M6 - Assessing students in classes


Assessment has two purposes in a university mathematics setting:

  1. Feedback on learning to the student; and to the lecturer on the learning process.
  2. Grading of students' knowledge, skills and attributes against a standard.

In Module 6 we will concentrate on learning and feedback; in Module 10 we will focus on assessment tasks for grading.

This module addresses how the teacher determines both the current level of student understanding, and the extent to which the targeted knowledge and skill development for the lesson have been achieved. This is comprised of: assessing student's current level of skills, identifying gaps and misconceptions, and then assessing the achievement by the students of the outcomes for the lesson. This is called formative assessment (assessment for learning).

Importantly, this module is not about assessment for the purpose of grades in the unit - it is assessing students during a lesson with the purpose of tailoring teaching, responding to need, and achieving a higher level of learner outcomes in each lesson.

Learning Outcomes

At the end of this module you will be able to:

  • design learning and formative assessment tasks to check whether the learning objectives of a class have been met
  • adjust learning activities on the basis of formative assessment results
  • design effective and efficient feedback to students in classes.

Module Structure

This module proceeds as follows:

Are we learning?

Consider this quote from a highly-cited paper:

“All classroom work involves some degree of feedback between those taught and the teacher, and this is entailed in the quality of their interaction which is at the heart of pedagogy. The nature of these interactions between teachers and students, and of students with one another, will be the key determinants for the outcomes of any changes” (Black & Wiliam, 1998, p 16)

Task 6.1 Monitoring learning

Think about a recent class you have taught (or participated in) and briefly note down the following - we will come back to there points during the module.

From the teacher's perspective:

  • List two ways that you checked whether the students were learning.
  • List two ways that you gave them feedback on their learning in the class.

From the learner's perspective:

  • List two ways that students checked whether they were learning.
  • List two ways that they received feedback on their learning in the class.

Assessment and feedback cycle

In this module we look at tasks and feedback as part of a cycle (see Figure 1), which is termed formative assessment (Sadler, 1989). Formative assessment is assessment that gives students feedback on their learning; it forms them or informs them. A very simple and familiar example is using the answers in the back of the textbook to check progress. Final examinations are generally not formative assessment- a final examination is an example of summative assessment, which will be covered in Module 10.

Figure 1: Assessment and feedback cycle

Examples of formative assessment:

  • Problem sheets with worked solutions or hints for solutions - drawing out the main ideas to be learnt.
  • The comments made to students during a tutorial about their solutions to problems.
  • An online test, where students find out immediately what they can and can’t yet do, and are allowed to re-sit.
  • Draft report. Students are completing a report for grading as part of the summative assessment of your unit. They hand in a draft report three weeks before the final report is due. You give them feedback on the draft - very much like refereeing a paper - and they make changes and submit the final report.

We now look at each of the components of the cycle in turn - tasks, feedback and reflection - before giving concrete ways to implement each of them.


In this context, the task can be anything designed to elicit a response, such as a verbal question to a group or individual:

  • Give me an example of a function that is continuous. Give me an example of a function that is discontinuous.
  • How does an eigenvalue help us?
  • What sample size is required? Why?

or a worksheet for students with answers. The student works through the question, looks up the answer - correct or not - then reflects on why the answer is correct or not.

These "tasks" are not for marks but are designed for you to gain an idea of how well the students have understood a topic and for the students themselves to get feedback on their understanding.

Good tasks and questions will help you improve your feedback and are strongly linked to how you organise your class. Good class organisation will help you engage your students in each stage of the cycle and achieve your learning objectives (see Module 3). Again, consider this quote from an influential author in the assessment literature:

“Tasks, particularly interactive tasks, create performances that can be and are judged or assessed, formally or otherwise. Good teaching therefore involves good task planning and good tasks afford plenty of opportunities for judgement and conversations about judgement.” (Knight 2006, p. 441)

As the cycle closes with reflection, we need to reflect on why students are responding in certain ways and adapt our questions (tasks) and teaching style accordingly.


Feedback is information that a student receives about his/her performance on a learning or assessment task. It can be given by self, peer, computer, tutor, lecturer or external organisation. Feedback can be to a group or to the individual, in written form or oral form.

Examples of classroom feedback:

  • Ask a student to write a solution on the board and then get the class to comment on it
  • Group feedback - a summary by the teacher of how the class has performed on a task.

Examples of feedback on work that has been handed in:

  • Use Camtasia or other screen capture systems to record written and oral feedback together (for an individual)
  • Podcast with a summary of main areas of difficulty (for the group).

Read this description and rationale for formative assessment and the feedback on it implemented by one Australian mathematics department:

Feedback can also be from the student to the lecturer. This can be informal, as in questions, or more formal, as in questionnaires to students (this will be discussed in Module 7).

Your role in feedback

A meta-analysis of over 800 education studies found that feedback was the second most influential factor affecting learning after alignment (see Module 8) (Hattie 2008).

As a teacher, your role is to provide the conditions that support student learning. Remember, it is the student who is doing the learning, but you can make a huge difference to how quickly and effectively this happens.

It is important not to take away someone's learning by giving them too much. Think carefully about the quantity and type of feedback that you will give - students need to make mistakes and puzzle over deep problems.

Students often feel that they don’t get enough feedback, while teachers are frustrated that the feedback they give seems to have little effect. One thing to consider is the type, rather than just the quantity, of feedback given. In the literature, it is emphasised that to be effective (and heeded) feedback must be not just an explanation or justification of a mark (Knight, 2006). Hattie and Timperley (2007) state that effective feedback answers three questions:

  • Where am I going? (Feed Up)
  • How am I going? (Feed Back)
  • Where to next? (Feed Forward)

and it must address process and self-regulation as well as current task performance. The literature also shows that feedback must be timely; this is one advantage of building feedback into your classes, over feedback supplied by marking work.

Task 6.2 The role of the teacher

Read the article Formative assessment: revisiting the territory (Sadler, 1998).

According to Sadler, what are the six things that highly competent teachers bring to what he terms the “evaluative act”?

Principles for feedback

  • Start with the positive.
  • Focus on only 2 or 3 points for students to work on.
  • Provide concrete suggestions on how to improve.
  • Make sure it is understood: ask the student to describe the feedback in their own words.


Reflection is an important part of learning as we fit what we are learning into our previous knowledge matrices. Reflection allows us to establish control and develop belief (from the ideas of Schoenfeld, Module 2). Threshold concepts often jar us into new ways of looking at mathematical concepts.

So one of the roles of the teacher is to challenge students to work with threshold concepts, incorporate new ideas, master new methods in a relational rather than instrumental way, and move to the next level in their thinking about mathematics. To do this the students need to have time to think and to reform their ideas. As a lecturer, you should be reflecting on the tasks and feedback that will prompt the student to deep learning of mathematics.

Reflection is the step that enables feedback to be effective. Information about a “gap” in understanding serves the role of feedback only when it is used to alter the gap (Sadler, 1989).

Ways to enable reflection in your classes:

  • Allow time for reflection - don't make the class so full that there is no time for thinking.
  • Less busy work - hone the number of examples so that students can complete the routine ones and still have time for the odd ones that will challenge their thinking.
  • Make sure there is time in the class for discussion on the key points/ideas.
  • Ask open-ended questions - Why? What's different about this one?
  • Ask for examples and counter examples - Give me an example of a differentiable function, give me an example of a function that is not differentiable. Being able to generate examples with particular properties is the domain of an expert mathematician.
  • Talk about yourself! How do you go about thinking of mathematics and attacking a mathematical problem?

Task 6.3 Incorporating reflection

Add the following to the things you noted at Task 6.1:

  • List two current ways that you provide opportunities for reflection.
  • Write down two questions related to your current teaching area that you can use to encourage reflection.

Now that we have looked at each of the components of the formative assessment cycle, we consider concrete examples and suggestions for implementing them.

Practical ways to assess students in classes

The ‘typical’ mathematics tutorial includes worksheets of problems (possibly chosen from a textbook), done by the students or by the tutor (hopefully in an interactive way!). Below we suggest some other things you might consider doing - some of which could be incorporated in lectures - and the things they might tell you about your students’ understanding and hence your teaching.

Methods to identify learning needs before the lesson

Often it is useful to set tasks before the lesson so that teachers can more accurately pitch the ensuing lesson to meet learner needs. For instance, students may be required to:

  • submit solutions to tutorial or prac questions
  • post to a blog
  • post to a wiki
  • respond to an online survey form with ratings for level of difficulty/understanding.

Assessing prior learning

Here are a range of tasks which the teacher can use to assess the prior learning (which can here mean the previous class) of students in a mathematics class:

  • brainstorm (individually or collectively)
  • describe
  • provide an example
  • draw a picture
  • explain how a concept is related to something else
  • why is this wrong?
  • construct a question that requires the prior knowledge to be demonstrated.

Each of these requires the students and teacher to interact to discuss understanding so that the students can ask questions and the teacher can assess understanding. Some of them can also be used during a lesson.

For more ideas, see the strategies used for interactions resulting from the Australian Universities Teaching Committee project on teaching large classes (Teaching and Educational Development Institute, 2006).

Assessing understanding during the conduct of a lesson

There is a range of ways in which students can indicate their understanding including:

  • thumbs up, thumbs down
  • explain concepts to each other
  • auto responders or electronic voting (e.g. clickers)
  • writing on a virtual wall (where computer equipped).

Some approaches allow more in-depth insight into student understanding than others - for instance, thumbs up/down only allows an instrumental understanding to be assessed, whereas written tasks can be used to assess a more relational understanding. Here is what one previous participant described:

I include questions on the tutorial worksheet of the kind "explain why ...". The tutors in my subject (which include me) have overheads I supply to put up which play the role of "the answer in the back of the book" and which they reveal gradually during the hour. But when they uncover the answer to an "explain why" question, it will say something like "convince your neighbour" or "explain your answer to the tutor".

Strategies for encouraging participation

To give feedback we need strategies to encourage students to respond to tasks. It’s hard to give feedback if students are reluctant to participate due to shyness or cultural background. The opposite is also true: a good student may answer too quickly. Here are some ways to control participation:


  • Direct. Ask a student to do the task without warning. For example - ask a student to write the solution to one of the tutorial questions on the board. This could be a homework question that they were supposed to prepare before class.
  • Some warning. Give a student 5 minutes to prepare an answer.
  • Prepared. Give a student a task to prepare for next week.


  • Small group. Students prepare a solution in pairs or small groups and present it to the class. This is useful for anxious students as they can test out their ideas and/or present on behalf of the group.
  • Class vote. Canvas the class for the answers to the task and vote on which is the best solution. This is useful for problems where there are multiple answers or methods of solution.

Other discussion methods can be found in Macquarie University's Guide to leading discussions (Wood, McNeill & Harvey, 2008).

At first you might think there is not much to ‘discuss’ in a maths class, so here is a suggestion:

  • Give students a set of objects to classify, and the rules for classifying them e.g. integrals and the technique to apply (by parts, substitution, trig identity); differential equations (linear, separable, neither, both); or abstract objects and a set of axioms. Make some of them subtle. This activity lends itself to think-pair-share; they have to convince each other of their answers.

Encouraging reflection on learning

  • Compare sets of exercises in different textbooks. For example, sample size is a threshold concept so take three texts where this is explained; ask students to work through the exercises and then explain which best helped their understanding.

Methods to check understanding at the end of a lesson

  • Short form at the end (three questions).
  • Ask students to write down for you what the main point of the class was.
  • Ask for a page with concepts they don't understand (minute paper).

These are examples of `Classroom assessment activities’ or CATs. Although you will collect student responses to reflect on them, the process of producing a written response has made your students reflect on their learning. The following links explain these CATS further:

Group tasks

Learning is not a solo activity. Many of us learn through interaction and collaborative learning methods. The Lawrence Hall of Science in Berkeley has a book of cooperative activities that are aimed at school level with some higher level tasks (Lawrence Hall of Science, n.d.). For our purposes it is the style of task that is useful for the context. These are group tasks where the participants have bits of the puzzle that they need to put together in a cooperative fashion.

One previous participant in this unit described a group task and the associated feedback:

At the beginning of the tutorial the students arrange themselves into groups of 4. Each group is given one copy of a set of questions ... where a “simple answer in the back of the book” can’t be provided, e.g. “Give two examples of applications where the scalar product is used”, or “Write a brief paragraph explaining what the Fundamental Theorem of Calculus says and how it relates to integration”. The groups work on these questions during the tutorial with the tutor circulating. They write up one set of solutions on their copy of the questions. Toward the end of the tutorial the tutor selects a group and marks one of their questions using the visualiser/document camera. The tutor will comment on the setting out as well as the correctness of the solution as well as stressing just what it is that a marker would be expecting of the students in that particular question. The tutor also prompts for alternative methods and approaches.

Using scratch cards (team-based learning)

Scratch cards are useful for class participation. They are multiple choice cards that are like scratch lotto tickets. You can use ordinary multiple choice. Here are a couple of videos that present this way of getting participation and feedback:

Authentic tasks (authentic assessment)

Authentic assessment presents students with real world challenges that require them to apply their relevant skills and knowledge. For mathematics we interpret this as assessment that models the knowledge, methods and practices of a mathematician. Although we often use simplified and shorter tasks in our teaching, one of our aims is to induct students into the discipline; therefore the design of tasks should reflect mathematical practices and beliefs.

Tasks for developing belief in mathematics students:

  • Use Schoenfeld's questionnaire (Schoenfeld, 1989).
  • Ask them what they think a mathematician does and how they work.
  • Black and white. Many people believe there is always a correct answer to mathematics. Give examples where this is not the case.
  • Assignments or tasks that require some trial and error.
  • Ask them what they think maths is in the first class, then use examples to challenge their beliefs as you move through the semester.
  • Ask about how they learn mathematics and how they will use it in the future.

When teaching in a service unit, this would require embedding the mathematics in the context and practices of the service discipline.

Examples of authentic tasks in service teaching are:

  • dealing with messy data
  • ill-defined problems
  • presenting results as a poster
  • running a mini-conference
  • peer review of solutions
  • using professional tools such as SAS, Excel, MATLAB, depending on the discipline group.

Task 6.4 Encouraging student participation

On the discussion board , post one or two questions or tasks that you use to encourage student responses and engagement in tutorials. Make comments on the questions posted by other participants.


Self-assessment is common in school mathematics, that is, students look up answers in the back of the textbook to check their solutions. This practice is also common in university mathematics, as is the provision of model answers and the marking of assignments. However, students need to develop their own strategies to check their own answers. Students can be asked to put a confidence interval (correct, not sure, incorrect or other appropriate scale) on their answers and say why.

Have a look at the new Review system at The University of Sydney (2011).

Another way to incorporate self-assessment is to ask students to mark their own work from a marking guide that you provide (not for grades). This means that you can give your time and attention to simply answering questions about what students don’t understand.

Peer assessment

Peer assessment is also a good way to model real practice. Typically people will work in a team in professional employment so results are regularly checked for accuracy. Generally when students work in teams or groups, peer assessment is taking place as they discuss what they are producing.

Peers can also give feedback in classes, say when one student presents at the board - again they could use a confidence interval and say why they believe the solution is optimal/correct etc.

Another idea is to ask students to swap work, and then mark it from a marking guide, as suggested for self-assessment. You can collect all work and redistribute it for anonymity.

As the teacher, you need to establish guidelines for peer assessment to be a ‘safe’ activity for everyone.

Task 6.5 Strategies for feedback

  • Review the list you made at Task 6.1.
  • Add three more ways that you will try to incorporate feedback after reading the ideas in this module.
  • Try them out and think about whether they work and how you can make them part of your repertoire of teaching tactics! Remember not all tactics work for all students or teachers, so you need to build up an armoury of approaches that work for you with your students.

Review and conclusion

The cycle of task - feedback - reflection is the basis of formative assessment.

Good tasks form the basis of learning supported by good classroom organisation that encourages participation and feedback. Reflection on learning is critical to identifying and correcting misconceptions and moing to deeper learning.

Let's challenge students to move to deeper learning of mathematics, to incorporating threshold ideas and to feel the excitement and challenge of our discipline.

In Module 10 we will revisit assessment and add the ideas of standards and marking as we move to summative assessment - that is assessment used in grading such as examinations. In the next module, Module 7, we examine how to collect evidence about your teaching and we consider action research as scholarly practice.

Relationship to Assessment Task Two

In Assessment task two, you are asked to plan a sequence of classes and describe how the learning will be supported, which is the focus of this module. For full details of this task, the submission date and the marking rubric, please consult the unit outline.


Updated: 24 Feb 2014