Teaching Bravery

This summer, I have been participating in some workshops about computer science, in preparation for teaching a new class in the fall.  In one session, we were referred to the TED Talk video from Reshma Saujani, the founder of "Girls Who Code".  In the talk, she referred to work by Carol Dweck, and what she said about teaching girls to be brave, not perfect, resonated with me and what I have seen in math and computer science classes.

At one point in the talk, Saujani talked about boys' responses to problems: "There's something wrong with my program" versus girls' responses to problems: "There's something wrong with me."  This may be over-generalized, but it highlights how girls, and, I think, many underrepresented groups in STEM fields, respond to difficulties.  As a white male, I automatically belong to the "STEM Club", so any difficulties I experience are not part of who I am; the difficulties are part of my process.  Unfortunately, those belonging to other groups not part of the "STEM Club" may start to believe something is inherently missing in their make-up or that STEM fields are not for them.  Just as unfortunately, those of us in the club can also start to believe this.  And it's nonsense!

Students tell me all the time that they're not good at math, and it's just not true!  Just because you have to work at something or just because you don't understand an idea quickly does not mean you are not good at it, or that you shouldn't try it!  Much of math, computer science, life even! is figuring out the next step based on limited information.  And even when you figure out your next step, you have to realize that it might not be correct and you have to do it again, and THAT'S OKAY!  Perfection is not expected nor encouraged.  Growth and improvement and moving forward are.

I teach because I want to help students embrace the struggle, because in the end it's not the math (or English or History, or ...) concepts that are the most important.  (Yes, I know, that's what's usually being graded, which makes it important, and that's my current struggle.)  Some of what I hope my students take from my classes is a willingness to work hard in the face of challenge and to wrestle with difficulty cheerfully.  With these, math (and life!) are infinitely more enjoyable.

"Success is staggering from failure to failure with no loss of enthusiasm."

The Personality Myth

I heard a piece on WBEZ's Invisibilia program, The Personality Myth, and the story resonated with me because its point was that personality is mutable.  The cells in our bodies are constantly being replaced, our brains are always being rewired, and our memories of events, even big important events, change over time.  What makes us who we are is not a fixed, unchangeable entity, but a mutable and constantly growing set of attributes that we actually have a lot of control over.

In the Invisibilia piece, a woman working with prisoners finds that her experiences, along with a conscious decision on her part, changed how she thinks about "good people" and "bad people": while there are amazingly good actions that people take as well as horrifyingly evil actions, people themselves, because of their mutability, are not so easily categorized.

All this reminds me of my reading and experience around growth mindset, and the saying "Success is never final and failure is never fatal; it's courage that counts."  What has already happened, whether good or bad, can be learned from and, as amazing, changeable people, we can choose the next steps on our path.

The implication for me and for my students is that regardless of past experiences with math, we have the ability to learn new strategies and techniques, and develop deeper understandings.  Right now, for example, I am in the middle of a two-week workshop getting ready to teach the Computer Science Principles class.  While I have some experience with CS, I am being asked to think in some new ways, and I am really enjoying this experience.  I'm not completely comfortable with the material yet, and the experience of learning new things has me both tired (brain work takes a lot of energy) and exhilarated (each new idea sparks lots of other ideas and questions for me).

At a deeper level, the Invisibilia piece reminds me that I have to be careful about categorizing my students.  Regardless of their past experiences and views about math, regardless of their apparent energy level for the topic, their gender, sexual orientation/identity, race, culture, year in school, or the thousand other attributes that make them who they are at this moment, my job is to recognize their humanness, honor what makes them unique, and help them determine the person they will become.

New Problems to Think About ...

Last night, I attended the January meeting of the Metropolitan Mathematics Club of Chicago (MMC), at which Steve Viktora talked about word problems through the ages,  He showed examples of word problems across cultures and times (going back 6000 years!), and highlighted some common themes he found.  Here's an example of a type of problem Steve called a "hydraulic problem"; these types of problems deal with some sort of public works, like building dikes, storing grain, or marking out land, and they appear in documents from Mesopotamia, Egypt, and China going back thousands of years.  This one is from the 4th millennium BCE from the Sumerian city of Shuruppog, and is the oldest known word problem:

A granary of barley.  One man received 7 sila [of grain].  What are its men?  [i.e. How many men can be given a ration?]  

Of course, to solve this, one needs to know that capacity of a granary was 2400 gur, and one gur was equal to 480 sila.  (Any Sumerian bureaucrat could tell you that.)

Steve presented other hydraulic problems.  This one got everyone talking, and there were a number of different, interesting, and elegant ways that people solved it.  There were also some good questions about what happens if the base is not a leg of the triangle.  I don't remember the  time or place for this problem:

A triangular piece of land [in the form of a right triangle] is divided among six brothers by equidistant lines constructed perpendicular to the base of the triangle.  The length of the base is 390 units and the area of the triangle is 40950 square units.  What is the difference in area between adjacent plots of land?

There were abstract problems from Islamic cultures, temple problems from Japan, puzzles from Medieval Europe, and many others.  They were fun to see, and we got to work on a few, which is always a good time.  (Yes, I like discussing and solving problems.  I am a Math Geek, but you already knew that.)

After Steve's presentation, a few of us were talking, and Carol said she was teaching a Geometry class, and asked the students to come up with four integers that could represent the sides of a rectangular solid and the length of one of its interior diagonals.  Nice problem, and if you are studying the Pythagorean Theorem and Triples, you can figure out that if the dimensions are 3, 4, and 12, the diagonal is 13.  One of her students said, "Sure, and if you pick and two consecutive numbers for the base, and multiply them to get the height, then the diagonal will be one more."  Carol thought that was an interesting hypothesis, and matched the expected solutions, but cautioned that it might not really be a general solution.  Then she did the algebra ...  So what do we have?  A way to find "Pythagorean Quadruples"?  I love math!  Paul added that when he was teaching the triples, he listed several out for his students:  3,4,5;  5,12,13;  7,24,25;  9,40,41;  11,60,61 ... and hoped they would see some patterns that could help generate more triples.  (There are all sorts of patterns here.)  Paul said one of his students noticed this one, which I had not seen before:  4 = (1/2)(3+5); 12 = (2/3)(5+13); 24 = (3/4)(7+25);  40 = (4/5)(9+41);  60 = (5/6)(11+61).  Okay, mind blown.  Did I mention that I love math?  MMC Dinners are always worth the price of admission.

Two weeks in

I just finished the first two weeks of school, and I have had more fun with this beginning than any I can remember.  Last year, I was too busy trying to figure out how teaching worked again, but this year, I am again comfortable with what I am doing.  (But not complacent!)  I've had the chance to collaborate with other teachers, and I have learned the names of all 100+ of my students.  All of my classes are fun; the students are pretty engaged (even after a rough quiz or two); and I'm happy with the climate for learning we are building.

I don't know too much about the students beyond their names, yet.  Some are really engaged, and vocal about their excitement.  One student blurted out in the middle of class, "This is really fun!".  Many others dive into the problems and seek out suggestions from classmates.  I am really thankful for their energy, enthusiasm, and willingness to engage with others they may have just met.  There are also a handful of kids who appear to be having some sort of difficulty, and others who have not said much at all.  I will need to spend next week attempting to reach out to them, so they don't get lost.

I'm looking forward to learning more about my students, and helping them see why I think math is fun, beautiful, and power.  (Yes, the noun is purposeful; more on that later.)    

First Day of School!

I get so excited at this time of year, I can hardly sleep!  This is the same feeling I used to have as a kid on Christmas Eve, anticipating the family visits, the food, and, of course, the presents.  Although there are no actual presents or special dinner tomorrow, the first day of school never gets old for me, despite the fact that I've had over 25 first school days as a teacher.  My wife tells me my inner teaching geek is showing, but I can't help wondering who my students are this year and thinking up ways to help them learn (and maybe even get excited about) math.

If you are reading this because you are one of my students, Welcome!  This is going to be a great year!  I really believe that you can get better at math this year, and most of my job is helping you be the best person, student, and mathematician that you can be.

If you are the parent or guardian of one of my students, know that I feel blessed to have the chance to know your child.  I know that as a core class, math is really important, and I will do my best to help each of my students succeed.  If you ever have a question, concern, or suggestion, please call or email me.

See you tomorrow!

What Am I Grading? (Part 3)

After talking to a colleague about my grading policy, I found that she has a similar grading policy, but states the weights of the categories a little differently.  Rather than 75% tests, she states that the homework and quiz grades together count as a test grade.  This works the same as my system, but her students seemed to respond more positively to the system.  (Am I giving away too much to my students by telling this?)

We also talked about how to cue students to important problems in the homework, and how to give students choice about which homework problems to focus on.  Additionally, we brainstormed ideas about how to have students practice writing about their thinking, and how we might provide feedback (not necessarily a grade) to this work.  Lots of ideas, but I am not yet sure which I will incorporate into my classes.  It was really exciting, thought-provoking, and satisfying to talk to another teacher about our work.  (Thanks, Cory!)  I think I was missing that last year.

Finally, I saw a really short article today, "Is grading killing learning for our students?"  which raises issues about grading and learning that I continue to worry about.  In particular, how do I create (or balance?) a culture of learning and a culture of performance?

As this year starts, I want to work on few things regarding grading:

• providing growth mindset messages to my students
• providing helpful and actionable feedback (not necessarily a grade) to my students about their work  (balancing learning and performance)
• making sure my expectations for good work are clear.

I'll have to revisit my thinking as this year goes along.

What Am I Grading? (Part 2)

At the end of last year, I talked to another teacher about the "standards-based" grading policy.  For this, students are given a list of the things they are expected to know and be able to do by the end of the unit, and instead of putting a grade on a quiz or test, the teacher gives them a score for each standard, which gets translated into a letter grade by the end of the quarter.  This is intriguing to me because I want the students to focus on the skills and understandings of math rather than on the grades.  I also tried this method of grading a number of years ago (almost ten years, I think), but the system seemed more confusing to students since they were not entirely sure how their grade was calculated.  (I also don't recall how I made this calculation.)  Now that we have an online grading program, parents and students can track their grades on a daily basis, and I'm really not sure how to make the calculations or how to make the calculations clear to the users.

Having some system through which it is clear to students (and parents) how well they are understanding the topics is important to me, so I am thinking about how to include aspects of standards-based grading into my classes.

One thing I tried over the summer, and plan to continue this year, is having a unit outline of four to six big ideas I would like the students to learn over the course of the unit.  I'm hoping to keep the number of these to four "skills" like "identify the important features (domain, range, vertex) of quadratic functions based on their graphs and equations", and one or two practices like "construct clear critiques of possible solutions".  I'm figuring that the practices will appear across multiple units, but the skills belong in only one unit.  The idea here is to alert the students to the important information I want them to learn, to provide me with reminders about what understanding I want to check (via exit slips, short ungraded quizzes), and to cue students to self-assess their understanding.  So I'm creating a set of "sample problems" to go with each big idea, and providing space for students to monitor their understanding over the course of the unit.  I'll still give graded quizzes (and allow retakes) and a graded unit test.