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.


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.


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.

Ten Mathematical Facts You Won’t Believe! Number Six Will Shock You!

by Sarah Lonberg-Lew

Mathematics and its history are rich with surprising events and results. Here are ten mathematical tidbits and stories you won’t believe! (And some of them you shouldn’t believe because they aren’t all true — see if you can figure out which ones are! The answers are at the end.)

1. Pythagoras (he of the famous theorem) and his followers were so upset by the discovery of irrational numbers that they drowned the man who discovered them. The Pythagoreans believed in an orderly universe and that everything in it could be described by simple, clean, whole numbers, or ratios of whole numbers. When it was discovered that using the Pythagorean Theorem to find the hypotenuse of a right triangle whose legs were each 1 resulted in a number that could not be expressed as the ratio of two whole numbers, the Pythagoreans were scandalized and poor Hippasus was drowned in the Mediterranean Sea for the crime of bringing this unpleasant fact to light.

2. The number of hours in one day is 4!.

3. If there are 50 people in a room, the probability that two of them will have the same birthday is almost 50%.

4. Mathematicians are still looking for the last digit of pi. Supercomputers have been working on it for years and have churned out trillions of digits, but so far the last digit has yet to be found. In university math departments around the world, people have placed bets on what the last digit will be. What do you think it will be?

5. Calculus and calcium come from the same Latin word, and calcium is a component of chalk, which is often used to do calculus!

6. The largest prime number that has been discovered is 282,589,933 – 1. It has 24,862,048 digits. It was found by Patrick Laroche in 2018 by running free software on his computer. You can participate in the search for the next one. If you find one that has more than 100,000,000 digits, you could win $150,000! (A prime number is a number that has exactly two factors, one and itself.)

7. Famous nurse Florence Nightingale was also a pioneering statistician who created a new way to make data visual. She used graphs called coxcombs (kind of like circle graphs but with more information) which allowed her to show detailed data about the causes of mortality of soldiers in the Crimean war and to document the positive effects of her efforts to improve sanitation.

8. If you could count one number every second without stopping, it would take you just over five days to count to one million.

9. The title of this column was a lie. There are only nine mathematical “facts” in this column (including this one).

The Answers

1. Probably not true. This legend has been around for a long time, and there was a man called Hippasus who was a fifth-century Pythagorean, but there’s no solid evidence that he was drowned for discovering the square root of two. Other stories report that he revealed Pythagorean secrets or that someone else was drowned for publicizing irrational numbers, but no one knows for sure if any of these stories are true. (https://plato.stanford.edu/entries/pythagoreanism/)

2. True! Okay… it may not be true if you’re reading that exclamation point as punctuation at the end of the sentence, but in mathematical notation, the exclamation point indicates a factorial, a special kind of mathematical notation that means to multiply the number by all the whole numbers less than it. In this case, 4! = 4 x 3 x 2 x 1 = 24. (And there is a period at the end of that sentence… go back and look.)

3. False! In fact, the truth is even more interesting. It only takes 23 people in a room for the chances of two of them having the same birthday to be 50%. If you have 50 people in a room, the probability is even higher – there is a whopping 96.5% chance that two people in the room will have the same birthday! (https://www.scientificamerican.com/article/bring-science-home-probability-birthday-paradox/)

4. False! In 1761, pi was proven to be irrational by Johann Heinrich Lambert. That means the digits will keep going forever. No end is in sight, nor will it ever be. Luckily, only a few decimal places are necessary to make calculations that are accurate enough for most applications.

5. True! In fact, calculus, calcium, and chalk all come from the Latin word “calx” which means stone. Calculus is a diminutive form that literally means “small stone.” The words calculus, calculate, and calculator all come from the Latin word used to describe pebbles used as counters. In medical contexts, calculus can also refer to a kidney stone or gallstone or to the plaque on your teeth! 

Credit: ITworld/Phil Johnson

6. True! Anyone with a computer can download free software from the Great Internet Mersenne Prime Search (GIMPS) and let their computer do the work. Since 1996, GIMPS has found 17 large prime numbers. The Electronic Frontier Foundation awarded prizes for the first prime with one million digits and the first prime with ten million digits. The next prize will be for the first prime found with one hundred million digits. (https://www.mersenne.org/ and https://www.eff.org/awards/coop)

7. True! You can see two of her original graphs here: https://understandinguncertainty.org/coxcombs

8. False! Counting one number every second, it would take you over 11.5 days to count to one million. Can you figure out how long it would take to count to one billion?

9. True! Don’t believe everything you read!

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.

A Revolution in Math Education – Why It’s Different This Time

by Sarah Lonberg-Lew

At the Adult Numeracy Center at TERC, we are big fans of the work of Stanford Professor Dr. Jo Boaler. Dr. Boaler has taken the groundbreaking work of psychologist Carol Dweck on “mindset” and built an organization aimed at revolutionizing the way math is taught. Her organization, YouCubed.org, promotes teaching math as an open, visual, creative subject, focusing on building strong conceptual understanding over ability to reproduce procedures quickly. Most importantly, YouCubed encourages math educators to work to cultivate a “growth mindset” in their students. Simply put, a growth mindset means the awareness that the brain can grow. With hard work and conceptually rich experiences, our brains can become better at math – something many people believe is not possible for them.

Dr. Boaler ends emails to YouCube’s subscribers with the words, “Viva la revolution” because this really is a revolutionary approach to math education, and in our own little corner of the math education world, we at the Adult Numeracy Center are a part of it. Many of our adult learners have been very hurt by a traditional approach to math education that focuses on speed and ability to memorize – and conflates those qualities with intelligence. They deserve a chance to realize their own mathematical potential and to reclaim what it means to be smart.

I recently had a conversation with another teacher about the distressing idea that attempts at math education reform have been going on for decades and that they are always met with resistance from teachers and parents. Ultimately, each new reform attempt fades away, only to be replaced with the next. People still talk derisively about the “new math” of the 1960s and how it took “easy” procedures for computation and made them unnecessarily complicated with the aim of having students understand why they were doing what they were doing and not just how to do it. One strategy for achieving this was to teach students to calculate in different bases (like base 2 or base 8) in the hopes that that would help them develop a really deep understanding of numbers and operations. This meant students were doing math that looked like nonsense to people who had learned “the old way” and was met with frustration by parents who could not make sense of their children’s homework. The things people said about the new math sounded very similar to the complaints about math education flooding the internet today. (See Tom Lehrer’s satire on the new math (below) and ask yourself if today’s parents and teachers could have written it!)

So I wondered, is what we’re doing now any different? Are we repeating history with this current attempt to reform math education? Happily, I found that the answer is that what we are doing now is different and new. There are two pieces that seem to me to be very different from previous attempts, and they give me hope.

One is that our focus now is not on showing students why they are doing what they are doing. That is only marginally more effective than teaching mnemonics for procedures. We now know that it is important for students to construct for themselves which of their strategies work and why. By beginning with an idea of the meaning of an operation, like subtraction, students have the opportunity to construct many strategies that they understand and retain because those strategies belong to them. Whether the traditional procedure we all learned in school is among those strategies depends on the student. Nobody has to be forced into being able to explain the idea of “borrowing” – they will either make sense of it or use other strategies. The important thing is that they will know when to subtract in the real world and be able to do it accurately.

The other very important difference between the “old” new math and the “new” math revolution is the idea of growth mindset and the real neuroscience that supports it. Cultivating a growth mindset in our students doesn’t just mean saying “Don’t give up! You can do it! I believe in you!” These are important messages, but more important is the idea of neuroplasticity – the ability our brains have to change how they work through effort and practice. This is a major paradigm shift and research has shown that when people develop a growth mindset, they approach their learning differently and become much more successful at learning math (or anything else!). Knowing that our brains are capable of growth empowers us to create that growth.

The current revolution may feel on the surface like old failed attempts at math education reform. Constructing understanding through visuals and flexible thinking can make the work look more complicated than traditional procedures on paper, but this is not the “new math” redux. This time we are empowering students to be their own sense-makers with the knowledge that they can grow their brains to think in new and powerful ways.

Viva la revolution!


Boaler, J. (2013). Ability and Mathematics: the mindset revolution that is reshaping educationFORUM, 55, 1, 143-152.


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.

Risk, Probability, and Parenting

by Melissa Braaten

(opinions in this blog are those of the author and not of the SABES Mathematics and Adult Numeracy Curriculum and Instruction PD Center)

PinkStock Photos, D. Sharon Pruitt [CC BY 2.0 (https://creativecommons.org/licenses/by/2.0)%5D

Parenting is hard. I just became a new parent a few months ago. I knew it was going to be hard before the baby even came the first time I picked up an instruction manual and learned how to strap a baby in a car seat. It seemed like every page had bold warnings in all caps about various ways you could do it wrong, each of which could result in “SERIOUS BODILY INJURY OR DEATH.” My husband had a similar experience assembling the crib.

“That was stressful,” he sighed when it was finally together.  “Did you know there are about a hundred ways a baby can die in a crib?”

Just today, I had a conversation with my mom that has become sort of common. We were talking about the baby and his impending teething.

“Do you have any of those teething rings that you freeze?” my mom asked.

“No, you’re not supposed to use those any more. They’re too hard for the baby’s gums.  The FDA is also trying to prevent people from using medication for teething, because it can be dangerous and cause a blood disease.”

“Oh. Well, you survived.” How many times have I heard this phrase in the last couple of months? It seems to sum up the feeling of bewilderment whenever I talk to someone from a previous generation about all of the things I have been advised to avoid (blankets, stuffed animals, baby powder, belly sleeping…) which might result in SERIOUS BODILY INJURY OR DEATH to baby. Well, you survived. And they are right. We did.

Keeping this baby alive and maintaining my sanity has had me thinking a lot about risk and the ways that we make decisions in the face of it. We all know that some risk is unavoidable, but we don’t always like to admit it. Risk enters the realm of randomness and uncertainty. Not all people — even mathematicians — are comfortable here. 

Generally, our evaluation of risk is based on two factors: the likelihood of an outcome, and how serious that outcome would be.

Something that has a relatively high likelihood of occurring and has potentially serious consequences are considered high risk. These are things that tend to be a little easier for people to agree on, and a little easier to legislate: for example, most states have laws about wearing seat belts, helmets, not driving while drunk, etc. There is plenty of evidence of serious consequences, and they occur often enough that we are willing to take measures to avoid them.

Outcomes with mild consequences are generally trivial, and we don’t spend a lot of mental energy worrying about them. It is in the other domain that things get interesting. Risks that involve unlikely, but serious outcomes seem to be far more subjective and controversial. Emotions play a big role. For example, the average American is far more likely to die choking on food than in a terrorist attack, and yet only one of these things has a huge place in our national consciousness (and budget).[1]

Hence the difficulty with parenting: even if it has a very low probability of occurring, SERIOUS BODILY INJURY OR DEATH to a new baby is terrifying, and means that many things probably take up more room in our consciousness than they really should. Does that mean I think hospitals and pediatricians should stop trying to prevent SIDS (Sudden Infant Death Syndrome)? Of course not. From 1990-2016, the rate of SIDS dropped from .13% to .04%;[2] something rare became rarer. Since there were about 4 million live births in 2016,[3] that decrease in SIDS means that potentially around 3,700 babies were saved in that year alone.

Nevertheless, the chance that my baby will die from SIDS is, thankfully, very low. After a while, I did stop staring at the monitor to see if he was still breathing. You have to sleep, and eat, and live your life, and drive to work, and somehow tolerate the fact that bad things could happen today—but they probably won’t. I won’t give my baby teething gel, but I don’t want to be too hard on those who do, either. After all, we did survive.

[1] https://www.businessinsider.com/death-risk-statistics-terrorism-disease-accidents-2017-1  The exact statistics on odds of dying from terrorism vary widely in different sources, mostly because it is hard to agree on exactly what qualifies as a death from terrorism.  But all the numbers I saw were still far, far less likely than choking.

[2] https://www.cdc.gov/sids/data.htm

[3] https://www.cdc.gov/nchs/fastats/births.htm

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.