Tag Archives: Literacy in Math

Representing Polynomials FAL & Open Card Sorts

Standard

After an assessment last week, it seemed to me what I was doing wasn’t sticking for my students with polynomials.  So let’s just scrap plan A.  Plan B – I pulled out my Discovering Algebra book, came up with a box-building data collection that lead into the FAL I have linked  below.

Formative Assessment Lesson – Representing Polynomials

Thursday, students were given a 16 x 20 piece of grid paper and asked to cut out square corners and create a box with the largest volume possible.  We combined our data as a class.  Recording the corner size removed, length, width and height.  Students were asked to observe the data and respond I notice…  & I wonder… and that’s where our class began on Tuesday.

We shared out our responses, some adding ideas as we continued the discussion.  Work with our data on TI84s – we saw a connection between our constraints 0, 8, 10 and the graph of the regression equation.  This was not new, during the discussion, a question was brought up about what values would result in a volume of zero.  Students were able answer that with confidence and a reasonable explanation.

The FAL pre-assessment confirmed my students weren’t quite ready for the full blown lesson.  With discussion of rigor and relevance the past few days, I wanted to offer students something engaging but not so over their head, it was a flop.

I backed up and did a bit of prep work yesterday – with the following discussions in class:

FALreppoly FALreppoly2

Following with whiteboards / sharing for this slide from FAL:  FALreppoly3

and a simple practice set to ensure they were on track.   FALreppoly4

 

We began class today with a quick check of the 6 practice – with a focus on similarities / differences.  Noting the double root of #5.

Prior to the actual FAL, I decided to use the same equations and graphs they were to match during the FAL, except I would have them do a card sort.  Originally, I had planned to ask them to sort cards into 2 groups.  While pondering how I could make it better, I recalled a colleague sharing ideas about open card sorts from a John Antonetti training she had attended.  So, this is what I did.

I told students I wanted them to sort the 11 equations – any way they wanted – they just needed to be able to share out their reasoning behind their choices.  After a few moments, I called on different groups and we looked at their sorts.  I should have snapped pics / documented their responses.  I was amazed – not that they did it – but how well they did it.   The things they were looking at – were much better than my original idea to sort in to 2 groups.  Students were asking students – why they put one in one group instead of another. Pausing after we had the cards sorted on the board – giving other opportunity to look others’ groups…some were obvious, others were not.   I even had groups who had the exact same sorts, but with completely different reasoning.  Wow.

FALreppoly5

 

At some point we began talking about “What does that tell us about the graph?”  Almost everyone was engaged and comments added to the discussion.  Next we went on to the graphs to sort.  Again, any way they wanted…just be ready to share reasons.

FALreppoly6

Most of the sorts were better than ANYTHING I would have suggested.  My eyes were opened – I could see their thinking.  And others did as well – it was obvious in the eye brows raised and head nods.  In both classes, there was one equation that never seemed to “fit in” the other sorts – but students were confident suggesting it belonged to a particular graph (& they were correct).

When I realized the sharing took more time than I had planned – I ran copies of the equations and graphs to send home with students and asked them to match on their own.  My plan is to put them back in their pairs for the actual pairing of the FAL.  They also had blank graphs for any without a match.

FALreppoly7FALreppoly8

I learned so much listening to my students today…  I am looking forward to the assessment of this standard.

I didn’t feel like I taught anything today…

…but I did feel like my students left with a better understanding…because I chose to step aside and give them the opportunity to share their thinking…

It was a great day.

 

 

Always, Sometimes, Never – #75FACTS

Standard

I’ll be honest, I’ve only truly dug-in to reading the first 6 FACTS of Keeley & Tobey’s book over the past 2 weeks.  Through KLN – Kentucky Leadership Network, I’ve explored several others over the past year.  But I’ve gotten very drawn in to processing the descriptions, suggestions given on the first 6 (by the way, they are listed alphabetically, didn’t know that until someone pointed it out in twitter chat).

This past week, of these 6, I’ve attempted some form of Agree/Disagree (#1), Always Sometimes Never (#3) and Comments Only Marking (#6) in my classroom.  I’ll share more later on A/D and Comments.

Last year, I began experimenting with the Formative Assessment Lessons from the MARS site.  Sorting Equations and  Identities lesson asked students to sort mathematical statements into categories – always true, sometimes true, never true.  Part of the task was to justify their choices.  After using this lesson, I realized students really struggled with these statements.  In fact, they hated them – moaning/groaning each time one would pop up.  Which said to me – they were having to think.  I began embedding them in lessons/notes – class discusses/questions – especially in assessments.  By the end of the year, students were “not afraid” to face ASN questions as before.

This week, I gave geometry students 15 statements about quadrilaterals/polygons, in which they had to answer ASN.  When they arrived in class the following day, I had areas of the room designated A, S, N.

Depending on the FACT, it may help to explain to students why you are using the new strategy.  Part of this discussion was that when someone makes a statement, it may seem true, but we should check it out to determine if in face it always applies, sometimes applies or never applies (page 57).  Through the activity, students were able to share counterexamples if they disagreed with another student’s statement.  Great discussion (even a few semi-heated arguements) occured!

Mathematical Practice – #3 Construct viable arguments and critiques the reasoning of others.

Were students engaged?  Definitely – from the time they walked in, they saw the A, S, N posted and KNEW what was coming.  Most were engaged during the activity.  At least those who didn’t want to think – had to at least choose an area to move to in the discussion.  I used my “name cards” to call on students to ensure everyone needed to be ready to share their justifications.

Were you confident/excited about using the FACT? Yes.  I’ve found a new love for always, sometimes and never statements – though I remember detesting them a particular college geometry course – now I realize what a great learning tool they can be.

How did use of the FACT affect the student-to-student or student-teacher dynamic?  I tried to allow students to share their own counterexamples – but when one was stuck, I would question – referring back to properties we had investigated, drawing figures on the board, presenting a what if… if needed.

Was the information gained from the FACT useful to you?  I realized some students still confused a few of the rhombus, rectangle, square statements.  Mostly, that students often only considered the “obvious” – but this activity was great because others were able to share their “what about…” with their classmates.

Would you have gotten the same information without using the FACT?  In the past, I would have likely made the same realizations but only after giving the unit assessment.  This FACT helped clear up some misconceptions during the learning process rather than at the “end of the learning.”

What added value did the FACT bring to teaching and learning?  Students had to think about their thinking, jusitfy their reasoning, could be critiqued by classmates’ thinking – great opportunities for discussion / sharing!

Did using the FACT cause you to do something differently or think differently about teaching and learning?  During the task, I was able to use student comments as a springboard for whole class discussion, pointing out examples that made it true and examples that made it false (great piece of learning to impact understanding of counterexamples).

Would you use this FACT again? Yes.

Are there modifications you could make to this FACT to improve its usefulness?  This FACT lends itself well to written work, whole class & small group discussions.  Follow up is key – probing students and guiding them to consider other examples – if not shared by classmates first.  Even after arriving at what seems to be class consensus, ask again – challenge their thinking – don’t settle for the first correct responses – ask why – let them justify their reasoning.

Literacy Strategies for Improving Mathematics Instruction

Standard

Wrapped up our twitter book chat this week over

#lit4math – Literacy Strategies for Improving Mathematics Instruction by Joan M. KenneyEuthecia HancewicLoretta Heuer .

You can read (here) how I am not a fan of writing and words.  Literacy – communication – its all the same, in my opinion;  You can read, write, speak – but its all to share/get information, right?  I do realize the importance of providing students with strategies that will help them succeed, to give them opportunities to write and talk about their thinking can be a key component in their learning to help expand their understanding of certain concepts.  I look at this chance to learn about literacy in math as a way I can learn with my students – to be open that words are my weakness – but by facing my fear – something I struggle with – I can help them realize words are not the enemy either.  I am able to help them learn this “new language” called math and share ways of conquering it !

Though this book did not end as strongly and wow! as it began, it was worth my time.  Chapter 1 really pulled me in, causing me to think about my classroom, questioning some of my strategies and left me craving more!  It showed me how students – who are not as math-minded – can struggle because they view concepts differently.  Chapters 2 and 3 – gave me tools / suggestions of ways I could provide students with opportunities to share – ways I could become more aware of their thinking – and prepare for their struggles.  Through our chats, I was able reflect how I could improve things I am currently doing – but also looking at new ways of viewing mathematical text and ideas (literacy really isn’t a 4-letter word).

The remainder of the book, well, I was diasppointed – but would still recommend at least a skim – because there are some key ideas – but mostly, some great articles/research mentioned you may wish to take a look at as well.

I’ve linked to catalog from Storify of our Twitter Chats – again, some good thoughts – good articles and links.  Also, take a look here, Teaching Statistics Blog offers some reflection with posts from reading the book in 2010.

June 11 – Chapter 1 Mathematics as a Language

June 14 – Chapter 2 Reading in the Mathematics Classroom

June 18 – Chapter 3 Writing in the Mathematics Classrom

June 21 – Chapter 4 Graphic Representation in the Mathematics Classroom

June 25 – Chapter 5 Discourse in Mathematics Classroom

June 28 – Chapter 6 Creating Mathematical Metis

All in all – it really boils down to becoming aware of those struggles students will encounter and being ready to help them bridge past that struggle.  Notice I didn’t say be the bridge – productive struggle is a good thing.  We must give them opportunities to read, write and share – expanding their understanding by listening to other learners.  When they write about their thinking – cognitive demand is much higher.  We must listen to their conversations – not always answering their questions, but providing them with questions that will move their thinking deeper.  When they talk, discuss, even argue over a solution – they have greater opportunities to build connections as opposed to a sit-n-get teacher centered classroom.

My summary:

 ‏@pamjwilson to get students actively engaged, the tchr must 1st be actively engaged- listen, question, be less helpful #lit4math