I have been studying magnitude and density: understanding the relationship between numbers. Magnitude refers to ideas like:
***1000 is ten times greater than 100.
*** One hundred is 10 times greater than 10.
***Ten is 10 times greater than ?
And what is next in my sequence?
Or a more primary example: If I put 6 here, where do I put 10? To the right or left?
Density refers to ideas about between. There are numbers between any 2 numbers you place on the number line. Density is an underpinning of understanding for work with fractions and decimals.
The tower task: This is not a counting task. I do not refer to the counting sequence.
Students build what I build. ALL my focus is more or less in reference to the height of the towers.
We begin by studying the relationship between the towers.Sounds simple hey?
Try asking questions like which is more? Which is less? How much less?
Never mind, the students, reflect on these questions as a teacher:
Are these the same towers?
Are they the same relationships?
Do you see how I have made 2 a referent?
Why did I do that?
How is 4 related to 5?
What is between 3 and 5? How is it related to 3 and 5?
What happens if I do not start at one?
Here I put a focus on 3. Do you see it?
Do you as a teacher understand why?
What kinds of tasks can I develop from this image?
The concept of 1 or 2 less, 1 or 2 more is a fundamental number sense idea. It is not resolved by simply setting up and addition or subtraction equations. It is a key relationship that students need to internalize. We will study how and why.
Then we move to teen numbers where the same relationships matter. How are teen numbers related may just be one of the most important gaps in number sense to help our students bridge.
Here I compare 2 sets of towers that represent ... what teen number?
Are the 2 sets representing the same teen number? Explain how you know. What relationship can you explain equal, not equal, more than, less than?
What equality or inequality can you record?
How about these 2 sets?
Do they represent the same teen number?
Can we put an equal sign between them?
What equations explain their relationship?
Magnitude and density are terms that refer to FOUNDATIONAL understandings.
Understandings every teacher at every grade should share!! We just do not use the towers at every grade.
An understanding or should I say misunderstandings of magnitude (size of numbers) and density (what's between numbers) underpin many of the common concerns teachers express at all grades.
Why are kids relying on counting by ones to find answers??
Why is subtraction so difficult to teach, to learn, to remember?
Why are teachers & parents pulling their hair out over multiplication and division facts!!
Don't even get me started on long division and work with decimals.
What is "place value" and why do our students not understand it?
THESE COMMON COMPLAINTS CAN BE CONNECTED BACK TO TEACHING FOR NUMBER SENSE, BEFORE WE FOCUS ON NUMBER EQUATIONS. Bridge the gap for all students by learning to embed a focus on NUMBER SENSE in all your teaching.
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