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.
Ten Black Dots
Creating A Ten Black Dots Book with your students.
Press Here
12 Ways to Get to 11
Activity Using Cuisenaire Rods
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:
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.
Subitizing-What is it? Why Teach it?
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.
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Reading |
Math |
Making Connections
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Making Connections
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Visualizing
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Visualizing
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Inferring
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Inferring
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Determining Importance
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Determining Importance
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Synthesizing
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Synthesizing
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Monitoring and Repairing Comprehension
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Monitoring and Repairing Mathematical Thinking
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Questioning
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Questioning
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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).
cc.- http://www.flickr.com/photos/steveritchie/6067642964/
Siena, M. (2009). From reading to math. Sausalito, CA: Math Solutions.
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
I remember the day my world fell apart in the face of new pedagogy in mathematics. I was working in a rural school division in Saskatchewan and was in the middle of a Cognitive Coaching workshop with 50 people that I did not know… and the facilitator asked for a volunteer to model a problem solving conversation.
After teaching for 14 years I had reached a point where I felt I did not know anything about teaching and learning. I ended up being a crumbled mess on the floor of that workshop – embarrassed at my emotional outburst and doubting my ability to teach, or facilitate, or guide, or whatever it was I was supposed to be doing.
That was six years ago.
So, how did I get past the Earth shaking moment when I lost my ability to believe I could meet the many challenges that I was facing?
So here I am six years later, and to be honest what I know now is not necessarily more than what I knew in the past – in fact, I know even more about how much I don’t know. But I no longer feel powerless. I know that there are people I can learn from and learn with. That has made all the difference.
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:
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.
Kindergarten Colours
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
N1.6- Estimate quantities to 20 by using referents.
N1.9- Demonstrate an understanding of addition of numbers with answers to 20 and the corresponding subtraction facts, concretely, pictorially, physically, and symbolically.
P1.1- Demonstrate an understanding of repeating patterns (two to four elements)
SS1.4- Compare 2-D shapes to parts of 3-D objects in the environment.
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:
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:
Making Math Meaningful to Canadian Students by Marian Small
Introduction to Problem Solving by Susan O`Connell
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:
A mathematician…
O’Connell, S. (2005) . Now I get it. Portsmouth, NH: Heinemann.
I haven’t always seen myself as an accomplished learner. In fact, I went through most of my elementary and secondary experiences with a sense of incompetence. I watched, somewhat helplessly, as my learning was unraveling all around me. I struggled with my vision of academic excellence and much of my time was spent trying to reinvent myself to fit a mold of traditional teaching and learning that left me hollow and standing alone.
This evening I came across a quote from Will Richardson’s new professional learning resource “Learning on the Blog” and it resonated with me to my very core.
“In many ways, our connections define us as learners, especially today when we can make so many connections online. And we are not just connecting to people; we connect to content and organizations and ideas. No one dictates what connections we should make or what networks we should join. We are driven to those complex choices by our own passion to learn. However we get there, we are active participants in the process, and the process itself is shared to deepen the learning. How are you connecting? How are you adding value in the context of those connections?”
It seems like such a simple idea… Learning is about making connections in a world that is becoming more and more connected. I have come to know that deep, authentic learning is made possible through connections. Connections continue to be a powerful model that supports, enhances, and continues to allow me to engage in my learning. In this vein of thinking, I have come to some important realizations that even I was surprised to find myself making.
In my role as an Educational Consultant I have spend a considerable amount of time making sense of supporting teachers, and subsequently student learners, in the intricate details of Mathematics. Sadly, a subject area that still continues to elude many of our learners, young and old. In our work we have relied substantially on The Strands of Mathematical Proficiency, from the professional resource Adding It Up: Helping Children Learn Mathematics, as we come to better understand what it means for our learners to be mathematically literate. My connections are far reaching as I make sense of this complex process of understanding.
In Mathematics, Productive Disposition is the belief in one’s ability and efficacy; view of mathematics as sensible, useful, and worthwhile. This is directly paralleled with how I aspire to include technology in my everyday teaching and learning. Being open, willing to take risks, transforming my current practice, and engaging my students in learning in ways I have never dared to dream or imagine.
When I think about Strategic Competence in Mathematics it is being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately. As I envision a learning community connected with technology, I see its authentic incorporation as a necessity. I need to be tenacious as I learn of not only it’s intricacies but also embrace the complexity. Higher order thinking allows for the fine balance of competence and confidence to be attained in the most meaningful of ways.
Comprehending mathematical concepts, operations and relationships is the foundation of Conceptual Understanding. What does that entail when it comes to the value of integrating technology? It is about engaging and navigating the world wide web in search of taking my teaching and learning to the next level. The opportunity to connect and collaborate creates the conditions for innovation to be a game changer for learning.
Procedural Fluency is the skill in carrying out procedures flexibly, accurately, efficiently and appropriately. My desire to engage in my learning in the most authentic of ways is made possible by web 2.0 tools. I see how technology can support, enhance, and differentiate the amazing opportunities already taking place in the classroom. Having a solid understanding of technologies has also afforded me a new found passion and zeal in learning. This confidence has captured a creativity I once possessed but I thought was long forgotten.
Lastly, Adaptive Reasoning is a capacity for logical thought, reflection, justification and explanation. The beauty of technology is that I can capture my thinking though this lens and share my synthesis globally. The secret to my success is the power of meaningful connections On the flip-side, when I encounter an obstacle that in the past seemed so insurmountable, I can connect with a community willing to problem solve along side me. Our strength comes from a mindset that showcases the power of collaboration.
It occurred to me that in my own learning journey I have had the privilege of growing my understanding by connecting my learning, which has afforded me the added value of strengthening that knowledge through collaboration with generous colleagues. It has taken me many years to feel confident in my gifts and talents as a learner, no matter what the subject might be. It is through this gift that I share my learning with you.
I look forward to hearing your thoughts.
