Dyscalculia Tutoring: The Hidden Intellectual Work Behind Great Maths Teaching

For a brief period during lockdown, millions of parents suddenly found themselves doing something they had never really expected to do: teaching their own children mathematics. Many were highly educated, perfectly numerate and completely comfortable with school maths themselves. Yet a surprisingly common experience was something like: I know how to do this. Why can’t I explain it to my child?

That question gets to the heart of something we routinely underestimate. Knowing mathematics and teaching mathematics well are not remotely the same thing.

This becomes particularly obvious when working with children who have dyscalculia or significant maths learning difficulties. If a child learns maths relatively easily, showing them a method and giving them some practice may be enough. When a child has struggled with number for years, however, ordinary maths tutoring can fail because the real problem may be buried several layers beneath the topic they are currently studying.

Excellent maths teaching therefore involves much more than knowing the answer or explaining a method clearly. It requires diagnosis, sequencing, representation, judgement and the ability to work out what another human being actually understands.

That is an intellectual discipline in its own right.

A wrong answer is the beginning, not the end

Suppose a child writes:

43 – 28 = 25

Most adults who are comfortable with maths immediately know that the answer is wrong. But for an experienced maths teacher or dyscalculia tutor, knowing it is wrong is the easy part.

The interesting question is: why did the child get 25?

Perhaps they subtracted the smaller digit from the larger digit in each column, so (8-3=5) and (4-2=2). Perhaps they do not properly understand that the 4 in 43 represents forty. Perhaps place value is reasonably secure but exchange is not. Perhaps they understand the mathematics but have half-remembered a written method. Perhaps their basic number facts are so insecure that the calculation overloads them. Or perhaps it was simply an isolated careless error.

Those are completely different problems and they require completely different teaching.

If I simply tell the child, “No, remember, you have to borrow from the four,” I may get the next question correct without repairing the underlying mathematics at all.

This is something I have learned repeatedly through years of specialist maths tutoring: every wrong answer contains information. The job of the tutor is not merely to correct it, but to interrogate it.

Dyscalculia tutoring often means finding where maths first stopped making sense

Consider a pupil who writes:

³⁄₄ + ²⁄₅ = ⁵⁄₉

It is easy to tell them that they need a common denominator. But why have they added the numerators and denominators in the first place?

Do they actually understand what a fraction represents, or are the numerator and denominator simply two numbers separated by a line? Can they see that ½, ²⁄₄ and ³⁄₆ represent the same quantity? Is their multiplication sufficiently fluent for equivalent fractions not to overload them? Do they understand the multiplicative relationships underneath the procedure?

The visible error may have appeared today. The underlying mathematical difficulty may have begun several years earlier.

This is particularly important with dyscalculia and persistent maths difficulties because mathematics is enormously cumulative. Fractions depend upon number sense, multiplication, division and proportional reasoning. Algebra depends upon arithmetic, equivalence and an understanding of mathematical structure. Percentages depend upon fractions and place value. Written arithmetic depends heavily upon place value and number fluency.

If one of those foundations is insecure, later mathematics becomes increasingly difficult to build.

Yet pupils move through school according to age and curriculum. They do not necessarily move through mathematics according to what they have securely understood. A Year 10 pupil struggling with a Year 10 topic may therefore not really have a Year 10 maths problem. They may have a Year 6 fractions problem, a Year 4 multiplication problem or, in some cases, a much earlier difficulty with basic number sense.

One of the most important jobs of specialist dyscalculia tutoring is finding that point.

Sometimes a tutor has to go backwards before going forwards

Once the underlying difficulty has been identified, another judgement begins: how far back should we go?

Too little, and the foundations remain unstable. Too far, and the pupil can become bored or feel patronised. The right starting point cannot simply be determined by age or by the chapter of the textbook currently being studied.

I have worked with older pupils whose mathematical difficulties ultimately required going back to extremely elementary ideas about quantity and number. That does not mean treating a teenager like a five-year-old. It means identifying precisely which mathematical structures are secure and which are not, and then rebuilding from the point where genuine understanding still exists.

This is why good dyscalculia tutoring needs to be highly diagnostic.

A worksheet cannot do this by itself. Neither can a generic sequence of lessons. The tutor is continually watching the learner, asking questions, testing hypotheses and adjusting what happens next.

Why simply “explaining it again” often doesn’t work

We also tend to overestimate the importance of explanation.

A clear explanation is valuable, of course, but sometimes another explanation is exactly what a struggling learner does not need. If the prerequisite mathematics is missing, explaining the current procedure more clearly may simply produce a more polished version of something the pupil still cannot understand.

Should I use Cuisenaire rods? Counters? A number line? An area model? A diagram? Pure symbolic notation? Should I demonstrate something first or ask the pupil to explore it? Which example should come first? When should the representation change? How much practice is required? When should I remove a scaffold? Has the learner actually understood the concept, or are they merely copying the procedure from the previous question?

These decisions happen constantly during specialist maths tutoring.

Often they happen within seconds.

The pupil answers a question. I update my hypothesis about what they understand. The next question is partly chosen because of the answer they just gave me.

Good mathematics teaching is therefore not simply information transmission. It is a process of continual inference under uncertainty.

Lockdown showed parents how difficult maths teaching actually is

This was one of the revealing things about home schooling during the Covid lockdowns. Parents suddenly had to cross the enormous gap between I understand this and I can get another human being to understand this.

A parent might demonstrate long division exactly as they remembered learning it. Their child stared blankly. So they explained it again. Still nothing. They tried another example. Still nothing. Eventually parent and child were both frustrated.

The parent may have understood long division perfectly well. What they did not necessarily possess was the professional toolkit for discovering why this particular child was not understanding this particular piece of mathematics.

That is not a criticism of parents. Why would they be expected to possess that expertise?

We do not assume that someone who can play the piano automatically knows how to diagnose why a beginner repeatedly tenses their wrist. We do not assume that speaking fluent French makes somebody an expert teacher of French.

Yet with mathematics there is still a strange assumption that someone who knows how to obtain the answer should therefore know how to teach somebody else to obtain it.

I suspect lockdown cured quite a few parents of that idea.

Dyscalculia makes the difference between knowing maths and teaching maths even clearer

The easiest pupils can sometimes make mediocre teaching look excellent. Show them the method, give them a few examples and they spot patterns, make connections and fill in missing conceptual steps themselves.

The real test of teaching expertise comes when that does not happen.

What do you do with the child who has been shown equivalent fractions repeatedly and still cannot see them? What do you do with the teenager who still has to reconstruct (7+6) every time they encounter it? What do you do with the pupil who can mechanically solve an equation but has almost no conception of what the equals sign actually means?

What do you do when a child has dyscalculia and fundamental numerical relationships simply have not become secure despite years of schooling?

Eventually, “explain it again” stops being a teaching strategy.

Something has to change. Perhaps the representation needs to change. Perhaps the conceptual step is too large. Perhaps the language is getting in the way. Perhaps considerably more practice is required. Perhaps the child has memorised procedures without understanding them. Perhaps what appears to be today’s difficulty is actually a prerequisite that should have been established years earlier.

Sometimes almost everything has to change.

This is where an experienced dyscalculia tutor earns their expertise: not when everything works, but when it doesn’t.

What should a good dyscalculia tutor actually do?

Parents looking for dyscalculia tutoring are sometimes understandably focused on techniques, resources or qualifications. Those things can matter, but I think there is an even more fundamental question to ask:

Can this person work out how my child thinks mathematically?

A good tutor needs to identify what is secure, what is insecure and what merely appears to be secure because the pupil has memorised a procedure. They need to understand how mathematical ideas connect and therefore know which earlier concepts might be responsible for a later difficulty.

They also need a range of ways of representing mathematics. Some pupils need to see and physically manipulate mathematical relationships before symbols become meaningful. Others need precise language, diagrams, patterns or repeated carefully sequenced examples. The representation should serve the mathematics and the particular learner, rather than becoming a gimmick in itself.

Most importantly, the tutor needs to be willing to abandon their planned route when the evidence in front of them says that the problem lies somewhere else.

That flexibility is one of the major differences between simply delivering a maths lesson and genuinely diagnosing mathematical learning.

Maths anxiety and confidence matter, but confidence cannot replace understanding

Children who repeatedly struggle with maths often lose confidence, and there is nothing mysterious about that. Imagine spending years sitting in lessons where other children seem to understand things that remain opaque to you. You put your hand up less. You become reluctant to take risks. Eventually you may begin saying, “I’m just bad at maths.”

Maths anxiety can then make the original difficulty worse. Anxiety can interfere with concentration and make already difficult mathematical thinking harder.

But this creates an important distinction.

Sometimes a child is struggling because they are anxious.

Sometimes they are anxious because they have been struggling.

Often both processes are happening together.

For a learner with dyscalculia or deep gaps in mathematical understanding, simply trying to increase confidence cannot substitute for teaching the missing mathematics.

In my experience, one of the most powerful ways to improve mathematical confidence is to find something the pupil genuinely does not understand and teach it successfully.

Find the missing prerequisite. Break the idea down. Change the representation. Give the learner enough time. Let them experience genuine success and then extend the idea slightly.

Eventually the pupil develops something much more useful than simply being told that they are good at maths.

They develop evidence.

They can do something today that they could not do a month ago.

Confidence built through genuine competence is extraordinarily powerful.

Why specialist maths tutoring is difficult to see from the outside

Experienced mathematics teachers eventually develop something difficult to describe: judgement.

When should I intervene and when should I keep quiet? When should I explain and when should I ask another question? Is this mistake important or incidental? Does this pupil need twenty more examples or none at all? Should I increase the difficulty, change the representation or go backwards?

There is no simple algorithm for these decisions.

And because experienced teachers and tutors make them increasingly fluently, much of the intellectual work disappears from view.

That is one of the strange things about expertise: the better somebody becomes at something, the easier it can look from the outside.

Society rightly celebrates mathematicians, scientists, engineers and people capable of operating at extraordinary levels of mathematical sophistication. But there is another kind of mathematical expertise that receives far less attention: the ability to take mathematics that has become incomprehensible to another human being and make it intelligible again.

For children with dyscalculia and serious maths learning difficulties, that expertise can be transformative.

The hardest part was never knowing the answer.

The hardest part is sitting opposite somebody who does not understand and working out, carefully and intelligently, what to do next.

That is excellent mathematics teaching.

And doing it exceptionally well is much harder than it looks.

Looking for specialist dyscalculia tutoring?

I provide long-term online maths tutoring for children and young people with dyscalculia, suspected dyscalculia, weak number sense and persistent gaps in mathematical understanding.

My approach is not simply to help a pupil survive the next worksheet or examination. I work to identify where their mathematical understanding has broken down and rebuild it from secure foundations, using carefully chosen representations, precise language and a deliberately sequenced approach.

If ordinary maths tutoring has repeatedly failed to get to the bottom of your child’s difficulties, you can read more about my online dyscalculia tutoring and how I work.