Math Reflection

Wowie Zowie, do I have a lot of questions.  I just looked over my blog posts from the last quarter, and I have far more questions about mathematics teaching methods now than I did at the beginning of the quarter.  I must be really good at math.

Another exciting result of my blog review's metacognitive process is that I now feel much more empowered in my ability to provide more equitable educative experiences for all students.  I see how group-worthy tasks in which students have designated roles and are assigned randomly creates access to opportunities for understanding for everyone.  I even know how to make a mathematical group-worthy task.

Technology's rising role in equitable education and the complete change of my conception of mathematics teaching technology has catalyzed, makes me really excited to be involved with technology and math education.  I am so looking forward to having all students bring in some data of something from their outside-of-school lives and understand it better mathematically.  I am equally excited to help guide every student in understanding math better to take back to use throughout their outside-of-school lives.

Finally, I am going to use manipulatives in my main placement.  They aren't used right now, but I have seen and personally experienced far too much in terms of manipulatives to ever convince me that they aren't invaluable tools for every single student to use at least sometimes.

I know the new mathematical teaching understandings I came to this quarter will not only help me be a better teacher, but they have definitely, definitely helped me become a better mathematics learner, a skill I want to pass on to everyone I meet.

Today's Math I.D.s for Tomorrow

What struck me most about the math reading and class this week was the connection to that and my experiences in 7th grade math classes in the fall, and the deeper understanding I have now about how much of math teaching is not actually specifically teaching math facts.  On top of that, how much not only isn't math facts, but is actually life skills.  Additionally, all of it, the math facts, deeper mathematical reasoning skills, and life skills all are learned through every individual's self-identity and emotional relationships with the subject.  Through this "healthier" (Leatham & Hill, p. 228) math class consciousness I myself even felt more comfortable and empowered to do the geometry puzzles and measurement algebra work with my group. 

As a result of this, I will make self-reflection and discussions of what mathematics really is about, how it really is relevant in students' daily lives, and even personal emotions, a major part of any math instruction I will do.  

In thinking about what mathematics students actually need to know and how my mind has pretty much been blown away in class this quarter thinking about all the potential of the math tools that could easily be in everyone's hand soon,  like motion and temperature probes for smart phones, measuring devices, the WolframAlpha site that seems to have every-everything, I have some questions.  Mostly my questions have to do with, what really are students going to need to be taught in class, what do they need guided experiences with in class, what are they being taught that is a waste of their 21st Century time?  And not only what are they going to need to be taught that I don't even know I don't know, but what of the same stuff did I learn  that they should learn even better than me.  I guess this last question is, if I had grown up with the technology that exists today, would I be using the mathematical knowledge I did gain in school much more regularly or effectively?  What in my life would be different if I was using mathematical thinking more regularly or effectively?  I think there is a solid chance I would really understand my world at a deeper level than I do already.  As an educator, I will do my best through professional development and any other means I can find, keep my own learning progressing with an awareness of what is possible in the present looking towards the future.

Differentiated iTouch Learning - Updated March Thinking

After this past quarter of utilizing the iTouch, I have greatly benefited from and seen its potential as an effective, yet basic tool.  Not that it isn't fully capable of being used for incredibly complex teaching and learning, it just took me a while to realize it's more basic yet useful potentials.  For example, it has been immensely helpful as a recording device during lessons, guided reading, and informal reading conferences, freeing my attention to other more important events to be able to return to the recording for analysis later.  Although this use of the iTouch does not directly apply to the students I was working with, I can clearly see how a student could use it in the same way.  Especially a student whose attention is more hindered than others in the act of writing notes, for example.  They could potentially benefit like me in having the ability to focus attention on other immediate matters and have the iTouch audio recording to return to at a later time. 

During the quarter I have developed a deeper understanding for the power of a small, hand-held skill builder, such as an iTouch with educational game apps.  From my peers I have heard examples of how an iTouch meant the difference between a struggling student having the opportunity to positively work on basic skills to catch-up to their peers, and a struggling student silently slipping more permenantly behind during in-class inaction. 

Through the mathematical methods course this quarter I have seen exciting learning experience potentials that can be realized through the use of iTouches, and how empowering it will be for every student to have this potential in their own hands.  I am really excited to see how it is going to integrate classroom learning with students' outside, real, lives, especially because they will be able to take this learning tool with them into the outside world, and of course, bring the outside world through hand-held learning tools.  The potential to make all learning authentic, relevant and meaningful to all students is fantastic.  I am very curious what parents and community members will have to say about the use of iTouches as learning tools as well as any learning ideas they have for students.

At the beginning of this quarter I think I was in search of much more earth-shattering ideas in using an iTouch and I underestimated the power of an iTouch as a basic, yet effective tool when used thoughtfully to enhance and encourage critical, equitable, pedagogy.  The future with them as learning tools is sure exciting.

Dynamic Mathematical Thinkers and Technology

After today's class I am really thinking about how I can't know what mathematics students are actually going to need to know in their lives.  But I do know I can help them practice mathematical thinking, that I can help this thinking be transferable across the great technologies that are constantly coming about, and help give them the self-confidence that they are mathematical thinkers that can and should use these new technologies for the betterment of their own lives and others'.  For example, if they can use a variety of programs to make a histogram, understand and analyze a histogram in meaningful ways, then making a perfect histogram by hand won't be of the utmost importance.

If in the not-so-distant future everyone has cell-phone probes to understand the world around them, will we be assessing if their understandings are logical?  With more technology, will mathematics assessments and materials be less hands-on with the real world or just more integrated?  For example, we worked with the shapes on a screen this afternoon, this is hands-on, right?  Collecting our own data and analyzing it with technology, that makes it all very "real-world" for students, right?   

As for implications for classroom practice, my thoughts from class today will help me keep my own reality-check for what students really could find useful in knowing and being able to do in their lives.  How will I keep perceived mathematical limitations off and away from every individual - probably by guiding them through mathematical practices with dynamic technology that keeps them as mathematically dynamic as the technology that will hopefully always be viewed as individuals' personal tools.

Mathematics with the Right, Slight Struggle

From class this past Monday, I learned that by doing something like posting a review sheet online for everyone to do two problems and comment on one another's answers, students are challenged in a way that makes them more responsible to one another and especially responsible for their own learning.  It also adds a slight element of struggle, which I've noticed the right amount of "slight" struggle can really help learning.  I learned there are great viable options for math journaling; there are super fun ways to incorporate learning about measurement, mean, and range; and, there can be quite high-order mathematical thinking involved when working with those elements and tanagrams.

A question I have right now is if the group task we did on Monday, making our tanagram giraffes twice as big, was a group-worthy task, right?  Because it helped a whole lot to have more than one of us working on it and because mathematical tasks involving changing size like that do come up in the real world? 

Implications for my classroom would be that I see the direct relevance of using math journaling, tanagrams, and fun activities, all in every grade, all advancing learning to deeper levels.  I think this seems so plausible because they all help me with my mathematical thinking as an adult and can see how they all help in the 1st/2nd grade classroom I'm in right now.  Even in this class of young students they have something called Math Notebooks where they respond reflectively to written mathematical prompts like journals where they explain their thinking.

Differentiated iTouch Learning - February Updated Thoughts

 

My thinking now is that iTouches are a fantastic opportunity for students to take responsibility for their own learning and construct it in their own ways to make it most meaningful to them.  I made a few more realizations and thought connections over the last few weeks.  One is that some students in my main placement (especially the ones I described in my last Tech post that could possibly benefit from iTouch differentiation) are disturbingly low on classroom learning motivation right now.  Another observation has been that when they get the opportunity to unleash their imaginations for a moment they are as engaged in the classroom learning environment as the others. 

If an iTouch was in every students’ hands for even a little time every day in connection with some classroom learning subject matter, they could be guided and encouraged to use a little freedom and creativity to make a digital story or interactive poster or timeline, anything, as partners, small groups, or the class.  There wouldn’t be a power struggle between the students and the teacher because the teacher would be putting concrete power in their hands. 

I got pretty excited over this BBC news article: Genes ‘play key role in classroom performance.  It explains that learning environment really only counts for about half of student achievement, the rest is due to our individual genes.  This strengthens the potential value of personalizing education with tools like interactive information technology.

New questions I have are what are the potential pitfalls that I am still unaware of?  I understand there are safety issues and the technology must be supported by effective teaching, that it must be used to support a cohesive community not separate everyone, but what else? 

What if not every student can hold an iTouch in his or her hands?

 Would it be a detriment to the students if they have one year with a teacher that incorporates tons of personalized learning with interactive technology, followed by years without any (could the experience discourage them in later more traditional-technology-lacking-unpersonalized education)?

Moving from Assessment to Instruction

What I have learned about my buddies needs, abilities, and interests:

 

Word identification in Isolation and in Context (Accuracy and Strategies)

Based on the Spelling Inventory done on 1/11/11, D.  was strong in using consonants, short vowels, and diagraphs and blends.  He could use some more support in his long and other vowel patters as well as with his affixes. 

Based on the oral reading of Amelia Earhart on 2/1/11,  D.’s miscue analysis shows that most of his errors have to do with word endings and vowel letter patterns, and although he is still able to self-correct about half of the time, his reading comprehension is probably limited because of these miscues considering that they change the meaning of the words about 80% of the time.

Fluency (Rate and Phrasing)

Based on the oral reading of Amelia Earhart on 2/1/11, D.’s fluency rate (as correct words/minute) was about 45 wpm, a very low percentile for a fourth grader according to the 2006 Hasbrouck & Tindal Oral Reading Fluency Data. Based on the NAEP Oral Reading Fluency Scale, D. would be somewhere between a Level 1 and a Level 2, because although much of his reading is word-by-word there are frequent three-and four-word groupings, but these are mostly awkward and unrelated to larger context of the sentence.  His reading does seem labored and difficult.  His reading pattern seems like a roller coaster in that when he first hits a word he is unsure about he slowly sounds it out and moves ahead very slowly, careful of each word, and then will suddenly zoom easily, as if on the downhill part of a roller coaster through the end of a long sentence and ahead into the next until he hits the bottom of another word that catches him and slows him almost to a stop, like the bottom of a roller coaster hill.  He is not yet fluent enough to read with expression.

Vocabulary (addressed in Comprehension below)

Comprehension (Content and Strategies)

After the oral reading of Amelia Earhart on 2/1/11, when asked to retell the story D. gave little information, but what he did say were big ideas in the story, that Earhart wanted to fly across the ocean in a time period around WWI when it was dangerous and that she had been a passenger on a plane which made her want to fly a plane.  He also retold how she was flying across the Pacific Ocean when she disappeared. 

During the cued recall of the story, D. was at the frustration level, only answering 4 questions correctly, though he answered more Implicit right (3) than Explicit (1), making me wonder if as he was slowly reading he was keeping up his comprehension to transfer to meaning that wasn’t verbal-English based.

D.’s comprehension is quite impressive considering his rating is at a frustration level.  I think that maybe because stories are so much of a positive part of his life, that he enjoys writing stories on his own in English at home and reads stories on his own in his first language at home, he is well practiced in linking text and meaning, he just gets slowed down in recognizing English text.  I think that because English and culture is relatively new for him it brings down his fluency and accuracy drastically.  He is often unfamiliar with vocabulary, not that he doesn’t understand the ideas he just doesn’t recognize the English names for them.  So when he does slowly but correctly identifies an idea he is able to translate or visualize it for himself so quickly that he has trouble going backwards in his knowledge to answer explicit retelling questions, but has made the meaning so strong for himself, does much better at implicit retelling.

Ownership including enjoyment

After the oral reading of “Amelia Earhart” on 2/1/11, and after the retelling and questioning, I asked D. if he thought reading this text was easy, hard, or just right.  He said, “Just right, I can read almost all the words.”  We had also read the “Early Railroads” text without time for the retelling, and he read it with about the same fluency.  For the Amelia Earhart text he was completely unfamiliar with the American history of women’s rights which I believe caused a little of his difficulty in understanding the full meaning of the story, and although he was more familiar with railroads he was completely unfamiliar with much of the vocabulary, such as “carts”, “engine”, “Thumb” as a name, and “locomotives”, making this story very difficult to comprehend.  I think if someone had translated or explained so he could translate for himself what much of the vocabulary was he would have had a much easier time comprehending this text.

When he was predicting about the text of Amelia Earhart he included that he had knowledge about the dangers of flying small planes from a video game he has, that based on the clues of the text he knew it had to do with someone using a plane to go on an adventure, and added that if he were to write a story about this he would have them go into a jungle, get lost, and when they make it back to the airport they wouldn’t be able to find their plane because some other people came to the airport and took it on an adventure of their own.  This leads me to believe that D. is immediately ready to make personal connections to the text and enjoys thinking about stories.  During the retelling, when he was describing what might have happened to Amelia Earhart he showed me with his hands what a crashing plane does. 

Between reading this story, his interest in thinking of his own story, his enjoyment of reading and writing stories on his own at home, I believe that D. really enjoys stories in general, especially when they include a main human character and then a surrounding out-of the ordinary story, such as animals or monsters interacting with the main character in unusual ways, or adventures. 

Context:  The informal interviews happened at D.’s desk in the classroom.  While there were others working and talking around us D. usually seemed to focus well on what just we were doing, although at the end of his oral reading he did stop to look at and listen to a student reading quite fluently behind us.  D. has started out every session seeming reserved and quiet, so I’ll start by chatting a moment about how much I enjoyed his written story I saw or about how I was thinking the past week about the interesting things he was telling me last time.  After a little bit of this I usually have asked an easy, open-ended question to him to hopefully help him get started talking with me more easily.  He is hesitant but positively tries very hard and doesn’t seem to let difficulty bring him down at all.  Even though the oral reading ratings had him at the frustration levels he always wanted to continue and never at least let on to me that he was frustrated.

AS A RESULT OF THIS KNOWLEDGE: 

Objectives I am considering for a lesson would be to focus on the vocabulary comprehension skill of using context and structural cues to gain new word meanings.  I wonder if this could be considered a “monitoring & fix-up strategy” and if it could be taught in a similar way.  For example, possibly I could find a text that has to do with an animal or adventure like he would enjoy, and we could go through a targeted small group instruction lesson.  For this I would explicitly explain we would be practicing understanding the text better by focusing on figuring out new vocabulary from our knowledge and the rest of the text, then I could model, he could read a little and then try with me there helping as guided practice, and then he could try on his own. 

If I were to do a lesson like this I would need a text he would be interested in at his instructional level, in which there would be unfamiliar vocabulary but that could be understood from previous knowledge and surrounding text.  I would need to be prepared with the vocabulary I thought he would have trouble with and ways to help scaffold this practice.

I am very open to other ideas of lessons though, and am just feeling like it is a little hard to decide on exactly what specific objectives would be most helpful for D. at this time because on page 51 of Reading Essentials Routman explains that for language acquisition students “need to hear, speak, practice, and interact with the rich language of stories and literacy”.  The broadness of this statement leaves me somewhat unsure on choosing specifics for my specific buddy when it seems that just about everything would be helpful for him.