Tag Archives: conceptual understanding in math

Place Value in Primary: Developing Number Sense

Place Value in PrimaryWow. I have had the most extraordinary summer. Truly extraordinary. And somehow between engaging in a series of remarkable, life-affirming adventures ¬†I have managed to write another teacher resource book… ūüėä

It’s all about Place Value (as I’m sure you’ve figured out!) and is intended for teachers of kindergarten through grade 2, with special accommodations for¬†those who teach in combined grades settings. ¬†There are 230 pages of developmentally framed lessons designed to address the diversity in our primary classrooms. Each one supports¬†students to represent and describe quantity, to compare and order sets, to use referents to estimate and to skip count. Lessons devoted to measurement — an ideal practical application of¬†place value in the world — are also¬†featured. Whole class lessons, centres tasks and games ¬†for practice allow students to connect these important concepts in a seamless way, and can be used both as a unit or¬†spread throughout the year to build and consolidate understanding.

Place Value in Primary: Developing Number Sense is available from my online store for $40 plus shipping. I hope you enjoy it!


(PS…¬†A companion volume for Grades 2 to 4 is in the works – expect it later this fall!)


Another New Resource! Multiplicative Thinking – From Skip Counting to Algebra (Grades 3 to 8)

cover Multiplicative bookFor those of you about to return to another school year, welcome back!

I am truly excited to announce the release of my newest teacher resource book: Multiplicative Thinking: From Skip Counting to Algebra¬†(Grades 3 to 8).¬†This book is¬†designed for¬†teachers of the intermediate grades and is focused on the teaching and learning of multiplication. This resource addresses multiplication deeply ‚ÄĒ what it means to multiply, when to use multiplication in problem-solving situations, as well as how to manipulate whole number, fractional and decimal factors using strategies like the distributive property.

Lessons on skip counting, patterns in the multiples, factoring, and on prime and composite numbers are included in this 220 page teacher resource. Algebraic thinking is explored as well, from T-charts and input-output machines to solving equations, from graphing linear relations and extrapolation to finding the slope of a line. Students engage with visuals and real-world problems involving proportionality, rates, discounts and taxes to build their understanding of multiplicative thinking and see its very real application to their everyday lives.

Each of the 40 lessons features a connection to prior knowledge, whole class and small group explorations of the Big Math Ideas, guided conversations about the mathematics with key vocabulary, opportunities for meaningful practice, tasks for consolidation and customized assessment tools. Skill building lessons are interspersed throughout the book, ensuring students recall and continue to practice the essential skills needed to apply multiplicative ideas.

And of course literature links and games for practice are ‚ÄĒ as always ‚ÄĒ included!

Multiplicative Thinking: From Skip Counting to Algebra (Grades 3 to 8) is available for $40 + $10 expedited shipping. To order, click here or on the link at the right. From there you can also order other titles, including Mastering the Facts: Multiplication, a resource dedicated to the teaching and mastery of the critically important multiplication facts. It’s a perfect complement to this new volume and one that can be used in advance — or concurrently — to build a solid foundation.

Thank you for your support. All the best for a remarkable school year!


Why Multiplicative Thinking? 

Multiplicative thinking plays an enormous role in elementary and middle school mathematics. So much bigger than simply knowing the facts ‚ÄĒ a critically important aspect ‚ÄĒ the ability to think multiplicatively is essential for success with almost every other mathematical concept, from ratio and proportionality to algebra. It is the operation most often used in ‚Äúreal life‚ÄĚ to make sense of large quantities, of taxes and discounts, of income per hour and kilometres travelled. It‚Äôs the operation we use when we figure out how much paint or carpet to buy or what a tank of gas is going to cost; when we convert currency for a holiday away or sort out how much to tip on a meal. No matter where we look, multiplicative situations abound. We can‚Äôt spend too much time on the teaching and learning of these critical concepts!¬†

In writing this resource, I have attempted to introduce multiplicative thinking ‚ÄĒ both the operation itself and the bigger concept of multiplicative reasoning ‚ÄĒ in a sense-making way. Through stories, models, pictures and words, students are introduced to the idea of multiplication as ‚Äúgroups of‚ÄĚ and as ‚Äúrows of‚ÄĚ. Problems are posed to support learners in connecting what they know about patterns in the multiples to proportional situations. The associative and distributive properties are introduced and applied. Algebraic concepts ‚ÄĒ input and output machines, graphing and exploring the rate of change in linear relations ‚ÄĒ round out the topic and provide a preview for multiplicative reasoning at the middle and high-school levels.¬†

Announcing…. New Cuisenaire Rod Resources for Grades K-3 and 4-7!

Cuisenaire rods in a pileI’ve said it before… ¬†Cuisenaire rods¬†rock. I was first introduced to Cuisenaire rods in 2006 by my dear friend and mentor John Van de Walle. In February of that year I was invited to join him in Spokane, Washington and to observe¬†as he did demonstration lessons in primary and intermediate classrooms. ¬†His problem-based lessons featured Cuisenaire rods, and while I watched the students responding to the openness of the tasks and the richness of the manipulatives¬†I knew¬†I was witnessing¬†something powerful. The proportional relationships between the pieces and the colour wheel connections make Cuisenaire rods both aesthetically pleasing and mathematically significant. I was hooked. It took some time and experimentation to figure out how to best introduce¬†the materials and to sort out what tasks and questions would promote thinking across the grades. But in the end, I’ve managed to compile a series of open-ended tasks, games, lessons and practice opportunities that are developmentally sequenced and laid out from Kindergarten to grade 7. ¬†I’ve compiled these lessons in 2 full-colour volumes ‚Äď a primary resource (grades K-3) and an intermediate resource (grades 4-7). ¬†All of the tasks have been tested¬†and refined to ensure they are classroom-ready and engaging for all! So it is with great humility (and no small sense of accomplishment!) that I announce the release of my latest resources ‚Äď Remarkable Cuisenaire Rods: Mathematical Tasks for Primary Classrooms and Cuisenaire Rods Rock: Exploring¬†Multiplication and Proportionality in Grades 4-7. These resources would not exist if it weren’t for the mentorship of John Van de Walle. ¬†He shaped my mathematical practice more than he knew in his all-too-short lifetime. And so I dedicate my student-centered and pedagogically grounded¬†efforts to his memory. Cuisenaire Rods K-3I hope you will enjoy the tasks and games, the investigations and the open-ended problems posed in these resources. They are intended to promote big¬†thinking in elementary ‚Äď from addition and subtraction to skip counting¬†and multiplication, from fractions to division and more…¬† The full colour primary resource, Remarkable Cuisenaire Rods: Mathematical Tasks for Primary Classrooms is intended¬†for Kindergarten through Grade 3 can be ordered by clicking here or on the image on the left. ¬†The resource is $40 plus shipping. Click on the link below¬†to preview lesson titles and the intended grade levels¬†for¬†the¬†tasks. Cuisenaire Rod Resource K-3 Table of Contents Cuisenaire Rods 4-7 The full colour intermediate resource, Cuisenaire Rods Rock! Exploring¬†Multiplication and Proportionality in Grades 4-7¬†is designed for intermediate students and can be ordered by clicking here or on the image at the right. This resource¬†is $50 plus shipping. Click on the link below¬†to preview lesson titles and the intended grade levels¬†for¬†the¬†tasks. Cuisenaire Rod Resource 4-7 Table of Contents As always, thank you for your support! Carole

Mastering the Basic Facts – Resources from Campbell River

I am very proud to announce the publication of two teacher resources called Mastering the Facts – Addition and Mastering the Facts – Multiplication. Created in collaboration with the Campbell River School district, these resources combine my work with the work of teacher leaders in elementary schools in Campbell River. We wanted to put together a set of lessons that could be used in classrooms across the province to help students truly master – and not just memorize – their basic facts.

Each book includes a set of lessons describing each of the facts, both addition and multiplication. Featuring strategies like “doubles”, “double doubles” and “bridging through tens”, the books make explicit connections to our WNCP curriculum and the mental math learning outcomes at each grade level. ¬†Each lesson features a direct instruction component and a “before-during-after” lesson frame. Guided practice, literature connections, real-world problems, independent practice (games) and fluency builders are also included. For each lesson, there is an assessment component to sort out what students know and can do. ¬†It evaluates in numbers, pictures, words, models and real-world problems the extent to which students have mastered the strategy – and also mastered the facts themselves. Games, line masters, literature links and assessment tools are included in the resource.

In addition to the teacher resources, a set of Addition Posters and Multiplication Posters are also available to describe – in pictures, and in students’ words – how each strategy is applied. ¬†There are 9 posters in the set, printed on 11″x17″ glossy bond paper.

Although the books and posters are being distributed through the Campbell River School District, I am available – and happy! – to do professional development with school staffs and resource teams around the materials. ¬†This might involve a series of sessions with time scheduled in between to apply and try the strategies, or an overview session to learn techniques and practice the games… ¬†Books for participants would be made available for the sessions so we could speak to the lessons explicitly.

I am very proud of these materials and of all the teachers who collaborated to produce them.



Click here to order!

Open-ended Problems for K-4

Hello all!

Last week I gave a session for teachers in Coquitlam looking to teach Рand assess Рproblem-solving. We talked about what made a good problem, both in terms of content and wording, then worked through some samples across the grades.  I have posted a selection of problems for you here, which I invite you to download and use with your students.

Remember that it’s important to collect students’ thinking in a variety of forms – numbers, pictures and words, to have them engage with models or manipulatives, and wherever possible to have students communicate their understandings about a concept by translating them to a problem of their own. ¬†The latter is no small task! ¬†:o) ¬†John Van de Walle’s diagram outlines the importance of not only including these representations but also connecting and bridging between them. ¬†Students learn deeply when they transform their learning from numbers to models, from words to pictures, from problems to numbers and back again…