Will This Be on the Test? #2

by Sarah Lonberg-Lew

Welcome to the second installment of our new monthly series, “Will This Be on the Test?” (If you missed the first blog, check it out here.) Each month, we’ll feature a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually.

There’s almost always more than one way to tackle a math problem. When teachers encourage students to try out different approaches, students gain confidence in thinking flexibly, experimenting, and finding strategies and solutions that make sense to them.

What about you? How would you approach this month’s problem?

Before you read further, take a few minutes to challenge yourself to come up with more than one way to approach this task. How many strategies can you think of? What visuals could you use to help you solve this? Also ask yourself, what skills and understandings do students really need to be able to answer this? To what extent, if at all, are memorized procedures required?

When you’ve given yourself enough time to really explore, read on to see some possible ways students might approach this problem using conceptual understanding and visuals.

1. They might rely on real world experience to estimate a solution to this. Students who has experience with landscaping, yard work, flooring, tile, etc., may have an intuitive understanding of about how big this backyard is and about how many square feet of sod it would take to cover it. It is important to connect the math that students learn in the classroom to the math they encounter in the real world. Without this connection, students may come to see “school math” as something separate from their lived experience. They may not realize that they can apply what they already know in a testing situation. Students who do not make this connection can end up panicking over trying to remember a formula instead of using their own mathematical intuition.

2. If the student has some understanding of the relationship between feet and yards, they might start by drawing a picture of the backyard with the dimensions shown in yards, and then divide each yard into feet to see that they should multiply 15 by 24 to find the number of square feet. 

3. A student might start by drawing a picture of the backyard with the dimensions marked in yards and then figure out how many square feet are in a square yard. Once again, if the student knows how many feet are in a yard, she could use this knowledge to visually divide a square yard into nine equal parts of one square foot each. She might then convert the area of the backyard from square yards (40) to square feet by multiplying 40 x 9. Alternatively, she might calculate total square feet in another way, like finding how many square feet are in each column and then adding all the columns together.

An important point here — even people who are experienced with working with linear and square measures (like feet vs. square feet) can confuse the conversions sometimes. It’s hard to remember to use a factor of 9 when converting between square feet and square yards, but to use a factor of 3 when converting between feet and yards. It’s especially tough in a high-pressure situation like a test. It’s also easy to overlook the conversion completely and quickly answer the question using the wrong unit. This is why it’s so valuable for students to have the confidence to take the time to make sense of a question instead of trying to rely on their memory of what they are supposed to do. Students who draw pictures and rely on their own reasoning are better able to bring their full mathematical power to bear on any question.

What can go wrong?

In contrast to students who confidently take time to draw pictures, consider strategies, or draw on their own experience, there are other students who are desperate to get through the questions. These students may resort to “number grabbing”, or quickly taking some or all of the numbers from a test item and arbitrarily applying to them the first formula they can think of. If they arrive at an answer choice, they pick it and move on. For students without strong conceptual knowledge of math operations, having answer choices can be a pitfall. The developers of standardized tests do not create the wrong answers for multiple choice questions at random; they choose “attractive distractors” — answer choices that will result from number grabbing or not thinking the problem through thoroughly. For example, in our problem above, a student might quickly choose to add or multiply the two numbers given in the problem and arrive, incorrectly, at answer choice A or B without really thinking at all.

Number grabbing is a panic response. For teachers, the best way to help students combat that panic is to nurture their confidence in their understanding, creativity, and flexible thinking. Students who are empowered to approach problems in ways that make sense to them instead of trying to remember how they are ‘supposed to do it’ are much better prepared to make use of all of their skills and abilities in testing situations and elsewhere. When students develop the confidence to think critically and calmly about test questions, they will also have the confidence to do so when they encounter math in real life.


Sarah 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. 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.

Will This Be on the Test? #1

by Sarah Lonberg-Lew

Welcome to the first installment of our new monthly series, “Will This Be on the Test?” Each month, we’ll feature a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually.

There are lots of good reasons to study math – it better prepares students for the numbers and relationships they encounter in life and at work, it builds flexible thinking and analytical skills, and students have to pass a test on it to move forward with their educational and career goals. The test looms large for students and is often their strongest motivation for coming to adult ed classes. For teachers and programs, the test occupies a place of great significance as well. Performance reviews for teachers and funding for programs may hinge on how many students pass the test and what scores they get. It’s no surprise, therefore, that directors, teachers, and students are inclined to prioritize preparing for the test over anything else. The good news is that preparing students for the test doesn’t have to be separated from deep and rich conceptual learning. In fact, deep and rich conceptual learning prepares students better than crash courses on procedures and test-taking strategies. This post is the first in a series that will look deeply at test-style questions and dig into how conceptual understanding, willingness to draw pictures, and flexible thinking can prepare students to excel on the test.

Here is an example of a question students might see on a high school equivalency (HSE) test. Before you read further, I challenge you to see how many ways you can think of to approach it. You may know a procedure to apply. Which approaches could be used that do not rely on memorized procedures? What skills and understandings do you think students really need to be able to arrive at the correct answer?

Here are some approaches students might take if they don’t have a memorized procedure:

1. A student might sketch rectangles with all the sets of dimensions including the original and eyeball them to see which one looks like it’s not similar to the rest of them.

2.  A student might start by drawing a picture showing a simple enlargement. The simplest way to enlarge a picture is to double both dimensions.

Based on this picture, a student might reason that there was only one proper enlargement with a 4” height and since doubling the dimensions gave one that is 4” by 6”, an enlargement that is 4” by 5” would not work.

 

3. A student might look at the ratio of the length to the width of the logo as a fraction and visually compare the fractions for each enlargement.  

4. A student might reason that you enlarge a picture by multiplying both dimensions by the same number and notice that answer choice A is the only one where the second dimension is not a multiple of 3. If she then checked by trying to set up equal ratios, her suspicions would be confirmed.

Does this mean that instead of teaching one procedure you need to teach four different ways to approach the problem? No! The purpose of seeing all these different approaches is to understand that when students have some conceptual understanding and the confidence to apply it flexibly, they can make sense of challenging test questions even if they haven’t memorized the procedure that the question is ostensibly targeting. To tackle this question, a student does need some intuition about what it means to enlarge a picture, but even with only that, she is prepared to make sense of the problem and persevere in solving it. In fact, worrying about which memorized procedure applies to this problem can make it harder!

Giving students rich learning experiences that help them draw connections between math concepts and the real world gives them the confidence to bring their strengths to unfamiliar test questions. When students know that there isn’t one single correct approach to math, they are free to find their own strategies.


Sarah 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. 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.

How to Open Up Word Problems (and promote digital literacy, too!)

by Heidi Schuler-Jones

In today’s classroom, we know that it is not enough to simply teach math content and skills. Students also need to develop facility with technology tools. Using a traditional word problem as a starting point, here are some ideas for doing so.

First, think about ways to ‘open up’ a word problem:
  1. Limit the information provided to students and instead have them find information based on their own interests.
  2. Allow students to make choices on figures, when possible, or recommend a range of choices.
  3. Instead of having one answer, provide a way for students to evaluate the information they gathered, organize and make meaning of the information, and communicate the reasoning behind the choices they made using a variety of methods, visuals, and models.
  4. Differentiate the same problem for multilevel classrooms by providing conditions for the types of numbers to use. For example, using less-friendly numbers for more advanced students or more friendly and common numbers for less advanced students. Other options: provide more or less information, offer more or less decisions, provide more or less steps (each of these can provide a push for more advanced students or support for those needing greater assistance).
Next, consider where digital literacy tools might be appropriate and useful.

Suggested digital literacy tools can extend beyond the basic calculator functions. Encourage students to use more advanced functions on their smart phone calculator apps and/or to become familiar with spreadsheet and graphing functions found in programs like Microsoft Excel, Google Sheets, and Desmos.

Now, let’s apply these tips to ‘open up’ a traditional workbook problem into opportunities for deeper, more active learning.
Traditional workbook problem:
John is interested in buying a used car for $15,000. He puts down 25% with the rest to be paid over a 5-year period. If his monthly payments are $197, how much will he pay altogether for the car?
Steps to Open Up
Word Problems
Closed (traditional)
word problem
Opened-up
word problem
Limit the information provided to students and instead have them find information based on their own interests.John is interested in buying a used car for $15,000.Check out the prices of vehicles at two different dealerships in the area. Decide on a vehicle that might suit your needs.
Allow students to make choices on figures, when possible, or recommend a range of choices.He puts down 25% with the rest to be paid over a 5-year period.Decide the amount you want to put down and the number of years you want to finance the vehicle. Decide the maximum monthly payment you can afford.
Instead of having one answer, provide a way for students to evaluate the information they gathered, organize and make meaning of the information, and communicate the reasoning behind the choices they made using a variety of methods, visuals, and models.If his monthly payments are $197, how much will he pay altogether for the car?Create a graph and/or in-out table to show the cost of the vehicle over the years that you have financed it for and be prepared to explain your choices.
Differentiate the same problem for multilevel classrooms by providing conditions for the types of numbers to use. For example, using less-friendly numbers for more advanced students or more friendly and common numbers for less advanced students. Other options: provide more or less information, offer more or less decisions, provide more or less steps (each of these can provide a push for more advanced students or support for those needing greater assistance).Conditions for more advanced students:
Estimate a reasonable answer first. Use two different down payment percents with a non-repeating fraction or decimal in each. Compare two different car deals on a trend graph.

Conditions for less advanced students:
Estimate a reasonable answer first. Use one of the following benchmark percents for the down payment: 10%, 15%, or 25%. Use the provided in-out table below to track the cost of the vehicle after the down payment (Year 0) and for each of the next 5 years.
Suggested digital literacy tools can extend beyond the basic calculator functions. Encourage students to use more advanced functions on their smart phone calculator apps and/or to become familiar with spreadsheet and graphing functions found in programs like Microsoft Excel, Google Sheets, and Desmos.Calculator
Smart phone (calculator
Calculator
Smart phone (calculator)
Amortization calculator or app
Web search
Spreadsheets and graphs
   *   Excel
   *  Google Sheets
   *  Desmos

Another way to open-up traditional problems is to remove the question completely and simply ask, “What do you notice?” and “What do you wonder?” Using our example above, the word problem would thus shorten to simply:

John is interested in buying a used car for $15,000. He puts down 25% with the rest to be paid over a 5-year period.

Students spend a minute jotting down the things they notice and wonder and then share these observations with a partner or the whole class. It generates critical thinking and reasoning about what information is needed before attempting to plug into a formula or calculating numbers that may not be relevant. It’s quite common for students to come up with the actual question themselves but in their own words, which helps to make sense of the problem. It also gives them practice developing test-like questions themselves based on their own understanding of the problem and the choices they make.

From those noticings/wonderings, students can follow the steps to open-up word problems as shown in the table above. Watch this video to learn more about this strategy: Ever Wonder What They’d Notice?: Annie Fetter


Heidi Schuler-Jones has worked in adult education since 2006. She participated in the pilot program of the Adult Numeracy Instruction – Professional Development (ANI-PD) in Georgia in 2010 and immediately found its techniques, methodologies, and research-based resources to be of tremendous value to her teaching and to the variety of students she saw daily. Heidi currently is a consultant for the SABES numeracy team where she facilitates trainings and works on course and curriculum development, including the Curriculum for Adults Learning Math (CALM). She also facilitates the Adults Reaching Algebra Readiness (AR)2 institutes for TERC. Heidi also is a LINCS national trainer for math and numeracy and serves as President-Elect of the Adult Numeracy Network.

What the Pandemic Has Taught Me to Value

by Melissa Braaten

If you had asked me a few months ago about my favorite tools to use in the math classroom, I would have talked about how much I love my square inch tiles and the value of group work.  I would have thought about how hard I work on my questioning techniques so I can check in with each group, try to assess where they are with the problem, and to provide just the right push to move them forward.  I am good at reading body language and the energy of an audience.

Then, of course, COVID hit and I was left sitting in my kitchen and wondering how on earth I was going to continue to teach math without any of the tools and skills I have come to rely on.  If you asked me previously about remote learning for math in adult education, I would have told you it doesn’t really work.  I would have thought the outcomes would be so minimal that it wasn’t worth the effort.

I can’t say I’ve figured out how to replace in-person instruction with remote learning, or how to use technology to make up for all the tools and skills I used to take for granted.  This has been a humbling experience (as I’m sure it is for many).  I’ve had to really rethink both my methods of instruction and my goals.  In the process, I’ve come to appreciate a few things that I previously undervalued.

Permission to Fail

I think most teachers (including myself) can be nervous about entering untested waters.  I spent the first month of the pandemic trying to teach as closely as possible to how I would in the classroom, using Zoom and trying to replicate as many of the activities and lessons as I could in that medium.  This worked all right, but only for a couple of students whom I could get interested and connected.  Most of my students I was not reaching at all.  But what could I do? 

I had a conversation with my director during which he told me that he expected us to try things that would fail.  There was no clear precedent for what we were trying to do, so we just needed to keep trying and learning.  I had considered mailing out packets and trying to teach math over the phone, but I had assumed that it wouldn’t work.  Having permission to fail helped me get unstuck and to rethink some of my assumptions.  In the process I was able to greatly expand the number of students I work with every week.

Did it fail?  It is slow, and frustrating, and I can’t say it works great, but students are engaging and are happy to be working on math again.

Connection and Engagement

In ordinary times, I wouldn’t be happy with teaching that produced such small academic outcomes.  But in these unusual times, I think there is value in connecting with students on a regular basis even if the academic progress is minimal.  The regular contact has value, and I think it is helping students to feel that they are still in school, and still working towards their goals, even though the pandemic has disrupted most of their timelines.  For some students, it may help them combat isolation and loneliness; for others, it preserves some small piece of normalcy and helps them hold onto their identity as a student.  This may help them in a small way now to cope with the strangeness of the current situation, and it should also make it easier for them to continue their studies when things start to open up.

And Last But Not Least: Color Coding

I’ll end with my most important takeaway.  If you are sending packets for students to work on remotely, color code them.  I decided at the last minute to add fluorescent cover sheets to each section I mailed to students, and when I am trying to work with students over nothing but audio calls, I am so thankful to be able to say, “Now pull out the green packet…”   Also, add page numbers, add letters to diagrams, and keep a master copy of everything for yourself.  I do not regret the time I spent doing this.  In the future, I think I will be even more meticulous about making sure that every student’s materials are the same colors.  This last batch, I made the mistake of using envelopes I had at home to hold shape sets.  Not all of the envelopes were white.  This has led to so much preventable confusion when I tell people to take out the white envelope and theirs is yellow.  There are so many things I used to take for granted when I could just hold something up in front of the class!

This pandemic has forced us all to try new things, move out of our comfort zones, and some days, to long for the things we used to do in our classrooms.  Best wishes to all as you continue to experiment, learn, and stay connected to your students and each other.  I also wish you all patience for the times when the envelope is yellow.  We’ll get through this.


Melissa Braaten is an adult education instructor at Catholic Charities Haitian Multi-Services Center in Dorchester, MA. Melissa has taught ASE and pre-ASE math and reading, as well as ABE writing, computer skills, and health classes. Melissa also is a training and curriculum development specialist for the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center at TERC. She has written several articles for Math Musings, the Adult Numeracy blog.

You’ve Heard of Digital Literacy. What about Digital Numeracy?

by Donna Curry

For too many of us in the United States, the definition of literacy is simply the ability to read and write. We supposedly include numeracy in that definition, but it is usually overlooked. Teachers will often say things like, “I’m a literacy specialist” or “I teach literacy.” When we hear those terms, we don’t assume that they are also teaching numeracy. In fact, “literacy” is often a code word meaning, “I don’t touch anything math-related.”

So, just as adult literacy is problematic because it doesn’t explicitly address numeracy, so is the term digital literacy. For some, digital literacy simply means being able to read and write using technology. But digital literacy should focus on so much more. Digital literacy can be defined as “the skills associated with using technology to enable users to find, evaluate, organize, create, and communicate information” (U.S. Department of Education, 2015). Clearly this definition implies more than just reading and writing, but does it include numeracy activities as well?

Finding, evaluating, organizing, creating and communicating information is what doing math is all about. It is not simply practicing computation for the sake of becoming faster and faster without even knowing when, where, or why the computations should be used. Think about the last time you used math. You may have had the information right in front of you, but if not, you had to go find it — maybe do a bit of research on price comparisons. You would have done some evaluation and organizing of those data in order to make an informed decision. And then you might have communicated your decision to your significant other, explaining why you thought Brand X was better than Brand Y. That’s what numeracy is all about, and the use of digital tools is a powerful way to make us all more numerate.

Here’s an example. With the fluctuating interest rates, you may have been considering refinancing your home. Did you do a lot of rote computation using pencil and paper? Possibly. Did you use a calculator? Maybe. But, if you were savvy enough to know that there are amortization tables galore available to you online, you could have made sound decisions without endless drudgery. In fact, using an amortization table allowed you to explore ideas that you would not have thought about otherwise, because it would have been too time-consuming and too tedious.

Making sound decisions is a major reason that we use math. Giving students similar experiences helps them realize that math is so much more than computation, or memorizing procedures for a test. Familiarizing students with digital tools gives them more resources to make life decisions. Give students a task: comparison shopping, deciding whether to lease or buy a car, or whether to rent or own a home, planning a room remodel, or investing money for the future. Then let them do their own research. Ask them to collect, organize, and evaluate what they find and then explain their decision in a class presentation.

Digital literacy is empowering, for us as teachers and for our students. If you’re a tech newbie yourself, you can start small. Invest in learning a couple of online math tools, testing out web conferencing with your students, or familiarizing yourself with some of the numerous Google tools that allow you to create shared, editable class documents. Not sure where to start? Check out the SABES professional development calendar for ideas or contact adultnumeracy@terc.edu with questions on digital numeracy.


Donna Curry is an educator, curriculum developer and professional development specialist with over 30 years of experience in adult education. For the past 30 years, she has focused on math standards development at the national level (Equipped for the Future National Standards and Standards-in-Action projects) and at the state level (including states such as Rhode Island, Washington, Oregon, New Jersey, Oklahoma, and Ohio). She has also worked on the National Science Foundation’s EMPower project and served as co-director for the NSF Teachers Investigating Adult Numeracy (TIAN) project. She co-developed and implemented the Adult Numeracy Initiative (ANI) project and Adults Reaching Algebra Readiness (AR)2. Donna currently directs the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center for Massachusetts and the Adult Numeracy Center at TERC.

Making Math Meaningful in the ESOL Classroom

This article also appears on the World Education blog here: https://edtech.worlded.org/making-math-meaningful-in-the-esol-classroom/

by Sherry Lehane

Math in ESOL Coursebooks

Did you ever start a home do-it-yourself project and suddenly realize you were in over your head? Maybe you didn’t have the carpentry skills or the right tools to do the project? This is how I felt many years ago when I attempted to teach math to my ESOL learners, many of whom had very little formal education. On the surface, it seemed easy enough. The ESOL coursebook I was using had math tasks embedded in the units such as totaling a bill, comparing prices, reading pay stubs, and creating a budget. I would model how to do the math, we would practice doing a few examples together, and then students would be able to complete the activity. There was one hiccup in this plan–the level of math thinking was superficial. There were many embedded math concepts that could be explored, but I did not yet have the tools to teach them. The tools I needed included a deeper understanding of the underlying skills and knowledge needed for problem-solving and strategies for teaching math to students with a range of math and language skills. 

The Intersection of Language and Math

ESOL coursebooks include math activities because they are part of the life skills that we encounter daily, and it makes sense to include them in thematic units. We can help our English language learners navigate situations that involve math by integrating math into our ESOL instruction in a way that empowers them. In order to accomplish this, we need to dive deeper into math by creating opportunities for students to explore math concepts by using concrete objects and visuals, sharing their reasoning by discussing what they know and listening to the explanation of others, and ensuring they are able to transfer their knowledge to the real world. This is where language and math intersect! Integrating math in the ESOL classroom has the benefits of teaching learners math, language, and life skills all at once.

As a starting point, ESOL instructors can begin by thinking about the math they encounter daily or they can refer to the math activities in their ESOL coursebooks. Here is an example modeled from an ESOL coursebook where students are asked to compare prices of food items: 

Typically, the accompanying tasks include reading labels and doing some computation such as: 

Which is the better buy? or How much cheaper is…?

In this example, students could use addition (2.99 + 2.99), multiplication(2.99 x 2) or division (6.99 ÷ 2) to determine the better buy. One way to extend learning and explore valuable math concepts is to allow students to first do the math and then share how they solved this problem. By listening to others describe their approaches, students engage in a productive discussion, work on articulating what they know, and begin to think critically — in this case, about why different operations work to solve this problem. This can lead to an understanding of the relationships among the four operations as the visual below illustrates. 

How would you explain the connection between division and subtraction? (Hint: Think about the connection between addition and subtraction as a starting point.) 

Traditionally, we think of finding the ‘right’ way to solve a problem, but in life, there are usually many valid options for solving a problem. The College and Career Readiness  Standards describe mathematical practices of numerate adults. One of these practices, MP1 Making Sense of Problems and Persevere in Solving Them, addresses the ability to solve problems and explain one’s reasoning (http://www.corestandards.org/Math/Practice/). 

Let’s look at another opportunity to expand learning using the same ketchup problem. Students might use the same operations such as division, but the algorithm, or procedure, might differ. Algorithms for division, multiplication, addition, and subtraction can be done differently in other countries. (Even math notation such as decimals, commas, and colons can have different meanings in other countries!) Discovering why different algorithms work is another teachable moment and lends itself to rich conversations and a deeper understanding of math algorithms. Here is a comparison of the U.S. and European algorithms for division. (Note: The European algorithm is used in many countries in Europe, South America, and the Caribbean.)

Differentiating for Math Ability

One of the challenges in any math class is differentiating instruction for math ability. Here are some ideas on how the Ketchup problem can be adapted for different math levels. 

More Accessible

To make the task more accessible,  change the numbers to whole numbers such as $2.00 and $7.00. 

More Challenging

For students who are ready for a more challenging task, you could add a step to this problem by adding a twenty-five cent discount coupon if you buy three or more bottles. Or you could modify the question by making Happy Ketchup a 22 oz. bottle. This would add ratios and proportional reasoning to the math task. 

Keep the Focus on Language Acquisition

In an ESOL classroom, a new language unit or theme is introduced with a personal question, visual, or situation that engages students in the topic and activates prior knowledge. The same can be done with math. Giving students a short word problem to do when they enter class is one way to engage them in the topic. Word problems that include math encourage students to talk about what they know and listen to the ideas of others.  This is also an opportunity for teachers to informally assess learners’ familiarity with the math concept.

Using the Ketchup problem as an example, teachers could post ads from flyers on the board and write up a word problem such as: You’re having a big cookout to celebrate the first day of summer. There will be about 75 people there. How much ketchup do you think you’ll need? Which is the better deal? 

Teachers can simplify the language or make the vocabulary and grammar more complex. The grammar focus can be adapted to whatever language structure you want to practice: comparatives, superlatives, question forms, conditional statements, and many more. This can be part of a shopping unit, a community event lesson, or a lesson about holidays and celebrations. As with any topic, think about the vocabulary, including math vocabulary, when planning the lesson. 

The next step is to decompose the tasks – a process with which ESOL teachers are familiar.  Both ESOL and ABE teachers need to determine what the embedded concepts are that students will need, and how they can explore them in order to gain a deeper understanding. This deeper understanding is how we can empower our learners and give them the language, math, and problem-solving skills they need in daily life, higher education, and the workforce. 

Continued Learning

Are you interested in learning more about strategies and resources for teaching math and numeracy skills in ESOL context? TERC, in partnership with the EdTech Center @ World Education, offers several online courses, including:

  • Mathematizing ESOL I: Integrating Whole Number Operations
  • Mathematizing ESOL II: Integrating Benchmark Percentages and Decimals
  • Mathematizing ESOL III: Integrating Ratio Reasoning

Contact sherry_soares@terc.edu for more information about the courses above and other professional development opportunities.


About the author: 

Sherry Lehane has worked in adult education for over 20 years, teaching ESOL and digital literacy to adult learners. She is a strong advocate of integrating numeracy in the ESOL classroom. In recent years, her work has focused on supporting teachers in integrating math and technology.  She has co-created several resources designed to help ESOL teachers integrate math including several online courses and ESOL math packets. In Rhode Island, she coordinates the activities of the Rhode Island Tech Hub for Adult Education, which is the professional development provider for the use and integration of technology for teaching and learning.  

How I Learned to Stop Worrying and Love Percents

by Sarah Lonberg-Lew

When I was about 8 years old, I went on a “date” with my best friend. My mother dropped us off at a restaurant with some money and we ordered burgers, fries, and ice cream sundaes, just like a couple of grown-ups. Everything went fine until we got the bill and realized we were supposed to leave a tip and neither of us had learned how to do percents! Luckily, the kind waitress helped us work it out. I remember her explaining what she was doing, but I don’t remember understanding it. What I did understand about percents at the time was that they were hard and complicated and I’d probably learn about them someday when I was all grown up like that waitress (who was probably a high-school student).

When I told my mother about it, she told me something that changed my relationship with percents forever. She told me that 1% was the same as one hundredth. In other words, if I broke a number into 100 pieces, each of those pieces was 1% of that number. It was as if she had given me the keys to the kingdom. With that one bit of information, I felt I could figure out anything I ever needed to know about percents. Figuring out a 15% tip? Just divide the bill by 100 to find 1%, and then multiply it by 15 to get 15%.

But in school, things got more complicated. I was taught different kinds of percent problems, each requiring a different procedure. Depending on which kind of percent problem I was doing I learned to convert the percent to a decimal and then multiply by the whole, or sometimes divide the part by the percent, or other times divide the part by the whole! I also learned how to translate percent problems into algebraic equations and solve them using the rules of algebra. I dutifully memorized all the procedures and became quite adept at useful things like figuring out what the whole was if 17.3 was 83% of it. I got so good at those procedures that, for a long time, I forgot that I already had the keys to the kingdom.

When I first became a teacher and taught students how to solve percent problems, I taught them the procedures I had memorized. My students had the same initial ideas about percents that I had held – that they were hard and complicated and only people who were really good at math could figure them out without a tip-card or an app. I’ve found that being able to calculate with percents is something that most adult students really want to learn because it does come up so often in their lives. Not being able to make sense of percents can feel frustrating or embarrassing.

I realized that teaching students three different procedures for three different types of problems only strengthened their mistaken belief that understanding percents was too difficult for the average person. It was when I began to learn about teaching percent concepts with benchmark fractions that I started to find my way back to a place of understanding. Students didn’t have to wait until they were ready to work at a sixth grade level before they could start reasoning with percents. Any student who could make sense of ½ could also make sense of 50%. Likewise, students who could reason about ¼ and ¾ could reason about 25% or 75%.

It turns out that the key concept my mom had shared – knowing that 1% was equal to 1/100  – was only one of several that helped turn percent calculations from an exercise in applying memorized procedures to one in reasoning. I thought of 1% as my friend. If I could find my way to 1%, I could find my way anywhere! Since then, I’ve made many friends in the realm of percents. 50%, who also sometimes goes by the names ½ or 0.5, is useful for quick calculations and estimations. Her siblings, 25% and 75% help me achieve greater precision without a lot of mental effort. And there’s an extended family that I have gotten to know through expanding my set of benchmarks (a process that takes time). Once I got friendly with multiples of 10% and 5%, I was prepared to estimate my way through any percent scenario. And if I need a precise answer, my first friend 1% is always there for me (as is my calculator when numbers get messy!).

The procedures I learned in school were efficient and accurate, but it required practice and memorization to develop fluency with them. Memorization can be a difficult and unreliable route for many learners, but reasoning is accessible to everyone. It took time for me to get to know the extended family of percent benchmarks that I feel so at home with now, but even with just a few benchmarks, students at any level can begin to approach percents through reasoning.

See It in Pictures

Want to see how my friends help me work out what the whole is if 83% of it is 17.3?

Let’s start with an estimate: 83% is almost the same as four blocks of 20% (and 20% is the same as 1/5 of the whole).

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If four blocks of 20% put together is 17.3, then I can divide 17.3 by four to find the size of one block of 20%. 17.3 divided by four is close to 4. So 20% (one block) of the number I’m trying to figure out is close to 4.

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That means if I multiply 4 times five (total blocks in the whole), I can figure out about how much 100% of the whole would be.

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Want to be more precise? Time to call on my friend 1%! I know that 83% is 83 blocks of 1%, so if I divide 17.3 into 83 pieces, each one of those will be 1% of the whole. This is calculator time. 17.3 divided by 83 gives me 0.2084 (rounded).

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That’s 1%, so 100% will be 100 times that or 20.84.

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(I’d better just check to make sure that’s reasonable… 83% of a number is most of that number, and 17.3 is most of 20.84. Looks reasonable to me!)


Sarah 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.

Rethinking Placement Testing

by Melissa Braaten

We all know that first impressions matter. Students start forming an impression of their program, class, and teacher from their first interactions during intake, which usually involves some sort of assessment for the purposes of class placement.

Many standardized intake assessments involve a student sitting by himself, answering traditional, procedural math questions that have only one correct answer.  This experience can reinforce the individualistic, procedurally-focused math experiences that students have had in the past, which often lead to math anxiety and a fixed mindset towards math.

      …Students from an early age realize that math is different from other subjects and that learning gives way to answering questions and taking tests—performing.  The testing culture in the United States, which is more pervasive in math than in other subjects, is a large part of the problem…[W]hy do some educators not realize their constant testing does more than test students, which has plenty of its own problems—it also makes students think that is what math is—producing short answers to narrow questions under pressure?  It is no wonder that so many students decide mathematics is not for them.

-Jo Boaler, mathematics educator and researcher, in Mathematical Mindsets

This was how I assessed students in the past, and I decided that if I wanted to  encourage students to embrace a new type of mathematical classroom and a growth mindset towards mathematics, I would start by rethinking the type of intake experience I wanted them to have.

Over the last two years I have been experimenting with a drastically different form of intake assessment and placement, and, at least anecdotally, I am happy with the way it has changed the culture of my classroom in the first months of the year.

Instead of a multiple choice, individually scored assessment, I have students work with a partner on a series of collaborative mathematical tasks (for example, one task was to arrange a series of fractions in order from largest to smallest). I encourage students to share their thinking with each other and with me, as I walk around and talk with different pairs. By giving them peers to work with, I am able to draw out more of their thinking and to reinforce the importance of peer collaboration. Some pairs engage with each other more than others, but on the whole it has been successful. Listening to the students as they work together or explain their thinking to me gives me valuable insight into the way each student approaches math, how well they are able to explain their thinking, and what type of conceptual understanding they bring with them. I am also able to probe with follow-up questions to uncover possible misunderstandings in a way that I would not have been able to do with a traditional placement test.

I learn more about my students with this type of assessment than I did before, but the primary benefit is what students take from the experience. They have a chance to experience math in a way that promotes thinking about concepts, collaboration with peers, and communication, rather than answer getting. Before they enter their first official class, they have an idea of what to expect, and how my class might differ from more traditional forms of math instruction that they may have experienced when they were younger. I surveyed a small sample of students who took this form of assessment, and all responded that they preferred this form of assessment to a traditional paper and pencil test.

Photo by mentatdgt from Pexels

In addition to the assessment itself, I decided to make placement collaborative between myself and the student. At the end of the assessment, I explain to the group the different levels of math that I offer and what types of concepts we will be working on in each.  I then ask students to write down for me which level they think is the best fit for them. After having just had an experience doing mathematics, I find that students are quite perceptive about what type of class they need. My own assessment of the student’s level from what I saw and heard during the assessment generally matches what students choose for themselves. When it doesn’t, I am almost always recommending a higher level than the student chose, which is usually an easy conversation to have. 

I like giving students the responsibility for leveling themselves, because I think it reinforces the idea that I want them to take the lead in making decisions about their learning. Having the opportunity to choose a level AFTER they have just done some relevant mathematics and heard a description of what to expect in the different levels gives students the information to make a good decision. Since I have been doing class placements this way, I find that students have more buy-in, especially when they are in the beginning level class, and I have eliminated potential power struggles.

This form of assessment does take time, since it has to be done in small groups, and it is more difficult to report the results (I write descriptive notes of what I observe from each student, but I don’t have a numerical score or grade that can be quickly compared). It is also far more demanding of my time than simply giving a roomful of students a paper test; I have to be listening, probing, and evaluating, often making decisions on my feet of how to respond or follow up on a student’s thinking. Nevertheless, I plan to continue to use and develop this type of assessment in my program, because it is easier to establish the type of classroom culture I want at the beginning of the year. From day one, I have the chance to influence students’ perceptions of what mathematics is really all about.

If you are interested in trying something similar in your classroom or program, the assessment tasks I have been developing will be made available soon, along with a training on how to use them. Check the SABES website for offerings from the Mathematics and Adult Numeracy Curriculum and Instruction PD Center to see all our current offerings as they become available!

Melissa Braaten is an adult education instructor at Catholic Charities Haitian Multi-Services Center in Dorchester, MA. Melissa has taught ASE and pre-ASE math and reading, as well as ABE writing, computer skills, and health classes. Melissa also is a training and curriculum development specialist for the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center at TERC. She has written several articles for Math Musings, the Adult Numeracy blog.

Using Desmos: How Can It Fit in My Classroom?

by Connie Rivera

I was sold on the idea of using free Desmos Classroom Activities as soon as I tried one myself. Why?  Because I discovered math ideas I didn’t already know just by working through an activity

Using tech tools in class can’t be technology for technology’s sake. Our instruction must be focused on the mathematical understandings we want students to develop. Only then should we search for the activities that help students discover those ideas. Desmos is a great source for facilitating deeper understanding of math concepts. Below is a way I’ve helped students develop a concept, beginning with paper and ending with Desmos. 

Triangles and Rectangles

First, I started with this printable activity which is an exploration of the relationship between Triangles and Rectangles.  

Here are some ideas students develop from Triangles and Rectangles:

  • Parts of a fraction must take up the same amount of space, but they might not begin as the same shape.
  • If I can prove an area is half of a shape, the other parts must total half the area.
  • I can flip and rotate images in my mind to consider a fit.
  • For a triangle to be half the area of a rectangle it’s within, it must share at least one side (base) with the rectangle.

In a broader sense, these realizations are about noticing a pattern, generalizing an idea, and visualizing.

Desmos Classroom Activity: Exploring Triangle Area with Geoboards

Next, I used Exploring Triangle Area with Geoboards with my students. Here are some step-by-step thoughts.

Follow the link above and find the blue “Student Preview” button in the “Screens” section. In order to experience the activity the way your students would, use the  at the bottom of the screen to close the “Teacher Moves” and “Sample Responses” the first time through. In fact, try working through the activity yourself before continuing to read this article.

Screen 1: Find the Area #1 Here are two examples of how students used the sketch tool to work through their reasoning.

As you can see from the screenshots, other students’ responses would show up on the screen as well — a feature that allows students to see other students’ thinking whether or not they are on the computer at the same time! Knowing that there’s an audience encourages students think about their answers more carefully beforehand. Afterwards, they can consider the accuracy of their answers and edit their responses if need be.

Screen 2: Find the Area #2 The problem on Screen 2 reminds some students of the triangle that didn’t work in the initial Triangles and Rectangles activity, leaving them to find new approaches. In this activity, we can see how the ideas that were beginning to develop in Triangles and Rectangles are explored from a different perspective and extended using Desmos.

In the examples below, we can see in the image on the left that the first student drew a square that encompassed her triangle and found its area (12). Then she marked smaller rectangles and found the areas of the white spaces (1.5, 1.5, 4). Lastly she subtracted the total of the white spaces from the rectangle’s area (resulting in 5 square units). 

The second student (the same student from example 2 in Find the Area #1) was, to my surprise, able to carry her visualization process through to this more challenging problem. She complained there weren’t enough colors to show her work, but her explanation made it clear that she could see each piece rotating, flipping, and moving to a new spot to create a square unit.  She was surprised that everyone couldn’t see things she saw easily. 

It benefits students to hear different ways that other students think through a problem. Finding an efficient solution method doesn’t result in the same solution method for everyone; it gives everyone a chance to find something that works for them.

Here you could pause the class (a feature for teachers in the dashboard) and ask students to share answers to the question, “When you were stuck, what idea got you unstuck?” Some responses might be:

  • I can flip and rotate images in my mind to consider a fit.
  • I can find the area of a rectangle holding the triangle, and subtract out the area of the outside pieces that I’m able to find the area of. This leaves only the area of the triangle I didn’t know (and now know).

Screens 3 & 4: Create a Triangle and Class Gallery undefined Screen 3 gives students a chance to create new problems of their own to challenge their classmates (and practice what they are learning from a different perspective). Screen 4 is space where students try out their classmates’ challenges. As a bonus, they can also see how others in their class have attempted to solve the same problem.

Getting Started with Desmos

Are you interested in using this activity with your students? Here’s how to get started.

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You can “Create an Account” using the green button on the top right. Next, click the green “Create Class Code” button on the teacher page of this activity. Now you have opened a class. Click “view dashboard”. Your code and how to get there is the first thing to pop up. Share your code with friends to try it out and learn how the dashboard can help you monitor a class. 


Connie Rivera teaches numeracy skills to adults of various skill levels, including court involved youth and English Language Learners. She is also a math consultant, providing math strategies and support to programs implementing the College and Career Readiness Standards for Adult Education (CCRSAE) in Connecticut and Massachusetts. As a consultant for the SABES numeracy team, Connie facilitates trainings and guides teachers in curriculum development. Connie is a past President of the Adult Numeracy Network, the adult affiliate of the National Council of Teachers of Mathematics (NCTM), and is a LINCS national trainer for math and numeracy.

Estimation – What Good Is It?

by Donna Curry

That’s a good question: what good is estimation? After all, aren’t tests are just looking for the right answer? Let’s explore this idea, but first try this quick test-like question:

Were you able to immediately find the answer because you knew that 2 x 80 is 160? Or did you get out your calculator to figure out the answer? If the latter, you might be one of many who could use some help with number sense – something that estimation skills help build.

Most of our students (and too many of us) have been taught to immediately pick up our pencils and DO something – anything – when we have a math problem to solve. We rarely first ask, “Should the answer be larger or smaller than the numbers that I’m working with?” or  “Is this a situation where something is growing?” Too often students ‘solve’ the problem then expect us to tell them whether the answer is correct or not. And, we often wind up asking the student, “Does this answer make sense?” I think we wouldn’t have to ask them if instead we taught them to reason and then do some estimation to determine what might be a logical answer.

Let’s look at some examples of how reasoning and estimation could really help students using this statement as a starting point:

Johnetta bought a $34.99 skirt that was half price. The sales tax was 5%.

There’s a variety of questions that could be asked about this statement. Some common ones could be: How much money did she save? What was the amount of tax? What’s the total price she paid including sales tax? If a student can reason and use estimation, it doesn’t matter what the question is. She could quickly determine a ballpark answer to a variety of questions, such as the following:

In the first situation, a student who can reason and estimate could determine that half of $35 is $17.50. If she knows 10% percent tax on $17.50 is $1.75, she would also be able to figure out that half of that (5% tax) is less than a dollar. So, $17.50 plus a little less than a dollar in tax will be a little under $18.50.

In the second situation, a student could use similar reasoning: an estimate of 10% tax on $35 is $3.50. So, 5% tax would be about $1.75.

In the third situation, it is fairly obvious what the answer should be . . . but ONLY if we can reason. We don’t even have to bother doing any calculations. Clearly the answer has to be more than $35, so there is only one possibility. No calculations needed to this typical test-like question.

And, what about the ubiquitous fraction problems? How does estimation fit?

Our students can never remember to find the common denominator so 9/11 makes sense if they’ve been taught a procedure. They might not remember exactly what procedure to use when, but they know they need to follow a procedure. If instead, they are taught how to reason, they can use estimation and eliminate the answer 9/11 immediately. 2/3 is more than half. 7/8 is close to 1. So the answer has to be at least 1 but less than 2.

But, if they don’t work out the problem and instead use estimation, how do we know they understand? In estimating, it is clear that the students understand about the relative size of the fractions compared to benchmark fractions. On the other hand, what do students really know when they follow a procedure that they have been told to memorize?

Rather than needing to do a lot of calculating, if students know that π is about 3, they can estimate the answer. Of course, this assumes that they know the difference between area and circumference. Perhaps they would if they could, in their heads, simplify π and focus on the different formulae instead. While some of you might be saying that they need to be more accurate, let’s talk about how accurate is accurate enough. After all, 3.14 is only an estimate. So is 3.14159. And so is 3.14159265359.

When we’re talking about estimation, we’re not necessarily talking about the traditional rule (which we CAN and should break when needed) that says we round up or down, based on whether the number next to the place we want to round to is 5 or more. Estimation can be more flexible than that. For example, a person might choose to round $4.34 up to $5 (rather than round down to $4) to make sure he has enough cash when he gets to the checkout counter. Or perhaps we want to round one number to make it more compatible with another number; for example, to estimate the size of the fraction 18/35, one might round the 35 to 36 because mentally it’s easier to compare the relationship of 18 to 36.

Are you thinking, “Wait, my students can’t reason like that!” Well, how about we make that our responsibility – to teach them to reason and estimate. We understand the pressure to “teach to the test” but think of teaching reasoning and estimation skills as a worthy short- and long-term investment. Students with the ability to reason and estimate have an advantage over those who are just taught to memorize, whether on a test or in real life.

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Donna Curry is the Director of the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center, a project managed by the Adult Numeracy Center at TERC. She has trained teachers nationally, taught and administered ABE classes, and developed adult education curricula and resources for over 30 years.