Math – Winter Quarter Reflections

As I approach the beginning of my full-time student teaching, I am excited and a bit anxious.  I have taught more whole class math lessons than any other subject, so I should be most comfortable there, right?  In my 5^th grade class, we use the Envision curriculum and it moves quickly with each lesson following the gradual release model (OLM) with demonstration, guided practice and independent practice.  Reflecting back over our intermediate math methods class, I know that I need to start doing more to augment the curriculum.  The frightening truth is the quote from the Lotan article on Group Worthy tasks that "one dimensional tasks require the same skills and the result is uniform success for some students and uniform failure for others."  The curriculum (at least how I have been using it), feels one dimensional and I need to focus attention on how to add to that.

This quarter, we have talked a lot about different ways to be “good at math” and the necessity of having multiple entry points into a math topic for our diverse populations of learners.  We talked about the importance of leveling the playing field so that socioeconomic, language and ability levels are blurred.  I want to be able to randomly group students so that they benefit from working with a variety of different students.  I am constantly trying to remember who I can put together and who I can’t.  Random groupings make a lot of sense and will simplify the process.

Manipulatives, virtual and real played an important role in our readings, class discussions and video reflections.  We have some manipulatives with the Envision curriculum, but they are mostly used for “games” at the end of the units.  I want to start integrating their use throughout the lessons to help get more of our students engaged in the learning.  I also found out that there are virtual manipulatives embedded into the on-line textbook.  We are all so busy getting familiar with the new curriculum and “getting through” the material that we are not using all the different modes we have to connect with our students.

I tend to be a planner, but I need to continue to get comfortable with the reality that sometimes things do not go as planned.  I need to be more flexible in my lessons, more flexible in my approaches and more flexible in my math thinking.  Our cohort has shown me that we all have different ways to approach math problems and now my students are showing me the same thing.  I need to get comfortable in being able to go where they lead and then bring them back as needed.  That journey can be uncomfortable but is how students create their own meaning.

Finding the ZPD for our students is one of the most important challenges that we have as teachers.  That continuum from boredom to anxiety is one that I see in my students faces everyday.  My own experience over the last two quarters tells me that I can be in a different place along the continuum on different days depending on the lesson, my background knowledge and a variety of other factors.  I know that some days, I was in the ZPD and that is what we want for our own students.  It won’t happen for every student everyday, but we need to be close enough to know that if not today, tomorrow.   

The more I know, the more I realize what I don’t know and that can feel frustrating.  I am going to try and relax and enjoy my student teaching experience.  This is what I have been waiting for!

ipod touch - final thoughts

     Perhaps the fact that I am not too broken up about giving back my ipod touch tells the story.  Quite frankly, it has not lived up the potential that I anticipated this quarter. I know that part of the issue is that I have been busy and rushed and I have not devoted as much time to trying to find ways to incorporate this tool into the school day as I would have liked.  Another issue is that the wireless in my classroom was not always fast or reliable (when I finally figured out how to use it).  A related issue is that the classroom computer usually could be used to check out google and with greater ease and visibility. However, for most of the quarter I have been hard pressed to incorporate using one ipod touch.



I have liked the audio recording device and wish I had a smart phone that could do that for me.  There are so many opportunities to use this feature that I might have to consider some alternatives.  The dictation and dictionary.com and google translate saw only "testing" in my hands.  I did use the Merriam Webster dictionary  (no internet required) last week and it worked well.  I used it to look up the Latin roots of equilateral, isosceles and scalene triangles.  It helped me give my students a way to help them remember what these mean and augmented my lesson on triangles. 


I am still very intrigued about the possibility of the touch just on sheer engagement alone.  My daughter thought it was the coolest thing that I was using a touch to help my teaching.  She is NEVER without it and as I have said before it is THE accessory of choice in her tween crowd.  I have also found that even in lower socioeconomic circumstances, it is something that kids own.  This device is powerful and I suspect as the price comes down and the availability goes up, access will not be an issue.  I think it is a tool to continue to think about in the classroom if everyone had one.  Only now, I will have to wrestle my kids to use theirs.

Complex Identities

OK, I admit it, I like 1970s worksheets.  I am comfortable with them anyway because it is what I grew up using in math classes.  It is good to come full circle and realize that our students will be at an advantage because we can give them a variety of options for learning and understanding geometry.  They can “play” with shapes on the computer or make drawings with a pencil on a piece of paper.  We can provide the definitions of the concepts first, or let them do some discovering before labeling the shapes.  We have more tools at our disposal and we need to get familiar with all of them and make them available to our students so there are as many entry points to math as possible.

The measuring activity was enjoyable too.  It was great to use a variety of everyday objects to “measure” the rectangle.  I appreciated hearing the different thoughts about why the graph of the measurements was linear.  It was a great way for me to deepen my own understanding of measurement, graphing, algebra and slope.  Nothing fancy required here, so it is very accessible.  This helps me as I have been concerned with the availability of some of the math software in our schools.

I really like the idea in the complex identities article where “helping our students become aware of their mathematical identifies can empower them to make that relationship more meaningful”.  My school is having a math week later this month and there was a staff planning meeting about it this week.  I proposed to my master teacher that our class have a math community building activity.  We would we talk about all sort of different mathematically related activities (the table in the article).  We then have an opportunity to have our students consider and talk about the activities that they are good at doing.

I will be reviewing some other math activities that we might include.  Some pictures of  the Fibonacci sequence and connections to nature are floating through my mind…..

ZPD and Engagement

It is good to have some practical examples of what the ZPD might look in a math class.  Boredom on one end when math is too easy and anxiety on the other end when math is too difficult.  The ZPD looks like a pretty narrow target and I believe it is going to take some practice to hit the bull’s-eye and engagement is critical!

Going to the computer lab was a good example of how we might engage some of our students in different ways.  Geometers sketch pad was a lot of fun and would allow some of our students a different entry point for geometry. I need to review my geometry vocabulary as I had my kite and rombus a bit confused.  Using DASL in conjunction with  Fathom 2 was a great combination for engagement for statistics and graphing.  We have already talked about how using real data can allow students to connect and create meaning in ways that “made up” data sets does not allow.  I think analyzing data is lot more important that being able to graph data and this is a brilliant way to enable that to be easy.

I did ask my high school senior and he has never used any of these tools.  He attends a pretty resource rich high school (Issaquah), so it does give me pause about the cost or feasibility of schools using these to engage students.  He does have the same TI graphing calculator that he has used since middle school.  Although he has never hooked it up to a probe to do motion analysis, he has done something similar with Loggerpro that he tells me is pretty cool.  My understanding Loggerpro allows a variety of sensors (thermometer, ph, etc)  to plug into hardware that connects to a computer for graphing.  So maybe it will eventually be at our fingertips, or smart phones.

He was good at math.....

I have been struggling to focus on math (and school) this week.  But today I realized that Tyler’s story is full of math connections.  Tyler struggled academically but was a passionate and successful lacrosse player.  His desire to play at the college level resulted in a switch to a small private school this year.  He had recently been accepted at a Division II school in California where he was going to play lacrosse.  Last week a car accident took his life and ended those dreams.

I remember the first time I met Tyler’s mom, it was at the parent meeting for lacrosse when our boys were high school freshmen.  We met in the coach’s classroom, he used to split his time teaching math and phys ed.  The coach was pretty intense and I remember he spit out a lot of statistics about his team.  Patty whispered to me that Tyler was not good at math and maybe the coach would be able to help him out.  I remember telling her to have Tyler check in with my son Aidan, he was a “math kid”.  That was before I knew all the ways you could be good at math.

Tyler had an amazing ability to see the entire lacrosse field and know where all his teammates and opponents were located.  He was a defender (like my son), but his talent for always being where the action was earned him a spot on the varsity as a freshman.  In addition, he was the guy to go to when you needed to restring your lacrosse head.  This is an art that requires patience, hand eye coordination and 3 dimensional visioning – a kind of puzzle.  This week, I learned that Tyler also knew all the starting NBA line-ups and their jersey numbers. 

He was a fabulous lacrosse player and he was good at math.  He will be missed.

Comprehension - next steps

S. is a confident 4^th grader who enjoys and identifies with being a reader and a writer.  She reads and writes at home for pleasure and she is supported by a family of readers.  S. reads different genres including historical fiction and adventure.  S. is fond of animals and her cats inspired the Fuzzball the talking cat story.  She has been known to create books to give as gifts and last year she gave her Mom a book for Mother’s day.  Her love of plays and acting contribute to her love of language.

The testing we have done to date, confirms that S. is a skilled reader.  On 1/25, we did a running record while she read The Mouse and the Motorcycle (grade 4.4 on the AR scale).  S. had 98% accuracy (based on 6 errors/300 words) and none of the errors affected meaning.  In addition, the QRI testing on 2/1 showed that S. read 139 words/minute putting her at the 75 percentile for 4^th graders.  In that passage on Amelia Earhart, she scored in the instructional range based on 7/8 questions correct.  She was also a 4 on the NAEP fluency rubric for qualitative measure of fluency and expression.  In her retelling, I scored it as a partial retelling (3) based on the 7 phrases she included in the recap of the story.

Regarding a lesson, I would like to take a deeper look into a story with S.  As Routman says “most students do not read deeply” and I have had glimmers that this might be true based on some superficial retellings.  In addition to the Amelia Earhart piece, S. has given us another retelling of a fictional story (Mouse and the Motorcycle) that was event focused as opposed to character focused.  I might base a lesson on character development and determine if S. can really connect with the characters and understanding why they behave as they do.  I would focus on inference to uncover if S. can read between the lines to understand a character’s motivations.  I have no information in this area.

My thought is to use Tovani’s double entry diary as an easy tool.  I would have S. fold a piece of paper down the middle and make two columns.  On the left would be specific quotes from the text with page numbers (alternatively it could be a summary of a section.)  On the right would be where she would write I infer…..  I would model how to do this out so that my teaching is explicit.  We could then do one together before letting S. go back to her seat to do more on her own.

Ipod touch revisited

Well, things have not worked out quite like I had planned regarding using my ipod in the classroom.  I thought I might be able to use some kind of combination of Dictation and Google Translate for R, for translating modifications to his homework.  It has not really been needed.  R. just does the math computation and skips the word problems, his English has improved considerably in the time I have been gone.  It also took me a couple of weeks to be able to access the wireless network in my classroom.  My teacher does not use it, so I needed to seek help from another teacher.

I did get the opportunity to use my ipod with H.  He was struggling to complete a science assessment that consisted of two short answer questions about labs from the previous week.  H. was inside during recess to get that completed as he struggled to get it done during class time.  My MT was out of the classroom and he turned it in to me with virtually nothing on the page.  I tried to coax him into giving it some additional time and he could not.  I then asked him if he might be able to talk to me about questions from the lab and he agreed.  I did record his answers on my touch.  Unfortunately, my MT told me that he did not have “voice recording” as an approved accommodation (he has a 504) and in addition, she is trying to get him to write and does not give him alternatives!  I ended up transcribing his answers to the assessment so that she could determine how much he understood from the lab.  She gave him a 2.5 (just below benchmark) for the assessment and felt it was helpful.

I have not used the touch for C, to extend science in any meaningful way.   I did use Google to get a quick picture of a plateau to assist my MT while she was teaching.  The concept was confusing given the pictures we had and so I found a picture and put the ipod under the doc camera. 

I have used the ipod to record the six science interviews I have done in class.  All the students were engaged and intrigued that I had this device as part of my classes.  It has allowed me to focus on getting to know the students and not have to worry about taking notes.  I will continue to look for opportunities to use the ipod touch in my classroom as a means of differentiation.  I will look to my cohort peers to see how they are using the touch.

Informal Geometry

Today's activities provided another opportunity for people to be "smart at math" in different ways.  The visual and tactile nature of folding paper into boxes and squares made geometry very concrete.  As I am starting to get used to, it may have been informal, but I felt less smart in math as than usual.  I am more familiar with drawing a geometric shape and learning all the rules about how angles and sides are related.  My stretching is continuing.

I was hoping we would discuss the use of creative writing in math class that was the subject of the Halpern article.  I thought that was a different type of example of how to make geometry (or any math) more accessible for students who have strong written expression skills.  As students get into Middle School, I often hear that students start being divided into those that are more humanities focus and those that are more math/science focused.  I read The Phantom Tollbooth last quarter and I felt it was a fiction book that would appeal to students who love words and/or numbers.  A bored boy travels in a toy car through a tollbooth to a place filled with adventures based on letters and numbers.  The Mathemagician rules the Kingdom of Digitopolis and a character called the Dodecahedron has 12 faces with different expressions.  It is another possibility for giving our strong writers an informal entry point into geometry.

Group Worthy Tasks

I loved the readings this week.  I got inspired while reading the article on Orchestrating Discussions, so I went online to join NCTM to be able to start reading the middle school journal regularly.  The bag of marbles problems is a marvelous and simple example of a group worthy task and gives me something to contemplate for the lesson assignment.  The article makes so much sense to me that I pulled out the five practices planning worksheets from last quarter so that I actually start using them in my main placement.  I like the idea of trying to anticipate all the possible ways that my students will solve the problem- especially the ways that they may solve incorrectly due to common misconceptions.  Monitoring, selecting and sequencing becomes a whole lot easier when I know what I am looking for.  As a new teacher, I am not as good "on the fly" as I would like to be so I want to plan as much as I can until my comfort level increases.  Making connections to the lessons goals and then extending and making generalizations are all things to strive for.  But it gives me a template and something to try and I like that.

The Lotan article on GroupWorthy Tasks is exciting and a bit daunting.  I know it is a lot of work up front, but the group work in the article really did help even the playing field for students from different racial, ethnic and social groups.  I loved the multiple entry points and opportunities for students to display "smarts" in different ways in response to complicated problems.  The frightening truth is the quote from the article that "one dimensional tasks require the same skills and the result is uniform success for some students and uniform failure for others."  We need to carefully craft group work to enhance participation and learning for all our students - it is worth the effort.  Our lesson plan will be an authentic opportunity to consider this goal.

Finally, I continue to struggle with manipulatives.  The Mira symmetry manipulative was a challenge to me.  I was slowly getting the hang of it, but I need remedial work.  I think I will need to make my way to Math and Things store sometime soon.  My experience with virtual manipulatives is even more challenging.  Just give me the algorithm!

Writing Analysis and Lesson Outline

Analysis of Student Writing

S. is an outgoing and talkative 4^th grader and clearly proud of her writing ability.  We briefly looked at several pieces in her writing folder before she selected one for us to review.  My analysis is based on S.’s response to a writing prompt about a talking animal.  S. wrote about her own kitten “Fuzzball” and she described what a day might be like if he could talk.  Her story described a series of events including eating breakfast, reading the comics, and washing dishes with the cat.  She engages the reader with her vivid descriptions and choice of language. 

Six Traits

S. has terrific ideas but instead of focusing on one, she has written about several vignettes – some more developed than others.  She is always on the general theme of a day with her talking kitten, but the ideas are not always connected and do not flow logically.  We are left with gaps and questions about the story.  (NREL .p. 2)  Reading the comics leads to a pillow fight with no explanation.  S. suddenly calls a friend and a fashion show is introduced.  These events have some detail, but are confusing because they seem a bit random.  Regarding organization, the story is a bit uneven.  S. starts with a very inviting introduction and I was eager to continue reading.  Subsequently, there are several ideas competing for focus and therefore the middle feels disorganized and somewhat fuzzy.  The story ends with a conclusion that is recognizable, but not completely satisfying for a piece that started so strong. (NREL p. 3)

S.’s strong voice and word choice are “powerful and engaging” and give her writing sparkle. (NREL p.5). Her words and phrases are thoughtful and specific; examples include “muted knock” by the kitten’s paw and “under the whistling voice of the tea pot”.  Vivid verbs include blurted, bounded and pranced.  Descriptions like “happy as a lark,” “thin, high heeled black shoes”, and “purple polka dot dress” all add color and drama to the story.

S.’s sentences always get the job done and in some cases are varied and energetic.  She does use dialogue in her story, although sometimes it is a bit choppy.  Her conversations succeed in introducing varied structure and length of sentences and helps move her story along. (NREL p. 6) Her fluency shows and conveys the excitement that she feels about her new found ability to communicate with her kitten.

S. has a very good grasp of standard writing conventions.  Evidence of S.’s ability to revise and edit for conventions were apparent in the marked up Fuzzball final draft.  There are a couple of small spelling errors (meant and licking) but I am not sure that they accurately reflect S.’s stage of development.  I suspect that with an additional review, the spelling would be corrected and the piece would be ready for publication. (NREL p. 7))    This is consistent with S.'s “Message in a Bottle” writing prompt that was published on the class bulletin board.  Spelling, grammar, paragraphs, punctuation, and capitalization were once again excellent.

Ownership

S. enjoys writing and she is confident and highly motivated to create stories at home and school.  In the Fuzzball piece, her delight in having her kitten talk makes the story silly but fun.  S. sees herself as a capable writer and she proudly showed us several writing pieces in addition to the Fuzzball story.  She was working earnestly on a pre-write about what one thing she would change at school.  Another completed piece was an “All About Me” story and she read us the clever conclusion “if you need more information, call 1-800- SAMI”.  S. enjoys acting in plays and I believe that adds to her love of language and dialogue. 

Spelling

In addition to spelling in context in her writing, S. did well on the spelling inventory.  She spelled all the words correctly except “celler”, “civiliazed” and “oppisition”. This would place her beyond vowel patterns on the developmental continuum.  Derivational relations (opposition) and suffixes (cellar and civilized) would be her stage of development in spelling.  In identifying words from a list, S. was very strong and tested at the independent level through the 6^th grade.  She insisted on continuing through all the word lists and she successfully identified many middle school and some high school level words.  She was eager and proud to know what words she had gotten correct.

In conclusion, S. is a strong writer whose voice and word choice shine.  Her writing is supported by her sentence fluency and correct use of conventions.  Two areas for improvement are ideas and organization.  S. has so many ideas that she would benefit by narrowing the focus of her writing.  Focus would enable her to elaborate on her main theme without getting sidetracked by random and distracting information.  The improved focus would help with her organization as transitions and clarity would improve.  For that reason, I would start a writing lesson on idea development and focus. 

Lesson Outline

Objectives

To brainstorm how to identify the main idea and narrow the focus for the tetherball story.

To identify overlapping and/or unrelated ideas in the tetherball pre-write.

Standards

EALR 1 Student understands and uses a writing process

1.1. Pre-write to generate ideas and plan writing.

EALR 3 Student writes clearly and effectively

3.1   Develops ideas and organizes writing

Materials

Pencil, post-it notes, new prewrite graphic organizer

Instructional Strategies

This lesson is designed as an individual writing conference and a follow-up .  Alternatively, it could be given as a small group lesson to students all working on narrowing the focus for their ideas.

I would ask S. to bring a copy of her tetherball pre-write.  I would have a copy of her Fuzzball story and start the conference by praising her terrific voice, word choice and conventions she demonstrated in the piece.  I would then gently point out that after an engaging introduction, the middle became a bit fuzzy with several competing and unconnected ideas (I would be specific).  I would then focus on the new project and explain how a pre-write helps to narrow a main idea before she starts her draft.  The goal being that the tetherball piece have a clear, well defined and narrow focus before starting the draft.

I would show S. a sample pre-write using the same prompt.  This could be from a previous year or one that I created for illustrating how to narrow topic focus.  I would demonstrate how after I finish a pre-write, I review with the goal to identify and narrow the main idea.  I would use a series of questions.  Are my ideas all different and distinct or do they overlap? Are they all equally important to the story I am trying to tell?  Are they related or connected in some way?  As I ask the questions, I would show how I could eliminate or combine repetitive ideas.  I would show how I could connect related ideas with so there is logical flow.  I would show how to identify the one idea that seems to be the most important in the story I am trying to tell – that is her main idea..

Next, we would turn to S.’s pre-write and together we would ask the first “review” question.  Are all my ideas different or do they overlap?  As S. identifies similar ideas, I would ask how she might combine or eliminate some of the information to help narrow the focus.  Although we did not spend a lot of time on this piece, my recollection is that S. had “listed” three reasons why she wanted to have more tetherball courts.  Yet all of the reasons were related to the main idea that students in different grades need to play on different courts.

When I was confident that S. could revise her pre-write on her own, I would have her go back to her seat to complete the revisions using the questions we used on the sample to combine, eliminate and revise her pre-write.  I would have S. come back when she felt that her revision focused on a narrow topic with all the other ideas connected to one main theme.

During the follow-up conference, I would have S. read her revised pre-write.  If the pre-write was focused on a main idea with related supporting information, I would have S. proceed to begin her draft.

Assessment

I will document how well S. can narrow the focus of the tetherball paper by comparing the revised pre-write to the original pre-write and checking for a main idea with a narrow focus and connected ideas.  As the assignment progresses, I will use the ideas section of the six traits rubric to assess the clarity and focus of her story. 

Differentiated instruction using ipod touch

R. is a student in my 5^th grade class who moved here from Japan at the beginning of the school year.  He arrived knowing very little English but goes to ELL each day – but not during reading time.  As I am returning to my 5^th grade, this is a subject area where R. continues to struggle.  The class uses a Houghton text and often has a “packet” of activities about the reading to complete.  The teacher works with a small group while the rest of the class works independently on the packet.  R. sometimes works on something from ELL or Rosetta Stone and I am wondering if there is an opportunity to help R. using the ipod touch and Google translate as demonstrated by Jane in our first class.  At this time, I do not have wireless access at school, but I am working on that.  As a first step, I would like to be able to get the homework assignment (usually modified) translated into Japanese and sent home by email to R.’s mom, who reads and writes English pretty well.  Right now R. usually brings home the same homework as everyone else and my master teacher tries to remember to write a note home about the modifications and what to focus on, but it is difficult with 31 students in our class.

H. is a bright boy who has significant difficulty with writing.  He attends special ed to get help with writing skills as he gets so frustrated during writing, he completely melts down.  I am wondering if letting him use the ipod and the Dictation app could be helpful as it gets his ideas down by speaking and then the written version could be sent to an email or Google doc for him to continue.  Getting started seems to be the biggest hurdle.

C. is a quiet and bright girl in my class who has is looking for opportunities to have her thinking extended in science.  C is asking for more difficult science experiments or opportunities to learn something beyond the Foss kits.  Is there an opportunity for me to use the ipod to help her get there?  Even using Google to research erosion and deposition from our lab last week might help extend the lesson for her.

Getting Kids to Do the Talking

I really enjoyed the practical article that was full of strategies and tactics to try in math class.  I think the teacher had a terrific personal goal of trying to change his own practice by 10% every year.  I think that is a manageable goal that ensures teachers are continually improving their own practice.  As a new teacher, I liked many of the ideas and plan to incorporate them into my initial classroom plan.  One of ideas that was brand new to me was how to handle students who say " I do not get this" when responding to a problem.  I think that "I don't get it" should be banned from classrooms.  I loved trying to get the student to ask a question when they are stuck so that others (peers, teacher) can help them get unstuck.

I am wondering how to implement some of the ideas in my class where math is very scripted.  We have direct instruction, guided practice and independent practice every day.  I know that there are opportunities for pair/share, group work and asking questions.  I just need to get comfortable with the curriculum so I can add to it in a way to help more students learn math well.

In my main placement (5th grade), we are just starting algebra.  I will be looking for manipulatives tomorrow!  I loved the tiles and think that they will be very helpful in my class.  The students were very confused last week with the introduction of variables.  Statements like n=3 were confusing and I think it is a big shift for students.  If we do not have manipulatives, I plan to take a look at the virtual manipulatives to see what I can find to supplement the book.

Math Thinking

In math class today, I learned that I am much better seeing patterns in numbers than I am seeing patterns in pictures.  It felt natural for me to write the numbers in a chart and figure out the general equation for the nth number.  However, it took a while for me to figure out how to manipulate the tiles to show the associated geometry picture for the garden equation.  It took me even longer to figure out how to show the picture for the snake equation.

I am trying to figure out how to be more flexible in my own math thinking so that I can understand and help all my students.  Heather had a great (and very fast way) to do the garden problem and it took me several times before I was on the same page and understanding her thinking.  I need to figure out several ways how to solve problems so that I become more flexible and get out of long held patterns.  Next time, I should start with the picture and not the numbers!

 Last quarter, Prof. Allison Hintz gave us some tools –including worksheets to help us plan how we might want to use students’ math thinking to show different problem solving strategies to our class.  Sometimes a particular sequence can help guide our students to move to a more efficient strategy.   As a new teacher, I would benefit from using the planning sheets or trying to solve problems in a number of ways to help me preview what I might expect to hear/see in class. It is powerful to share students’ math thinking, but I think it needs to be facilitated well by a flexible thinker.