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.



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.


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.

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.