Zero continues

Hey Everybody,

Here is an interesting site, to continue Rebecca's debate on ZERO. I finally have a bit of time to check it out and it is awesome, thanks to Paula Blower, one of the secretaries at East End!!

http://en.wikipedia.org/wiki/o_number

Have a great summer!!

Origami

Hi Everybody,

Now that we can relax for a while, I must say that I was more nervous than I thought I was on Wednesday. I tried hard not to think about it because I needed to pay attention to what I was trying to learn from Maggie, so when my time came I never even looked at the notes I wrote to myself!! I was overcome by the memories of my family in Indonesia and sharing about my Dad was not part of my presentation!! But I was glad that I did. So, when Maggie was singing her song I was also thinking of my family in Indonesia, including my daughter who has been there since January, teaching English at a university in my hometown. She will be back at the end of July.

Anyway, what I was trying to do now is ask you all to think back before the presentation and what you wrote on that bright yellow paper!! Please think about what you wrote (without divulging anything), unless you so incline, and find out how you think you can connect Origami to Math! What are the things that you saw or did during the time you "manipulated" the paper into a boat/crane, that you think can be useful to teach Math? Do you think that students can "get it" by using this kind of manipulative?

In the past I have used Origami workshops at ESL conferences to help LA teachers understand what their students have to do in order to understand English. As I said, they told each that after about 10 minutes of listening and trying to follow directions in a different language, they had a headache. I used these workshops to get "free" admission to the conferences:):):) Now I intend to expand Origami into Math because in the past it never occured to me that what I consider mental therapy could be a great tool to help students learn Math concepts, at the same time taking the product home, which I think is instant gratification.

So, when you have a minute, please tell me what you got out of that, a short post reflection to help me find more "angles" and "hooks" to connect Math to Origami!! Believe me, I was plenty nervous, and I had to sit down to calm my nerves but you all were such great and supportive audience, and I really appreciate it. It was hard to stand in front of you and try to remember what I was supposed to say:):):) I am glad that it was not too hot that day, or I would have been "sweating."

There is also another recent connection to Math that I found. I taught five elderly ladies, to (each) make a pinwheel bag at Whole Foods (WF) when we finished the coaches meeting on Thursday, and they paid handsomely so I can get more paper!! Actually, that was my true "connection" to Math, the dollar value:):):) They want me to continue working with them!! AND, when the marketing director at WF saw me do that she asked me to teach Origami at WF because they have sushi classes there!! Sound delicious huh....eating sushi and doing origami.....hmmm........!!

Maggie, I don't know how you can so calmly teach so many adults. You always looked so relaxed and composed.........My hat off to you Maggie!!

I look forward to hearing from you all.........:)

From Zero to Infinity

The debate and sharing of ideas regarding "zero" as a number has been quite interesting. I appreciate all the comments, musings and the effort to share. My kindergarten teacher mind is always thinking about how to make concrete connections to language that we use for mathematical concepts. Zero is a word, but what I find interesting is how students explain it's meaning in their minds. "It means nothing!" "There is nothing there (questioning look on students face)." "It's invisible." "It's like when my glass is empty and I haven't got any more milk." This last comment by a kindergartener is one of my favorites because it is connected to a real experience. It is very concrete.

The continued debate about zero jogged my memory about another student conversation about number words and math language. Two kindergarten boys were hotly debating which was bigger google-plex or infinity. For them it was a matter of whose idea was winning. As they argued a little girl in my class walked up to the boys with her hands on her hips. “You guys should know that both of those [meaning google-plex and infinity] are words. Google-plex and infinity are too big to count so we use words for them.” Wow! I was amazed. I couldn’t have said it any better.

MTM academy has reaffirmed for me how important it is to really listen to my students. It is so easy to get swept away by our curriculum expectations, etc. and forget to hear where the students are coming from. So….. “To infinity and beyond.”

Sum Help: New Search Engine for Mathletes

http://online.wsj.com/article/SB124516890985419379.html

My husband is teaching himself Trigonometry. We have interesting math conversations.

He brought home an article from The Wall Street Journal today. Wolfram Alpha is a new search engine and people are debating mastering algorithms when "computational power is always at hand."

Check it out! I say it is a website that can be useful tool , just like a calculator is (at times) in my own classroom. I wonder where this search engine will take us.

Rita

"But I thought this was math!?"

My math class just finished reading "Chasing Vermeer" by Blue Baillett. It is the first year that I have done it (reading a novel in math). I teach a high group of 4th graders and I felt that it was the message that I wanted to send them away with: Be curious, be diligent, be bold. The characters are all of the above and more. The story includes a lot of logic, patterns, codes, and pentominoes. It took three weeks to read, but I balanced my class with more "traditional" math and left out the weekly challenges. I used the IGETIT approach, and let the kids discuss and figure things out, and the classes were perfectly unpredictable. It worked out better than I envisioned, (better next year, though) and suprisingly, I had no complaints from parents. Talk to me next week if you are interested in more info. I think the book is best for 4th-6th grade.

zero

(I am posting this for Rebecca)

We had a big argument in the classroom today (kindergarten/First Grade)  It started by saying that a person couldn’t be less than 1 year old because they couldn't be 0.  I asked a student about her dog who is 6 months. Some of the kids figured out that 6 months is 1/2 way between birth and 1 year.  Others were stuck on the idea that when you were born you were 1-year old- they didn’t get that the 1st birthday signified 1 year of life.  So we talked to Mrs. Nelson about her baby, when he was born, how old he is now and when he will be 1.  Eventually everyone pretty much accepted that you are not 1 year old when you are born and that you are officially 1 when you have been alive for a year ( I didn’t mention some cultures start counting life from conception so they actually turn 1 aprox 3 months after being born - We had enough problems already!).  Anyway…at that point I was feeling like we could move on with life but I was oh-so wrong........The next issue to arise was:  "0" is not a number.  I was thinking that 0 represents an amount, an amount of nothing but it does signify that there is nothing there and either side of 0 is a number representing positive or negative units of measure.  Several students who go to another teacher for math insisted that Zero is nothing and since it is nothing it is not a number.  They told me "Mrs. Linevitch says 0 is not a number".  I told the kids I wasn’t sure and we could gather information and try to figure this out.   I went on line because there is sooooooo much I don’t know about math and found a lot of contradicting information.   I need help!  Is Zero a number?  Why do some sources say yes and others no?  I have polled staff around the building and have found some believe 0 is a number while others do not.  I am so confused - has my belief in 0 as a number been wrong all my life?  Help!
Rebecca

Chunking

I recall Maggie replying to what I wrote, long time ago on the blog, with a question about how I transfer methods to teach Literacy into Mathematical implementation, and, I was stumped. I was always under the impression that teaching Math is nothing like teaching Literacy. Since I was stumped by Maggie's question I have been thinking about it and, at my age, I feel that learning is a never ending process. Well, I don't want to waste whatever time I have left:):):)

A few days ago, in one of my third grade Read 180 groups, (not even working on Math), there was a sentence containing a six digit number: 330.000 of something. The student couldn't read the number and he just said thirty three, and just kept saying that three times. I had to do a "mini" Math lesson in the middle of a reading program, without getting my "tucked away" home made Place Value chart. So I covered the three zeros and asked him to read 330, and he did beautifully. I uncovered the rest of the numbers and asked him to read the whole six digit number but he lost it again.....and said three hundred thirty........over and over again. I empathized with him, because some of the other four students barely had their eyes attached to their sockets. They knew how to read that. However, I have always told them that giving someone the answer without giving them a chance to think is like robbing/cheating them from the learning/thinking process. They have been great about not blurting the answer......now!!

So I HAD TO get my chart......and asked him to put those numbers on the chart, noticing that the 330 is under the thousand "roof." Bingo......he could read the numbers and he was so proud of himself! To me the process of covering the zeros resembles chunking in Literacy, reading a long word by identifying the shorter words in it, seemed to be an AHA to me but perhaps to others it is not news:):):)

I had to share this with you all. I was too lazy to get my chart to begin with, at times being lazy has its own advantage:):):) Hopefully this makes sense to you. Thanks for taking the time to read.

Origami and Fraction

Origami has always been a big part of my life for as long as I can remember, growing up in Indonesia, but I never thought it is a great way to help learn Math, specifically fractions until recently. Last week, when one of my iSucceed Math students was working on fractions asked me about whether each of her fingers is one fifth or is it one tenth! I asked her what she thought of that. She was convinced that because she has five fingers on one hand that one finger represents one fifth of her whole hand! When I asked her what characteristics fractions have, she looked at her hand/s. I was looking for the equal part of course. She then told me that one finger is part of her whole hand. This student is from Ethiopia and has always been so brave to ask questions without worrying what her peers may think of her. She is also in my Read 180 program. I love her curiosity!!

I wished that I had some of those round fraction manipulatives!! I could have used other manipulatives but I decided to take some colored 8.5 X 11 paper and all five of them came to the table, they were also curious and perhaps wanted to take a break from their computer work. I asked them to fold it in exactly half, then into another half, and into another half and open it. I then asked them to draw lines on the foldings and asked them to count the parts and write the fraction in each of the spaces. They saw the equal size of each spaces and finally told me that one of the rules of a fraction is to have equal size for each part.

I asked if her finger is one fifth of her hand.......she said no....... with a smile:) and added that her fingers are not of equal size!! I believe that she ate the mango! She didn't only help herself, but I feel that she also helped the others to remember. I will certainly review with her after the vacation.

Unfortunately I can not get to the national library of virtual manipulatives on these computers because the computer programmings are taking all the memory.........:) I know that in the nlvm there is a section for fractions. I agree with Shawn that it is a very good site for students to learn Mathematics.

NCTM conference

Is anyone planning on going to the NCTM Conference in Boston next October?
http://www.nctm.org/conferences/content.aspx?id=18223

triangles

I was so excited working with triangles this week as I was trying to come up with a way to have kids discover the rule that the interior angles of polygons add up to the number of sides minus 2 times 180. With my math team we started with octagons and then tried other polygons. We made triangles from the center of the octagon out to each angle. The had done enough investigation to know that the interior angles of a triangle equal 180, so they used this knowledge. Of course they first just multiplied the number of triangles by 180 and then we talked about the angles around that center point. Brilliant! 

With my 4th graders, some of whom were having trouble distinguishing acute and obtuse angles, I went to the wedge. I provided them with 4 in x  4 in squares and we used them to measure angles. Smaller than the wedge was acute, larger than the wedge was obtuse. This was far more effective for most than the paper patty protractors we had made. They hadn't had enough experience for the protractors to make sense. Now they are working on investigating different triangles like we did on Thursday. I had several kids come up and ask me, "Is it possible to make a right equilateral triangle?" I am amazed at how successful all of my kids are using the 4 x 4 wedge and creating right, isosceles, scalene, acute, obtuse and combinations of these triangles. The excitement continues......

web site for ten frames

In our 1st grade group at the muster yesterday, I mentioned an interactive website that has a ten frame activity on it. I use it on the Smartboard, but it can also be used with a PC or laptop. It's NCTM's Illuminations site and offers several activities to support what children K-12 are doing in the classroom. There are numerous ideas under lessons and activities, as well as additional ones under Web Links that look promising for all grade levels. For my first grade colleagues, the five and ten frame is under activities, pK-2. My students love to do these during their choice time, as a rotation during Everyday Math Explorations, or at the end of the day waiting for the bus.


http://illuminations.nctm.org/

Angle Vocabulary

There was a question at the April coach session about naming angles that are greater than 180 degrees, but less than 360 degrees. On the "ask dr. math" website, they list the name as a "reflex" angle.

While there, as usual, I stumbled across something new I didn't know. I knew that 90 degrees is a right angle, 180 degrees is a straight angle.........

....two angles that add up to 90 degrees are complementary, two angles that add up to 180 degrees are supplementary............do you know what the name of 2 angles that add up to 360 degrees is?

Check out this link for the reward of an answer:

http://mathforum.org/library/drmath/view/63015.html

Subscribing to the blog

Hi everyone,

I went to the very bottom of the main blog page. At the very bottom, there is a place to subscribe to: Posts (ATOM).

I clicked that link and made it a subscription to "Live Feeds" as a bookmark.

I added it to my bookmark toolbar and it shows me the list of recent posting. When I click on one of the posts, it opens up the blog for me. After I have read something the 3 arcs become a capital B.

I did not find a way to send it to e-mail. If anyone does find that, I'd love to know how.

~Shawn

As I read the Early Childhood article on algebra I am gently reminded that teachers need to do more than patterning in the early grades. I see lots of repeating and growing patterns in primary classrooms, but does the algebraic thinking stop here? Young students need to have many experiences representing and analyzing mathematical situations and structures. As a fourth grade teacher, I see students who don't truly understand the concept of equality, the idea that the equal sign means balance and not "the answer comes next."

They also need experiences in quantitative relationships. The other day I gave my students a problem: There are some kids and some dogs on the playground. There are 22 heads and 68 legs. How many kids are there? How many dogs are there? More than half of my students had no idea how to think about this problem. I don't remember much of my algebra learning and I know there is probably a way to solve this problem using algebra, but I want my students to be able to make sense of what they are doing, not use someone else's rule........so we muck around a lot with manips, tables, pictures and it starts to make sense.

So as I think about my kiddos all being ready to take algebra in 8th grade, I think not. Some may be ready before then, some later, but I am hoping with lots of early experiences in the primary and intermediate grades, they will all meet success.

Book Reviews

If you are thinking of buying a book online to add to your mathematics library, but are unsure of the contents, you can post your query on the blog and ask if anyone has it and what they think. If no-one has it, I'll order it and write a review for the whole group. That may help to ensure your money is well spent. Just a suggestion. I am ordering Family Math Night by Jennifer Taylor-Cox as Linda has expressed interest in it. When it arrives, I'll post the review for all.

How the Brain Learns Mathematics by David Sousa

This book is published by Corwin Press- ISBN 978-1-4129-5305-4 - Sousa takes the reader through how the brain develops number sense to how teachers can recognize and address mathematics difficulties.  Each chapter ends with a section called "Reflections" which helps the reader process the information.  A fascinating read!

Link to Rita's algebra article

Here is the link to the article Rita introduced us to on algebra in the early years.  Copy and paste the following into your browser to access it.

http://journal.naeyc.org/bti/200301/

Teaching Math and the Brain

I thought I would try to get into this conversation, although I am not too sure how to BLOG, so I will just muddle through. I have read a good book, How the Brain Learns Mathematics, by David Sousa. He explains how the brain has a special area called a number module - where number symbols are hardwired - this is located in the parietal lobe.

Now, the language module is in a different place in the brain and it is where the words are stored, including all our "math" words  - I wonder if how we teach these two concepts help link these two parts of our brains -  Now, I am just writing this because I think it is really interesting. I hope to join in other conversations with you.

Marcy Emberger  

Basic Facts - A Compromise

Coaches... I can't paste the contents of the email I sent here (about a solution for sharing the basic facts presentation) without retyping all of it.  So I would ask you to simply refer to that email and post your comments here.  Thanks!

I have heard from Karen and David so far.  

Basic Facts Reaction to Maggie e-mail

Providing us with a video allows us to concentrate on remembering the content, and delivering the message in a compelling way.
I am not sure I can get 3 hours (3, hopefully consecutive) with the other math teachers at my school.

Presentation Tips

For your upcoming MATH MOMENTS session, you were emailed a document called 'Presentation Tips' for you to consider as you prepare to deliver this first session.  You have put a lot of effort and thought into developing this session and my hat goes off to each of you.  Please let us know, by comments to this post, how your plans are coming along, what logistic issues you may be having, brainwaves to share, what worked and what didn't etc.  This team is gelling nicely and each of you has a lot to share.  I can't wait to hear!

Algebra from the Primary Perspective

Okay? You want me to comment on Algebra?, was my first reaction to our assignment. Before I muddle forward with any discussion regarding the subject of Algebra let me layout some background information. My "eighth grade experience" with Algebra was in the 10th grade and it was 40 years ago. Having taught either preschool or primary grades, mostly kindergarten, in the last 30 years, my contemplation of Algebra has been limited to thinking about my own children and their various experiences with Algebra. Out of 4 of our children the youngest who is now 24 took Algebra in the 8th grade. A comment he once made during his eighth grade year has stuck with me and echoes in my mind whenever I work with students and the idea of equations. My son said,
"Teachers should really use blocks to help kids understand Algebra." Incidentally, my son enjoyed math a great deal, tried it as a major in college, and finally settled in the geosciences.

Blocks! What do they have to do with the article "Recalculating The 8th Grade Algebra Rush? Before I contemplate that question I want to focus on a small quote from the article that struck me as the heart of article and is also very connected to my thinking about blocks. The last paragraph states,
"It would be better to think of Algebra as we do swimming: something everyone should learn, most importantly learn well. Get everyone into the pool as soon as possible. But let's not mark them as having passed the course until we are sure they can swim several lengths without drowning"
I am also a swimmer and past teacher of swimming. Of swimming I can say: not everyone learns swimming at the same rate, over the same length of time, through the same methods of teaching, or with the same amount of practice. H-m-m-m, that sounds familiar; Teaching Methods 101, differentiating instruction.

This quote illustrates a developmental perspective in that any given skill, in this case swimming or Algebra, each need time to develop, have an individual component for each learner, and in the case of either, to be useful to the learner they have to be firmly connected to prior knowledge (learned well). From the perspective of a primary level teacher where student thinking is very concrete and connected to real objects that may eventually be represented by equations, blocks and other math manipulatives become an integral mode of teaching. Blocks are a concrete medium through which students can develop theories, test their thinking and make conclusions. Unit Blocks, the large, blonde colored blocks that need their own shelf in most kindergarten rooms, can be used to build equations: block size A = 2 of block size B or A = 2B. Similarly, Cuisinare Rods also give rise to exploring equivalent relationships. From my experience, concrete exploration with materials that can be manipulated and assist in evolving mathematical thinking is a key developmental component that lays ground work for Algebraic understanding. The reason why my son's comment has stuck with me is that it illustrates how the cultivation of well-based new learning has a connection to prior knowledge and experience. In his case to all the years he played with blocks and legos. Starting with block and other manipulatives, a concrete model to bridge mathematical knowledge to the abstract world of Algebra can be built.

So what do all my ramblings have to do with "the 8th grade rush?" I believe learning is developmental and experiential, not everyone will be ready in the 8th grade for Algebra. Whether any given student is developmentally ready for abstract thinking (Piaget- formal operations) and to learn Algebra will vary from individual to individual. Experientially I wonder what role the primary teacher plays in providing concrete investigative experiences on which bridges can later be built for learning Algebra and learning it well. Thinking of "getting everyone into the pool as soon as possible, what do you think our role, if any, as primary teachers is regarding the development of mathematical thinking that leads to learning Algebra well whether it be in the 8th grade or later?

Re Staff Development with Basic Facts

You have been sent (via email) a link to an article in the Washington Post about teaching algebra.  What is your opinion of the argument presented in this article?  Please justify your response (tell why you believe what you believe) and give concrete examples where possible.  This is a timely topic;  algebra is receiving widespread attention as a 'must have' in every curriculum K-8).  We should know what the issues are.  This is just one.

Please consider this the original post (your thoughts should be posted as comments).  

Cheers!

Maggie

Number Knowledge Test

Per request at the coaches' meetings, here's the website for this assessment of a child's developmental level for math understanding:
http://clarku.edu/numberworlds/nw_TestInfo.htm

After reaching the Number Worlds website, click the button marked "Assessment." You will need to download each of the four components listed. (It is free!)

This test was designed especially for K and first grade. I've also used this with struggling second-graders. It is an individually-given normed test that takes about 10-15 minutes. The score is given in the form of a chronological-age equivalence.

As schools are headed toward more data-driven decision-making, we often ignore the developmental differences in students at the younger ages. Here is a way to include that information into the equation.

Make Your Own Bingo Cards

At the coaches meeting this week, I mentioned this great Bingo site, and was asked to post the address, so here it is.
http://www.educationalpress.org/#PickWorksheetType
You can type in the items for the card, select the number of cards you want, and the program automatically scrambles the spaces so every card is different. The cards can be 3x3 (9 items) or 5x5 (24 items and 1 free space). I always make 5x5 and enter 24 items, so that each card will have all the items.
Some hints when you go to the site:
Under Game Boards, choose Bingo boards
Options: enter # of boards you want, choose 5x5, choose font and size (I usually use "school" for correctly made numbers)
Delete the "top 10 words" and enter your 24 items in list form.
Unclick "center space free", this puts the free space in a different place on each card.
I make the "calling cards" separately, or cut up one of the Bingo cards if being used with a small group.
I usually print these on my home printer which can take card stock. You can also print them on regular paper, then run those through the school copy machine, which usually can take card stock.
If you have trouble with this, you can email me at debra_butterworth@cape.k12.me.us or use the blog.

using colors with 10 frames to help students with addition strategies

I've had some good results in teaching addition strategies to my special needs students using colors with 10 frames. I use one color of counters on the top 10 frame (and outline the frame that color) and use a different color of counters on the bottom 10 frame (also outlining that frame with that color). When I try to teach my students to add two numbers by first making a 10, I cue them to tell me how many of the top color there will be (to make the 10) and then how many of the bottom color. Once they identify how many of the bottom color, they can quickly tell the total. So for example, if I want them to figure out 8 and 4 more, I show them the 8 blue counters on the top frame and then have them try to visualize how many more blue counters they need to make the 10 and then how many yellow counters they will have to make 4 in total. Using the colors really helps them to visualize this process and helps us to communicate about it. I also glued the 10 frames on a magnetic board and I glued magnets on the counters so I can place the frames vertically in front of the students as a demonstration.

decimals

Happy New Year! I am working with one 4th grader in particular who is having difficulty grasping the concept of decimals -tenths and hundredths specifically. We have used the base ten blocks where a flat is one, a long= 0.1, and a cube = 0.01. This alone has caused confusion because they used to represent something else! One of the biggest challenges is getting him to realize that the ones place when looking at a decimal is the same ones place when reading whole numbers. We have discussed decimals in terms of food, rainfall, measuring to give them a real life connection. At times, he seems to be getting it but then falls back on subsequent days. we have also related tenths to more common fractions such as fourths and halves. Furthermore, he has difficulty writing the decimal correctly. He often puts the decimal in the wrong place. Any ideas as to what prerequisite skills I may be overlooking or any other suggestions?
Thanks,
Karen

Does anyone have any suggestions for working with students with memory issues?

I am using the K Investigations curriculum along with a variety of other activities.  Over 1/3 of the group are ELLs.   Most of the students are really getting the concepts.  I have one student, however, who is an English speaking student and  a first grader (from another class) that has issues with memory.  Two days a week this student works 1:1 with an Ed Tech during math.  This student is receiving Special Ed services and I am following his IEP but don't feel I am being effective.  We follow a routine using the calendar, taking attendance, counting jar, .... But each day this seems like it is new information for him.  Some days he can count objects 1:1 to 20 and other days he misses one or several numbers along the way or recounts the same objects multiple times.  Patterns are challenging for him too. He can follow an AB pattern I have made but cannot start his own.   One day he can tell me a square is a square and name several attributes and the next day he will call the square a triangle and be happy with his answer.   I have visual aids around the room but they don't seem to work.  Any suggestions????