In my continuing reflection of Peter Mattock's books (Visible Maths & Conceptual Maths) both are providing me with an excellent opportunity to reflect on the fundamentals of maths and I would highly recommend them to teachers as a source of inspiration and discussion.
I'm conscious though, that my intended audience is not one of mathematical educators, (though I hope they would find this interesting) but made up of parents and/or mentors of learners who may not have a wide experience of maths education.
So, in this blog I'm focusing on how number lines could be used to help us support the development of learners understanding.
So, lets consider the simple number line ...
We can illustrate values (amounts) and operations on this line.
By exploring these illustrations with learners and asking them to illustrate their thoughts and reasoning we can seek to develop a better understanding of the number system, help to visualise problems and also the nature of maths itself. The number line is another way of representing a mathematical situation, other examples of representations could be counters or bars as well as the abstract numbers and symbols that we are aiming to get the learners to become comfortable with.
I recommend listening to the conversation between Debbie Morgan & Steve McCormack Using number lines in Key Stage 1 | NCETM in it Debbie highlights the need for the learners to appreciate numbers with respect to other values on a number line (in this case 1-10).
The position on the number line represents the value, a concept I feel cannot be taken for granted. So, matching the position of let's say FOUR with the symbol and name "four" should perhaps also be combined with manipulatives and/or other representations the learners have met before.
Learners may not immediately grasp the concept of four on the number line or appreciate it's size. But the same image alongside counters ...
... and alongside a bar of an appropriate length,
can help the learners appreciate the meaning of the position of the numbers on the line.
So, asking learners questions such as ..
TRUE or FALSE , 7 is larger than 8 ?
What number is bigger than 3 and less than 5 ?
I'm thinking of a number larger than 5 but less than 8, what could this number be?
and also asking them to give their reasoning, can be supported by manipulatives but eventually the learners will be confident in just using the number line without these additional guides to a sense of size
The concept of magnitude from the left mark of zero in turn (at the appropriate point) introduces them to measuring lengths with strips of paper and eventually rulers.
It's also perhaps worth noting that the learners need to develop the idea of counting from a given number and not always from zero, so "what's one more than...?" and "what's one less than ..?" are useful early problems to pose.
Giving way to "What's the difference between?"
For example
"Which number is bigger FOUR or SIX?",
"Can you explain why?"
"I'm thinking of two numbers that have a difference of two, what could they be?"
The vertical arrows (some use dots and/or circles around the numbers but I prefer the preciseness of an arrow) so far have indicated a value, but what about an operation?
Augmentation
If this represents the mathematical operation THREE add FIVE is equal to EIGHT, then the operation of "add five" can be reversed by rotating the arrow to become "take-away five".
I'd be tempted to use bars to represent the two values THREE and FIVE then swap the order of them around to arrive at the commutative FIVE add THREE.
As such, I wouldn't use this diagram (below) with younger learners and would only introduce it later as they become more proficient and comfortable with the use of number lines.
In a sense the idea of partitioning and aggregation does not as easily lend itself to the concept of an "operation" such as augmentation and taking away that can be represented by the horizontal arrows. But aligning arrows (or vectors) such as this nose to tail is an important concept that needs to be addressed in time. (I'm tempted to leave this for a later blog perhaps alongside negative numbers).
Finally, I've used horizontal arrows to represent adding or or taking away. Many teachers will use curved or "bouncy" arrows to represent the same thing. This is understandable as it mimics the act of counting on (perhaps with a finger) but the purist in me prefers the straight rather than curved line.
An example of "bouncy" counting on in ONES (above) and "skip-counting" (below) in THREES which will be addressed in multiplicative concepts.
