My name is Jennifer Brokofsky and I am an Instruction Consultant with Saskatoon Public Schools. I am first and foremost a teacher who strives to learn alongside my students, and colleagues. I enjoy expanding my understanding in all areas of education but am particularly passionate about mathematics, science, and the transformational use of technology in and outside of our schools. I believe that students are the heart of all we do, and the most important people in our learning environments. My job is always to facilitate their learning by creating the conditions for inquiry, and wonder. I look forward to connecting and collaborating with educators near and far who share my passion for learning and children.
The opportunity to move the learning outside of the classroom can often serve to engage students in mathematics. Math can come alive and be seen in the world beyond the four classroom walls. One way to move math out of the classroom is with math trails. Math trails are like scavenger hunts with a math spin to them. As students move along the “course” they have opportunities to explore, observe, discuss, and connect with math in the world.
To get you started on designing your own math trails the NCTM has a great article called Designing Math Trails in the Elementary Classroom. They have ideas on how to select a site, on- site activities, classroom follow through activities, and cross curricular connections.
The Canadian Math Trail is a great site for ideas about math trails created across Canada by teachers.
Spring is finally here and the weather has been beautiful. I often look longingly out the window and think about how much I want to be outside. But why not? Why not go outside?
One of the beautiful things about math is that it does not need to be confined by the four classroom walls or a desk. It lives in the world beyond our classrooms… so why not take the learning outside? Last week at Mayfair School I noticed students working outside with a variety of math “tools”. When I went out to inquire into what they were doing their teacher, Ms. Nancy Barr shared with me her outdoor measurement stations. She said that she needed her students to measure “big things” so that they can expand their understanding of distance, estimation and measuring. So Marvelous Math Measurement Day emerged. Students worked through stations that included: measuring the length of the city block, measuring the distances they could throw various objects, and measuring their bodies. As I walked around enjoying the day, the math, and the learning I heard conversations filled with laughter, mathematics, and enthusiasm. It truly was a Marvelous Math Measurement Day.
There are some great print and online resources out there for supporting students subitizing skills. I have highlighted some of them below along with a website you can visit for ideas on extending the learning beyond the book.
Subitizing is the ability to instantly see how many in a small collection of items without counting. Dots on a die, shapes on a playing card, number of fingers held up on a hand, are all examples of subitizing in action. In order to subitize successfully students need to see the whole as a collection of objects as well as the individual units. Subitizing is considered to be a fundamental skill for supporting students understanding of number and ability to perform number operations.
In the primary years students should be given regular and consistent opportunities to subitize in order to build their skills, improve number sense and lay the foundation for future mathematical learning. In kindergarten numbers to 5 should be focused on for instant recognition. Once students are familiar with familiar representations of 1 to 5, larger collections can be used to encourage students part-part-whole thinking. For example, on the card below students may instantly recognize a three and a four and then add the numbers together to know that there is a collection of seven dots on the card.
As the collections get larger students can be encouraged to use their estimation skills to think about “how many” and “how do you know”. Our Saskatchewan Curriculum refers to this fundamental skill through several outcomes from Kindergarten to Grade 2:
Kindergarten- NK.2 Recognize, at a glance, and name familiar arrangements of 1 to 5 objects, dots, or pictures.
Grade 1- N1.2 Recognize, at a glance, and name familiar arrangements of 1 to 10 objects, dots, and pictures
Grade 2- N2.1 Demonstrate understanding of whole numbers to 100 (concretely, pictorially, physically, orally, in writing, and symbolically) by:
representing (including place value)
describing skip counting
differentiating between odd and even numbers
estimating with referents
comparing two numbers
ordering three or more numbers.
This video is an excellent example of a kindergarten teacher who is using Quick Images to build on her students subitization skills, and create opportunities for mathematical conversation.
Reading in Mathematics? When I was a student in elementary school these two subjects were not only separate but almost complete opposite. Today however, my perception has changed and I see more similarities between the two than differences.
First of all, there IS reading in mathematics…reading of textbooks, word problems, literature, the whiteboard…the ability to read supports the student’s ability to take in information, comprehend problems and creating meaning.
Secondly, reading at it’s very essence is thinking. It is the interpretation of a set of symbols (letters and words), and using our understanding of the symbols to create meaning. This process must involve thinking. Likewise, mathematics is the interpretation of a set of symbols (numbers, objects, representations, letters, and words) to create meaning, and gain understanding. This process must also involve thinking. Since both subjects are looking to strengthen thinking it only makes sense that we use the strategies and supports for strengthen student thinking, and comprehension in reading to strengthen thinking and understanding in math. Creating consistency between the strategies can foster students ability to make connections and allows them to build on an existing foundation within a new context.
In reading we use the Super 7 Reading Strategies to support thinking and comprehension. In mathematics these same strategies can be built upon to support mathematical thinking comprehension.
Text to Self
Text to Text
Text to World
Math to Self
Math to Math
Math to World
Creating a mental image to help construct meaning
Creating a mental image to help construct meaning
Reflecting on reading
Reflecting on mathematical thinking
Determining topic and main idea
Determining author’s message
Using knowledge of narrative or expository text features/structures
Determining what is given in the problem
Determining what we are being asked to discover
Using existing knowledge in mathematics to solve new problems
Reviewing, sorting and sifting through information leading to new insight as thinking evolves
Reviewing, sorting and sifting through mathematical problems and information which leads to new insights in math
Monitoring and Repairing Comprehension
Monitoring understanding and knowing how to adjust when meaning breaks down
Monitoring and Repairing Mathematical Thinking
Monitoring understanding and knowing when to stop and adjust (when thinking breaks down)
Identifying where thinking broke down and trying another solution
Clarifying meaning by asking questions before, during and after reading to deepen comprehension
Clarifying mathematical thinking by asking questions before, during, and after solving problems to deepen understanding
Asking questions of others about their strategies for solving problems.
So next time you are explicitly teaching comprehension strategies to your students in reading consider the possibility of expanding on those strategies in mathematics. As Maggie Siena (2009) so eloquently puts it “we can become more effective teachers of mathematics by drawing from our successful experiences with teaching literacy. It’s the art of lighting two candles with one flame” (p.2).
Exploration by it’s very nature is a step into the unknown. Armed with no map, no set of steps, no set path, the explorer embraces the excitement of discovery with a willingness to get lost in the adventure. Imagine for a moment exploration that was planned, with every destination known, every step predetermined…it would not be as much fun, it would not belong to the explorer, and it would not be exploration.
Mathematics is a place where exploration is not only possible…it is necessary. Through exploration and play we can breath life into the learning of mathematics, as we open up opportunities for imagination, creativity curiosity and wonderment. These qualities can carry the mathematics beyond the textbook, the worksheet, the drill and practice. They can make mathematics come alive. This exploration needs to belong to the explorer and the explorer in every learner needs to be given opportunities to discover and create in mathematics. There may be times when we would want to give students the map with the route laid out and support them on reaching the destination but there should also be times where students face a problem or situation and need to reason your way to the other side, without the road map. In those times it becomes about finding their own way, embracing the different but equally valid paths/solutions of others, and truly discovering not only the science of mathematics but it’s artistry and creativity as well. This exploration does not need to happen in isolation. Explorers can join resources and thinking power to help navigate the journey and share in the excitement of discovery.
In this video Dan Meyer‘s describes the problems facing mathematics education when we take all of the exploration out of the subject.
1. Lack of initiative- Translation- I don’t want to do it!
2. Lack of Perseverance- Translation- It’s too hard! I give up!
3. Lack of Retention- Translation- I don’t know. Blank look.
4. Aversion to word problems- Translation- Can I just have a sheet of numbers.
5. Eagerness for a formula- Translation- Can you just show me how to do it?
By purposely and deliberately creating an environment where mathematics exploration and discovery flourish we can harness the inner mathematics explorer in our students.
Exploration is really the essence of the human spirit. Frank Borman
An anchor on a boat can hold it in place, prevent it from going adrift, off course or lost. In the same way anchor charts can hold thinking. They can help learners get back on course by reminding them of discussions, activating prior knowledge, and point them in the right direction. In mathematics anchor charts can be powerful instructional tools that support and enhance student learning.
Communication is an important process in our Saskatchewan Mathematics Curriculum. To effectively communicate in mathematics students need to represent their mathematical thinking concretely, pictorially, symbolically physically, verbally, in writing, and mentally. With so many ways to communicate it is important that teachers support students in learning how to effectively communicate. Anchor charts provide an opportunity for teachers to model thinking in writing, support and record the learning of the class, and build mathematical vocabulary. By supporting communication in your mathematics classroom you can help students to clarify, reinforce, and modify their understandings of mathematical ideas.
To realize the potential that anchor charts can hold it is important to consider a few things when using them in mathematics:
Students will use them more if they have helped create them. To prevent them from becoming wall paper co-construct them with your students.
Refer to them frequently during instruction and practice. If a student asks a question which can be answered on the chart refer them to the chart.
Focus on one mathematical concept per chart. Too much on a chart can limit its effectiveness and create visual overload.
Use mathematical vocabulary. These charts will become a jumping off point for mathematical vocabulary building, discussions and writing.
Don’t limit yourself to text. Pictures, graphic organizers, models, even actual manipulatives can be taped to the chart to support the concept. Just remember to not overload the chart. White space can make learning easier.
I Can…. Chart
Thanks to Holly Portas from the Saskatoon Mizbah School for sharing her grade 2 Pattern Anchor Chart.
Math Wall in Progress
Thank you to Laurel Clark from Forest Grove School for sharing her Grade 1 Math Word Wall.
Thanks to Erin Broughton from Lester B. Pearson School for sharing her co-constructed colour charts. These are a great way to study attributes and sorting rules.
Steps to Solve Word Problems
Thank you to Debbie Durand from Forest Grove School for sharing here problem solving anchor chart.
Connecting Ideas- Grade 2 Linear Measurement Concept Map
Co-constructed concept maps can be created as students learn about each idea of a concept. As the unit develops so does the map and with it student understanding.
In Saskatoon Public Schools the Picture Word Inductive Model has become a powerful approach for teaching and learning in Language Arts, Science and Social Studies. The rich photographs provide an amazing opportunity to look into a different place, culture, and experience and then connect that to the learning in those subject areas. But can it be used to support teaching and learning in Mathematics? I believe that it can. In mathematics one of the ideas teachers and students strive to achieve is the ability to connect the math in the classroom to math in the world. The PWIM photograph can provide a “window” into the world and this “window” can be opened to mathematics.
For example, in this Grade 1 photograph of a market math is everywhere. There is math in the street, in the produce, in the shapes, in the people…everywhere. As I look at the photo through a math “window” I can see several mathematical outcomes that could be explored and extended.
N1.1- Say the number sequence, 0 to 100
Students can count vehicles, fruit, boxes and people and discuss why would we want to know this information.
N1.6- Estimate quantities to 20 by using referents.
How many bananas do we think might be in the box?
N1.9- Demonstrate an understanding of addition of numbers with answers to 20 and the corresponding subtraction facts, concretely, pictorially, physically, and symbolically.
Can we (teachers and students) use the context in the photograph to create and solve problems involving addition and subtraction?
Example: How many white boxes plus how many brown boxes= total number of boxes?
P1.1- Demonstrate an understanding of repeating patterns (two to four elements)
Looking at the men’s shirts what is the pattern we see? Can we recreate this pattern?
SS1.4- Compare 2-D shapes to parts of 3-D objects in the environment.
What are the shapes we see in the photograph? How do the shapes compare?
Using PWIM photographs to extend mathematical concepts and foster connections to the real world can create a rich and powerful learning opportunity for students. By looking at your mathematics curriculum and your photo the window to mathematics in the real world can be opened.
In Saskatchewan our provincial curricula are built around one central core…inquiry. Inquiry is a philosophical approach to teaching and learning that empowers students to explore and discover. Through inquiry students are active participants in the creation of understanding and knowledge. They are asked to be curious, to wonder, to question, to reflect, to share and to think
In Mathematics inquiry is often seen through problem solving. By collaboratively solving and creating mathematical problems students construct and deepen their understandings of concepts, strengthen their strategic competence, and develop their logical reasoning skills. As students engage in problem solving they need to attend to 4 important stages in the inquiry/problem solving process:
Stage 1: Understand the Problem- This very important stage is often overlooked in the classroom. It is often assume that students understand the problem in front of them but this assumption can by costly. Without a solid understanding of what they are being asked to do, students are often stumped before they start. Time spent carefully dissecting a problem and ensuring understanding can be critical to success.
Stage 2: Make a Plan- Just like the construction of a building, the construction of a mathematical solution requires forethought and a plan. Having a carefully considered plan of attack can help students construct their solution successfully. Problem Solving plans or strategies can include:
Acting it out
Using a model
Drawing a picture
Guessing and testing
Looking for a pattern
Making a chart or a graph
Making an organized list
Using logical reasoning
Stage 3: Carry out the Plan -In the words of Nike - Just Do It!
Stage 4: Look Back - This stage involves careful reflection, checking to see if your answer makes sense, and considering the solutions of others. Considering the solutions of others and comparing them to your own provides an opportunity for understanding and learning to be collaboratively constructed in the mathematics classroom.
For more information on Problem Solving in Mathematics check out these great resources avaliable in the Saskatoon Public School CMC:
September…in teaching it is a month of new beginnings, many possibilities, new crayons and community building. September community building lays the foundation for learning in the classroom, develops relationships, establishes norms and provides an environment where learners can thrive. Community building is often seen as not specific to any one subject, however the process of establishing a community of learners that is subject specific can lay the foundation of learning within each area. In mathematics September provides an opportunity to create a Community of Mathematicians. This community of mathematicians can create positive interdependence within the classroom, promote interactions, build group skills and allow for the possibility of group processing (O’Connell, 2005).
One way to build community amoung mathematicians is to construct with students an anchor chart about the question “What does a mathematician do? This can create rich discussions and some possible answers such as:
listens to the ideas of other mathematicians
encourages other mathematicians
knows what to do in mathematician’s workshop
takes time to take all the information he (she) has and puts it all together
stops and thinks
writes to remember her (his) thinking
shares his (her) thinking with other mathematicians
O’Connell, S. (2005) . Now I get it. Portsmouth, NH: Heinemann.