaction time, do the math.

Let’s look at the data.

Suppose you want to visualize the pattern in the distance traveled while reacting versus the speed of your car.  Do I travel the same distance while I’m reacting no matter the speed or does the speed influence the distance traveled just while reacting?

• What does this pattern tell us about reaction distance traveled vs. speed?
• Can you find the mathematical model for these data?
• What is the slope? What is the meaning of the slope?
• Is this direct variation?
• Which of our learners can find success with this?

How about the pattern in the distance traveled while braking versus the speed of your car?

• What does this pattern tell us about braking distance traveled vs. speed?
• Can you find the mathematical model for these data?
• What is happening with the slope?
• Which of our learners can find success with this?

Now, how about the pattern or relationship between the total distance traveled while stopping the vehicle vs. the speed?

• What does this pattern tell you about the total braking distance vs. speed?
• Can you find the mathematical model for these data?
• What is happening with the slope?
• Which of our learners can be successful with this?

I don’t want to give away the mathematical models; I want you to have time to consider and think about the mathematical models.  If you need or want a hint, please leave a comment below and I’ll write you back.

Phases of the Moon…Middle School Connections with Trigonometry and Science

My learners struggle to read and interpret graphs for meaning.  It makes me wonder…How are we teaching them to read and interpret graphs? When our learners get to precalculus, are they adept at reading graphs for meaning so that they can concentrate on mathematical modeling?  Wouldn’t it be advantageous for a new-to-precalculus or new-to-physics learner to already have a context with which to identify when presented with periodic data?

What if we integrated the ideas of plotting points and interpreting graphs with some earth science?  We are not going to have middle school students model this data, but we are going to have them interpret the data and label the graph.  We are going to expect them to connect the math and the science.  I’m pretty sure that there’s a great connection to periodic poetry too.

I believe that somewhere in 6th or 7th grade science we teach our learners about the phases of the moon.  The geometry is awsome.  (Take the Lunar Cycle Challenge.)  In pre-algebra and algebra, we work on plotting points on the Cartesian coordinate plane.  What if we practiced plotting points and pattern-finding by plotting real data?

In terms of diagonstic assessment, ask:

1. Can you name the 8 traditionally recognized phases of the moon?
Then wait…6th and 7th graders know this, which means that older learners will need a little time for recall.  Wait time is critical.  Full moon and blue moon almost always occur.  Generally the vocabulary will come back to any group of learners.  This is a great place to integrate teachnology.  Let them find the answers.
2. Is there an order to these phases?  Are there any patterns?
Ask your learners to sketch a graph of what they have described.  You might want to check their understanding of the vocabulary and the images.
3. Are the phases identifiable when not in order?
To reinforce the geometry, you can use the Lunar Phase Quizzer.

1. Can we find data for the percent of the moon showing every day?
2. If we plot this data, will there be a pattern?
3. How much data should we plot to see the pattern?

Let’s look at the data for January and February 2011 from the United States Naval Observatory.

Questions to ask to check for numeracy, geometry, and understanding of the vocabulary:

1. Is the moon waxing or waning on January 1, 2011?  How do you know?
2. Is the moon crescent or gibbous on January 1, 2011?  How do you know?
3. Sketch the moon’s illumination on January 1, 2011.  Check using Today’s Moon Phase.

Don’t you think that these are great TI-Navigator formative assessment questions?  You can repeat those three questions about any date in January and/or February until you have consensus.

Let’s discover if there is a visual pattern in this data.

This graph always gets a big WOW! if you are using TI-Nspire and can see the points animate into place.  Questions to ask to check for graphical interpretation and connection between the graph and the earth science:

• When was the full moon in January of 2011?  How do you know?
• When was the new moon in January of 2011?  How do you know?
• When was the first quarter moon in January of 2011? … the third quarter moon? How do you know?

Again, you can check using Today’s Moon Phase.

Deeper questions connecting to writing and interpreting inequalities with connections to more vocabulary:

• Name one day in January of 2011 that the moon was waxing?… waning? … crescent? … gibbous?  How do you know?
• Over what days in January of 2011 was the moon a waxing crescent moon? … waning gibbous?  etc.

Pretty good stuff if your students are struggling with writing inequalities, particularly compound inequalities.

Now for patterning…

• Will February’s data look like January’s data?
• How are these data similar?  How are these data different?
• What would the graph look like if we graphed the percent of the moon showing vs. the number of days since December 21, 2010?  (In other words, February 1 would be day 32.)  Would the pattern continue?

How cool is that?

• Can you identify all of the above questions for February?
• Can you predict the day of the full moon for March?

Please use Today’s Moon Phase to check!

Here are the burning questions for me…

• Can middle school students plot these points?
• Will they find connections between the math and the science?
• Will these connections help them understand how to interpret graphs?
• Will these connections help them understand the moon and its phases?

If  our students would start off in trigonometry and physics understanding the connections between the math and science and could interpret what they see, would they be more likely to find success modeling the data?

If you teach trigonometry or physics, there is a clear path from the graphical interpretation to finding a function that models this data.

TI-Nspire Resource Files
• Phases of the Moon Diagnostic Assessment
• Phases of the Moon PublishView document
• Phases of the Moon Data .tns file

From my webpage….

During every month, the moon seems to “change” its shape and size from a slim crescent to a full circle. When the moon is almost on a line between the earth and sun, its dark side is turned toward the earthThe moon’s cycle is a continuous process, there are eight distinct, traditionally recognized stages, called phases, which are ordinarily adequate to designate both the degree to which the Moon is illuminated and the geometric appearance of the illuminated part, to the extent that Moon visibility has relevance to everyday human activities.

In this activity, we will investigate the fraction of the moon seen each day for a month and then for a year.

1. Identify the eight traditionally recognized stages of the moon’s cycle.
2. Find the approximate period of the moon’s cycle.
3. Extract the fraction of the moon showing from the United States Naval Observatory, for the days of the year.
4. Set up a scatter plot of the fraction of the Moon showing in January versus the day of the year.
5. From the data or the graph, determine the amplitude and the vertical translation.
6. Find the cosine function that fits this data.  Identify the phase shift for this function.
7. Find the sine function that fits this data.  Identify the phase shift for this function.
8. Edit the data to graph the fraction of the moon showing in January and February versus the day of the year.
9. How well do your functions fit this extended data?
10. Determine which day of the year corresponds to today’s date.  Predict the phase of tonight’s moon.
11. Check the accuracy of your prediction using Today’s Moon Phase.
12. Take the Lunar Cycle Challenge.

Being Part of the Club…Being Labeled…

﻿

Have you ever been excluded from something?
How did/does it make you feel?

Do you carry a label?
Was it of your own choosing, or were you labeled by others?

I teach children, bright, smart, sometimes discouraged children about algebra.  They are labeled…they label themselves.  The course of study I’m responsible for is called Algebra I.  Algebra I, not Regular Algebra I, not Intro Algebra I, we study Algebra I.  We label them as Algebra I students rather than Algebra I Honors students.  They call it Dumb Math.

There are few things sadder to a teacher or parent than being faced with capable children who, as a result of previous demoralizing experiences, or even self-imposed mind-sets, have come to believe that they cannot learn when all objective indicators show that they can. Often, much time and patience are required to break the mental habits of perceived incompetence that have come to imprison young minds.
~ Frank Pajares,
Schooling in America: Myths, Mixed Messages, and Good Intentions

They call it Dumb Math.  They perceive that they are incompetent.  It is sad to me.  They are bright young learners.  We learn a full course of Algebra I.  We study linear, quadratic, and exponential functions.  We solve all types of equations including equations involving right triangle trigonometry.    We analyze graphs and identify intercepts and other critical points.  We use technology to learn. If you would like to see the details, they can be found here: and Second Semester Essential Learnings.  For those of who speak algebra, it is a good, solid algebra course.

I struggle during this time every school year.  It is part of my job to label my students, and this label could be carried with them throughout their high school career.

A couple of disclaimers…

1.       I do not have the first honors course on my transcript anywhere, and that is okay.

2.      My biggest goal for each child every year is to help them see their talent and build their self-efficacy.

I struggle when I receive this email.

Dear Faculty, Your Honor and AP Recommendation forms are due to the office no later than March 4, 2011. Thanks for your help with this important process.

It will arrive twice, once for the handful of 9th graders that are in Algebra I, and then again for my 8th graders.

Now, even I don’t think we should mix our weakest and our strongest math students together.  It can be just absolutely demoralizing for both sets of children, particularly when we do not know how to differentiate our lessons any more than we are already doing.  Everyone seems comfortable differentiating lessons and homework, but are we ready to differentiate assessments?

I’m wondering if we have to have the label.  We already differentiate by tracking our students.  There is no honors label in 8th English.  Part of me knows that we have to have the label.  They cover more algebra topics in Algebra I Honors.  We do not learn about fractional exponents, completing the square, and other topics.  Maybe it is the label “Honors” that I am discouraged about.  Would there be the same press or desire to enroll if we called it Advanced Algebra I or Accelerated Algebra I or Algebra 1.5?

There seems to be this urban legend that our students won’t get into college if they are not in Honors Math.  Have we ever had a student not accepted into college?  I don’t think so.

Our process this year for making these recommendations is much more data driven.  It is not about having a 90 average.  It is about how much you have learned and the depth of your learning.  Our formative assessments provide the data path about accelerating learning to the “honors” level.

We hope that our learner serious about moving to the honors track will provide evidence of their ability to meet the additional challenges and responsibilities of the more rigorous course.  Of course, this evidence is still in the form of assessments, but at least we are making a move to differentiate.  We have more to learn, and that is okay.