Tag Archives: TI-Nspire Navigator

Swing Even If You Miss; Stepping Up To The Plate Is Not Enough

Is a swing and a miss better than no swing at all?  I used to coach my learners to step up to the plate, just step up.  You can do it.  But now I realize that just stepping up is not what it takes to learn.  It is a great first step, but it is just a first step.  Will you ever hit it out of the park if you don’t swing?

Today I got to co-teach Algebra I with D. Dietrich with her learners.  It was day two of Stopping Distances.  After reviewing what we accomplished during the last class, we stopped to ask them to write a complete sentence about what the graph of reaction distance vs. velocity represented. 

When I asked for volunteers, no one responded.  Yikes!  Now, they all had written something.  Finally, one sweet, brave learner volunteered.  It was a sentence that needed some coaching.  We talked about the structure of the sentence as well as the need to be more specific about units.  Since the distance is in feet, what are the units of velocity?

We collected this sentence with our TI-Nspire Navigator system.  The screenshot above allowed the entire class to have multiple representations of the sentence.  They could see it as it was being read.  The screenshot is evidence of a swing.

I asked for another volunteer…  …  …  …  So, I finally asked the one child that was making eye-contact with me why he didn’t want to read.  “Because I know it is not perfect.” Isn’t this what learning should be about?  Can’t we have a culture where it is okay to not be perfect?  How will I ever learn if I am not brave enough to share my thoughts with others?  Being in school should be about drafting and making revisions, shouldn’t it?  Have we built a culture where we are afraid to try because the only correction I get is criticism?  (double-Yikes!!)

We edited and coached and then tried again for the second graph, braking distance traveled vs. velocity.

We got, without question, the best sentences of the day for the second writing. The power of feedback and revision was demonstrated.  Unknown to me, the learner that volunteered for the second writing rarely speaks in class and never volunteers.  Isn’t that great?  She found the confidence to write and share.

But, as the facilitator of the learning, do I know how many of these learners actually took a swing?  Not really.  I love that one student wrote his sentence on the screen of his calculator.  But it is the only one I could monitor.

Which makes me wonder…how many times are my learners telling me they understand when they have not actually taken a swing?  Are they so conditioned to tell us that they understand that they are not testing their understanding and seeking feedback?  WOW!  That is a problem!

We stopped and talked about trust.  We talked about risking being wrong to grow and learn.  But, again, were they just agreeing with us because they know it is what we want to hear?  Would they take action…to swing and possibly miss?  Would we celebrate the swings and give quality, safe feedback for a miss?  Risky for everyone, right?  Trust is the key.

Yesterday we wrote the equations for both of the graphs above.  y = 1.1x and y = 0.05x², where x is velocity in miles per hour and y is the distance traveled in feet.

New learning for today…How do we find the model, the equation for the data, of total distance traveled in feet based on velocity in miles per hour.  Important questions must be asked.

  1. How will we calcuate the total distance traveled based on the given data?
  2. What pattern does this data follow?
  3. Is the pattern exponential or quadratic?Okay, so let me stop here and say that the question exponential or quadratic is an important question.  If you can’t answer it, how will you know what type of model to write?  This is Algebra I; we are not using regressions.  We are hand fitting data.Aren’t the results interesting?  Q for quadratic, E for exponential.

Before you boo, a bunch of learners came without their technology or had login issues.

But notice, even when anonymous, one learner did not take a swing. It was a 50-50 shot at being right and this learner would not take a swing.

New and MUCH more interesting questions are now possible.

  1. Is the majority always right?
  2. Can we listen to an opposing view and try to understand their reasoning?
    (Shouldn’t Democrats and Republicans try this occasionally?)
  3. Can each side make a reasonable argument for why they made their choice?
  4. Are you willing to consider that the other side might be right?

The minority view, quadratic functions, explained first.  Then a member of the majority party raised her hand and said “I voted exponential, but I can now give another reason why it is, in fact, quadratic.  Is that okay?”  WOW!  We stopped and voted again.

Progress!  But, what do we do about the 2 remaining e’s?  Remember, the majority is not always right.  More discussion ensued.  The two e’s identified themselves, AB and SS, and explained why they were now in the quadratic function camp.  Also notice, they everyone took a swing this time around.

Let me stop here and say that this is where and why I prefer using my TI-Nspire Navigator over Poll Everywhere and the SMARTBoard clickers.  While it appears to be anonymous, I can click on Poll Details and see which learners need intervention if they don’t self-identify during the lesson.  I can also see what is on every learner’s graph which I will show if you keep reading.

Now that we established that we are pretty sure the function to write is quadratic, we again returned to the data.

How was the data in the total column computed?  “Everyone knows, Ms. Gough, that you just add the reaction distance to the braking distance, duh!” Eyes rolling as only 14 year-olds can do.

So, if that is true, write the equation that fits the total distance traveled in feet based on the velocity in miles per hour.  You have all that you need.  You can do it.  Take a swing.

No one got it right the first time. Some took a swing and missed; some got close; some just stood at the plate.  At least two students were still in the dugout – maybe the parking lot; Caswell and St. Cloud were absent the day before.  They did not ask for the file.  St. Cloud was working to catch up; Caswell was just observing.  And we absolutely would not have known if we didn’t see their calculator screens.

Vuckovic looks like he took a swing, but no.  He connected his scatter plot; he had not attempted to construct a function.  Gibson has tested a function that is close, but does not fit very well; he took a great swing which we celebrated.

So we asked again, how was the data in the total column calculated?  Can you use this information to write an equation that will fit the data?

Before class was over, everyone took a swing and got a hit.  It was very collaborative.  We then took the model and began to interpolate and extrapolate.  If I was in a car going 65 miles per hour, how much distance would I need to not crash?  Easy, but we needed to talk about how to document our work and thinking so that future physics teacher would know how we arrived at our answer WITH units.

If I am a stunt car driver and know that I have a football field’s length to stop my car, what should my top speed be when I hit the brakes?  This question was not so easy.  Learners fell in to three camps.

  • Camp 1:  I substitute 100 in for y.  They got feedback quickly from their peers that 100 was in yards and the units were feet!
  • Camp 2:  I substitute 300 in for x because that is where I substituted 65 before.  They also got quick peer feedback that x was in mile per hour and to pay attention to your units!!
  • Camp 3:  I substitute 300 in for y, but what do I do now?  Unfortunately, in this class, they got no peer feedback.  We facilitators were informed to say the least!  Not one child connected this to using the quadratic formula to solve.  (triple YIKES!) Good formative assessment that they are not connecting the skill to the application.

Once they realized that they could use the quadratic formula to solve, they were successful.  We could monitor the swings, and lack of swings, using our Navigator.  By now, everyone was swinging because they knew we were supporting their effort and learning.

The big take-away for me:  Formative assessment that offers feedback and support in the moments of learning are critical for success and confidence.

The big take-away for my learners:  You can do this; we will help!

You will not hit it out of the ballpark if you do not swing.  A swing and a miss is so much better than no swing.  Step up to the plate; dare to swing.  Miss; swing again.  You can do this; we will help!

Turnpikes, Toll Roads, Express Lanes

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?

Which leads to more questions:

  • 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?

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?