Identifying and Addressing Problems in Student Understanding

Last week, I received the following message asking for advice. Anyone in education has their own thoughts about how students learn, what can make students learn more effectively and why some students don’t make the expected progress or come to Upper Secondary/High School with the basic mathematical skills or motivation to learn.

Obviously, this discussion could easily be a PhD thesis, there are so many different aspects that could be explored and without any empirical evidence all of the suggestions I have offered are based on my observations, but I have supported with academic research where appropriate.

At our homeschool we find the students that have come to us all have very poor maths skills. Even the ones who joined us as Grade 10 students have huge gaps in their maths.

We have found :
A. Most don’t know their timetables by heart
B. Most can’t do word problems
C. A lot can’t do basic things eg Identify prime numbers in a series of given numbers
D. Most struggle with fractions
E. Most can’t do percentages
F. Most can’t work out:
If 500g of meat cost $100 how much will 200g cost.

On your experience as a teacher (when u were teaching) do you think your students faced the same problems?

The reason I ask is because, we are thinking we need to do something to address the fact that a vast majority of students hate math. We want to turn them around. However, we need to identify the base problems so we can come up with a solution.

What are your thoughts on this?

On your experience as a teacher (when u were teaching) do you think your students faced the same problems?

First of all, I think many of the issues you have stated are fairly common for students throughout both Primary and Secondary, particularly word problems and contextual problems to apply mathematics in real situations. There could be a number of reasons for this, but generally non-native English speakers find it difficult to put the word problem into a mathematical problem that they can then solve arithmetically or algebraically. Also, I believe that students quite often have never really been guided to ‘how to problem solve’ effectively. Below is Polya’s ‘Steps to Problem Solving’ that guides students with some key questions about the processes involved when approaching problems, particularly word problems.

As Secondary teachers it is easy to point the finger and blame their prior experiences at Primary or Lower Secondary. As teachers we have to understand that quite often students come to Grade 10-11 with preconceived ideas and beliefs about certain mathematical concepts. Sometimes, as pointed out in the question, even some of the most basic arithmetic principles are a struggle for some students. This is a concern, and one that authors like Jo Boaler (, are campaigning to address by encouraging students to understand concepts and understanding rather than memorisation of methods. By the time students come into Grade 10 some students have key misconceptions that are identified when it comes to the examination style questions or when students are required to apply their skills in context.

The reason I ask is because, we are thinking we need to do something to address the fact that a vast majority of students hate math.

My personal thought about this is that many students come to hate math because of the way in which they are introduced or taught the topics. A lot of the time, especially with weaker students, they will question the relevance of learning topics. As teachers we obviously want our students to enjoy the subject they are learning, but one thing I have noticed is that students there is a strong connection between students’ mindset, understanding and therefore their attitude. Students quite often hate math either a) because of the way they have learned maths in the past or b) because they cannot do the mathematics we are teaching them and do not have the motivation/belief that they can do the mathematics.

However, we need to identify the base problems so we can come up with a solution.

By identifying the topics that students struggle with, you have already started to come up with a solution in terms of what to teach. The difficulty that you now have is to ‘change the mindset’ (Carol Dweck) of the students, to give them the belief that they can do mathematics and to help them to see some of the key concepts in a way that they will be able to understand and apply them.

The difficult part is to find the balance between students enjoyment of mathematics through creative exploration, developing a positive mindset with the urgency of upcoming examinations and requirements to understand the content in the examination style questions. In my opinion, students sometimes need to go back to basics, to learn to enjoy mathematics before they can be pushed further and are willing to explore mathematics for themselves. Even with things as simple as times tables and number topics like fractions and percentages, students may need to be introduced to the topic in ways that they will see as relevant, or maybe they can visualise it in other ways.

These questions cannot be answered in a simple blog post like this, but as I said at the beginning, I can only offer my opinions based on observation and what I have read in academia. Without knowing individual students’ backgrounds or the context, it is easy to speculate, point the finger and make blanket comments about students whose individual cases are not known.


Assessing without levels

Belmont Teach

We recently held the first in a series of voluntary curriculum conferences for mid-leaders to share their ideas about what might influence the design of our new post-levels curriculum. Ideas that were shared during our first meeting:

  • Designing a new English curriculum and post-levels assessment system from scratch (which you can read all about here)
  • An Ethic of Excellence (which you can read all about here)
  • Using cognitive science to inform curriculum design (which you can read all about here)
  • Assessing without levels

assessing without levels 1 The chance to break free from using National Curriculum levels for assessment offers us real opportunity:

  • The opportunity to provide our students with formative feedback that means something
  • The opportunity to create an “Ethic of Excellence” – where excellence is expected and everyone can improve and aim for excellence
  • The opportunity to develop a curriculum that instils a growth mindset – no glass ceilings or self-labelling by students…

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Why Radians?

Teaching Calculus

Calculus is always done in radian measure. Degree (a right angle is 90 degrees) and gradian measure (a right angle is 100 grads) have their uses. Outside of the calculus they may be easier to use than radians. However, they are somewhat arbitrary. Why 90 or 100 for a right angles? Why not 10 or 217?

Radians make it possible to relate a linear measure and an angle measure. A unit circle is a circle whose radius is one unit. The one unit radius is the same as one unit along the circumference. Wrap a number line counterclockwise around a unit circle starting with zero at (1, 0). The length of the arc subtended by the central angle becomes the radian measure of the angle.

This keeps all the important numbers like the sine and cosine of the central angle, on the same scale. When you graph y = sin(

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Teaching: if you aren’t dead yet, you aren’t doing it well enough

Othmar's Trombone

So another World Teachers Day has come and gone. All the build-up, all the excitement, and it just seems to go by in a flash. One minute we’re all hanging our stockings up in the classroom ready to be filled with gifts from our generous pupils, the next minute we’re all sick of spending the week eating leftovers from the big World Teachers Day feasts laid on for us by our families and friends.

I love all of the traditions of World Teachers Day: chugging a yard of tea, the enormous full-sized teacher-shaped chocolate cake (bagsy the heart – it’s the biggest bit!), marking under the mistletoe, pinning the grade on the lesson observation (blindfolded, of course), being allowed to go the toilet, the Airing of Grievances, the singing of teacher carols (“Mark! The Herald Angels Sing”), the Returning of the Glue Sticks, and – the best bit –…

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