All posts by adultnumeracy

The Misbegotten Trifle: What Cooking Fails Have in Common with Math Learning

by Sarah Lonberg-Lew

In a Thanksgiving episode of the popular nineties sitcom Friends, the character Rachel attempts to make a traditional English trifle. The layers include: ladyfingers, jam, custard, raspberries, beef sautéed with peas and onions, bananas, and whipped cream. As she lists them, the other characters’ reactions go from appreciative to incredulous to disgusted, and the mention of the beef layer gets a big laugh from the studio audience. Upon investigating, her friends discover that the pages of the magazine from which she got the recipe were stuck together and she had in fact made half a trifle and half a shepherd’s pie.

The scenario gets such a big laugh because most of us know it is absurd to have beef sautéed with peas and onions in the middle of a dessert, but Rachel clearly did not. If she had, she might have questioned the recipe or even dared to modify it on her own. As it is, it turns out she did find it a bit odd, but she had more faith in the recipe than in her own intuitions. (Click here to see the “trifle scene.” Warning: The clip contains some PG dialog.)

Sadly, this is exactly the situation many of our students are in. Lacking conceptual understanding or a desire to seek it, they rely almost exclusively on procedures, believing that faithfully executing steps, practicing and memorizing them, is the only way to reach their math goals. They don’t even pause when they achieve an absurd result because they have no concept of what kinds of answers make sense.

The reasons they believe so strongly in steps are not insignificant. For one thing, many have never known that there can be any more to math than following steps. For another, it really feels like it’s working in the short term. By teaching students steps to memorize and then giving them practice problems that require nothing more than applying them, both the teacher and the students can feel successful in the moment. However, when it comes to long term retention, knowing when an answer is reasonable, or the ability to solve non-routine problems or apply their learning outside of class, the time and energy invested in memorizing steps fails to serve our students in any useful way. In fact it is counter-productive because they have missed the chance to use their time to develop genuine understanding and they end up having to “learn” the same material again and again. (See “What Community College Students Understand about Mathematics” by Stigler et. al. for an in depth explanation of research supporting this.)

I recently had an opportunity to observe a student struggling through finding half of 544. He knew the steps and diligently used long division to divide by 2, even going through the procedure twice to be sure. Both times he arrived at an answer of 322 (having made the same mistake twice) and said he felt confident about the answer because he had followed the steps (he really said that!). We then talked a little about what half means and I asked him if he could find a way to check whether 322 was really half of 544. He multiplied it by 2 and was completely stymied when it did not come out to 544. His steps had failed him and he had nowhere else to go.

Instead of reviewing the steps or combing through them for his mistake, I lent some context to the question, asking how he would manage if we had $544 to share equally between us. Immediately his energy and attitude changed – I could see in his face that he understood the task and had an idea of how to approach it. He began talking about what kinds of bills the money could be in, and I kept pace with his thoughts by drawing pictures of the bills as he talked. When he saw them in front of him, he drew loops to divide them into two groups, even deciding to go to the bank to make change when the bills couldn’t be evenly divided. When I asked him this time what reason he had to feel confident in his answer, he pointed to the two groups, showing me that they were both the same and that he had “given out” all the money. Using his conceptual understanding of division, he arrived at the correct answer with confidence and was proud of his hard work. I followed up with another “finding half” problem and then asked him to see if he could use his strategy to divide a number into three equal parts which he did with relative ease. By this time, the strategy was becoming a little more abstract – just numbers in boxes instead of pictures of bills– and eventually even the boxes fell away. It was especially fascinating to watch him because his approach evolved into something that I could not have taught him. It was his own invention born of his own understanding.

It is not just our students who suffer from the misconception that math is all about steps. Most of us have also been socialized to believe that the key to success in math is being able to follow directions. I remember my tenth grade math lesson on completing the square (a strategy for solving quadratic equations). My teacher drilled us over and over again on the sequence of steps and said, “Keep your pencil moving! Don’t think!” I was overwhelmed and lost and despite many other successes in math in the years following, I believed for a long time that I would never understand completing the square. I just couldn’t remember all those steps. When it came time to teach it in my own classroom (during my stint in K-12) I did only marginally better with my students than my teacher had done with me, giving lip service to explaining the algebraic steps and trying to be more gentle and compassionate with them as they struggled to memorize them. It was even longer before I made sense of the method for myself and saw that it was in no way beyond my ability.

As teachers, we need to stay strong when our students beg us to just give them the steps – when they say, “I know if you just show me how to do it, I’ll understand.” I have had students who have insisted that memorizing steps is “how they learn best,” but even those students who are good at memorizing benefit more from understanding. For all of us there is a limit to how much we can remember – but not to how much we can understand.

We have to take the long view and know that even getting a perfect score on a worksheet is not the same as understanding. Taking the time to develop conceptual understanding can feel slow and our students are usually in a hurry to move forward with their academic and career goals. But learning math takes time just like any other kind of meaningful learning. For our students, the difference between spending their time learning deeply and spending it memorizing steps may very well be the difference between progress toward their goals and one more failure in one more ABE class.

The misbegotten trifle came about because of a blind reliance on following steps. The consequences of such an approach for our students may be far worse than an unpalatable dessert!


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 adult education programs in both Gloucester and Danvers, MA. Sarah’s work with the SABES numeracy team includes facilitating trainings and assisting programs with curriculum development. She is also an actively involved member of the Adult Numeracy Network.

 

 
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Teaching Math to ESOL Learners: A Reflection

For this installment we’re featuring a special guest blog by Sister Margaret Lanen of Notre Dame Education Center in South Boston (http://www.ndecboston.org)

When I was asked to teach a math class once a week this year, I was most willing. I taught algebra, geometry and advanced math in high school for many years and thought it would be easy to teach basic math to new English language learners.  Little did I think that the language used in math would be such a stumbling block to ESOL students! I thought it would not be very difficult to teach math vocabulary and processes. What a surprise!

The students were nervous about their lack of math skills when they began the class. I started with the basic operations of addition, subtraction, multiplication and division. It didn’t take long before I realized how many words like minus, sum, times, and divided by were new vocabulary to many of the students. Trying to solve a division problem during the first weeks (especially with some of the Haitian students, who use a different division procedure than we use in the U.S.) made me realize that teaching math to ESOL students was not going to be so easy. Students would tell me they knew the answer, but they couldn’t explain it in English. It is challenging to get the right vocabulary and language to express mathematical thinking in another language.

Luckily, our program director had agreed to invite folks from the SABES PD Center for Mathematics and Adult Numeracy to come to our site for math professional development. When I participated in the workshops, I realized how different my experience as a high school math teacher was from teaching math to adult language students. The trainer introduced a whole new way of teaching math which was hands-on. This was brand new to me.

She began with the staff working together on basic mathematical concepts. The emphasis was always on how we got the answer. Staff worked in groups placing fractions on a specially created number line. We had many activities which included using scissors, graph paper, string, magic markers, rulers, etc. We were learning by doing.

How did this affect my teaching of math? Well, now in every class I always include a group activity to give the students a hands-on experience of the skill I am teaching. It is a challenge for me to plan the class. For example, I knew that the concept of pi would be difficult to get across. With string, rulers, cans of different sizes, and scissors the students measured the circumference of the can and the length across the center. The three groups found that the ratio of the two measurements were the same. This experience gave the students a visual measure of pi as a number a little bigger than 3.

I do feel a sense of community as the students work together to help one another. I’ve learned I need to be patient and give the students time to get started with an activity and decide what they are going to do. Group work often illustrates that there are different ways of getting an answer.

One big fear of some of the students when they started the class was that they would fail math. The best part of teaching this math class is that the students do not need to pass a test to succeed. They come to see the many ways math can help in their daily life. I bring newspaper ads and they determine how to take 25% off the cost of an item at the store. We learn to read graphs such as the history of immigration in the United States. They use graph paper to decide how much fencing they need for a garden or what is the best plot size for the vegetables. Many math concepts develop from these activities.

When I hear the conversation of students in their groups helping one another to understand a math concept, I feel excited. My sense is that the students are learning math and feeling more confident and I am learning, too.

What Does It Mean for Math to Be Relevant to Adult Learners?

by Melissa Braaten

When I am trying to prioritize my curriculum and to build buy-in from my students, I am always trying to think about how to make the math relevant to them. But what does relevance mean?  Is it relevant for someone to learn math that they might use someday in a career they are considering? Are the specific algebraic procedures what they will use in the future, or the deeper algebraic reasoning and problem solving that they develop in the process? Is a particular math concept relevant if a student will not apply it in practice for years?

Some of the challenges faced by making math relevant involve time gaps between learning and application. In addition, research has shown[1] that the math used in careers is often specific to that career, and methods are “idiosyncratic” and learned on the job. In this article, I will describe different types of relevance that I aim for as an adult basic education math teacher when designing and prioritizing my curriculum, and which types of so-called “relevance” seem to bolster the argument that “school math” is something that real adults never use.

Immediate relevance to adult and family survival

When I think about immediate relevance, I am thinking about math that can be applied to contexts that adults are engaged in already, and in which they already have a felt need for more mathematical understanding. A few examples of this would be math that involves:

  • shopping and consumer awareness (math skills: understanding percentages as used in sales and taxes, proportional reasoning to determine the best deal)
  • financial literacy (math skills: basic operations for budgeting, decimals and percentages to understand banking, credit, savings, and loans)
  • home and appliance use and maintenance (math skills: reading gauges, understanding measurement systems, types, and conversions to purchase and calibrate products for the home, car, etc.)

The list could go on, and will also vary a bit from person to person depending on the contexts in which they are currently operating.

This type of relevance has (in my experience) the greatest effect on student buy-in. In addition, increased numeracy skills in these areas can help families materially improve their lives. When planning a unit, I try to include at least one application which will fall into the category of immediate relevance for most of my students.

Relevance for expanding horizons

This next category will vary quite a bit from student to student. There are many applications for math (especially the math content taught at levels GLE 1-8) than can be applied immediately by most adults to improve or enrich their lives, but which may not already be a felt need because the person has not previously been engaged in that context (possibly because of lack of math skills!). An example for many of my students would be home improvement. All kinds of simple home improvement and repairs require skills of estimation, measurement, numerical reasoning, fractions and decimals, or geometric concepts of area, perimeter, and volume. I have found that many of my adult students, who often lack some of these math skills, also avoid engaging in this context. Having gotten by so far, they may not come to me with a felt need for these skills, but I am hoping the experience may actually expand the types of activities they feel comfortable exploring or engaging in.

Here are some other examples of contexts which tend to be expansions for many of my students (they can immediately apply these skills, but may not have previously thought about doing so):

  • Banking and personal finance, such as comparing and opening accounts, building credit, saving for retirement, or filing taxes (math skills: numerical reasoning, positive and negative numbers, deep understanding of decimals and percentages, linear and non-linear growth)
  • Making choices based on risk or probabilistic reasoning, in contexts such as medical decisions, investment choices, or even understanding weather predictions (math skills: probability concepts, ratios, data and sampling)
  • Evaluating statistical and data based claims, which are often present in the news and necessary to follow policy arguments

These are just some examples. Often, the skills that tend to fall under immediate relevance are those that pertain to day to day survival, while the skills listed above may not be necessary to survive, but are certainly important for adults to fully and independently thrive and participate in society. At my current program, many of my students are legally homeless or dealing with significant economic instability, so the applications of math to expand their horizons might not feel as immediately important to them, but I want to give them the opportunity to begin to see what else is out there, and to know that they too have the ability to reason through and participate in these contexts. This is a way in which adult numeracy education can invest in long-term change in families and communities.

Timely relevance to educational next steps

Many math skills are also important for academic reasons, because they are built upon in later units or because the skills are important for a different discipline (data analysis in science, for example). Relevance to educational next steps, be it the next unit or the next level or to achieve a certain certification (high school equivalency or Accuplacer level), is an important reason to teach math concepts and should not be overlooked. Credentials matter to the job market and our students know it.

While I plan all of my units with future units, test preparedness, and interdisciplinary applications in mind, I do this with two important caveats. The first is that while this type of relevance is necessary, in adult basic education I believe it is also not sufficient to justify teaching a certain topic. There are so many topics in the College and Career Readiness Standards for Adult Education (CCRSAE) levels A-C with both immediate relevance and expanding relevance as well, that to leave out the first two does a disservice to our students. Academics for most adult students is not an end in itself but a means to a better life. Including the first two types of relevance along with preparing students for the next educational requirement allows them to start to use math to better their life now, instead of seeing the payoff always in the distant future.

The other caveat for me is that these connections must be timely. When I am planning levels and units and sequences, I would like students to apply their math skills to other academic contexts within a six-month window, whenever possible. Without timely use and retrieval, those math skills become something forgotten in one more notebook, gathering dust.

“Math I never use in real life”

When math is considered one of the most important skills for financial and career advancement in our society, how is it that so many adults (including me!) still feel that they never use much of the math they learned outside of school? How can we teach to avoid making math so “irrelevant”?

I don’t have a simple answer, but a couple of suggestions:

  1. Intentionally weaving in applications of math with applications of the first two types of relevance helps adults see that numeracy is something they can benefit from now and in the future.
  2. Advanced math with specific career applications should be taught later in the educational sequence, when students are already committed to a specific career path and are going to use the math in that career in a timelier fashion.
  3. Prioritizing the development of mathematical practices (such as the Standards for Mathematical Practice in the CCRS), number and operation sense, and proportional and algebraic reasoning over specific procedures.[2] These words describe thinking patterns, foundational concepts, and strategies for thinking that contribute to numeracy at every level and are relevant to any career context. Knowledge of specific procedures degrades over time if they are not used frequently, but can be easily relearned or figured out if solid mathematical reasoning is already in place.

It’s not easy to make decisions about what to teach when there is so much math and so little time. Since most adults do not learn math for its own sake but as a means to achieve other life and career goals, it is important that we teach them math that is both useful and relevant. In the process, we may even affect some long-term change in attitudes about what mathematics is all about.

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[1] See National Institute for Literacy, (2010) Algebraic Thinking in Adult Education, Washington, DC. p 3. Available at https://lincs.ed.gov/publications/pdf/algebra_paper_2010V.pdf
[2] “Adults often say they have never used the algebra they learned in school.  That may be true for the rote aspects of manipulating symbols, but they likely are using the mathematical reasoning and problem-solving aspects of algebra unconsciously.” From National Institute for Literacy (2010), ibid.

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Melissa BraatenMelissa Braaten is an adult education instructor at St. Mary’s Center for Women and Children 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.

What’s Wrong with Right Answers

by Sarah Lonberg-Lew

NOTE: This blog is a companion piece to an article Sarah wrote earlier this year called “What’s Right with Wrong Answers?”

As much as students seem to hate word problems, there is no way around the fact that they are a necessary part of the curriculum. Setting up and solving word problems is a vehicle for learning to analyze information and reason logically. It is an essential skill for taking high school equivalency test and for success in college and real world situations requiring math —this in spite of the fact that the contexts for these problems are often silly and occasionally absurd. The funny thing is that students who groan when I bring up word problems also suddenly appear to care very deeply about how many more plums Joy has than Jonathan (even though they have never met Joy or Jonathan and plums are out of season). When I pose such a problem in class, I hear a chorus of answers, some of them right and some of them wrong. What seems to be most important to the students is knowing as soon as possible who came up with the right answer and what it is. This concerns me because what really matters is the process of problem solving and not the answer itself. However, I’ve seen students stop thinking once they know the answer. They write it down on their papers and move on to the next problem even if they have no idea how it was obtained.

This makes sense considering their histories with math education. In a traditional math classroom, students can get the idea that their purpose is to find the right answer, present it to the teacher, and thereby accumulate stickers or check pluses or “A”s. They are just as happy to get the answer from the teacher or a classmate as to rely on their own reasoning and intuition. The former is usually easier.

Because of this, focusing on the correct answer in math class seems counterproductive to me. Learning doesn’t come from knowing the answer; it comes from finding it. If I shut down the discussion of a problem as soon as I hear the right answer, I also shut down learning for all those students who didn’t find it themselves — and possibly even for those students who did. For example, a student who comes up with a right answer because she remembers from high school that similar-looking problems are to be tackled using a certain procedure hasn’t really learned anything from the exercise, either.

Sometimes, in order to take the focus off the right answer I leave an exercise unfinished. Maybe we’ve worked hard on a problem and arrived at the conclusion that we have to multiply 207 by 48 to get the answer. At this point I feel that we have done the important work of problem solving. We’ve figured out what computation will get us the answer, and the answer itself is not that important. Even though there would be some satisfaction in finding the actual number, the tension of leaving it unfound can help to make the point that the value is in the process instead of the solution. Even when we do math in authentic contexts, the process is more important than the answer. Knowing how much a fictional driver will pay to lease a car is not useful to a student. Knowing how to figure out the cost of leasing a car, on the other hand, is very valuable.

One sad consequence of a classroom culture that values right answers above all else is that students can start to become like “Clever Hans”. Clever Hans, the student of a mathematics teacher in early twentieth century Berlin, could answer all kinds of questions correctly. Clever Hans was a horse. He tapped out the answers to questions with his hooves and amazed people with his mathematical skills and knowledge. It wasn’t a hoax. Clever Hans could answer questions correctly even when people besides his trainer asked them. However, his conceptual understanding was eventually found to be severely lacking. He wasn’t able to get the answer if the questioners themselves didn’t know it. That’s because Clever Hans wasn’t reasoning at all. He was reading the faces of the people asking the questions and they were communicating to him (whether they wanted to or not) when he had arrived at the correct answer. Clever Hans had learned something very interesting in developing the ability to read those cues, but he hadn’t actually learned what he appeared to have learned. As teachers, we also can be fooled into thinking our students have learned what we were trying to teach when they come up with right answers and we don’t look beyond them.

The researchers who worked with Clever Hans found it impossible to control the micro-expressions that the horse was reading. The only way they could keep from communicating the right answer to him was to put up a screen so that Hans could not see their faces. That is an impractical solution for the classroom and our students are cleverer than Hans. We may not be able to control communicating when we have heard the right answer, but we can work to create a classroom culture where we focus more on the process than the outcome. One thing we can do is make sure we always ask for the reasoning, whether the answer is right or wrong. We can train ourselves to say “why?” instead of “yes” or “no”. My students want to hear from me, the authority, whether their answer is right or wrong, and it frustrates them when I refuse to answer. But when I push them to explain their reasoning, they usually figure out for themselves whether they had it right or wrong. I hope that by being consistent in asking for reasoning, I will help my students learn that their math education is not about coming up with the number that will satisfy the teacher, but about thinking deeply and feeling proud of their hard work.

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sarahlonberg-lewSarah Lonberg-Lew has been teaching and tutoring math in one form or another since she was in the tenth grade. 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 adult education programs in both Gloucester and Danvers, Massachusetts. Sarah’s work with the SABES numeracy team includes facilitating trainings and assisting programs with curriculum development. She is the Massachusetts state representative for the Adult Numeracy Network.

Changing How Problems Are Presented

by Connie Rivera

I am an avid numeracy blog reader. As I read about the experiences and ideas of others, I find I’m challenged to think deeply about decisions I am making while teaching in my own classroom. The blogs I read inspire me with new ideas on a regular basis.

Lately, I’ve noticed ideas from different sources that are all ways to change how problems are presented.  All of these teaching strategies produce a slightly different way of investigating a problem, even if it’s a basic word problem. Because each strategy comes at the problem from a different angle, students learn something more profound than if they had all the information laid out at once (which doesn’t happen in real life!). Instead, students are forced to look deeper into the problem or to develop new problem solving skills. These skills are useful for solving problems that appear in other academic disciplines as well as in real life, when learners need to research possible options and choose the best course of action.

The ideas I’ve come across have me connecting to so many other things that I know about teaching adult numeracy. When I read about them, I think about Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them. All of these ideas increase opportunities for my students to make sense of problems before solving them. I am also reminded of Universal Design for Learning and how considering multiple means of representing the problems make the learning more accessible for my multi-level classes, which include English Language Learners.

All improvements start with making just one change to our instruction. Try some of the example problems in the linked resources below to get a feel for what’s possible. Can you think of a problem you already use in class that you can adapt using one or more of these strategies?

Strategy 1: Makeovers

Text book questions provide so much information that they are no longer a “problem” (in the sense of Mathematical Practice 1) for students. Makeovers consist of blocking out most of the information, usually given in a textbook, until students make sense of what’s there and ask for the information they need.

 

Strategy 2: Numberless Word Problems and Graphs

When all the information is provided to you, it’s easy to glance at the numbers, and perhaps key words, and jump to conclusions. Numberless problems force you to make sense of the problem before solving it. With this strategy, ask students for a solution pathway based on the information without the distracting numbers. Only after they’ve made sense of it do you plug the numbers in and the students solve the problem.

 

Strategy 3: Notice and Wonder

Notice and Wonder is an approach to problem solving that allows you to make observations about a situation, and then ask, research, and answer a mathematical question about which you are curious. I find this approach especially useful with my English Language Learners.

 

Strategy 4: Problem Solving Scenarios

These scenarios from NCTM’s The Math Forum and Dan Meyer lend themselves to using a Notice and Wonder approach to ask a question.  Present a problem scenario without a question, then let students ask the questions.

 

Strategy 5: Reversing the Question

Beginning with the end is an idea for showing students a visual or calculations and asking, “What questions gave these answers?” or, “What are the questions these calculations find out?”

 

Strategy 6: “Open Middle” problems

Open Middle problems appear simple and procedural but actually involve deeper thinking. They have:

  • A “closed beginning” – start with the same initial problem.
  • A “closed end” – end with the same answer.
  • An “open middle” – multiple ways to approach and solve the problem.

You can create this style problem yourself, but among the great collection of procedural problems, you will find some in word problem style such as the one in the link.

 

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connie riveraConnie 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 (CCR) Standards for Adult Education. As a consultant for the SABES numeracy team, Connie facilitates trainings and guides teachers in curriculum development. Connie is President of the Adult Numeracy Network and a LINCS national trainer for math and numeracy.

What’s Right with Wrong Answers

by Sarah Lonberg-Lew

I used to feel awkward when my students came up with wrong answers. I didn’t want to embarrass or discourage them, so I felt that the kindest thing to do was to say, as briefly as possible, “That’s not quite right – try again.” I even avoided using the word “wrong”. However, in my rush to save my students from potential bad feelings, I missed an opportunity to find what was good and valuable in their reasoning. And when students reason, there is always something valuable in their reasoning, even if it is wrong.

Recently a spirited and unexpected discussion took place in my classroom when I posed a question that was looking for an even number and to which the answer was 50. About half of the students spoke up indignantly, saying that there was something wrong with the question because 50 was not an even number (and I felt very gratified that my students felt comfortable challenging me on that). Instead of shutting down the discussion by explaining why I was right, I immediately backed down from my assertion that 50 was even to give space for them to explore the question. I asked them for their reasoning and it was interesting and thoughtful, though ultimately incorrect.

The main line of thinking seemed to be that there was only one real number in 50, and that was 5 – the zero didn’t count. Since 5 is odd, 50 must be odd as well. When pressed to dig deeper into this idea, the students said that even numbers had to end with an even number and that zero was not an even number. This got us thinking about what it means for a number to be even (way off topic, but very worth the detour, in my opinion). We arrived at quite a conundrum when we came up with two meanings for an even number that seemed to contradict each other. One student decided that 50 was indeed even because she counted by 2s and hit 50. Others felt sure that the last digit had to be divisible by two and this was not the case with zero because there was nothing to divide. This felt a bit thorny, but we were eventually able to agree that zero could be fairly divided into two parts because each of the two parts would be the same size.

At this point one of the students engaged in my favorite type of incorrect reasoning. She said, “If 50 is even because it can be divided by 2, and 50 divided by 2 is 25, doesn’t that mean 25 is even also?” I love this reasoning because it isn’t really incorrect at all, it’s just incomplete. This student was stretching, looking for patterns, making a conjecture. She was being mathematically adventurous. She just stopped before reaching the essential step of investigating her conjecture. I encouraged her to explore her idea and she quickly figured out for herself that it was not correct, but I also jumped on the opportunity to call out what an excellent question it was. By stretching and conjecturing, she was showing the first part of sophisticated mathematical thinking. The second part, asking whether the conjecture was true in this case and whether it will be true in every case, is the part she hadn’t mastered yet. But, what better time to talk about the nature of mathematical reasoning than when student has already come part of the way just by following her intuition?

Throughout the CCRSAE we find calls for students to apply and extend understanding, and this is actually something students seem to do naturally, although not always correctly. When students present faulty reasoning, the fault is often that they have made it only to the conjecture phase and stopped there. By calling out the positive parts of students’ reasoning and taking the opportunity to model the rest of the process, we can boost their confidence and also help them move further along their path to using the third standard for mathematical practice: Construct viable arguments and critique the reasoning of others.

At the end of the discussion, those students who had advocated for the wrong answer or advanced incorrect reasoning felt energized and like they had learned something valuable. I even noticed one of them practicing labeling numbers as even or odd in her notebook days later.

When a wrong answer that a student has worked hard for is simply swept under the rug, it can really hurt. It teaches that there was nothing valuable in the effort and that student may very well decide that it is not worth it to try so hard if she thinks her chances of getting the problem right might be just as good if she just guesses. And she wouldn’t feel so bad about making a wrong guess because she hadn’t really been trying in the first place.

By finding and celebrating what is positive in students’ reasoning, by celebrating even those links that show logical but incorrect thinking, I hope I am making the terrain of reasoning and risk-taking less intimidating to my students. When they can feel good about working hard on a problem, even when they get it wrong, they will be less afraid to try.

These days I have a new approach. I strive to create a classroom culture where we are not ashamed of wrong answers because they mean that we have taken risks and thought deeply, and because they provided us with opportunities to learn. Now I am not afraid to use the word “wrong.” I come right out and say, “That was excellent reasoning! Unfortunately you arrived at a wrong answer, but there were a lot of parts that made sense. Let’s see if we can figure out where it went wrong.”

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sarahlonberg-lewSarah Lonberg-Lew has been teaching and tutoring math in one form or another since she was in the tenth grade. 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 adult education programs in both Gloucester and Danvers, Massachusetts. Sarah’s work with the SABES numeracy team includes facilitating trainings and assisting programs with curriculum development. She is the Massachusetts state representative for the Adult Numeracy Network.

Differentiating Instruction: Some Quick Adjustments for Math

by Donna Curry

Every math class, no matter how you try to ensure some homogeneity, is a mix of levels. Some students are good at decimals, others only have operations with whole numbers down pat, while some students can reason and estimate well but struggle with procedures.

So, what can you do? Nope, the answer is not to completely individualize instruction where students work independently on decontextualized skills. Instead, you still want to encourage them to work on more challenging situations, working together to persevere, and trying different strategies as they productively struggle.

But, since there is a fine line between productive struggle and frustration, especially for our adult learners, you might need to sometimes ease up a bit on a task you have given students. Here are a few quick strategies you can try to make the task a little less daunting:

Simplify Numbers. If the situation involves fractions or decimals, you may suggest that students use whole numbers instead. Or, if they are trying to work a problem that involves a more ‘messy’ large number (such as 456,780), have them use a less intimidating number such as 456,000 or even 500,000, depending on the comfort level of the student.

Limit Data/Information. Think before you ask students to choose a coupon out of a flyer or look for a specific bit of information in a chart or on a menu. Is there so much information on the page that it is overwhelming? If so, simply make a copy of the page and white out some of the information. You want to still include enough bits of data that the student will need to make some choices, but the choices could be from three or four bits of information rather than an entire page of options.

Begin with Familiar Contexts. Many of you already use this strategy when students struggle. For example, when you ask, “What is half of 7?” and students give a blank look, you probably come back with, “What is half of 7 dollars?” Students light up and often readily respond. Build on those more familiar contexts, such as money, to get students started while remembering that, ultimately, students need to tackle situations that are unfamiliar to them.

Limit Steps. If the task requires that students find information, decide on the percent off, determine the exact cost, and figure out how much change they should receive, think about how overwhelming this might feel to someone who has fragile math understanding. Think about what aspects of the task you want to be sure the student gains experience in doing and then remove some of the other steps in the task. This does not mean to always minimalize the task; rather you need to be sensitive to how many steps the task requires and whether it would benefit a student if she had fewer steps to undertake.

The point in differentiating instruction is NOT to always give students easier (or harder) tasks, but rather to adjust instruction to nudge them a bit beyond what they are capable of now without seriously overwhelming them. Not challenging students is as harmful as overwhelming them, so you want to tweak tasks as needed to have students struggle but ultimately succeed.

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Donna Curry is the Director of the SABES PD Center for Mathematics and Adult Numeracy, a project managed by the staff of the Adult Numeracy Center at TERC. She has trained teachers nationally, and has taught and administered ABE classes for over 30 years.

Utilizing Correct Mathematical Language

by Pam Meader

The Career and College Readiness Standards have shifted the way we teach mathematics to our students.  One change is the importance of using correct mathematical language for both our students and our teachers. In Mathematical Practice 1, students need to make sense of the mathematical language in order to problem solve. With Mathematical Practice 3, students need math language to discuss their ideas or critique their classmates’ ideas. With Mathematical Practice 6, students need to use the mathematical language in writing as well as in their heads.

Last week, the listserves were lit up with a new research article Supporting Clear and Concise Mathematical Language: Instead of That, Say This by researchers Elizabeth M. Hughes, Sarah R. Powell, and Elizabeth A. Stevens.1 These researchers feel that language plays an important role in learning mathematics. They suggest that teachers may not interpret mathematics as a second or third language for students when in fact, all students are mathematical language learners. That resonated with me as I think of all our ELL/ESOL teachers and their approach to learning vocabulary.  In their classes, vocabulary isn’t a list of words to memorize but rather a list of words to experience and understand through role play, etc.  The same is true in a math class. New vocabulary should be addressed as the learning experience happens through using manipulatives, in mathematical inquiries, or through discussion. What the article revealed was that we should be using the correct math terminology when the situation appears in the classroom. The authors gave many suggestions of when teachers might “say that, not this”. I will discuss a few of them by relating their ideas to an adult education class.

One suggestion was to stress the difference between number and digit when teaching place value. For example, 235 is a number, while the 2 is a digit in the hundreds place. This could be done by using base ten blocks where the student represents the number 235 using 2 hundred blocks, 3 ten rods and 5 unit blocks. They could see the visual representation of the number 235. They could also see the value of the digit 2 as 200, the digit 3 as 30, and the digit 5 as 5 ones. You could also represent the number 235 on a number line to show how close it is to 200 or 300. The point is that the words number and digit can have some conceptual understanding if utilizing manipulatives and number lines.

Other suggestions centered on fractions.  Too often we refer to the parts of a fraction as “the top number” and “the bottom number”. This suggests that the fraction is composed of two different numbers instead of digits that together are one number. This results in confusion for students. I can remember one of my students claiming that 3/4 and 4/5 were equal. When I asked her to explain why, she said that because 3 + 1 = 4 and 4 + 1 = 5, it was the same increase of 1 to arrive at 4/5, therefore the fractions were equal. Clearly the student was looking at each part separately and not thinking that 3/4 itself was a fractional number. The researchers suggest that we refer to the parts of the fraction by their correct names, numerator and denominator. I can remember hesitating to use these words with a low level math students for fear I would sound too technical. The point is again to introduce these terms more conceptually. Use fraction strips so students can develop understanding of what the numerator and denominator represent, and illustrate fraction as a number by locating it on a number line.

While both of these researchers based their work on elementary students, I find their suggestions very applicable to the adult education classroom as well. One detail they stressed was that most tests (such as the HiSET in adult ed), utilize more sophisticated math language. Students exposed to this language conceptually in a problem-based classroom develop a deeper conceptual understanding and perform better on standardized tests.

These are just two of many examples the researchers shared. To get the full report go to: http://bit.ly/2dUVkLH and request a copy.

1 Hughes, E.M., Powell, S. R., & Stevens, E.A. (2016). Supporting Clear and Concise Mathematical Language: Instead of That, Say This. TEACHING Exceptional Children, Vol. 49(1), 7–17.

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SAMSUNGPam Meader, a former high school math teacher, has taught math in adult education for over 25 years. She is a math consultant for the SABES PD Center for Mathematics and Adult Numeracy professional development initiative for Massachusetts. Most recently, she helped co-develop Adults Reaching Algebra Readiness (AR)2 with Donna Curry. She is a national trainer for LINCS and ANI (Adult Numeracy Instruction). Pam enjoys sharing techniques for teaching math conceptually from Basic Math through Algebra and has co-authored the Hands On Math series for Walch Publishing in Portland, Maine.

If Only I Could Remember! (and how Coherence can help)

by Donna Curry

From my elementary school history class, I remember, “In 1493, Columbus sailed the big blue sea.”  From my science class, I memorized the colors of the spectrum because I remember ROY B GIV (Red, Orange, Yellow, Blue, Green, Indigo, Violet). And, from my math classes, I can still recall PEMDAS – Please Excuse My Dear Aunt Sally… or was that PFMNS (Please Forgive My Niece Sally)?  Let’s see: (6 + 4)/5 + 3(2) – 1. Parentheses [6 + 4], then Fractions [10/5], then Multiplication [3(2)], then I do my Negatives [6 – 1] and last I do my Sums [2 + 5] for an answer of 7.

OK, maybe it isn’t PFMNS, but that makes just as much sense as PEMDAS (also known as Pink Elephants Destroy Mice And Snails!). Students who come from other countries may have been exposed to similar math mnemonics like BEDMAS or BIDMAS or even BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction). Heck, creative types have even come up with rap songs to remember the order of operations!

These strategies are ways to remember the order in which to do calculations:  First, tackle what’s in parentheses, then any exponents, then multiplication and division, and lastly, addition and subtraction. But, if I use the mnemonic, and especially if I’ve only learned what the letters stand for and not the understanding behind those words, I’m likely to solve a problem like 7 – 3 + 4 incorrectly.

Why? Well, with PEMDAS I’d probably get an answer of 0, because I see that addition has to come before subtraction. After all, that’s the rule—Aunt comes before Sally! But that’s all I know about it. As you can see, PEMDAS may have its place, but not when used as an isolated memory tool without any conceptual understanding behind it.

Coherence Across the Levels

I started thinking about all the tricks and mnemonics we teach our students when I read about coherence, one of the key shifts in the College and Career Readiness Standards for Adult Education (CCRSAE). Coherence is about making math make sense. Put another way: “Mathematics is not a list of disconnected tricks or mnemonics. It is an elegant subject in which powerful knowledge results from reasoning with a small number of principles such as place value and properties of operations. The standards define progressions of learning that leverage these principles as they build knowledge over the grades.” [See www.achievethecore.org]

Wow, “a small number of principles”? That’s not what most of our students think about math. . . and definitely not what I thought about it as I was learning all my rules and procedures, tricks and mnemonics.

The CCRSAE are actually quite coherent (although this is sometimes a bit challenging to visualize). However, you can get a better sense of what coherence looks like if you download our handy overview which was specifically designed to show that coherence. And since we’ve been talking about order of operations, which involves operations and properties of numbers, let’s see what coherence looks like across levels.

LEVEL A
Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) (1.OA.3)
LEVEL B
Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. (3.NBT.2)

What if we started showing our students early on that they could use parentheses to organize what they are doing when they decompose? Take for example the problem:

31 – 15 = (30 + 1) – 15

Using the associative property of addition, and we can show 30 + 1 = (20 + 10) + 1 = 20 + 11 to explain how borrowing works.  Another approach could be to create an easy-to-subtract number in place of the 15. For example, by adding 5 to the 15 we get 20, but to keep the subtraction situation the same (the same difference between the two amounts), we have to also add 5 to 31 and get 36, and see that 36 – 20 = 16. But how does this work? We are using the additive inverse property (a – a = 0) this way:  31 + (5 – 5) – 15 = (31 + 5) – (5 + 15) or 36 – 20, an easy subtraction problem. While the addition procedure of (-5) + (-15) = (-20) may recall algebra, it is intuitive to explain that subtracting 5 and then subtracting 15 is the same as subtracting 20 all at once.

Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16. (Distributive property.) (3.OA.5)

The distributive property is a way to solidly begin to teach the order of operations. Discovering which operations work with the commutative and the associative property is also part of the initial understanding of the order of operations. If students were exposed early on to the distributive property, maybe they wouldn’t have to learn about Dear Aunt Sally since they would realize that they could either do what’s in the parentheses first, or not:     7(3 + 4) = 7(7)   or   7(3) + 7(4)

If either of these processes is correct, and if we taught our students this important principle, why would we tell our students that they have to do what’s in the parentheses first?

LEVEL C
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 x (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. (6.EE.3; p. 66)

If we want our students to be successful at Level C (much less Levels D and E!) we have to build on what they should have been exposed to in Levels A and B.

Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. (4.NF.3b; p. 62)

If we built on the properties with whole numbers, students could more readily deal with fractions. For example, this is similar to the earlier whole number example where numbers were decomposed and then regrouped based on the associative property:

Example:          3  1/3 – 1  2/3

Let’s look at alternative equivalent representations of the 3  1/3 in the problem:

3 1/3 = (3/3 + 3/3 + 3/3) + 1/3   or   (3/3 + 3/3) + (3/3 + 1/3)   or   (3/3 + 3/3) + 4/3

If we choose to represent the 3  1/3 as  2  4/3, the new problem (2  4/3 – 1  2/3) is now much easier to solve, and I did it using number properties rather than a seemingly new strategy for borrowing with fractions. Here is another way to use equivalent fractions to add or subtract mixed numbers.  Since 3  1/3 = (3/3 + 3/3 + 3/3) + 1/3, we know an equivalent representation is 10/3. Similarly, 1  2/3 = (3/3 + 2/3) = 5/3. So, 10/3 – 5/3 = 5/3 or 1  2/3. In this method, there is no borrowing at all.

LEVEL D
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. (7.NS.2a; p. 71)

As you can see, each progressive level builds on previous understanding of some core properties of number and operations. The CCRSAE help guide us in what to teach when and even provide some examples of how to teach those standards. I challenge you to closely review the CCRSAE yourself. Create your own ‘progressions’ to help you see what grounding students should have before you try to add on a new skill to their often weak foundation. Think of ways to begin to infuse your typical teaching with core basic principles. Ask students to explain why an algorithm works so maybe they will actually remember the rule or procedure later.

A version of this article first appeared in The Math Practitioner (V18.3. Fall 2013).

Editor’s Note: You can read more on one teacher’s view of all the rules we teach – and what happens when they get to develop here.

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Donna Curry is the Director of the SABES PD Center for Mathematics and Adult Numeracy, a project managed by the staff of the Adult Numeracy Center at TERC. She has trained teachers nationally, and has taught and administered ABE classes for over 30 years.

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5 Great Reasons to Teach Number Lines

by Melissa Braaten

I am an adult education teacher, therefore I never have enough time with my students. I want my students to be able to problem solve a wide range of mathematical problems, but I also want to ensure that they leave me with a deep conceptual understanding of the material they have studied. To this end, I find that I am always trying to prioritize my math curriculum to figure out which core concepts or big ideas will make the most of our precious and limited time. Recently, I have discovered that one of my favorites is teaching number lines.

I decided to cover a short unit on number lines with whole numbers with my level A (GLE 1-4) class this past spring. I figured it would be a quick topic, but as we started exploring number lines together, the material proved to be mathematically rich and highly relevant.  We ended up spending an entire eight-week unit on number lines with whole numbers, and I still think it is one of the most valuable units I have ever taught.

Here are some of the reasons that time spent teaching number lines is time well spent:

1. Number lines are a great modeling tool for visual learners. Part of our unit had us exploring different operations on the number line, and many students had some real “aha” moments. For example, number lines help to demonstrate some different ways of thinking about subtraction.

Donna was born in 1974 and Carlos was born in 1992.  How much older is Donna?
(Subtraction as comparison rather than “take away”)

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When we drew multiplication on the number line as repeated jumps of a certain size, some students were amazed to see how they could visualize closely related division facts:

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6 “jumps” of 3 is 18. (6 x 3 = 18)

How many 3’s fit in 18? (18 ÷ 3 = 6)

If you share $18 with 6 people, how much does each person get? (18 ÷ 6 = 3)

2. Number lines provide excellent fluency practice with all operations.  Students created and filled in number lines following the rule of “equal spaces (jumps) have equal values”, which required them to compute constantly in order to follow this rule. Different sized intervals allowed for easy differentiation (some students worked with intervals of 2, 5, and 10, while others challenged themselves with jumps of 250, .50, etc.).  Students also discovered that they could divide (that most dreaded operation) to break a long interval up evenly, but that their division could be easily checked with other operations (Am I still adding up by 5s?).

3. Number lines prepare students to think about signed numbers and signed number operations.  By the end of our number lines unit, my very early level students were ready to start conceptualizing negative numbers, with their understanding of the number line as a visual model. We looked at what would happen when you kept taking equal jumps below 0, and their familiarity with the left to right  or up and down orientation helped them understand why negative numbers “appear” to grow backwards (why -1 is greater than -100, for example). We also used number lines to look at the “difference” between high and low temperatures and why this difference is so large when we have numbers on opposite sides of zero. When I connected this to the (half-remembered) rule for “switching the sign” when subtracting a negative number, one student remarked, “Wow, that actually makes sense now.”

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4. Number lines are an important preparation for coordinate graphing and scale. One student attended both levels of math concurrently in the same cycle, going from number lines in level A to a unit on linear algebra in level B. Teaching the two in tandem, I came to appreciate how important an understanding of number lines and equal increments is to becoming fluent with coordinate graphing and axes. While my more advanced students struggled constantly with correctly labeling their intervals, the student who had been working with me on number lines connected them easily to her coordinate axes. Her greater fluency with number lines meant she was able to devote more energy to thinking about which scale would be appropriate for the task at hand.

5. Number lines have numerous and immediate applications to adult life. Oven dials.    Thermometers. Time lines. Analog clocks. The further we went in our unit, the more applications began to appear. I also came to appreciate (once again) how powerful adult learning can be. During our unit, I purchased an outdoor thermometer and put it in the window of the classroom. I quickly discovered that many students were not comfortable reading the thermometer if the dial was not pointing to a labeled mark; they were not sure how to figure out, for example, what the mark halfway between 40 and 60 would stand for. (Some voted 45 and others voted 50. It led to an illuminating discussion.) I thought about how long it took before I learned to use an electric drill: not because it was too hard, but because it becomes habitual to avoid things we don’t know how to do. Now I pick up a drill whenever I can, and every time it makes me feel a little proud.

There are many other (mathematical and practical) reasons to work number lines into your curriculum.  Visualization (in this case, using number lines) is a big idea that can be traced through the CCRSAE beginning with whole number operations (at level B) and touching topics including fractions, decimals, data, measurement, coordinate geometry, rational numbers, through the inclusion of irrational number approximations (level D). The idea is not to teach every possible application of number lines in one unit, but instead to weave them into appropriate units at the appropriate level for your students.  Big ideas in mathematics are ideas that keep coming back, illustrating the overall COHERENCE of mathematics as a field of knowledge.


Melissa Braaten

Melissa Braaten is an adult education instructor at St. Mary’s Center for Women and Children 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.

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