In this lesson, the class extends their understanding of the strategy of bridging and applies it to subtraction. This lesson follows a similar format to the previous lesson, beginning with a whole-class session to review the strategy of bridging with subtraction, which they initially learned earlier in the year. This is followed by a session of guided practice during which the class completes a series of practice equations, again using multiple representations to practise and consolidate their use of the strategy. They move on to independent practice, during which the teacher checks student work and provides additional support as needed. During the lesson, the teacher notices that the students’ basic facts to 100 are not as secure as they could be, so she concludes the lesson with a whole-class session for some additional practice of these number facts.

Introducing bridging with subtraction

The pedagogical strategies you will see in this video include:

• Using concrete, visual, and abstract representations to support conceptual and procedural understanding
• Offering fully worked examples to reduce students’ cognitive load and support procedural learning
• Recalling prior learning to strengthen the connections between existing knowledge and new learning
• Checking student understanding at every step to support mastery
• Using word problems to help students apply their mathematical knowledge
• Using correct mathematical language to ensure clarity and build students’ understanding
• Modelling the use of materials to support students’ conceptual understanding
• Asking lots of questions to allow the teacher to check students’ understanding

In the first part of this video, the teacher reviews the prior learning of the class by revisiting several important concepts and procedures from the previous day’s lesson. She briefly reminds the students that they had been learning about and practising bridging with addition in the previous lesson, and goes on to introduce the focus of today’s lesson, bridging with subtraction, beginning with a simple problem, 12 – 4, to support the students to recall the strategy (which they learned earlier in the year). Then she reviews the principle they had used throughout the previous lesson to do with place value knowledge, asking ‘if you know 12 – 4, what other facts do you know?’ Then, the teacher revisits the question of order that they had explored in relation to addition in the previous lesson in order to ensure that the students understand that, while they can change the order of the addends when solving an addition problem, the same rule does not apply to subtraction problems. Finally, they practise turning a word problem into an equation, as they had done the day before.

During this opening part of the lesson, the teacher also explicitly teaches an important procedure for using number lines when solving subtraction problems (which is that you start from the right-hand side), and she models how to use the materials by exchanging a hundreds block for ten tens blocks to allow for partitioning.

Practising bridging to solve subtraction problems

The evidence-informed practices enacted in this video include:

• Solving word problems to enable students to apply mathematical principles
• Guiding student practice and checking for understanding to develop mastery of the new learning
• Working through problems in small steps to develop procedural skill
• Providing multiple models to support conceptual understanding
• Asking students to explain their work to strengthen their learning
• Using mathematical language to consolidate students’ conceptual knowledge
• Monitoring students’ independent practice to help build fluency
• Using scaffolds to support students to build their knowledge

The teacher begins this part of the lesson by explicitly reminding the students about the kinds of problems they are working on and the strategies they will be using (bridging through 100). The class begins by working in pairs and small groups to solve the subtraction problems set by the teacher, who then invites several students to share the different ways they reached the solution. She asks ‘does it matter how many groups we partition into?’, and affirms the students’ negative response, reminding them to check their work by adding up the ‘jumps’ on the number line.

After working as a class on a series of practice problems, the teacher invites the students to pursue some independent practice. The students design their own problems by selecting numbers from two lists, while the teacher moves among the groups, checking each student’s work and offering support where needed (see Video 3) or additional challenge where appropriate. As the students are working in small groups on their practice problems, the teacher acknowledges the ‘maths conversations’ she can hear and affirms the students’ use of maths vocabulary. This helps to contribute to the students’ developing mathematical identities, and reinforces the value of using mathematical language.

In the final part of this video, the teacher brings the class together on the mat to spend some time working on their number facts to 100. She has noticed during the previous session on bridging through 100 that the students’ knowledge of these is not as strong as it could be, so she uses physical materials (Numicon blocks) and visual representations (number lines and a number trio) to scaffold the students’ basic facts to 100.  This is also in preparation for the following day’s lesson on bridging with decimals, for which the students will use their number facts to 100.

Guiding student practice in bridging with subtraction

In this video, you will observe the teacher:

• Guiding student practice to strengthen their conceptual understanding and procedural skill
• Providing scaffolds to support student learning through difficult tasks
• Working in small steps to reduce students’ cognitive load
• Using concrete representations to build conceptual understanding
• Supporting students’ understanding of mathematical language to consolidate their mathematical knowledge
• Obtaining a high success rate in student work to build their fluency

This video includes two examples of the teacher guiding the practice of individual students who require some additional support. In the first example, she works with a student who has only been living in Aotearoa New Zealand for one year, and whose first language is not English (indeed, she spoke no English when she arrived in New Zealand). The teacher notices this student attempting to use an algorithm to solve a subtraction problem, according to methods that she had learned in her previous school. However, her conceptual knowledge is not quite strong enough to support her use of the algorithm (she makes a place value error and misaligns the columns). The teacher uses manipulatives to consolidate the student’s understanding of place value, then represents the equation visually on a number line. She validates the student’s use of algorithms, but reminds her that they rely on a sound understanding of number.

In the second example, the teacher uses materials to support a student whose basic facts knowledge needs a little consolidation. When the student starts to guess rather than recalling her basic facts to 10, the teacher uses Cuisenaire rods to support her working. The same student then asks to check her work by doing ‘the adding one’, which the teacher rewords using mathematical language as the ‘inverse operation’. She guides the student through the process of using the inverse operation to check her working in the subtraction problem, affirming that she has done it correctly, and reminding the student that she has used her prior knowledge (by remembering that she can use the inverse operation to check her work).