Less Is More: Writing Formative Assessment That Actually Informs Teaching

1. Introduction — Less Is More

We are constantly bombarded with messaging that centres around the statement:

Formative data captures are important. You should be using formative data captures to inform your teaching.

This is all well and good, but rarely (if ever) will you hear someone outlining/explaining how to write formative assessment that informs your teaching. In this article I'm going to discuss formative data captures in mathematics, what everyone gets wrong when it comes to designing these resources, and what you can do to create resources that will do two very important things:

1. Inform your teaching
2. Develop student understanding (and give them the ability to take ownership of their learning)

2. Formative and Summative Assessment Require Different Thinking

Before diving into formative assessment, it is important to distinguish the difference between it and summative assessment. Too often I have seen other teachers create a mid-unit quiz, give it to kids, mark it, give it back, and call that formative data. If you don't actually do anything with the data, it's not doing any informing at all! The point of FORMative assessment is to inFORM the learning of the student, or inFORM the teacher of their teaching.

Broadly speaking, summative assessment aims to establish student achievement against a set of learning expectations. That's it. It's a way to say

"You were taught this. You were expected to learn this. Did you learn this? If so, how well did you learn it?"

Don't get me wrong, I'm not slandering summative assessment, it has it's place, however it has a very different purpose to formative assessment.

Formative assessment on the other hand is designed so that insights can be made about student progress and understanding. As I beat you over the head with earlier, it's designed to inFORM teachers and students of their current level of understanding/ability. If no inFORMation (okay I'll stop now) is gathered from the process of formative assessment, than the assessment wasn't formative at all, and at best, it was practice.

Now that we understand the different purposes, we can start considering what makes an individual formative assessment task/question effective.

3. Writing Effective Formative Assessment Questions

A. Begin with a specific learning intention

• What precise mathematical understanding are you trying to establish?
• Avoid trying to assess an entire topic when you're only interested in one concept or skill.
• Consider what successful understanding would look like in a student response.

Poor Example: I want to see if my students can do fraction arithmetic.

Good Example: I want to see if my students understand the need for a common denominator when adding/subtracting fractions and I want to see if my students are able to get a common denominator between fractions.

In the poor example, a teacher would just make a worksheet with a bunch of fraction questions. It could include addition, subtraction, multiplication, division, simplifying, improper fractions and mixed fractions. It would be difficult to determine misconceptions or gaps in learning as there is no structure or clear goal from the assessment. This sounds like a good way to get students to build fluency in their fraction arithmetic, but not a good way to inform teaching and learning.

In the good example, a teacher would make a worksheet with just a few questions. They would have a clear goal in mind (the two concepts mentioned above), and the questions would be designed to be obvious which part of the process students don't understand.

B. Anticipate misconceptions before writing the question

• What mistakes are students likely to make?
• What might those mistakes reveal about their understanding?
• Can you construct questions or distractors that distinguish these misconceptions?

A wrong answer becomes much more useful when it provides clues about why a student might be struggling.

An example of a misconception to tease out when it comes to adding fractions is to be on the look out for when students have added denominators as well as numerators. Knowing this, when you provide students with solutions you can also provide them with common wrong answer, and why they're wrong. This is how you get kids reflecting on their own understanding, instead of just being right or wrong. E.g. If a question is

$$\frac{1}{2}+\frac{3}{5}$$

don't just give students the answer of:

$$\frac{5}{10}+\frac{6}{10}=\frac{11}{10}$$

Give them the above as well as:

If your answer was $\dfrac47$ then you forgot to get a common denominator first. Once you have the common denominator, you only add the numerators.

C. Remove unnecessary obstacles

• Is the language clear and age-appropriate?
• Are difficult calculations distracting from the concept being assessed?
• Does the question contain unnecessary context?
• Are students being asked to demonstrate several unrelated skills when only one is being assessed?

In our fractions example, it is unnecessary for me to be writing the question as

Mark wants to eat 1/2 of a pizza, and Dave wants to eat 3/5 of a pizza. If they are wanting to eat the same pizza, is this possible?

Mathematically, this question is identical in skill, however it also assesses whether students can interpret the question as fraction addition and conclude that the answer is impossible as 11/10 is more than a whole pizza.

D. Select the minimum useful number of questions

• Could one question establish the necessary understanding?
• Would a second question help distinguish between competing misconceptions?
• Is a third question necessary to check whether understanding transfers to a different representation or context?
• At what point do additional questions stop contributing useful evidence?

This one is so important. We need to distinguish between practice lots of questions to build fluency in a skill, and practicing questions to inform our learning and understanding. As the title of the article suggests: Less is more. That doesn't mean there's some magic limit of three questions. It means that every question should have a reason for being there.

4. A Practical Example — Turning Ordinary Questions into Diagnostic Questions

Stage 1: Start with a conventional question

For example:

$$\frac{1}{2} + \frac{3}{5}$$

Typically, students will either get this question, or they wont. That gives you some information, but not necessarily why students are getting it wrong unless you take the time to analyse their working and look for patterns.

Stage 2: Redesign to expose misunderstandings

Suppose I gave students this question but with multiple-choice options. These options could be:

a. $\dfrac{4}{7}$ — to determine which students are just adding both numerator and denominator.

b. $\dfrac{4}{10}$ — to determine which students are finding the correct common denominator, but are not adjusting numerators.

c. $\dfrac{3}{10}$ — to identify students who might be applying the multiplication rule for fractions instead of addition.

d. $\dfrac{11}{10}$ — to determine which students are getting the correct answer.

With this question structure, not only am I finding out about who is getting the question right (on track) and who is getting it wrong, I gain insight into which misconceptions might be the most prevalent and in most need of correction. You might find that your students are either getting it right, or they're selecting (a). This gives you a useful starting point for figuring out who needs a reteach, and specifically what they might need to understand conceptually. Of course, selecting an answer doesn't prove why a student chose it. You might need to ask them to explain their thinking before you can be certain.

Stage 3: Consider whether further evidence is needed

The initial question addresses whether a student can find a common denominator, and correctly add the fractions, but it doesn't cover every possibility. Not only do we want students being correct, we want them to be efficient mathematicians. Another question which informs something slightly different would be:

$$\frac{2}{5} + \frac{3}{15}$$

This is an excellent follow up question, because it can give us an indication of which students recognise 5 as a factor of 15, and which students use the more general procedure of multiplying the denominators to find a common denominator. You could even go one step further here and instruct students not to simplify their answer. This way you will get students who are in one of three camps:

1. Students who get it wrong (diagnostics in previous question)

2. Students who get $\dfrac{30}{75} + \dfrac{15}{75} = \dfrac{45}{75}$ (using a valid, but less efficient common denominator)

3. Students who get $\dfrac{6}{15} + \dfrac{3}{15} = \dfrac{9}{15}$ (recognising a more efficient common denominator)

This is a great teachable moment, as you're able to begin differentiating for students. You can give the first group some direct instruction which targets their misconceptions. You can get the second and third groups to compare methods and explain to each other why both answers are correct, but one approach is more efficient. It's important to emphasise to your second group that they've done nothing mathematically wrong in their answer. Choosing the larger denominator doesn't necessarily mean they understand the concept less well, but it's a good chance for them to develop their efficiency even further.

Stage 4: Would it be helpful to break the skill down into simpler components?

Suppose we designed a formative assessment that was built to capture information about whether our students could add fractions or not. This is helpful, but we would then need to diagnose what the underlying misconceptions and gaps in knowledge are. These can often be built into the assessment itself and become a reference for students. Let me explain.

Consider our goal of figuring out: Can my students add fractions. There's nothing wrong with the above techniques, however these mostly identify conceptual misunderstandings rather than procedural gaps. Lets create a series of questions that can address this and give us even more information about our kids.

Q1: Determine the lowest common multiple between these pairs of numbers. E.g. 2 and 5 = 10 (multiply them together) or 4 and 12 = 12 (recognise 4 as a factor of 12).

• This question allows us to determine whether kids even understand the concept of a common multiple, and whether they have the math skills to find one. Students need to be able to find a common denominator to add fractions with different denominators. It doesn't have to be the lowest common denominator, but recognising the lowest one can make the working much more efficient.

Q2: Express a fraction with a different denominator. E.g.

$$\text{What's the value of } x \text{ if } \frac{3}{5} = \frac{x}{20}$$

• This question allows us to determine whether students understand the concept of "multiplying by 1 in the form of a/a" in order to manipulate the expression while keeping the value the same.

Q3. Add fractions with the same denominator. E.g. $\dfrac{2}{7} + \dfrac{3}{7} = \dfrac{5}{7}$

• This question allows us to identify if students understand that they are ONLY adding numerators, and not denominators once they have a common denominator.

Q4. Put it all together with our initial question of $\dfrac{1}{2}+\dfrac{3}{5}$

• This is the question we really care about, where we can see whether students can put all the individual skills together.

This sort of question development allows us to identify cracks in the road, and patch them up before asking kids to drive it safely (not my finest analogy). That's four questions rather than one, but each question has a specific job. Building a formative assessment this way is more time consuming, but much more valuable for the student, particularly if you take the time to link the importance of the questions together in the final product.

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5. Write With the Next Teaching Decision in Mind

• If nearly everyone demonstrates understanding, what comes next?
• If a substantial proportion of the class shares the same misconception, what should change?
• If only a small group struggles, what support could be appropriate?
• If students produce unexpected responses, how might you investigate further?

Keeping these guiding questions in mind allows you to decide whether you need to adjust your teaching sequence depending on the outcome, or whether the evidence tells you to keep going as planned. Experienced teachers do this automatically as they gather informal formative data. Consider those times where you're walking around the class and every second student seems to be getting the same question wrong, and in the same way. In these moments it's best to put a halt to the working, spend a couple of minutes at the board outlining the misconception and the correct method/understanding, and then move on. When you're a beginning teacher, this is hard to do, as you likely have a very regimented plan and are focusing on a hundred other things (like behaviour management!), and so building this ability to adjust into your planning can be a big help.

The formative "assessment" you develop doesn't even need to be a printable worksheet. It can simply be an exit ticket question you write on the board with 15 minutes to go in the lesson. The students could write their answers on mini whiteboard, or send you an email of their answer, or you could look at their books, or they could go into a corner of the room labelled a,b,c,d which matches the multi-choice answer they think is correct, or probably 1000 other ways of answering the question. The technique for this is largely unimportant, the important part is that you find out their answers and use this information to inform the next phase of teaching.

6. Assessment for the Teacher Versus Assessment for the Student

We have already touched on this in the article, so I will be brief here. Mostly I want to address the difference in how the feedback from a formative assessment is used by a teacher vs how a student would use the information.

A. Formative assessment to inform teaching

For teachers, formative assessment is used to inform the future sequence of teaching. There is absolutely no point in plowing on with adding mixed fractions or adding negative fractions if the students don't understand the concept of adding fractions in the first place. Your formative assessment gives you feedback on student progress, understanding, and misconceptions.

If you don't use this feedback to make decisions about your teaching moving forward, you're wasting your time with the assessment.

With that said, if students collectively demonstrate they understand the math, a valid decision is to simply move forward with the teaching sequence. Moving forward with no evidence is the mistake.

Another benefit of formative assessment is that it allows you to identify students who are struggling OR students who are excelling. In both cases, these outliers can be identified and allow you to make differentiation decisions to best support them. Often simply knowing which students are falling behind and which students are flying ahead is enough for you to support them in these directions. You can also leverage the knowledge/ability of the top students through structured peer explanations, where students compare their thinking and methods. This can help students who are struggling, while also getting those who understand the mathematics to articulate their reasoning.

B. Formative assessment to inform students

Getting the outcome where students reflect on their own learning and understanding usually takes a lot more thoughtful planning than it will take for you to get something out of a formative assessment. You're a professional educator, whereas students are in a phase of their lives where the most important thing is who's dating who, or what the latest influencer has said about the Kardashians (are they relevant anymore?). Needless to say, they are the metaphorical horses that need to be led to water - they're not going to sniff it out themselves.

With this in mind, a simple heuristic to follow is: If I wasn't delivering the lesson, and it was a substitute teacher with no content knowledge, but armed with the instructions/solutions, would student be able to learn from this assessment? If the answer to this question is no, consider adjusting it so the answer is yes.

Reconsider our example from before where along with the correct solution, the common wrong answers were provided and an explanation for why they were wrong. This is a simple and powerful starting point for getting students to think about their own learning. But I'd take it one step further: get students to identify the mistake they made, correct their original working, and then attempt a similar question to see whether they've actually addressed the misconception. Now they're not just being told they're wrong. They're using the assessment to work out why they were wrong, and what to do next.

7. When should you be using formative asessment?

In a perfect world, every lesson. Lets face it though, you don't have the time to prepare all that. This is why experienced teachers develop the ability to formatively assess on the fly, like it's a super power. You should be thinking about tracking your students' progress and understanding like Mad Eye Moody, with constant vigilance!

With that said, considered and purposefully created formative assessment has a particular advantage: you can design questions around the misconceptions you're expecting, and think about how you'll respond before the students even answer. That's not to say assessing on the fly is less valuable. Sometimes an unexpected misunderstanding and a quick response to it can be exactly what the class needs.

To give some practical advice, you can consider formative assessment in three distinct stages:

1. Before instruction.

Colloquially known as a "pre-test", a formative assessment can be given to students before a learning sequence in order to determine what they already know, and what they have no idea about. Despite being about a bigger unit of work, you should still aim to craft these such that the results from the assessment give you an insight into where students have strong foundational knoweledge vs gaps. This should not be used to rank kids on "how many they got correct" but rather guide your teaching sequence for the upcoming learning. No students should be expected to know something you haven't taught them yet, however you could use these assessments as an opportunity to identify knowledge about a skill that's fundamental to the new skill being taught. An example of this might be before a unit where you teach the distributive law and expanding brackets e.g. $3x(x+4y) = 3x^2 + 12xy$ you first assess whether students understand how to multiply algebraic terms e.g. $3x\times4y$.

2. During instruction.

Consider those moments where you're teaching something. You get halfway through a procedure and the facial expressions you're witnessing are either sheer terror, "lights are on but nobody's home" face, or (one of my favourite memes of all time)

It's during these moments where you've just received the feedback we usually work so hard to get. The students have missed something, and it's time for you to investigate, begin asking questions.

3. After instruction.

This has been the majority of the focus of this article. Whether it's informal observation of students practicing, or a strongly considered and well planned worksheet for students, you have a lot of tools at your disposal. I won't go into them again, but hopefully you have a good idea of what to do at this stage.

8. Conclusion — Better Questions, Better Information

The next time you prepare a formative assessment, rather than beginning by selecting ten questions, consider starting with just one.

Ask yourself:

What exactly do I want to learn about my students' understanding, and what question would give me the clearest evidence?

Then do something about it.