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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.

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.

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!

Reference:

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

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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.

In Defense of Guess-and-Check

by Sarah Lonberg-Lew

Problem solving is a lot more than reading a short story problem and performing one or two calculations. Real problem solving is what happens when you can’t see a clear path to a solution and have to be creative. There are many great problem-solving strategies: look for a pattern, examine a simpler case, draw a picture, model with an equation, etc. Also on this list is the humble guess-and-check, aka guess-check-and-revise. For some reason, when people learn fancier strategies like writing and solving equations, guess-and-check gets relegated to the status of beginner math and students are often in a hurry to leave it behind. It might be used as a last resort, but it isn’t real math – some students even see it as cheating. Guessing just isn’t as good as figuring something out. Even teachers sometimes see it as not real problem solving and pass on to students the unfortunate and discouraging message that they have just gotten lucky in solving the problem using guess-and-check but still need to learn a proper way of tackling the problem.

But guessing blindly and applying the strategy of guess-and-check are two completely different animals. When teachers and students dismiss guess-and-check as problem solving by luck, they are not seeing the sophisticated reasoning and understanding that must be brought to bear to use this approach which has as much right to be called a strategy as any other. Consider the following problem:

Tori has gotten the following scores on her last four math tests: 79, 86, 92, 88. What does she have to score on her fifth test to have an average score of 85?

Before you start writing an equation to find the answer, also consider why you might pose a problem like this to your students. What is it that you want to know about what they know? Is the important thing that they be able to abstract the definition of an average into symbols, or that they understand how averages behave and what they mean? Although it may be marginally faster to write and solve an equation (assuming one is already skilled at that), consider the reasoning and possibly even learning that can take place when you approach this with guess-and-check.

Let’s put ourselves in Tori’s shoes. She wants to get exactly an 85 average – no worse and no better. She decides to explore what will happen if she scores an 85 on the next test. This is reasonable because the scores are all fairly close to 85 already. Maybe she can hit the average by aiming right for it.

Testing out her guess, Tori adds up the five scores to get 430 and then divides by five to get an average of 86. That’s close! And Tori has just demonstrated that she knows how to find an average. Her guess got her close, but the answer was a little higher than what she was aiming for, so a second guess is needed.

Because 85 was too high, Tori decides to try a lower number. She wants to lower the average by one, so she tries lowering the guess by one (that’s reasoning!). With a guess of 84, adding up the five scores gives a sum of 429 and dividing by five gets Tori to an average of 85.8.

Huh…. that didn’t result in the change she expected, but she also may have just discovered something about the structure of the situation and of averages in general – making a small change to one number makes an even smaller change to the average.

Next, Tori decides to try a much lower number. (Maybe if she can get by with a pretty low score, she can hang out with friends instead of studying the night before the test!) This time she tries 75. She arrives at a sum of 420 and an average of 84. Oops! That pushed the average too far in the other direction.

From here I’ll leave Tori to continue on her own. She knows now that 84 was too high and 75 was too low and I have confidence that she’ll hit the solution within a few more guesses. She’ll also have not only practiced with finding averages, but also seen how changing the numbers affects the average. She’ll have both made use of the structure of averages and deepened her understanding of it. She may have even noticed that the numbers she was trying contributed different amounts to the sum and if she wanted a sum that was going to give a result of 85 when divided by 5, there was a particular sum that she should be aiming for.

This is actually more thinking and learning than will be done by a student who knows how to model the situation by writing and solving an equation. There’s nothing wrong with that approach, either, but that student hasn’t really engaged in problem solving, only performed an exercise.

In order to use the strategy of guess-and-check, students must at least understand the structure of the problem. Without that understanding, they cannot check their guesses or make improved guesses. So, when they successfully navigate a problem with this strategy, neither they nor their teachers should chalk their success up to luck. Instead, students and teachers should appreciate the hard work and reasoning that goes into solving with guess-and-check as well as the learning that can result from it.

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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.