Mathematics seems scary…

…but does it have to be?

Hello all. Welcome to the 111th edition of TEPS Weekly! 

Talk about learning mathematics in school to anyone and you will always get a mixed set of responses. Most of them will be groans from people who’ve had a tough time with it. And a lot of them also feared it. This is common across age groups, cultures and sometimes countries. Why is this the case? What is happening currently in the Indian classrooms that is causing so much fear around mathematics? 

• Mathematics is an abstract concept that students are not able to visualise intuitively in concrete terms. This can lead to confusion and fear because they can’t make the relation between the number of objects and the abstract symbols.

• A lot of the practice of mathematics is focused on memorising formulae and not on understanding the HOW and the WHY. For example, we learn that we have to add with ‘carry-over’ and subtract with ‘borrowing’. But WHY do we do that? When concepts are unclear, students will not have the confidence to face different types of problems leading to fear and anxiety.  

• The focus often is on the final answer without enough emphasis on the process, which means either you get the answer correct or you get it wrong. If you get the answer correct, you are good at it; if you get it wrong, you are bad at it. This creates a fear of failure in students.

• Building procedural fluency (applying the method to solve a problem quickly and accurately) is key in mathematics. However, focussing on fluency or speed when students are learning may not work, and may be anxiety inducing for students. 

• We have the 3rd and 4th issues due to the way that performance is evaluated in summative assessments. Though summative assessments need to assess whether a student is doing things fluently and accurately, this cannot be the focus when students are learning. 

So what can we, as educators, do to remove (or at least reduce) the fear of mathematics in students?

1. Use authentic problems to make mathematics more meaningful and relevant for students.

Authentic problems:

• have a real purpose: You’re not solving a problem just for the sake of it, but there is a genuine problem to solve.

• are relevant to learners: The context of the problem should be student-friendly and familiar to them. 

• foster discussions: There should be discussions around the strategies the students used as each one could arrive at the answer in different ways. 

Ms. Priya introduces the concept of volume to her 5th grade students by giving the following authentic problem:

The school needs a sandbox for the preschool kids on their campus, and the school needs to decide how much sand will be needed to fill up the sandbox. 

Here students will work on an authentic problem because:

a. The need of the school to build a sandpit is real, and the students will understand the relevance of mathematical topics and concepts in life. This gives the problem a real purpose and relevance for the students.

b. The students will need to measure the length, breadth and height of the box and then find out the volume needed – this will lead to discussions among the students.

2. Promote conceptual understanding over memorising.

This is about understanding the WHY behind mathematical concepts. When students understand the reasoning behind a concept, they are less likely to feel lost and overwhelmed when faced with new and challenging problems.

Mr. Iqbal is teaching grade 6 to find the area of a triangle. He writes the formula on the board: Area = ½ × base × height. Area of a triangle is taught after students know that the area of any quadrilateral is base x height.

Instead of directly using the formula and asking students to memorise it, he draws a quadrilateral with a diagonal to show that a quadrilateral is made up of two triangles. Then he shares the lengths of the sides of the quadrilateral and asks students to calculate the area of the quadrilateral. If the sides of the quadrilateral are 4 cm and 6 cm, then the students will find the area to be 24 cm2. 

Then Mr. Iqbal points at the diagonal and asks, “If I look at the figure ABD, then what will it be?” Students understand that the figure is a triangle and it is half of the area of the quadrilateral. So to find the area of any triangle, they have to halve the product of the base and height.

When students understand why the formula works, they feel more confident using it — even if the triangle looks different. Instead of panicking, they trust that the rule still applies, because they know where it comes from.

3. Create a safe space for students to learn from their mistakes

Mistakes are a huge part of learning because they help our brain form new and stronger connections. The brain is exposed to multiple new things in one day. It forgets most of these. However, when one of the things that it has forgotten is brought back for the brain to notice, there is a higher chance of the brain remembering it and remembering it for longer. Mistakes in procedural mathematical operations are rules that students have forgotten. Allowing students to practise, make the mistake and get the mistake pointed out helps with creating deeper learning for students. 

Students need to build accuracy and fluency in mathematics. That is what learning in mathematics is, and student performance is tested on these bases. However, while building their expertise, students need space to make mistakes and correct them. So students should not be reprimanded for their mistakes, but instead given a chance to find their errors and learn from them.Ms. Asha gives her second grade students a few minutes to solve an addition problem  (28 + 17 = ?) independently in their notebooks. She walks around and notices that some of them have written 35 instead of 45. Ms. Asha encourages one of  the students to explain their thinking out loud.

Ms Asha: “That’s interesting! Can you show us how you solved it?”

Rahul: 2+1=3 and 8+7=15 but we can’t write 15 here so it’s just 5.

Ms. Asha: Okay, That’s correct, we can’t write 15 there! But, why is that?

Rahul: (looks a little confused) Because it’s too big?

Ms Asha: So can we also not just write 1 and leave out the 5 instead?

Rahul looks confused again.

Ms. Asha: When we get 15 as an answer, which digit should go in the ones place?

Rahul: The 5 goes in the ones place.

Ms. Asha: Right. Now, what about the 1? Where should the 1 go?

Rahul: I don’t know.

Ms. Asha:  Remember, the 1 is in the tens place. What should we do with the tens?

Rahul: We should add the 1 to the 3 in the tens place.

Ms. Asha:  Exactly! Now, what is 3 plus 1?

Rahul: It's 4!

Ms. Asha: So can you tell me what the answer will be now?

Rahul: 45!

Ms. Asha helps form a mistake-friendly culture in the classroom in this way, and so her students do not have to hold themselves back from engaging with mathematics problems due to the fear of making mistakes.

4. Use various assessment methods that deprioritise speed, and not just standardised, timed written tests.

As we mentioned earlier, summative assessments will check for student fluency and accuracy in mathematics. However, it is important not to do that while the students are still learning. For this, formative assessments should focus on levels of fluency or accuracy. The assessment should also focus on whether the student has understood the concept behind an equation or a formula. Students should be encouraged not to only solve practice problems, but also engage in verbally explaining concepts and procedures. This can be done through:

• Projects (like the sandbox one)

• Journals with student reflection on their learning 

• Classroom discussions. 

And while the students are trying to figure out the sandbox problem by discussing among themselves, the teacher should go around listening to the students and asking questions that will help her understand the thinking and understanding of the students. Here is an example of what the teacher hears:

Raj: I think we just measure the length and width.

Kala: But the sand goes all the way down. Shouldn’t we measure the height too?

Mohan: Yeah, maybe we need length, width, and height?

Teacher: That’s a good observation. Why do you think all three are important?

Kala: Because the sand fills the whole box, not just the bottom.

Teacher: Exactly! 

From this interaction, the teacher understands that taking only length and width into consideration shows a low conceptual understanding of space and measurement. But the suggestion to measure height shows that there is an understanding that volume requires a third dimension.

5. Instil a growth mindset in students.

   
   

   All the suggestions mentioned so far work the best if there is a culture of growth mindset among the students. So it becomes imperative that the teachers instil this belief in each student. What is a growth mindset? In short, it is the mindset that allows one to say, “I don’t know, but I can figure this out.” This means that the students are not limited by what they know. They understand that there is a lot more to know, and only by trying and (sometimes) failing will they succeed.

As a teacher, it is important to keep reinforcing the idea that a mathematics problem that seems difficult to solve will get easier with practice. Say things like:

• “It’s okay if you don’t know it yet. That’s why we are learning it together.”

• “Remember how tricky the process of addition felt when you first started? And now it’s easy for you to do. Similarly, this new skill will also get easier with practice.”

• “What strategy did you try first? What else could you try? Let's explore it together.”

6. Provide opportunities for students to discuss their feelings about math.

Journaling can be a powerful tool for students to reflect on their learning and express their feelings around it. While journals can be used in different ways, they help manage fear and anxiety around mathematics in two ways:

1. Being able to organise one’s thoughts on their learning and identify what is easy for them and what is difficult for them can itself give a sense of clarity and confidence to them. 

2. Putting down their feelings on paper may free up the working memory of the space that the anxiety was taking, which can now be used in solving the problem. 

Prompts can look like: 

• Today I felt ___ in mathematics class because…

•  One thing that confused me was…

Mathematics is one of the secondary skills that we learn as human beings. Our brains did not evolve to learn mathematics. So it is understandable that students find it difficult or are afraid of it. That is why it becomes crucial for the teachers to create an atmosphere of practice and learning instead of making it a relentless pursuit toward accuracy and fluency. Yes, those are the goals, but the environment in which a student pursues those goals makes mathematics enjoyable or scary.

---

If you found this newsletter useful, please share it.

If you received this newsletter from someone and you would like to subscribe to us, please click here.

Edition: 4.18