• Statistics Tables: Roulette and the Gambler’s Fallacy

This series of statistical examples is intended to inform students about the statistics and psychological ploys that casinos use on table games to make them more informed consumers. The aim of this series is to provide real-life examples of what students often only see as calculations on a page, or a topic in a textbook. This series is not intended to be a “how to beat the house” or any other sort of get rich quick scheme. If I had a way to beat the house, don’t you think I would be doing it and not giving up the secrets? Overall, it is good to think of casino games as good forms of entertainment, and you are encouraged to treat them as such. If you have or know someone who has a gambling problem, please use resources, and reach out to a professional for help.

Students often have a hard time conceptualizing independent events when calculating probabilities. The standard textbook example is to provide a probability matrix of events A and B along with their complements and have students compute the equation P(A|B)= P(A). Instead, let’s take the students back into the casino, this time to the Roulette table. The game is rather simple: a spinning wheel, with separated landing pockets distinguished by number and color, and ball are set in motion in opposite directions with the ball coming to rest in one of the 38 for an American Roulette wheel (or 37 for European Roulette, or 39 for Triple Zero Roulette wheels) numbered pockets. This example stays away from betting and different techniques and instead focuses on the simplistic spinning of the wheel to create the event. Any of the options that get the player closest to 50% on the Roulette Wheel can be used: Red/Black, Even/Odd, or 1-18/19-36. These options are all not 50% because of the inclusion of the 2 green 0 and 00 pockets (neither red/black, or even/odd, or between 1-18/19-36). The number and color of the pocket the ball lands in can be bet on using the different “outside spots” on the table (not a truly 50% bet since there are 18 red and 18 black and 38 spots on an American Roulette wheel 47.368%).

“Do the right thing for every student, every time.”

Callie Daniels has lived by this motto since she first heard it as an undergraduate education student.

Now, after 30 years as a higher-ed math instructor, Daniels understands how truly important that advice is — and has taken her time to share her teaching knowledge in a new webinar.

“Math is challenging, and some of our students are barely hanging on.”

She likens struggling math students to cowboys in a rodeo, holding on to their horses’ saddles for dear life.

“It’s hard to know what their needs are going to be when they get to us,” Daniels says, “but if we can determine the right thing and just do it, then that’s the best we have to offer our students.”

Her statements highlight a key dilemma for educators: How can you continuously offer your best to students while avoiding burnout?

“MyLab uses your time wisely and your students’ time effectively.”

Author Callie Daniels knows that when higher ed math instructors have the right tools at their disposal, it’s much easier to meet students where they are.

Engaging, interactive resources like MyLab Math and eTextbooks can help you empower learners and more easily identify and address your higher-ed math students’ needs.

In her 30-minute on-demand webinar, Daniels explains how to tailor MyLab Math and eTextbook resources to your unique teaching style and objectives

• Making the math of finance relevant to students’ lives

A recent survey sponsored by Inside Higher Education and College Pulse found that over 75% of undergraduate students will have student loan debt upon graduation. Of those students nearly half of the respondents do not know what their monthly payments will be. In the same study, about 25% of the students reported having credit debt and about 15% reported having car loans.1

Finite mathematics texts often include a chapter on the mathematics of finance, and for decades these books have covered topics such as amortization of consumer loans with an emphasis on home mortgages. Although mortgage loan examples are helpful because they often last thirty years and can involve large amounts of accrued interest, as the number of first-generation college students increase, a growing number of students do not come from families that paid a mortgage for their residence.

To make the mathematics of finance more relevant to students’ lived experiences, we emphasize examples that involve student loans, auto loans, and credit cards in Mathematics with Applications and Finite Mathematics.

Student loan examples

Even at a public university, the average amount of student loan debt in 2021 was \$30,030 for a bachelor’s degree. At an interest rate of 2.75%, that leads to a monthly payment of \$286.52. Over the course of 10 years, the total interest paid on the loan will be \$4,352.40.

Due to the rise of interest rates since the pandemic, the interest rate for student loans will be 5.50% for the 2023-2024 year,. To demonstrate the impact this will have on monthly payments, an instructor could ask the class, “With a current interest rate of 5.50%, how much does the monthly payment increase on the same amount borrowed of \$30,030 on a 10-year payment plan? How much total interest will accrue over the course of the payment plan?” The answer shows that the monthly payment increases by \$39.38, which may not appear to students to be a significant increase per month, but the overall interest paid over the course of the loan will more than double, to \$9,078.

Auto loan examples

The changes in the U.S. economy have also affected interest rates for auto loans. The Board of Governors of the Federal Reserve System reported that, in November 2016, the average rate for a 6-month new auto loan from commercial banks was 4.05%. In May 2023, the same group reported an average interest rate of 7.81%. How do these changes affect monthly payments and total interest payments?

To make this even more interesting, and perhaps more relevant to students’ lives, nerdwallet.com2 reported in August 2023, average auto loan interest rates by credit score and whether the automobile purchased was new or used.

• Improve Math Test Scores by Asking the Right Questions

Every college math instructor has been there. The students have been actively engaged in class. They’ve completed their homework (for the most part). The majority have even turned in the test review that you provided. Yay! Then you grade the tests. Questions were left blank, many scored shockingly low, and several students left sad notes in the margins. Some did well, but so many failed that the bell curve is upside-down! How is it they learned so little?!

Then, we dive into the ice cream to ease the pain (or maybe that’s just me).

Well, put the ice cream back in the freezer, my friends, because there is hope! A few tweaks to the way you design test exercises could potentially improve test scores and right that bell curve, not by lowering standards, but by more accurately assessing student knowledge by asking more focused test questions.

How many levels of cognition are you assessing?

One of the challenges that college math students face is that most math exercises require several levels of cognition and a variety of mastered objectives. Consider the exercise: “Solve 5𝑥(𝑥−2) = 3𝑥−2.”

• Statistics Tables: Craps for the Normal Distribution

This series of statistical examples is intended to inform students about the statistics (and how it relates to the psychology) casinos use on table games to make them a more informed consumer. This series is not intended to be a “how to beat the house” or any other sort of get rich quick scheme. If I had a way to beat the house, don’t you think I would be doing it and not giving up the secrets? Overall, it is good to think of casino games as forms of entertainment, and you are encouraged to treat them as such. If you have or know someone who has a gambling problem, please use resources, and reach out to a professional for help.

Often students do not have a concept for the Normal Distribution when it comes to the sampling chapter, and the Galton Board is used to give students the visual reference in the classroom for discrete random variables showing a normal distribution when enough observations are dropped through the board. Unless students are soon to be contestants on The Price Is Right and are faced with Plinko, or NBC’s game show The Wall, they are not likely to encounter the board in their life outside the class. A more accessible way for students to see the normal distribution and understand the importance of sample size is the Craps Table.

The casino game of Craps is simplistic: the act of throwing two dice and summing up the showing faces is the experiment in the game of craps. While using the full casino game with payouts and their corresponding probabilities creates a valued learning activity, this activity focuses only on the act of rolling the dice. To emphasize the previous chapters (discrete random variables categorization and visualization)  the image below shows the number of ways the dice total can occur:

• Guided Notes: A Road Sign to Success for Video-Based Learning in Math Courses

We’ve all been there. We sit down to watch a movie. The storyline is a little slow in the beginning and our minds start to drift. As we try to bring our attention back to the screen, we decide we’re a little hungry. We get up and go to the kitchen to make a snack, all the while telling ourselves that we didn’t need to pause the movie because we can hear it in the kitchen. We return to the living room and settle-in to watch the movie as we munch on our pizza rolls and soda. The food rouses little Gizmo from underneath the couch and she sneaks out to investigate the enticing aroma. We offer her part of our snack and give her a little scratch behind the ears. Before we know it, we are involved in an all-out tug-of-war with a 10 pound ball of fur. The movie is long forgotten.

Now replace the movie in this scenario with the carefully constructed video lessons that you have created for your students so that they would be eager to delve into the latest lesson on solving equations or factoring polynomials. The truth is that most students endure video lectures as a means to an end but struggle to stay engaged enough to absorb the material. It would be easy to say that this is just an issue for developmental or freshmen level students. The harsh reality is that it is true at all levels in all subjects. I witnessed first-hand as my son, who was finishing a master’s degree in Biosystems Engineering, struggled to stay awake while watching online lectures for a required statistics course which was only offered in an online format. He would stop every ten minutes, literally take a lap around the house, and then sit down to try and watch a few more minutes.

While there is no universal solution to this difficulty for students, we can supply them with tools which will help to mitigate the time lost to distracted viewing. When I created full lesson videos for my online students several years ago (pre-covid), I included colorful guided notes to help them stay engaged with the material. Using PDF files deployed in our learning management system, I supply my students with word-for-word, picture-for-picture materials that match the video they are watching. I have strategically placed blanks and empty boxes on these guided pages, so that the student must fill-in-the-blank as they watch the video. If their mind begins to wander, they will miss a blank or box and will have to rewind to get the needed information. Sometimes the blanks are words that are being said in the audio. Sometimes the boxes are specific letters or numbers that are relative to the problem being shown. It is important to include three keys for creating and successfully implementing guided notes in your course: Color, Active Learning, and Grading.

• A conversation with Lone Star College - Kingwood math professor, Mari Menard

In March 2023, the Pearson Math & Stats team had the pleasure of speaking with Lone Star College–Kingwood Mathematics professor, Mari Menard. In the conversation below, hear how she came to teach math after a failed attempt at Medical Tech school, and a few other lessons regarding teaching higher education that she has learned over the years. She also talks a little about the features in MyLab Math she likes the most, and why she changes things up every semester. We hope you enjoy the conversation.

Pearson Math & Stats Team:

What made you want to become an instructor?

Mari Menard (MM)

That's the funniest thing. When I first started my career, I thought, “I’m out of high school, what now?” I was going to go into the medical field, and what I soon found out, it wasn't for me. Some of the classes I was going to have to take over again. So, I dropped out and decided to come back to college a year later. My mom was the one who told me to do math. However, I could not multiply in grade school, as I had to go to what they called Resource Math. And only there did I learn how to multiply.

So, when I went back to school, my college advisor asked me if I had taken trig or any of the other classes. I told him, “No, I don't really even know what trig is”. His feeling was, “Well, then that is where we are going to start. And if you don't do well in calculus, then I need you to really rethink your degree plan.” So, right there I thought to myself, I'm going to show you. I am going to really show you.

From there, two main things happened. First, he ended up being my calculus instructor (and there were several other classes I took with him). Then, second, when I was in my graduate degree program, I graded for his calculus students, which was was interesting. As I was working, we learned (especially in calculus), it's a good idea to get a group of people together to study. Inevitably, I would be the one that would be at the chalkboard. I was answering the questions and the students that were there to study would be the ones asking me questions. And well, I was thinking I am pretty good at this. At first I had thought I would be at a high school. But then I was like, “You know, no, I don't really think I want that.” And so I've never taught anywhere but in college. (laughs)

I like to say I've never left college since I returned in 1992. I always say people retire after they teach and they still end up teaching. I think my main thing was when I was helping other students when we were studying Cal 1, Cal 2, Cal 3...all the way through differential equations...I was the one always at the board working the questions and answering questions.

Good

Pearson:

Go back to before you became an instructor. What in life led you to want to do med school?

MM:

So the degree plan was called ‘medical technology’, and I was studying to be a Medical Technologist. I'm not sure if you know what those are, but I worked in a lab at a hospital. My mother's a nurse. My brother's a nurse. My brother-in-law is a nurse. So, people in my family were in the health field and I loved working in the lab. I was a phlebotomist for several years. I drew people's blood in the hospital. So, the person in charge of the laboratory where the blood and other bodily things go is the Medical Technologist. They do lab work, microbiology, and things like that. I thought, well, that sounds like me. I would love that. So, I was working in the lab and I come to find out there's certain things I cannot handle, and one of them is mucus. Mucus and I do not get along. (laughs)

This is one thing a lot of students don't realize; you should get involved in the career path you're thinking about. I was so happy I did because I saw how the lab worked. And it wasn't just these random blood samples and people, you know, it was people's bodily fluids coming our way. Or from people that, for example, lost a leg; there would be body parts. And the smell of formaldehyde. And if you can't handle that stuff, then the lab is not the place for you. And that's how I soon learned it was not the place for me.

Pearson:

That's great. Thank you for indulging us.

MM:

Oh no, that's perfect, because people wonder, “how do you go from one to the next?” But it is also why I have a minor in biology. (laughs)

Pearson:

What courses are you currently teaching, and are there any that you've taught in the past that you don't teach now but want to teach again? Or are you kind of good with where your career is taking you and these are the ones that you enjoy the most?

MM:

I think so. I've only taught at the college level before, never at a university, and always at a 2-year college. I lead into that because most of your colleges were just on the freshman and sophomore level math anyway. So, when the developmental education thing was going on that was when I started, and am now in my 22nd plus year of teaching.

I used to love teaching pre-algebra because they (students) would follow me. So, I would teach pre-algebra, and then they would follow me to introductory and intermediate college algebra and so on. And you can see them growing and doing well.

In Texas, we now have corequisites, and it's six courses total, and they're taking two math classes. Which means, developmental math students are taking their developmental math class and their credit level class at the same time. They're trying to do math in both classes. And if they don't understand, the developmental course should be first so that it helps them with the credit level. So, I like corequisites and the credit level, provided it's a cohort of students. I could have students like I have in a credit level business math class. I have corequisite students in there, but I also have students that don't need corequisites, so it's called ‘co-mingled’.

But, currently, I teach what we call business math. I love it. It's by far one of my favorite classes to teach. It was the hardest to find corequisite content for because you want to find word problems since they encounter a lot of word problems in that class.

And I'm currently back teaching trig and precal, which I love too. We're going to do trig identities next class. I said I can do these forever. I could do every trig identity that I come across and still do more because I like them so much. And the students often say that’s not good because if you like it, then we know we're not going to like it. But, I’m thinking you don't know that.

The one (course) I wish I taught that I haven't taught in a while is the business calculus class. But I guess the best way to say the reason for that is the students. They don't appreciate what that class shows. It shows the whole purpose of business. It doesn't just get into the revenue and the cost and all that stuff, but it shows what happens as things change. Just tiny little changes and what it can do to the business. But for a lot of students, unfortunately, I don't know if it has to do with COVID or just with all the technology that’s available, they just say why can't we just use our calculator? And, well, your calculator can't tell somebody if the slope is negative, and what that means in the context of their business. If you're saying, “Oh, I'm good, but your cost is constantly going up, and if your profit is constantly going down, how are you good? And if you don't even understand the difference between revenue, cost and profit? Then how are you good?” And so, yeah, I try to get them in the finite class. By that I mean the math for business class before they get to the business calculus class so that they have the understanding of how important it is regarding all these aspects of it. And it is a word problem which they hate. But life is a word problem. (laughs)

Pearson:

What is one best practice that you use that you think works really well and you would want to share with others, whether it's in a classroom setting, working in groups, or working one-on-one with a new teaching technology?

MM:

It's kind of funny because every semester I change things, which I guess is one of my best practices. I'm always asking for student feedback. Not how I teach, but what I use in terms of resources or what I use to calculate their grade. So, here’s an example...

Previously, I used MyLab Math homework as a bonus option. The minute students hear bonus they think, “oh, I don't have to do it!” So, then none of them were doing anything. Of course, when they take a test they wonder [if there’s any bonus point opportunities], but by then it's kind of too late.

Last semester I used homework as a bonus, where I had discussion boards in our online learning platform as a graded assignment in the face-to-face class. And one of my students at the end of the semester said it makes no sense that we're doing a discussion board and a face-to-face class. I asked them, “what if I use it as bonus?” And she said, ‘’yes, because then it's something that's not going to hurt us. It can only help us.” So I asked her, “what about the homework?” And she replied, “that's the stuff you need to grade, because if we don't know what we're doing, then by the time we take the test or do the practice test or do quizzes and MyLab Math work, then we haven't learned anything.” She, of course, was a student that did really well. She was doing all of the things, you know. But I even had students that didn't do the homework, so the homework needs to be part of the grade.

And I thought, “Hmmm, how do I do that?” So, I made homework for some of my sections but not all. When we teach 30 sections, you can't have homework for every section. I usually base it on anywhere between two and five sections; it just depends on the course. I designated MyLab Math homework for one, and then I tell them it's over sections, let's say sections one through five. I provide the media options (which I love by the way) and then there's questions that they'll work on. What I tell them is if you can't do these and I have to help, I will turn the example off, because I think they just try to find a shortcut way to compare them and then just put the answer in there. I also tried giving them an unlimited number of times for each question, which I've determined was not a good thing.

Students love to circumnavigate me, and try to find an easy way to solve what I've done. So what they're doing, I fully believe is, they're just hitting the reiterations until they see a question they've already done and they're not really learning it. They're just trying to regurgitate it. Which isn't going to help them. And this is why I'm still getting students making hundreds on the homework but making a 20 on my test.

I think then my best practice is realizing that change is not a bad thing. I always tell people I learned that through COVID. Change is never bad, especially if it's going to improve things. Or not necessarily improve, but enhance what I already do.

Pearson:

Do you feel the pandemic helped students, that they think more conceptually, and that you are able to use content like the pandemic within the classroom and relates it to their day-to-day, and how? Also, was there an increased interest in that topic or were you caught up in noticing that there were a lot of students falling behind?

MM:

Unfortunately, I think that's what happened. So, I just had a test last week and I have never had what happened happen before. I think it was five students, and with three of them, the minute they saw their test, they were like so when am I going to be able to do a retest? One of them e-mailed me and said they weren't feeling well. They wanted me to send the test. I guess so he could take the test at home? I think some of that behavior has come out of COVID, in terms of what students expectations are, and I like to say what they can get away with. I think they're relearning just as we are.

Like now, for me, I am much more mellow about things like the student that wanted me to e-mail him the test so he could take it home. I laughed for a full day. I mean, I just laughed. Because, who does that? I mean, who does that?! Essentially, he wanted me to e-mail him the test so he could take it, and this was a trig student. So after I laughed about it, I referred him to the syllabus and how I offer makeups, and that you can take the makeup test at the testing center here at the college. The test was on Thursday and I gave him through yesterday to take it again. But, I haven’t heard back from him, and he didn't come to class today. Oh, and another student, he slept through the exam. So, you know... (laughs)

So, is it that the students have really changed? I don't think so. I've experienced all kinds of things in the 20 years I've been teaching. But are they a little more interesting as to what their expectations are? Definitely! Where are they getting this idea they can do retakes? Well, I fully believe that that's what happened in high school. Because they were just trying to make it through high school, you know, and I understand that. But it's now college and I've even learned some of the universities now are going back to what it was like pre-COVID. And it's taking some of the other colleges a little longer.

Pearson:

In your opinion, what is higher education going to look like in the next two to three years? Is it a little bit of revisiting the past moving forward, while also trying to reuse what you've learned about what their expectations are around bonus work, regular homework, test retakes, etc.?

MM:

My thing is if they want to take a test at home, then take an online class. Some instructors are allowing students to take tests at home. But for me, I have an online trade class. Their tests are all taken through MyLab Math. There's no testing center. They can take it through a date range. You have to submit your work, as there are regular expectations. So moving forward, I'm not stuck on if this is how I'm going to do it for the rest of my career scenario. But, again, I'm constantly changing, which I think is stupid on my part sometimes because then I have to work at things every semester. (laughs)

Currently, I have no videos to use, which is irritating me a little. At the moment, I am just using publisher videos, such as Pearson's videos and all the other things that students have resources for. I had planned on doing that, making videos and everything but... This semester there's been no time between fall and spring. So, am I going to make some videos over the summer? Heck yes I am!

But for me, it's always just actively asking students how things went. I think I have things set up pretty good in terms of this is one of my favorite things. I've learned to say certain things to students because students will say, “when am I ever going to use this?” And I say, “let me just tell you. In this world, we all usually will have a job, and the requirements of your job are pretty clear.”

So for me, my expectations, are that when you continue on, if you need to learn whatever it is I'm supposed to teach you in this class, you've got it. But, I also want to make sure that I teach student learning outcomes. So you may not like linear programming or probability or set theory or trigonometric in identities. But, it's my responsibility to teach you. I'm going to do that to the best of my ability. I provide you with things to help you along the way, learning from videos, lectures, and notes. I have booklets. I have PowerPoints. I have all the things, you know, homework, and quizzes, to see how things go. And if you have that learning, then I will put down a check mark and I'm doing my job. Then I can move on.

Pearson:

Finally, are there two or three big things you think everybody should use or the reasons you use it? Such as the videos you were talking about creating yourself and having those inserted into the lesson(s), or is that something you would like to be able to do so that it's interchangeable with the content that Pearson provides?

MM:

I love the ability to change or to do what I want to with MyLab Math.

For instance, I insert my logo, which some say who cares? The college, you know, they're making my theme, the colors, the layout and changing the names of things, integrating it with our learning management platform and all the resources. So, one of the ones that most folks, and they call it different things based on the publisher, but they'll have guided notes or something like that. I've actually taken them and made them my own. So, I still put Pearson on the bottom, and I also use the Beecher Book, and I use the precal Sullivan book.

Students don't need a whole bunch of stuff. Sometimes the problem is they get too much stuff. They'll go out and find YouTube, Khan Academy, and various other things to try and learn one thing. And so now they've seen it five or six different ways. And I wonder how is it helping you? That's just going to confuse you. So I have Pearson. Nice and sweet. I’ve got somebody's other way in which to do it, and I have my way. From there, I include a quiz within it. So, I like that idea! Then Pearson is a just a great resource scenario, such as the pooling option. Don't get rid of the pooling option! (laughs) That by far is my favorite things. I do a practice test and their test, and I believe there's a strong correlation with how I have the practice test set up to the grade they actually make on my test.

The Role of the Affective Domain and ‘Soft Skills’ in Collegiate Math

Although students are reaching college-level math classes sooner due to many of the recent acceleration projects, they often arrive without developing many of the student skills that would improve their chances for success. One way we can help our students to develop those skills is through the use of prompts and assignments that focus on the affective domain.

I like to use a series of prompts in my classes, because they are powerful and flexible. They can be used for writing assignments, discussion board prompts, or to start face-to-face classroom discussions. The in-person discussions can start with student volunteers sharing their responses, or they could use a think-pair-share format. They fit well in corequisite support classes, as well as the main credit bearing course. Finally, the instructor could devote as little as 5-10 minutes of class time to these prompts or use an entire class session.

I focus on topics that I feel are most beneficial to my students, helping them to be more successful in my class as well as becoming a better learner in general. I like to start with prompts related to developing a growth mindset, because I believe that helping students reset their mindset regarding mathematics is an important first step on the path to success.

I then follow up with prompts related to time management. Our students are over committed, yet lack skills for scheduling, organizing, or prioritizing tasks. Many students struggle with procrastination as well. These prompts lead to discussion and the discovery of strategies to improve time management.

After growth mindset and time management, I like to add prompts for goal setting using S.M.A.R.T. goals, reflection, and other study skills. You can tailor your choice of subjects to the skills you feel are most important, or to build up skills that your students need the most help with.

Lastly, I have assembled a set of 30 prompts that are included in the new third edition of Interactive Statistics, including notes for using them in class. Please reach out to me if you have any questions on using affective domain prompts in your classes, or if you are looking for feedback on strategies that you are using to help your students.

Try a prompt in your class with one of these samples.

• Visualize Calculus. It’s a Revolution.

Many virtual resources exist in MyLab and elsewhere to help students see concepts of calculus. For example, we have embedded almost 800 interactive figures in our Calculus eTextbook (Briggs, Cochran, Gillett, and Schulz) in MyLab. The figures available in our interactive eTextbook have been used successfully by tens of thousands of teachers and students to master calculus concepts, such as finding the volume for a solid of revolution. However, an interactive figure on a device screen is not helpful for every calculus student and is likely useless for a visually impaired calculus student.

The challenge of helping every student learn mathematics has weighed on me throughout my teaching career; that is one reason I have been passionate, obsessed perhaps, about creating interactive visualizations. However, I have had students for whom the interaction of a figure on screen has not been sufficient for them to understand the concept being visualized in the figure. For these students, a physical 3D object to feel and manipulate would have been helpful. Furthermore, 3D objects have great potential to help us teach calculus to students with significant visual impairments.

I have embarked on a new project this year to create a large number, say 100 or so, of 3D printable objects for calculus, beginning with solids of revolution and continuing through multivariable and vector calculus. For example, the following image is a 3D solid used for approximating the volume of the solid of revolution created by revolving the area in Quadrant I beneath the curve around the -axis using cylindrical shells, with .

For each 3D object, an image of the solid, an STL file, and the Mathematica source code used to create the STL file will be accessible from within any course in MyLab using our current calculus materials. The 3D objects can be sliced and printed on your own 3D printer or easily sent to an online 3D printing service.

Watch the webinar >