Ranked-choice voting: How does it work?

by Donna Curry

Imagine that it’s the end of the school year and you’re planning a class celebration with food and drinks. You ask your students to choose which beverage they’d prefer: juice, water, or soda. The students respond as follows:

So, you decide to go with soda for everyone since more students preferred that to juice or water.

During the party, you realize that less than half of the students drank any soda! They either drank from the water fountain or didn’t drink anything. What you didn’t realize is that maybe those few who wanted juice may have preferred water to soda. But how could you have known that? That’s where ranked-choice voting could have come in handy.

There’s a myth that ranked-choice voting means some people’s votes don’t count, but that’s not accurate. In this system, even if a voter’s top choice doesn’t win out, his or her next-favorite choice might. Had this voting method been used in the class party example, all of your students would have had a chance to vote for their favorite beverage as well as their second- and third-favorite (if they had wanted to).

Had ranked-choice voting been used, the results might have looked like this:

As you can see, 5 of the 12 students chose soda as their first choice. But, the majority of students didn’t want soda – 7 of the 12 chose something else instead (water or juice). In ranked choice voting, you have to have a majority to ‘win’, even if it’s only to choose a beverage. You can also see that not everyone voted for a 2nd or 3rd choice. That is absolutely fine! What you cannot do is vote for a choice more than once. So, even if a student absolutely loved soda and nothing else, she could only vote once for soda.

In ranked-choice voting, if no first-choice option wins a majority of votes, the option that received the fewest votes is removed from consideration. In this case, juice was the ‘loser’ because it received only three votes. But, the people who voted for juice still get to have a say in the result if they indicated a second-choice option. Now we can compare the 2nd choice votes with the 1st choice votes.

If you tally the first choices along with the second choices of the ‘loser’, you see that there are now seven votes for water. Since seven is more than half, the majority – people who are OK with drinking water – wins. If you had used this process, you would have discovered that most of your students would have preferred to have water instead of soda and your party would have been more of a hit.

So, to reiterate a few key points about ranked choice voting:

  • The winner has to be an acceptable option to a majority of those voting. It’s not enough for the ‘winner’ to just have more votes than the second-place option – the winner has to get more than half of the total vote. That means, the winner has to receive at least 50% + 1 vote.
  • Everyone’s vote counts. Sometimes a person gets to have a result he or she can accept even if it’s not his or her top choice.
  • People do NOT have to vote for more than one option, but they also cannot vote multiple times for their favorite option.

The beverage example shows that sometimes the option with the most votes is not always what the majority of people really want. There are several times in recent U.S. history when a presidential candidate did not get the majority of votes. For example, in 2016, neither Hillary Clinton nor Donald Trump received the majority of votes in twelve states because a third-party candidate received some of the votes that would have gone to one or the other.

Probably the most significant example of how ranked-choice voting could have had an impact on the presidential election is in 2000 when almost three million voters in Florida chose Ralph Nader over Al Gore and George W. Bush. Had those three million voters had a second option, we may have had a different president. Whenever we have more than two political candidates in the same race, often people have to make the hard choice between the candidate that they personally prefer versus the candidate who has the best chance of getting the most votes. In a ranked choice voting system, people wouldn’t have to make that hard decision.

Most recently, the January 2021 run-off election for two Georgia senate seats could have been a lot simpler – and quicker – had ranked choice voting been used. (As a reminder, because neither candidate received a majority in the November 2020 election, per state law there had to be a run-off.) If those voters whose top choice came in third in the November election had had a chance to indicate their second choice, those votes could have been tallied to help give one of the two leading candidates enough votes to win, eliminating the need for a separate run-off election.

As with any election system, ranked-choice voting has pros and cons. [Consider all of the issues regarding the electoral college system, especially in the 2020 election year!] To read more on the pros and cons, read the New York Times article New York City Voters Just Adopted Ranked-Choice Voting in Elections. Here’s How It Works.

To help your students gain a better idea of how ranked-choice voting works, you might have them participate in an activity similar to the one illustrated in this short, easy-to-understand video How does ranked-choice voting work?


Donna Curry is an educator, curriculum developer and professional development specialist with over 30 years of experience in adult education. For the past 30+ years, she has focused on math standards development at the national level (Equipped for the Future National Standards and Standards-in-Action projects) and at the state level (including states such as Rhode Island, Washington, Oregon, New Jersey, Oklahoma, and Ohio). She has also worked on the National Science Foundation’s EMPower project and served as co-director for the NSF-funded Teachers Investigating Adult Numeracy (TIAN) project. She co-developed and implemented the Adult Numeracy Initiative (ANI) project and Adults Reaching Algebra Readiness (AR)2. Donna currently directs the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center for Massachusetts and the Adult Numeracy Center at TERC.

Will This Be on the Test? #5

by Sarah Lonberg-Lew

Welcome to the first installment in 2021 of our monthly series, “Will This Be on the Test?” Each month, we’ll feature a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually. (You can find links to the earlier installments in the sidebar list on the right.)

I learned a lot of rules in math class, and when I started teaching math, I taught a lot of rules. It was easier to teach students to memorize the rules rather than help them fully understand the concepts (especially when I didn’t always understand the concepts myself). When I taught math to high schoolers, I remember teaching a chapter on the rules of exponents. There was a separate section of the chapter for each rule and I taught them one at a time, so that students could memorize and practice applying each rule before moving on to the next. We spent at least a week on that chapter, covering a new section each day. For some reason, though, everything always fell apart in the last section of the chapter where the problem types were all mixed together and students had to figure out which rule to apply or sometimes whether to apply more than one rule.

There was a time in my teaching career when, if a student had asked me about the problem below, I would have said, “You have to know the rules to be able to solve this.” I don’t say that anymore. How about you? Could you figure out a way to approach this without knowing “the rules”?

Before you read further, take a few minutes to challenge yourself to come up with more than one way to approach this task. How many strategies can you think of? What visuals could you use to help you solve this? Also ask yourself, what skills and understandings do students really need to be able to answer this?

Before we get into some different approaches, let me be clear that if students do not have a grasp on the idea that a letter can stand for a number, then this problem is most likely out of reach and they should not spend time on it. However, if a student has that concept and knows what an exponent means, they do not have to have memorized a set of rules to be able to reason through this. (And while letters representing numbers is an abstract concept, it is an understanding that students can begin building early on.) Here are some possible approaches.  

1.Try it with a number. Just because the problem is written with variables doesn’t mean it has to be solved that way. One important understanding about equivalent expressions is that they will have the same value no matter what the value of the variable is. This means that a student can change x to a friendly number, like 2, and work out a problem like the one in our example:

24 • 25

16 • 32

512

Now the student only needs to figure out which answer choice has a value of 512 when x = 2.

Note: Even with a small number like 2, the value of the expression is going to be pretty big, so students might want to use a calculator. Choosing 1 is not a good idea. Why is that? Could your students see why?

2.Write it out the long way. A student who knows the meaning of an exponent can write equivalent expressions to x4 and x5 like this:

x4 = x • x • x • x

and

x5 = x • x • x • x • x

The student can now multiply the longer expressions together like this:

Now the student can turn their answer back into an exponential expression to match one of the answer choices.

3. Figure out the rule on the fly. One of the most powerful tools in a mathematician’s toolbox is being able to figure out how complicated things work by thinking about simpler things. A student who is empowered to think of themself as a mathematician may realize that they have the power to discover or recreate the rule by looking at simpler cases. A student might do a few quick experiments with expressions similar to the one in the problem to find out what is going on:

  23 • 22 = 2 • 2 • 2 • 2 • 2 = 25 (Check by hand or with a calculator – that works!)

  23 • 23 = 2 • 2 • 2 • 2 • 2 • 2 = 26 (Check by hand or with a calculator – that works, too!)

Hmmm… the exponents tell me how many 2s will be in the long expression. It looks like adding the exponents gives me the total number of 2s, so I can add the exponents to get the final exponent.

The rules in math were not made up by some high mathematical council that decided one day to confuse generations of students by making the rule that when you multiply expressions with exponents you add the exponents. You add the exponents because it makes sense to do so when you understand how exponents work. The reason that students jumble up the rules or misapply them is that rules are abstract concepts and abstract concepts must be understood, not memorized, to be useful to students. Teaching rules without reason, especially when those rules are not what you might guess at first, disempowers students because they just have to accept what they are told even if it doesn’t make sense to them. Teaching that rules are formalizations of reasoning and patterns empowers students to discover rules and even solve problems without them — and they are freed from the burden of relying solely on memorization.


Sarah Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency.

Sarah’s work with the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center at TERC includes developing and facilitating trainings and assisting programs with curriculum development. She is the treasurer for the Adult Numeracy Network.

Finding the Genius

by Pam Meader

In my previous blog, I mentioned being consumed with watching countless webinars on topics of diversity, equity, and inclusion. Recently, I attended one sponsored by SABES with Dr. Gholdy Muhammad, author of Cultivating Genius, An Equity Framework for Culturally and Historically Responsive Literacy. Her thoughtful words helped me to better understand approaches for addressing diversity, equity, and inclusion in our math classrooms.

Her first approach is to view each student as a genius, and ask ourselves, What do they bring to the math classroom that makes them “shimmer”? So often we have already made assumptions about students by looking at tests scores or allowing our implicit biases on race or ethnicity to influence our perceptions of certain students’ abilities and motivational level. As Dr. Muhammad says, “We won’t start our stories with terms like at risk, defiant, disadvantaged, unmotivated, comparing students to other ethnic groups, etc.

Educational doctrine in the United States has been designed and shaped by white educators and intellectuals. The work and talent of non-white scholars has historically been ignored, resulting in the marginalization of Black, Indigenous, and other people of color and a biased and incomplete educational experience for all American students. Dr. Carol Anderson, author of White Rage, suggests that an exclusionary history erases the work that everyone helped to make our nation a nation. She uses the analogy of a choir to illustrate this idea. Suppressing all voices is like creating a choir with only sopranos. If the only songs picked are one that sopranos can sing, it leaves a lot of beautiful music ignored. She then says to imagine a choir with basses, tenors, and altos that provides an array of music that all can sing. Likewise, our curricula and lesson plans should strive to be more inclusive.

As Dr. Muhammad warns, we have a system that has not valued all our lives and in fact, has made many to feel inferior and have less options. If our curricula isn’t helping everyone, we have to change. To address these inequities, she suggests following three pillars for developing a culturally relevant education. The first pillar is developing academic success, which she refers to as intellectualism — what students gain as a result of our instruction and learning in the classroom. The second pillar is developing cultural competence, by which students are made aware of their historical roots and feel included in the classroom environment. Developing social-political consciousness is the third pillar. This means bringing the outside world in and applying it to curricula. Math teachers can do this by contextualizing lesson content to include the lives and problems of all our students. As Dr. Muhammad suggests, don’t just teach to students’ math levels — teach to their life levels.

Teachers also need to thinking beyond CCRS standards. Dr. Muhammad suggests that a standard indicates there is a stopping point for learning, when learning should be a pursuit for life. She challenges us to think about five learning pursuits as we plan our lessons:

Cultivating identity. We need to think about how our instruction helps our students learn about themselves and others who are different from them. I suggested some activities in my recent blog with some ‘getting to know you’ activities at the beginning of a course.

Cultivating skills. This is probably the area where we feel the most confident as we ask ourselves how our instruction will help students learn the skills and standards in our math classes. Our task is to expand these skills to be life skills for all of our learners.

Cultivating Intellectualism. We should ask ourselves how our instruction will help all students learn new knowledge and concepts. Dr. Muhammad suggests putting this knowledge into action by understanding new topics, concepts, and ideals.

Cultivating criticality. This means implementing the pursuit of diversity, equity, and inclusion by having our students understand power, inequality, oppression and social justice in relationships. In a math classroom, that might be by looking at data and graphs on various ethnicities or conducting surveys on community problems. One of my algebra students collected data on homeless and marginalized people and housing. From her research, she presented her analysis of the problems to the city council and why housing needs were not being met. I feel this is what Dr. Muhammad means by putting their knowledge into action.

Joy. As a math teacher I want my students to find joy in learning math and find it a viable tool for making decisions and utilization as a life skill. To many of our students, math and joy may seem like exact opposites, but when lessons are crafted with these all five pursuits, true joy can be realized.

For some of us, making our teaching practice and classroom environments more diverse, equitable, and inclusive can seem overwhelming. This is true of any shift in long-held thinking and behavior, and it’s normal to struggle a bit. However, every effort moves us closer to the goal of DEI awareness and implementation. If we start small by first looking inward at our beliefs and biases and taking appropriate steps to change, I believe we can meaningfully grow our teaching toolkit and mindset.


Pam 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 Mathematics and Adult Numeracy Curriculum & Instruction PD Center professional development initiative for Massachusetts. She helped co-develop Adults Reaching Algebra Readiness (AR)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.

Will This Be on the Test? #4

by Sarah Lonberg-Lew

Welcome to the fourth installment of our monthly series, “Will This Be on the Test?” Each month, we’ll feature a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually. (You can find links to the earlier installments in the sidebar list on the right.)

This month let’s dive into one of my favorite types of problem to see on a test – an algebra story problem. I love this kind of problem because I think it’s accessible to almost anyone who has the confidence to play with it. The context makes it accessible. I always encourage my students to look for those problems where there is a story they can understand. With that toehold, they often can figure it out. What about you? Can you figure out this month’s challenge without using procedures you memorized in algebra class?

Before you read further, take a few minutes to challenge yourself to come up with more than one way to approach this task. How many strategies can you think of? What visuals could you use to help you solve this? Also ask yourself what skills and understandings do students really need to be able to answer this?

When you’ve given yourself enough time to really explore, read on to see some possible ways students might approach this problem using conceptual understanding and visuals.

1.A student might reason (proportionally!) that if 4 pencils and 4 erasers cost $3.20, half as many pencils and erasers would cost half as much. That means 2 pencils and 2 erasers cost $1.60. Now the student might compare the cost of 2 pencils and 2 erasers to the cost of 3 pencils and 2 erasers (a picture really helps):

If the only difference between the two pictures is a single pencil and the difference in price is 30 cents, the pencil must cost 30 cents! But wait – the question didn’t ask for the cost of a pencil! This is a great start, but it’s always important to make sure you’ve finished the problem, especially on a test. Once they know the cost of a pencil, the student would have several options for figuring out the cost of an eraser.

2. Alternatively, a student might reason that they could figure out what 1 pencil and 1 eraser cost together by dividing the cost of 4 pencils and 4 erasers by 4. The combined cost of a single pencil and eraser is $0.80. Knowing what they cost together means that you can find the cost of one if you know the cost of the other. This paves the way for a pretty efficient guess-and-check.

  • GUESS 1: If the eraser costs $0.10, then the pencil costs $0.70. This means that the cost of Jason’s purchase would be 3($0.70) + 2($0.10) = $2.40. That’s way too much. (Jason only spent $1.60.)
  • GUESS 2: Since the first guess was way off, the student might jump to answer choice (c) or (d). Let’s try c. If an eraser costs $0.30, then the pencil would cost $0.50. This means that the cost of Jason’s purchase would be 3($0.50) + 2($0.30) = $2.10. That’s closer, but still too much.
  • GUESS 3: At this point there are only two possible answer choices. Trying either one will get a student to the correct answer, either directly or by eliminating the other possibility.

3.Speaking of guess-and-check, that’s almost always a workable approach to a situation like this. Even if a student can’t write an equation to find the cost of a pencil if they know the cost of an eraser, they can probably work it out by thinking about the real items. For example, a student might guess that the eraser costs $0.20. Then they might look at Jason’s purchase: 3 pencils and 2 erasers cost $1.60. If the erasers are $0.20 each, they would account for $0.40 of the cost, leaving the remaining $1.20 as the cost of 3 pencils. This means that the pencils must cost $0.40 each. Now the student could check to see if these two costs work out in the larger purchase. 4 pencils at $0.40 each and 4 erasers at $0.20 each would total $2.40. This doesn’t match Irita’s cost, so the guess was wrong.

A student might even keep track of their thinking in a table like this (based on starting with Jason’s purchase and then checking with Irita’s purchase):

That was a lot of work to eliminate just one answer choice, but remember that there will be questions on the test that are not within students’ reach at all. It’s okay to spend some extra time on a problem they have a good chance of getting right. (Also, try not to worry about the fact that students will probably not make a neat table with everything labeled precisely when they are under pressure in a test situation. Tables are an excellent way to keep your thinking organized and if students practice with and feel confident using them in your class, they will be more likely to make use of a quickly sketched table in a test situation. Even a reduced table where students just keep track of their guesses and the cost of 4 pencils and 4 erasers will help them navigate a complicated situation like this.)

As teachers, we are charged with getting our students to pass the test and that is often what’s driving our students as well. But focusing on the test often brings with it the mistaken idea that there is one right way to solve each problem. A student who sees this problem as a prompt to remember and apply a procedure may freeze up and miss the chance to score some points. On the other hand, a student who values their own ability to reason about quantities and relationships, especially in the familiar context of money, may make sense of this in their own creative way. Teaching students to reason flexibly and creatively, to be problem solvers, and to value their own prior knowledge and experience will not only prepare them for the test but also help them grow as numerate citizens and critical thinkers.


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

I Found the Answer! (Now What Do I Do with It?)

by Donna Curry

I was using a recipe that I had gotten from my cousin in Germany. The recipe called for 200 mL of milk. I wanted to know what that was in cups so I Googled the information. Here’s what I found:

To calculate 200 Milliliters to the corresponding value in cups, multiply the quantity in milliliters by 0.0042267528198649 (conversion factor). In this case we should multiply 200 milliliters by 0.0042267528198649 to get the equivalent result in cups: 200 milliliters x 0.0042267528198649 = 0.84535056397299 cups.

Now, I’m not afraid of decimals (not even ones that are as exceedingly long as the ones above). But, even though I’d found out what the answer was, I realized that I didn’t have a measuring cup that measured in decimals, never mind to the hundred-quadrillionth place! So, the answer itself was useless.

But, I did have enough number sense to know that 0.845… was pretty close to 8/10, or its reduced form 4/5. And, although my measuring cup also doesn’t measure in tenths or fifths, I know that 4/5 is close to 1. So, I knew I needed a little less than a cup of milk for the recipe. I could have also used a slightly different approach to get the same answer. If I had started my calculations by shortening the conversion factor several places to 0.004, I would have come up with 0.8 which would make more sense to me than the answer provided by the online calculator.

This volume conversion experience made me think. Using number sense, I was able to think of these unwieldy numbers in more manageable terms. I was able to connect them to benchmarks that were easier to conceptualize. But what about the many adults who have not been taught to use flexible thinking in math to associate something new with something already understood? The conversion factor procedure may have provided an answer, but how well can the average person understand the concept of a number like 0.84535056397299? What does it mean in everyday terms? How does it relate to the actual problem we’re trying to solve? In short, do we know what to do once we have an answer?

It reminded me of how often we think of math as simply a bunch of procedures to be followed: Don’t worry about making any sense – or being useful in real life – just follow the procedure. When there’s an absence of conceptual understanding, we are prone to:

  • choosing the wrong calculation method because we don’t understand what is being asked or how the amounts relate to one another;
  • not being able to recognize whether our answers actually make sense
  • not being able to “see” our answers in alternative, relatable situations or representations that make sense to us

Is this what we want for our students – to just follow procedures (often being told to memorize them, not even having the luxury of Googling a particular procedure) without any understanding, any estimation, any reasoning about whether the answer makes sense or not?

P.S. After doing some estimating based on the information I found on the internet, I pulled out my measuring cup only to discover that one side had gradations in cups . . .  and the other side had gradations in mL. I guess I should pay closer attention to the math tools that I use. At least I was able to check my work another way!


Donna Curry is an educator, curriculum developer and professional development specialist with over 30 years of experience in adult education. For the past 30+ years, she has focused on math standards development at the national level (Equipped for the Future National Standards and Standards-in-Action projects) and at the state level (including states such as Rhode Island, Washington, Oregon, New Jersey, Oklahoma, and Ohio). She has also worked on the National Science Foundation’s EMPower project and served as co-director for the NSF-funded Teachers Investigating Adult Numeracy (TIAN) project. She co-developed and implemented the Adult Numeracy Initiative (ANI) project and Adults Reaching Algebra Readiness (AR)2. Donna currently directs the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center for Massachusetts and the Adult Numeracy Center at TERC.

Will This Be on the Test? #3

by Sarah Lonberg-Lew

Welcome to the third installment of our monthly series, “Will This Be on the Test?” (You can find the first two installments here and here.) Each month, we’ll feature a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually.

There are few topics that provoke anxiety and frustration in learners and teachers like fractions. Students are often carrying the weight of having tried to learn fractions operations so many times and never being able to remember the steps, let alone understand them. Teachers are often carrying a similar weight – how many times can you go over those same procedures? Why do you have to keep reteaching this topic? It’s tempting to spend one or two class periods teaching students how to do fraction operations on a calculator and be done with it. After all, students are allowed to use calculators on most HSE tests.

With that in mind, here is this month’s problem:

Before you read further, take a few minutes to challenge yourself to come up with more than one way to approach this task. How many strategies can you think of? What visuals could you use to help you solve this? Also ask yourself, what skills and understandings do students really need to be able to answer this? To what extent, if at all, are memorized procedures required? Would a calculator be helpful here? (For a bonus challenge, try to find the “attractive distractors” in the answer choices. What mistaken strategies might students pursue that would lead them to these incorrect answers?)

Before you read further, take a few minutes to challenge yourself to come up with more than one way to approach this task. How many strategies can you think of? What visuals could you use to help you solve this? Also ask yourself, what skills and understandings do students really need to be able to answer this? To what extent, if at all, are memorized procedures required? Would a calculator be helpful here? (For a bonus challenge, try to find the “attractive distractors” in the answer choices. What mistaken strategies might students pursue that would lead them to these incorrect answers?)

When you’ve given yourself enough time to really explore, read on to see some possible ways students might approach this problem using conceptual understanding and visuals.

1.Estimate! A student who takes the time to make sense of this problem and think about which answers are reasonable will quickly narrow the possibilities down to two answer choices. It takes a little time and effort to figure out the relationships in this problem, but it’s worth it. A student might do a quick sketch of the fraction in the problem and use it to figure out what they have and what they are looking for:

Three of the answer choices are bigger than 60, but answer choice E is more than twice as big and the sketch shows that the whole circle is only a little bit bigger than the shaded part, so the answer must be C or D.

2.Think about and draw ratios. A student might interpret the fact that 4/5 of the applicants were accepted as meaning that four out of every five applicants were accepted. In other words, for every four that got in, one did not. A student might try drawing all the applicants. Using quick symbols, it doesn’t take as long as you might think. How many Xs and Os are in the picture? A student could count them one at a time, count by fives, or think of this as an array and multiply the dimensions.

It’s also possible that after drawing the first few sets, a student might start to see some patterns and regularity and ask themself, “How many of these will I have to draw?” Since the student already knows that there are five “people” in each row, answering that question will be a quick way to get to the total. (Here is a an example of when a calculator might be helpful. Since there are four accepted people in each row, and 60 accepted people total, the number of rows can be found by dividing 60 by 4. Then the total can be found by multiplying the number of rows by 5. Calculators, used judiciously, can help students use their time well and stay focused in test situations.)

3.Draw a Singapore strip diagram (also called a bar model). Singapore strips are great for making sense of parts and wholes. If students are comfortable drawing and reasoning with them, they provide a way of making sense of the structure of the problem that makes answering the question fairly intuitive.

To solve this problem with the diagram, a student would figure out how many people each box represents. (Four boxes represent 60 people, so how many people are represented by a single box?) Since the total number of people who applied is represented by the whole bar or five boxes, once a student knows how many people each box stands for, they will be able to find the whole in just one more step. (For a similar example using a model like this, see our blog, “How I Learned to Stop Worrying and Love Percents”.)

What about the calculator?

What did you decide about the usefulness of a calculator in addressing this month’s problem? What would your students do? Solve the problem without a calculator? Use a calculator judiciously for some quick computations? Grab the calculator and start doing whatever fraction operations they can think of? A calculator can be a great tool when used strategically, but only teaching students how to do fraction operations procedures instead of teaching how to reason about fractions will not serve them as well on the test or in life.


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

Beginning Steps in Addressing DEI in a Math Classroom

by Pam Meader

So much has happened in so little time. First the COVID-19 pandemic hit and shortly after school buildings were closed and classes went virtual. Getting up to speed on the latest technologies and transforming face to face class lessons to virtual was a huge task for teachers. The killing of George Floyd followed, and our nation was ripped apart again over racial inequities. Teachers are grappling with how to make changes in their practice and classroom culture, to promote awareness of systemic racism and issues of inequality. It can feel overwhelming.

However, we cannot let ourselves to be so overcome by the enormity of these issues that we give in to inaction or hopelessness. In his now-famous “butterfly effect” principle of chaos theory, Edward Lorenz theorized that even a small variation in conditions, like whether or not a butterfly flaps its wings in an Amazonian rainforest, has the potential to later impact the course of something as powerful as a tornado in Texas. So it is with teaching. Small steps in the right direction help us gain momentum; this momentum energizes us, moves us forward, and ultimately can result in powerful change in ourselves and for our students.

As teachers, I believe there are two major things we can do to begin to address diversity, equality, and inclusion (DEI) in our classrooms. First, we all need to become informed of black, brown and indigenous histories — histories that were truncated, inaccurate, or altogether missing in the textbooks most of us read as children. I suggest reading some books on the history of suppression in our country or perhaps arranging a virtual book study and discussion group with colleagues. Two books that had a profound effect on me were How to Be an Antiracist by Ibram X. Kendi and Caste: The Origins of Our Discontents by Isabel Wilkerson. These books illuminate the backstory of the racism and inequality that persists in society to this day, challenge us to become aware of and examine our own biases, and make conscious choices on how to move forward in our thoughts and behavior.

That brings us to the second thing teachers can do: taking action by implementing changes in our classrooms. Small changes in lesson planning can begin to address DEI issues in the classroom. I have been watching many great webinars recently on this topic, and as I have, some themes have started to emerge for me. One is the idea of identity. Dr. Gholdy Muhammad, author of Cultivating Genius: An Equity Framework for Culturally and Historically Responsive Literacy, suggests thinking about how your instruction can help students learn about themselves and others that are different from them. She posits that if students see themselves in the curriculum, they are more motivated to learn.

Here’s an example of how this could look in a math classroom. When I taught at Portland Adult Education in Maine, my math classes were very diverse. Students from African countries and Asian countries were learning alongside students from the U.S. We always started our first day of class by learning where each student was born and by placing pins on a world map to display everyone’s country of origin. Then we would do interviews during which students would share information about their favorite foods, hobbies, interests, etc. We would then represent the data from the interviews as bar graphs and circle graphs. We’d also compose sentences using language like more than half, less than a quarter to talk about the data we collected. If new students joined the class, we would interview them, collect their data, and discuss how it would affect our graphs or our sentences.

There are several other areas where teachers can make changes to promote a culture of DEI. For example, in her recent STEM Forum on Equity and Inclusion talk Recognizing Student Brilliance, Faith Moynihan recounts asking herself what her students needed to be successful. She came up with four concepts. The first is that all students need to feel safe which means they need to feel valued and that their knowledge will be accepted. Safety in a math classroom is important especially to explore interesting problems, struggle with concepts, or share ideas in a group. If the brain does not feel safe, no new learning can occur (Jensen, 2005; Hammond, 2014). Starting classes with some group building, using a math autobiography to get to know your students better, and math journaling offer opportunities to build trust and safety in a classroom.

The second concept was utilizing interesting tasks for your students. Moynihan suggests thinking about what makes a task worthy of students’ time. What she discovered was that personal lessons were engaging lessons, which speaks to the “identity” factor that Dr. Goldy Muhammad talked about. In the words of researcher Emily Style (1988),“Education needs to enable the student to look through window frames in order to see the realities of others and into mirrors in order to see her/his own reality reflected.” For example, at the beginning of a course teachers could ask students what they hope to learn and why. What occupations are they seeking? What daily life challenges are they facing? Tailoring your math lessons to their pertinent needs and interests shows respect for the time and effort students are putting in to their education.

The third concept is creating opportunities for students to express their thinking in different and interesting ways. No longer is raising a hand in a math classroom enough. Teachers can utilize varied activities and tech tools such as group chats, journaling, number talks, online visualization and collaboration applications and more to promote communication. Moynihan herself reflected on a Desmos activity in which she could see that some of her student’s answers were not correct. Rather than just pointing out they were wrong, she asked them to explain their thinking and, in the process, this illuminated some deep thinking by her students. Developing metacognition (the awareness and understanding of one’s own thought processes) is a powerful tool for students and teachers alike.

Having students feel their thinking is valued is the last concept. Consider a time when your students came up with different answers to a problem. If you just give the correct answer to the problem, you are demonstrating that the focus is on getting the right answer and put no value on the thinking process involved in finding the answer. Those that got it right feel great, but you have left out many who could have shared their thinking. Those students instead feel their thinking is not valued or worse, that they can’t do math at all. What I have done in my classroom is group students by similar answers and then ask groups to defend their positions. As students listen, many change their minds or “critique the reasoning of others” (Common Core Math Practice 3).

A perfect example of this is the activity “My Favorite No”. First, the math teacher starts with a warm up and has students put their work on a notecard. She quickly sorts the correct answers from the wrong ones and picks a wrong answer to present as her “favorite no”. The students are asked to comment on what was done correctly in the problem. This affirms the value of the thinking of the student who provided that answer, who can then feel that while her answer might be wrong, she was awfully close to getting it right. This approach not only builds confidence, it helps students persevere in productive struggle.

My Favorite No video

The suggestions given here are small steps that could easily be adopted in a math classroom. Being in virtual classrooms makes this more challenging, but group talks could be designed using features like breakout rooms. Journaling can become a habit of mind and part of work to be shared with the teacher or colleagues using attachments in emails or shared cloud documents. The whole idea is to start to look at your class equitably, recognizing the various identities and brilliance each student can offer. As teachers we need to look inwardly and identify implicit biases we might hold and start to address these, recognizing that, in the words of Ibram K. Kendi, it might require “…a radical reorientation of our consciousness.”

References

Hammond, Z.L. (2014). Culturally Responsive Teaching and The Brain: Promoting Authentic Engagement and Rigor Among Culturally and Linguistically Diverse Students. Corwin Press.

Jensen, E. (2005). Teaching with the Brain in Mind, 2nd Edition. ASCD.

Kendi, I. X. (2019). How to Be an Antiracist. One World.

Lorenz, E. (2000). The Butterfly Effect. In R. Abraham & Y. Ueda (Eds.), The Chaos Advant-Garde: Memories of the Early Year of Chaos Theory (pp. 91-94). World Scientific. https://doi.org/10.1142/9789812386472_0007

Muhammad, G. (2019). Cultivating Genius: An Equity Framework for Culturally and Historically Responsive Literacy. Scholastic Teaching Resources.

Wilkerson, I. (2020). Caste: The Origins of Our Discontents. Random House.


Pam 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 Mathematics and Adult Numeracy Curriculum & Instruction PD Center professional development initiative for Massachusetts. She helped co-develop Adults Reaching Algebra Readiness (AR)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.

Will This Be on the Test? #2

by Sarah Lonberg-Lew

Welcome to the second installment of our new monthly series, “Will This Be on the Test?” (If you missed the first blog, check it out here.) Each month, we’ll feature a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually.

There’s almost always more than one way to tackle a math problem. When teachers encourage students to try out different approaches, students gain confidence in thinking flexibly, experimenting, and finding strategies and solutions that make sense to them.

What about you? How would you approach this month’s problem?

Before you read further, take a few minutes to challenge yourself to come up with more than one way to approach this task. How many strategies can you think of? What visuals could you use to help you solve this? Also ask yourself, what skills and understandings do students really need to be able to answer this? To what extent, if at all, are memorized procedures required?

When you’ve given yourself enough time to really explore, read on to see some possible ways students might approach this problem using conceptual understanding and visuals.

1. They might rely on real world experience to estimate a solution to this. Students who has experience with landscaping, yard work, flooring, tile, etc., may have an intuitive understanding of about how big this backyard is and about how many square feet of sod it would take to cover it. It is important to connect the math that students learn in the classroom to the math they encounter in the real world. Without this connection, students may come to see “school math” as something separate from their lived experience. They may not realize that they can apply what they already know in a testing situation. Students who do not make this connection can end up panicking over trying to remember a formula instead of using their own mathematical intuition.

2. If the student has some understanding of the relationship between feet and yards, they might start by drawing a picture of the backyard with the dimensions shown in yards, and then divide each yard into feet to see that they should multiply 15 by 24 to find the number of square feet. 

3. A student might start by drawing a picture of the backyard with the dimensions marked in yards and then figure out how many square feet are in a square yard. Once again, if the student knows how many feet are in a yard, she could use this knowledge to visually divide a square yard into nine equal parts of one square foot each. She might then convert the area of the backyard from square yards (40) to square feet by multiplying 40 x 9. Alternatively, she might calculate total square feet in another way, like finding how many square feet are in each column and then adding all the columns together.

An important point here — even people who are experienced with working with linear and square measures (like feet vs. square feet) can confuse the conversions sometimes. It’s hard to remember to use a factor of 9 when converting between square feet and square yards, but to use a factor of 3 when converting between feet and yards. It’s especially tough in a high-pressure situation like a test. It’s also easy to overlook the conversion completely and quickly answer the question using the wrong unit. This is why it’s so valuable for students to have the confidence to take the time to make sense of a question instead of trying to rely on their memory of what they are supposed to do. Students who draw pictures and rely on their own reasoning are better able to bring their full mathematical power to bear on any question.

What can go wrong?

In contrast to students who confidently take time to draw pictures, consider strategies, or draw on their own experience, there are other students who are desperate to get through the questions. These students may resort to “number grabbing”, or quickly taking some or all of the numbers from a test item and arbitrarily applying to them the first formula they can think of. If they arrive at an answer choice, they pick it and move on. For students without strong conceptual knowledge of math operations, having answer choices can be a pitfall. The developers of standardized tests do not create the wrong answers for multiple choice questions at random; they choose “attractive distractors” — answer choices that will result from number grabbing or not thinking the problem through thoroughly. For example, in our problem above, a student might quickly choose to add or multiply the two numbers given in the problem and arrive, incorrectly, at answer choice A or B without really thinking at all.

Number grabbing is a panic response. For teachers, the best way to help students combat that panic is to nurture their confidence in their understanding, creativity, and flexible thinking. Students who are empowered to approach problems in ways that make sense to them instead of trying to remember how they are ‘supposed to do it’ are much better prepared to make use of all of their skills and abilities in testing situations and elsewhere. When students develop the confidence to think critically and calmly about test questions, they will also have the confidence to do so when they encounter math in real life.


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

Will This Be on the Test? #1

by Sarah Lonberg-Lew

Welcome to the first installment of our new monthly series, “Will This Be on the Test?” Each month, we’ll feature a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually.

There are lots of good reasons to study math – it better prepares students for the numbers and relationships they encounter in life and at work, it builds flexible thinking and analytical skills, and students have to pass a test on it to move forward with their educational and career goals. The test looms large for students and is often their strongest motivation for coming to adult ed classes. For teachers and programs, the test occupies a place of great significance as well. Performance reviews for teachers and funding for programs may hinge on how many students pass the test and what scores they get. It’s no surprise, therefore, that directors, teachers, and students are inclined to prioritize preparing for the test over anything else. The good news is that preparing students for the test doesn’t have to be separated from deep and rich conceptual learning. In fact, deep and rich conceptual learning prepares students better than crash courses on procedures and test-taking strategies. This post is the first in a series that will look deeply at test-style questions and dig into how conceptual understanding, willingness to draw pictures, and flexible thinking can prepare students to excel on the test.

Here is an example of a question students might see on a high school equivalency (HSE) test. Before you read further, I challenge you to see how many ways you can think of to approach it. You may know a procedure to apply. Which approaches could be used that do not rely on memorized procedures? What skills and understandings do you think students really need to be able to arrive at the correct answer?

Here are some approaches students might take if they don’t have a memorized procedure:

1. A student might sketch rectangles with all the sets of dimensions including the original and eyeball them to see which one looks like it’s not similar to the rest of them.

2.  A student might start by drawing a picture showing a simple enlargement. The simplest way to enlarge a picture is to double both dimensions.

Based on this picture, a student might reason that there was only one proper enlargement with a 4” height and since doubling the dimensions gave one that is 4” by 6”, an enlargement that is 4” by 5” would not work.

 

3. A student might look at the ratio of the length to the width of the logo as a fraction and visually compare the fractions for each enlargement.  

4. A student might reason that you enlarge a picture by multiplying both dimensions by the same number and notice that answer choice A is the only one where the second dimension is not a multiple of 3. If she then checked by trying to set up equal ratios, her suspicions would be confirmed.

Does this mean that instead of teaching one procedure you need to teach four different ways to approach the problem? No! The purpose of seeing all these different approaches is to understand that when students have some conceptual understanding and the confidence to apply it flexibly, they can make sense of challenging test questions even if they haven’t memorized the procedure that the question is ostensibly targeting. To tackle this question, a student does need some intuition about what it means to enlarge a picture, but even with only that, she is prepared to make sense of the problem and persevere in solving it. In fact, worrying about which memorized procedure applies to this problem can make it harder!

Giving students rich learning experiences that help them draw connections between math concepts and the real world gives them the confidence to bring their strengths to unfamiliar test questions. When students know that there isn’t one single correct approach to math, they are free to find their own strategies.


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

How to Open Up Word Problems (and promote digital literacy, too!)

by Heidi Schuler-Jones

In today’s classroom, we know that it is not enough to simply teach math content and skills. Students also need to develop facility with technology tools. Using a traditional word problem as a starting point, here are some ideas for doing so.

First, think about ways to ‘open up’ a word problem:
  1. Limit the information provided to students and instead have them find information based on their own interests.
  2. Allow students to make choices on figures, when possible, or recommend a range of choices.
  3. Instead of having one answer, provide a way for students to evaluate the information they gathered, organize and make meaning of the information, and communicate the reasoning behind the choices they made using a variety of methods, visuals, and models.
  4. Differentiate the same problem for multilevel classrooms by providing conditions for the types of numbers to use. For example, using less-friendly numbers for more advanced students or more friendly and common numbers for less advanced students. Other options: provide more or less information, offer more or less decisions, provide more or less steps (each of these can provide a push for more advanced students or support for those needing greater assistance).
Next, consider where digital literacy tools might be appropriate and useful.

Suggested digital literacy tools can extend beyond the basic calculator functions. Encourage students to use more advanced functions on their smart phone calculator apps and/or to become familiar with spreadsheet and graphing functions found in programs like Microsoft Excel, Google Sheets, and Desmos.

Now, let’s apply these tips to ‘open up’ a traditional workbook problem into opportunities for deeper, more active learning.
Traditional workbook problem:
John is interested in buying a used car for $15,000. He puts down 25% with the rest to be paid over a 5-year period. If his monthly payments are $197, how much will he pay altogether for the car?
Steps to Open Up
Word Problems
Closed (traditional)
word problem
Opened-up
word problem
Limit the information provided to students and instead have them find information based on their own interests.John is interested in buying a used car for $15,000.Check out the prices of vehicles at two different dealerships in the area. Decide on a vehicle that might suit your needs.
Allow students to make choices on figures, when possible, or recommend a range of choices.He puts down 25% with the rest to be paid over a 5-year period.Decide the amount you want to put down and the number of years you want to finance the vehicle. Decide the maximum monthly payment you can afford.
Instead of having one answer, provide a way for students to evaluate the information they gathered, organize and make meaning of the information, and communicate the reasoning behind the choices they made using a variety of methods, visuals, and models.If his monthly payments are $197, how much will he pay altogether for the car?Create a graph and/or in-out table to show the cost of the vehicle over the years that you have financed it for and be prepared to explain your choices.
Differentiate the same problem for multilevel classrooms by providing conditions for the types of numbers to use. For example, using less-friendly numbers for more advanced students or more friendly and common numbers for less advanced students. Other options: provide more or less information, offer more or less decisions, provide more or less steps (each of these can provide a push for more advanced students or support for those needing greater assistance).Conditions for more advanced students:
Estimate a reasonable answer first. Use two different down payment percents with a non-repeating fraction or decimal in each. Compare two different car deals on a trend graph.

Conditions for less advanced students:
Estimate a reasonable answer first. Use one of the following benchmark percents for the down payment: 10%, 15%, or 25%. Use the provided in-out table below to track the cost of the vehicle after the down payment (Year 0) and for each of the next 5 years.
Suggested digital literacy tools can extend beyond the basic calculator functions. Encourage students to use more advanced functions on their smart phone calculator apps and/or to become familiar with spreadsheet and graphing functions found in programs like Microsoft Excel, Google Sheets, and Desmos.Calculator
Smart phone (calculator
Calculator
Smart phone (calculator)
Amortization calculator or app
Web search
Spreadsheets and graphs
   *   Excel
   *  Google Sheets
   *  Desmos

Another way to open-up traditional problems is to remove the question completely and simply ask, “What do you notice?” and “What do you wonder?” Using our example above, the word problem would thus shorten to simply:

John is interested in buying a used car for $15,000. He puts down 25% with the rest to be paid over a 5-year period.

Students spend a minute jotting down the things they notice and wonder and then share these observations with a partner or the whole class. It generates critical thinking and reasoning about what information is needed before attempting to plug into a formula or calculating numbers that may not be relevant. It’s quite common for students to come up with the actual question themselves but in their own words, which helps to make sense of the problem. It also gives them practice developing test-like questions themselves based on their own understanding of the problem and the choices they make.

From those noticings/wonderings, students can follow the steps to open-up word problems as shown in the table above. Watch this video to learn more about this strategy: Ever Wonder What They’d Notice?: Annie Fetter


Heidi Schuler-Jones has worked in adult education since 2006. She participated in the pilot program of the Adult Numeracy Instruction – Professional Development (ANI-PD) in Georgia in 2010 and immediately found its techniques, methodologies, and research-based resources to be of tremendous value to her teaching and to the variety of students she saw daily. Heidi currently is a consultant for the SABES numeracy team where she facilitates trainings and works on course and curriculum development, including the Curriculum for Adults Learning Math (CALM). She also facilitates the Adults Reaching Algebra Readiness (AR)2 institutes for TERC. Heidi also is a LINCS national trainer for math and numeracy and serves as President-Elect of the Adult Numeracy Network.