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Building on learners' thinking : Post 16 Level 2 : Mathematics-specific Pedagogy

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Post 16 Level 2
Building on learners' thinking
Question 1 of 3

1. How confident are you that you can plan mathematics learning activities that enable learners to build on their existing knowledge and understanding?

a. Activity One


Imagine watching a learner doing the following tasks.

  1. Draw a poster to show everything you know about the circle.
  2. Do this calculation: 3.75 + 1.23 - 2.41 =
  3. What is the same and what is different about these charts?
    Problem 1

What might you learn about the learner’s knowledge and understanding of each topic? How could you use what you saw to help plan further activities?

Prompt 1.

You might find it helpful to do the tasks yourself as though you were a learner.

Prompt 2.

  • What will my answer tell someone about my knowledge of this topic?
  • Will my answer show whether I have understood the topic fully?

What other methods could you use to gauge how much learners already know and hence plan activities that will take their learning further?


In order to gauge how much students have learned, it is not enough to assess their knowledge and skills at the end of the topic or course. We also need to find out what they know at the start of each topic so that we can identify more specifically the knowledge and skills they have gained during the topic or course and so that we can plan our lessons based on their prior knowledge.

There are a variety of methods which assess students’ prior knowledge and skills. Some methods are direct measures of students’ capabilities entering a course (for example, diagnostic tests). Other methods (e.g., students’ self-reports, inventories of prior courses or experiences) are indirect measures. Here are a few suggestions of ways you can gauge students’ prior knowledge:

  • Asking learners to draw a poster showing everything they already know about a topic or idea ( for example “Everything you know about circles”, or “Everything you know about straight lines”), enables them to show you what they know and any gaps can help you plan further activities to help them. If you ask appropriate encouraging questions, you will find that they know much more than you thought, at first, about the topic. They may even surprise themselves at what they know and remember. However, if you ask them to complete a test, they may underperform through anxiety as a result of failing in the past at similar tests.
  • Diagnostic tests are sometimes used to assess students’ relevant background knowledge. However, it is important not to rely solely on what is essentially a snapshot, but use them in conjunction with other forms of assessment. Such tests should not be graded. They can help you and your students gain an overview of their preparedness, identify areas of weakness, and adjust the pace of the course. The diagnostic assessment should include tasks or questions that test students’ capabilities in the knowledge and skills that they will need to succeed.
  • Prior knowledge self-assessments ask students to reflect and comment on their level of knowledge and skill across a range of items. Questions can focus on knowledge, skills, or experiences. The advantage of a self-assessment is that it is relatively easy to construct and score. The potential disadvantage of this method is that students may not be able to accurately assess their abilities. An example of such a self-assessment might be:

    How familiar are you with stem and leaf diagrams?
    • I have never heard of them or I have heard of them but don’t know what they are.
    • I have some idea what they are, but don’t know when or how to use them.
    • I have a clear idea what they are, but haven’t used them.
    • I can explain what they are and what they do, and I have used them.

Teachers can use other activities to quickly gauge students’ comprehension and help them plan further sessions on a topic.

Some examples are suggested below.

These will give you immediate feedback about the entire class’s level of understanding, not individual learners.

  • Minute Paper: Ask your learners to identify the most significant things they have learned from your lesson. Give learners one to two minutes to write a response on an index card or paper. Collect their responses and look them over quickly. Their answers can help you to determine if they are successfully identifying what you view as most important.
  • Muddiest Point: This is similar to the Minute Paper but focuses on areas of confusion. Ask your learners, “What was the muddiest point in today’s lesson/your homework?” Give them one to two minutes to write and collect their responses.

These will give you feedback about individual learners:

  • Posters/Concept maps: Ask the learners to draw a poster telling you everything you know about a topic or ask them to draw a concept map. For example, you might ask them to tell you everything they know about pie charts. This is also a good homework exercise.
  • Group/pair activities: Many group activities such as Tarsia jigsaws, matching multiple representations, and always true, sometimes true, never true cards, will help you gauge the understanding and knowledge of students in a non-confrontational way. Why not ask your learners to prepare a Tarsia jigsaw or write an exam question for homework? Then the next lesson, everyone can do everyone else’s jigsaw or exam question!

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