An Idea for After the AP Exam: Who Takes Calculus?

My last post was about the documentary Counted Out. I am hoping that the Pennsylvania Council of Teachers of Mathematics will be able to hold a virtual showing of this film in August or September. If that happens, I'll be sure to announce it on my blog.

If you are wondering which students are counted out in today's math classrooms, I encourage you and your students to read this blog by Just Equations: (Not Yet) Hidden Figures: Preserving Data for the Future of Education.

The blog states:"One of the motivations for this blog's focus is the knowledge that this data source and other Just Equations has relied on may not contiue to be available, given recent cuts to the U.S. Department of Education." Data related to the demographics of students who complete various math and science courses can be found in this chapter of a report called High School Mathematics and Science Course Completion. If I was still in the classroom, I would probably show my AP Statistics and AP Calculus students a few of the graphs and ask them to tell me the story shown by those graphs. The graphs I found most fascinating were the following:

Figure 2: Percentage of public and private high school graduates who completed selected mathematics and science courses in high school, by race/ethnicity: 2019

Figure 4. Percentage of public and private high school graduates who completed selected mathematics and science courses in high school, by school type: 2019

Figure 5. Percentage of public and private high school graduates who completed selected mathematics and science courses in high school, by percentage of students at their school who were eligible for free or reduced-price lunch: 2019

After discussing the story of the graphs, we would look at the report itself, how the data was gathered and the conclusions that were drawn by the authors. Finally, I would have the students reflect on what surprised them the most about the data and encourage them to share their new understanding with parents and other adults in their sphere.

Note: If you want to learn more about Just Equations and their work to support equity in math education, I would encourage you to read their report called The Mathematics of Opportunity: Rethinking the Role of Math in Educational Equity. This report would be a good companion to accompany a discussion about the documentary Counted Out.

Counted Out: An Urgent Call to Action

Last Wednesday, I saw a screening of the movie Counted Out (www.countedoutfilm.com/) at Lafayette College in Easton, Pennsylvania with about 70 professors and teachers. After watching the movie, I felt hope. Hope is something that is often in short supply these days. The movie blends interviews with mathematicians and math educators along with classroom observations. It also includes chats with students, both children and adults, who have felt the fear, shame and anxiety that is experienced by many during their K-12 schooling. Each story they told was something that has been almost universally felt by students at some point on their math journey.

My earliest memories of math involve drilled timed tests on addition and subtraction facts in second grade. I remember not being able to get the answers fast enough before the disembodied voice on the record player moved onto the next question. Tears and frustration frequently accompanied those tests. Eventually it got better. Was it because the teacher took me aside and told me it is ok to not be fast at math as you are learning? Was it because my parents drilled me on my math facts at home? Was it because eventually I just learned them because I had no choice?

The answer to all of these questions is “no”. I distinctly remember learning the math facts because I saw patterns. I saw that the sum of 9 plus a single digit was a number that started with a 1 and ended in a number that was one smaller than the single digit. For example, 9 plus 7 would end in a 6. The answer was 16. I did not realize it at the time, but I was essentially regrouping and rewriting 9 + 7 as 9 + 1 + 6 = 10 + 6 = 16. I also figured out that it was ok if I didn’t memorize 8 + 5, since I knew 8 + 4 and could add one more. It felt like I had discovered a secret to solving these that I thought might be considered cheating. I didn’t share my discoveries with anyone.

Focusing on just getting the answer and thinking of doing mathematics as a linear path with one right answer is what is causing many students to be “counted out”. The movie emphasizes that new teaching methods must be employed to allow students to have access to math and more options. In addition, math has typically been used to divide children into groups resulting in many students seeing themselves as incapable of math simply because of how they were tracked in math class. 

I did walk away from the movie with a feeling of hope, but also a feeling of being overwhelmed. Helping students and adults see the urgency of why math needs to be viewed differently, essentially changing the views of the masses, involves working with larger systems. How do we get state departments of education to understand that there is no quick fix to challenging problems? How do we get all stakeholders to see that math eduction in many areas has not advanced into the 21st century? How do we get current educators to understand that the textbook they used to learn Algebra 1 in the 1980’s is no longer sufficient for the math needed today? How do we get to the place where it is recognized that all students are capable of learning math when we give them the tools to explore, play, and discover? How do we get everyone to see that access to quality math education is crucial to promoting equity, democracy and a sound economy?

I don’t have the answer to these questions, but I am hoping to host a virtual screening of the movie with the Pennsylvania Council of Teachers of Mathematics in August or September. Stay tuned for details.

Polar Function Intersection?

Although spring is around the corner, polar functions crossed my mind last weekend as temperatures fell into the teens. And when mathematicians think of cold, we think of polar functions. Consider the graph shown. How many points of intersection do you see?

Most people would say there is one point of intersection on the vertical axis. But what if I told you the two equations were the following? How many points of intersection are there on the domain of 0 to 2pi?

When you try to solve this system of equations, you get -2 = 1 + sin𝚹 or -3 = sin𝚹. Since the sine ratio can only vary between -1 and 1, this equation has no solution. So why does the graph show a point of intersection? Or does it? Watch the video below and decide for yourself.

I encourage you to play around with Desmos at desmos.com/calculator. Here are three things for you to explore. 

1. What if r = 2 instead of r = -2?
2. Notice that I added the term 0 times sin𝚹  to r = -2. What happens if that is removed?
3. Can you create a pair of polar functions that look like they intersect in 2 or 3 places, but don’t actually intersect.

Share your observations in the comments. 

The World Fertility Rate: A Relevant Mathematics Topic

I have decided to begin to blog again. As someone who is very task oriented, I was always excited to be able to check of "write blog" from my To Do list, especially in the days when I blogged each day for one full school year. I really have no idea how I found time to do that. It must be that I created a 25-hour day. As of September of 2024, I am no longer in the classroom and I wondered what topics would be in my future blogs. You'll need to read future blogs to find out! 

Besides the near daily question of "Did I miss anything?" from students returning from an absence, the second most asked question high school math teachers hear is "When will I ever use this?" This is often heard for the first time in Algebra classes, where math begins to seem like an exercise in algebraic origami. For those of us that have an innate love of math, (I wore a red t-shirt that boldly declared “I Love Math” when meeting my colleagues for the first time at Champlain Valley Union High School in Vermont.) we don’t see the need for all math to be relevant. Math is beautiful and should be studied to improve reasoning and thinking. But sadly, it seems as if much of the United States feels that math must be relevant to be worthy of being a part of the curriculum.

And this leads me to the topic of today's blog - The World Fertility Rate: A Relevant Mathematics Topic. As I was looking at a daily news email a few days ago, I clicked on a link with the following headline:

Charting global fertility rates, which have declined by half since 1965

I won’t go into the all the details of what I discovered and ruin the fun for you. But I did find the following two graphs to be quite interesting and learned that the “Global Replacement Fertility Rate” of 2.3 children was reached in 2015. This is the fertility rate at which the population says constant. You might be wondering why it isn’t 2, but that is because we need to take into account the deaths of the female population before the end of their childbearing years. 

If I was using these graphs in my classroom, I would give students 5 minutes to look at both and write down what they notice and wonder about the graphs. Then, I would have them share their ideas with a partner for about 5 minutes. Next, we would debrief as a class for about 10-15 minutes. From our list of wondering questions, students would be asked to choose 1 or 2 of the questions and work in groups to try to find the answer to those questions. 

Possible questions could include the following: 

1. Why is China’s fertility rate lower than the fertility rate in the US now than before 1980?
2. If this trend were to continue, when would a global average fertility rate of 2.3 be reached?
3. Do countries with higher (or lower) fertility rates have common characteristics?

  

The data, graphs and analysis can be found at https://ourworldindata.org/global-decline-fertility-rate.

If you use this graph with your students, post how things went in the comments. Math educators sharing and learning from each other makes the world brighter.

Random Rendevous with Desmos

Over a year ago I heard statistics professor Allan Rossman speak during a virtual session. The problem he shared was easy to understand and potentially challenging to solve. 

Here's the scenario:

Two friends agree to meet for lunch.  They agree to wait for 15 minutes for the other person to arrive. If they don't arrive in that 15 minute window, then they leave. The friends arrive at a random time between noon and 1 PM.  What is the probability they will see each other?

Allan demonstrated how to solve this problem using a simulation in R and I decided to try creating a simulation for the problem using Desmos.  I presented my Desmos version of this solution during a 15 minute Epsilon Talk at Moravian University during the spring semester.  This blog will take you through that talk.

Let's consider the following times.  Do they work or not?

To help us think about all the different random rendevous pairings, we will use the following definition. 

Let (A, B) represent the minutes after noon that Person A arrives and the minutes after noon Person B arrives.

Example: Person A arrives at 12:15 PM and Person B arrives at 12:56 PM. This would be represented as (15, 56).

Here's an image that shows many ordered pairs like (15, 56). Which pairs correspond to meeting for lunch?

Remember that there is a 15 minute window were each person will wait for the other person. This means 

and we can color code our Desmos graph, where red dots mean that the two friends met and blue dots means the friends did not meet. Is there a pattern that emerges?

I invite you to go to this Desmos calculator page and use the slider to investigate adding more random ordered pairs.  What shape is formed by the red dots?  

Other Desmos simulations can be seen here and here.  The first simulation creates a histogram with the width of each bar being 15 minutes. The second simulation uses a uniform distribution and a ticker and was created by Andrew Knauft. 

Of couse any good question leads to more questions...here are some others that I invite you to explore:

• How long should the friends wait if they want the probability of meeting for lunch to be 50%?

• What if they wait different amounts of time?

• What if they arrive randomly in a 30 minute window of time?

• What if more than 2 people want to meet for lunch?

Please share your answers to these questions or your own random rendevous questions.

Post 4 of Wally's Math Books: Explore Like a Mathematican

About a week ago I listened to a podcast called "Can we help kids learn to love math?" which featured math educators Christopher Danielson (@trianglemancsd) and Sara VanDerWerf (@saravdwerf). One of the key takeaways from this is that math is NOT about calculating fast. In fact, we destroy kids views of themselves as mathematicians when we tell them that being 5 seconds slower at a calculation is bad. The truth is math is about exploring and playing - things kids enjoy. Math is about noticing, describing and generalizing.  These things can happen naturally when students are given time to play with math.   

So what does playing with math have to do with Wally's math books?  If you look at my grandfather's college math textbooks from the 1920's, you'd think that math is all about solving things in one way, using theorems and performing calculations.  However, I did find glimpses of what mathematicians actually do in this book - the making observations, looking for patterns and drawing conclusions part of mathematics. Here is one such question in the chapter called "General Theory of Equations":

                    

I graphed these three functions on Desmos.  It was easy to see that the functions were just vertical transformations of each other.  However, I was curious about what happens with the x-intercepts.  So, I created a table which included these functions and a few others

Wow!  The sum of the x-intercepts is 3.  This leads to a whole list of questions.

• Why is the sum of the roots 3? 
• Is it related to coefficient of the quadratic term, which is -3?  
• What would happen if I changed that coefficient? 
• If I changed the coefficient to -4, would the sum of the x-intercepts be 4? 
• What if I added a linear term to the function? How would the sum change?
• What about the product of the x-intecepts? How is it related to the coefficients of the cubic polyonomial?
• How is the sum and product of the roots impacted if the polynomial is an even degree?

Noticing, wondering, generalizing...thanks to Wally and his math book for giving me something new to learn about polynomials and giving me an opportunity to think like a mathematician.

Post 3 of Wally's Math Books: The Solution to the Grapefruit Problem

In my previous post I noted that one of the Wally's textbooks was explicity written to cover all of what was on the New York Regents exams, which still exist today - although the ones for January 2022 have been cancelled due to COVID-19.  What about these exams is so enduring that thay have lasted since 1865?  The exams have been cancelled multiple times more than once during the pandemic and that makes me wonder.  Are these exams really all that important? And if not, why administer them at all?  But I digress.

Here is the problem I had posted previously.  Did you try to solve it or give it to some students to try to solve?

Since I am a math teacher, I decided to approach this from an algebraic perspective. But then I got an equation that requires the quadratic formula and so I switched to using Desmos.  After you students try to solve it, see if they can follow my paper/tech combo solution.

Post 2 of Wallly's Math Books: Fundamental Laws and Operations

This is the second post of a series of blogs devoted to my grandfather Wally's college freshmen math textbooks. One of the books Wally used was called Advanced Algebra. It was published in 1925 and in the preface it states the book is designed for students who have had one or two years of college algebra.  In addition, it states "The syllabi of the College Entrance Board, of the Regents of the State of New York, and of practically all states have been covered." This makes me wonder if there are college math texts today that take the New York Regents or Common Core Standards into account when designing their textbook.

In the previous blog post, I encourgaed people to try the following problem.  In addition to posting on twitter, I linked the blog post to the "Higher Education Learning Collective" Facebook group and was told by some that this problem was too easy to be a college level problem.  Here was that problem.

Problem 1: Prove that "The sum of the squares of any two unequal real numbers is greater than twice their product.  That is if a ≠ b, a² + b² > 2ab."  

Here is essentially the algebraic proof as given in the textbook: 

Given: a ≠ b, we know that a - b ≠ 0.  Squaring both sides gives  a² - 2ab+ b² > 0 and that means that a² + b²  > 2ab.

Here is a way to visually see that the sum of the areas of two squares is more than double the area of a rectangle formed by the sides of the squares by using Desmos - a much more dynamic and visually appealing proof.

And here's a link to the original Desmos page that I used to create the video above: https://www.desmos.com/calculator/r6p2hpdplf

Problem 2: This problem comes from pg. 28 of the Chapter 1 Fundamental Laws and Operations. Although it doesn't appear to have been assigned to my grandfather and his classmates, it seemed like a interesting problem to solve. In fact, it is problem that I could see myself giving to my MATH 120 (Math for Teaching) students on the first day of class just to see what problem solving strategies they might use.

How would you solve this problem?  Are there any generalizations you can draw?  Feel free to post in the comments of this blog post.  We'll look at at least two approaches to this problem in my next post.

Post 1 of Wally's Math Books: The Syllabus + Problem 1

As I opened up my grandfather's math books - Advanced Algebra by Edgerton and Carpenter, published in 1925 and Plane Analytic Geometry by Barnett, published in 1927 - I saw a piece of paper folded up.  I carefully opened it and realized it was the course syllabus! How exciting to see how a syllabus from the fall of 1927 compares to the current syllabi I write for the math classes I teach.  Before you read my observations about this document, I encourage you to look at it yourself.  Zoom in.  What do you notice? What do you wonder?  Feel free to add a comment to this blog post.

What I Noticed 

The paragraph at the top of the paper contains many instructions found in a typical syllabus. 

1. Time Spent Outside of Class - It says that each assignment "merits two hours of study and review". My syllabus says "You are expected to spend at least 7 hours outside of class working on problems or studying for this class." 
2. Number of Classes Per Week - Although I don't see how long a class is on this syllabus, the class meets 4 times each week.  This would translate into 8 hours of work spent outside of class.  My class meets 3 times each week for 70 minutes each class period.  So, it is possible that the amount of in class time may be similar.
3. Ask Questions and Get Help - It is clearly stated that the instructor Professor Waltz wants the students to ask questions during class and outside of class.  There is no mention of specific office hours.  Maybe office hours were a construct that became standard after 1927. 
4. Saturday Help Sessions - There is time and a location for help.  Saturday at 9 AM would not go over well for most students. But then again, it is a time when all of them would be available.  We currently offer virtual help sessions 11 hours a week and every day of the week. except Saturday!

What I Wondered

1. What problems should be done? The list of problems was outlined, but what did 5n, 2n mean?  As I looked in Wally's math books, I noticed that he had checkmarks beside certain problems.  Every 3rd problem or every 5th problem. So, this notation was to show multiples of a given value.
2. Assessments Where there quizzes or exams?  It says "Final Tests" on class meeting #62.  Since it says "an average of A or B will exempt from final tests", there must be something that is being graded.  But what?
3. Where are the problems found? There are two books listed and it looks like Plane Analytic Geometry book is used starting on October 31.  However, it is not clear that Advanced Algebra is the first book.  And does October 10 - 97(5, 8, 10) mean page number 97 and problems 5, 8 and 10 was the assignment on October 10th?  Maybe...but neither book has a page 97 with those problem numbers.
4. What is up with the margins? What is the arithmetic that Wally did on the side?  It looks like these may be his grades in the class and he is averaging them.  Perhaps he is trying to be sure he is exempted from the "Final Tests".

What did you notice and wonder about the syllabus?

Problem 1: This is not a problem that was assigned, but whas listed as a theorem on page 70 of Advanced Algebra.  It states "The sum of the squares of any two unequal real numbers is greater than twice their product.  That is if a ≠ b, a² + b² > 2ab."  Feel free to create your own proof and upload it in the comments or on twitter.  Make it an algebraic proof or a proof without words.  I have already started to work on a way to visualize this with Desmos and will share my solution in the next post.

Introducing the Series: Wally's Math Books

It's been some time since I last did a blog post. Looking back on my previous posts makes me realize that I miss writing about math and teaching math. It's a kind of therapy for me. Even if no one likes my post or comments on it, it is there for me to look back on later. And I have a sense of accomplishment when a post is done, without the need for peer review or endless edits. (Speaking of peer review...an article I wrote on The Global Math Department was recently published in the NCTM Journal Mathematics Teacher: Learning and Teaching PK-12.  The article appeared in the "Ear to the Ground" section and can be accessed here.)

Now back to the regularly scheduled blog post...I have decided to do a series of posts related to my grandfather's math textbooks from his freshmen year of college at The University of Pittsburgh.  Those posts will start in a week or so and I plan to do a post each week.  I figure that I blogged each day of the school year in a #teach180 blog during the 2017-2018 school year and I can meet a goal of blogging once a week.  The textbooks are from 1927 (nearly 100 years old) and inside I found what appears to be a copy of the syllabus! Now if only I could figure out what problems were assigned.  If I can figure that out, I may work through the exact same problems my grandfather did.  If I can't figure out the syllabus, then I'll select problems that have some of Wally's annotations written beside them.

The books are seen in the image at the top of this blog along with some notes that he left for me.  Walter Rupp (or Wally) was a chemical engineering major and his first course in his freshmen year involved books in Advanced Algebra and Analytic Geometry.  These courses laid the foundation for him to graduate at the age of 20 and become a successful engineer for Esso (now Exxon-Mobil).  He helped to create oil refineries in the Caribbean in the 1930's and some of his work led to patents of oil refining processes.

Summertime: Math Playtime with Desmos Art Project

Children learn so much and are so engaged in learning when they have time to play, experiment and fail.  Yes...failure is a part of the learning process, a challenging concept for many parents grounded in the "my-child-is-better-than-your-child" culture.  But without time to play, experiment and fail, I would not have learned the following items: HTML (which helps with writing this blog), embroidery stitches (learning chain stitches prior to YouTube can be challenging), and how to use Audacity to create audio files from the video files for the Global Math Department podcasts. Given time to experiment and play, there is much to be learned!

Image from needlenthread.com

As a high school math teacher for over twenty-five years, parents would sometimes ask me, "What should my child do over the summer? Khan Academy? A specific workbook? Take a test prep course? Enroll in a math class so they can AP Calculus and not fall behind their peers?"  I would often turn the question back on them and ask, "What is your child interested in doing this summer?"  Often this led to a discussion of what colleges or majors the parents wanted for their 15-year-old child, without the child being present for the conversation.

In each of these conversations with parents, I wanted to respond, "Give your child time to play, experiment and fail this summer. Ask them what they would like to try to learn and encourage them to share their failures and successes with you."  Of course, parents still crave some sort of structure and want their children to continue to do some math over the summer.  If you are a parent looking for this structure or you are a high school teacher that has been told to give a "summer math assignment", you can give an this optional creative task of "Desmos Art Project (Summer 2021)."  A special thank you to Julie Reulbach for initially creating a version of this Activity Builder for easy curation of the student art projects and to Javier Cabezas for creating the modifying colors screen in the activity.

Note to parents and teachers new to Desmos: You will need to create a free account at desmos.com to see the activity and then "assign" the activity to your child or students. Click on the triangle to the right of the "Assign" button to create a "Single Session Code". Click on "Create Invitation Code" and "View Dashboard". Then, you can copy the code to give to your child or students.

For examples of art projects from the most recent Desmos art contest, go to https://www.desmos.com/art and encourage your child or students to submit their work in any future Desmos art contests.

Student #106745: You Greeted Me By Name

All of my classes are being held live synchronously over Zoom this semester. One day I said to my MATH 120 (math for elementary teaching) students that I noticed they all showed up to class every day.  This wasn't necessarily true of my other classes.  Then I commented that I thought this was remarkable, especially considering the class is on M, W and F from 4:15 - 5:15 PM.  (Yes, it is a struggle to want to attend class on a Friday at that time.)  I also told them that I noticed that many of them kept their web cameras on.  

What happened next made me realize that connections can happen in a virtual environment.  They told me that I am one of the few teachers that greets them by name when they come to class.  I hadn't really thought of it before.  I use the waiting room and click on student names to admit the students into class. Their name is displayed which makes it very easy for me to say "Hello, NAME. Welcome to class."  Or on a Friday, "Happy Friday, NAME. I hope you have some plans to relax this weekend." Or to a specific student, "Hi, NAME.  Did you get to go for a bike ride today?" And on our last day of class, "Welcome to the final class of the semester, NAME."

If knowing my students - what motivates them, their view of math, how they evaluate their own learning - matters to me, then it makes sense that I should start by learning their names.  Learning names is not my strong suit, and in a face-to-face classroom, I seat students alphabetically by first name with a seating chart to help me learn their names a bit more quickly and easily. Luckily, I teach at a small college and with maximum class sizes of 25, it is a bit easier to learn student names and learn a bit about the students themselves.

You may think that calling students by name is trivial, but my students told me it matters. This makes sense. Our names are closely tied to our identity.  If we are called a name we don't want to be called, we often correct the individual.  For example, I go by my middle name and not my first name. I will change my Zoom name to Leigh, if I notice Zoom has me logged in with my first name, Sidney.  

Names are important not just in life, but in remembering those who are no longer with us.  I encourage you to either watch poet Billy Collins read "The Names", which was dedicated to the victims of 9/11 or read the NY Times online article called "Those We've Lost", a series designed to put the names and faces to the victims of COVID-19.  Understanding thousands of lives lost in both of these tragedies has a different impact on you when you hear, read and think about the indivdual names of the people impacted and their families.

Thank you to my students for helping me to remember that names do matter.  If I see you on campus at some point and I don't recognize you, please greet me by name.

Thank You Murphy's Law

We switched to online learning very rapidly in March of 2020. Classes were in person on a Friday and moved online on a Monday. Murphy's Law came into play - it turned out that I was to give one of my exams of the semeter on that Monday. I pushed the exam off by one class day to give me time to get the exam put on Canvas. Unfortuantely, it isn't as easy as it looks. My images did not show within the exam and students still had to do part of their exam on paper and scan and upload it. The exam was in two places and grading became a nightmare. For consistency, I like grading the same question across all student papers at the same time. I can see common errors and score student papers in a similar way for those errors. Canvas does not let me do that, which is a major drawback. Plus, most students uploaded their work in 4 - 6 image files. Grading took about three times as long as it did before and my feedback was limited since I could not type in math formatting.

 

This fall classes are still being held virtually via Zoom. This and my disasterous issues with assessments in Canvas, led to me to decide to use Desmos Activity Builder for quizzes in my Functions and Derivatives class. It has worked really well and the students adjusted to it quite easily. Several even commented that they liked taking the quizzes in Desmos Activity Builder during a mid-semester check-in survey. Doing quizzes in this way has been so much better that I don't see myself going back to paper quizzes even when we return to face-to-face learning. 

Why would you want to do assessments in Desmos Activity Builder? I can think of four main reasons: they are easy to create, support a variety of student responses, it is easy to manage classes and you have the ability to quickly give students feedback. I'll comment on each of these. 

1) Easy to Create: I can easily embed images within my quiz without having to upload them some place else, link it and check a box to make sure students can see it. I can also design a question that allows for student written input. An example of such a question can be seen in the student work below. Plus, it is very easy to copy one quiz and modify it to create a second version of the quiz.

2) Variety of Student Responses: In Canvas, I can ask a multiple choice question, but would need to ask a separate follow up question to have students explain their choice. In Desmos Activity Builder, that can all be done within the same question. Students also have the ability to type in their answers in math format by clicking on the keyboard. Formatting of math answers is not so easy in Canvas. Here's an example of a question involving rational functions with the math keyboard for students to easily enter their solution.

3) Managing Classes: In September of 2020, Desmos added a "Manage Class" option. This makes it easy to assign activities to classes of students. Why would you do this? If you don't use the "Manage Class" option, it is challenging to see who is taking the assessment and who is not. You might see there are 24 students logged in and you should have 25. Who isn't taking the assessment?  With "Manage Class" all your students names are already listed on the dashboard and you can easily see who has logged in and who has not. Plus, there is easy integration with Google Classroom. For more information about this feature, watch this brief video from Desmos. 

4) The Ability to Quickly Give Students Feedback:In March 2020, Desmos released its Feedback feature within Activity Builder. This allows teachers to give feedback to their students, including the ablity to type feedback with math formatting. As an added bonus, I can easily grade a single question for every student and then move to the next question! Students can see that they have feedback when they log into student.desmos.com. As a teacher, you can see if they have read their feedback by looking at the teacher dashboard. The grey triangle means a student has read the feedback and the green triangle means the student has not read the feedback.  Thinking this might be something you want to try? I have two recommendations. First, if you are new to Desmos Activity Builder, I recommend going to learn.desmos.com/activities to get started and learn.desmos.com/create to see how to create your own activities. Second, start small with just one or two slides of student input. Use Desmos Activity Builder as an exit ticket or warm-up. 

Interested in an activity to getting started?  Here is one of the quizzes I gave a few weeks ago.  If you have an account at teacher.desmos.com, you can copy and edit this quiz on Polynomials, Rational Functions and Limits. 

Thank you, Murphy's Law, for giving the impetus I needed to broaden my use of Desmos Activity Builder.  If you would like to learn more about Desmos and Desmos Activity Builder, I invite you to join me for my online seminar with BER called Making Best Use of Desmos to Strengthen Your Math Instruction (Grades 6 - 12).  Click on this link to see information about dates and how to register.

Learning Math is Social: We Are in This Together

Note: I haven't blogged recently and these thoughts are a culmination of ideas that have been bouncing around in my head over the past few weeks.

When I first saw the rules posted online for a local high school back in July, I shook my head in disbelief. The rules have changed somewhat (masks always required in class), but initially it said something like this:

Students can take off their masks when seated as long as they were facing the front of the room. If they get up to move around the room, they must have their mask on. And teachers were told: Students will not to sit in groups during class in order to maintain social distancing.

What many school administrators, department of education leaders and school board members don't realize is that high school teachers have advanced their teaching methods beyond the 1950's. Classrooms are configured with tables or desks in pods. Learning is social and we want our students to work together. Even after teaching the same content for ten years, my students make new observations or connections and amaze me with what they are able to conclude. This would NOT happen, if students worked by themselves. Or at the very least, only the student that thought of the new idea would benefit from it.

Even within my synchrounous virtual classroom this fall, I use breakout rooms on a daily basis and students work together for 10-15 minutes on course content. They post in a common Google doc or move things on a Google slide or write on a common Google Jamboard. And then we debrief as a class after they return from their breakout rooms. Learning together is really that important for my students. Learning is social and teachers need to be given the tools to help their students learn together while being socially and often physically distanced.

Image from @NeONBRAND

Even as teachers work to modify their lessons, they get new restrictions being thrown at them almost daily. An AP Statistics colleague of mine said he will have 2 days of face-to-face instruction and 2 days of asynchrounous virtual instruction per week. That doesn't sound bad, right? But then he has also been told that new content cannot be taught on the asynchronus days. That could make it a bit challenging to teach the entire curriculum. But it could be done, maybe. But wait...there's more. He will only have half of his students in front of him on the face-to-face instruction days. The other half of his students will be working virtual on those days. This means each day, he needs to prepare 2 different lessons for the same class. And because of social distancing, he will need to create or find new ways to do things that would normally have students do while working beside each other. This is the stuff of teacher nightmares.

Will teachers survive teaching in the pandemic? Maybe some will resign, as happened in a district near Buffalo, leaving nearly 80 virtual teacher vacancies. Those who wish to survive will work with their colleagues. Some will reach out via social media or email or by attending free webinars, like this one offered by The Global Math Department on using Delta Math for Distance Learning: https://www.bigmarker.com/GlobalMathDept/Using-DeltaMath-for-Distance-Learning.

I recently shared a pacing guide for AP Statistics with a teacher who made a plea for help over an AP Statistics Teacher Facebook group. This led to an email exchange, where I offered the following advice for starting the school year strong.

1. Be up front with your students - teaching this way will be very different than what you or they are used to.  If something is unclear or is going wrong (I can't tell you how many times I was talking for 15-30 seconds before a student would unmute themselves and tell me I was muted), they need to tell you and you thank them for sharing.
2. Let your students know, even with 28 years of experience, this is your first time teaching AP Stats.  Tell them to think about something they learned for the first time (like riding a bike or learning to dive) and the fact that it did not go well.  They had friends to help them learn and you have friends to turn to for help with teaching AP Statistics.
3. Tell them that your goal is for them to succeed, and that you will work along side them, in a physically-distanced sort of way, for this to happen.

Although my students work together in breakout rooms and I can pop in their room to check on them, I miss the collective buzz and energy of math conversations. When I am in the same physical space as my students, I can quickly assess if I need to clarify an idea or if there is a pre-requisite idea that needs to be addressed. I can walk by 8 groups in less than 5 minutes and gather real-time data. I yearn for the day when I can be with my students again.

BONUS NOTE: If you want to experience how I use breakout rooms and teach with Desmos, join me for the virtual PD session offered by BER called, "Making Best Use of Desmos to Strengthen Math Instruction (Grades 6-12)" Depending on which session you choose and the time zone you live in, you might not miss a full day of school. Dates in November, but there will also be virtual sessions in the spring. Feel free to ping me on twitter @mathteacher24, if you need more information.

The Writing on the Desk: An Apology for my Silence

Please note: The language I am sharing in this post was used by one of my students over twenty-five years ago. The purpose of this post is to reflect on a personal exprerience from may past and make suggestions for what educators can do moving forward.

 The year was 1995 and I was teaching math at Ames High School in Ames, Iowa.  It was my

first year as a full-time teacher. I didn't make it a habit of checking my desks for student writing.  Perhaps that particular day there was a pencil on the floor or a piece of paper and I went to pick it up near the desk.  I can't recall those details.  However, I do recall seeing something written on the desk and I bent over to read it.  The words I read were burned into my memory.  It said, "Mrs. Nataro is a n***** lover." I was shocked by the words I saw, naievely thinking my community was safe and free of racism.  It has taken me until now to realize that what I did next contributed to the underlying racism that was in present in my classroom.

Image from pixnio.com

A flood of thoughts raced through my mind, "What do I do? How long had this been written on the desk?  How many students had seen it?  Was it written on other desks?"  I recall quickly walking around the room and checking each desk.  After I saw it was only on the one desk, my thoughts changed to "I can't let other students see this. I need to remove this vandalism from the desk immediately."

I can't recall where I found the products I used to clean the desk, but I do remember scrubbing vigorously and checking several times to make sure the words could no longer be seen.  I had erased the words and thought I had solved the problem.  The desk was clean, but the problem was still there.  I had removed the vandalism, but I did not address what had led to that vandalism.

At the time, I thought I had done the right thing.  I was protecting my black students from being exposed to that vile language and I was preventing those words from being viewed and used by my white students.  I now understand that by not saying anything to administrators or my students, I contributed to the racist undercurrent in the school and my classroom.  And if you were a black student or black teacher at Ames High School in 1995 and you are reading this post, I am sorry that I did not speak up.  I am sorry I did not recognize what my silence meant.

So, what can we do as educators?  I am not an expert on this, but offer a few thoughts.

1) Train educators with what to do and say in response to racism and intolerance.  Train us to listen and train us on the ways we can support marginalized students.  Posting platitudes around the school like "we care for our students" and "your education matters to us" is meaningless without the tools teachers need to promote change and have difficult conversations.

2) Find groups that work to actively support anti-racists practices in education and share your experiences and expertise within those groups.  If you are looking for some ideas, check out the link to the recent statement posted at The Global Math Department.

3) Don't relegate teaching of black history or marginalized groups to one month of the year or a specific week or a single assembly to check off a box. Doing that is like saying, "Look here. We didn't forget about you. Problem solved."  This doesn't come close to doing the hard work that needs to be done.

Dorms Closed for Coronavirus: My 90 Second Announcement

Two weeks ago I was on spring break. Had you told me that I would be teaching online and not likely to see my students again for the remainder of the semester, I would not have believed you.  But here we are.

For the last week, I have been more exhausted than I have been since my daughter was born.  You know the good old days when you child would not fall asleep easily and then you would lay down in their room and fall asleep on the floor?  I adjusted to using Zoom for teaching my college classes. I learned how to do breakout rooms. When the students were taking their test in Canvas and they couldn't see the images, I scrambled and posted the images in the chat. The students who were absent received individual emails to check in on their health and well being each day.  Some are struggling.

I have always said I am a teacher first and that my subject area is math.  Three-fourths of my students were scheduled to take a test tomorrow.  About 90 minutes ago, the governor of Pennsylvania said that non-life-sustaining businesses are to be shut down as of 8 PM tonight.  My college students need to be out of their dorms in 24 hours.  Some have already gone home, but not all.  I know they are confused.  They want to do their best.  They feel conflicted.  They are emotional.  They are worried.

Here is what I sent to them through our class announcements about 10 minutes after the President sent the email saying that college students must leave their dorms in the next 24 hours and go home or to a place they consider safe.

My college went from Tier 1 to Tier 5 of our response plan in less than 10 days.  I am still shellshocked.  For all the other teachers out there, accept it.  Your teaching won't be the same.  Your students aren't the same.  They won't remember any (or very little) of the content that you might teach them over the next 2-3 months. Lower your standards now. 

The tests don't matter.  The syllabus doesn't matter.  You won't be able to go at the same pace. You are human.  Your students are human.  Listen to them and care for them.  Listen to your body and care for yourself.  Spend time with your family and care for them. 

Think About a Person Who is Good at Math (Class Day 1)

On my first day of class, I wanted to set the tone that all students are capable of learning mathematics.  This can be a hard sell in a "Business Calculus class", especially with juniors and seniors that have been removed from a math class by 4 or 5 years.  To help students see that the characteristics of a person who is good in math are actually achieveable by everyone and within their control, I took a lesson from the playbook of Howie Hua.  He sent out a tweet a few days ago about this activity.

Step 1: Instruct students to think of someone they know who is good at math.  The person could be a family member, sibling, friend or classmate.  But it had to be someone they knew.

Step 2: Think about the characteristics that person has that helps them to be good at math.

Step 3: Have students work together to compile their list of 5 characteristics.

Step 4: Display the lists in the room and ask students what they notice about the lists.

The lists generated by one of my classes is posted below.

We had to unpack what "Calculated" meant in the first group.  They described it as being organized and thoughtful about how they approached problems, a "calculated approach".  We also discussed "thinking outside the box".  This is partly seeing things from multiple approaches and can be improved with practice.

In my second class, we put the responses on paper and looked at them under the document camera.  That class had words like focused, organized, hardworking, determined and diligent.  I pointed out that these are characteristics that can be developed by anyone and that means they were all capable of becoming better math students.

I also told the story about my first math quiz grade in college - it was a 50%.  And I mentioned that I went to speak to the professor right away about that grade and then worked hard to understand the material in the course, looking for patterns in the problems, doing extra problems and working with friends.  The lowest quiz grade in that class was dropped and that was my lowest grade. I ended the class with a B+, because I put in the extra effort and got help when I needed it.

Today I had a student find me to check his work and he told me that exercise in class made him realize he should start working early and it motivated him to come in for a little help.  I also had another student email me about coming in for help.  She admitted that math has been very challenging for her in the past, but she wants to work.

So, thank you Howie for a great opening day activity!  And if you don't follow Howie on twitter you really should.

Thoughts on my Last Lesson

Today marks my last day at Kent Place School in Summit, New Jersey.  In August, I will be teaching full time at Moravain College, a much shorter commute from my home in Easton, PA. This marks the end of my 26th year of teaching.  You may think that after teaching for that length of time that I had nothing to learn about teaching, about students or about myself.  However, I have continued to grow as a teacher and I have gained insights in each of these areas this year .

Thoughts on My Last Lesson

I'll known for a while now that teaching is more than just understanding a subject.  It is about the students in front of you at that precise moment. Yesterday was the last day of school.  It was a half-day.  No exams, no regular classes.  Just final good-byes and one last chance to have students think and perhaps learn something new. Faculty were asked to design an activity and students would sign up to attend the 30 minute session of their choice.  The one I led was "The Case of the Stolen Jewels".  Seven volunteers were actors in a mini play.  The players were the cook, the chauffeur, the maid, the butler, inspector Euler Toots, Lady Shmendrick and the narrator.  Jewels were stolen from the mansion and the thief dragged his feet through the snow to throw off the authorities.  The testimony did not fit what was on the map.  However, I asked the students if they could tell from the map alone who stole the jewels.  (Can you tell?  Hint: Think Euler path.)

Map for "The Case of the Stolen Jewels"

I did this session three times and each time the student interaction was different.  Not only the interaction of the students with each other, but my questioning of the students.  Usually when I teach a lesson multiple times, my last time is the best.  Reflecting on how the lesson went the previous time(s), I can anticipate student questions and the direction of the lesson better.  However, my first time of leading "The Case of the Stolen Jewels" was the best yesterday and I think it was a result of my questionning.  I believe there are three main questionning techniques that can either open up a lesson or shut it down.

Number 1: Type of question  

Consider the following two questions. "We said you could trace a path if you start and end at the odd vertices.  What other questions might Euler have investigated related to odd and even vertices?" versus "Do you think it is possible to trace if we have three odd vertices?" 

The first questions directs students to focus on the vertices, but doesn't suggest any specific changes to make.  Any answer to this question is open to exploration. This was the question I posed to my first group and we had a lively discussion related to changes they suggested - 3 odd vertices, all even verticies, all odd vertices, etc. They were thinking like mathematicians. 

The second question removes the mathematician agency from the students.  It is a yes/no question and leads students in a very specific direction.  And since it definitely has a right answer - it can be traced or not - students hesitate to answer because there is a chance of being wrong and in the eyes of many students, even on the last day of school, wrong is bad.  This second version of the question was asked to my third group of students and it definitely changed the atmosphere of learning.  I had to follow it up with multiple questions and the discussion overall fell flat.

Number 2: Wait Time

This year I became better at wait time.  When I would ask a question, I would often restate the question - either word for word or slightly revised.  This gave students who hadn't fully heard the question the first time to hear it and it gave some more time for students to think.  Also, I did not go with the first person to raise her hand as I would have in the past.  Doing this often rewards the fastest thinkers and leads other students to think "I don't need to think about this question, because she will call on Susan. Susan knows all the answers."  Instead, I would wait until many hands were raised and often call on students who participate less often.  And, even if those students say something that isn't fully correct, we unpack it as a class.  Early in the year, students learn that mistakes are a valuable part of the learning process. You can learn so much from doing something wrong.  In response to mathematical misunderstandings, you can often hear me say things like, "I hadn't thought of that. Thank you for sharing that idea." or "Let's think about that some more.  There is something we all can learn from what you just said." or "That is an interesting thought.  Let's see what happens when we do that."

Number 3: Share with Others First

Not all students are risk takers. Think back to your days as a student. How many of you would say "I loved to share my thoughts that could be wrong and incomplete in front of 15-20 of my peers for them and the teacher to critique."  Students prefer to participate when they think they are mostly right and will be validated for their correctness.  So, how do we get students willing to take risks and share?  When I ask a question and see that no one wants to take a risk to answer it, I say "share some thoughts about that question with your group for a minute."  Then, I walk around the room and listen in on the conversation.  Sometimes, I will even tell specific students that I'll be calling on them to share their idea.  More students are willing to become risk takers after that minute of sharing with others first.

This summer I will be leading/co-presenting at several different workshops or conferences. Often the pace of the workshops is frantic, but that generally doesn't lead to deep understanding.  Sharing, risk-taking and reflecting will be a large part of my workshops this summer, as it has been in my classroom this year. Be sure to visit my blog in the fall to hear about my reflections in the college classroom and if you are a teacher, be sure to take some time for yourself this summer to relax and reflect.

Even Tech Has Its Limits: Learning from Students

One of the things I absolutely love about my teaching situation this year is that I have students that are very curious. They ask "what if" questions, make connections between concepts and notice things that I, even after twenty-six years of teaching, haven't noticed.  This type of noticing happened in Advanced Algebra on Friday and reminded me that math tech can have its limits.

Here is what happened. Do students really need to know about end behavior or behavior at x-intercepts for polynomial functions?  They can just plug the function into Desmos and see what it looks like.  But because I let my students discuss ideas and they can share them freely without fear of "being wrong", one of my students commented the function f(x) = x3(x+3)(x-4)  (shown here) seemed to have many

x-intecepts around the origin.  Because we had talked about behavior of x-intercepts of polynomials based on their factored form, she knew that there should only be x-intercepts at 0, -3 and 4.  Yet Desmos was telling her something different.  So, we decided to zoom in to verify that the function really only had one x-intercept at the origin.  And this is what we saw, after zooming in and tracing along the graph. It looked as though my student was right! The graph showed that several of the values close to zero had y-coordinates of 0, but we knew from the factored form of the polynomial that this could not be.  Desmos was lying to us!!! How could we trust it to do our math correctly?  I could tell that several students confidence in Desmos was shook.  At this point, we decided that  Desmos had to round the output to show it on the graph.  And based on what rounding convention Desmos used, it rounded the result to zero.  The students' understanding of polynomial behavior helped us understand what was really happening near the origin.  Incidentally, f(-0.002) is not zero, but it's pretty darn close to it at 9.598 x 10-8. Letting students share what they notice and wonder can be risky.  You don't know what they will say all the time and sometimes they may surprise you or throw you a curve ball, where you don't have a satisfying answer.  But letting students see that you are on a learning journey with them can definitely have unintended positive consequences, where even you as a teacher can learn something new.

Bonus tech tip: If students want to use the Desmos app on their phones, tell them to download the  "Desmos Test Mode" app and use it when doing their math homework.  If the select "Start Test", they can work in a distraction free way - no notifications of any kind will pop up on their phones!