Math Week 9: Transformations through Reflection

This quarter we have studied so many aspects of mathematical learning.  We have talked about using manipulatives in order to root abstract concepts in concrete ideas and understanding.  Also, we have been involved in the group work process and exploring what it means to have meaningful and effective group work.  We have also interacted with a variety of technological tools to use to assist in mathematical learning such as iPod Touches, computers, and graphing calculators.  Throughout all of this, we have found ways to reach all students by offering multiple entry points for learning math.

It was difficult learning about teaching without having time to see different classrooms.  However, the videos that we watched were beneficial glimpses of good teaching practices and I tried to incorporate various methods of math teaching into the classroom I'm in.  I find that the most difficult part of math is accurately assessment students' abilities.  Some students are very good at showing their work and arithmetic, while others are superior at problem-solving and logical, creative thought.  One test certainly doesn't seem to fit all.  What are other ways of assessing student's math knowledge so that we can cater our teaching to suit their needs?

I have greatly valued the time I have been able to spend with small groups of students and individuals in the classroom.  A couple of times so far, I have had students write on a slip of paper what they would like to work on.  Then, I sorted the students into groups suiting their interests.  Being able to work with these small groups on particular mathematical concepts seems to be greatly benefiting them.  I got one comment from a student who said, "why can't we do math like this all the time?"  I also think that the one-on-one mini-conferences I've had with students has been beneficial for them.  In these cases, I had students write a question to me and in the mini-conference, we went over the answer to that question and any others they had.  When I see the results of the unit assessment next week, I hope to find evidence that the students benefited greatly from both of these learning approaches.

I believe that the integration of math with other subjects is critical for the growth of intellect and morale.  I've seen so many math classrooms (particularly at the middle and high school levels) where math is an isolated subject.  Thus, students either develop the belief that they are good or bad at math; there seems to be no middle ground.  If math was more integrated, we would be able to draw upon our students' skills and interests and connect new learning with background knowledge.  Not only would this significantly increase learning, it would also revive math to be a magical and mysterious subject, rather than a mundane and tedious one.

Impossible?

Okay, this is totally cool.  These guys obviously didn't develop the "that's impossible" complex when they were growing up!

Tech Post 3: Upgrade!

The use of iPod Touches in the classroom strikes me as best when used with individuals.  The reason for this is not because the applications are more suitable for individuals, but rather because of the physical constraints that the size of the iPod Touch dictates.  Now, with the 5th graders I see, pair work would work with the iPod Touches only if the students were in same-sex pairs.  The screen size is small enough that the students would need to be cheek-to-cheek, and the boy-girl pairs I see working together are reluctant to even talk to each other.  I'm sure that this differs class to class and grade to grade.

I have learned a bit about the iPod Touch, since I haven't used them much before.  I realize now that some applications only work when you have connection to the internet while others work without a connection.  For example, the Dictionary.com app works when connected while the Merriam-Webster Dictionary works while offline. Also, the battery life of this little device is quite impressive.  I've only had to charge my iPod Touch once this quarter, and it retains its charge even when not in use.  Basic as this seems, it is critical for the functionality in a classroom setting.  Knowing that these devices can be used for prolonged periods without charging will greatly increase their usability in the classroom.

I am still wondering about the process of getting iPod Touches (or any other technology, for that matter) for the classroom.  Where do we apply for grants and/or who do we apply to?  I realize it was a long process to get the iPod Touches for us to use.  Will it be easier or more arduous to obtain these resources for use in a public school?

Math Week 8: Math + Philosophy = X

Today in math class I learned about different ways to teach geometry, how to think about graphing backwards, putting emotion into math, and having the right to fail.  Woah, that's a lot.

The backwards graphing caught most of us off guard.  We're so used to graphing with units in mind.  We've been trained and been marked down on tests for not labeling the units on the axes.  So today, when we had to plot data points to show the linear relationship in measurements, we felt like the earth had switched its rotation direction.  What makes present-day math teaching and learning so linear?  Why aren't more teachers allowing their students to explore different facets of math and offer different access points for learning?  Is it fear of failure?

This society also is very set in the mindset that we cannot let students fail.  We can see this on innumerable levels and understand why this is so ingrained in our culture.  I like what we talked about in class in terms of the lessons that we learn from failing.  Often, we have made our most life-changing choices when faced with these situations.  This is juxtaposed by the philosophy that in order for us to keep desiring learning, we need to be successful.  There is a balance between these philosophies; the question is where?  I suppose the answer is that it depends on the individual. 

Ahhh...refreshing!

Check out this clip by the talented Demetri Martin.  His comedy relates to attention span, percents, self-image, and obedience...everything we see in schools!

Jokes.com
Demetri Martin - Doors
comedians.comedycentral.com
Demetri Martin Stand-Up Demetri Martin Jokes Demetri Martin Person Videos

Math Week 7: Techie Transformations

Today, I learned how to use some of the basic features of Geometer's Sketchpad.  This is a great, simple program that allows you to draw shapes.  In the process, you learn how to informally prove that a shape is what it claims to be, as well as explore the concepts of definitions of shapes.  This is a tool that we can use in our classrooms, on the computer or Smartboard.  We can also setup example problems, like the one we did in class today, and have our students interact with the software, solve, and discuss concepts.  I wonder how this software could be used as a method of assessment also.  It reminds me of when I used Solidworks in a class I had and took tests by building models using the program.  I think this type of assessment would be yet another access point for different learners to demonstrate their knowledge.

We also learned about all the add-on gadgets for graphing calculators which can provide ways to collect data and analyze graphs.  I like how this interactive process is engaging for the whole class while also helping students turn abstract ideas into concrete understandings.  As we wondered in class, when do we think technology will transform cell phones into graphing calculators?  After all, why have a bunch of gadgets when they can all be combined into one?

Math Week 6: Many Ways to Learn

This week in math, I learned about how using visual tools and manipulatives (such as Tangrams) allow the students in our classes to see how different people have different skills.  Some people have a mind for visualization of concepts while others thrive using logical thought processes.  Ideally, I think, it would be most beneficial to be able to offer the students a variety of modes of learning so that each student can access the learning in his or her own way.  What's the balance between encouraging all students to try different methods and making sure they are satisfied with the learning experiences they are having?

Also, we talked about how journaling is a great way to assess student understanding in a pressure-free way.  In this high-tech age, we can use blogs instead of journals so that students can keep their physical journals with them for reference at all times.  Also, as teachers, we can subscribe to our student's blogs so that we know when they are updated.  As we continue talking about different ways to assess students, I think it's so important to provide a multitude of opportunities for formative assessments so that we can successfully help all of our students learn.

Many people are still flustered by the idea of integrating math with other subjects.  We've talked about this topic so many times that I'm thinking this concern is not going to be resolved until we start delving into teaching.  Right now, I think that integrating subjects really is going to be smoother than we think.  This is one of those theories that will probably pay-off in the long run (students will learn more) even though the evidence that it's working probably isn't clear up-front.  I guess it's our job to be both daring and logical by collecting that evidence which shows that students are learning more through integrated methods.  (Then, we can write one of those scholarly articles that will be read by all those in the profession!)

Literacy Post: All About D!

What have you learned about your buddy’s needs, abilities, and interests? 

D is an enthusiastic learner who considers himself exceptional at math and science, but only average at writing or reading.  When it comes to literacy, though, D does say that he enjoys reading mysteries and some fantasy books.  However, it does not sound like D has found motivation to do reading outside of what's required for school.  

Looking at the results of D's various comprehension assessments, it seems that D needs the most help in this area.  He reads aloud with clarity, confidence, and intonation, but only with average/below average speed.  In retellings, D omits or confuses the main points and supporting details.  What's interesting, and I will investigate this further tomorrow, is that D's "comprehension" significantly increases when prompted with the right questions.  Thus, there is the possibility that D's comprehension is at or above grade level, it just does not appear so during the standard method of assessment.

D's writing shows his creativity, sense of organization, and ownership of his work.  However, as detailed in my previous analysis, D's convention errors detract from the meaning of his piece.  Thus, D is well on his way to becoming a clear, fluent writer, but needs to work on his grammar.

As a result of this knowledge, what learning objectives and materials are you considering using for your lesson? 

Based on the data I currently have, I think a comprehension lesson focusing on identification of main ideas would greatly benefit D.  This would provide a clear purpose to motivate D's reading and set clear expectations of an end product to help D focus on the text.  In finding main ideas, D would have to sort through supporting details and "discard" them so that he does not veer off-track or misinterpret the text.

Another possibility would be to help improve D's fluency.  It is possible, as we learned, that if a reader reads too slowly, it becomes harder to comprehend the text.  As we were told, the reader might forget what the beginning of the sentence was about by the time he gets to the end of it!  So, perhaps a fluency lesson would also provide D with the skills he needs to improve comprehension.

Tech Post 2: iPod what?

I don't know if I'm alone here, but somehow I don't think so.  Unfortunately, I have not been able to use my iPod Touch in any classroom except for the purpose of recording conferences and interviews.  For these purposes, the iPod Touch has been a great assistant.  It's fantastic to actually get to engage in talking with students rather than being bogged down by note-taking. 

Nevertheless, I have not used any of the apps or other features with my students.  I think the main reason behind this is that with only 1 iPod Touch available, it is impossible to have our students engage in this style of learning.  I could demonstrate with it under the doc cam, but a computer does the same job, only better.

Consequently, my opinion about the use of iPod Touches to assist with differentiated instruction has not changed since my first post on this topic.  I still think that there is a lot of potential for these devices to assist in learning; but, because I haven't seen this technology in action, I have not developed any more specific ideas or questions about using iPod Touches for differentiated instruction.

One theoretical question that I have is about where the line is drawn between complex instruction and differentiated instruction when it comes to utilizing technology.  Is the benefit of the personalized instruction provided by each student having an iPod Touch worth the minimization of group work that results?  Would a different tool (such as a laptop, with a bigger screen) lend itself better to fostering social and emotional learning?  This makes me wonder whether or not iPod Touches are the best available technological tool for enhancing classroom learning.

Math Week 5: Can you prove it?

"It's a square because it looks like one" suddenly became an unsatisfactory answer!  How awesome!  This is when we math teachers insert a tiny "not drawn to scale" under each diagram and thereby stretch our students' mathematical thinking and visual perception.  In math class this week, I learned about the developmental stages of geometry learning.  I find it intriguing how there is a correlation between these stages and the ability for us to perceive geometry by trusting our logic and creativity rather than our eyes.

I am still left wondering about the use of the paper folding to build a conceptual knowledge of proofs.  In order to verbally express your reasoning, you need to have a certain vocabulary.  For example, we used the terms "linear pair", "alternate interior angles", and "perpendicular bisector".  So, would a lesson like the paper folding fall sequentially after learning terms and before formal proofs?  Would this type of activity be an effective learning tool for understanding terminology?  I'm curious about the different modifications for this paper folding activity so that I can use this hands-on, engaging activity to assist in the learning of different geometry concepts.

Now that I know there are distinct stages of developing geometric reasoning, it has become a lot clearer to me why some students seem to grasp geometry concepts quickly and can manipulate them while others are seemingly confined by the strict parameters of what they see.   I will utilize this new knowledge to help my students sequentially build their understanding of geometry.

Math Week 4: Mira, Mira, on the Wall!

In math this week, I learned what miras are!  No matter how exciting these tools were, though, it was incredibly sad to be reminded of all the math tools are stuck in closets, not being used.  And thus, I made a mental note and vow to always search the school for stuff before purchasing it.  I guess you could call this my personal school budget plan.

I also learned today that purposeful group work is very different than just working with other people.  Group work needs to be engaging and be a beneficial learning experience for all the group members to make it worthwhile.  Group roles are an excellent way to accomplish this.  Yet, the thing to remember is that group roles need to be monitored and upheld by the teacher, otherwise the students won't adhere to them. 

Even though a specific structure to group work might allow for greater learning opportunities, some part of me wants to think that all group work has benefit to it. Not only does it help kids learn how to interact with another person or people, it also then teaches accountability (when you do activities such as partner sharing and being held accountable for your group's work).  So, do all group work activities truly need to meet the "group worthy" criteria?  Or can group work sometimes just be about teaching students how to interact and resolve disputes?

Literacy Project 1: Writing Analysis and Lesson Plan

Overview

D (abbreviated for confidentiality) is an enthusiastic 4^th grade learner.  His passions are math and science, although he says he also enjoys reading mysteries and other stories that make him think.  An analysis of D’s writing and spelling follows.

Analysis

Meaning of the piece:

In the piece of writing I analyzed, the author is responding to a prompt about a talking pet.  Even though this writing was initiated via a prompt, it would appear from D’s enthusiasm that he enjoyed writing this piece.  D chose to write about a talking parrot because he loves birds.  He wanted to convey to the reader a sense of adventure and suspense through his development of the story and share his love and amazement of birds with the reader.

Six Traits:

Ideas:

In this piece of writing, the topic is fairly broad since it is a “bed-to-bed” story, spanning the length of one day.  However, you can definitely tell that the author has chosen this duration because of the nature of the prompt (one day with a talking pet).  The author did stay with his topic and only included main points and details related to his interactions with the talking pet, which makes the story focused, even though it deals with a broad topic. 

The author also does a consistent job providing details to support his main ideas.  For example, the author writes that he was walking too slowly for his pet parrot, so the parrot picked him up and flew him around to save time.  The rest of the story is filled with details chronicling the day’s adventures.

Organization:

One of D’s strongest elements in his writing is his organization.  He has a clear and provocative opening paragraph and a paragraph to conclude the story.  From the first sentence, the reader wants to hear what happens next.  D writes, “I was sleeping all cozy and warm, when I heard a noise under my bed.”  D takes his time to setup the characters and setting of the story in the first paragraph before transitioning to more action-oriented scenes in the body of the story. 

In conjunction with a solid introduction and conclusion, D makes an appropriate choice in paragraph breaks and structure to suit the purpose and audience of the story.  He uses transition words such as “finally” to make paragraph transitions smooth and chose to break the paragraphs at points related to shifts in time and content.  D’s organization makes the story flow well and very understandable.

Voice:

D takes a risk with his choice of voice, using a very conversation tone with phrases that D uses in his own daily conversations.  Through reading this piece, the reader can tell when D is annoyed, “I said no way Jose!” and when the D is yelling, “I said PUT ME DOWN”.  Through his voice, D reflects himself in his writing.  D chose to write in the first-person so that the reader can follow the author’s adventures with his talking parrot throughout the whole day. 

D provides personifying details about the parrot and includes many action-oriented descriptions that make the reader become engaged in the story.  D stays focused on the topic, chooses interesting details to include, and writes with his own natural and engaging language. 

Word Choice

Looking at D’s 1^st draft compared with his final draft, a reader can see that D attempted to insert colorful language to provide more vivid descriptions in the story.  However, although the word choice it at the right level, it sometimes seems out of place.  For example, D writes, “There we were at the cheerful kitchen.”  D added in “cheerful” to help us visualize the appearance of the kitchen, but this descriptive word seems out of place and perhaps unnecessary to enhance the story. 

In conjunction with D’s choice of voice, the word choice remains very basic and is full of conversational phrases.  For example, D writes, “He just drop me on my bottom hard”.  There are only a few fine moments where the author’s word choice enhances the story.  For instance, “My parrot refused me” gives the reader a clear image of the parrot being demanding and stubborn.

Sentence Fluency

While D consistently includes smooth transitions between paragraphs (see Organization) he does not always include transitional words that would assist the reader in knowing how sentences interrelate.  When reading his own story out loud, D’s verbal expressions helped the listener understand the sequence and flow of the story.  However, upon reading the story without D, the reader questions how some sentences relate.  For example, D writes, “When he [the parrot] was carring me I saw the stairs.  I thought there was no way he’d carry me down the stairs, and he didn’t.”  The sequence of events in this quote make sense.  However, the choice of sentence breaks and lack of transitions makes the meaning of the sentences difficult to understand.

In this narrative, D often uses the word “I” to begin sentences since the story is about all of the actions and feelings that he is experiencing.  However, he does use the occasional transition word such as “So”, “Then”, and “Finally”.   Through the revision process, it appears that D inserted some varied vocabulary to begin sentences, although most of his sentence beginnings remain repetitive.

Conventions

In this writing piece, D struggles with many errors in grammar that affect the meaning of his story.  Most commonly, D switches between the past and present tense and inserts unnecessary or omits necessary verbs for creation of a complete sentence.  Here are a few writing samples that demonstrate these errors:

“I scream!  I was afraid he is going to drop me…” 

“I decided that better get him some food soon.”  

“We headed toward the kitchen and that’s was my day began the day all animals could talk.”

From talking with D, it sounds like leaving off the “ed” at the end of words happens in speech and reading.  Thus, this error is carried into his writing.  In analyzing the symbiotic relationship between speech and written language, Cambourne writes about how the natural development of speech translates into written language (p. 28).  Thus, this relationship in literacy development may be the reason behind D’s grammatical errors.  If these verb errors are stemming from D’s speech, making the necessary changes so that his writing is grammatically correct will be very challenging.

In terms of punctuation, D consistently shows appropriate beginning and ending punctuation.  However, his comma and quotation usage are lacking or incorrect.  This sometimes affects the fluency and interpretation for a reader.  For example, D writes, “He [my parrot] was jumping on my face and said, get up.  It’s time for breakfast.”  In this example, the lack of quotation marks makes it hard to distinguish narration from dialogue.  However, with moderate editing, this text would read smoothly and be ready for publication.

Ownership of writing:

D shows that he has ownership of his writing in a few different ways.  First, he chose his own details to suit his interests, like a parrot and flying.  Also, he used his own voice and expressions to show that it was really him in the story.  D was proud to share his story with me as well as show me the illustration that he did to accompany his story.  He is very motivated to do his work, which includes spelling, writing, reading, and other subjects and is proud of his accomplishments, even if he doesn’t receive very high grades.  Finally, D is very willing to do a lot of revising before writing his final draft, which shows that he is able and willing to manipulate his work with a purpose.

Spelling Development:

It seems that D does not hear or speak words accurately all of the time.  It seems that this is impacting his spelling mostly by the fact that he often leaves off or unnecessarily adds suffixes to words (most frequently “ed”).  For example, he spelled fortunate “forchened” and confident “confinded”.  In his writing piece, he wrote “I was freak out!” even though the expression either requires an “ed” or “ing” at the end of “freak”.  Because this error remained unchanged between his rough and final drafts, I don’t think D knows that he is saying the expression incorrectly.  This kind of mispronunciation is extending itself to both spelling and reading other words as well as resulting in past- and present-tense grammatical issues. Templeton writes about spelling and the benefit of learning patterns within words to assist in spelling development (p. 104).  One skill that would help D improve his spelling is listening and looking for the patterns (specifically, ending patterns) in words.

Works Cited

Cambourne, B. (1988). An alternate view of learning. In The whole story: Natural learning and the acquisition of literacy in the classroom (pp. 28-42). New York, NY: Ashton Scholastic.

Templeton, S. & Morris, D. (1999). Questions teachers ask about spelling. Reading Research Quarterly, 34(1), 102-112.

Lesson Plan

Objective: Student will become aware of different verb tenses and understand how to write in the past and present tense.

Standards: EALR 3.3.5: Knows and applies writing conventions appropriate for the grade level.  For example, Grade 3: Uses consistent verb tense.

Materials: Pencil and Paper

Instructional Strategies:

Explain the objective of the lesson to the student.

Provide 2 sample paragraphs demonstrating use of the past and present tense when writing a narrative. 

Talk with the student about the differences they notice between the two passages.  Lead the student to some of the key patterns (such as “is” vs. “was” and suffixes such as “ing” and “ed”).

When the student is comfortable recognizing some of these patterns, direct the student to write a paragraph about something he did yesterday. 

Review the paragraph with the student, first having the student read the paragraph aloud, then asking questions to guide the student to make any revisions to verb tense so that everything is consistently in the past-tense. 

When the student is satisfied with his consistent use of the past-tense in his paragraph, direct him to write the same paragraph in the present-tense (as though he were doing yesterday’s activities right now).

Review the paragraph with the student, first having him read the paragraph aloud, then asking questions to guide him to make any necessary revisions.

Bring the lesson to a close by having the student verbally summarize for you the key points from today’s lesson.  Instruct the student to keep these two paragraphs as examples of the past and present tense for his reference.

Assessment: When reviewing this student’s future writing (formally or informally), look for consistent and proper use of the past or present tense. 

Is your kitchen making you fat?

I know what you (might) be thinking!  This is a load of you-know-what.  Well, I thought the same thing, until I watched it.  And, in light of us being teachers, and observing lots of teachers who eat on-the-run or don't eat or have health problems, I think this video is rather apropos.  So, here's to your health and making good eating choices.  Bon appetit!

Tech Post 1: Differentiated Technology

iPod Touches are a fantastic, relatively-affordable way to provide differentiated instruction.  Below are a few ways that I see iPod Touches could be used to help some of the students in my classroom.

For students who are advanced readers, the iPod Touch provides access to an Amazon Kindle app, on which all of the copyright-free classic literature is available for free!  Students will be able to access more literature through this electronic medium than is available at their school (and even local) library.  Also, the iPod Touch has a free app that accesses Dictionary.com, allowing students to look up words quickly and build their vocabulary as they read.  Having access to this literacy-development tool would be purely beneficial as long as students are reading on the iPod Touch at appropriate times during the day.

Many students need a starting place for organizing their responsibilities.  Thus, instead of the soon-to-be "old-fashioned" paper planner, students could use the organizer on the iPod Touch.  They have the options of creating voice memos for themselves, typing brief reminder notes, or using the calendar feature.  For this to be truly effective, the students would need to be issued their own iPod Touch for the year and be responsible for taking it home, charging it, and treating it with care.   The iPod touch offers a great opportunity for students to develop their organization and accountability skills.

The iPod Touch has many free apps that offer reference material, such as math formulas and explanations for how they're used, how they originated, and examples of the formula in action.  These reference materials could be used to help support students who thrive with visual aids.  Also, this kind of iPod Touch tool could help students learn how to make the most of their resources and develop independent learning skills.

The iPod Touch could also be used as a keyboard/typing instrument.  Many students who have physical trouble writing cannot get the assistance they need to scribe, and they do not have access to a computer to type.  If students had iPod Touches, they could use these to assist them in completing assignments which would otherwise require hand-written work.  One disadvantage of using the iPod Touches for this purpose would be that typing is difficult and there is no Word document format for the iPod Touch.  However, using this tool to help students complete work would probably still be more beneficial than having students sit around when the rest of the class is doing hand-written work.

Some students prefer to not speak in front of the whole class to offer their opinion.  Thus, having iPod Touches would provide a way for students to participate in surveys and other real-time problem-solving without having to speak in front of the class.  Ideally, other strategies would be used to help all students feel comfortable sharing their ideas with the class.  However, this is a great interim activity that would help all students participate by using a different medium.

Math Week 2: Manipulating Questions ≠ Questions + Manipulatives

Today, I learned all about the art of questioning.  As math teachers, we should never give answers or tell students that they're right...they'll shut off their brains.  Make students work together to find solutions.  Give them the tools they need to problem-solve and check their own work so that they build confidence in math ability. 

What else did we explore?  Make use of manipulatives whenever possible!  We learn every other subject by starting with concrete information and examples and move to abstract thoughts and concepts.  Why do we teach math the other way around?  So, by using manipulatives as tangible, concrete examples, we can begin to shift math teaching to align with the way the human brain is accustomed to learning.

I'm still wondering about a few theories.  If students rely on the use of manipulatives on tests, then how will they solve the problems on standardized tests?  How do we build a community that doesn't judge a student because they still need to use manipulatives to solve problems?  Also, we can be very "sly" to use tricks like think-pair-share so that students are held accountable for their group members' success, but that still doesn't negate the social issues that middle school students are going through and the feeling that students are at differing abilities if they use different math problem-solving strategies.

Math Week 1: ...Say what?

I love hearing how different people solve math problems!  It always seems that no matter how many different ways you come up with to solve a problem, there are always other people who have figured it out using another method.  In today's class, I learned that we need to build a supportive, patient classroom community so that all of our students are comfortable sharing their mathematical thinking.

I am still questioning the use of cards/randomness to call on students to share their ideas.  I liked the way that this tool was used at the beginning of each activity, to break the ice, and then substituted with "does anyone have a different way of solving this?" so that we didn't see too many repeat answers.  Nevertheless, even though our classroom environment is very supportive, there still seems to be nervousness when it comes to sharing our mathematical thinking with our classmates.

The activities that we did today were extremely engaging and inspired exploration and learning.  This type of activity will be a great tool to use in our future math classrooms.  The bonus is that these activities truly cover all of the state standards!