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.
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.
Ultimately our goal as teachers is to have students use mathematics independently in a way that demonstrates a deep understanding. On the journey to mathematical understanding teachers take the wheel for a while as they demonstrate and teach the lesson, then they often get out of the car and ask the students to keep going. However, this independent practice is often too quick of a jump for students, coming before they have a true understanding of the math concepts and skills. Students can manage to keep the car on the road for a while, especially in the early grades where they can “get by” with partial understandings and some support from teachers as they go. Where the lack of understanding with independent driving stops forward motion is in the later grades where a partial understanding of the mathematics will become a barrier for learning and student confidence. Their car can no longer remain on the road, and often the students no longer want to take the wheel or even stay on the journey. By attending to a gradual progression toward independence teachers can support students where they are and help them confidently take the next step on their learning journey.
The Gradual Release of Responsibility Model, first developed by Pearson & Gallagher (1993), is a research based instructional model that outlines the process necessary to promote independent application of skills and understanding. In this model the teacher gradually decreases his or her support as students’ demonstrate success and if necessary, increases the level of support when students are struggling. This model is used constantly in the classroom as a way of helping teachers identify where students are and as the basis for future, targeted instruction.
Teachers are very skilled in I do it (I’ll drive) and You do it alone (You’re in the vehicle alone) phase, but it is the middle two phases We do it and You do ittogether where we can focus more attention. “We do it” is like sharing the wheel with the students -sometimes the teacher drives and sometimes the students do with the teacher remaining in the car. In the “You do it together” phase the teacher is also in the car but is only a passenger. The drivers are the students who share the wheel as they move their learning forward. The middle two phases consist of a lot of student discussion and collaboration…it can be messy, noisy and has the potential for students to get off task. However, it also has the greatest potential for helping students become truly independent with a richer, more complete understanding of the concepts and skills.
Here are some ideas of what these two phases can look like in mathematics:
This chart is far from complete but it can be a start to professional thinking and discussion. I believe that if we, as teachers take the time to focus our instruction on these two phases we can not only foster independence but understanding as well. If we can remain in the car longer we can help guide the journey, model the way and ultimately support students moving their own learning forward. We need to support them as they learn to drive, and then know when they are ready for us to get out of the car.
Like with any model there is danger in getting “locked” in to a linear model such as this one can appear to be. In life and student learning nothing is ever cut and dry, black and white. I think the power of the model comes from the thinking it can open us up to as professionals. Are students ready for independence? How much support do they need to experience success? From me? From their peers? These questions can lead to actions that can turn a student learning challenge into a success.
Recently I have had the pleasure of connecting with teachers who are new to teaching math and were looking for suggestions for setting up their classroom to support Mathematics. On their own they had explored the internet, the teacher stores, other colleagues classrooms looking for ideas about how to create an environment that fosters student mathematical thinking, working, collaborating and independence. Their questions have led me to reflect on my own classroom set up in the past and how I would set it up today. Here is what I would need in my classroom to support mathematics:
Math Area/Corner- Just like how every classroom needs to have a class library to foster student literacy I believe that every classroom needs to have a math area to foster mathematical literacy. This area, be it a shelf, bookcase, an area on the floor, can house the “tools” students will need to access during Mathematician’s Workshop. These tools would include manipulatives (both purchased and natural items), containers to store them in and graphic organizers that students might need (10-frames, 100 Chart, Base Ten Chart, blank Venn Diagrams…). I also like having several small baskets in this area which I call math tool boxes. These baskets can house all materials which students will need for the day and save time in the distribution of manipulatives in the classroom. As an added bonus I find the use of “Tool Boxes” helps foster student independence during cleanup and makes it easier for them to keep their manipulatives contained while they are working.
A Math Tool Box- very handy for distributing manipulatives
Math Books- These would be a basket or several baskets of literature, both fiction and nonfiction, which has a mathematical slant or theme. The goal with allowing students to access mathematical literature is to allow them to extend and expand their mathematical thinking, foster connections and allow them to see mathematics in a variety of situations. For some ideas about possible math books to include check out my post on My Favorite Math Books.
Math Wall- This wall would contain mathematical words, definitions, sentence frames, representations anything to foster understanding of mathematical vocabulary and student independence. I would also include on the wall co-constructed anchor charts about math concepts students are currently working on, photos of math in the world, student work and norms for mathematician’s workshop. For more information on Math Walls check out my post Why Math Walls?
If we truly value math in our classrooms then math has to have a place in our rooms as well. In so doing students will see its importance, be able to access supports, and explore the mathematical concepts. By having a place for math language, math tools and math literature and by supporting students use of them we can start to create a math rich classroom.