All posts by adultnumeracy

Rethinking Placement Testing

by Melissa Braaten

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

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

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

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

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

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

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

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

Photo by mentatdgt from Pexels

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

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

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

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

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

Using Desmos: How Can It Fit in My Classroom?

by Connie Rivera

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

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

Triangles and Rectangles

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

Here are some ideas students develop from Triangles and Rectangles:

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

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

Desmos Classroom Activity: Exploring Triangle Area with Geoboards

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

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

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

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

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

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

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

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

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

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

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

Getting Started with Desmos

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

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


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

Estimation – What Good Is It?

by Donna Curry

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

Integrating Statistics in the Content Areas

This post originally appeared in the December 2018 edition of Reflect & Connect: The Reflective Practitioner, managed by our colleagues at the SABES ELA Center. This is a companion blog post to “The Case for Statistical Literacy Across Content Areas”.

by Melissa Braaten

Interested in incorporating statistics into your content classes, but not sure what this might look like in practice? 
There are ways to get students involved with collecting and interpreting their own data, as well as lots of room for making use of available data to enrich your content classes—and your students’ statistical literacy!

Giving Students Experience with the Statistical Cycle
I recently had the opportunity to work with a class in which I taught the math, ELA, science, and social studies content. I decided to invest a lot of energy early on talking about statistics and data, in hopes that this would lay a foundation for later work in social studies and science.

Frequently, math instruction in statistics focuses only on data analysis—that is, reading and creating graphs and learning how to calculate measures of central tendency, such as mean, median, and mode. While these are certainly important, the scope needs to be much broader to help students become truly “statistically literate.” 

I started the unit by exploring the statistical cycle in its entirety: posing a question, deciding on and executing a plan to collect data, analyzing the data, and finally reflecting on the results to see if they do, in fact, answer the question, and with what degree of confidence.

We undertook our own survey project in which students designed a survey to collect data that they felt would be useful for the staff of our adult education program. They debated different wordings of questions and discussed the best way to distribute and collect the surveys. 

All along, the students took a very active role. They decided they could get more responses if they offered the survey in English, Spanish, and Haitian Creole, so volunteers stepped up to translate. Other students went into classrooms one at a time to distribute, explain, and collect the surveys.

After the surveys were collected, groups of students discussed ways to tabulate the data (while I encouraged them to come to a consensus about how they would deal with irregular or unexpected responses). They then created several types of graphs and compared what impact each type of graph had on the impression the viewer had of the data. Finally, they had to decide to what extent they had answered the original question.  

In many cases, they found that their original wording did not quite give them the answers they wanted (piloting questions first can be helpful in understanding how they will be interpreted!), or that they may have needed a larger sample to feel more confident. While my students may not have changed the world with this project, they certainly got an authentic experience of what data collection and analysis is really like.

Data and Statistics in Science
Later in the year, we studied scientific methods and ran our own experiment to determine the effect of material on distance: how the type of paper an airplane is made of affects the distance that airplane will fly. The steps typically associated with scientific methods closely mimic that of the statistical cycle, and students seemed to grasp the idea fairly quickly. 

While conducting the experiment, the messiness of real data collection showed up again: some of the airplanes hit people or objects, and they had to decide what to do with those trials. They discovered that two different people timing the same event on their phones will be off by a little bit—how do they deal with that? Was it okay to open the window in the room, or would that affect the “flying conditions”? My students had a lot of fun with this project and did a great job discussing and working through the challenges that arose.

The statistical method (and the classic “controlled experiment”) are a great place to explore data and statistics, but this is not the only way data is used in the sciences. 

In an upcoming unit on the human body, I am going to teach a lesson on vaccines.  Students will hear arguments on both sides of the childhood vaccine debate and will also look at data showing the rates of smallpox in different countries over time, comparing before and after vaccines were available as well as before and after they were made mandatory. 

Looking at data collection in the context of health care will be interesting, as controlled experiments are not always possible for ethical reasons. For example, we cannot ethically assign children to not receive vaccines, since there is evidence that unvaccinated children are at a higher risk of contracting certain diseases.

Later on, I will be teaching a unit on climate and climate change, which presents another challenge for data collection: how do you run a controlled experiment on a system as large and complicated as a planet, with changes that take place over a long period of time? When we look at how climate scientists collect and interpret data, we will see them taking advantage of “natural experiments,” such as climate information before and after the Industrial Revolution, when human production of CO2 increased dramatically.

Data and Statistics in Social Studies
The study of history relies on a lot of quantitative data, but also presents its own set of challenges for statistical analysis. We can’t go back in time to collect data that no one recorded, so historians sometimes have to use other types of data as a proxy. For example, a historian might use voter registration rates to make inferences about political participation, or school enrollment numbers to make inferences about education and literacy levels.

Raw historical data can also help generate questions. In an upcoming unit on U.S. immigration, I will have students look at a graph of U.S. refugees over time, which shows not only the total numbers, but also breaks the total down by continent of origin. Looking closely at the graph should generate some questions about what was happening in the world at different points in time, such as What was going on in Europe in the 90’s that led to so many refugees to the U.S.? What happened in 1976, 2001, and 2017 to cause sudden drops in the number of refugees? Looking at data can get students curious, since data never tell the full story.

Some Techniques for Presenting Data
When presenting data or statistics that students have not generated themselves, how the data are presented can make a different in student engagement. Two techniques I use a lot are notice/wonder and graph reveals.

In notice/wonder, you give students time to make observations about a piece of data, asking “What do you notice?” I usually ask them to write down two or three observations.  There are no wrong answers. Then I ask them “What do you wonder?” and have them generate two or three questions that come to mind about the data. This can work to introduce a piece of data, or even to spark an investigation, if students want to try to answer some of their questions.

When I find an interesting graph, I often combine notice/wonder with a graph reveal. In a graph reveal, I create a slideshow that starts with a version of the graph without any labels or context.  I ask students what they notice and wonder, and what information they would like to have. Each slide adds a little more to the graph (scales, axes labels, and last of all, a title). At each stage we do a quick notice/wonder. This works well for very complex, rich data displays since students don’t have to process all the elements at once, and creates a lot of interest because they want to know if their predictions are correct. (I originally encountered this idea on the blog Teaching to the Beat of a Different Drummer, in a post titled “Trick or Treat!”)

If you are interested in learning more about how to incorporate statistical literacy into your adult education classroom, check out the SABES website for some upcoming courses on data and statistics from the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center. This is a topic that deserves a place in every content area, not just the math classroom.

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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.

The Case for Statistical Literacy Across Subject Areas

This post originally appeared in the October 2018 edition of Reflect & Connect: The Reflective Practitioner, managed by our colleagues at the SABES ELA Center.

by Melissa Braaten

In the academic sphere, statistics can sometimes seem like the odd one out. Most of us associate statistics with math, although it is far more dependent on context for meaning than other domains of math.

For example, a graph of population data with the context removed is just a meaningless squiggle.

Now consider the same graph with its intended labels.

(For even more information, view the original chart.)

In fact, statistics is a discipline of its own that uses math as a tool to help us gain a deeper understanding of other contexts that matter to us, such as science, social policy, finances, and health. Therefore, although it requires certain mathematical and quantitative reasoning skills that could be taught in a math class, it also requires integration into other content areas to develop full “statistical literacy.” For adults to be statistically literate, they need to be able to ask questions about the world, consider different contexts and how data is available in each, and to interpret results based on appropriate methodology and background knowledge.

Different fields of study have different ways of collecting and interpreting data.  Physicists may run controlled experiments in a lab; climate scientists may compare historical and present day measurements; social scientists may collect surveys. These context-specific applications of statistics can be taught in content classes to enrich students’ understanding of statistics, as well as the content matter.

For example, while teaching a unit about the U.S. Civil Rights movement, my class looked at voter registration data before and after the Voting Rights Act of 1965. The dramatic (sometimes up to tenfold) increase in the rates of black registered voters in Southern states after this legislation helps to tell part of this country’s history. In addition, it demonstrates how data can be used to support inferences about historical realities, such as the effectiveness of voter suppression against African Americans before the Voting Rights Act.

Many of our most pressing and sensitive social issues involve long-term accretions of cause, such as racism and climate change. We cannot compellingly demonstrate the existence of systemic injustice with any one incident, as other factors could have been at play. We cannot convincingly argue that any one storm or season of odd weather is evidence of climate change. However, statistics allow us to validate these realities, to a high degree of certainty, when long-term patterns of data show trends that cannot be explained by chance. While emotion and personal stories certainly play a big role in how people approach these topics, greater statistical literacy could help more Americans understand why researchers are so convinced that these issues are real and require collective action.

In adult education, we want our students to be able to engage in their personal, professional, and civic lives in a deep and meaningful way. We want students to grapple with the big issues of the day and contributing to the conversation. This is one of the reasons for the instructional shifts in the College & Career Readiness Standards for Adult Education (CCRSAE) across content areas. When it comes to ELA, the shifts emphasize the use of textual evidence in reading, writing and speaking. Our adult students need to be able to analyze textual and academic arguments, and to cite evidence from those arguments to form their own. In the “content-rich nonfiction” texts that we are using to build knowledge, arguments based on data and statistics are common, and statistical literacy is a must.

Please visit us again soon for Part 2 of this blog (Integrating Statistics in the Content Areas).
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Melissa Braaten
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 PD Center for Mathematics and Adult Numeracy at TERC. She has written several articles for Math Musings, the Adult Numeracy blog.

Making the Most of Word Problems

by Sarah Lonberg-Lew

The Argyle Sweater - tas_c100116.jpg

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

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

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

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

… and the list goes on.

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

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

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

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

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

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

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

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

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

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

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

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

… and so many more.

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

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

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

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