Fosnot Institute Day 3

What a wonderful day. These 3 days have reaffirmed my belief in how I teach math. I have been given the tools and an extra jolt of horsepower to continue my path to having students talk, communicate and become mathematicians. I am also convinced that 2.0 apps and how they promote collaboration and publishing students work fits nicely into my new plans. I also love voicethread. Two years ago I was happy when kids went into paint and created an image. Last year slideshare, bubbleshare and slide.com came into their posts and math work. This year the posts will be kicked up a notch when students see that they can create movies and talking slides using the newest apps. Wow what will next year... wait next month... wait tomorrow's newest apps bring to their work. Now on to day 3 of the Institute.

Fosnot Day 3

The math is not in the model to be seen but in the learners head. The learner is constantly trying to figure out and understand. Use real context this will give the opportunity for authentic learning.

So you teach fractions here are some strings for you
Think about a half? Elicit responses from your community. Many will come out but the first one I want you to think about is money.

First string think about $$



String 1 (coin strings)

1.) ½ + ¼ = (3/4 75/100)
2.) ½ +1/10 = 6 dimes 60 cents
3.) 2/10 + 1/5= (how many people thought of a nickel)(good for /100)
4.) 4/10 + 2/5 +1/4=

This is a very early string for fractions. These are landmark fractions. Landmark fractions are easy because you are dealing with whole numbers instead of dealing with fractions. Link to percents because of the $1.00. Get rid of the fractions and make it contextual.

String 2 (clock)



Draw the clock to represent the fraction
Calculate the minutes. You do not need to show the equivalent fractions because the clock model is there you do not need to show the equivalent fractions
1.) ½ + 1/3
2.) ¼ + 1/3 (anyone get 7/12)
3.)1/6 +3/12 +1/3
(1/6 is the kicker here) 1/6 is 10 minutes… start to look at the whole.. There was some breakdown here. The 1/6 was causing problems… get the community to provide the proof for the breakdown. This is key keep the kids talking and doing the explaining. Knowledge can be generated out of a communities talk. 4) 10/60 + 1/6 +1/4 +7/12

This problem was generated after the community was having issues. Free flow. Getting bogged down in some lcd’s
Pedagogy 10/60 and 1/6 are = this should be a good scaffold
When doing strings think on your feet and go with the flow… but not too much
This string again gets you away from fractions and into wholes… then you can get back into the fractional form.
Scaffold the landmark fraction(ability to use whole numbers) and coins.


String 3 (you choose the strategy)



Look at the numbers and choose coins or clocks
1.) ½ +1/3 (clock because of 1/3 $)
2.) 1/3 +1/4
3.)¼ + 1/5 ($$)
4.) 4/5 – ¼ ($$ because of the /5)kids will start to see which fractions are clockable and which are $$able

String 4 (double numberline)



Pick any number you want to be a guide on the track (numberline)
1.) 1/3 + 1/7 =
2.) ½ +1/4= Community will choose numbers to be the numberline total

(need to add the other aspects of the string) Kids will start to create common denominators What others numbers are easy numbers Lets pick a simpler number…. This gets to the common denominators. Work to prove conjectures. Name them after the student that came up with the idea. Use other examples to prove or disprove Double open numberline paves the way to common denominators…… Push to show equivalence
Multiplying fractions

String 5 (Use an array to demonstrate)



1.) 1/3 x 1/5 left with 1/15(outer inner boxes)
2.) 2/5 x 1/3 (the whole remains the same one more piece doubles)
3.) 2/5 x 2/3 (double doubles) (use the playground context to be the building block)
Inner and outer rectangles show the algorithm
4.) 3/5 x 2/3 You can use the previous array (shows the pattern)
5.) 2/5 x 3/3 Curious leap here (new array) what happened to the inner rectangle. It shifted 2 by 3 becomes 3 by 2)
Can we swap the numerators to make a friendly question
6.) 4/7 x ¾ = 3/7 x 4/4

Division of Fraction
Use a ratio table see photo Use the ratio table to explain yours is not to reason why just invert and multiply



The use of the ratio table was brilliant. It should be the model used to bring understanding to why we invert and multiply fractions.

Many times we take part in activities and professional development that is lacking in so many ways. These 3 days out of my summer break were stimulating and exciting. I can hardly wait to use my knowledge of 2.0 apps and push the limits of what I learned this weak. Oh Yeah you can annotate and draw while doing a voice thread. That is too cool for school.

For those of you who want to get the Fosnot material here is the web address

Contextsforlearning.com

The material I was being taught is from Fractions Decimals and Percents.

Fosnot and Dalk Mathematizing Institute Day 1

What a terrific day I had today. With the exception of the fire alarm that interrupted the afternoon session great learning was had by all. Here is my recap...

Fosnot Institute Day 1

Starting to use math with context. It is important for students and teachers to learn math with a context. Allow students to be mathematicians learning and explaining. For too long teachers have been the be all know all.

Here is the context. The teacher poses this question to the class.

A rectangular lot in neighbourhood A is 50m by 100m. Of this lot ¾ of it will be a playground. Of this playground 2/5 will be blacktop.

A rectangular lot in neighbourhood B is 50m by 100m. Of this lot 2/5 of it will be a playground. Of this playground ¾ will be blacktop.

Which Park has more space for blacktop?

In pairs you now attack this problem. As a teacher you stand back and take notes on the conversations happening between the students. This thinking time is important for the mathematical ideas to take place. A push or prompt needs to be held in instead of giving that helping hand.

Here are some photos of our finished work

It was fun to work with a partner and talk math. Together we worked out the problem and were chosen to speak for the math congress. Hmm not bad for my first day back thinking about math.

Following this work time students post their work which had been done on chart paper for a gallery walk. During this time students are encouraged to post notes using post-its on the other pieces of chart paper.

Students need to be trained to have good gallery walks. Choose similar and different solutions. Teachers need to use guided questions to start this process.

· What was done similar to your solution. Is it clearer on this paper?

· What do you not understand on this solution. Is there something missing?etc

Next to the congress. This is a part of the lesson where students become the teachers. You could call it double learning. Students are reinforcing their learning when they are teaching the rest of the class.

The teacher based on the gallery walk chooses examples that will further the learning process. This does not necessarily mean the best examples but examples that add more context to the topic.

During the congress it is important for the teacher to remain on the side lines. Instead of asking Do you get it? The teacher needs to ask;

How many people and put in there own words what this group has said?

Giving a group a second chance to explain a topic will give them another chance to reinforce their knowledge. The second time around concepts are easier to explain or at least seem to be more coherent.

During this congress after the group presented there was time for a pair talk. Are 2 fifths equivalent to 4 tenths. Questions that arise during the congress are the avenues to deeper contextual understanding and avenues to further discussion and scaffolding.

It is not the presenting groups responsibility to explaining the new topics arising from the congress. Other students take turns explaining using their own words and pictures on the assignments hanging up throughout the room.

If students get bogged down in these instances the teacher then jumps in and tries to rephrase the topic. (pictures can be a powerful manipulative)

The congress develops a sense of community in the classroom. It is important to recognize the importance of math to students. Celebrate questions and explanations explaining to students the mathematizing they are doing.

Side note Create a classroom space for working and congresses. Find a way to separate the two pieces of the problem.

From the congress here were some things I heard… Lets prove it.

Of means multiply…(use pictures that are out there in the congress)

• to prove it you need to disprove it. Find a math sentence that uses of in a different way other than x.
• two of 5 or 2 out of 5
• groups of means x?

The use of arrays proves this …?

Facilitator needs to stop this and give this as homework journal. (blog it)(move it back to the individual level instead of the group)

Multiply numerators and denominators the demomenators give the number in the gird(array)the numerators multiplied gives you the amount of the whole.

Communicative Property 2 times 5 is the same as 5 times 2

All the above were ways in which we took the contextual problem and stretched it further our understanding.

Take away the AHHA’s and kids will never want to be mathematicians.

Petagogy… make the kids do the explaining. Take your time and have them do the explaining…double learningl.

In the afternoon we broke up into two groups. We were with Maartin D talking about the landscape of learning. This landscape takes into account

• Building the landscape of learning.

• What do children really think and do….

• What is it that I as a teacher want my students to talk about

• Building context…what content can I steal….

• What is the order

• Big ideas take time to create

• What models are used

In their supplemental material that is provided Fosnot and Dalk give many examples of teachers and students interacting in mini-lessons and math congresses. This gives educators a chance to see this mathematizing in action. The CD's have the ability to cut and paste video lessons and parts of video lessons into your own customized clips which you can then add to your landscape of learning for the unit you are creating. We went through this task in the afternoon and I was pleasantly surprised with the ease at which we started to create our own landscape to improve our learning experience.

As you watch the many different video clips that are provided you will be able to deduce for yourself,

• What is the role of the teacher
• What are the children saying and the math behind it(this is how you neeed to see the footage we watch)
• What is the big idea being discussed in the video blip?

·

As we watched the first video clip it was clear that the students were doing most of the teaching and the learning. The teacher was merely a guide to facilitate the learning.

You want the students to start to use the model with a context to understand a variety of different situations.

I went home with a positive attitude and a desire to return the next day and continue my journey into mathematizing and the joy of seeing students be key instruments in their own learning.

Message of day one... Context is important, kids need to be mathematicians.

Mathematizing

One of my first blog posts was on Mathematizing. Now I will be having a 3 day institute on implementing Mathematizing ideas into my classroom.

Catherine Fosnot and her co-author Maarten Dolk( are coming to Winnipeg to do the workshop in person. This is an amazing opportunity. The authors ideas include the idea of mathematizing. This is when students become mathematicians and talk about the math that they are learning. This is a perfect way to use 2.0 tools. Part of being a matematician is to publish your work.

Mathematizing is solving problems, posing problems, playing with patterns and relationships and proving their thinking to fellow mathematicians. It was so fun to see these ideas in action in the classroom video clips from New York.

There is a diference between activity and genuine Problem solving. So we need more than "Hands ON" Discovery Learning.

Classrooms become communities. Children meed in groups and as a class to present and talk about solutions to common problems. There is "no wise one". convincing arguments are made to the group. Knowlege emerges in a community of discovery.

Doing math is like climbing a mountain. You take it one step at a time. Sometime you can take many steps before seeing the vista and all its beauty. For students to continue to climb and enjoy the journey they need to undertake this journey themselves. Only then will they continue to climb instead of staying at one leve

Many techniques were used with the young learners. They were always placed in pairs and the pairs were carefully selected. You did not want to put the best student with your worst a A and Z pairing. You wanted to do a A and C parining so that there was a difference but not a vast gap.

The students would receive a large problem on a large piece of paper and then in pen put their answers to the question. The teacher would choose examples of the student work and have the students present to the class. The students would lead the discussion and ask quesitons to the presenters. The teachers role was to facilitate the conversation to hit petagogical ideas.



Teaching was done in the form of mini lessons to activate strategies and ideas. Learning was always group based and fostered math language skills.

When we have math students talking and writing about math they conceptualize what they are doing. This allows them to understand the math deeper than if they were just drill and practicing.

I am going to create a wiki and do blog posts about this course. Her books are a must read for teachers who teach elementary math or middle school math.