Mathematically Speaking

 Authentic Connections

Last week I was able to take my students snowshoeing.  We were all excited to explore our “new shoes” and see what it was like to walk on top of the snow.

As we walked around outside my students started noticing signs of animals in the snow.  This is an opportunity for me to ask questions.  “What do you notice about the tracks? Are they close together or far apart?  This dialogue invites my students to think deeply about what they see and make connections. One of my students noticed the tracks were symmetrical.  Then I have an opportunity to ask this child to explain; How do you know? What is symmetrical? Mrs. D.; “I see animal tracks. We need to try to figure out what animal made them.” So we are able to take a photo of the tracks with an iPad.  This gives us an opportunity to capture something we notice and are curious about.

Now when I go back to the classroom I am able to ask the question: “I wonder what type of animal track this is.” My students offer using Kindle as a way to help us find out what animal made the tracks.

Nature is full of so many wonderful things.  Just by being outside my students are able to experience an appreciation of what is under their feet, around them and above their head.  The fresh air is great for our bodies and gives us all an opportunity to enjoy a new experience together.

As my students took off their snowshoes they started helping each other and then discovered ways to stand their snowshoes up.  This was wonderful.  One of my students said; “Mrs. D. we are making triangles with our snowshoes.  Our snowshoes have 3 sides and 3 corners.”  Another student said; “Mrs. D. our snowshoes our 3 dimensional.”   As a teacher I am always excited when my students make authentic connections through their experiences.  As a teacher of young children this is a nice example of how important it is to weave learning into a variety of experiences that give children an opportunity to make connections, reflect and share what they know.

Thanks to the Vermont NEA and a donation from my school I was able to get snowshoes.  I hope we get more snow!

Colorful, Foam Dice

As the weather became nicer and school coming to an end we decided to take our mathematical thinking outside.  I paired up my students, gave them a pair of colored, foam dice and some chalk.  I asked them to create number patterns, number equations and anything else they could think of.  What fun!

Everyone was engaged and happy.  As they rolled the dice they were able to run and jump to see what their numbers were.  I enjoyed listening to the laughter and excitement as my students began writing equations on the front sidewalk entrance of our school.

As I walked around and listened to the conversation I was struck at how the children were genuinely interested in creating numbers with their partner.  One pair even decided to guess if they rolled an odd or even number while others were invested in the throwing of dice in the trees and running to see what awaited in the grass.

Afterwards we sat together and shared our ideas and what this experience might of helped us understand.  Here were some of the ideas shared….

1. I was able to add my numbers fast.
2. I made negative numbers.
3. We subtracted our dice because it challenged our thinking.
4. Numbers are fun.  We created patterns.
5. We liked the chalk because we could change our thinking.
6. It is fun learning outside Mrs. D.

Reflecting on this experience it reminds me of the importance to weave play, movement and fresh air into our learning opportunities for young children.  Thinking now about foam dice with letters and numbers.

Counting Order, Number Recognition and Podcasting

As the new school year begins I am thinking about how the common core becomes a part of the work I do each day with my students.  I am thinking about this, but I am also thinking about “how to” create learning opportunities that will help the common core come alive and become enriching and engaging for my students as well.

So I started thinking about number recognition, counting order and how the iPad might be helpful.  These skills are important and I wanted to think about how I could keep engagement high as well as “how to” provide opportunities for my students to reflect and share what they are learning.

At a work station last week some of my mathematicians were putting number cards in counting order.  Afterwards we started  counting forwards and backwards.  One of my students was so proud of her work that I asked if she would like to share this.  She did!  So we created a podcast of her counting forwards and emailed it to her mom.  Just by having her listen to her podcast of herself counting, others were intrigued and wanted to try also.  Now we are all creating podcasts of ourselves counting forwards and backwards.  The excitement and interest in this type of activity is very important.  My students are now not only very interested in learning these new skills, but are excited to do so! The skills number recognition and counting order are part of the common core.  Just by offering a different way of thinking about “how to” show what we know, my students become interested and motivated to not only count forwards, but backwards.  Now we have opportunities to share with each other and a larger audience.  

Kindergarten has really been amazing this year!  My students have been thinking deeply about their math thinking and exploring many mathematical concepts like number recognition and counting order.

During a share session my students have an opportunity to share not only what they created, but have an opportunity to ask others what they notice!  Mathematical conversations now begin!

Comparing and Contrasting with Air Server!

As my students and I continue to experiment with Air Server we are discovering how collaborative it can be.  During our number corner time other mathematicians are able to share their mathematical ideas along with the leader.  Air Server allows us up to 4 students to share their ideas.  When this happens my students are then able to compare and contrast their ideas.  For example we were writing numbers that come before and after a certain number.  Students share their ideas and then others have an opportunity to share how they know their idea to be true. 

Later during our number corner time we also explore the date of the day.  For example; the 18th.  Using Doodle Buddy and the stamper feature on the iPad my students are able to create different ways of representing this numeral.  When they appear, students then have an opportunity to share how they know they have 18.  Some may have a group of 10 and 8, so 10 and 8 makes 18.  Another response might be a group of 4 and 14, so 4 and 14 is 18.  The collaboration is powerful because all my students are participating and not only making mathematical connections, but are justifying their mathematical ideas.

What a wonderful way for young mathematicians to compare, contrast and collaborate about their ideas.  Learning is fun, but when you have cool technological tools, it is even more fun and learning is constant!

*At times Air Server pops in and out.  Still working on fine tuning this technology.  What is wonderful about this, is that my students are also problem solving and thinking critically about why this might be happening and what can we do!  It’s all about the connection! *

Multiple Representations

A resource was shared with me that was found on twitter about a possible math opportunity for St. Patrick’s Day.  This is not a holiday I celebrate in my classroom, but I was intrigued by the opportunity to engage the children in a mathematical project using Lucky Charms to create a graph.

One of my mathematicians brought in a large box of Lucky Charms for us to explore.  Wow!  Right away the children were intrigued by the charms and what were we going to do with them.  The fun began……

Each table had a sorting tray and 1 bowl.  The children began sorting the different types of charms and talking about their features; color, size, shape.  The oats were put into separate bowls.  As the children were sorting the charms I had a great opportunity to walk around and listen to their mathematical decisions for why they were sorting things the way they were.  Some of the children noticed charms with similar shape, but the color differed..  So right away the process began of how to find a group/category for the charms.  For the most part the children really seemed to work together collaboratively and wanting the same thing, the goal was to sort all of the charms.

As the tables of children finished another opportunity appeared..  How are we going to count our charms.  The children began counting their different groups by 1’s.  Then I noticed a child counting smaller groups of charms and I asked “how are you counting your charms?”, the child said, “by 5’s”.  That was all it took for other children to notice other ways that they could count their charms.  I observed children counting by 2’s, 5’s, 3’s and 10’s.  The charms made it very easy for the children to group and manipulate. I continued to walk around and ask children how they were counting and can you show me how you know you are counting by 2’s, etc.  I was so impressed with the children’s ability to vocalize how they knew what pattern they were counting by.  One child said, “I am counting by 2’s because each charm has a partner, see?”  Then the children continued to line the charms up in rows and columns.  I asked if this was a faster way of counting(more efficient way) than counting by 1’s.  Immediately the child responded, “yes, because you skip numbers and it is in a pattern.”  Wow!  Lots of opportunities here  to extend this child’s thinking…

The children then worked with a partner and brainstormed how we could represent the number of charms in our graph.  After taking private think time and having a mathematical dyad with their partner, the children had great ideas. 

We decided as a class to glue the charms in ten frames.  This was really wonderful for the children as it provided a nice visual for them to see what the numbers were as they constructed the frames and then it made it so easy to count when we were adding the totals.  As a class we have been using 10 frames all year, so this is a way the children are already familiar with when it comes to representing numbers.  The other way we represented the number of charms in each column was by writing the actual number of the charms.  I then wanted to expand their thinking a bit and asked everyone to take some private think time to think if their was another way we could represent each number.  The ideas were shared and we decided on tally marks.  This was a great activity.  The children were making connections while doing this activity with a partner.  “Hey, each group of tally’s is 5 we can count by 5’s to find out our number”.  As a teacher I see lots of ways here that all children can participate.  The lower numbers were better suited for some mathematicians, while higher numbers that needed tally marks presented a nice challenge to a few of my mathematicians.

Using iPads the children were able to take photos and make podcasts about the different ways they used the Math Habit of Mind, Multiple Representations.  The iPads provide additional opportunities for the children to share and reflect about their thinking.

Math Habits of Mind In Kindergarten

Our K-8th grade professional teaching staff is part of an exciting collaboration with the Teachers Development Group. This collaboration was inspired by our administrative staff 2 years ago.  The main idea behind this initiative was to help our students become stronger and more confident in the area of mathematics.  After we all completed a 5 day intensive math class we were then put into collaborative teaching groups with our colleagues to begin our work around  teaching mathematics using best practices.  This was the beginning of my new learning and inspiration around the what, why and how I teach mathematics in Kindergarten.

One of the most powerful things I have learned is how amazing young children are in their thinking around mathematical concepts.  In Kindergarten I began developing a culture that not only had examples and artifacts of our learning, but ways for children to begin to use “math talk”, which is the language I began modeling in explicit ways for children to see and began to practice in their own understanding of the concepts we are exploring. Through daily, explicit modeling through our daily number corner, math dyads and other mathematical work stations the children began to apply their understanding in meaningful ways throughout the day which has helped to build self-confidence in all of the children. What is important to understand here is that I did have to add something new onto my already full plate, but rather this was an opportunity to learn some new tools and a different way of thinking about what I was already teaching. This is one example of  where I started to see how rigor and relevance applied in my teaching and how vital it is and has become in my daily teaching practice.

MHOM have been instrumental in helping not only myself, but my students begin to develop a deeper understanding of rich, cognitive math concepts.  One of the MHOM we have been exploring is Changing My Thinking.  The idea here is that it is important for children to understand that it is okay to make mistakes.  When you make mistakes, you develop a deeper understanding of their thinking.
Look at the student examples in the photographs.  Here you see 2 Kindergarten children applying their understanding of this MHOM.  These children are exploring the idea of changing their thinking to understand what they are doing.  A great example of making sense out of a simple math task.

I Changed My Thinking

Making Connections, I Changed My Thinking

I always tell my students that mathematicians change their thinking and that it is okay.  What a nice way for children to begin to build self-confidence in becoming not only a strong math thinker, but also strong risk takers who are confident in their ideas.

Another MHOM we have been spending a lot of time on is the idea of conferring.  This has been a huge challenge for me as a teacher.  Young children confer all day with each other about many different things.  Conferring about math ideas and understandings is different.  I started thinking back to the beginning of the year when I started introducing the many different MHOM to my students.  I have large posters with mini cards under the poster for children to get anytime they see, hear or see one of these MHOM being used.  Having that in place I have already established a culture of mathematics in Kindergarten because children are beginning to use “math talk” around their ideas.  The next step for me was thinking about how to teach in an explicit way, the language I wanted to expose children to and to help them begin to develop this for themselves.  So I created 3 phrases and posted them in my classroom as a reminder to everyone about how to begin talking about our math ideas. These phrases have been great “talking starters” for children to begin to use to help support them to talk/confer/generalize about their math ideas.

Open Ended Phrases That Encourage Children To Share Their Ideas…

Now as I travel around from table to table during our math work station time I begin to listen to some of the conferring that is already happening.  Some of what I have heard is; I see a pattern here  because…. that’s how I know, I made a connection with you because I noticed that too, I changed my thinking because I liked your idea better because…..  As a teacher this is where it gets tricky because I want to be able to record all of what I am hearing and have time to have meaningful math discourse with the children.  This is a huge task.  I have recently set up a math notebook that I use 2 days a week where I can check in with students and have some meaningful math discourse about their mathematical ideas through conferring.  This is a beginning for me.  This will not be my only tool, but for now a start.

Technological tools that will be helpful and vital here are the iPad, FLIP camera, podcasting and of course my SMARTboard as they all offer ways for children to reflect and share with others about their mathematical understandings.  More on that later…..

Exit Tasks

During our math workshop time we continue to focus on many of the MHOM(Math Habits of Mind) as well as improving our mathematical dyads and our understanding of exit tasks. The children use digital cameras, iPads, our document camera and SMARTboard to highlight our discoveries and understanding of math tasks.

Small Cooperative Work Around Patterns

We use exit tasks during our math workshop time to help check our understanding of the math tasks we are doing.  Sometimes we do our own math task and we also do exit tasks as groups too!  These are some examples of some group tasks that we did to show our understanding of patterns.

An ABC Pattern The children noticed that depending on where you start/look at the pattern it can be different. Wow!

 The children brainstormed the paragraph above with me after we completed a small group exit tasks around our understanding of patterns.  Each child signed the document which is now on the wall in our classroom.  It is a great artifact to have available to remind us about what an exit task is as well as how it shows us how deeply we think about math tasks in Kindergarten.

When it comes to creating exit tasks I have been thinking of how technological tools can also be used to capture children’s thinking around math tasks.  In the examples above I was able to photograph the task cards using my document camera and then insert the images into a SMART notebook page where the children could manipulate and interact with the math task.  This way the task is connected with what the children are using.  At this time of year the children are already so comfortable using a variety of technologies that the emphasis is on the learning, not the “how to” of the tool itself.

The iPads are so easy to use as well.  Children can create their understanding by using an app called; “Doodle Buddy” to draw attributes, numbers, etc. Once they begin to feel comfortable with the tool they begin to see how the tool and app can be adjusted to help clarify their understanding.  For example; I am working on helping children develop their understanding of the concepts before and after.  When asked to draw the number that comes before/after a number, the children write what they understand and then notice the size of the number and can adjust the line size and color to make it clearer or different.

The iPad offers an easy way for the children to take photos, video and even create a podcast in regards to their understanding of the math task.  I also love the easiness of being able to email these recordings to myself and parents so they can also be a part of what their child is doing.

I am still learning about exit tasks and how to make them more of a daily practice within my math workshop time.  I love exploring with different technologies to help me identify and enhance the key/high level math concepts that are important to highlight when working with deep, high cognitive level mathematical ideas.