Mathematical structures is the way the pattern is organized.
What is the biggest mathematical structure used in number? (The base 10 system)
The place value strategy is that a number is multiplied by 10 when it moves to the place value on the left, and divided by 10 when it moves to the place on the right. 2 tenths multiplied by 10 is 2 ones, 2 tens (20), 2 hundreds (200), 2 thousands (2000) etc. Each place increases/decreases by 10.
Another mathematical structure is the hundred board. The number patterns (multiples 2, 5 and 10) are organized into a 10 x 10 grid. A difference of 10 as you go down the rows and a difference of one as you go across.
Danger of 'key words'
Students look for verbal clues when solving word problems, for example 'more' usually (but not always) suggests addition and 'less' usually (but not always) suggests subtraction.
Incorporate problems where these words suggest the opposite of what they usually do. For example:
- Lisa had 14 books which was 2 less than Frank. How many books did Frank have?
- Lisa had 14 books which was 2 more than Frank. How many books did Frank have?
One way to ascertain if students have developed structural thinking is to ask them to draw a hundred board from memory.
This diagnostic task was conducted with a Year 2 student.
Other similar diagnostics that you could do with your students is to give them a partially filled hundred board, and students need to fill in the missing numbers. Another diagnostic to check for structural thinking is to give students a partially filled hundred board hat has been broken up into a jigsaw. Students need to put the pieces together to complete the hundred board.
Other types of thinking that students demonstrate is additive and multiplicative thinking.