# Addressing Math Misconceptions: The 4-Stage Process To Identify And Correct Misconceptions, Tested In Over 700,000 Lessons

At Third Space Learning, we teach around 7,000 online math lessons every week as part of our 1-to-1 math intervention programs. That means 7,000 opportunities for students to make mistakes, and for tutors to correct them.

Mistakes come in all sorts of different forms. Some can be as simple as not reading the question properly – something we’re all familiar with! Others may arise from not understanding what is being asked, understanding the question but making a calculation error, or forgetting the method to find the answer.

But sometimes a student will make a mistake that suggests a deeper lack of understanding of mathematical concepts, which can’t be solved simply by providing the correct answer in its place – a misconception.

### What are misconceptions in mathematics?

Consider the following math problem: two students are asked to word out ‘0.3 x 30’.

Student A answers as follows, showing their working:

Student B answers like this:

Both answers are wrong, but Student A has made a mistake where Student B has made a misconception.

Student A has made a simple and relatively common multiplication error: 3 x 3 = 6 instead of 9. Their fundamental understanding of multiplying decimals by powers of 10 appears to be sound.

Student B seems to have applied the ‘trick’ method of multiplying by 10 to the equation, and simply added a zero to the end of the correct initial calculation. This betrays a lack of number sense and understanding of the concept of multiplying decimal numbers by powers of 10 – a mathematical misconception.

Read more: Math Tricks or Bad Habits?

### Why is it important to address misconceptions?

Calculation errors are commonplace and usually the result of ‘slips of the mind’ that are easily rectified.

Misconceptions are different. If we only address them by quickly correcting the errors, they can recur and build over time, leaving young mathematicians with significant gaps in their math understanding that can carry on into middle school, high school, and further.

As an example, in Third Space Learning’s online 1-to-1 5th grade program, one of the most commonly reported misconceptions is thinking that any fraction with a larger denominator is a larger fraction.

In the example below – taken from one of the lessons in our online 1-to-1 curriculum – a student might initially believe that 7/11 are larger than 5/6 in this case.

This misconception actually comes from a 2nd grade topic; it is a gap in their knowledge that they have carried almost to the end of elementary school.

When misconceptions like these arise, our tutors need to be able to address not only their result, but also the misconception itself; this is the only way to ensure that any gaps in students’ understanding of math concepts is filled.

#### Why Third Space Learning’s math interventions are a great way to remediate student misconceptions

We include a module in our tutor training program specifically on addressing misconceptions and equipping tutors with the tools they need to help their students overcome conceptual misconceptions.

Our programs cover every aspect of effective tutoring – from delivery, to effectively imparting subject knowledge and problem solving skills, to student encouragement techniques and promoting student voice – ensuring they are able to provide the best teaching possible for our intervention students.

Moreover, our tutors are ideally suited to address misconceptions; they work one-on-one with students in sessions lasting up to one hour, where the student is their sole focus.

Tutors and students can work through problems in a way that is difficult for teachers in classrooms of around 30 students, ensuring that each young learner’s specific misconception is addressed.

### How our tutors tackle misconceptions

With thousands of lessons under our belt and plenty of experience identifying and addressing misconceptions, we’ve identified a distinct process for identifying and tackling misconceptions, comprising four stages: preparation, investigation, addressing, and assessing – and this is what we train our tutors to do.

N.B. While we refer to what our tutors do in each of these stages, most of the content here can also be used by class teachers/TAs, etc.

#### Stage 1: Make sure you’re ready to identify misconceptions when they arise.

Before trying to fix a misconception, you need to be able to spot one. You can make this easier on yourself by:

• Reading up on common misconceptions for the topic you’ll cover before teaching.
• Establishing an environment that encourages students to articulate their thinking and feel comfortable with making mistakes.

As part of our online 1-to-1 interventions we have students sit for an initial Diagnostic Assessment consisting of a series of multiple choice questions on various math topics.

Each question includes one correct answer and three distractors, carefully designed to highlight specific misconceptions a student may have about each topic. This means tutors have an awareness of every student’s potential misconceptions before they even start teaching, and are prepared to address them if they appear during the lesson.

We ensure our tutors can correctly address misconceptions by first ensuring that they’re grade level and subject experts. Not only do we hire from applicants with a STEM background, we provide them with further curriculum-aligned training to develop a strong subject knowledge base from which to base their tutoring.

We also provide tutors with in-house CPD on spotting and correcting common misconceptions, supplemented by concept-related videos in their online training portal, to refresh their memories ahead of lessons. They also spend time analyzing and peer-reviewing lessons to help them pinpoint those moments where things start to go wrong in a student’s understanding.

#### Stage 2: Investigation

This step is about identifying when a misconception has been made. In order to establish this, our tutors:

• Identify the initial mistake that was made (e.g. the wrong answer to a word problem).
• Facilitate a discussion about the mistake, focusing on having the student explain their thinking e.g. by asking questions such as “How did you come up with that answer?” and “Why do you think it’s correct?” This clears up whether the error was a simple case of ‘slip of the mind’, or a misconception.
• N.B. Our tutors also encourage students to explain their reasoning even when they get an answer right; sometimes misconceptions can occur without affecting the specific problem being solved.
• Use further questioning to find the specific knowledge gaps the student has; this ensures they are addressing the correct misconception(s).

1-to-1 tutoring means this stage – often one that can take time in a full 30-student classroom – occurs quickly and easily, ensuring misconceptions can be properly addressed in a more timely manner.

#### Stage 3: Addressing the misconception

It’s here that tutors actually set about ‘filling’ the knowledge gap. The exact method for this will depend on the topic; for example a misconception about rounding may be best dealt with by using a number line to visually present the concept. Some of the more common misconceptions in elementary school math are explored below.

Where a misconception is relatively small, we encourage our tutors to deal with it immediately. However, it’s important to adapt to each situation based on the resources available.

If a child expresses a misconception during the warm-up activity for example, and the related topic will be covered in the main body of the lesson, the tutor may model the correct answer and highlight that a misconception has been made.

They will then address the misconception more thoroughly when it comes up later in the lesson. This way the student is aware of their misconception and the fact that it will be addressed, but valuable time is not spent on something that may well be dealt with in the natural progression of the lesson.

#### Stage 4: Assessing Learning

In the same manner as any new topic being learned, tutors will gauge a student’s understanding of the material (and therefore whether they have addressed the misconception sufficiently) through assessment and progress monitoring

In the case of misconceptions, assessment is simple enough: ask the student a similar style of question to the original, or else ask them to re-explain the concept you’ve just been discussing back to you.

If the misconception remains or if the student expresses confusion, it is important to once again investigate their knowledge gaps related to this topic; we want to establish roughly how much time we will need to dedicate to teaching this.

### Three Common Misconceptions and How To Address Them

#### Misconception 1: Using addition to solve subtraction

We spend a lot of time teaching addition and then spend much less time teaching subtraction. With this in mind, it’s no wonder that students often revert to using addition when faced with a subtraction question.

One of the best ways to overcome this misconception is by teaching the part, part, whole model; where the underlying model of both addition and subtraction are clear.

This is much better than rushing through using examples with bigger and bigger numbers, as it means children really understand how subtraction and addition fit together.

It is much better if a child really understands that 5-2=3 and how this relates to 3+2=5 and 5-3=2 and 2+3=5 than if a child can do equations with numbers to 20.

#### Misconception 2: Understanding money through 1-to-1 correspondence

One-to-one correspondence is one of the key math concepts we teach at the early elementary level, and is central to many of the topics we teach in elementary school.

But when we introduce money we run the risk of confusing children – the idea that one coin could represent 10 of another can be very difficult for students to grasp.

One practical solution is getting children to see how heavy, awkward and fiddly carrying lots of 1 cent coins around is. We can then show them how larger denominations of coins are a helpful solution to this problem – one 10 cent coin being much easier to carry around than 10 1 cent coins, for example.

This can even be represented visually; during our online interventions tutors may use images of coins in much the same way we might use real coins – after all, 25 1 cent coins take up a lot more space on a screen than a single 25 cent coin!

#### Misconception 3: Confusing square numbers with multiples of 2

The superscript 2, or exponent, in square numbers is a recipe for confusion; students can very easily fall into the trap of assuming this means multiplying by two. Address this head on and use cubes to show 3 x 2 as well as 3², and discuss how these are different.

In our one-to-one online interventions our tutors make use of pictures of cubes for much the same purpose, as well as using multiplication grids etc.

It’s important not to rush through this: repeat the discussion for lots of different examples to ensure students are secure in their understanding.  This technique of showing non-examples as well as examples is another really good way of building deep understanding within a new schema.

Misconceptions can grow to become serious hindrances to students’ math progress if they are left unaddressed, but it is often extremely difficult (if not outright impossible) for class teachers to cover every student’s misconceptions in class.

That’s why we’ve trained our tutors to be able to identify and rectify math misconceptions when they encounter them during one-to-one interventions. We help plug gaps in student knowledge to ensure they can make progress in class and beyond!

Do you have students who need extra support in math?
Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor.

Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way.

Personalized one-on-one math tutoring programs are available for: