We’ll still be posting our monthly features and our popular “Will This Be on the Test?” series, but the blog will be hosted within the Adult Numeracy Center at TERC website where you can find lots more information about our professional development work and resources. In other words, one-stop shopping for your adult numeracy needs!

I have been in the field of education all of my adult life (and spent most of my childhood in school!). I’ve had about 40 years of experience working in adult education, including teaching in prisons, family literacy programs at libraries, workplace environments, and storefront community-based organizations. I feel like I’m fairly knowledgeable about teaching and learning. So, while I would never call myself an expert, I would say that I do have expertise in adult education.

However, no matter what our background, in reality we are always moving back and forth along a continuum, from novice to advanced. With the COVID-induced shift to remote instruction this past year, I have to say that I quickly moved from advanced to novice in regards to my understanding of how to teach adult learners. I knew how to engage learners in face-to-face situations, and I had facilitated online courses and webinars, but I had never tried to work with adult learners remotely, especially those with limited digital access. Reading up on how to teach remotely and learning how to use technology tools is helpful, but can only get me so far. Nothing replaces the actual experience of teaching in a remote situation. Despite all my years of experience in education, I have had to acknowledge that in this area, I am a novice.

For educators, the idea of not being the expert in our classroom can be downright disconcerting. There is a general perception that the teacher should always be the most knowledgeable person in the classroom. After all, our job is to help our students make progress towards their educational goals. A big part of their success involves the acquisition of new knowledge and skills, which has to come from somewhere, right?

In some ways, we do have more expertise than our students. Most of us have been trained as teachers, so we have an idea of what to teach and when and how. Many of us have specific content knowledge in areas like math, literature, and language acquisition. Often we have spent many years in the classroom working with hundreds of students over the span of our careers. There’s a lot that we’ve learned through our training and experiences, and there’s a lot we can offer to our students. But to believe that we always must be the expert or source of all knowledge is an unfair expectation to put on ourselves. This belief can even stunt our professional development by causing us to cling to teaching the way we always have, using the same procedures, activities, and materials, because doing so is comfortable. We may even insist our students learn things “our way”. By doing so, we can remain unchallenged experts. On the other hand, trying new things or admitting that we don’t know the answer to a question can feel very uncomfortable or powerless. Teachers aren’t used to being in the novice seat.

The good news is, it’s possible to be a teacher while still being a learner. We must remember that our students are adults. They are consumers, workers, and parents. We sometimes forget that they may have more experience and knowledge than we might in certain areas. This is especially true today. Some of our students are working on the front lines, maybe manning the checkout counter in a retail job, working in health care facilities, or serving customers in restaurants. Some of them have had to develop strategies to make $200 a week go a long way. Many juggle their own homework while helping their kids do theirs (and in the process have learned how to use course management systems). Still others are growing their own side business or are doing gig work.

As adults ourselves, we may have many life experiences in common with our students. For example, we all have experience with handling our personal finances. We could certainly teach students how to budget and save. However, we need to remind ourselves that our students may have different circumstances, priorities, and life experiences than we do. They are the experts of their own lives. Before jumping in and telling them what they need to do to handle real life math challenges, we should slow down, listen, and learn.

As teacher-learners, we need to appreciate what our adult students have done, what strategies have brought them success, and what is different about their lives than ours. We can’t expect students to always think the way we do, learn the way we do, or choose to problem-solve the way we would. Moreover, it would be paternalistic of us to think they should. Everyone needs a chance to shine as an expert. Be willing to share the stage. Create opportunities for students show what they know, express what they feel, and help others in the class (including the teacher) learn things they didn’t know before. Encourage student input on the direction of the class. Find out from them what’s most critical to learn right this minute, especially during these trying times. All the adults in a classroom should feel empowered to shape their educational experience.

And so, fellow “experts”, let’s embrace the role of learner. In doing so, we become better teachers.

Donna Curry is an educator, curriculum developer and professional development specialist with nearly 40 years of experience in adult education. Her work 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 Teachers Investigating Adult Numeracy (TIAN) project. Most recently she co-developed and implemented the Adult Numeracy Initiative (ANI) project and Adults Reaching Algebra Readiness (AR)^{2}. Donna is the Director of the Adult Numeracy Center at TERC and currently directs the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center for Massachusetts.

Welcome to the latest installment of our monthly series, “Will This Be on the Test?” (You can find links to the earlier installments in the column to the right.) 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.

Let’s dig into some data this month. What skills and understandings do you think of when you think of the data domain? There’s a lot more to it than calculating mean, median, and mode! Making sense of representations of data is important for not only the math section of the test but also for science and social studies. And on top of that, it’s one of the most useful numeracy skills for adults to have in the real world.

Here is this month’s challenge:

Before you read further, allow yourself to bring your full mathematical reasoning power to bear on this challenge. 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 reallyneed to be able to answer this?

Before you read the possible approaches below, be aware that every single one of them relies on the important first step of understanding what the chart is telling you. If a student does not know how to read a circle graph, they will not be able to make sense of this task at all! It is essential to understand that each wedge of the circle graph tells what percentage of total sales came from each product. Even though we are inundated every day with representations of data, it is important not to assume students know how to make sense of them.

Here are some possible approaches:

1.Estimate! Look at the relative sizes of the wedges for T-shirts and hats. Seeing that the wedge for hats is smaller than the wedge for T-shirts won’t help much because that only tells a student that less money was made from hats than from T-shirts, and all the answer choices are less than $1,600. However, if a student fine-tunes their estimate a bit, they may make some progress. The wedge for hats appears to be a little less than half the size of the wedge for T-shirts. (Looking at the numbers confirms this – 15% is a little less than half of 40%.) Which answer choice(s) could reasonably be described as a little less than half of $1,600?

2.Find one percent from another. A student who has built a mental bank of benchmark percents and is comfortable comparing them to each other can figure out how much 15% of sales by starting with what they know about the 40%. There are a lot of different ways to do this, jumping from one benchmark to another. Here are a few possible paths:

(a)40% >> 20% >> 5% >> 15%. Starting with 40%, a student might reason that half of that is 20%. We know 40% is worth $1,600, so half of that is worth $800. Therefore, 20% of the total sales is $800. From there, she could figure out 5% of the sales by dividing 20% by 4. $800 divided by 4 is $200. The last link in the chain is to get to 15% from 5%. How does 15% compare to 5%? What operation is needed?

You can take the illustration from here. (Hint: How many blocks of 5% does it take to make 15%?)

(b)40% >> 20% >> 100% >> 10% >> 15%. This chain starts with the same first step as the last one, but a student comfortable with the benchmark 20% might go from there to figuring out the total sales (the whole). If 20% of the whole is $800 (see reasoning path (a) above), then the total sales must be five times as much as that, or $4,000. From there, a common strategy for finding 15% of a number is to find 10% (or one-tenth) and then add on half of that. In other words, 10% + half of 10% = 15%.

(c)40% >> 10% >> 15%. Combining ideas from the two chains of reasoning above, a student could jump from 40% to 10% by finding one-fourth of 40% by dividing 40% by 4. Then they could use the strategy of adding half of 10% on to 10% to get to 15%.

(d)Quite a few other possible paths! Students who understand the relationships between percents, fractions, and wholes can move around very flexibly in the world of percents. Starting from a benchmark of 50% (or ½ ) and building from there, students can get to a point where they can navigate to almost any percent with confidence.

3.Good old proportional reasoning. Percents are a special kind of ratio and therefore students can bring proportional reasoning tools to bear on them. As in the chains of moving from one percent to another above, there are several paths a student could take setting up proportions to get to the answer.

Use the known percent to find the whole and then use the whole to find 15%. In our example, use one proportion to find the total sales, then use the total sales to find the $ made from hat sales:

(Note: This is not the only way to set up these proportions to find the total sales and then the money earned from hat sales. And once the proportions are set up, there are many ways to reason about them!)

A more direct path is to compare the known percent and the unknown percent in a proportion.

(Again, there is more than one possible way to set up and solve this proportion.)

Did you notice that reasoning about this data task involved a lot of number sense and proportional reasoning? One thing I love about the data domain is that it brings together elements of other domains in a concrete and applicable context. (Geometry is no stranger to the data domain either – representations of data use area and angles to show differences in the sizes of categories.) Teach data deeply and conceptually! We all need to understand it and it makes so many kinds of reasoning real and relevant.

SarahLonberg-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.

In math, seeking patterns is a helpful strategy. Once we think we’ve found a pattern, we can make a rule or generalization, or sometimes even a formula. In some life situations, however, we need to be more cautious about making rules when we think we see a pattern.

Finding patterns helps to make learning easier and faster, but it can also lead to biases. All of us have biases of some kind. Some of them are innocuous (a preference for certain colors, music, food, clothes) and some are even helpful (avoiding certain things or situations that resemble negative experiences we’ve experienced). Biases like these are about our own selves, not about judging others, and they may be implicit or explicit.

Our biases, or prejudices, are part of who we are and they often protect us. Biases are not negative until we act on them to create discrimination. And it’s not just explicit biases that can be harmful. It is often the subconscious, implicit biases that can cause damage if we are not aware of them.

For example, let’s imagine you are a teacher who has just received a class roster with the following last names: Soares, Rivera, Schuler, and Curry. What would be your immediate reaction to the names?

Would you assume that you have at least three new students from cultures other than that of the United States? Which of them might not speak English as a first language?

What would be your gut reaction to a name like Soares? Do you think that maybe Rivera can help you translate for students whose first language is Spanish? And, what do you expect from someone with a name like Schuler? Which of the students are you already thinking might be better at math? Do you assume that ESOL students might be able to do simple math computation, but definitely not word problems?

Do an internal check-in to see whether you have already made some assumptions about who these students are and what their abilities may be based solely on their names. Have you linked these new names with your past experiences (patterns that you’ve noticed over time)? Or have you made assumptions based on media representations or the attitudes of your friends and family? Did you prejudge without even realizing it?

We don’t just make judgments based on names. We also make snap judgments of our peers. How often do you prejudge based on position or education without even realizing it? Do you assume that someone with a formal math education is a good math teacher? Do you assume that the ESOL teacher in your program can’t effectively teach math?

The best way to break some of those patterns, those implicit biases, is to get to know the students and your peers. They are each unique individuals whose identity is not limited to being a student in your class or the teacher in the next room, but also an individual who may or may not affirm some of your initial assumptions. Create opportunities to find out about each individual from a personal perspective. See each person as a unique individual who has something positive to contribute. Appreciate what they can share.

We math teachers might think that we have to focus solely on the content, but this isn’t so. We should focus on the student. Think about teaching students math, not teaching math to students. What you believe is how you will focus your efforts. Get to know Soares, Rivera, Schuler, and Curry. In doing so, you’ll learn that only one of them —Curry— had an immigrant parent and spoke another language fluently as a young child.

Welcome to the latest installment of our monthly series, “Will This Be on the Test?” (You can find links to the earlier installments in the column to the right.) 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.

Last month we looked at a question about a snail crossing a garden. Modeling the question with number lines or Singapore strips made it very accessible. (If you haven’t read that column yet, I recommend you do so before continuing with this one.) What if the fractions involved in the question were a little less friendly? Would the same visuals and conceptual strategies still work? This month I invite you to challenge yourself to solve a slightly thornier variation using visuals and conceptual understanding.

Here is this month’s challenge. The snail is back, but the numbers have changed.

Before you read further, allow yourself to bring your full mathematical reasoning power to bear on this challenge. 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?

Here are some possible approaches:

1.Estimate!The snail has been traveling for 2/3 of an hour (or 40 minutes) already and is only going to travel for another 10 minutes. Drawing a quick sketch can help a student think about how much farther it looks like the snail would get. About how far do you think the snail would get in 10 more minutes?

The snail is already just over halfway across the garden, so answer choice (c) can be eliminated because it is less than half. Answer choice (a) is more than half, but not by much. From the sketch, it looks like the snail will be well past the halfway point in another ten minutes, so answer choice (a) is not likely. Answer choice (b) is more than one, and it doesn’t look like the snail is going to make it past the end of the garden in only more 10 minutes. There are still a couple of possible answers left. Which one would you choose just from eyeballing the picture? What fraction understandings were used here?

2.Use a Singapore strip diagram (also called a bar model or tape diagram). A student might start with a bar representing the 4/7 of the garden the snail has already crossed. Notice that it bears a resemblance to the sketch above; both show 4/7.

This is where it gets a little trickier. The snail traversed those four blocks in 2/3 of an hour. A student who understands that unit fractions (fractions whose numerator is 1) are single pieces and non-unit fractions are groups of pieces may reason that those four blocks are covered in two chunks of time that are each 1/3 of an hour. In other words, the snail covers two blocks every 1/3 of an hour, or every 20 minutes.

If we know how many more blocks the snail will travel in 20 minutes, how many blocks will it cover in another 10 minutes? What total fraction will that make? (The diagram above doesn’t show the final step of adding on the last 10 minutes of travel.)

3.Reason proportionally. This task is about something moving at a constant rate. In other words, the distance the snail covers is proportional to the amount of time it has been traveling. A student might reason that the 10 minutes the snail has yet to travel is one-fourth of the 40 minutes it has traveled so far, so the snail would cross an additional fourth of the distance it had already covered. Thinking about finding 1/4 of 4/7 of a garden might make your head spin, but looking at the picture of 4/7 represented as four blocks out of seven makes it easier to see that 1/4 of 4/7 is one block.

Once again, we found that visual and conceptual approaches carried us through without ever needing to “choose and apply an operation.”

This question of course can also be answered using fraction operations, but it takes a few steps. First, divide the distance covered by the time it took to cover it to get the rate: 4/7 ÷ 2/3 = 6/7 of the garden per hour. Then, multiply the rate per hour by the fraction of an hour the snail adds onto its journey (10 minutes). Ten minutes is 1/6 of an hour so 6/7 x 1/6 = 1/7 more of the garden that would be covered in the extra time. Finally, to determine the total amount of garden covered, add the new distance to the original distance: 4/7 + 1/7 = 5/7, which is answer choice (d). That was three operations and not a very intuitive path through them. (It’s also not the only possible operations approach.)

Even though the operations approach is valid, it tends to elicit the I’m-never-going-to-remember-all-these-steps! response from students. Having to do three different fraction operations to get to the answer can make a student want to run the other way. On the other hand, giving students visual and conceptual tools prepares them to chart their own path through problems in a way that makes sense to them.

SarahLonberg-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.

My name is Sarah. (If you don’t believe me, check the byline.) On my first day of high school, I met four other girls named Sarah. It has often been my experience to not be the only Sarah in a group, especially a group of people close to my age. I’ve had to add descriptors to my name to be able to be uniquely identified. I’ve been “short Sarah,” “Sarah short hair,” “Sarah L,” and on more than one occasion I’ve been “math Sarah.” If you also have a common name, you can probably relate.

Just how common is my first name? There is a fantastic internet tool that can answer that question for me with a wealth of interesting data. It is WolframAlpha, the computational search engine. If you are looking for quantitative information of any kind on the internet, this is a great place to look. One exciting thing this site can do is provide current and historical data about the popularity of a name. For example, here are some things I learned about my own name:

About 1 out of every 303 people in the United States is named Sarah.

There are about 909,700 Sarahs living in the United States right now. That means I may not be one-in-a-million, but I’m close!

The most common age of a Sarah in the United States is 30.

WolframAlpha also showed me data that helped shed some light on why I so often have to qualify my name in large groups. This graph shows the popularity of my name over time:

When I was born, my parents didn’t know that my name was getting more popular. Looking at this graph, I wonder if a lot of other parents might have been similarly unaware. Can you guess about when I was born?

The same site also provided a graph showing the age distribution of people with my name in the United States:

What do you notice about the two graphs? What do you wonder?

There are many mathematical avenues to explore within the name results at this website. Beyond looking up your own name or your students’ names, you can also compare multiple names. For example, here’s a graph that compares the popularity of my name with the name Donna (the name of the esteemed director of the Adult Numeracy Center):

What do you notice about this graph? What do you wonder about? One thing I wonder about is whether the Ritchie Valens 1958 hit song “Oh Donna” had anything to do with the name’s popularity. What do you think?

The trends in naming babies can reflect what is going on in the world as well as what is popular in music, literature, movies, or TV shows. Here’s a striking example – this graph shows the historical popularity of the name Barack:

One interesting thing to note on these graphs is that the scale on the y-axis is not always the same. The peaks for the graphs of Sarah and Donna are at over 1.5% of babies born in a given year. The peak for Barack is at about 0.0035% of babies born that year.

A few notes about exploring names with WolframAlpha

It’s important when looking at data to know where it comes from. WolframAlpha gets these numbers from the Social Security Administration.

If you have an uncommon name, WolframAlpha may not recognize that you are looking for name data. You can fix this by adding the word “name” to your search.

WolframAlpha may make assumptions about your search term that affect the results. The site lists the assumptions it made in a blue box at the top of the results. You can modify those assumptions if you need to.

How you can use this resource with your students

Have them look up their own names or compare their name to a classmate’s. Are they surprised by what they find? Or does it confirm their lived experience?

Do you have students who are thinking about baby names? They may want to look up the names they are thinking about to see how common or unique certain names are.

Challenge students to draw connections between the different ways that information is presented in the search results. For example, WolframAlpha told me that the most common age of a Sarah in the US is 30. Where can I find evidence for that in the graphs?

Click on the “more” button at the top right of the history graph to see even more graphs. What does each graph show? Why are the shapes similar or different? What is the y-axis measuring on each graph?

Ask your students what they see in their search results. They may surprise you!

There’s so much math you can explore with this engaging and personally relevant data. Clicking on links in the results may also bring you to interesting (although possibly not useful) information. For example, I discovered that the combined weight of all the Sarahs living in the U.S. is about 63,682 metric tons! What will you discover about your name? Go have fun!

SarahLonberg-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.

Welcome to the latest installment of our monthly series, “Will This Be on the Test?” (You can find links to the earlier installments at the bottom of this post.) 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.

I have heard an argument made that it’s not worth spending time teaching fractions in adult ed classes. One reason I’ve heard for this is that students are mostly in class to prepare for the test, and they can use a calculator on the test. Isn’t it more sensible and efficient to just spend one class period teaching them how to do fraction operations on a calculator and then moving on to higher level topics? After all, fractions are difficult and frustrating and it seems like you are always reteaching them anyway. Why not take the easy way out?

Unfortunately, taking the easy way out often doesn’t pay off in the long term – and in this case, it doesn’t pay off in the short term either. It doesn’t pay off in the long term because students miss out on real learning and sense-making that can help them make sense of the real-world math they will encounter. It doesn’t pay off in the short term because there’s more to solving test questions about fractions than just doing operations. High school equivalency tests are designed to test students’ mathematical reasoning, not their ability to use calculators.

Here is this month’s challenge. Imagine you are a student who is expert in doing fraction operations on the calculator but that is all you have learned about fractions. Would you be able to figure this out?

Before you read further, allow yourself to bring your full mathematical reasoning power to bear on this challenge. 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?

Here are some possible approaches:

1.Estimate! The snail has been traveling for 3/4 of an hour already and is only going to travel for another 15 minutes (or 1/4 of an hour). Would it get a little further? A lot further? Twice as far? A quick sketch could help a student get a handle on what kind of answer might be reasonable:

The snail is already more than halfway across the garden, so answer choices (a) and (b) don’t make sense because they are less than half. Answer choices (c) and (d) are both more than one, and it doesn’t look like the snail is going to make it past the end of the garden in only more 15 minutes, so those don’t make sense either. That leaves only one possible answer! (What fraction understandings were used here?)

2.Use a Singapore strip diagram. A student might start with a bar representing the 3/5 of the garden the snail has already crossed. Notice that it bears a resemblance to the sketch above: both show 3/5.

A student who understands that 3/4 of an hour means three groups of 1/4 of an hour, will be able to recognize that if the snail traverses three blocks in 3/4 of an hour, it is covering one block every 1/4 of an hour. (This is a non-trivial understanding and is important enough to have its own standard in the College and Career Readiness Standards for Adult Education: Understand a fraction a/b with a > 1 as a sum of fractions 1/b. (4.NF.3))

Filling in the fact that the snail covers one block every 1/4 of an hour will show the student the total fraction of the garden the snail will have covered after 15 more minutes:

3.Reason proportionally.This task is about something moving at a constant rate. In other words, the distance the snail covers is proportional to the amount of time it has been traveling. A student might reason that 15 minutes is one-third of 45 minutes so the snail would cross an additional third of the distance it had already covered. A student who understands that 3/5 is three groups of 1/5 can recognize that 1/5 is one-third of 3/5, so the snail will cover an additional 1/5 of the garden.

Did you notice that none of these approaches involved that old standard, “Choose and apply an operation”? Even after building a deep understanding of what is going on in this task, it may not be obvious which fraction operation, if any, is the ‘right’ one. It turns out that you can get the answer to this task using a fraction operation, so a calculator could be helpful to you if you can figure out which operation makes sense for the situation.

(Psst… want to know which operation it is? You can find the answer by dividing 3/5 by 3/4 because that is dividing distance by time which gives you a rate. Since the rate is in fractions of a garden per hour, the answer to this division gives you the total fraction traversed by the snail in one hour, which is what the question was asking for! This is also a good option, but the approaches explored above are probably more accessible to students.)

SarahLonberg-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.

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 2^{nd} or 3^{rd} 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 2^{nd} choice votes with the 1^{st} 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.

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.

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:

2^{4} • 2^{5}

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 x^{4} and x^{5} like this:

x^{4} = x • x • x • x

and

x^{5} = 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:

2^{3} • 2^{2} = 2 • 2 • 2 • 2 • 2 = 2^{5} (Check by hand or with a calculator – that works!)

2^{3} • 2^{3} = 2 • 2 • 2 • 2 • 2 • 2 = 2^{6} (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.

SarahLonberg-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.

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)^{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.