This lesson is the first in a series of five practising the strategy of bridging. The teacher introduces the strategy and then takes the class through a systematic programme of guided and independent practice.  Her pedagogical approach is highly interactive, and she engages the students’ interest and attention by drawing on their existing knowledge and connecting it to the new learning, asking many questions, and encouraging them to talk to their peers about what they think and know. In the following videos you will observe the teacher using a range of strategies for effective teaching and learning. You will also note how this systematic approach supports the students to build confidence and fluency in using the mathematical strategy being learned.

Introducing bridging with addition

The following evidence-informed pedagogical strategies are evident in this lesson:

• Reviewing prior learning, which  activates students’ existing knowledge and strengthens recall
• Working through new learning in small steps, checking at every stage to ensure students understand what they are learning
• Asking lots of questions to encourage students to explain their thinking and connect their new learning to their existing knowledge
• Using multiple representations to increase conceptual understanding, develop procedural skill, and increase flexible thinking
• Modelling mathematical procedures to reduce cognitive load and support learning
• Using word problems to help students learn to apply their mathematical knowledge

This video begins with a review of foundational maths knowledge that the students will use throughout the week’s lessons, namely the principle that knowing how to solve 8 + 7 also allows students to solve similar equations such as 0.8 + 0.7 and 80 + 70. The objective here is to remind students that they can deploy their existing knowledge of number and place value, and apply it to the week’s learning about bridging. Reviewing prior learning and activating students’ existing knowledge is one of the principles of effective, explicit instruction.

During this part of the session, two students offer different strategies for solving 80 + 70. The teacher reviews and explains both methods, and ensures that the whole class understands them before moving on. The teacher then goes on to introduce the strategy of bridging with addition. She asks lots of questions to make this part of the session highly interactive, as well as to constantly check that all students are participating and keeping up with the concept and procedure being introduced.

She then gives the students a word problem to solve, and models the process of extrapolating the mathematical equation from the wording of the problem. She invites the students to solve the problem using either manipulatives (in this case, base tens or place value blocks) or by writing the problem out using a number line on a mini whiteboard. This is an example of using multiple representations when solving problems – in this case, the students can use either a concrete representation (the base tens blocks) or a visual representation (the number line). The teacher concludes this part of the session by representing the problem as balancing equation, which is an abstract representation (as it uses only mathematical symbols). The use of multiple representations supports students to increase their conceptual understanding, strengthen their procedural knowledge, and think more flexibly when completing mathematical tasks. It is particularly important for neurodivergent students, but is a beneficial approach for all students. It also allows the teacher to quickly assess students’ understanding by reviewing how they have used the manipulatives or what they have written on their whiteboards.

Practising bridging with addition

The strategies you will observe in this video include:

• Guiding student practice to support mastery of concepts and procedures
• Asking lots of questions to encourage students connect their new learning to their existing knowledge and to check students’ understanding
• Using mathematical language to ensure clarity of understanding
• Reviewing prior learning to remind students to use their existing knowledge and apply it to the current problem

In this video, the teacher sets the class a series of problems of increasing difficulty. During this part of the lesson, she moves around the room to check in with different groups of students, asking lots of questions, and checking the students’ understanding at each step. She also reflects the students’ thinking and procedural knowledge back to them. Sometimes, she rephrases what they have said using specific mathematical vocabulary (for example, using the word ‘partition’ when describing the steps in bridging, or the phrase ‘is equivalent to’ in relation to the equals sign in an equation).

At several points she refers to using existing number knowledge to solve a problem, which reinforces to the students the idea introduced earlier in the lesson that they all have prior mathematical knowledge that they can apply to the current problem. For example, she encourages them to use their number and place value knowledge when she invites them to estimate whether or not solving the problem 600 + 445 will require them to bridge through 1000.

During the session the class pauses to explore whether changing the order of the numbers in an addition problem makes a difference. The class asserts that it does not, so the teacher invites them to ‘prove it’ using a number line. Exploring a mathematical question in this way empowers the students to see themselves as mathematicians.

Throughout this session, the teacher can be observed guiding student practice by working through problems with them in short steps, checking for understanding at every stage by asking for agreement from the class. She is also able to use the students’ work on whiteboards and with materials to see at a glance whether or not they have understood and completed the mathematical procedures correctly. There are also examples in this video of the teacher explicitly demonstrating to students how they can check their own working when using a number line. In this way, she supports them to assess their own work.

Guiding student practice in bridging with addition

During this video you will observe the teacher:

• Using multiple representations to strengthen the students’ conceptual and procedural understanding
• Working in small steps to help students master new learning
• Checking students’ understanding to ensure they have learned the new material
• Scaffolding difficult tasks to reduce students’  cognitive load
• Ensuring a high success rate to consolidate the students’ learning

In this video, the teacher works with certain students individually or in small groups to guide their practice through some of the more difficult addition problems. As in the previous videos, she works through each problem in short steps, asking questions to ensure the students are actively involved, to elicit their thinking, and to check their understanding. In one example, the teacher uses base ten blocks to work through a problem with a student who is finding it a little difficult to explain her thinking using a number line. They then move from the concrete to a visual representation by also showing the working on a number line. Finally, the teacher invites the student to assess her work by checking whether 22 + 70 makes 92 using an abstract representation by just working with number names, rather than using concrete or visual representations of the numbers. Using multiple representations in an integrated way like this helps to strengthen students’ conceptual and procedural understanding.

Note also that the teacher allows plenty of time for the student to think and answer – in this instance, she does not answer immediately, but answers correctly, and can explain how she knows her answer is correct using the principle that the class reviewed at the beginning of the lesson (that 80 + 70 can be solved using the same strategy as 8 + 7). It is important to remember that some students may process information more slowly, but this does not necessarily mean that they are not capable of reaching the correct answer.

With another group, the teacher scaffolds a difficult task by explicitly modelling how to partition numbers using the base tens blocks. This is a form of guided practice design to support the students’ understanding of how to bridge when starting with a number that is not a multiple of ten. Finally, she supports a student who has used a less common method of solving a problem on a number line to check his work by adding up the different numbers into which he has partitioned one of the addends in the equation.