Teaching Manuals :: more information

• Review facts learned in previous teaching: A short pre-test (a sample is given) shows whether pupils are ready to start with Lesson 1. If further practice is needed, there are games and activities at the beginning of Unit 1 to reinforce their learning; this is especially important for those pupils new to Structural Arithmetic to familiarise them with Stern’s materials.
• Discover the structure of two-place numbers: A good understanding of positional notation, also known as place value, underlies all operations in our number system. Working with the Dual Board gives pupils practice and insight into the role of positional notation. Pupils learn to build up units (in the form of blocks) and transfer them into tens, and, conversely, they practise breaking down tens into units. The unhurried approach, through games and activities with the Stern materials ensures a thorough grounding. There are some simple applications with money and a group activity test to consolidate learning.
• Learn to solve for x: Although we usually associate ‘solving for x ’ with algebra at a much later stage, this simple introduction, where finding x is simply ‘finding the missing number’, has two purposes: to introduce pupils to the new terminology, and also to enable them to develop further their cognitive thinking by encouraging them to ‘turn their ideas around’ through games and written exercises.
• Learn to analyse word problems: Problem-solving is a real test of understanding, which pupils often find tricky, not through lack of mastery of addition and subtraction in this case, but because of their difficulty in interpreting the words of a problem and translating them into the language of arithmetic. In Structural Arithmetic, pupils are helped to overcome this by acting out the problem, which gives the teacher new insights into the level of understanding of their pupils. The Teachers Manual facilitates the teacher’s role by identifying the three different types of word problems in addition and subtraction, and showing how to guide the pupil’s thought processes successfully through each type. Gradually the pupils no longer need blocks and other aids; they are able to see the number relationships and solve the problems in their minds or by writing the examples on paper.
• Learn the value of coins: Pupils are introduced to money in Unit 3 as an application of place value in teen numbers, breaking down and combining coins of low denomination: 10, 5, 2 and 1. Later, in Unit 7, money is applied to adding and subtracting two-place numbers up to 100, so that all coins up to and including the pound, dollar or euro are used. All pupils, especially older students who have learning difficulties, benefit from and enjoy learning this valuable life-skill.

  Note: Money is presented in three separate currencies: Pounds Sterling, US Dollars and Euros. In the Teachers Manual (3) lessons are covered in  Pounds Sterling. For Euros  and  US  Dollar  currencies,  Lessons  and  Pupils  worksheets are found in the money section of the Resources CD which accompanies Book 3.

- Adding 9, adding to 9 also adding 8, adding to 8:
Up  to this  point,  results of addition facts have been contained within 10, or within the  same  decade. Extending  the  results  further into the next decade brings the  introduction of the  concept  of bridging,  which  enables calculations  to  cross over  this  'boundary'.  Pupils  can  easily see the effect of adding 10 to a number when a 10-block  is  added  to  a  number  block  in  the  20-tray. Continuing in this way, by using  a  9-block,  they soon realise that adding 9 (or adding to 9) gives a sum 1 less than  if  the same number had been added to 10. They enjoy discovering that they are  able  to  work this out by logic, and to generalise the results to other numbers, rather  than  by  laboriously  counting  out nine dots. A similar process is shown with adding 8, or adding to 8.
- Combinations that make 11 and 12   
In  Unit  12  pupils  build  the  staircase in the 20-tray beyond 10 to 11 and 12, and explore the combinations that result in them. When children know the components for addition, they can work out the corresponding subtraction facts.
- Doubles and Neighbours
Using  the   doubles  and   neighbours   board  enables  pupils  to understand more relationships  in  addition:  the  relationship  between  the  addition  of two identical numbers -   doubles  -  say 7 + 7, and  that between the addition of a number with  the next higher number -   neighbours   - say 7 + 8.  This  section covers all doubles and  neighbours  results  from  11  to  20, and is very clearly demonstrated by using the board.
- Related subtraction facts for each group
Pupils  working  with the Structural Arithmetic programme are encouraged from the beginning  to 'turn  ideas around in their minds'. By working with concrete materials, they  have  a  clear  visual  impression of how to do this when learning the addition/subtraction facts. In  this  way,  they  expect  to see addition as 'doing' (performing an operation),  and  subtraction  as 'undoing'  (the inverse of addition). (Later, they see a similar inter-relationship  of 'doing'  and  'undoing' between multiplication and division).  There  is  plenty  of  practice  in  the form of activities, games and written work to reinforce learning.

          Later,  this  teaching is developed so that pupils learn to add within a decade

          without  carrying   1-ten.  To illustrate this: 4 + 6 = 10 easily transfers to "14

          needs what to make 20?", or "24  needs what to make 30?" as pupils

          complete each decade. Similarly, in subtraction, using the number track and

          blocks, and without  the  cumbersome   process   of   borrowing/regrouping, 

          pupils, - knowing that 10 - 6 = 4 - can see that 20 - 6 = 14 and 30 - 6 = 24.

          After covering the idea of bridging from one decade into another, pupils learn

          not  only  that,  for  example, 9 + 8 = 17, but 19 + 8 = 27, and 49 + 8 = 57

          in other decades.

• Understand regrouping (carrying and borrowing, or exchanging) in addition and subtraction: The Structural Arithmetic programme aims to bring meaning  to  the  procedures  and  vocabulary  associated  with  this important concept, which many pupils find confusing. Starting with two-place numbers, the Dual Board helps clarify the teaching, giving pupils a 'hands-on' experience of exchanging 10 ones for 1 ten in addition and 1 ten for 10 ones in subtraction. As pupils gain confidence, the process is broadened to three-place numbers. Plenty of practice in the form of games, activities and written work consolidates this topic.