IB Mathematics Internal Assessment

Reflection in the Maths IA: write it throughout, not just at the end

Most students save reflection for a paragraph before the bibliography, and most of those paragraphs say the same thing: the model was good, the results were interesting, and there could have been more data. That earns very little. Reflection is thinking you can see happening inside the mathematics. Here is how to build it in as you go.

What reflection is, and what it is not

Reflection is criterion D. In my own words: it asks whether you stop and think critically about your work, rather than simply reporting it. Did the result make sense? What does it tell you about the question you asked? What would you trust about it, and what would you not? Where did your method help and where did it get in the way?

Notice that none of those questions can be answered in a closing paragraph written after the fact. They are answered at the moment a result appears or an assumption bites. A reader looking for reflection is looking for a mind at work across the whole document, and a single summary cannot fake that.

Six places to reflect as you write

Rather than adding a section, look for these natural moments. Each takes two to four sentences, and each is tied to something specific in your work.

• 1. After your first result. Before you move on, ask whether the answer is the right size and the right sign. Check it against a rough estimate, a special case or a value you already know. Then say what that check told you.
• 2. When you choose a method. Say what you weighed and what you gave up. Every method costs something: accuracy, simplicity, generality, the ability to explain it. Naming the cost is a reflective act.
• 3. When you make an assumption. Do not just list assumptions in a box at the start. At the point you use one, ask how much the answer would move if it were false. If you can test that, do so.
• 4. When something does not work. A poor fit, a strange value, a method that will not converge. This is your best opportunity. Say what you expected, what you got, and your best explanation for the gap, then follow it up.
• 5. When you compare two approaches. Set them side by side and say which you trust more and why. A preference with a reason attached is reflection. A preference without one is not.
• 6. When you interpret the answer. Translate it back into the situation you started from. What does the number mean for the person, the team or the system you were asking about? Would you act on it?

Reflection is not the same as limitations

Listing limitations is one form of reflection, but it is the weakest, and it is often done in a way that costs nothing: "the sample was small" and "real life is more complicated". These are true of everything and tell the reader nothing.

A useful limitation is specific and consequential. It says which part of your answer it damages, roughly how badly, and whether you did anything about it. Compare "my data had errors" with a statement about which readings you distrusted, why, what happened to your result when you removed them, and whether your conclusion survived. The second is a small piece of mathematics as well as a piece of reflection.

Ask "so what?" three times

A quick way to deepen any reflective moment is to keep asking what follows from it. Suppose your model predicts something a little off. First: how far off, and in which direction? Second: why might it be off in that direction, so what assumption is responsible? Third: given that, how far should anyone rely on the prediction, and for what purpose?

Most weak reflection stops after the first question. Strong reflection reaches the third, where you are judging how far your own conclusion can be trusted. That judgement is the heart of the criterion.

Keep a working log so there is something to reflect on

The students who reflect well are almost always the ones who kept notes. A running list, dated, of what you tried, what surprised you and what you changed is enough. Three habits make it useful:

• Write down your prediction before you run a calculation, however rough. Then you have something to compare the result with.
• Record every change of direction and the reason. Half the reflective content of a good exploration is a change of mind.
• Note any moment you were confused. Confusion you resolved usually turns into a paragraph worth keeping.

When you come to write up, you are not searching your memory for something to say. You are choosing the best moments from the log.

Where it links to the other criteria

Reflection overlaps with personal engagement and with Use of Mathematics, and that is useful rather than a problem. A visible change of method shows your own decision-making and shows critical thinking about the mathematics. Testing an assumption does the same. You do not need to write separate material for each criterion. You need to make your judgements visible, and the same judgement can serve several purposes.

The difference is one of emphasis. Engagement asks whether the work is yours. Reflection asks whether you can step back from it and assess it honestly. A student can have plenty of the first and almost none of the second, by loving a topic and never questioning their own answer.

Common mistakes

• One paragraph at the end. It is too late to influence the mathematics and reads as an afterthought.
• Praise instead of judgement. Calling your work "accurate" or "successful" without saying how you know.
• Reflecting on the process instead of the mathematics. "I learned a lot about time management" is not what the criterion is after. Reflect on the ideas, the results and the methods.
• Nothing ever goes wrong. A document in which every step works first time either hides its process or did not look hard enough.
• Reflection that changes nothing. If you say a method is weak and then carry on using it, ask why. Better still, do something about it.

A check before you submit

Take a highlighter, or a colour in your word processor, and mark every place where you evaluate, compare, question or interpret. If the marks all sit in the last page, you have summary reflection. If they are spread through the document and each one is attached to a particular result or decision, you are in good shape. Then read each marked passage and ask whether it could appear in a different student's work. Where the answer is yes, make it more specific to yours.

One caution. The thinking has to be yours, and it has to be honest. I can help you notice what you have already questioned and are not yet showing, but nobody can supply the judgement for you, and a reader who reads submissions every year can tell.

More like this: all Maths IA guides. Related: personal engagement in the Maths IA