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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.

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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.

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.

How a Health Diagnosis Made My Math Skills All Too Relevant (And Why Math Education Is Critical for Public Health)

A huge international study of adult literacy and numeracy skills (Program for the International Assessment of Adult Competencies, known as PIACC) showed that in the U.S., 30% of adults had numeracy skills at or below level 1, which means they could only perform the most basic, single step, whole number operations.1,2  This could impact the lives of adults in many ways, but one that has recently caught my attention is the role of numeracy in health care, and specifically diabetes management. I am a math educator and professional developer, and I am familiar with how often I use math and mathematical reasoning every day, but it has never been more explicit than when I was diagnosed with gestational diabetes (GDM) in the last trimester of my pregnancy.

Pregnant_Woman_Holding_Apple

First, I needed to understand the probabilities. After the diagnosis, I was a little shocked since I think of myself as a generally healthy person and diabetes does not run in my family. How did this happen to me? I had been craving a lot of fruit to get me through the hot summer months, so part of my mind thought the extra apples must have thrown my sugar out of whack. I did a little research and learned that between 2-10% of pregnancies in the US develop gestational diabetes3, with the risk going up for mothers over 25 years old (a benchmark I passed a while ago.) So to put things in perspective, the risk of someone my age getting GDM is probably between 1 in 10 and 1 in 20. Unlikely, but not rare.

All throughout the pregnancy, I was faced with risk: 1 in 2500 chance of cystic fibrosis4; 2% chance of heart defect; less than 1% chance of Downs Syndrome.5  Almost an infinite number of other potential consequences to me and baby, although most of these risks are very small. I pictured each one in my head like a bathtub filled with marbles, with only a few bad marbles, and I had to keep drawing from the bathtub over and over. My chances for any one of these things happening was low, but the fact that after all these different draws I got one bad marble…not too hard to believe.

Soon I learned about just how much proportional reasoning I was going to have to do every day.  To manage the diabetes, I had to go on a very strict diet that called for eating a certain number of grams of carbohydrates at specific intervals six times a day.  Each “meal” had a prescribed target of carbs I was supposed to hit (not too high or too low), and if I wanted to eat more than one type of food for the next three months, that meant a lot of math.  Every meal involved planning.  I had to consider the serving size for each carbohydrate-containing food (which is pretty much everything from the bread, fruit, or dairy food groups) and adjust it to get the number of carbs I wanted so that the meal would add up to the target.

For example, in order to plan a reasonably normal lunch:

  • I first check the bread to see how many grams of carbs there are in each slice, after subtracting fiber. Two slices of wheat bread give me 23 grams of carbs.
  • Then I make a kale salad to eat all week, and add enough fruit so that if I divide it into four servings, each will have 6 grams of sugar. This amounts to most of an apple and 1/16 cup of low-sugar cranberries.
  • The soup I bought contained 11 grams of carbs for 6 ounces. I wanted to get that up to 15 grams. I mentally divided 11 grams by 3 (a little less than 4 grams, since 12 grams/3 = 4 grams), and reasoned that 2 ounces of soup contains about 4 grams of carbs, so adding two extra ounces would give me about the serving size I needed. Fortunately, I also knew that 8 ounces is one cup, so that was easy to measure.

All of this involved knowledge of how to work with ratios, estimation and mental math, knowledge of measurement units and unit conversions, and comfort with fractions: all math taught in upper elementary and middle school, and precisely the math that most low-numeracy adults are lacking.

In addition, I had to deal with elapsed time, another concept that some adults struggle with. My six meals had to be spaced 2-3 hours apart, with blood sugar readings to be taken 1 hour after each of my three main meals of the day. Depending on my agenda for the day (meetings, travel, errands, and other interruptions), I had to plan, sometimes down to 30 minute intervals, when each of these things had to happen. If I woke up late or missed a meal time (which sometimes happened), my last meal of the day might have to occur at 10 p.m. or later. A few times I had to set an alarm to wake up and eat that final snack. Skipping a meal would have put me at a deficit of carbs for the day (which could cause my blood sugar to fall too low) but eating too soon after dinner could have caused it to spike. Planning was essential.

In 90% of cases, women with GDM go back to normal after giving birth, so I like my marble jar odds on that one, although I will be at increased risk of developing diabetes type 2 later in life.  If I am lucky enough to avoid it, this will only be a short, three month foray into a mild form of the disease that an incredible 9.4% of the US population lives with.6 The percentage goes up (12.6%, or about 1 in 8) for folks with less than a high school education,7 who are also less likely to have the math (and other) skills to manage the disease. The math skills needed to manage diabetes are not advanced, but they need to be deep, flexible, and fluent to allow a person to use them on a day-to-day basis. This experience has brought home to me once again the importance of the work we do in adult numeracy for health equity and justice, and for giving people tools to improve their quality of life.

[1] Understanding Literacy and Numeracy. Centers for Disease Control and Prevention. 2016. https://www.cdc.gov/healthliteracy/learn/UnderstandingLiteracy.html
[2] PIAAC Proficiency Levels for Numeracy. National Center for Education Statistics. https://nces.ed.gov/surveys/piaac/numproficiencylevel.asp
[3] Gestational Diabetes. Centers for Disease Control and Prevention. 2017. https://www.cdc.gov/diabetes/basics/gestational.html
[4] Cystic Fibrosis: Carrier Testing. ARUP Laboratories. 2016. https://www.aruplab.com/files/resources/branding/Brochure_patient_cystic.pdf
[5] Down Syndrome: Trisomy 21. American Pregnancy Association. 2015. http://americanpregnancy.org/birth-defects/down-syndrome/
[6] New CDC report: More than 100 million Americans have diabetes or prediabetes. Centers for Disease Control and Prevention. 2017. https://www.cdc.gov/media/releases/2017/p0718-diabetes-report.html
[7] Ibid.

Some Help Hurts: Our Responsibility to Our Students

by Sarah Lonberg-Lew

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

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

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

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

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

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

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

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

Calculator Use in Adult Education

by Melissa Braaten

Calculators are something that both teachers and students seem to have strong opinions about. Some cling to them like a magical tool that will solve all of their math woes, while others blame them for the decline in mathematical fluency. I don’t find either of these attitudes helpful. A calculator can be a useful and powerful tool, but only in the hands of someone who knows how and when to use it. As a teacher, I try to be strategic in terms of when and how I allow calculators to be used in class, and I also work to engage students in making their own strategic decisions about when and how to use the calculator and other mathematical tools at their disposal.

Like all math teachers, I would like to avoid students using the calculator in place of mathematical reasoning. For example, some of my students come to me having learned in the past that they can find fractional parts by multiplying the decimal form by the whole. For example, if asked to find one fourth of 500, they would multiply .25 X 500.  I always affirm the correctness and usefulness of this method, but I also strongly encourage students to connect ¼ with the operation of dividing the whole into four equal parts, which is a far more visual and intuitive understanding of one fourth that many of my students lack. In a unit on benchmark fractions, for example, I may still provide calculators, but I will ask students to solve the problems without the use of decimals, forcing them to think about the fractions, what they mean, and how they can use division and multiplication to make sense of the fractions themselves. For many, multiplication by a decimal is not something they intuitively understand, but rather a “trick” that they have been taught.

When deciding when and how to use calculators in the classroom, I consider my class objectives.  For example, when beginning a unit in my GLE 1-4 class on division, I had some students who could use the long division algorithm quite fluently, others who wanted to learn it, but no one who really understood when it was appropriate to use division when faced with a simple scenario or word problem.  In that case, I asked students not to use either calculators or the algorithm while we explored visual models and concepts of division as equal sharing.  For that end, slowly passing out the quantities into different piles and discovering different strategies for doing so was more important than efficient calculation of an answer.  I wanted my students to develop operation sense, an understanding of what division looks like and how it relates to other operations, so they could go on to identity partitive (sharing) situations and develop methods for mental estimation of division problems.  Without these underpinnings, even students fluent with the algorithm would not be able to yield it well.

At other times, the calculator is a powerful tool in the classroom.  Many students, even in my more advanced level classes, are not completely fluent with arithmetic facts, and the calculator can be part of the scaffolding that allows them to explore more advanced concepts.  A student who is not fluent with her multiplication facts may still be able to explore concepts like area or rate, and the calculator can help her keep up with the class.  Disfluency with basic facts in adults can have many sources (lack of basic number sense, learning challenges, etc.), but most of them are not quickly remedied.  While I want to make sure to give all my students a chance to build their number sense and fluency, I don’t want someone with a learning challenge to be held up forever because he can’t remember basic facts.  He can work on his fluency with certain activities, and use the calculator or other tools at other times to help him access more advanced concepts.

An all or nothing attitude towards calculators can have one more unfortunate effect, and that is to rob adult students of the opportunity to become their own arbitrators of when calculator use is appropriate.  As a numerate adult myself, I am always weighing the pros and cons of different methods of working with numbers: Are the numbers friendly or is an estimate sufficient?  I use mental math.  Are the numbers less familiar and I need a high degree of accuracy? It might be worth grabbing the calculator. Am I working with many numbers that I could easily lose track of, or that would be too inefficient to calculate one at a time?  Sounds like a case for a spreadsheet![1]

Excel.png

This type of strategic thinking is described in the CCRSAE Standard for Mathematical Practice 5, which is summarized as “Use appropriate tools strategically.”  As a teacher, I can help students become more strategic in their use of calculators by explicitly asking them to reflect on different types of problems and which method is most appropriate.  I also need to give them opportunities in the structured classroom setting to make those determinations themselves, with some guidance from me.  For example, if I see a student using the calculator to divide or multiply by 10, we will probably have a conversation about whether she sees patterns in the results and whether this might be a calculation she could do in her head.

Like any other tool, calculators are only as helpful or harmful as we make them.  To help our students develop the math reasoning and application skills they need to be “college and career ready,” we need to use this tool strategically in the classroom and teach our students to do the same.

[1] Pencil and paper algorithms are the method I use least in everyday life – ironic that we often spend so much time on them in math class.

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Melissa BraatenMelissa 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.

Scale Diagrams as a Testing Strategy and a Teaching Opportunity

by Melissa Braaten

The end of the calendar year is the season for HSE testing, so I’ve had a lot of students looking for test prep recently.  Every practice test seems to include at least one Pythagorean Theorem question with an accompanying diagram, like the one below:

Sarah has to drive from her home to the post office, then on to the grocery store before she returns home.  She will travel on the roads shown below.  How many miles does she drive altogether?

Picture1

It did not surprise me that my students mostly didn’t recognize that this problem could be solved with the Pythagorean Theorem, since I had not yet taught them this.  I was surprised, however, at how few students attempted to estimate the missing side in order to come up with a reasonable distance for Sarah’s trip.

While not all diagrams on tests and in textbooks are drawn to scale, many are, and this can be a valuable reasoning tool.  (Often, diagrams that are not to scale are labeled “not to scale.”) I decided to try to assess students’ propensity and ability to estimate a missing side on a scale diagram. I gave 16 adult basic education students (of all reading and math levels) the problem below:

Picture2

 

Estimate the length of the side with a ?, and explain how you came up with your estimate.

 

 

Of the 16 students given this question:Picture3

  • 5 gave no answer at all, either leaving it blank or writing “I don’t know.”
  • 2 gave answers that were not reasonable and appeared to have been the result of a calculation (one student wrote “I added both sides.”
  • 4 wrote reasonable amounts, then added “I don’t know” or “Not sure” or in one case, erased the number.
  • 5 wrote reasonable amounts and gave either no explanation or a reasonable one.

I offered all of my students one hour of instruction on estimating with scale diagrams. I gave them sticky notes and showed them how to use the length of the labeled sides (marked on the sticky) to estimate the length of the unlabeled sides. Three weeks later, I gave them the same trapezoid question again.

Of the 14 students who answered the post-assessment question:

  • 2 out of the 14 (14%) did worse.
  • 7 out of the 14 (50%) showed improvement: 3 went from no answer to a reasonable one, 2 went from unreasonable answers to reasonable ones, and 2 went from reasonable answers they doubted to reasonable answers without doubting.

The results of this informal data suggest that a relatively minor intervention (1 hour of instruction) led to improvement for a significant number of students in their ability or willingness to use proportional reasoning to estimate the missing sides on scale diagrams. This could help them both with test taking as well as geometry instruction, since the ability to estimate reasonable lengths can help students make sense of geometric shapes, quantities, and formulas.

The relatively high incidence of students who were able to come up with a reasonable estimate, but felt the need to qualify that they didn’t know or weren’t sure, suggested to me that many students didn’t so much lack the ability to estimate proportionally, but were not aware that this was something they were “allowed” to do in math. A few of the students who got unreasonable estimates did so by improperly adding or multiplying, probably remembering that these are things they were often asked to do (for area and perimeter, perhaps) and did not recognize or were not bothered by the fact that the resulting numbers made no sense.

As teachers, we should let our students know that it is appropriate and encouraged to use spatial and proportional reasoning to estimate with scale diagrams. This will improve their comprehension of geometry concepts, push their proportional reasoning, possibly lead to better test taking, and hopefully, above all, reinforce the idea that mathematics is about making sense of our world.

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Melissa BraatenMelissa 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.