Tag Archives: math teaching

Making the Most of Word Problems

by Sarah Lonberg-Lew

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Whether they’re called word problems, story problems, or problems in context, they usually amount to the same thing – a short story about a character who for some reason needs to know how long it will take to do a task or how much it will cost to go on ten rides at the fair. Why do our students need to know how to answer these questions?

The clearest reason seems to be that they will have to do so on the high school equivalency test that is standing between them and their next step, be it college or career. To that end, it makes sense to teach students a strategy for finding the answer quickly and with a minimum of fuss.

As a result, many teachers choose to have students memorize a mini-dictionary of key words:

more = add
less = subtract
of = multiply
each = divide

… and the list goes on.

There are two problems with this approach. The first, and relatively minor problem, is that it doesn’t actually work.

The word “more” can indicate subtraction as easily as addition in a word problem, for example:

Jack has five more apples than Jill. Jack has eight apples. How many does Jill have?

The word “each” can mean multiply as easily as it can mean divide. Contrast the following problems:

Suzy gave five apples to each of her four friends. How many apples did she give out?

Suzy had twenty apples and gave the same amount to each of her four friends. How many did apples did each friend get?

Test makers know the key word dictionary, too and will likely present students with problems for which these key word shortcuts don’t work. This is not because they are trying to trick our students or make them fail, but because the purpose of word problems is not to assess how well the students have memorized the key words. It is to assess their ability to reason.

And this brings me to the second and much bigger problem with this approach: it wastes our students’ time and deprives them of the opportunity to learn something valuable. Using key words to solve tidy word problems may help them score some points on the test, but that’s where its (already questionable) usefulness ends. It’s a shame to invest so much time and energy on such a short-sighted goal. Instead we can choose to capitalize on the need to learn to solve word problems as an opportunity to develop confidence and skill with mathematical reasoning and critical thinking.

One good way to push students to reason about word problems instead of trying to find the answer as quickly as possible is to remove the question and maybe even some necessary information from the problem. For example:

Jack and Jill went apple picking. The orchard charges $8 per person for admission. A half-peck bag costs $10 and a one-peck bag costs $18.

Think of all the questions your students could ask and answer with this simple scenario, like:

  • How much would it cost for each of them to pick a half-peck of apples?
  • How much would it cost for Jack to pick a half-peck and Jill to pick a whole peck?
  • If they have $50 between them, how many bags of apples can they get and in what sizes?

… and so many more.

By asking and answering questions, students are really engaging with their math. They are reasoning about how the quantities in the problem relate to each other and what role each of them plays. They are choosing and using operations to answer a question that they understand instead of following a translation code that they have memorized. And the skills that they are acquiring through this process are transferrable and therefore worth spending their time on.  (See https://adultnumeracyatterc.wordpress.com/2017/02/16/changing-how-problems-are-presented/ for more ideas on how to present word problems and more in ways that get students reasoning.)

My students want to pass their high school equivalency tests and I want that for them too, but I want more for them as well. I want them to be able to reason about the real problems in contexts that really do come up in their lives and that aren’t presented in tidy packages with key words. I want them to be able to make sound decisions about how they invest their time and money so they really can choose the phone plan that’s best for them or figure out how many classes they can take in a semester and still make rent and feed their families. And I want them to be savvy consumers of the quantitative information that comes at them every day so they can reason confidently about the real and messy numbers in their lives.

Word problems are sometimes silly and contrived, but we are stuck with them and we can slog through them without learning anything or we can use them as a way to develop strong reasoning in our students that will serve them beyond test day.

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sarahlonberg-lewSarah Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency. She teaches in an adult education program in Gloucester, MA. Sarah’s work with the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center at TERC includes developing and facilitating trainings and assisting programs with curriculum development. She is the treasurer for the Adult Numeracy Network.

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Some Help Hurts: Our Responsibility to Our Students

by Sarah Lonberg-Lew

A student joined my class in the middle of April and told me she absolutely had to achieve her high school equivalency by the end of June. “I can. I must. I will,” she said to me. She is willing to do whatever it takes – get a tutor, watch videos about algebra on YouTube, get her high school-aged daughter to help her. She has grit and determination and has been told that this will get her to her goal.

Another student has been with me a bit longer. She has a traumatic history with math education, as so many of our students do, and her mistaken beliefs about what she is capable of (she thinks she is not a math person) consistently interfere with her learning. She also has grit and determination. She is going to succeed at math no matter how painful it is.

My heart breaks for these students. They have worked hard and failed and now they are here trying again and still laboring under the lie that the only thing needed for success is hard work. And if they fail again, who do they blame? They think it must be their fault for not working hard enough. I don’t know how to break through this lie – it is so entrenched, and it serves the status quo so well. If students’ failure can be blamed on their lack of effort, there’s no need to change anything in the way we are teaching. We just need students who work harder.

It isn’t just the students who believe the lie. Teachers and directors believe it too, and with the best of intentions, they become cheerleaders for the students – praising their inordinate effort, giving them extra worksheets, setting them up with tutors, sending them to math websites. It’s like a conspiracy. Everyone involved wants to believe in these hard-working students and support them with extra help and resources. The students ask for more and the teachers provide it. But the students don’t know what they need in order to be successful at math. They think they can learn algebra, geometry, fractions, percents, negative numbers, statistics, exponents, polynomials, functions – you name it – all at once,  just by going online and following examples until they’ve memorized it all, or by sitting down with someone who is willing to show them the steps over and over until they stick. That is not how learning happens and we should do everything we can to prevent our students from wasting their time on a fool’s errand like that. And it isn’t just their time that is wasted. How long can they go on like this before they finally succumb to the even bigger lie that math just isn’t for them?

One thing our hard-working, deceived students often ask for is practice tests, and teachers are happy to oblige. After the test is taken, student and teacher conspire in a plot to have the student learn to answer every single question they got wrong. Both parties are happy to set their sights on passing the test, as if that means the same thing as learning. But even if a student can memorize their way through to a high school equivalency certificate, even if they exit a program feeling successful, all we have done is push back that wall they will eventually hit when they, their college instructors, or their employers realize they have not really learned math. When we give students false success, we are still setting them up for failure  –  it’s just that the failure will come later when we are not there to see it and not there to support them.

Students generally do not appreciate the scope of all the math there is to learn, nor the idea that concepts build upon each other and some are prerequisite for others. Those determined students with grit are willing to learn anything and want to learn everything. Our responsibility to our students is not to give them what they want, but to give them what they can handle, building concepts coherently and helping them learn how to learn. Allow them to struggle, but be sure that struggle is productive. It is wonderful that so many of our students come to us with the willingness to work hard. They are trusting us with their time, their futures, and their self-images. We owe it to them to guide their effort in useful directions, even if those directions are contrary to what they say they want.

If I had an ambition to run a marathon (I don’t!) and I hired a coach to help me prepare, I would expect her to know better than I what level of training and exercise were appropriate for me. If all she did was encourage me to run as fast as I could for as long as I could, I not only would end up unready to run a marathon, but would likely end up injured as well. Even if I was motivated to run for six hours every day (I’m definitely not!), a coach who supported me in that course of action instead of guiding me through a training regimen that built my strength and endurance would not be helping me. Desire to achieve and willingness to work hard are not enough. Our students need thoughtful, considered guidance from teachers who know the terrain better than they do.

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sarahlonberg-lewSarah Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency. She teaches in an adult education program in Gloucester, MA. Sarah’s work with the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center at TERC includes developing and facilitating trainings and assisting programs with curriculum development. She is the treasurer for the Adult Numeracy Network.