Friday, May 9, 2014

Teacher for a day!

As a teacher, I am always looking for ways to create opportunities for my students to showcase their abilities as well as demonstrate their knowledge. Designing projects and activities that develop higher-level thinking around a chosen topic is just one way to achieve this.  Ideally, it's the students who create these projects and activities themselves. Just recently, the second grade math students were lucky enough to have the opportunity to learn something new and fascinating - yet this time, it was from one of their peers.

A student approached me with an interest in Fibonacci numbers.  She was clearly interested in the concept and was eager to share this new information with her classmates.  We discussed the material and after assessing she had a firm grasp of the information, I suggested that the following day she teach a lesson to the class explaining Fibonacci numbers.  We talked about a structure for her lesson and what she would need to do to prepare.  It was clear she was excited, and I knew this would be a valuable learning opportunity for all.  After all, in order to teach something effectively, one must have a firm understanding of the material, and feel comfortable explaining it to others.  What better way to extend one's knowledge than by having the opportunity to teach?

The next day the student arrived eager and ready.  She wrote out the classroom morning message, outlining what would happen in math that day. She had even prepared hand-outs for the class as an extension to her lesson.  As she began her lesson, her classmates listened attentively, asking questions and offering comments when appropriate.  When it was time for them to complete the classwork, she circulated around the room, offering help and encouragement to her peers.  I was amazed at the depth of knowledge that was gained from this experience.  Beyond the mastery of the mathematical material, this type of activity can instill confidence and a level of engagement that comes with pursuing a genuine interest.  The other students clearly benefited as well - watching a peer stand up in front of the class and direct a lesson was fascinating, engaging, and inspiring.   After the lesson, we discussed the experience and I encouraged everyone to find an exciting mathematical concept or area of interest to present to the class.  There were lots of ideas generated, and I'm hopeful that others will be inspired to follow suit.

morning message

Fibonacci numbers are circled 

the prepared hand-out 

helping friends 

the proud teacher 

fully engaged students 

working together 

completed work, corrected by the teacher!  

While it’s unlikely that you have ever heard a person say, “that worksheet changed my life,” most people have an assignment from their childhood that they remember with pride because it was meaningful to them. More often than not, that memorable assignment was one that allowed them to express themselves and follow their passions. It's been a joy to watch the second graders find these passions and pursue them.   Other passion projects in the works: a math newspaper, origami tutorial, game making 101, prime numbers, and multiplication charts!

Monday, April 28, 2014

The second graders approached our recent unit on fractions with their usual enthusiasm and curiosity. We explored the concepts of one whole, one half, one fourth, one third, as well as more complex fractions such as one eighth, two thirds, and three fourths. This was a unit where real life examples were useful and provided context for the students to think about these ideas. Half of a cookie, one quarter of a pizza, one third of an apple - these were all opportunities for the students to visualize what these fractions would look like in real life.

After much exploration, the students made their own fraction books, an activity that provided additional practice and an opportunity to further display their understanding of the concepts. Each book was unique and detailed - some students wrote and illustrated stories exploring the concepts, others drew complex pictures of various fractions, and some even posed questions and answers to their readers. Creating these books allowed students to move beyond simply accessing this new information. Possession of facts cannot be the bulk of learning. What is an important skill is the ability to sift through abundant information, identify what is valid and meaningful, then use it to create meaning and express it. Writing and illustrating a book related to our topic, and presenting it to their peers, illustrated that deeper learning had occurred.  This was another example of the importance of student creation.

A student's book 

Hard at work!

Making books 

The second graders have also been extremely interested and motivated to create their own math games to play with friends. Guidelines have been established to allow for personal creativity, yet still extend and broaden students' knowledge of the particular unit we may be studying (first fractions, now more complicated addition). Far beyond filling out answers on a worksheet, these assignments allow for individual talents and personality to shine through and provide students an additional way to illustrate their understanding of mathematical ideas.

Math draws much of its meaning by relating to life in the world outside the classroom. Children seek to understand the world by inquiry and investigation. Giving students ample opportunities to develop sound investigative skills at an early age is essential to nurturing their ability to think critically and mathematically as they get older. Activities such as creating books, playing and developing math games, and other hands-on activities provides a wonderful and critical opportunity for learning.

More books! 

Math can even provide inspiration for creating new words!  

Students create their own math game 

More addition games 
Quizzing one another on math facts 

Practice makes perfect!  

Thursday, March 20, 2014

Games, graphs, and origami

Amazing things happen when children are given the freedom to explore, notice, and create things of their own choosing.  The unit we just finished, "How Many Tens? How Many Ones?" allowed students to explore our base ten number system, specifically in relation to 100.  This meant addition and subtraction up to 100, counting by 2's, 5's, and 10's, skip counting, and the exploration of early multiplication and division concepts.  Students all worked on mastering these ideas, but something equally important developed as well - passion to pursue one's own mathematical interests.

Making origami 

It all started with curiosity about multiplication. Students began to notice certain patterns in the numbers and answers they were getting to problems.  This led to further exploration, and for some, a personal journey into new material.  One student began work on a prime number chart.  Making a chart of all the prime numbers, testing and retesting findings, led this student to amazing discoveries along the way.  A few students became passionate about origami - folding paper to make intricate and elaborate 2 and 3-dimensional structures.  We counted the number of shapes and angles, and students were able to view their objects in a new, mathematical way. Students also worked together to make a math game, adding and subtracting, and solving math problems as they moved along their game board.  Some students took favorite subject matter and decided to make thoughtful and detailed graphs.

Exploring multiplication 

Brainstorming ideas for a graph
This chance for free exploration of math concepts is critical for children to embrace math and enjoy the experience.  Giving children a voice in the activities they wish to participate in allows them to make discoveries on their own.  They notice patterns, make connections, and gain a deeper level of understanding when they have invested in their own area of interest and have opportunity to freely explore ideas.

This is a wonderful article about this very idea. Looking at these photos, and spending time with these inquisitive and highly motivated children during math, clearly illustrates the benefit for free play, games, cooperation with peers, and choice. Math connections can be made everywhere, children just need the freedom to explore and find inspiration from a rich and social mathematical environment.

Double checking work 
Counting coins

Playing a newly created game
The graph takes shape

Exploring the concept of 100 
We brainstormed a list of ideas 

How many more cubes do we need to get to 100?

Sunday, February 9, 2014

Favorite pets, birding, and more!

We recently completed our unit called Pockets, Teeth, and Favorite Things.  The unit focused on data collection.  The term data was explained to the students as the facts or information we collect about people and things in our world.  There was ample opportunity for real world connection to clarify ideas in this unit.  The students surveyed their friends about their favorite pets (or other favorite things), sorted, and then represented this data in ways that could be understood by others.

We started the unit with a brief survey of everyone's favorite weekend activity.  The students wrote down their favorite activities and put them on our large classroom chart.  Once everyone's activity was present, it was time to organize our data into a meaningful way that could be easily understood.

Placing data on our classroom chart

The very beginnings of our class data collection

In subsequent math classes, we made graphs of the data.  The students noted that it was much easier to interpret and analyze the data once it was clearly organized in some way.

We also talked about Venn diagrams and how to sort data by different attributes.   The students made their own Venn diagrams and sorted data by two attributes.  Partners took turns guessing attributes and sorting information.  This early work in classification provided experiences in considering only certain attributes of an object while ignoring others.

Sorting objects by their attributes

Using Venn diagrams to help sort the data

More sorting and discussion about attributes

Noticing differences in data

To further connect this unit to real life, we took advantage of our beautiful surroundings
and went outside on our monthly bird walk.  This time even the 3rd graders joined us on our adventure.  Students keep track of what birds we see outside, and back inside we enter the data on the ebird website and incorporate it in our classroom bird graph.  

Outside on our bird walk

Now it was time to represent our data clearly.  Students used a variety of representations: line plots, bar graphs, pictures, pop up charts, and other creative means to show the amount of birds that were spotted.  It was wonderful to see how each student approached this task with gusto and enthusiasm.  The data we had collected held real meaning for them (these were birds we saw on our school property, after all!), and they each worked to their fullest potential to display the data clearly.

Beginning to represent our data

A variety of representations were used

During this unit, students were introduced to various frequency distributions in which each piece of data is represented by one symbol (e.g., an X, a picture, a sticker, a square, or even a sticky note).  In using this kind of representation, students have to think through the meaning of two ways numbers are used in describing data:  Some numbers indicate the value of a piece of data; other numbers indicate how often a particular data value occurs.  The Second graders were able to describe the data by considering the frequency of each value, the mode, and the highest and lowest values.

By participating in an entire data investigation from start to finish, students encounter many of the same issues encountered by statisticians as they decide how to collect, keep track of, organize, represent, describe, and interpret their data.  All this while having real life experiences to tie things together - interviewing friends, talking about their favorite things, and bird watching!

Sentiments from a Second grade mathematician