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.

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)
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?

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.

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.




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.





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”)


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:

Blog 19_Picture 2

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.”

Blog 19_Picture 3

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.



Math Journals

by Pam Meader

About 10 years ago, various documents such as the NCTM Standards and recommendations of the National Research Council were urging more math communications. I decided that I was going to incorporate some writing into my math classes. I wasn’t sure how to start, so I tried a variety of approaches in all my math classes. However, new research has recently been published that lends clarity to the type of writing that occurs in math classes and its benefits.

With a grant from the National Science Foundation, a task force was convened to investigate mathematical writing in elementary school and the implications for math stakeholders. The researchers began their work in October 2015 and released their findings in a report entitled Types of and purposes for elementary mathematical writing: Taskforce recommendations.”[1] While this research related to elementary school students, I found many of the suggestions relevant to what I was trying to do in my adult education math classes. What the task force discovered was that mathematical writing was important, but that its purposes were not so clearly defined. I had discovered that I, too, had many purposes for incorporating writing, so I could relate to that finding. Some of my purposes included determining if the student understood a particular math concept, how the student was feeling about math, what discoveries or frustrations they were encountering, and where they saw the math they were learning connecting to their daily lives.

Through their research, the task force found two primary goals of introducing writing in the math classroom. One was to have students reason mathematically, and the other was to have students communicate ideas with mathematics. To illustrate reasoning and to be able to communicate mathematically, the task force suggested four types of writing: Exploratory, Informative/Explanatory, Argumentative, and Mathematically Creative. Of the four types of writing, I found most of my students’ writing fell into the first 3 recommendations (exploratory, informative/explanatory, and argumentative). Exploratory writing is when a student tries to make sense of a problem, situation, or their own idea. In my classes, that type of writing occurred more often in non-journal exploratory activities where I would ask my students to explain what happened or make conjectures. Most of my students’ journal writing illustrated informative/explanatory writing where students describe or explain a mathematical concept such as explaining what area and perimeter have in common and how they are different. Argumentative writing is when a student can construct an argument or critique an argument. That type of writing clearly illustrates the Common Core’s Mathematical Practice 3, Construct viable arguments and critique the reasoning of others. I found I used argumentative more with test questions where the student had to determine if a particular assumption was correct and then defend or critique the situation. An example would be: “All squares are rectangles. Do you agree or disagree with this statement?”

The last type of writing, mathematically creative, is defined by the task force as the ability to think creatively and document their mathematical ideas that extend beyond the expected or intended outcome of a task, situation, or problem.” Part of thinking creatively is to think flexibly and to make connections within and beyond mathematics. At first, I thought my journals would be too structured to allow for this, but that wasn’t the case. Within the journals tasks, my algebra students were able to connect and make abstractions to the math concepts they were writing about. For example, in explaining how the number line was constructed to include both positive and negative values, one student wrote that the number line was like playing the piano where the right hand were the positive values and the left hand were the negative values.

The task force also considered that 1) writing will develop as a student moves through various levels; 2) that the audience can influence student’s mathematical writing; and 3) that mathematical writing can take on many forms. I can attest to the first and third suggestions with my journey in using math journals. My lower level students’ writing might be to just fill in simple statements like Today I struggled with ________, while my higher level algebra classes had to respond to a journal prompt, give an example, discuss where the math connected to their lives or other disciplines, and provide a reflection of their learning. The third suggestion of writing taking on many forms I also utilized with journals that included pictures or diagrams to explain their reasoning.

The one area the task force didn’t address—but which I have found very important with adult learners—are their feelings about math and how they persevered through their struggles. The reflections in my students’ journals told me much about them: when the light came on, when it went off, what strategies they were now using to solve problems, and where they were feeling more or less confident about mathematics. Many adults come to our classes so afraid of math and failure that the reflection piece can be therapeutic for them. I had one student confess to me that it took her 10 years to have the courage to take a math class but through journaling she became very excited about math and the connections she made. To this day, she emails me new math websites she has encountered.

In closing, I would urge you to try incorporating math journals as part of your teaching routine. The quote I leave you with is from a 19 year old student I had in my class a few years back. She hated math and did not want to be in a math classroom, but writing in a journal helped her with math. She said, “I think the most successful part of this course, for me, are the journals. Numbers and formulas tend to float about rather haphazardly in my mind. The act of binding mathematical concepts and numerical processes with words brings me to a far greater level of understanding and clarity. Once I have written about something it becomes very easy to visualize and helps me to see connections between various areas of mathematics. Writing makes it whole for me, so to speak.”

[1] Casa, T.M., Firmender, J.M., Cahill, J., Cardetti, F., Choppin, J.M., Cohen, J., Zawodniak, R. (2016). Types of and purposes for elementary mathematical writing: Taskforce recommendations. Retrieved from http://mathwriting.education.uconn.edu


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.



The Value of Sorts, Matches, and Clipping to a Number Line

by Connie Rivera

In my class, I regularly ask students to:

  • Match to each other cut outs of different representations such as: a graph, table, or equation matched to a situation; an array matched to an expression; or a drawing matched to a fraction, decimal or percent.
  • Sort into categories shapes, graphs, or different visual and symbolic representations of math concepts.
  • Place on a number line, whole numbers, fractions, decimals, and percents as well as representations of these as groups of objects, time, money, and parts of a whole.

I encourage communication between students during these categorization activities by asking them to state their reasoning out loud to a partner or their group before adding or moving any of the pieces. Stepping back, watching, and listening allows me to quickly assess students’ level of understanding as well as spot their misconceptions. After I’ve given students a chance to correct their own and each others’ mistakes, I might ask a few questions:

  • Are there any pairs that could be combined because they are equivalent?
  • Are you happy with… the distance between these? Are there any you’d like to rearrange?

Here’s the hard part for me – if they still don’t catch their mistakes, I don’t correct them. Most of these activities have pieces that can be taped together and posted to look at later. The number line for example, lends itself to being added to over time as their understanding refines with new lessons. As Jo Boaler describes in What’s Math Got To Do With It?, brain research tells us that students have two opportunities to “learn from a mistake.” The first learning opportunity occurs when a student makes the mistake, whether or not they are aware of it. The second learning opportunity comes from catching that mistake themselves later on.

These experiences are for students at different levels, working together to learn. A description of a solution can appear to make sense when we hear about it, but turning to explain it to someone else is how we know we really understand. These activities allow our learning to start with a more hands-on, concrete and pictorial exploration before moving to more abstract representations. Through the exploration, students make connections between the visual, symbolic, and quantitative representations a concept. Sorts, matches, and clipping representations to a number line enable all students to use these connections to make sense of math.


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.

Creating Open Questions

by Melissa Braaten

“Open questions” are getting a lot of attention these days in math education, and for good reason. Unlike more traditional “closed” questions, which have one right answer, open questions allow for many possible correct answers, and/or many possible ways of approaching the problem. Open questions often invite deeper mathematical reasoning than closed questions, more closely replicate how the math is used in real life, and can be an excellent strategy for differentiating instruction in an adult education classroom.

Let’s look at a few common types of open questions.

Type 1: Create a mathematical argument


Which of the following decimals is different from the others and why?

.30                              .21                              .67                              .05

When I presented this to my students, they came up with a variety of ways to argue that one of the decimals didn’t fit:

.30 doesn’t fit because it is the only number without hundredths

.67 doesn’t fit because it is the only number greater than one-half.

.05 doesn’t fit because it is the only number with no tenths.

.05 doesn’t fit because it is the only number with a placeholder zero.

There are probably others. This activity provided some great review for my students to consider different fraction and decimal concepts they had learned, and gave them an opportunity to argue a mathematical point.


Type 2: Planning with parameters


Alexis spends 32 hours a week at work, 4 hours at the gym, and 6.5 at her volunteer placement. Create a possible 1-week schedule for Alexis.

These types of open questions do a great job of replicating real life applications of mathematics, because adults frequently encounter situations in which they have to create some sort of plan (a budget, a schedule, a nutritional plan, an exercise routine, a floor plan, etc.) within certain parameters. There are many possible ways to satisfy the requirement, but they require mathematical reasoning and critical thinking because not all plans will work, and some will be more realistic than others.


Type 3: Two (or more) missing variables

This type is easy to create from textbook style word problems by removing a piece of information.

For example, a traditional style word problem about ratios might be as follows:

Bread is on sale, 4 loaves for $8.  How much would it cost to buy 7 loaves?

To open it up, you can remove one of the numbers given, and insert blanks for all the missing quantities.


Bread is on sale, 4 loaves for $8.  You can buy _______ loaves for ________.


Bread is on sale, 4 loaves for ________.  You can buy 7 loaves for ________.


Bread is on sale, ________ loaves for $8.  You can buy 7 loaves for _________.

Now there are two variables, so the students have to find a pair of numbers that work.  (If a whole class does this, you can put the pairs in a chart, and suddenly you have an algebraic in-out table.)

I have come to love giving these as warm ups, because I can challenge fast students to try some less familiar numbers, and everyone gets to contribute.  It also really draws students’ attention to the relationships between the numbers in the problem, rather than sending them looking for the correct set of steps to use.  In many cases, it also lends itself to an algebraic exploration of the problem (in-out tables can become equations and even graphs!)

There are many more types of open questions out there, and I encourage you to browse through some that teachers have written (www.openmiddle.com has some great ones) and to practice writing your own!


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.

The “Right” Formula?

by Connie Rivera

I was a participant in a training recently where we were given the tools to “discover” one of the not-as-common formulas provided on the Formula Sheet for the Hi-SET test. After some exploration experiences, we were asked to write an equation that showed the relationship between the variables that could be used to find the area of any such shape. Many of the other teachers already knew the formula and were able to link together the ideas to explain how to get the most common representation of the formula (the one on the Formula Sheet). Because I did not know the formula, the formula I created represented what I saw as I worked through patterns and constants in an example problem. My version made me feel accomplished and like I had a deeper understanding. It made sense to me and I could explain how to get there, making a composite of more common shapes. However, when the others at my table saw such an unusual equation, the general reaction was, “That can’t be right!” My tablemates spent some time re-arranging my equation and learned that it did actually work. It sparked a conversation about the “right formula” and there was a sense at the table that I hadn’t really gotten the answer right.

As someone who struggled through math the first time around, and who was told to follow a formula that I couldn’t “see,” I now think, “Why?” What do we accomplish by insisting that all students do things the same way? I think that permission to see math flexibly is part of what freed me from some of the anxiety that math still brings all these years into studying it. If one student sees the perimeter of a rectangle as l + l + w + w, another (L+W) + (L+W), and someone else sees 2x +2y, as long as they can explain the meaning, recognize it in other forms, and know how to use the formula to find perimeter or a side when the perimeter is known, does it really matter? When a student creates a formula, they “own” it and connect to it. It has meaning. To be honest, I still can’t “see” the formula on the Hi-SET Formula Sheet and so I still can’t tell you that formula without looking it up. It doesn’t make sense to me that way. However, I can still re-create the formula I created and find the area of that shape even though the session was over a month ago. So, not only is this the way it made sense to me, but it was something I retained because I connected to it.

When I attended college, I planned my schedule to avoid math. It wasn’t until I was a GED Prep teacher that I started facing it again. Because I knew it was my weakness, I attended a math PD session and it ended up changing my life. It opened my eyes to the way it can feel to receive math instruction. I hope my engineer father never finds this blog, but I can tell you that being told tricks to memorize a formula that didn’t make sense to me (like “Cornbread are round, pi R squared!”) feels very different than being guided through a sequence of math experiences and being asked generative questions to draw a formula or a solution out of me. I do not feel the same sense of accomplishment and deepened understanding when using traditional formulas like those on the Formula Sheet to “plug and chug” – substituting numbers from a word problem into the formula without thinking.

Have you ever noticed that many students who can use A = π to tell you the area of a circle can’t use the area to tell you the radius or diameter? It’s not just that they don’t know the steps to manipulate a formula and isolate a variable. It’s more than that. They don’t “see” the meaning of each variable and the relationship between those parts. I wonder what would happen if we started by leading students to discover pi as a relationship between two variables rather than thinking of it as just a number that never ends?


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 in Connecticut and Massachusetts. As a consultant for the SABES numeracy team, Connie facilitates trainings and guides teachers in curriculum development. Connie is President-elect of the Adult Numeracy Network and a LINCS national trainer for math and numeracy.

Why Do We Need Adult Education Classes?

by Donna Curry

The other day I was online, looking for an explanation about why a particular math procedure works. I knew how to do the problem, but I wanted to know why it worked. It took me quite a while to discover the answer, even though I tried some of my favorite go-to sites right away.

In the process of looking for an answer, I realized that there are MANY sites that I could go to for help if I didn’t remember the procedure, or algorithm. There are plenty of YouTube videos, each explaining the same procedure using different casts of characters. And, of course, there’s Khan Academy where you can find out how to do just about any procedure you want. So, if there are so many websites readily available to teach “math”, why do we still need adult education math classes?

That’s a question that needs answering, especially when federal and state dollars are used to fund programs. I think the answer is that those of us in adult education try to do more than just teach procedures that students have already been “taught” ad nauseam – sometimes from second grade on. What we can do that Khan doesn’t is to help students develop the conceptual understanding that underlies all those rules that students keep forgetting. I think we can also provide concrete examples from their own personal lives of math in action so that there is a link between “school math” and real-life math.

For example, think about the skill of dividing fractions by fractions. I know how to do that – it’s a piece of cake! Simply invert the second fraction and then multiply across. If I forget, I can go to myriad websites where I can be readily reminded of the procedure. But, if it’s such a simple thing, why do students still struggle?

Here’s where the role of an adult education instructor is critical. In a math classroom, a teacher can engage students in conversations about what it means to divide fractions – in fact, what it means to divide at all! (We make assumptions that students know what the four operations mean, but many of our students only know how to follow procedures related to those operations.) A teacher can begin a discussion about what it means to divide by using whole numbers (the conceptual understanding for division is the same for fractions as it is for whole numbers). She may pose the question, “What are we asking when divide 12 by 4?” (We are asking how many clumps of 4 are in 12; or we could also be asking how many will everyone get if we share 12 things among 4 people.)

If we understand that 12 ÷ 4 is asking how many 4s there are in 12, then we can ask the same thing with fractions and whole numbers. 2 ÷ ¼ is simply asking how many fourths are there in 2. Obviously, since ¼ is much smaller than 1, there are several of them in 2. There are four in 1 whole, so there are eight in two wholes. The same understanding holds true for fractions divided by fractions: ½ ÷ ¼ is still asking how many fourths are in ½. If the teacher has previously done a lot of work helping her students understand benchmark fractions, students know right off that there are two fourths in ½. Do the procedure yourself and see if that reasoning isn’t correct. And, she can actually help her students understand this idea by using visual representations.

half _d_fourths

In fact, the teacher can do for students what the ubiquitous procedural explanations can’t do – she can help them learn to reason. So, to continue from the example above, if a student is faced with this problem: ⅔ ÷ ½, he can reason about the situation (thanks to the teacher’s modeling of how to do so many, many times in each class). The student can think, “Hmmm, this is asking how many ½s there are in ⅔… I know that ⅔ is larger than ½, but smaller than 1. So, I know that I could get at least one ½ out of ⅔ of something and I’d have some left over. So, I know it’s more than 1 but less than 2.”


Then, and only after there has been some reasoning about the situation, the student can use the procedure to find the exact answer: ⅔ ÷ ½ = or 1 ⅓. The student will know that this is a reasonable answer because he has already figured out that the answer had to be “one and some more”.

These are the kinds of conversations that can go on in an adult education classroom that you will not find on the web. It’s these conversations, this style of teaching and learning, that make adult education teachers invaluable.




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.

Commit to Changing Just One Thing (every semester)

by Connie Rivera

A participant in a recent training told us that she felt overwhelmed by being given so much new information during the course and not knowing where to start. It made me remember the presentation and activity where I was first exposed to math as “understanding and making connections” rather than “memorizing and following the rules”. It hit me like a powerful wind that blew me off course. It made me look at a whole new, colorful view of the world that I’d never noticed. Once I realized that there was another way, I was overwhelmed thinking about everything I needed to change. I felt like now that I knew a better way, I should be using it in every classroom moment. But, as with other resolutions, it wasn’t realistic of me to think I could change all at once.

I went back to class and taught that same activity to my students and watched them have the same ah-ha moment I had experienced as a participant in that session. I loved the feeling of watching my students make connections so much that I included that activity in my instruction every semester for years. Meanwhile, there were still many other topics I was teaching with the passive approaches I had been shown as a student because it was the only way I knew. But I still wanted to find out all I could about this “understanding and making connections” way. I participated in more professional development. I began to think about “Why?” when I was planning my lessons, so I looked far and wide to find resources to answer that question. Slowly, I collected more great ideas and incorporated those. Each time I did, it crowded out those passive lessons and the workbooks of empty procedures and tricks I used to rely on.

Whether the change worth making is weight loss, the couch to a half marathon, or learning to teach conceptually, complex changes don’t happen all at once. On any of those journeys, the important thing is t18 sizeo START. Commit to changing one thing, and then do it again every semester.

What will your One Thing (every semester) be? Here are some ideas:

  • Ask students, “How do you know?” when they give an answer.
  • Make sure all students participate and that when they do, they say a second sentence. Complex reasoning isn’t shared with others in just one sentence.
  • Teach a new, powerful activity that exposes misconceptions and makes connections.
  • Teach one new concept for understanding.
  • Spend time getting comfortable using a new tool (square inch tiles, area models, pattern blocks, snap cubes, etc.) and try an activity with them in your class.
  • Find the math in one every-day activity and begin from there rather than from a workbook.


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 in Connecticut and Massachusetts. As a consultant for the SABES numeracy team, Connie facilitates trainings and guides teachers in curriculum development. Connie is President-elect of the Adult Numeracy Network and a LINCS national trainer for math and numeracy.

Bedtime Math Helps Kids (and maybe adults, too!)

by Martha Merson

Do you read to your kids or grandkids at bedtime? If so, you may be interested in a fun new twist that some families have incorporated  into their nightly routines — one that may have academic benefits.

At least two studies have found that kids whose parents engaged them in math-related story  and puzzle activities performed better on tests. Both studies focused on families using Bedtime Math (www.bedtimemath.org) resources. Bedtime Math, a New Jersey nonprofit organization, offers free kits for starting math clubs for kids, as well as daily math puzzles within a story context via its website and tablet app. The math stories deal with all sorts of themes and topics, ranging from snowmobiles, to chocolate, to eight-legged animals, and feature kid-pleasing sound effects. Stories are brief and are leveled for children of different ages.

In an initial 2011 study conducted by Bedtime Math itself, children participating in a New Jersey Boys and Girls Club six-week summer program did one of their math problems daily. The kids were tested before and after the six-week program. The study found that most of them didn’t exhibit the typical “summer slide” in their math scores (kids usually slide back about two or three months in their math scores after the summer off from school). Rather, 72% of the kids in the program did better.

This study was followed by a larger, formal 2015 study by the University of Chicago**, in which 587 Chicago metro area first graders in both public and private schools and their parents used Bedtime Math over the course of one school year. This study found that “doing math at bedtime had a significant effect: Children who used the app two or more times per week outpaced peers whose family rarely used it.” One of the study’s authors, Dr. Sian Beilock, commented, “It’s like they’ve had 3 months more of math instruction.”

The study, originally published in the journal Science, suggests that children who used the app with their parent(s) may have experienced growth in math skills partially because of the influence the app had on their parents’ feelings about math. Prior research indicates that a children’s feelings toward math are colored by their parents’ own view. If a parent has negative feelings about doing math, the child may feel likewise. There is evidence from this study that through regular Bedtime Math activities, many parents became less anxious about math over time. The researchers hypothesize that this is why the greatest math gains were observed in students whose parents described themselves as math-phobic at the start of the study but who used the app at least once a week. As parents became more comfortable with math, so did their kids. Families that were already comfortable with math had to use the app at least twice a week to see a similar academic benefit.

Researchers admit that they don’t know for sure the why behind the positive effect on kids’ math skills. Is there increased engagement by kids due to the game-like novelty of math apps? Is it the increase in general positive feelings towards math at home? Or maybe math is better received in a story context? The answer is still unclear, but the research may have some interesting implications for adult learners. Might parents who are enrolled in adult education classes experience math gains themselves if they spend time doing math activities with their children? Or will such interventions only help students at a certain age or level? Certainly, there is evidence that incorporating more math into family time has positive consequences for both parents and children.

So before you kiss your kids goodnight and wish them sweet dreams, you just might want to do some math!

**To read more about the University of Chicago study, including a graph comparing results for different groups, please see:
http://news.sciencemag.org/brain-behavior/2015/10/bedtime-problems-boost-kids-math-performance     and     http://blogs.edweek.org/edweek/curriculum/2015/10/students_with_math-anxious_parents_may_benefit_from_free_app.html



Martha Merson has worked in adult basic education since 1988. She is currently the project director of the NSF-funded iSWOOP project. She was also the PI and project director of the Statistics for Action project. She is one of the co-authors of the EMPower materials has also contributed to the development of Mixing in Math (activities and training resources for afterschool leaders and librarians serving elementary school aged children). Prior to her time at TERC, Merson taught all levels of adult basic education learners and trained volunteers for the Center for Literacy in Philadelphia and then provided staff development on math, social studies, science and writing through the Adult Literacy Resource Institute in Boston.

Encountering Resistance

by Melissa Braaten

When you walk into a vacant position at a new school, replacing the last beloved math teacher, you are an unknown quantity, and the unknown can be suspect.

I was surprised at how intense the resistance was when I started teaching at a new school this year. Math is my specialty, and I had been teaching in a similar program for five years, successfully implementing a math curriculum for adult students, which emphasizes conceptual understanding, beginning with real world contexts, and building on prior knowledge. I was used to students feeling confused by math class that was different than when they were in school, or the period of transition that is sometimes required to help students build confidence in their own reasoning. But usually, students came to appreciate the emphasis on reasoning, problem solving, modeling, and defending and evaluating each other’s thinking.

I decided to begin this school year with a review of “benchmark” fractions, decimals, and percentages like 1/2, 1/4, 3/4, and 1/10. Students who were returning to the class had studied decimals the previous year and were quick to point out that they could use a calculator to multiply by a decimal to find the needed portion. When we discussed how to find a quarter of a whole, for example, they knew they could multiply by .25. I affirmed that this was true, and a reliable method to find one quarter. And I watched carefully.

Sometimes these students could complete the work with their method, and sometimes they could not. Often they would accidentally divide by .25 on their calculators, and come up with a quarter that was much larger than their whole — four times larger, in fact. When I pointed this out to them, they were not sure why this was so. Furthermore, they could not explain why their result did not make sense, instead punching the numbers into the calculator again to try to get a result that I would deem correct.

I expected some of this, and I tried to get a few of these students to let me draw some models and number lines to help them “see” what was going on. Other students shared their (generally simpler, more intuitive) method of dividing the whole by four. I expected that all the students would eagerly adopt these techniques, especially since they were clearly getting mixed results with what they “knew” about multiplying decimals.

Instead, resistance was strong. Several students refused to try any modeling or other techniques. They continued their hit or miss work with the calculator, and complained loudly that they were wasting time on something they had already learned.

They were frustrated, and I was frustrated. However, if there is one thing I have learned working with adults, it’s that you can’t force anyone to do anything. I decided to be patient, allow these students some latitude in how they approached their math, and to look for opportunities to build relationships. I also decided to take their complaints seriously, even if I disagreed with them. Here is how things have played out:

Students complained that what we were learning was not on the HiSet. I took this complaint seriously and went through several practice HiSet tests, pulled out questions involving benchmark fractions, and adapted them for my class. Although all of the questions used only halves or quarters, they required a deep understanding of the concepts. When students made comments about how hard the problems were, I expressed confidence in their ability to reason them out, and took the opportunity to explain that problems like this require a lot of practice thinking critically, which is why I was planning to have them do just that in class.

As I remained patient and allowed these students to make choices about how they approached their work, one finally opened up, explaining how she was frustrated with herself because she felt like she should understand the material better, having studied it previously. This made me grateful that I had affirmed the method these students were using, because I now realize how it represented a lot of previous effort and real learning on their part. What this student had said gave me a new perspective on the situation, and I thanked her for that. I also told her that reviewing a concept is not starting over, it about deepening and making the knowledge more permanent.

Recently, one of the initially resistant students agreed to meet with me after school to go over some work. This time, as I used a number line to explain how to find the whole when given 75%, she said, “You know, I guess the picture really does help.” I then watched her use the number line to think through several more problems on her own. Success – not because she was using the method I taught her, but because it did help. That was the kind of victory we could both appreciate.


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.