- Intermediate Phase (Grade 4-7) Classroom: General
Throughout the later grades, it is envisaged that the teacher will work with small groups of children in a focus group at the front of the classroom. The rest of the class is productively engaged in seatwork activities. These groups should be determined by the children’s stage of development; children who are at a similar stage of development should be in the same group. Each group should consist of between 6 and 8 children and should meet with the teacher in a focus group at least 3 to 4 times a week.
In the Intermediate Phase, the children typically complete pages in the workbooks in preparation for their focus group session. Because children have “seen/done” each page previously they should be able to make a start. The focus group session is used to reflect on the activity.
- The focus group is at the board.
- The data projector is displaying the NumberSense Workbook page that the focus group is working on.
- The remaining groups of children are sitting at their desks.
- Each child has:
- A “thinking book” (a jotter, not a whiteboard) in which to make notes and/or calculations as needed
- Their NumberSense Workbook open to the correct page
- A pen and/or pencil (no eraser) with which to write (calculators are allowed where necessary/appropriate)
The teacher starts the lesson by quickly telling each group what their task is for the day.
Typically this will include:
- Telling the groups in which sequence they will be coming to the board during the lesson.
- Telling each group what they must do when they are working independently. It is a good idea to write this on the board. Typical work assignments include:
- Working through the activities on a particular page with a special focus on one or more of the activities that the teacher will want to discuss when the group comes to the board. In other words, the group is preparing to participate in a discussion on these activities.
- Working together to do corrections on previous pages – this means that the teacher has marked the books and identified activities that need to be revisited.
- Working together to check each other’s work when they have completed the activities on a page.
- The teacher works with one group (the focus group) at the board at a time; typically for 15 to 20 minutes.
- During this time the groups that are working independently should work together on the tasks set for the day.
- The groups working independently may not:
- Be disruptive, and/or
- Interrupt the teacher to ask for clarification, etc. If they get stuck and cannot help each other they must leave the question, go on to the next one and be ready to ask the teacher for help when it is their turn.
- When the teacher has finished with the group at the board she will look at the list of groups and activities on the board and call the next group to the board. The group from the board moves quickly and quietly to the vacated seats and gets on with the work that has been set for them.
- While the groups are swapping seats, the teacher quickly visits each of the other groups and checks that they are still working and/or clarifies any quick questions they may have. The teacher should ensure that he/she does not get distracted.
- The cycle repeats itself with the teacher aiming to see at least three groups at the board each day.
What happens with the group at the board?
The teacher typically does three things with the focus group: Manipulating number, discussion and setting homework.
Manipulating Number (4 to 5 minutes)
The manipulating number routine (mental mathematics or “Up and Down the Number line” activity) is highly structured. The teacher focuses on one or more of the following in a number range that is appropriate to the group’s developmental state:
- Multiplication facts
- Doubling and halving
- Bridging 10s, 100s and 1000s
- Completing 10s, 100s and 1000s
- Arithmetic with multiples of 10, 100 and 1000
- Single-digit arithmetic
Throughout the manipulating number activity there are two things that are critical:
- The teacher asks questions in a deliberately sequenced way to reveal the underlying structure/pattern, and
- The teacher asks the children to describe how they worked out their answer. She then asks the other children if they understand the explanation and to demonstrate that they understand the explanation by doing a similar question in the same way, etc. This reflection is at the heart of the success of the manipulating number activity.
Discussion (6 to 10 minutes)
This is a key part of the session and involves one or more of the following (in order of priority):
- Discussing a task that the group has completed in time for the session together (the group was primed to prepare the task). In this discussion the teacher will ask questions such as:
- What did you have to do? How did you do it?
- Was there somebody who did it differently? How did you do it? Will your approach always work?
- Have you done something similar before? If yes, how was that task similar and different, and how did that task help you to do what you did with this task?
- Did you notice any patterns/relationships that helped you to do this task?
- What have we learnt from this task?
- What mistakes did you make and why did you make those? What can we learn from those mistakes?
- Introducing a new idea, vocabulary or definition, etc. The teacher should anticipate the need for these sessions by looking ahead in the workbooks. It is important that the discussion of the new ideas, vocabulary or definitions are limited to new ideas, vocabulary, or definitions needed to complete activities in the workbook and do not involve the teaching of “methods”.
- Making sense of a calculation strategy. Throughout Workbooks 13 to 24 there are calculation strategy pages – typically every 5th or 6th page. It is very important that these pages are not used to investigate all the different possible ways of doing a calculation but rather are focused on doing the calculations in the way that the page suggests. This forces children to make sense of (interrogate) and understand the method and in so doing to increase their range of calculation strategies. Note, it is not necessary that the children adopt the strategy from the page but it is important that they use the strategy for the duration of the page and are able to explain how they are doing so.
- Review of previous work that the teacher has realised (through her marking of the workbooks) is causing difficulty or confusion.
- Looking ahead at a task that the group is about to tackle and ensuring that the children understand what is expect of them. Because the activities tend to repeat themselves throughout the workbooks, there will seldom be a completely unfamiliar task.
Setting homework (2 to 3 minutes)
Setting homework and tasks for the next time that the group is working together will involve questions similar to those above, such as:
- Have you done something similar before?
- If yes, how was that task similar and different, and how did that task help you to do what you did with this task?
Monitoring of children’s work
The teacher needs to establish a routine by which she monitors the children’s work in the workbooks. The more regularly she does so, the better. At a minimum, she should look at the workbooks of the children in the groups that are coming to the board on the day/afternoon/evening before they do so. This will enable her to anticipate any problems that need discussion (see separate discussion on marking workbooks in the FAQs section). It also makes good sense for the teacher to keep a record, in her mark book or elsewhere, of the page and other progress that each child/group is making. If groups are moving too slowly (less than about 7 pages every 10 days), they will not complete the workbooks assigned for the year and they will effectively be “falling behind”.
In the Intermediate Phase, there are four different types of Workbook pages. Three of the four page types are found in the NumberSense Workbooks and the fourth is found in the NumberSense Companion Workbook.
- NumberSense Workbook
- Practice pages
- Calculation strategy pages
- Concept development pages
- Pre-algebra patterning
- Ratio, rate etc.
- NumberSense Companion pages
Each of these pages has a specific teaching approach associated with it.
- Intermediate Phase (Grade 4-7) Classroom: Practice Pages
In each NumberSense Workbook there are a large number of pages where the sole purpose is to provide practice on concepts and skills with which the children should already be familiar. The pages typically include activities that provide practice with:
- Other familiar mathematics concepts.
- Assign pages to children to complete
- They should be able to do the work on the pages without any explanation of what is expected of them.
- The pages could also be used as homework exercises
- Mark the work
- You could mark these pages yourself or have the children mark each other’s work etc.
- Review any issues that arise with the students
- Intermediate Phase (Grade 4-7) Classroom: Calculation Strategy Pages
Calculation Strategy Pages
Foundational to all mathematics is a sense of number, which includes the ability to calculate fluently and flexibly – strong mental arithmetic. Beyond Grade 3, the calculation strategies that were developed in the earlier years are no longer sufficient for the increasing number range in which we are working. The calculation strategy pages provide children with exposure to an increasing range of mental mathematics strategies.
See the Resources tab for a list of the Calculation Strategies in Workbooks 13-24.
- Introduce the calculation strategy
- Introduce the first problem
- Children close their workbooks and solve the problem
- Discuss the different strategies
- Identify the desired strategy
- Children work independently to complete the calculations on the page using the desired strategy
- It is important that the children record enough on their pages to be able to describe how they used the strategy
- Review the children’s use of the strategy to complete the calculations on the page
- Introduce the calculation strategy
- Intermediate Phase (Grade 4-7) Classroom: Concept Development Pages
Concept Development Pages
It goes without saying that throughout the workbooks we are supporting children in developing mathematical concepts (knowledge, skills etc.). Key concepts that are being developed in the Intermediate Phase include:
- The fraction concept
- Pre-algebraic thinking (patterns and patterning)
The methodology of the NumberSense Programme is to use situations or problems to reveal mathematics rather than to tell children how to do certain procedures. In this spirit, concepts are developed gradually and over a long period of time (along well-defined developmental trajectories), with children developing an increasingly rich and deep understanding of the concept.
- Students work on the page in anticipation of a discussion with the teacher
- They have seen pages like these before and they should be able to “get started”
- Students must anticipate that they will be expected to explain what they have done, discuss what they have noticed and contribute to a discussion
- Discuss the work with the students
- The discussion is where the learning takes place – the mathematics is “revealed”
- It is critical that the teacher has a deep sense of what she/he wants the students to “notice” (learn) from the page and leads the discussion accordingly
- Students finish off the page
- Intermediate Phase (Grade 4-7) Classroom: Companion Pages
The NumberSense Companion Workbook pages deal with:
- Space and Shape (Geometry)
- Data Handling
In the Foundation Phase, learning in these areas was largely activity based with children learning through play. In the Intermediate Phase this “free play” shifts to more “focused play” (investigations).
These pages don’t have as well defined a “step-by-step approach” as the calculation strategy and concept development pages do. That said they are more like the concept development pages and follow the general pattern of:
- Children work on the page using resources as needed
- Teacher provides ongoing support to the children
- Teacher concludes the page(s) with a discussion that reveals the mathematics to be learned