When done well, mathematics assessment can lead to greater effectiveness for teachers and greater mastery for students. When done poorly, assessment can have a detrimental effect on both teaching and learning.
On this page, we’ll look at:
- The various types of mathematics assessment
- Different assessment tools and strategies
- Equity in assessment
- Some common assessment pitfalls
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 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.
- 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.
What are some different assessment tools and strategies
One key type of assessment is the daily assessment that teachers gather when working with their pupils. This formative data shows how deeply pupils are learning and how well they are understanding a concept on any given day, says Adam Gifford, a Maths — No Problem! series editor, author and trainer.
It isn’t the whole assessment picture, but it’s a good starting point when building links between assessment and Continuing Professional Development (CPD).
According to a study on educational practices by Barak Rosenshine at the International Academy of Education, daily review was part of a successful experiment in primary school mathematics.
Teachers in the experiment were taught to spend eight minutes every day on review. Teachers used this time to check the homework, go over problems where there were errors, and practise the concepts and skills that needed to be practised until they became automatic.
Students in these classrooms had higher achievement scores than did students in other classrooms.
The most effective teachers in the studies of classroom instruction understood the importance of practice and they would begin their lessons with a five- to eight-minute review of previously covered material.
Teachers are encouraged to undertake daily assessment using prompts, questions and suggestions to collect information and then act on that information accordingly. Any recorded information including workbooks and journals also provides assessment data to add to the overall picture.
The key to conducting good daily assessment is preparation, says Gifford. Knowing in advance what you need to do to support your learners will help you get the most out of every lesson, as well as help to make your formative assessment more responsive to your pupil’s needs and more in-line with the results of tests and yearly checkpoints.
Daily assessment becomes even more powerful when we ask ourselves the following questions in advance of the lesson:
- Which questions can I ask to assess whether a learner can carry out these examples?
- How can I use the Concrete Pictorial Abstract approach for each of these examples?
- How will I best aid the transition from one known idea to an unknown idea in the context of this lesson?
- Does the task I am using offer opportunities to assess these areas?
If you’re unable to answer any of the questions — don’t panic! This is an opportunity for professional development to help grow your assessment skills.
These conversations may also tie in with the school’s more formal assessment approach of yearly checkpoints and testing.
Yearly checkpoints and testing
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.
When used alongside the Insights tool, results can be analysed at an individual, class and school level.
Using Insights can provide a breakdown of achievement by cognitive domains; knowing, applying and reasoning. This level of data allows teachers and school leaders to make decisions on day to day teaching, school-wide focuses and to inform a professional development programme.
Links between daily and yearly assessment approaches and CPD
But even with a steady stream of formative and summative assessment, our data is only as good as how it’s used, says Adam Gifford.
Accepting assessment happens everyday in our classrooms between the teacher and the learners, this data should give a clear message to the teacher about how well the children are understanding new ideas. Yearly assessment checkpoints allow the teacher to see how well these new ideas can be applied over time.
If yearly checkpoint assessments indicate a discrepancy between what is seen in the classroom and what is seen in the results data we should be asking a key question quickly.
Using maths journals
How should you assess using maths journals
Peter Marriott, a teacher at St Bridget’s C of E Primary School in Wirral, says sometimes a child writes something in their journal, and you think they have the basics right but they could go a bit further.
“The fact that they’ve written something in their journal, that journaling itself is telling me, ‘I think I could push them a bit further.’ So within the lesson, I’ve assessed that child straight away, and I can do something about it there and then, which is great.”
He compares that to historical teaching methods.
“You gave the child their maths book to work in. Here’s a load of questions for say, column method addition. But it’s just column method addition calculations on a page. That doesn’t tell you they can use column method addition to solve a problem. Whereas if I posed a problem to them and they use column method addition independently. Brilliant. I know that kid’s at the right place.”
Rumours Activity for Assessment
A rumours activity involves presenting learners with a piece of maths chit chat that they have to unpack and investigate to test its validity. Rumours are usually short statements that are deliberately chosen to engage learners in sustained discussion in order to improve their learning.
DuFour’s 4 critical questions
The golden rule of formative assessment is it needs to have a purpose and the information gathered needs to inform the learning.
To prevent formative assessment from merely being a formality, planning for formative assessment outcomes by using Richard Dufour’s 4 critical questions can be useful. These can be scaled back and adapted to the context of an individual lesson and thought about as an effective formative assessment planning tool.
What are DuFour’s 4 critical questions?
- What do we want all students to know and be able to do?
- How will we know if they learn it?
- How will we respond when some students do not learn?
- How will we extend the learning for students who are already proficient?
Being clear about these questions and planning for how you will respond to these outcomes during lessons will help ensure you are meeting the needs of all learners in real time and the response to formative assessment is more effective.
Throughout a maths lesson, it’s your role to monitor children’s understanding of the learning to ensure any misconceptions are being immediately addressed.
At the conclusion of the maths lesson, it’s important to check how secure learners are with the objective and then use this to inform further intervention work if necessary.
Assess, Diagnose, Act
There are layers to assessment data that need to be analysed and interpreted in detail at the individual, class and whole-school level. The framework Assess, Diagnose, Act may be helpful to support assessment across the board and reduce the opportunity for error.
Pre-Topic Diagnostic Assessment
Using pre-topic diagnostic assessments can be a useful way for teachers to learn about children’s starting points, says Tom Oakley, maths adviser at Cambridgeshire County Council. That’s different from using “cold tasks,” where children are tested on content they haven’t learned yet. Rather, it means assessing prior curriculum content. After all, just because something was taught doesn’t mean that it was learned.
A diagnostic assessment helps you to find out what is, and what isn’t, well understood; and which aspects will need addressing before new content is taught.
A key feature of any diagnostic task is that it should go beyond revealing which questions a child can or can’t answer; it should also be clear what the child is thinking so the teacher can ascertain what the gaps are and how to address them.
Just doing a diagnostic assessment isn’t enough. The results of the assessment must inform future teaching and learning and, ideally, that includes addressing gaps and misconceptions before moving on.
What could a pre-topic diagnostic assessment look like?
A pre-topic diagnostic assessment could involve:
- Asking multiple choice questions
- Undertaking a test based on prior content then discussing the answers to identify whether incorrect responses were simply mistakes or misconceptions. One option is to use an ‘end of block assessment’ from the previous year
- Using a practical dialogic activity that’s designed to discover potential misconceptions and elicit key vocabulary
Conceptual versus instrumental understanding
If a child is struggling with a mathematical idea, it’s important to check if the issue is with their conceptual understanding, or with their understanding of the procedure and convention.
This idea of different kinds of understanding in mathematics was developed by British psychologist Richard Skemp, whose research suggests there are two kinds of understanding needed for understanding mathematics: conceptual (or relational) and instrumental understanding.
Conceptual mathematics: focuses on establishing connections, building understanding over time, applying concepts to other problems, and gradually increasing complexity.
Instrumental mathematics centres around rote learning, memory, rules and correct answers.
Conventional mathematics is understanding that a convention is a fact, name, notation, or usage which is generally agreed upon by mathematicians.
Expressions are a useful example of conceptual understanding versus instrumental and conventional understanding.
If a child is struggling with expressions, it’s important to check whether it’s an instrumental or conventional issue, where they may not be applying the order of operations correctly. Or is it a conceptual issue? Do they have issues with using the operations to find the total?
Developing an awareness of the different kinds of understanding will help you effectively identify why a student may be struggling with a concept.
If it’s an instrumental or conventional issue, the solution will likely be exposure and explicit telling. If the issue is conceptual, then tailored interventions with concrete resources may be necessary.
Equity in Assessment
High-quality mathematics assessments must focus on equity concerns to ensure that all students are well served by assessment practices, according to the authors of Measuring What Counts: A Conceptual Guide for Mathematics Assessment. This fundamental concept is embodied in the educational principle that assessment should support every student’s opportunity to learn important mathematics.
To meet the equity principle, tasks must be designed to give children a sense of accomplishment, to challenge the upper reaches of each child’s mathematical understanding, and to provide a window on each student’s mathematical thinking.
Just as good instruction accommodates differences in the ways learners construct knowledge, good assessments accommodate differences in the ways that students think about mathematics and display mathematical understanding.
Although all students are to be assessed on important mathematical concepts and skills, the equity principle implies that assessments must be sufficiently flexible to allow all students to show what they know and can do.
International comparative studies, for example OECD 2015 and Museus et al. 2011, have frequently identified achievement gaps between majority and minority students, according to the 2018 study Assessment in mathematics education: responding to issues regarding methodology, policy, and equity.
When it comes to assessment, children with English as an additional language “often struggle with some of the reasoning questions because they’re wordy,” says Louise Hoskyns-Staples , a test development researcher who, while working with the Department for Education, developed the 2016 national test framework.
Maths — No Problem! addresses this concern in its online assessment tool Insights.
Hoskyns-Staples, who authored the Assessment Papers for Maths — No Problem! says “where there’s been some bias against those children, we ’ve gone back, and reworded the questions, to make sure that they don’t present any additional difficulty.”
What are some of the pitfalls of assessment
Ideally, a school’s Assessment for Learning framework will produce their accountability metrics at the same time, says John Dabell. But the nature of reporting requirements and the fact schools have limited resources sometimes means compromises.
Educators often say maths assessment is messy or that it’s too difficult to readily access the data needed to improve teaching.
But assessment in maths is essential for successful learning. If teachers don’t know what children have learned, they won’t know how to adapt their approach.
The quality of assessment training is England is inadequate and generally only of limited help to teachers, according to the 2017 report Testing the Water by LKMco and Pearson, which also found that:
- Only a third of classroom teachers in England feel ‘very confident’ about assessment
- One in five classroom teachers would not know where to look for information on assessment if they needed it
- Under half of teachers received training in assessment as part of their initial teacher training
Complex and confusing
Tim Oates, group director of assessment research and development at Cambridge Assessment, says
the fundamental thing in assessment is being extremely clear about the object of the learning.
“In assessment terms, we call this the construct: what is it that I want a group of children to take away from this learning experience — the really fundamental thing?”
“Too complex a system of gathering evidence, a lack of precision in what it is that we’re looking for, just develops unmanageable systems in which there’s a lot of noise and a lack of clarity in terms of direction from the school and teachers to pupils as to what they should be learning. That’s when evidence gathering becomes too complex, that’s when it becomes all confusing."
Assessed too much
Some practitioners feel there’s too much focus on summative assessment and not enough on formative assessment.
“Our formative assessment needs to be more often,“ says Jonathan Evans, Teacher at Manley Park Primary School, Manchester. “Certainly in terms of thinking of Assessment for Learning, we should be doing that each and every day. I guess my personal opinion is that children in primary school are probably assessed too much, too young. We put too much importance on summative assessment, where actually, we need to be looking at how children are doing day to day and taking away some of the external pressures that exist on schools.”
Where accountability fails
No amount of statistical analysis will turn apples into oranges or vice versa, says Mirkka Jokelainen. Individuals are affected in many ways by the environments in which they live and learn. Some of these environments are more conducive to learning than others, regardless of what happens in school.
That means that some children, and schools, have to work harder than others to achieve the expected standard.
Systems of accountability are good at identifying the groups of pupils, schools, and areas that are underperforming, but bad at determining underlying causes or finding solutions.
Too often, a child who doesn’t fit on the curve gets blamed for not being ‘resilient’ and ‘persistent’ or not having enough ‘grit’.
This is unfair to children who are navigating environments and situations that many adults would find tough.
The result? Children are often failed by the very accountability frameworks that exist to protect them and ensure their success despite the challenges.
Well done on making it to the end of our Assessment Guide.
We’ve looked at the various types of assessments, and taken a deep dive into many of the most-effective strategies for assessing learners in mathematics.
We’ve also discussed equity in assessment, a hot topic in current educational circles, and looked at some of the pitfalls associated with assessment.
If you’d like to learn more about assessment, we recommend checking out the following links:
- NCETM’s Primary Assessment Materials
- Example assessment questions in the DfE’s latest non-statutory guidance
- Maths — No Problem! best practices for gathering assessment data
- Dr. Yeap Ban Har explains how to assess students during singapore maths lessons
Also, don’t miss our other Ultimate Guides:
- The Maths — No Problem! Ultimate Guide to Maths Mastery
- The Maths — No Problem! Ultimate Guide to Early Years