For the 2019-20 school year, the “thinking together” series of blog posts will focus on the curricular content in the mathematics curriculum. The “thinking together” series is meant to support professional learning and provoke discussion and thinking. Each month we will zoom in and focus on one curricular content area with examples from K-12 classrooms in Richmond.
The curricular content is the “know” part of the know-do-understand (KDU) model of learning from BC’s redesigned curriculum.
The curricular content develops and builds over time. Each grade level has core curricular content knowledge and these are reflected in the big ideas for each grade level. There are five big ideas that reflect five strands of curricular content – number and number operations, computational fluency, geometry and measurement, patterning and algebraic relationships and data analysis and probability. A sixth content area in mathematics, financial literacy, is new this curriculum.
The curricular content, along with the curricular competencies, comprise the legally mandated part of the curriculum, now called learning standards. This means that both curricular content and curricular competencies are required to be taught, assessed and proficiency/learning achievement is communicated to students and parents/guardians.
“Understanding what counting is for is the starting point of an outburst of numerical inventions. Counting is the Swiss Army knife of arithmetic, the tool that children spontaneously put to all sorts of uses. With the help of counting, most children find ways of adding and subtracting numbers without requiring any explicit teaching.” (Dehaene, 1997, p.122)
Counting is considered a number concept and is connected to understanding of our number system, place value, multiples and other relationships between numbers. Within the learning trajectories research from Clements and Sarama (2014), and the critical learning phases work of Kathy Richardson (2012), the following stages are considered in the development of counting:
- Counting Objects (one-to-one, stability, checks by recounting, cardinality, estimates, counts out a particular quantity)
- One More/One Less (knows one more/one less without counting, recognizes when a number sequence is out of order)
- Counting Object by Groups (counts by moving in groups, knows quantity stays same even when counting by different groups)
Clements & Sarama
- Counter and Producer
- Counter Backward from 10
- Counter from N
- Skipcounter by 10s
- Counter to 100
- Counter On Using Patterns
- Counter On Keeping Track
- Counter of Quantitative Units/Place Value
- Counter to 200+
- Number Conserver
- Counter Forward and Back
(the names of these stages are descriptive of the counting occurring, for more information visit the learning trajectories website)
Although these stages focus on whole number counting and number understanding, similar stages of development can be seen in parallel tasks when counting by fractions, decimal numbers or integers.
The skills and concepts involved in counting are developed over time and through multiple experiences:
- correct sequence of number names
- one-to-one correspondence: saying one number name for each object counted
- cardinality: the last number said is the quantity counted
- stability: the quantity of a group does not change if the objects are rearranged (also related to conservation of quantity)
- relative size: more than/less than
- make connections between number names, quantities and symbols
- counting forwards, backwards and from any starting point
- base-ten structure: how can I count or organize by tens and ones to find out how many?
There are many instructional routines that support the development of counting across the grades.
Counting Around the Circle
Counting around the circle is essentially having students count in sequence, taking turns to say the next number in the sequence, one student at a time. The starting number can be changed, the direction of count and the type of count can also be determined to practice specific skills and concepts. Norms can be put in place so that students feel supported by asking a neighbour, or having time to count ahead so they don’t feel “on the spot” when it is their turn if they are unsure of the number they need to say. An example of counting around the circle would be to begin with the number 81 and count backwards by 2s. The count could be recorded on a chart/whiteboard while the students count so the count can be discussed after the circle.
A math game related to this routine is “Buzz” where students sit in a circle and a number of the day is chosen, for example “4”. Every time a multiple of four should be said, a student says “buzz” instead. For example, 1, 2, 3, buzz, 5, 6, 7, buzz, 9, 10…
Choral counting is a routine that involves having students count in unison to a preplanned counting sequence. As students count together, the teacher records the count in rows and columns providing a visual and symbolic connection to the oral counting. After counting together, the students look at the recording of the count to notice patterns and relationships.
Stenhouse Publishers have an online choral counting tool to plan choral count, including counts with fractions and decimal numbers. It can be accessed HERE.
Counting collections is a routine that emerged out of the research done with CGI (Cognitively Guided Instruction). In essence, students (usually in pairs) choose a collection and count it in multiple ways and record their count (quantity and process) in a way that makes sense to them. Students may begin counting collections by 1s but then continue to develop their understanding of counting by counting in multiples such as 2s, 5s, etc. Grouping tools such as cups, plates and ten frames are often used as part of the counting process. Intermediate students can count items that are already grouped like a box of eight crayons or think about counting with decimal numbers as they count dimes or quarters.
The following are some blog posts on our district blog about counting collections:
As we think about how counting and number concepts develop over time, we might consider the following questions:
What would you identify as core content around counting and understanding numbers at the grade level/s you teach?
What curricular competencies are connected to the curricular content of counting and number concepts?
How do we support students’ development of counting, paying attention to the different concepts and skills involved with counting? What assessment techniques will give use the information we need?
What opportunities are there for your students to apply/transfer their understanding of counting to authentic contexts and problems?
Learning and Teaching Early Math: The Learning Trajectories Approach by Douglas Clements and Julie Sarama (2009, 2014)
How Children Learn Number Concepts: A Guide to the Critical Learning Phases by Kathy Richardson (2012)
Choral Counting and Counting Collections by Megan Franke, Elham Kazemi and Angela Chan Turrou (2018)
The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene (1997, 2011 – revised and updated edition)
Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3 by Jessica Shumway
Number Sense Routines: Building Mathematical Understanding Every Day in Grades 3-5 by Jessica Shumway