What is Assessment?
In a nutshell, we can define assessment as the process of gathering data to better understand the strengths and weaknesses of student learning.
Assessment can be split into three types:
- Assessment for Learning
- Assessment as Learning
- Assessment of Learning
They may look almost identical on paper, but they’re distinct and serve different purposes. Let’s examine each in turn.
Assessment for Learning
Assessment for learning is also known as formative assessment. It describes any of the various methods teachers use to help determine how well pupils understand the maths being taught.
It’s the kind of assessment that benefits the learner and is used as a tool to support learning outcomes.
Formative assessment answers the question: where are the pupils in their learning and, knowing this, what should be done next in teaching?
It can be thought of as the day-to-day, between-the-desks, or real-time assessment.
Formative assessment can be both formal or informal and may range from teacher observations to tests. It often contains data in the form of numbers, descriptions, comments and checklists.
Here are some examples of the types of formative assessment:
Asking skillful questions throughout the lesson will help teachers check for depth of understanding, identify progress toward learning goals and diagnose root causes of incorrect answers.
These tips for asking good questions were condensed from a list produced by Steve Reinhart, a maths teacher in Wisconsin, U.S:
- Never say anything a student can say
- When tempted to tell students something, ask a question instead
- Ask questions that require more than recalling a fact
- The student should be able to learn something from the question, and the teacher should be able to learn something about what the student knows or doesn’t know
- Use more process questions than product questions
- Product questions require short answers and don’t give much information about what a student knows; process questions require students to reflect, analyse and think at much higher levels
- Replace speaking with sets of questions
- Very often, classroom time involves too little active thinking and discussion, resulting in a low percentage of students paying attention
- Be patient
- Increasing wait times to five seconds or longer can produce more and better responses
Skillful questioning may sound like a mundane skill, but it’s actually essential.
Asking questions with yes/no or right/wrong answers can stunt the development of children’s metacognition and communication abilities, explains Alex Laurie, a primary maths teacher in New Zealand who has extensive experience in the Singapore method. That’s because these ‘lower-order’ questions don’t give students an opportunity to reflect on how they reached an answer.
Instead, by focusing on one rich problem and using higher-order questioning techniques, the learners’ responses will reveal various levels of understanding as well as how they approach maths problems in general.
Bloom’s Taxonomy, named for American educational psychologist Benjamin Bloom, is a learning model that consists of three domains — cognitive, affective and psychomotor — each with different levels of thinking, ordered from the simplest to the most complex.
As you move up the levels, the thinking skills become more sophisticated. For example, remembering and understanding are lower-order skills, while evaluating and creating belong to the higher-order.
Here’s how to encourage higher-order thinking:
- Ask questions that give learners a starting point, for example:
- How could you sort these?
- How many ways can you find to…?
- What happens when we…?
- What can be made from…?
- How many different [ ] can be found?
- Ask questions that stimulate mathematical thinking, for example:
- What is the same?
- What is different?
- Can you group these objects in some way?
- Can you see a pattern?
- What do you think comes next?
- Is there another way?
- Ask assessment questions, such as:
- What have you discovered?
- How did you find that out?
- Why do you think that?
- What made you decide to do it that way?
- To develop learners’ metacognition skills, ask final discussion questions like:
- Who has the same answer/pattern/grouping as this? How do you know?
- Who has a different solution? How is it different?
- Have we found all the possibilities? How do we know?
- Have you thought of another way this could be done?
Listening to your learners
Finally, we sometimes forget that really good questions in the classroom don’t always come from the teacher.
Questions from the children can give you valuable information about their learning. Allow the children to question each other and you.
Dawn Copping, Headteacher at Shaw Primary Academy, says when it comes to classroom assessment, there are dozens of methods you can use to check in on your learners and move everyone forward. But after months of disrupted learning, making sure assessment is both efficient and effective is a top priority.
The best evidence is hearing from the children. Create the opportunity to talk to them about what they’re learning.
You could ask them:
- What do they know now?
- What didn’t they know?
- How do they know they’re making progress?
Introducing plenary questions or activities
Plenary questions or activities check whether learners are ready to move on at the end of the lesson. In mastery programmes, learners complete independent work that progressively requires a deeper level of understanding. This carefully structured work allows you to easily gauge how well a child understands the lesson’s objective by how much of the independent work they are able to complete.
Implementing quick check routines
Some teachers choose to implement a quick check routine where children use a signal, such as a thumbs up, to show their level of confidence and help with the pitch of the lesson. This quick scan can determine next steps, like whether the majority of the class can go ahead and complete independent work while those still unsure can continue in a guided setting with the teacher or peers.
Assessment as Learning
Assessment as learning — another type of formative assessment — is when pupils actively monitor their own maths learning, ask questions and use a variety of strategies to decide what they know, what they don’t know and what they partly know.
It’s the process of developing and supporting pupil metacognition, encouraging pupils to be self-reflective and to ask questions about their learning.
These comments from a teacher in a 2017 study show how a mastery approach, in this case using the Maths — No Problem! programme, shows that developing a classroom learning environment that embraces struggle and mistakes is intertwined with collaborative learning, formative assessment and development through metacognition as a self-regulated learner:
“It’s that idea that you are building a maturity as a learner.”
“Mistakes actually become a central process. Now in our lessons we’ll be quite glad when somebody makes a mistake because it’s something to run with. And you’re also modelling that those skills of self-checking — so the child’s talking out loud about what they’re doing and they might make a mistake, they’re self-checking and self-regulating which is actually a really important skill.”
Why are metacognitive skills important?
Put simply, metacognition is the ability to think about one’s thinking.
Research suggests that there’s a direct link between learners’ metacognition and their achievement in mathematics. When you compare the behaviour of high-achieving and low-achieving learners in problem-solving tasks, you often find key differences.
- Explain things to themselves as they work through examples
- Try to construct relationships between new processes and what they already knew
- Tend to infer additional information that wasn’t directly given
- Follow examples step-by-step without relating it to anything they already knew
- Don’t try to construct a broader understanding of the procedure that would enable them to generalise it to new situations
As teacher and former Ofsted inspector John Dabell explains, children have a vital role in taking responsibility for their own learning and in supporting the learning of their peers. One of the foundational principles of formative assessment is to help children become actively engaged in accurately assessing their own understanding and taking action.
Teachers can create amazing lessons by engineering discussions, scaffolding learning and providing guidance on the next steps in learning, but pupils need to be able to calibrate, self-assess and reflect.
Teachers and pupils can co-create learning goals to guide their next steps toward peer assessment, self-assessment and reflection.
Maths curricula all over the world emphasise the importance of developing mathematical fluency, conceptual understanding and reasoning skills in their learners. However, these skills are not always easy to assess. Cue the humble maths journal, says Alex Laurie.
Maths journals give children the opportunity to reflect on their learning and consolidate their understanding.
But what is a maths journal? And what does a good maths journal look like?
Maths journals are exercise books — grid, lined or plain paper — learners use to record their methods, explanations and ideas for solving problems and learning concepts.
Journals give teachers insight into a student’s thinking and understanding of mathematical ideas, allowing them to address misconceptions as they surface.
What should they look like?
- Add the date and title
- Every journal entry needs to begin with the date and a title.
- Add a learning intention
- Often in maths, we begin the lesson by letting children know what mathematical concept they are going to learn that day. We give them the learning intention or the ‘We Are Learning to’ (WALT) to record at the top of their page
- A journal title is similar to a WALT but instead of telling learners the mathematical concept, the journal title comes from the learner and will reflect what they think they have been learning during the lesson
- Set expectations
- A good journal title should reflect the mathematical idea the lesson has focused on. Journal titles offer insight into a learner’s level of understanding of the concept
- Take the following maths journal title examples from Maths — No Problem! NZ Textbook 2A, Chapter 7, Lesson 6 (the Explore task)
- Subtract from 10: This might suggest the child has understood the mathematical concept and they are able to use formal mathematical language
- Taking away logs: This might suggest the child has understood the mathematical concept but they are using informal mathematical language
- Amira and her logs: This might suggest the child is not making the link between the mathematical idea and the context of the problem
Maths journaling can play a major role in children becoming actively involved, giving them an opportunity to focus on their learning journey and articulate their own understanding. It enables them to embrace mistake-making and think critically.
According to Roger Hitchin, head of Singapore maths and drama at Wellington Prep School.
creative journaling encourages learners to develop models and stories and show the full range of their understanding.
Creative journals are also a safe place to explore misunderstandings or stretch more advanced learners who have mastered a concept.
There are five different types of maths journals, says Helly Douglas.
- Descriptive questions ask children to describe the methods they have used or explain a concept. For example, “How do you multiply a two-digit number by another two-digit number?”
- Evaluative tasks ask children why they chose a method and how helpful it was to solve the problem. Children must justify the choices that they’ve made.
- Creative journaling lets children develop models and stories to show their understanding. For example, “The answer is 25. What could the question be?”
- Children can explore a problem and record their findings, such as, “I buy a toy in a shop. I get 20p change. What coins did I get?”
- Children show you what they have learned and understood in formative journal entries. This is an opportunity for them to reflect on what they found challenging and why.
In short, journaling is a tool students can use to communicate their knowledge and feelings about mathematics, helping them to consolidate their understanding. It’s also a tool for teachers to evaluate student learning and growth.
Assessment of Learning
Assessment of learning, or summative assessment, is the process of testing individuals in order to determine their understanding of maths. It can be used to grade a child’s rank in class and as a comparison to peers.
Summative assessment sums up what has been learned so far. It asks the question: have the pupils learned or mastered what has been taught? This often brings to mind quizzes and tests and numerical data, but can take many forms. Summative assessment is often seen as a tool for Assessment for Accountability.
While summative assessments are normally delivered at the end of the learning process in a traditional exam setting, they can also be used formatively as an opportunity for upgrading learning.
Their effectiveness, though, really depends on the nature and quality of the feedback. The results shouldn’t be used to find fault, but to work with formative assessment to deliver the correct learning environment for every child.
How summative assessment is used
Some of the uses of summative assessment include:
- To plan future learning goals and pathways for pupils
- To provide evidence of achievement to pupils, teachers and parents
- To provide data for the wider community and outside groups
- To offer a snapshot of learning
- To identify where additional resources are most needed
When assessing for accountability, targets based on predefined metrics and numbers take centre stage, says Mirkka Jokelainen, a product manager at Oxford University Press. It can support improving learning outcomes but what’s printed on a pupil’s report is sometimes prioritised over other information that teachers can use to take positive action in their classroom.
We like to pretend assessment for accountability doesn’t happen. No one wants to think of six-year-olds taking tests just so that someone somewhere can have an impressive graph on their slideshow. Yet the reality is that most schools have to produce information that shows leadership or external stakeholders they’ve reached their targets.
Adam Gifford, strategic primary lead in the NCETM’s Maths Hub pilot programme and a Maths — No Problem! series editor, author and trainer, says summarising a child’s learning is a difficult task.
A high level of skill is needed to write questions that can help form an idea of a child’s understanding. Summative assessment is sometimes viewed negatively on its own, but works well as part of a comprehensive, interlinked assessment programme.
Assessment in Insights
Maths — No Problem! assessment papers are designed to provide achievement data in different content domains; number, addition and subtraction, geometry, measures, statistics, multiplication and division, and fractions, decimals, percentage and ratio.
But when used alongside the Insights tool, results can also be analysed at an individual, class and school level.