What did I learn in math class this winter quarter?

I learned a ton about math this winter quarter! Here is a list of big ideas that stuck out to me:

• It is important for students to explore and experience math. This involves moving from concrete to abstract.  Manipulatives can be a great tool to for students to experience and understand concrete understandings first.  A deeper understanding and connections overall in concepts and formulas can be better grasped this way.    

• Manipulatives we used in class that I had not used before included: probes and a graphing calculator (I have to find my graphing calculator - good to know it's not obsolete), beans, algebra tiles, flying frogs (cotton ball and paper clip), poker chips, a square piece of paper, and tangrams.  I also learned about virtual manipulatives and http://www.nctm.org is a good resource for that.  I also saw how a website like gapminder.org  can allow students to see math as a part of their history and lives.  Understanding graphs stood out with using gapminder and a gallery walk interpreting other group's graphs. 

• Technology is changing the way we think about math.  Teachers need to keep current and experiment themselves with how to integrate society trends into the classroom.  With website like http://www.wolframalpha.com/, this changes the approach to teaching math.  Students being able to verbally explain their thinking will be more a part of the assessment of student understanding.

• There are different ways to be good at math.  It is important to have multiple entry points for students to be able to experience success in math.  Creative writing, using tangrams, and logical thinking are some different areas.  Math has a strong emotional connection and it is important to know your students and try to decrease the math status is a classroom.

• I learned it is not only important for math teachers to explain math well but it possibly more important for the students to “explain things so well that they can be understood” if students were truly going to learn mathematics. This reminds me of what was said in class about how kids will learn more from each other than they will from you- the teacher. Asking good question that produce thoughtful thinking and fruitful answers is skill needed for teachers.  A teacher's role now is more of a facilitator.

• Group work!  Creating a culture of productive group work in math was added to my teacher tools.  Participation is not optional.  I learned the strategy of having group members responsible for each other's learning.  I like this idea of teamwork.

Lastly, in addition to learning about math this quarter, this class was great because of the discussions we had about the human relations aspect of teaching as well.  "At the heart of teaching is care."  Showing kids you care and teaching kids to care for one another will make the world better in the future.  (I shared this theme with my first graders and even they looked at me with a funny first grade glare, I know they felt the encouragement).  I like the reminder about the engagement part of teaching and how teachers often have to be actors and actresses when we teach.  I am appreciative of the experiences shared about the realities of being a new teacher, diversity and judgement, and the humor and frankness that was also incorporated to our math class.    

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Scratching the Surface with iPod Touch Project

My views using the iPod Touch to support differentiated instruction is still the same from my last update. I used the Touch for Student C mainly because he is not able to do the same work the whole class is doing. I have continued using the Touch for math with student C because when he comes back to class, there is still ten to fifteen minutes left during math and the rest of class is usually doing independent work. With the Touch, Student C seems to be always engaged using either a numbers recognition app or simple arithmetic. I came to find out, he uses one at home too.

Finding the time to incorporate and experiment with the Touch was the hard part. After reading some of my comments from my last update- yes, I think using the Touch for transitions would be a great idea. With my class the only concern I have are the students who I foresee rushing through certain assignments just to use the Touch. This has been the case for some students when using their Netbooks. They rush through writing or math so they can go onto Dreambox or Coolmath.com. Even with their own personal laptop to use, the students are now preferring to use the large computers. There was an interesting discussion in class about the small size of the Touch and perhaps an iPad maybe would be of better tool to use for instruction.

For one of the my classes earned parties, the class chose to have an electronic games party. Students brought in their little Nintendo DSi and their parent iTouches. For students that could not bring any, they were able to use their Netbooks. There were about four iTouches that the students brought from home. This made me think about another discussion in class about how technology is a big and growing part of our society, and with this being part of culture, how then are we going to integrate learning with technology? Let me tell you, every single one of the students were on task that day! It was also interesting to be able to read an email from one of the parents in concern with the electronic handhelds. This parent did not like the idea of students bringing in high end toys in a public school because she thought it promoted materialism and student's that did not have these types of games would feel left out. This added another food for thought with the use of technology- getting parents on board.

Personally, I am all for the use of technology and integrating it into the curriculum. I like the way the iTouch adds another entry point to learning. I think I was only able to scratch the surface with my ideas on how to incorporate the iTouch for different students. Someone should write a textbook about this. It is also a great teacher tool to record student interviews, record how students read and assess their fluency rate, and record a lesson given. All of these can be used to better assess students, share with parents, and also help a teacher assess themselves.

Emotions in Math and www.wolframalpha.com

1. What did I learn?
From the article reading this week, I learned again how children and adults have strong emotions when it comes to math. It is interesting to see that just because a child is good at math still does not necessarily mean that he/she likes math. It is also encouraging to read that a student who may not be good at math but continues with it, still could end up in a career using math all the time. It is important as teachers to know your students and try to find an area in math where a child can succeed to build confidence. It was also interesting in the article, to see where other children perceive their peers to be good at math and compare that with the reality of their internal perception.

Another interesting discussion we had in class involved remembering that "children have a right to fail," and we also talked about how teaching is also about human relationships.

I also learned about www.wolframalpha.com. Wow! And what an impact this type of website has for teaching. It changes the idea of handing out worksheets for homework.

2. What do I have questions about?
I'm wondering how to incorporate homework and assignments when there are websites, like wolframalpha.com out there for students to get answers and also the "show your work" part can be found.

3. What are the implications for classroom practice?
Again, technology and the web is impacting classroom learning. Now, teachers can say "watch this or research how to solve this" and then lets discuss what we found. Explain your thoughts and thinking will become more involved in assessment if answers to math problems can be easily found on the web.

Creating Opportunities to Learn from Each Other

I recently watched Adora Svitak again on TEDTalk and I am reminded of how fascinating it is to hear kids explain their learning and understanding. I usually learn something new when I ask students to explain their thinking? As I start to teach more lessons, students continually amaze me with their creativity, intuition, and instinct. Adora is making the point that learning should be reciprocated in the classroom. Students are the leaders of tomorrow and it is important to provide opportunities for students to lead and succeed. High expectations is a big idea that has presented itself over and over in this program and I agree. I am thinking of ways to create more leadership opportunities in the classroom. I am also amazed at how much students love to help, show, and teach one another. When a student has a light bulb moment, there needs to be a way a student can share in their discovery and celebrate in their learning with others and/or at the same time acknowledging the classmate who taught them. Learning is reciprocated and appreciated in each way.

Technology and Math

What did I learn?
This week I re-learned about the Zone of Proximal Development by Vygotsky. I was reminded about the tension in learning and our goal as teachers is to reach the "zone" where students are not bored nor are they filled with too much anxiety where they feel they cannot succeed. I am also reminded about the engagement part of teaching and how we often have to be actors and actresses when we teach. At the heart of teaching is care. Showing kids you care and teaching kids to care for one another will make the world better in the future. Good food for thought for Vaelentine's Day.

I also learned about about dynamic geometry in the computer lab. We went to the computer lab used Geometer's Sketchpad and Fathom. Sketchpad is quite fun and a very engaging way to understanding properties of quadrilaterals. Fathom was a great technology tool for students to not only graph but also understand the meaning. The big idea today was looking at how technology is shanging the way we think about math. Just a seed was planted today on how technology allows children to actually EXPERIENCE math. (Just like how video games can feel more life like, engaging, and now, get up out of your seat and actual dance to the music, this is where math is headed with technology.)

2. What do I have questions about?
I just wonder about the future and what it will look like with the iPods, smart phones, and all the new computer technology. I think it is exciting and great for teaching and learning. But, I also wonder about about the allocation of resources. Some kids will be exposed to the new technology and some will not. How will the new technology affect the achievement gap?

3. What are the implications for classroom practice?
Teachers wil need to keep up with technology and experiment with what's out there to find the best ways to use it in the classroom.

Measurement, Tangrams, and Multiple Entry Points

1. What did I learn?
I learned about activities involving measurement and also tangrams. Measurements do not have to be boring. "Flying frogs" with a cotton ball and paper clip is a way better lesson to discuss measurement, incorporate some physics, mean, and range. We also played with tangrams and did an activity called "The Giraffe." While I discovered being spatially intelligent may not be one of my strengths, I had a lot of fun trying to figure it out. Multiple entry points and showing kids there are lots of ways to be good at math was a big idea we discussed again. Maybe a child cannot understand abstract algebraic equations but they can do a tangram puzzle easily. Confidence and success in math is wonderful since math and emotions are often connected. Incorporating writing and creativity is another entry point for students to display their strength in math. My table group added how you could also make a poem, write a song or rap, or even a visual representation to make math concepts more interesting for students.

2. What do I have questions about?
How do you apply a tangram activity to real life and the students' lives?

3. What are the implications for classroom practice?
Math does not have to be straight out of a textbook and just involve worksheets. It could be fun and interactive. Multiple entry points should be a conscious effort when designing lesson plans. Teachers will need to stay current with technology to understand how blogs and/or voicethread can be incorporated in the classroom learning. This may take some risk but I think it is a risk worth taking.

iPod Touch Project: Differentiated Instruction (Update)

I chose to use the iPod Touch for student C, who has an IEP and gets pulled out for all subject areas: math, reading, and writing. It was hard to find the time to incorporate the iTouch since this student is in an out of the classroom a lot. However, I did have time during "Choice Time" to have this student use a letter recognition app. He was very engaged and learning too! I was also recently able to sit in during his parent teacher conference with the special education teacher there as well. The special education teacher said more practice with number recognition will help. I thought, iPod Touch would be a great way to get practice. I have been searching for a good app and excited to see how this works out and also assess if his number recognition has improved because of its use.

I think the iPod Touch is a great tool to have for this student. Since student C gets pulled out of class throughout the day, when he returns, he sometimes has to sit for a little while with nothing to do. For example, during math he comes in when there is about ten minutes left of whole class math time. This would be a great way for this student to have differentiated instruction. This idea just came to me, so my goal is to try this out this coming week.

In the beginning I wasn't sure how to go about using the Touches. But, the ideas are now starting to come especially for student C since he is not able to do the assignments the whole class is doing yet. The difficulty I am finding is finding a good app to fit exactly what I want. An app maker for teachers would be nice.

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I have also used the ITouch for science and literacy to record conversations with students. For science class, I recorded interview questions, and this was very useful to transcribe what was said and not have to write furiously to get everything down. In literacy class, I also used it to figure out fluency rate since the recordings are timed, and I was able to go back and listen to assess miscues and analyze student reading. Being able to record student voice as they read is wonderful and makes assessment a lot easier and more accurate.

If I am able to and have the time, it would be interesting to teach a lesson on expression using the ITouch. Each student would have a passage to read and the objective would be to read with expression. After recording all students readings, combining the recordings (I don't know how to do this but the idea sounds nice), a read aloud composed of student voices for one book could be done as the finished product. This certainly could be done with one student and one short book, and then shared with the class as a multimedia product with snap shots of the pictures/text in the book and the student's voice.

Literacy Writing Prompt for 2-18-11

What have you learned about your buddy’s needs, abilities and interests?
Interests
My fourth grade buddy loves to play football at recess. He also likes the Seahawks, soccer, scary and mystery books, fast cars (Ferrari, Lamborghini), and playing video games (World of War Craft). His favorite color is purple. One of his goals is to read more at home instead of playing video games. He said it is hard for him to find a book of interest at home and there is limited resources. My partner and I find our fourth grade buddy very sweet to work with; we are curious to know why his desk is right next to the his teacher's instead of with a group?
Abilities and Needs
- When my partner and I did the spelling assessment, he really enjoyed spelling. When we had time left in out next sessions, he would always want to wrap up with some spelling. Words ending in “ed,” long vowel, other vowel patterns, consonant doubling, inflected endings, and syllable junctures are learning targets of focus. I would place this writer in the Within Word Pattern and Syllables and Affixes spelling stages (Bear & Templeton, 2004).
- During the word bank exercise, this student did well, read the words fairly quickly, and finished all the words. He self-monitored when he did not know the word and attempted to sound out the letters. According to the assessment, he is able to read independent words at about a 5th grade level.
- Comprehension is the area that I would like to focus more on. When our buddy read from his own text book, he had a hard time retelling the story. It seemed he was more concerned with reading than comprehension. He had to go back to the text often and re-read to recall events. However, this student was aware of words that were unfamiliar to him. When we read a book of our choosing, Goosebumps, which is at about a fourth grade reading level, he said the text was too easy. And he was able to retell a good share of the details. When we read "Amelia Earhart", this student was able to recall some events in his own words; however, when we asked him the questions that were pre-written, our buddy had difficulty. From this assessment, he was at the frustration level and at the twenty-fifth percentile in fluency rate (2006 Hasbrouch & Tindal Oral Reading Fluency Data). This assessment was surprising to my partner and I because during our other sessions, his rate in regards to fluency was fine. Our buddy did not have any background knowledge on Earhart and did not seem to have an interest in the story.

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

I would like to focus the next lesson on comprehension strategies. Is he making mental images in his head as he reads? Is he able to use background knowledge? Is he able to ask questions as he reads? Is he aware when things do not make sense when he reads? I think explicitly sharing one or two of these types of strategies will help in his reading comprehension because this student does have the ability to read fluently with out many miscues and self monitor. For materials, I would like to find a book he would enjoy reading about (i.e. fast cars, football) and a text that is still at fourth grade reading level. The learning objective would be focusing on one of these strategies to improve cognitive awareness of meaning in text.

Geometry and Using Creative Writing in Mathematics

1. What did I learn?
In class this week I learned how to incorporate geometry understanding with a simple square piece of paper. With this paper we talked about the properties of squares, triangles, rectangles, lines, and angles. For example, how do you know a square is a square? This type of exercise forced us to use mathematical terms and use them to discuss geometry concepts aloud. Another layer of assessment is used when math terminology is used in conversation. Math terms like any subject is sort of like learning a new language, and is ofter best understood verbally. "Let kids experience geometry" is a big idea learned. In a similar fashion, the article we read "Using Creative Writing and Literature in Mathematics Classes" by Halpern $ Halpern showed great examples of students' writing using math vocabulary and creative writing. I was surprised with the creativity and had not thought of using creative writing and math as an interdisciplinary tool.

2. What are the implications for classroom practice?
Math should be discussed and talked about in a classroom. Kids should practice talking about math vocabulary in conversation and in print. Writing practice can be made relevant to student lives with their current learning in math. In order for students to do this successfully, teachers should also ground themselves in the same way and do the homework they assign their students. Teachers need to do this in order to teach beyond a visual or basic analysis level, and prepare them for deeper understandings in the next grade.

3. What do I have questions about?
What are other ways students can experience geometry?

Graphing Data and Group Work

This week in math, we had a gallery walk and looked at everyone's graph that told a different story from www.gapminder.org. It was interesting how people represented data and how different people interpreted the information. What was clear to some people was not always clear to others. "Graphs should stand on their own" was a big idea and reminder that was shared in class.

I like the idea of groups producing a finished project, showcasing their work with a gallery walk, and also using sticky notes to ask questions. By asking question in an anonymous way, it provided an avenue for more feedback and constructive criticism. This type of feedback could be incorporated easily in a classroom.

Group work and collaboration lessons are great if it is done well with roles and part of the classroom culture. It appears a lot of upfront work on the teacher is in dividing groups, establishing roles, and making sure everyone is accountable for their part. Once these norms are established, I like how the students can run with the project themselves and the teacher is now more of a facilitator. Can this be done with primary students too?

A relatable YouTube video to open up a discussion or in this case math group work is a great idea. I like this type of hook idea. It shows relevance, humor, and thinking. I would definitely like to incorporate something like this in my lesson plans. I have embedded the video below to keep as a reference.

Project 1 – Analysis of Student Writing and Lesson Plan, Comments from Peers, and Final Reflection

Overview
This student’s writing piece is in response to a prompt about finding a bottle with a rolled up piece of paper inside and then telling a story about what happens next. The student has put some thought and good detail in the story describing his feelings of enthusiasm when he finds the bottle and also adds his best friend’s name into the story showing a personal touch. There is an introductory setting of a very hot day on a sandy beach, and intrigue with the idea of a treasure map. Certain elements happen in the story leaving the reader wondering, why? However, with more additional clarity and build up in the middle section of the story, it is a great start to an eventual ending which includes discovering sixty grand and a purchase of a Lamborghini.

6 Trait Assessment (NWREL, 1992)
(1) Ideas: Support is attempted, but doesn’t go far enough yet in fleshing out the key issues or story line. The reader is left with questions. More information is needed to “fill in the blanks.” After the author and his friend discover a treasure map in the bottle, the author then writes, “then we got scared and whined like we were 2 months old…” The reader is left wondering what happened that they would be scared. The idea of feeling frightened was a great emotional addition but support was lacking. The treasure map was leading them to money, but details and the adventure in the journey was missing.
(2) Organization: The paper has a recognizable introduction and conclusion. Transitions sometime work, at other times, connections between ideas are unclear. Pacing is fairly well controlled, though the writer sometimes lunges ahead too quickly or spends too much time on details that did not seem pertinent to the story. “And” and “there was” is used frequently for transitions; however, the student does show awareness of organizational flow with the use of “then” and “next.” In the middle section, the student adds a paragraph about him and his best friend getting a bite, not feeling good, “drunk” some milk, and falling asleep. This detail could have been eliminated. Instead more aspects of what was on the treasure map could have been added.
(3 & 4) Voice/Word Choice: The writer does connect to the audience through the use of natural engaging language. The writer does take risk by the inclusion of personal detail that reveals the person behind the words. Striking words or phrases often catch the reader’s eye and linger in the reader’s mind. While the student can elaborate better on ideas and edit some sentences for print instead of talking, the voice of the author shows strong potential. Here are some examples:
- “…I wake up to find Im like burning hot like Id be in a oven at 200 degrese…”
- “…help me open this dum bottle…”
- “… whined like we were 2 months old.”
- “…lead me to 60 grand I was happy that I bought a lubergini and I still rember the day I found a bottle in the sandy beach.”
(5 & 6) Sentence Fluency and Conventions: More work will be needed in sentence fluency and conventions to make it easier on the reader to keep track of the message. Parts of the text invite expressive oral reading; but many sentences without proper conventions ramble on. Punctuation is often missing and the reader must read once to decode, then again for meaning. Commas, periods, and even exclamations will add to voice and fluency. Some word choice: “and” for transitions, “there was”, and “like” were repetitive. There were also some verb tense issues: “wake” instead of woke and “drunk” instead of drank.

Spelling
Misspelled words included: alomost, dwn, latter (later), roolled (rolled), peice, boottle (bottle), awesme, coul (cool), dwn, opend, dum (dumb), figuerd, fel, checkd, wonderd, and labergini. The writer is sounding out words to help aid in his spelling. Words ending in
“ed,” long vowel, other vowel patterns, consonant doubling, inflected endings, and syllable junctures are learning targets of focus. I would place this writer in the Within Word Pattern and Syllables and Affixes spelling stages (Bear & Templeton, 2004).

Summary
This writer has a clear introduction and conclusion. Elaboration of ideas in the middle will add to meat of the narrative. In addition, taking out some unnecessary detail will help the fluency for the reader. The writer shows potential for strong voice with the use of his descriptions and ideas. Lack of proper conventions is limiting the fluidity of this writing piece. More detail on the “why” part for sentence meaning will help the audience understand. I would recommend a conference discussing organization, elaboration and/or focus on ideas, sentence fluency, and conventions for learning and growth in writing.

Lesson Outline

Objective: Student will learn to organize, edit, and elaborate on ideas. We will be asking the questions: who, what, where, when, and why. A specific focus will be in answering the question, “Why is this happening in the story?”

Standards: GLE 1.3.1: Revises text, including changing words, sentences, paragraphs, and ideas.

Materials: Notebook paper and a pencil

Instructional Strategies:
This lesson will occur in a personal writing conference.

First I will ask the student to read his story aloud and tell me what he thinks so far.

I will then tell the student all the excellent ideas and thoughts on his paper already. (Celebrate student's writing from Routman, 2005)

Then I will remind him to keep his audience in mind and ask him if there is anything he thinks that will help the reader follow along with the story. I will also ask him if he thinks there is anything detail we should take out. I will take notes on the notebook paper of the ideas we discussed.

Then I will touch base on the elements of: who, what, where, when, and why. The focus will on the “why” part. For example, as a reader, I want to know what is scary, and why is it so hot, what is on the map, and why do you buy a Lamborghini?

After discussion, I will have this student work independently to make edits and add additional detail to add to his already wonderful start. The thoughts we discussed and notes taken will be given to him for reminders.

The next day, I will follow up with another conference. We will revisit his work to see if edits of elaboration were made and how he felt about his story now. Then together, we will discuss the next phase which if ready, will be a look at conventions.

Assessment:
I will compare first and second drafts and see if his ideas were expanded on and if as a reader I come away with fewer questions. I will also assess his new draft with the Six Traits Rubric with a focus on ideas.

Learning phonics on the iPod Touch at age 3

iPod Touch Project: Differentiated Instruction

I can see using an iPod touch being beneficial for all students in a classroom. In my first grade, main placement, I have seen students using netbooks for reading e-books. It takes a little bit of time for students to learn how to properly get their netbooks, login, and put them back; but after the prep work is done and the technology is working, I have seen full engagement in the classroom and a first grade classroom quietly engaged and reading. Similarly, this can be done using the iPod Touch, and one of the benefits with the Touch is its portability and reading can happen anywhere. Some apps (i.e. Learning A-Z) provide a way for students to read books at their ability level which is a great way to differentiate instruction for differing reading abilities.

Blog prompt: How might the iPod touch support differentiated instruction for three students in my main placement? What are the limitations for each and what supports will be needed for each student?

First grade Students A and B: Both are very bright students; they are both in high levels in reading and math, however both are often off task. Student A wanders around a lot and has a hard time focusing. Student B talks often and also has a hard time focusing. With the use of a iPod Touch, and its interaction capabilities with sound, video, and interactive keyboard, I think the pure engagement and entertainment of the technology will help them stay focused. The iPod touch should be seen as a privileged and hopefully motivation to stay on task. These students often need more challenge and the different math levels available in some of the math apps can progress students to their next level. Making sure these students are using appropriate apps and using the learning tools suggested to them will be the extra support that is needed. For example, if they are playing a non-educational game instead of learning game then this would be a loss of the privileged.

First grade student C: Is a special needs students who is at a slower rate developmentally. Student C needs occupational therapy to help with fine motor skills, particularly with hand writing and is said to be below kindergarten level with academics. Using an app that reads stories aloud can help with reading. Using a phonics app can help with letter and sound recognition. Instead of a teacher writing ideas on a piece of paper for the student based on what the student says, perhaps the student can learn to use an app that records his ideas through his voice and then transcribes what he says. Then he can write what the app has transcribed. This would have to be modeled and a quiet area will need to be provided. But the benefits would be a move toward independence in writing. A stylus pen could help with handwriting and hand motor skills. Also with the iPod Touch, this student could practice his letter recognition with his finger. I also know learning visually would benefit this student a lot and it would be exciting to see how much technology can add to Student C's learning and growth.

Multiple Representations

In class this week, I learned how to show students positive and negative integers in different ways: using beans and a large number line down the middle of a classroom. I like the idea of getting the students out of their seats and doing math. Moving from concrete to abstract was a big idea I learned. Another fundamental understanding I learned to associate is the definition of subtraction - which is addition of the opposite. I never used algebra tiles and now at least I know what they look like. I learned it is important for teachers to understand different strategies and present students with different multiple representations of the same math problem. Just because I learn best one way does not mean my students will see the solution the same way. Lastly, I am also starting to see how technology is changing math - for instance today we took a look at the website: www.gapminder.com.

Using math manipulatives, talking about your thinking, asking questions, drawing pictures, using a number line, etc. should all be a part of the classroom norm. I remember math class consisting of being my math book, notes, and homework from the book and a worksheet. There was hardly any talking or discussion with my classmates. As a new teacher, I am going to have to play with algebra tiles, geometric shapes, and beans for example to be comfortable showing students how to use them as a strategy and discover mathematical rules and properties.

Questions: It seems inquiry based approach and students being able to discover their own genearlizatinos are good foundations for students before given them the formula or the "how to" but are students really doing better in math with this type of inquiry based approach? Will the attitudes about math change?

Boarder Task, Sneaky Snake, and “Never Say Anything a Kid Can Say!”

This week I learned about the importance of quality in math problems (sometimes in lieu of quantity) and how exploration can be great way to foundational thinking before going to straight to the algorithm. The cooperative activities like Boarder Task and Sneaky Snake in class incorporated algebraic thinking and sharing in different strategies that I had not been exposed to before. This type of activity also met standards which was remarkable.

Reading the article “Never Say Anything a Kid Can Say” by Steven C. Reinhart, I learned it is not only important for math teachers to explain math well but it possibly more important for the students to “explain things so well that they can be understood” if students were truly going to learn mathematics. This reminds me of what was said in class about how kids will learn more from each other than they will from you- the teacher. Asking good question that produce thoughtful thinking and fruitful answers is another big idea presented in the article. I also learned to keep in mind that good questions take time, but a key principle is trying new things little by little so it becomes a norm. I like the quote on the definition of lecture: “The transfer of information from the notes of the lecturer to the notes of the student without passing through the minds of either (p. 480).” Ahh yes, this is one of the reason I disliked lecture so much in my past schooling.

The article also presented a lot of implications for classroom practice. A productive math lesson involves the teacher’s role as more of a facilitator and listener then a lecturer. The classroom environment should make students feel comfortable sharing, taking risks, and discussing ideas. “Students should understand that all their statements are valuable to me, even if they are incorrect or show misconceptions” (p. 480). I like the idea of providing a risk free environment and not using questions to embarrass or punish. I remember often feeling embarrassed in class if I did not produce the correct answer or if I did not understand right away. Another big idea in the classroom is for everyone to understand that “participation is not optional.” If students cannot answer a question, then students know then they are required to pose a question to the class to help in their understanding. Group work becomes more than finding the right answer but the members in the group are responsible for each other’s learning. This adds more accountability and hopefully more discussion on ideas and concepts. Instead of sitting back, listening to rules, memorization, worksheets, and tests this type of lessons makes math more alive via discussion and collaboration.

Questions:
I like the cooperative group work concepts. But for slow learners, sometimes the time allotted for understanding and doing group is still not enough. Wouldn’t this cause group work frustration? And for the slower learner, it may make him/her feel they are not fast enough, and then reinforce what we are trying to prohibit- that I am just not good at math. Realistically, is there really away to stop the status in math?