# Student Reflections on Sequences

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To end class today, students were asked to compare/contrast Arithmetic and Geometric Sequences.

Leave it to me not to snap a pic of some of their thoughts.  Many you would expect…a starting value, n-1, repeated operations, etc.

One student made the statement that “they both involved positive operations.”

Me:  Hmm?  What do you mean by that?

S: “I see addition and multiplication as positive operations.  Even when we were subtracting, I defined it as adding a negative value.  When I was dividing, I defined it as multiplication by a decimal or fraction factor.”

Hmm.  That thought has kept coming back all through the day…not sure why I keep pondering on it.

# A Reflection on My First Weeks #MTBoSblaugust

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Basically, the beginning of the year goes something like this –

• Week 1:  Team Building / Communication (the beach towel, handcuffs) & Four 4s to review Order of Operations Some Sol Lewitt to intro Notice/Wonder and Creating their own questions.
• Week 2: Barfing Monsters Stations Day 1, Barfing Monsters Days 2/3 and an intro to Visual Patterns
• Week 3:  We began each day with some WODB, then Speed Dating – Problem Solving with Patterns/Sequences to intro/focus on Standards of Mathematical Patterns, Mathematical Models (NAGS) and finally someone asks – is there a quicker way to do these sequences?

We ended the week with Arithmetic Sequences…and tomorrow will be Geometric Sequences.

At the end of class on Friday, I asked them to reflect on which of the 3 options they preferred.  In 5 different classes, not one method was favored any more than another…almost an equal split.  I could tell a difference in students who prefferred the algebraic to numerical – no big surprises there.  When asked why they preferred continuing the pattern, several said it just made sense to them and they understood it.  Though while sharing preferences, several said they preferred numerical –  however, they realized it was not the most efficient one to choose sometimes and could see the value in the algebraic models.  Option A: an=a1+(n-1)d      and Option B:  y = the terms BEFORE the starting value +(common difference)x

After a first run with AP Statistics last year and listening to students describe their patterns the first week, I like beginning with the starting value in my equation model.

Looking back over my examples, I often chose a term number that allowed us to continue the sequence to confirm the rules worked.  I need to give them outlandish term numbers to give them a real reason to turn to the equation models, I suppose.

Even as 9th graders, some are simply not confident enough or are not quite at the level they need to intuitively see the algebraic, YET.

When asked why I felt so strongly about beginning with patterns/sequences.   After taking Jo Boaler’s course a few summers ago and working with repeat Algebra I students last year, I saw it made Algebra accessible.  Students intuitively like patterns.  It allows them to get their footing and share their thinking which allows me to play off their ideas to intro some of the concepts I am responsible for.

The next unit will be an Intro to Functions then Linear Functions (modeling).  I feel like Systems provides a reason/context for Solving Equations and Inequalities.  I hope that I play off of inverse operations enough to build a good conceptual understanding for solving equations, at multiple types / function connections.

At that point, I would like to take a slight break and do Probability prior to Polynomials which I will use to “build” quadratic functions, then some work with Exponential Functions and end with solving quadratics with connections back to our graphical models.  I am interested in what others think of this sequence of topics.  This is sort of the the order we had planned when I was transferred to Geometry / Algebra 2 back in 2012 just as we were getting our footing with CCSS.

I am very happy to be back in 9th grade and Algebra I.  I always felt like I had a more positive impact on students at this age/level.  My first 3 weeks have been enjoyable in the classroom – its going to be a great year.

A snapshot of a few students after paint wars prior to football game Friday night…

# Lagging Homework #MTBoSBlaugust Post 22

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So many ideas from Make It Stick and reading @henripiccotio and thoughts from @steve_leinwand regarding homework.

Here’s my first attempt.

Week 1 in class, I began with a look at Sol Lewitt to develop student questions, spent time with students doing Number Patterns, Open Questions like the Four 4s, Barfing Monsters to develop a sense of sharing ideas, tell me what you see, notice, wonder.  I attempted to build a space that allowed them to share their own thinking.

Yesterday, I passed out a booklet with 4 problem sets for the entire week, here are a couple of them.

Needless to say, a work in progress.  Each set contains 2 problems similar to our current in class work, 4 problems similar to last week’s work and 2 open ended questions (yes, they are a bit lacking, more reflection than open questions).

Today, I took some time before class to discuss self-monitoring on their practice.  This is the system we will attempt to use.  At the top of the sheet are #s 1-8.  When they complete a problem, they circle it.  If missed but a mistake, 1 slash.  If an issue, the draw railroad crossing, which requires us to pause and look both ways…proceeding with caution.  If correct, solid circle.

We discussed differences between a mistake (something they can correct on their own, they know how to do it) and misconceptions  (where I step in if there’s a gap in their understanding).

The idea is to have a system I can flip through quickly and only pause at questions with “issues” without having to check all 32 problems.  I have no idea how this will go.  I explained my reasoning to students with the understanding we can adjust as we go along.

Today I placed answers on overhead.  Asked students to put pencils down, check with colored ink, marking the correct answers in margins for later reference.  As I walked around to spot check, it was a quick way to see common issues and address immediately with entire class.

I plan to give it a few weeks, reflect and adjust if needed.

I’m hoping I can be brave enough to stick this plan through. In class engage, explore, explain.  Next week elaborate/enhance with focused practice.  Following week evaluate/extend.

I’ll let you know how it goes.

# One Good Thing…

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Part of an email I received today…

Thank you for the time you spent on me! You reinforced my belief to always challenge myself, one of the reasons I am currently enlisting in the Air Force.

This email both filled my heart with joy and emptied it to sadness.

Yes, there were times of frustration due to lack of effort or care.  But then again, I’ve never walked in their shoes, or faced life challenges like theirs.  I could not begin to understand the frustration they surely felt sitting hour after hour, nothing to truly engage them intellectually.

So many paths this child could have chosen.  Trouble sometimes met them along the way.  Other times they chose trouble.

I did my best to hold them accountable.

I cried silently last May when they didn’t walk the graduation line.  I felt our school had failed.  I felt I had failed.

The email was to let myself and a colleague know they completed the required credits.  They finished the game of school.  They graduated.

There is nothing I could have taught this young, brilliant mind.  But I took the time…to say you matter.

And today, I realize this student is a starfish…

# Jelly Blubbers & End of Week Reflection #MTBoSBlaugust Post 20

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This is  my second round with AP Statistics.  My feet are at least on the ground – still much to learn but not that blinded feeling of last semester.

This week students read Chapter 12 on Sampling Techniques (BVD, 3rd ed.).  We began class by a quick survey of which technique you felt was going to be easiest to remember and which was most difficult.  Interesting results – they felt systematic sounded the most difficult.  I don’t have a picture of their tallies.

Not sure where Jelly Blubbers came from – but @druinok had shared them with me.  Its a great illustration to help students have a concrete model of each technique.  While leaving class, a student thank me for the activity because it helped them differentiate between them.

The stratified colony was a GREAT illustration for students to see how each strata was homogenuously grouped – it made sense when they saw it!  After going through the entire set of sampling techniques, the following day, I asked them to collect more samples, a couple using each method.  I felt it was important for them to run the collections on their own – me available when questions arose.  It was a quiet time, they were able to process “connect the dots” for each method.

Our discussion went back to our initial feelings of which one was most difficult/easy for them to remember.  Funny how most of them changed their minds with systematic and felt it was quite easy, once they “saw” how it worked.

Class ended Friday with a 2-minute reflection grid of their week.

+ Something I’ve learned…

? Something I still have a question about…

! Something I don’t want to forget…

“lightbuld” – an A-HA moment…

No big surprises.  There are still questions about parameter vs. statistic, how to differentiate between cluster & stratified.  I plan to sit down this afternoon to find an efficient way of addressing their questions.

Its a small group of students.  Sadly, I lost several who took the class who were interested in Nursing because of scheduling conflicts with an MNA class.  But I love the mix of students I have – varied life experiences, varied interests, varied attitudes about their math abilities.  I look forward to this (ENTIRE) year with them.  My first round with stats was a semester block last spring – it felt rushed and cramming, but now I will have opportunity to let them develop their thinking and understanding of concepts.

It was a great week of school.

# Class #choosekind Challenge #MTBoSBlaugust Post 19

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Thursday evening someone tweeted a link to this challenge.

Find out how your class can become Certified Kind this school year!

Last summer, @mathymeg07 shared Wonder as a great, quick read right before school started back.  I couldn’t agree more.  If you haven’t read it.  Read it.  It lends itself to a younger middle school audience, but the story and lessons are great for anyone age 10-100!

I showed the challenge to my classes and on Monday, each class will have a labeled #choosekind jar.  They will place post-its describing #choosekind they observe from their classmates on our “Be a Bucket Filler” board.

Each day, our class Kindness Ambassador will count the post-its, get a marble for each kind deed to drop into their kindness jar.  Our Friday tweets (they can live tweet @pamwilsonmath or paper tweet) were to share a way they could #choosekind or what they might observe someone else doing to #choosekind

When a class fills their jar, I will email a picture to the campaign to acknowledge their efforts.  My goal is to empower these students to know their actions and words have weight and they can impact their classrooms, school, community and world.

# Barfing Monsters Day 2 & Day 3 #MTBoSBlaugust Post 18

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Here’s the version of their documents/ideas we used in class this past week Day 2& Day3.

After discussing a few of our friends issues from Day 1 – Students were asked to work alone on Day 2 – where Blurpo was burping up graphs.  After a while – I asked students to turn and discuss their responses with a neighbor – discussing any differences they may have had.  We then had a whole class debriefing utilizing desmos.  It was a great way to introduce them to desmos.

The sliders really helped when students commented the parabolas – one was wider than the other and others argued they were the same graph, only transitioned down 2 – which made it appear wider at a certain point because the original was “inside” the translated one.

I also had some pipe cleaners to demonstrate the width actually held the same.

What I like most about this day was the 3rd graph, they had to provide the burped up version – and the last graph, where they were given the burped version and had to describe the “eaten” graph.

I used this to share how their brains were processing the patterns to provide structure to apply the pattern to a new questions.  And how being given a “backwards” problem required their brain to think in the other direction as well.  When many of them continued the original pattern and was wrong, they realized their mistake and was able to correct it.  Our brains just grew!  Twice!  We talked about how it was not a misconception (they didn’t understand it) but a mistake (not paying attention) that they were able to correct on their own- not needing me to tell them “how” to do it.

Day 3 is a perfect intro to visual patterns.  Students were given the choice to work on their own, in pairs or a small group.  Linking blocks were available for those who wanted to “build” Spikey’s patterns.  I enjoyed observing their different approaches to building/drawing the patterns.  It was fun listening to their discussions of how to continue the patterns or figuring out how to find the number of blocks required to build the nth pattern without actually building/drawing it.  Again, the power of SMP at work.

These are some of the strategies I saw/heard along with some of the equations a few developed.

It was a great way to allow them to see how others were viewing the patterns.  And when asked which method was better? Silence.  Finally, a student says, well I liked mine best until I saw ___’s and it makes more sense to me now.  But we all agreed there were multiple approaches and the one we should choose is the one that our brain sees.

We only used different equations to show they would result in the same number of blocks needed for a given step.  I didn’t do a very good job of connecting their equations to the methods used to build the patterns – something I definitely want to improve in the future.

Again – I want to shout out to @cheesemonkeysf and @samjshah (was @mathdiva77 in on this as well?) – thanks guys!