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.

 For the lesson, ask

  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: First Semester Essential Learnings 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. 

 

 

Students Request Common Formative Assessment

I remember making a note to self about coming to office hours that afternoon. After thinking for a few minutes about my question, I decided to ask here in class instead of waiting until office hours. Hearing the “oohhs!” of my classmates, I was relieved to hear some of my peers had the same question.
~
MC (from the 2/8/11 journal)

The opportunities to reflect and ask questions are incredibly important.  How often do our learners leave class without asking their question?  For whatever reason, questions go unasked and unanswered. From our friend, Grant Lichtman, we embrace the idea from The Art of Questioning chapter in The Falconer: What we wished we had learned in school

 “Questions are waypoints on the path of wisdom.”

Have we ever stopped to ask “What do you need or what would you like to learn or work on tomorrow?”  If we want more student-directed learning, wouldn’t a great first step be to involve them in the planning of tomorrow’s lessons?

Yesterday, I sent out a #20minwms tweet at the 2o minute mark in class, but I also used exit cards as an additional opportunity for my learners to help inform our next steps in their learning.  I chose 4 to tweet as evidence of learning from class.  These learners from Algebra I said:

If you read my previous post, you will share in my excitement for GW.  I was very pleased with the notes and ideas on all of their exit card.  Then on my way to dinner, one of my Synergy 8 learners asked a great question.

I explained my version of the day’s exit card and the following conversation happened.  To be clear, @TaraWestminster and @fencersz are two of our Synergy 8 learners while @bcgymdad and @danelled111 are my teammates.

  

While I’ve been considering how to promote more student-directed learning for quite some time, I have to admit that I have never considered student-directed common formative assessment.  Isn’t it great that these two bright young learners feel they have a voice in their learning and assessment?  Won’t you join me in applauding their advocacy for themselves and other learners?  Doesn’t this show that we are a community of learners?  We learn from each other; we focus on learning and questioning. 

It is about lifelong learning!   

_________________________________________

Lichtman, Grant, and Sunzi. The Falconer: What We Wish We Had Learned in School.
     New York: IUniverse, 2008. 35. Print. 

Enrichment…Intervention…Benefits of the 4-Point Rubric

Our team has been immersed in creating formative assessments, assessments for learning.  We have been developing these assessments based on the work of Tom Guskey, Rick Stiggins, Jan Chappius, Doug Reeves, Bob Marzano and many more.

We have tried to convert

Level 1: beginning,
Level 2: progressing,
Level 3: proficient or
Level 4: exceptional

to kid-friendly, kid-understandable language.  We have been saying…Level 3 is the target; it is where we want you to be as Algebra I learners.  Think of Level 1 as what should have learned as 6th graders and Level 2 as what was learned as 7th graders.  Level 4 is a blend of Algebra II and Algebra I Honors.  Level 4 is the stuff that you will see later in your math career; it 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 progression of the images and ideas speak to me and to the 2 teachers and 4 students that worked with me on this after school.  We start off seeing the ocean, but we are only willing to get our feet wet.  We are getting our feet wet.  In the kiddie pool we can experiment with getting soaked.  We are comfortable in the water but need and want lots of support.  In the deep end of the big pool we can swim confidently.  Back in the ocean we can maneuver without as much support.  Lifelong learning and teamwork tell us that there will always be more to explore, and we will always need to be careful in the deep end.  We won’t abandon all of our support and safety.

There are multiple ideas and benefits to these formative assessments.

  • Our learners have a much clearer way to gauge their success on meeting the standard for each essential learning.  They self-assess their level with these formative assessments; they have immediate feedback on what they should know and where they are in the process.
  • Questions are much clearer; we now communicate using a common language.  No longer to we field “I don’t get it.”  I cannot say this with enough emphasis.  We NEVER hear “I don’t get it.  I ‘m lost.”  We are asked “I am at level 2, will you help me get to level 3?”  “I have learned that I’m at level 3; how do I get to level 4?”  Even better, QB dropped by after school and said “Ms. Gough, I understand the distributive property, but I’m still having trouble when I multiply two binomials, can you help me?”  And the follow-up question was “Okay, now tell me does this show good work?  Am I communicating my ideas?  And are my conventions good too?  How is my organization?”  WOW!
  • Learners are motivated to level up.  Differentiation is not only possible it is motivating.   MR – very quiet, hardly every speaks unless called on – started talking to me in the hall when she was 2 classrooms away from me.  “Ms. Gough, I got the level 4 problems last night!  I had to use your work on the webpage, but I now understand and can do it myself!”  ER said “Me too.  I’m now at level 3 because I could work with your work.”  Both learners feel success even though they are not working at the same level.
  • Homework is differentiated based on level.  Students now have some choice in their homework.  We post our homework in levels; you can see it on the table of specifications in the document below or on our webpage.  One of my teammates, @bcgymdad, says it best.  He asks his learner – we all do now – to try the first three problems from the next level.  “Say you are at level 2; start with the level 3 homework.  If you struggle too much with the first three problems, then drop back to level 2 and do that homework.  But try, try to level up.  Challenge yourself; you can do it.  We will help you.”
  • Intervention and enrichment are now easier and often self-directed.  Our learners how have choice in their learning and direction for how to improve.  We are very clear.  Everyone must get to level 3 in order to be proficient in our course.  We are offering enrichment and intervention at the same time.  When we individualize face-to-face one-on-one instruction, we answer the learner’s questions.  They tell us where they are and where they want to go.  With intervention, we sometimes have to direct their questions, but at least we are doing that on an individual basis rather than in whole group discussion.

Here’s the specific formative assessment my learners are discussing.

The benefits to the teacher, the lead facilitator of learning, seem huge.  The benefits to the learners seem huge too.  I’d love to know what you think.  We, my team, would love to know what you think.

As @thadpersons asks:  What speaks to you about this?  What do you want to know more about?

Being Slow…Mindset…2nd Chances…Learning

On the drive to school yesterday morning AS (age 6) explained to me that she was the slowest in her class.  It was very matter of fact.  “I am the slowest in class, Momma.  I finish last every time. It takes me longer than everyone else.”  With a very heavy heart, I explained that I thought it was okay to be last.  Learning is not a race.  Everyone learns in their own time.  AS persisted “But, Momma, I am always last.  I am slow.”  I asked her why; why did she think she was slow?  “I like to read what I write.  It is fun, but it takes a long time. I like to look at it to figure out the words.”

Rule Three from The Talent Code by Daniel Coyle is SLOW IT DOWN. 

“Why does slowing down work so well? The myelin model offers two reasons.  First, going slow allows you to attend more closely to errors, creating a higher degree of precision with each firing – and when it comes to growing myelin, precision is everything.  As football coach Tom Martinez likes to say ‘It’s not how fast you can do it. It’s how slowly you can do it correctly.’ Second, going slow helps the practitioner to develop something even more important: a working perception of the skill’s internal blueprint – the shape and rhythm of the interlocking skill circuits.”  (p. 85)

 We still take a lot of heat from our colleagues about 2nd chance tests.  It makes many people, teachers and parents, uncomfortable. 

About our version of 2nd chance tests: 

  • Our learners take the test; we mark (not grade) each problem as correct or incorrect, and return the paper to the child without a number-no grade yet. 
  • Their job is to find, correct, and identify errors.  We ask them to categorize an error as either a “simple mistake” or “needs more study”. 
  • We also ask them to complete a table of specification and determine their proficiency on the assessed essential learnings. 
  • After all problems are corrected, students write a reflection about their work. 
  • Armed with the experiences of teamwork, feedback, and self-assessment, students are given a 2nd Chance test and are tested on only the problems missed during the first testing experience. 
  • The final test grade combines the correct work from the first test with the work from the 2nd Chance test. 
  • Yes, it is completely possible to bomb the first test and end up with a 100 in my grade book.  

My assumption is that this discomfort comes from how non-traditional – radical – this concept comes across.  Just because it is different does not make it a bad idea, does it?  The discomfort comes from gut-reaction or theory rather than practice.  Shouldn’t you try it?  What do learners say?

Here’s what some of my learners say.

“If you give your best effort the first time around, you will have learned more in the process and the second time around will be less stressful therefore making the hard work the first time more rewarding. I think that the second chance test is a very valuable learning technique. Even after that unit is complete, it shows you where you need to improve before you start building on those concepts. So far this year, I have seen great improvement in my learning from my previous years in math. This year it has all started clicking, and I am excited about the new units to come.”
~CM

“Before we jump into a new chapter, our class usually takes a formative assessment to tell us where we are and what we know before we actually start learning from Mrs. Gough. I take these seriously because I think they really do help. If I can see where I am in the beginning and then where I am in the end, I can see how much I’ve learned and accomplished.”
~ MC

In Mindset, Dr. Carol Dweck writes

“When people believe their basic qualities can be developed, failures may still hurt, but failures don’t define them.   And if abilities can be expanded – if change and growth are possible – then there are still many paths to success.” (p. 39)

More from my learners:

“Taking formative assessments and tests is something that I think is very important. I give my best effort, and work to learn from my mistakes. The second chance test is something that I think helps us actually learn from taking tests and making mistakes, rather than just getting tested on the material. Math has become one of my favorite subjects this year, and I have worked to learn from all my mistakes.
~VB

“I think that first chance tests and formative assessments are amazing because I can first understand my level and see what to work on and then really learn the material on the test to do better on the second chance. I do well in groups (except for the occasional random moments), and I love working in groups instead of taking notes the whole time. By helping others, it also helps me understand what I am doing wrong or just what I am supposed to do.”
~ HA

I feel the same as Daniel Coyle in the epilogue of The Talent Code when he writes

“Mostly though, I feel it in a changed attitude toward failure, which doesn’t feel like a setback or the writing on the wall anymore, but like a path forward.”

One more quote from our learners

“Overall, I feel as though I have done a pretty good job so far, but there is no one who can stop me from really stepping it up to an unbelievable level. The rest of the year I am going to fix any flaws I have, and show everyone what I can do when I REALLY put my mind to something.”
~ LM

In case this has been too broad for you, let’s go deep.  Here is one learner’s story from three perspectives.

From my perspective…

“GW came to me feeling that she is not very good at math and that she hasn’t been encouraged to like math.  She seeks an advocate and coach.  I strive to support GW as she becomes empowered to take control of her learning.  She is learning that it is great to struggle to learn; it is worth it to struggle to learn; and through the struggle she finds success.  Success leads to more confidence and more success.”   

From GW’s perspective…

“When I started out in math I had a really hard time and math was a definite challenge for me and my first test grade didn’t make it any easier. I was “in a hole” as my parents would tell me and I had to dig myself out. I started to go to extra help a lot more often and made solid B’s on my midterm and exam grades. What helped me through this process was the support. Support from not only my family but from Mrs. Gough and the faculty that really encouraged me to do my best.”

 From GW’s parents’ perspective…

“GW quietly got way behind in math first semester.  Partly due to an inner voice telling her she did not do well in math and partly a lack of commitment and time management. GW had given up.  Mrs. Gough communicated to us that GW needed to demonstrate the deep practice method on all homework. With our support and encouragement (not hands on help) GW began to do the deep practice on homework and began to “review and preview” every night. Our emphasis was ‘the process’ not the letter grade.

Her great success is directly attributed to the teacher/student relationship that Jill forged. Through encouragement (emails), support (office hours), an emphasis on deep practice and patience, Jill taught GW to try and try again, make the mistake, work through it, and get to the answer. Through perseverance, determination and resilience GW moved from failure and “not being good at math” to more than just passing. For us the 80 on her final exam was an A+ in effort, team work, student/teacher relationship, and determination.”

There are many take-a-ways for me…

If I can see where I am in the beginning and then where I am in the end,
I can see how much I’ve learned and accomplished

 It’s not how fast you can do it. It’s how slowly you can do it correctly.

I have worked to learn from all my mistakes.

There are still many paths to success.

This year it has all started clicking.

I am excited about the new units to come.

There is no one who can stop me from really stepping it up to an unbelievable level.

Try and try again, make the mistake, work through it, and get to the answer. 

So here’s to being slow, making mistakes, and trying again.  It’s about learning content and skills.  It’s about learning persistance and determination.  It’s about learning.  Period!

Time is a variable.

Learning is the constant.

_________________________________

Coyle, Daniel. The Talent Code: Greatness Isn’t Born : It’s Grown, Here’s How. New York: Bantam, 2009. 217.  Print.

Dweck, Carol S. Mindset: the New Psychology of Success. New York: Random House, 2006. 39. Print.

Connect Abstract to Practical…Meaning to Mechanics

Square-foot gardening, heard of it?

 

Hold that thought for just a minute…We’ll come back to it I promise. 

I am conflicted about what I am teaching.  I want to say my team is conflicted about what we are teaching, but I think it is really me.  I worry that our teams are on pedagogy auto-pilot; we teach what we’ve been teaching the same way we’ve been teaching it, which would be fine IF they were learning it, but they are not – not everyone.  It is not relevant, and they know we are going to teach it to them again next year.  What is the point in learning it now?  (Totally over-simplified but it makes my point.  Forgive me, please.)

We don’t know how to find balance between by-hand skill and skill with technology.  It is such a hard question and so risky that many choose not to consider it.  We think “I learned it this way; this is the safe way; I understand this.  Since I don’t know what is right concerning technology, I choose to ‘do no harm’.”

I’m supposed to teach a unit on operations with polynomials.  I am searching for context.  Why do we need to learn to add (subtract, multiply, divide) polynomials?  Who cares?  Where is the application of this skill? 

Is the entire history of what can be taught in algebra based on what was possible to learn before technology teachnology?  I can and need to factor polynomials if I want to graph them.  Isn’t that the number one reason to teach factoring?  BUT, don’t I now have a calculator that will do that for me? 

Not only do I have a calculator that will graph any polynomial, I could have a calculator that will do the algebra for me too.  Interesting…Is it important to factor big-hairy polynomials or is it important to know what the factors tell me about the polynomial…to make a connection from the algebraic to the graphical to the numeric? 

Isn’t it a forest and tree thing?  If I can’t factor, I can’t see why I need to factor.  If I don’t see why I need to factor, why do I need to learn to factor? 

My current conflict is about adding polynomials.  If I don’t see why I need to add polynomials, why do I need to learn to add polynomials?  I want my learners to know how and why to add polynomials, but how much is enough by-hand?  How much time should be devoted to the mechanics of adding polynomials?  In the face of technology teachnology, shouldn’t we focus on meaning rather than mechanics?

For example, a great learner question is about adding like terms.  Why can’t you add x and x^2?  Why aren’t they like terms?  Aren’t they both x ’s?  If learners don’t understand the why will the ever care about the how? Do we stop to think about why you can’t add x and x^2?  Can we give an understandable explanation? 

I can use my calculator to show that you can add x ’s together; you can add x^2 ’s together, and you can’t add x ’s to x^2 ’s.  Boom.  Done.  But, this technology does not tell us why.  Why can’t you?  What reason – meaning – prevents these variables from being “like” terms?

 

This brings us back to square-foot gardening…Doesn’t this picture illustrate why x can’t be added to x^2?  Think about it…

 

Traditionally, each box in the picture is 4’x4’ and is subdivided into 1’ sections.  You plant in each section.  How many x ’s can you see?  How many x^2 ’s can you find?  Can you see why x can’t be added to x^2?  Can we use this image to give meaning to x and to x^2?  Aren’t units (shout out to all science teachers) critical to this understanding?

If we want art and design, meaning and service, reason and understanding, integrated studies and PBL, and so much more, don’t we need to connect the abstract to the practical?

Why would anyone garden this way?  Is square-foot gardening more or less efficient than gardening in rows?  Can you grow more or less in squares?  How can you efficiently irrigate this type of garden and recycle at the same time?  Would a 4’x4’ garden improve your heath, your lifestyle, feed the hungry?

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Mel Bartholomew – Introducing Square Foot Gardening

Frequently Asked Square Foot Gardening Questions
 

SpinPost – PBL

Seeking brightspots and dollups of feedback about learning and growth.

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