(Other) Wordy Problems
In which, we learn that maths is stupid but adults are stupider.
A typical issue that young learners face in school-level mathematics is being unable to decipher how to approach word problems.
Word problems. Problem sums. इबारती सवाल. Call it what you want.
Here’s what a typical word problem looks like –
Lata bought a plot of land for ₹50,000. She wants to keep 1/4th of the land to herself and divide the rest between her two children in such a way that both her daughter and her son get an equal share. Assuming the price of the plot remains same, what will be the price of the land that goes to the daughter’s share?
Okay, maybe this one was a little complicated. Here’s a simpler one –
Pintu had 4 gulab jamuns. His friend Sweetie took 1 gulab jamun from him. How many gulab jamuns does Pintu have left?
One could argue that Pintu still has 4 gulab jamuns considering the relationship dynamic between him and Sweetie… but that’s for arts students to figure out.
What’s the problem?
It’s fairly well known that most learners in primary and upper primary classes struggle with parsing language. Yeah they can solve your questions about information recall and grammar but give them an “unseen passage” and ooh does a shiver slip down the helicopter mother’s rotor in the night.
Your typical elementary school child (Class 1 to 8) will do just fine if you give them a problem like –
$$\text{4 + 5 = __}$$
But this would have the kid stumped –
$$\text{Joseph has 4 dogs. One of them gave birth to 5 puppies. How many total dogs now?}$$
You can probably see that this is quite literally the same question as before with just some context slapped on top of it. Why is this harder than the other one?
Glad you asked.
Why is this a problem?
There could be a bunch of reasons. I am only listing a few that I have encountered in my experience.
1. Arithmetic is taught without context.
Most kids encounter mathematics with absolutely no context. You are taught $\text{2 x 3 = 6}$ before you learn (if at all) what ’times’ even means. You memorise that the area of a circle is $\pi r^2$ without ever finding out what significance $\pi$ holds and how it was even arrived at. Or that a circle can be approximated as a polygon with many, many, many sides.
I will let Paul Lockhart supply the exasperation here –
But what about the real story? The one about mankind’s struggle with the problem of measuring curves; about Eudoxus and Archimedes and the method of exhaustion; about the transcendence of pi? Which is more interesting — measuring the rough dimensions of a circular piece of graph paper, using a formula that someone handed you without explanation (and made you memorize and practice over and over) or hearing the story of one of the most beautiful, fascinating problems, and one of the most brilliant and powerful ideas in human history? We’re killing people’s interest in circles for god’s sake!
— A Mathematician’s Lament, Paul Lockhart
2. The learner is never allowed to struggle.
We want answers now and we want them right. You get them wrong, we hit you. You take too long, we hit you.
Take a goddamn break, dude. The whole joy of mathematics and literally any discipline ever is of struggling with a problem and arriving at an answer that works. (What even is ‘right’?) The ancient Greeks, Chinese, Arabs and everybody else took centuries to arrive at shit that you are trying to unload into a 7 year old’s head.
Lessons learnt from a struggle stick around forever. Simply think of how you learned to walk or ride a bike.
3. The learner is never allowed to create.
Man don’t learn when he solve other problem; man learn when he solve his own problem.
Reading a book on swimming is never going to help as much as actually trying to paddle your feet to stay afloat in a pool is. All the problems in mathematics textbooks (all textbooks rather) are somebody else’s problems.
Yeah, Joseph’s dog gave birth to puppies, very droll, how quaint - but why do I care how many puppies he has? I am rather concerned about what Joseph is going to do with all those many dogs in his house. Is he going to sell them? Is Joseph a breeder? These questions are so much more interesting to me than counting dogs. By the way, mom said yesterday that I could buy a chocolate pie if I didn’t pee in bed for a week. How much longer do I have to hold my pee in?
4. Language is more important than you think.
This should be somewhat obvious to you, my dear adult reader. But unfortunately you probably realised it far too late in your life as well. We focus way too much on priming our children with times tables and finding prime factors and too little on letting them read, speak and create. Even in language classrooms, the priorities are stacked towards getting them to speak “proper English” rather simply getting them to speak.
You are free to imagine the consequences of these barbaric acts but let me lay out for you how this affects learning mathematics. You see, each discipline, much like each situation, has its own “register”. Yes, you are learning mathematics in the English language (say) but the English that you see and use in a maths textbook is not quite the same as the one that you would use elsewhere.
For example, look at the usage of the word “factor” in the following statements –
• Her discipline is an important factor behind her success.
• The Rhesus factor of the child returned negative.
• The factors of a prime number are 1 and the number itself.
• Pay-scale of an employee is factored into her benefits.
Same, but different.
If you’re having a moment of realisation right now, well, take your time. Better late than never, am I right? Don’t worry, it’s not entirely your responsibility to fix it. There are people already working on it. Don’t let that stop you from doing your bit though.
How do we fix this?
First, you must memorise these three principles. I mean really. Do it. Read them a bunch of times. Understand them. And make sure you know them so well that they are almost your own ideas.
Okay, here goes –
- There is no easy fix. Or we would have fixed it by now.
- No single solution can fix it. We need complexity.
- Each child has her own needs. One size does not fit all.
Got it? I don’t want you getting inspired from this article and trying to set up an NGO to fix the world’s maths curricula. Or worse, a startup to make 6 year old Aryabhatta actualise his name.
Like I said, there are people already working on fixing this machine. You just relax. Please.
Let’s look at the possible solutions now.
Note: I am writing this as techniques that could be used directly to help an individual (or small group) of learners. This isn’t a policy suggestion.
1. Let them read
Rhymes, stories, puzzles, magazines, video game walkthroughs – just let the kid read. The more s/he reads the more likely it is that s/he will encounter the usage of the same word in a variety of contexts. (See ‘factor’ above?)
This helps in two ways. One, the child will be less afraid of encountering text. Two, s/he will be more familiar with language and will thus be more likely to be able to wrestle the meaning out of the text.
How do I do it?
There’s tons of reading material for kids out there. I know you can just buy it on Kindle or show them a video on YouTube but seriously, if you really love this kid, buy them a paper book. Inundate the little dude with literature. Don’t force them to stick with a book. Let them figure out what interests them and what doesn’t. If they sit down to read the labels on a shampoo bottle, well so be it. All reading is good reading.
Aside from this, you can build some flashcards with kids on which they write all the new words they encounter in the maths textbook. Behind each card then they write where they encounter them, an example question or a statement.
How do I know I am doing it right?
If the kid is able to read the word problem without struggling, you’ve done well. Yeah, they might not yet figure out how to solve the problem or perhaps might not even get what the problem is all about - but at least they can read it. And if they can read, they will soon decipher what it means as well.
Do not expect immediate results though.
Give it at least a few months. If you start this ‘programme’ with a Class 1 kid, you will see some benefits in Class 2-3 but the mega-benefits will start coming in in middle school when all her classmates will struggle with 4-syllable words and she’ll blaze through.
Pro Tip: Let the child read the maths textbook like it’s just another book. Kids don’t ever get to see the maths textbook as what it is - a book. They always see it as something to be feared, or worse, something that invites trouble and punishment.
2. Let them talk
Yeah, I know kids can be annoying and repetitive but that’s just how they learn, man. The only way out is through. What I mean by letting them talk is more specific than just letting them rattle whatever’s on their mind (although there is some merit to that as well).
Usually the conversations that adults have with kids are about information recall. What is your aunty’s name? How old is your friend? Where did we go last summer? What we need to focus on instead are reflective questions. The hows and whys. Things like - How did you make this lemonade? Why do you like going to your friend’s place? What will you do when your cousins get here?
Questions like these make you reflect about your actions (in the past) or plan about your actions (in the future). They get you thinking about emotions - your own and that of others. This whole exercise helps kids figure out the intricacies of language syntax and boosts their confidence about their own language abilities.
How do I do it?
Ask nice questions nicely.
Do this in tandem with the writing exercise prescribed ahead. I’ll detail it there.
3. Let them write
This one comes backed up by research. The aim here is to get learners to write reflective short pieces about their mathematical efforts. Here are some example prompts –
- Explain to an absent classmate the formula for area and perimeter of a rectangle, with three examples of how to use them.
- Explain the Pythagorean theorem in your own words. Provide several examples where this theorem can be used.
- Your younger sibling will be in this class next year. Your responsibility is to inform your sibling about the prerequisites for learning ratios and proportions. How will you explain it to them?
- Explain the angle-side relationship of triangles. Describe two types of questions that could be asked on a test.
A meta-analysis study showed that “Writing summaries, research reports, arguments, self-reflective journal entries and even narrative stories all seemed to work most of the time.” So feel free to switch up the response format if things start to get boring. Kids could have a lot of fun writing a dialogue between a square and a rectangle as each boasts to the other about its amazing geometrical properties.
Granted, such an exercise would work better with middle schoolers and older, but even primary school kids would benefit from engaging in an interesting and reflective exercise like this. Writing is like a muscle, isn’t it? So even if it is hard at first, persistence eventually pays off.
How do I do it?
In everything that we do, there’s what you do, how you do it, and why you do it. Think of some fun prompts that would make the child recall what was done, how it was done and why it was done that way. Take inspiration from the examples above.
How do I know I am doing it right?
In most cases, the simplest marker of success is the child’s improved confidence. You let the kid express herself and she’s going to be a happy child. More specifically, the child will likely be able to verbalise where she is getting stuck in solving a problem better, instead of putting out an unrevealing “I don’t know”.
Do not expect immediate results.
The purpose of writing and talking is to get kids to express what’s going on in their hearts and brains. Once they learn this bit out, life is going to be a whole lot easier for them. And this takes time. But it’s worth it.
4. Give them context
Make the problems interesting, challenging and worth solving. Instead of a bland textbook statement, add some local context to the problem. Give it a relatable setting. What is the child interested in? What would she rather do if she didn’t have to sit through your boring maths class? Talk about that!
Here’s an example –
- Textbook: Anand wants to erect a fence around his rectangular farm. What will be the length of the fence he will have to erect?
- Contextbook™: You have to run along the boundary wall of this park in front of your house. How many steps will it take you to cover the whole distance?
You’re still talking about perimeter in both instances but one is about doing a task that you have no interest in doing, in an unknown land to some unknown person’s farm. And the other is about a game that you like to play, in a place that you know very well.
How do I do it?
You’ve really got to know the kids that you are working with to be able to do this well. Or you should just know kids in general well enough to be able to make an educated guess. The easiest thing to do is to get the context from the kids themselves. Just ask them about what they like to do, what cartoons they watch, what games they play, what the names of their friends are.
Something as simple as replacing the names of the characters in a word problem with those of a learner’s friends can suddenly make it a lot more accessible.
How do I know I am doing it right?
Is the child still listening to you? Yes? Congratulations. Kids won’t pay attention to you if talk in an alien context.
This can give you immediate results.
Getting kids interested in an activity pays off immediately. Maintaining that interest is tough of course but we are here for small victories.
Pro tip: Combine context with writing and get kids to come up with example problem statements themselves. That way, they build ownership over the task and deepen their understanding of it.
5. Break it down
This one is the most intuitive and the easiest. I mean, I am pretty sure you’re probably doing it already.
- What is a word problem? It’s simply a short story with a missing fact.
- What is the child’s problem? The child is unable to comprehend the story.
- What do you do then? Break the story down into smaller chunks and feed them to the child gradually.
Take the problem about erecting a fence around Anand’s farm. Here’s the full statement – Anand wants to erect a fence around his rectangular farm. What will be the length of the fence he will have to erect?
Here are the chunks of information in this statement –
- Anand.
Who is he? - Anand’s farm.
Where is it? - Anand’s farm needs a fence.
Why does it need a fence? - Anand’s farm is rectangular. So it has a length and a breadth.
- The fence will go all around the farm’s boundary.
- How long a fence will he need? It’ll have to be as long as the farm’s boundary, of course.
- How long is the farm’s boundary? Depends on its shape.
- It’s rectangular. What is the perimeter of a rectangle?
- What is the perimeter of anything?
What is the purpose of life?
You get it. I shouldn’t have to spell it out anymore. This is one of those tactics that we all naturally do with kids. The length of this pseudo-socratic dialogue depends on your patience and the child’s reverence for you.
It’s pretty silly but yeah maybe it will work sometime, somewhere with someone. You are going to do it anyway. Carry on, you wayward child.
6. Let them struggle
Don’t just give them the formula straight away. You are taking away the whole joy of achievement from them. Give them a problem and let them struggle with it for a while. If you’ve got enough time, let them struggle indefinitely. Otherwise a day or two is okay.
What’s important here is that the problem should be worth struggling for! Compare the two prompts –
- What is the area of a circle with radius $r$? What will be the ratio of the area of this circle and a square of side $2r$?
Or
- Cut a big square out of a piece of paper. Measure how much area it covers. Can you now make a circle that covers exactly the same amount of area as this square?
Which one do you think is worth struggling for? (Ask the kids)
In conclusion
Word problems are hard because maths is stupid. And because adults don’t care for language. Adults are stupider than maths.
Here, read this –
Grown-ups love figures. When you tell them that you have made a new friend, they never ask you any questions about essential matters. They never say to you, ‘What does his voice sound like? What games does he love best? Does he collect butterflies?’ Instead, they demand: ‘How old is he? How many brothers has he? How much does he weigh? How much money does his father make?’ Only from these figures do they think they have learned anything about him. If you were to say to the grownups: ‘I saw a beautiful house made of rosy brick, with geraniums in the windows and doves on the roof,’ they would not be able to get an idea of that house at all. You have to say to them: ‘I saw a house that cost $20,000.’ Then they would exclaim: ‘Oh, what a pretty house that is!’
— The Little Prince, Antoine de Saint-Exupery
Good bye. Good luck.