Volunteer Tutoring

A practical example from Secondary Mathematics

by Martin Richards, Maths teacher, coach and author

Volunteer tutoring

I am a Maths Teacher and Certified Coach. I live in Gothenburg, Sweden. 

Although I retired from teaching some years ago, I still feel the need to talk with students about their learning, especially in Mathematics, so I am grateful that, once a week, I can go to the local Library where a mix of volunteers: University students, parents and pensioners, are ready to help young teenagers with their Mathematics homework. 

I worked for many years as a Math teacher. I blended student-encouragement techniques from the world of coaching, with problem-solving techniques from the world of Mathematics. I found that combining the two sets of techniques was a delightful balancing act which has resulted in powerful learning for every student that I have worked with. 

This Story

The following story of working with one of the teenagers illustrates how an educator chooses between two skill sets, sometimes supporting the student, sometimes explaining the subject. 

Coaching supporting the student

Teaching explaining the subject

The act of choosing is a dynamic one, reflected in the changing relationship between the teacher, the student and the subject. 

Progress is measured in how well the student performs in solving problems on their own. 

My example here is from Mathematics, and I am certain that choosing and balancing apply to other subjects too.

A brief background of the student

She was about 13. Let’s call her Lisa. 

She was here with her mother, who had put Lisa’s name on the list of students waiting for help. 

Her mother came up to me and showed me the list asking me, by flapping the sheet of paper with gentle insistence, to help her daughter as soon as possible. 

By this time I think they had been working for half an hour, maybe forty minutes, with the Maths problems in the homework. 

I recognised Lisa and her mother. They were often here, and they were always struggling with Mathematics. I like working with Lisa because she is teachable. She listens, she tries. She doesn’t let mistakes prevent her from trying again and again; and in the end, she gets it. Always.

Lisa’s mother did not speak very good Swedish, at least not with me. She made herself understood by energetically smiling, waving her hands and pointing. I think she did the same when helping her daughter with her homework. Helping her daughter was done with love that shone through every nod and smile, and every gentle nudge of her hand on her daughter’s arm, urging her with these simple gestures to have another go, read it again, think again and never give up. I believed I could see in her actions a mother’s hope that her daughter will have a better chance of a good life in this country.

It’s strange isn’t it, but a good life often boils down to being good at Maths, and English, (and in this country Swedish as well). I felt like the richest man in the World. I’d always been good at Mathematics, my native language is English and I had learned, after a couple of years of struggle, to speak Swedish.

Diagnosing the challenges: for the client and the mathematical problem

When it came to Lisa’s turn, her mother pulled up a chair beside her daughter and invited me to sit down as though this was her living room, and invited me to help her Lisa with the Maths homework that was displayed on a computer screen. 

It’s not just Algebra

A quick glance told me everything I needed to know. 

It was algebra. Equations. For some people, algebra was like a foreign language and equations feel like earthquakes in their bellies. 

Equations spark fear in all those who have not yet mastered them because they know equations are the grammar of Mathematics and that without equations they will never master it, and that increases the risk of them not having a good life.

Paper explanations sometimes are no help at all

On the screen the teacher’s reasoning - the way the equations were to be solved - was colour-coded. 

The equation was written in black ink, and the thoughts of the invisible mathematics teacher were written in red ink. To the initiated, this coding was blindingly obvious, and I took in the meaning of the entire page in a single blink. 

Lisa was not one of the initiated, and for her, the page was covered in illegible scribbles, coded beyond her understanding.

Speaking in the language of my client and using metaphors to create a conducive and positive learning space together

Tears can be a good sign

Lisa looked as though she had been crying, with the wet tears only recently wiped from her face. 

The expression on Lisa’s face suggested that she understood next to nothing, so I was interested in what she did know, what she did understand. So I asked, “What’s this all about?”

“I want to understand this,“ she said pointing at the computer screen.

I could’ve hugged her. She was not the kind of student who moaned, “I don’t understand, I can’t do this, Why do we have to do this?” or make any other kind of complaint that would’ve blocked her learning. 

No, she wanted to understand, and she was prepared to struggle and struggle and get it wrong until she understood it and got it right. 

She was lovely to work with. This is why I came to the library every week, I was aglow knowing that within a few minutes, Lisa would understand, and I would have played my part in her understanding.

A little Mathematical jabber jabber

I knew that Lisa would need to use additive inverses and multiplicative inverses to generate additive and multiplicative identities. 

But such high-level mathematical jabber would not serve her. I needed to find a language that she would understand, she was after all 13 years old and admittedly not very good at mathematics, yet.

A metaphor is needed

So we began swimming in the paddling pool. I asked, “Can you swim?“

“Yes,” she said.

“How far?” I asked.

She didn’t answer but indicated that she could swim quite a long way. I think she was having trouble finding a number that would represent how far she could swim. Or maybe she was wondering why we were talking about swimming.

“I can swim,” I encouraged, “only a little bit, up to my knees,” I admitted, slapping my knees with my hands. “But that’s paddling.” We laughed at the funny word and my attempts at acting out paddling.

“And then you get into deeper water, up to your chest,” I said, showing the level of water with my right hand against my chest.

“Can you swim in deep water like that?” I asked.

She nodded, glanced at her mother who was also probably wondering why I was talking about swimming and allowed that yes she can swim in deep water.

I pointed at the mathematics on the computer screen, “Deep water,” I said solemnly, “Very deep water.”

Ah, she understood that we were going to learn to swim in deep water, but we were going to start by paddling. My metaphor was accepted. We began paddling.

Basic steps

“What’s the opposite of four?” I asked, “No, I mean what’s the opposite of ‘add four’? I emphasised the word add, showing the four fingers of my right hand being added to the palm of my left hand. 

“What’s the opposite of ‘add four’?”, I removed the four fingers from my left palm.

“Take away,” she said, correctly identifying the action.

“And the opposite of ‘add four’?” I insisted on the four.

“Take away four,” she replied.

I beamed at her answer and rattled off a series of ‘opposites questions’:

“What’s the opposite of ‘add three’? What’s the opposite of ‘add five’? What is the opposite of ‘add -2’?”

The last question foxed her for a moment. I mimed -2 by taking two right-hand fingers off the palm of my left hand. 

“What’s the opposite?” I asked miming the two fingers coming back onto the palm of my left hand.

“Two?” She tested.

“Add two”, I corrected.

“The opposite of ‘take away two’ is ‘add two’, I complemented the description of my hands using simple mathematical terms. 

“The opposite of ‘add two’ is ’take away two’. They are opposite.” “The opposite of -2 is +2. The opposite of +2 is -2.” I repeated using the correct mathematical terms, and added, pointing at the computer screen, “Deep water.”

I pointed out where the instructions showed +4 in black being cancelled to zero with the -4 in red.

“What do you get,” I asked, ”if you have four,” l showed the number four with my fingers, “and take away four?”, I mimed the four fingers being taken away.

“Nothing”, she said.

“Yes, nothing, or zero.” I drew zero in the air and then pointed at it on the computer screen. “Plus four take away four is zero.” 

I looked back at Lisa to check if she was still with me in the deeper water. Her eyes were focused on the screen. She was in deep water. And swimming.

Diving into the pool of equations

There were six examples of equations being solved, coded on the screen in red and black. For each equation, there was a red number that cancelled a black number. 

Some of the black numbers were positive, some of the black numbers were negative. The positive black numbers were cancelled by negative red numbers, the negative black numbers were cancelled by positive red numbers. 

Slowly, the coding was revealing itself as pairs of opposites carefully chosen to create a zero. I said nothing whilst I pointed out these opposite pairs and allowed Lisa to come to her own conclusion about how the pairs were working together to create zeros.

Thus Lisa was equipped with additive inverses, i.e. the opposite number, and the additive identity, zero. 

There’s more to solving equations

However, solving equations is more complicated than that. They require the careful use of additive inverses AND the careful use of multiplicative inverses, in that order. Additive first, then multiplicative. 

We were going to swim in deeper water.

This next step would require Lisa to understand multiplicative inverses. So we started swimming. 

I pointed to where a 4X was being cancelled to X by a red division sign and a red four. 

“What do you get if you divide 4 by 4?” I asked, tenting the four fingers of my right hand with the four fingers of my left hand.

“Eight” Lisa replied.

Failure is the mother of invention

I hemmed my disapproval and disappointment at my failure and tried again. 

“What do you get if you have four cakes, shared by four friends?” I said naming my right hand ‘cakes’, and my left hand ‘friends’, and matching the cake fingers with the friend fingers as before.

The change of metaphor was jarring to Lisa as it was well out of the water we had been paddling in, but I needed to get back to her primary school mathematics where division was often described as sharing between friends. 

So I asked her mother, “What do you get if you share four cakes with four friends?” It was a crude manoeuvre since I knew she didn’t speak Swedish, but it had the desired effect.

Lisa rescued her mother, “One each,“ she correctly replied.

“One,“ I emphasise the word wiping my index finger vertically down through the air, almost as I had done with circling the zero earlier.

I pointed back at the computer screen where the 4X in black was cancelled to X, by using a red division sign and a red 4. “So what do you get when you divide 4 by 4?”

“One,” Lisa triumphed.

I pointed out all the examples where multiples of X were being cancelled to the multiplicative identity by using its multiplicative inverse. I said nothing until I had pointed out all of the examples. Lisa followed, her widening eyes and deepening smile revealed she was successfully decoding. Then I read them out as questions: 

“What do you get if you divide 8 by 8?, What do you get if you divide 5 by 5? What do you get if you divide a quarter by a quarter?” 

In every case Lisa triumphed, grinning, “One”.

Basic swimming lesson complete

Thus Lisa was equipped with a multiplicative identity, one. And she had identified that the action, the multiplicative inverse was the same as dividing or sharing.

Now we were ready to swim around in the deep water and solve some equations.

I chose the first example on the screen, it was clear enough. There was only one additive inverse to consider, and only one multiplicative inverse to consider. 

We calculated the first, and then the second. Within minutes Lisa had arrived at the correct answer. She beamed as the doubts in her heart and fear in her belly transformed into excitement and anticipation at being able to solve the next equation. 

It’s funny isn’t it how close fear and excitement are. 

They are but one step apart. 

One moment you’re standing in fear,

take a step and the next moment, 

you’re standing in excitement.

The second example had two choices for additive inverse, so I let Lisa choose: 

“Which one would you like to take first, this one or that one?“

Lisa pondered for a moment. It seemed like doubts were beginning to cloud her mind.

I encouraged her, “It doesn’t matter which one you choose first. We are going to do both anyway.”

She picked one, I asked for the additive inverse, “What do we need to add or take away to get zero?“ 

Lisa replied correctly. She picked the next one. I repeated the question, “What do we need…” 

Lisa interrupted me with the correct answer. She was swimming in this deep water like a buttered dolphin.

Finding the Mathematical Balance

There was one more thing that I needed to include in the equation solving strategy. The equation had two sides, a left side and a right side. The left side of the equation was separated from the right side of the equation by an equals sign. 

We had identified what to add or subtract in order to get zero. We had identified what to divide by in order to get one. Once we had identified the right number to add, subtract or divide by, we needed to do the same on each side of the equation, in order to maintain the balance of the equality.

I mimed the left-hand side of the equation in my left hand being balanced with the right-hand side of the equation in my right hand as though I was weighing marbles. 

“What we do here,” I said, showing my left hand, “... we must do there, to keep the balance,” I showed balance being restored with my right hand being brought level with my left hand.

Lisa simply accepted my description. Perhaps she remembered it from a previous maths lesson? 

Either way, she was happy to do on the right as she did on the left, or to do on the left as she did on the right. The arithmetic was simple, and she was getting things right time and time again. The water was getting deeper. She was swimming.

Now we are swimming

We tackled the third example which had a combination of inverses to be applied. I began by writing out the question on a fresh sheet of paper and asked Lisa:

“What number should we look at first?” allowing her to decide which of the two additive inverses to work with first. 

She made her choice; and calculated correctly. I hesitated, waiting for her to say that I should apply the same arithmetic to both sides of the equation, but she didn’t. So, I wrote the first line of the equation again, with the additive inverse only on one side, leaving a gaping hole on the other side where the additive inverse should also be applied. Lisa tapped the gap with her finger and told me what to write there.

A little coaching from the side of the pool

We continued with me asking: 

“What should we look at? What should we write here? Where else should we write it, and what does that make?”

As we progressed through the equations, I asked Lisa fewer and fewer questions. 

At the sixth equation, I passed the pencil to her and let her write her own answers. 

She began writing out the equation, and after a few lines had solved it correctly. I leaned back a little. She wrote out the seventh question. I looked over at her mum whose wide eyes revealed how totally amazed she was to see her dolphin daughter swimming in deep water.

“Right,“ I said, “I’ll leave you to it.”

Outcomes from mathematics coaching

Lisa’s mixture of relief, gratitude, and delight at being able to solve equations lit up all three of us. 

It was fire born of an odd transaction. All three of us had gained so much simply by giving. Lisa‘s mother had given her trust in an elderly stranger. Lisa had given her trust too but also used her determination and willingness to learn. I had had a lot of fun coming up with strange metaphors and mimes to describe complicated mathematics in simple terms.

Her mother offered her thanks. She was so grateful that it was embarrassing. After all, I had gained at least as much from this conversation as she and Lisa had. 

I just said it was “No problem.” It felt like a poor way to end the conversation, but I could not think of anything else at the time.

I knew that Lisa would remember this lesson, and be better able to solve equations because of it. 

And Lisa’s real name is something else. I actually don’t know what it is. I’m not her teacher, I’m not going to give her a grade, nor write a report. And besides, her mother was there all the time so they didn’t need a written report. There are many Lisa’s at the Library every week, and too few volunteers. 

My learning and insights

I am grateful for the opportunity to tutor students. 

I will come back to the Library next week and every week that I am able. 

I will continue to share with students who want to learn, my love of mathematics, and my ability to mime and explain strategies. 

• I believed early on in the interaction with Lisa, that she would need more encouragement than explanation. I imagined that her frustration had built up over the 30 minutes she had been struggling on her own, and it had possibly started during the lesson which had introduced the techniques for solving equations some days ago.

• Essentially, all I wanted to say was, “You will learn to do this gradually, just like you learned to ride a bike or swim.” I think that’s where the swimming metaphor came from.

• I also took it for granted that the paper explanation, with its red and black coding of the teacher’s thinking on how to solve the equation, was, at the moment, of very little use to Lisa. That is why I started with some basic questions which I believed she could answer correctly.
• 
  
• I used my hands and fingers to demonstrate the numbers so that I could avoid, at least to begin with, writing anything on paper which resembled the equations in the explanations, because of the risk of scaring Lisa.
• From a mathematical technical point of view I knew that the so-called inverses and identities for the operations of addition and multiplication had to be very clear to Lisa prior to tackling the solution of the equation. 

• When my hand-waving examples distracted Lisa into giving a wrong answer, I knew I needed to change to a different metaphor. I used the fact that Lisa’s mother was sitting right next to her. It was a crude manoeuvre, but it worked. In a classroom setting, I would have used one of Lisa’s friends or classmates in a similar way.

• It was when Lisa was able to correctly answer a whole series of questions about multiplicative inverses, that I realised we had caught up with the previous lesson and that we now needed to move onto balancing the equation.

• Regarding the balance of coaching skills and mathematical skills, I alternated between sometimes encouraging Lisa to come up with her own answers, and sometimes giving her information that she needed or telling her what strategy we would be using.

• We were, for a while, standing with one foot in the scary land of Failure, we took a few basic steps together and landed in the land of Success when Lisa was able to not only reply correctly but also predict what was needed to solve the equations.