Teach 180: The Day Before Christmas Break (Day 75)

As part of the Desmos Fellow's weekly challenge, I made the following to celebrate the Christmas season.  When I shared it with my students, I told them that I created the graph and many wanted to know how the stars were "blinking".  Designing activities and lessons to make students wonder why something makes sense or how something works is what I strive to do on a daily basis. I look forward to continuing to reflect and blog on January 2.  Now for a much needed break!

Teach 180: Comenius Independent Study Projects (Day 74)

https://www.famousbirthdays.com/people/john-comenius.html

From 3:30 - 4:45 this afternoon students met with me and other members of our independent study committee to describe their progress on their Comenius Independent Study projects.  Although we only met with five students today, by the end of the week we will have met with all sixteen independent study scholars.  Comenius Independent Study projects have been an optional part of the curriculum for at least 20 years.  I have been a mentor for some scholars with math projects, but this year the projects students chose to do are more focused on English and history with a few science projects.  Many of the students spoke rapidly and were very animated in describing the work for their project.  Others had not made as much progress as anticipated.  The next round of check-in meetings is in March and presentations happen in April.  Each year I am impressed by the work and initiative of our Comenius scholars.

Teach 180: WODB (Day 73)

I love it when I reach into my folder from last year for a particular chapter and discover engaging activities that I had forgotten about. "Which One Doesn't Belong"? is a great way to open up discussion at the beginning of a class.  Or it can be used to help break up a long class.

This year, I used it as a class opener.  We had just finished a unit on polynomial functions.  Each table of students was asked to identify which one they thought didn't belong.  Usually students don't pick up on the line-type (dotted vs solid) or the color of the graph, but these students did.  So next, I asked for a mathematical reason for why a function wouldn't belong.

Here are some answers my students gave me.

Top Left: Has same end behavior on both left and right.  Others rise on one side and fall on the other for end behavior.
Top Right: It is the only function where the graph crosses at x = 0, instead of touching and turning.
Bottom Right: It is the only graph that is not in the fourth quadrant.

The bottom left was the most challenging.  What about the graph makes it different than the others?  One of my students said it was the only one to have a minimum in the 4th quadrant, but that wasn't true.  Someone else suggested that it was the only one to have a x-intercept of 3.  But that wasn't true, either.

We finally decided on the following.  The bottom left function doesn't belong, because is the only one that has both positive and negative y-values when x is positive.  Can you find any other reasons why the bottom left function doesn't belong?

Teach 180: You Can Lead a Horse to Water (Day 72)

At the beginning of the year, I had a sense of which students would struggle for me in my classes.  Some teachers had used the words "lazy and arrogant" to describe some of the students I have this year.  However, I was seeing the students differently.  There is one student in particular.  I had a conversation with him on the second day of school at opening channel.  We talked about which colleges he was applying to and what his future aspirations were.  As I talked to the student, the description of "lazy" did not once cross my mind. 

Several weeks passed and I noticed that he was not doing his homework sufficiently and this led to low test grades.  Something that this student seemed to expect.  However, I noticed that he knew answers to questions when I called on him in class and he wass using vocabulary correctly to describe concepts.  He was able to answer "why?" and "how do you know?" questions.  

It is at this point that I pulled him aside and told him that I thought he was capable and could do well.  It was a matter of him believing it.  It didn't matter how much I believed it.  It was up to him to believe that he was capable of doing well in math class.  Part of getting good at something is consistent practice.  Not only doing the work when you feel like it, but working each day and gradually you get better.  Bringing up a low grade in the class will not happen overnight, but with consistent effort it will happen. 

http://www.thehorse.com/articles/33982/your-horses-water-sources-things-to-conside

On Friday, the student met with me before school to review for his test.  His test was first period.  At the end of the day, he stopped by to see me to see if I had his test graded.  I did not, but I did grade it later that afternoon and emailed him the good news about his test grade.  His hard work was paying off!  

The saying of "you can lead a horse to water, but you can't make him drink" is definitely true.  There are times when I have worked hard to get students to understand concepts, but ultimately it is the decision of the student (especially when the student is in high school) to learn.  The student needs to have a growth mindset and believe that they are capable.  They have to understand that failures show areas where growth is needed and that a failure is not a reflection on their self-worth.  I hope that this student continues to recognize that it is his effort, his daily effort, that is leading to his success. 

Teach 180: Conceptual Understanding in Calculus Revisited (Day 71)

In Calculus today, I had students work in groups to discuss questions related to the conceptual understanding of first and second derivatives. (If you recall from a previous blog, many of my students did poorly on this section of their last quiz.)  Students were given the graph of the second derivative and asked questions like, "Where is the original function concave up?" and "Where does a maximum occur?"  Some students understood quickly, but other students still had trouble understanding that what they were viewing was a graph of the derivative and not a graph of the actual function.  At one point, I realized I could have written the question at the left better.  It should have read "on what interval(s) is the derivative negative".   Some students were thinking in terms of ordered pairs or points and not intervals.  By the final question in our set of 5 questions, almost all the students were able to explain how the various parts of the graph of the derivative were related to the graph of the original function.  Why did students finally get it?  I think a main reason was students were discussing the answers.  Students who understood the concept worked hard to justify their thinking to their classmates.  They wanted them to understand it, too.

Teach 180: The Two Hour Delay (Day 70)

Teachers are in a caring profession and we frequently look out for others, often at the neglect of our own health and wellness.  Today we had a 2-hour delay and rather than getting extra work done for school or getting caught up with housework, I chose to take care of myself.  I slept for an extra hour and then did a cardio workout in my basement. (A graph of my steps is at the right.)

That helped me to be ready to take on the day, reschedule a missed meeting, meet with four students, give a test, attend a faculty meeting and attend an optional meeting called "Faculty Fellows".  This month's topic is facilitating class discussions and faculty read excerpts from "A Classroom Revolution: Reflections on Harkness Learning and Teaching". 

In case you are wondering how my Calculus class did with the activity I described in my previous blog, the first two classes were dropped today due to the two hour delay.  There was no Calculus class today, but I will be doing the Plicker activity with them tomorrow.  Tune into tomorrow's entry to see how we did.

Teach 180: Assessing for Conceptual Understanding Fail (Day 69)

What do the following numbers represent?

Yesterday I gave my students a quiz to test conceptual understanding of first and second derivatives.  The maximum possible score on the quiz was 16.  You can see that a few students had a strong conceptual understanding of these ideas, but many didn't.  What type of conceptual difficulties did they have? 

Let's look at the back of the quiz first. (Note: The front of the quiz was low level vocabulary questions and calculating the second derivative of f(x) = 1/x to determine concavity.)

The other questions I asked were:

• Based on the first derivative, when does the original function have tangent lines with negative slopes?  Explain how you know from the graph.
• Based on the second derivative, does the original function have an inflection point?  Explain how you know from the graph.
• Based on the second derivative, when is the original function concave up?  Explain how you know from the graph.

For some reason students thought that the graph on the left was the original function and spoke about derivatives of the parabola.  When some analyzed the second derivative, they said that there was no change in concavity, because the graph had a constant slope and since the slope was positive, the function was always concave up.  Others talked about the function having a minimum at (0, -3).  Although the first derivative has a minimum at that point, the original function does not have a minimum at that point.

Based on what students wrote, you can see that they have been in a calculus classroom.  However, they had a hard time making the connection between the calculus ideas (concavity, inflection points, extrema) and the graphs of the derivatives.

Tomorrow, I will be starting class by assigning students to small groups of 2-3 students with one of the top scoring students in each group.  Then, I'll have them do some similar questions using Plickers.  Having students discuss the solutions with their peers should help some of them to get a better conceptual understanding of what is being shown on the graphs for the derivatives.  I also have a derivative matching activity and I'll probably use that on Friday or Monday.

I called this blog post "Assessing Conceptual Understanding Fail", but actually it helped me to uncover some of the incomplete understanding students have.  Learning about what my students know/don't know and understand/don't understand and modifying my teaching based on that is a win.

Teach 180: Assessing for Conceptual Understanding: The Quiz (Day 68)

Assessing for conceptual understanding can be challenging.  My students in Prob/Stat are used to explaining their thinking or interpreting their results.  They get good at writing sentences to justify their answers.  After seeing students struggle on today's Calculus quiz, I know that I need to do a better job getting my calculus students to show they have a conceptual understanding of the first and second derivative as it relates to a specific function.  In addition, they need to learn how to verbalize their understanding. (Note: I haven't actually graded the quizzes yet, but my perception was that they were struggling. Stay tuned when I blog more about this topic tomorrow.)

Here is one of the quiz questions from today.

Here are some predictions relative to this question.

1)  Students will say that there is no maximum, only a minimum at x = 0.  They are correct that the derivative has a minimum at x = 0, but that doesn't mean that the original function has a minimum at x = 0.

2) Students will explain that at x = -1 and x = 1, the derivative is 0.  However, they won't be able to explain how the graph of the derivative is related to the sign of the derivative.

3) There will be some other wacky stuff.  Not sure what, but based on some of the questions students tried to ask me during the quiz, I am confident that there is some other wacky stuff that students wrote about.

I hope you'll return to this blog tomorrow to see how the students did and what I think I need to do to get my students to improve in this area.

Teach 180: Conceptual Equity and Access in Calculus (Day 67)

Algebra is a gateway to higher level math.  If a student can't do algebra, they are bound to have difficulty in higher level mathematics.  Students that typically have problems in calculus have problems, because they have issues with their algebra.  However, we can teach conceptual understanding and allow students access to calculus even though they struggle with getting all the details of the algebra right.

NCTM's Position Statement on Access and Equity in Mathematics Education states "creating, supporting, and sustaining a culture of access and equity require being responsive to students'...knowledge when designing and implementing a mathematics program and assessing its effectiveness."  In addition, we need to be sure we are "acknowledging and addressing factors that contribute to differential outcomes among groups of students".  So, equity and access doesn't just mean addressing cultural, gender or socioeconomic differences, but addressing differences in the levels of mathematical understanding that students bring with them to class.

If learning calculus is built on algebraic manipulation, students who are deficit in algebra skills won't be able to do calculus. So, how can we get those students to build a conceptual understanding of calculus?  Do we allow algebra to be a barrier to building conceptual understanding?

Here is where technology can help to bridge the gap. Consider the function: f(x) = (2x - 5)^1/3 + 1 and identify its extrema, inflection points, intervals where the function is increasing, intervals where the function is decreasing, intervals where the function is concave up and intervals where the function is concave down.  We could do all of the work of calculating the derivative and second derivative by hand and consider the sign of the first derivative and the second derivative.  Lots of algebra.

Let's look at the graph of the first derivative instead.

What does this graph tell us about the function?  Are there any places where the derivative is undefined?  Is the location where the derivative is undefined a maximum, a minimum or neither?  Describe how you know this from the graph of the first derivative.

Now let's look at a graph of the second derivative.

What does this graph tell us about the function?  Do we have any places where there is an inflection point or change in concavity?  Is the function ever concave up?  Is it ever concave down?

Now that students have analyzed the graphs of the first and second derivative.  Have them make a sketch of the graph.

This analysis allows all students to gain conceptual understanding without being held back by their errors in transposing numbers, their errors in simple arithmetic calculations, or their errors in algebra.  I am not proposing that we ignore the weaknesses in student's algebra skills. What I am proposing is one way to build conceptual equity and access in calculus.

Teach 180: Teaching on a Day Off (Day 66)

Today I took a professional day in order to participate on a College Board panel.  The meeting started at noon and it was in New York City.  So, it was about an hour and a half drive. (With the accidental detour into Queens, it was a bit longer.)  I left school shortly after 9 AM, after teaching my first period class. 

Why did I teach my first period class on a day off? 

My first period class was 2 classes behind my other Prob/Stat class and I am worried about completing the content prior to the AP exam due to the change in our bell schedule this year.  To make sure my first period class did not fall 3 days behind the other section of Prob/Stat, I chose to teach the class today instead of having my sub teach the class.

While at my meeting, I learned about the rollout of the Pre-AP program and AP Insight resources.  We also discussed challenges related to equity and access to AP courses and how to make professional development more meaningful and useful for teachers.  I'm headed back to the College Board office on Saturday morning for more meetings that will last until 2 PM.

Teach 180: Sometimes You Change Mid-Lesson (Day 65)

Today in Prob/Stats we were continuing work with the binomial random variable.  What would be the average number you would expect to get correct on a 10 question multiple quiz with 4 answers each?  Since you are guessing, the probability of getting a correct answer is 0.25.  The average number correct is 1/4 of 10 or 2.5.  Although this calculation makes sense, I had a feeling my students weren't buying it.  So, I changed my lesson. 

I decided to have students take a mock quiz.  Try it with your students.  It's quite fun, actually.  Tell them to number their paper from 1-10 and write down answer choices of A, B, C or D beside each question.  As they do this, you make up an answer key.  Next, have students check their own work as you read the answer key.  I had students hold up their fingers to show how many they got correct and quickly totaled 33 total correct out of 13 students or an average of 2.54.  One student got them all wrong and one student got 5 correct.  After that we calculated the binomial probabilities for 0 correct and 5 correct.  These probabilities and the mean of 2.5 for this binomial distribution made more sense after this mid-lesson change. 

Teach 180: The Non-Teaching Stuff (Day 64)

Teachers do a lot of non-teaching stuff as part of their job - chaperoning trips, supporting student fundraisers, attending theater productions, etc.  Now that we are officially into December, I am in the midst of entering athletes, updating records and entering entries for...swim meets!

When meets officially start, I spend about 4 hours per home meet and 1 hour per away meet on the swim computer, which translates into about 40 hours each swim season.  My compensation this year will be about $300 or $7.50/hour.  Why do I do this?  I don't really have extra time on my hands and the pay is clearly not the reason.  I do it for two reasons and I think these reasons probably resonate with many teachers.  I do stuff like this, because I enjoy helping others and it is hard for me to say no to those who ask for my help.  However, I am thinking of cutting back some in the next few years and this year I created a 20 page manual to train the next Moravian Academy swim statistician.  Hopefully, by the end of the season, I will have trained one or two of the student managers in the nuances of the swim computer.

Teach 180: Discussion to Construct Understanding (Day 63)

Today in PreCalculus class, we were beginning to work with polynomials. I showed students
the following graph in Desmos and asked them what did they notice.  If they had to describe it to someone over the phone for the person at the other end of the line to draw, what would they say?  The conversations I heard at first started with general statements like, "it wiggles in the middle" and "it looks like a parabola that someone dented and bent in the middle".

 "It wiggles in the middle could look like a lot of different graphs," I replied to the one group.  "Can you be more specific?" 

At that point students started talking about x-intercepts and they also noticed that sometimes the graph crossed the x-axis and sometimes it touched the x-axis.  The student who noted the parabola shape on the ends was starting to hint at end behavior.  After a bit more discussion, we decided to play around with the exponents and to see how that impacted the behavior at the x-intercepts.  We also talked about what we could do to change the end behavior of the function.

It is important to note that even before my students put pencil to paper we were discussing concepts and playing around with ideas, dynamically.  At times students were discussing ideas in groups and other times we were putting our ideas together as an entire class.  It takes time and sometimes patience to have students learn this way.  But ultimately, I believe it leads to deeper understanding.  Students come to realize that if they can't recall a concept (like, does it fall on the left and the right for end behavior), they have a way to reconstruct that concept, because they constructed it initially.

Teach 180: The Stalled Start (Day 62)

Today I started my 8 AM class with a Desmos card sort.  Students were to match quadratic and cubic functions with their derivatives.  We did a quick review of what the derivative of a general linear function would be, but this didn't lead to student success with the activity as fast as I thought it would.  An example of a matched derivative and function can be seen below.


Although students worked with pairs, the first 20 minutes of class felt like starting a car up on a cold winter morning after a week of it being idle.  (With no class on Friday due to our rotating schedule, this is probably an accurate analogy.)  We had many mismatched pairs and I had to work with individual groups of students to explain how the graph of the derivative matched with the graph of the function.

So, would I do this activity again?  Most definitely.  However, I think I would create a slide with just 3 matches in the future and have our class work on that together - asking for verbal justification of why students paired certain graphs together.  This would be the equivalent of letting the engine and car heater run for about 5-10 minutes before driving the car.  Perhaps this would have led me to not having a stalled start.

Teach 180: My Voice (Day 61)

To a teacher a voice is a valuable thing.  We can't really do our jobs without it.  This week, I have been losing my voice.  It usually happens around this week, because I have my church Christmas concert and my voice gets quite a workout.  Usually things are fine as long as I don't get a cold.  Unfortunately, my daugher had a slight cold and throat issue before Thanksgiving and she shared it with me. (Thanks, Cassie!) This blog will be a progression of my voice during the day.

8 AM: I don't teach for the first few periods.  I am planning on writing two quizzes and doing lesson plans for next week.  I don't plan on talking much.  My Yeti cup is filled with tea and honey.  I have a chat with a colleague about using our online grade reporting system and visit the Upper School Director for more tea and a chat about Christmas break plans.(Fridays is Dylan's open door day to his office.)

10:15 AM: It's advisory period and we have class meeting first.  Luckily, I don't have to talk during this time.  When class meeting is over, I take my advisees to my classroom and we play banana-grams.  We don't have to talk much to play the game.  Another voice saver.

11:00 AM - 3:15 PM: Lunch and three classes. Unfortunately, I need to do quite a bit of talking in Prob/Stat and PreCalculus.  In Prob/Stat, we are starting a new chapter.  There are basic concepts to be explained related to mean and standard deviation of a random variable.  There is really no way to make this lesson, where students teach each other the concepts.  In PreCalculus, we reviewed two examples involving completing the square to determine the vertex of a parabola and did a Kahoot.  (My random question of the day (RQD) was "What is the name of Mrs. Nataro's blog?")  The day ends with a test in my second section of Prob/Stat and my voice is saved from talking too much.  However, in that class I had multiple students come up and ask me questions during the test.  Based on their questions, I am slightly worried about the grades on the test.  Time will tell.  I'm planning on grading them on Saturday morning.

I am now at home and I am working on finishing my blog.  How's my voice doing?  Not so great.  I am guessing it will be about a week before I am back to 100%.