Friday, December 15, 2017

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.

Thursday, December 14, 2017

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.

Wednesday, December 13, 2017

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.

Tuesday, December 12, 2017

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.

Monday, December 11, 2017

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.


Friday, December 8, 2017

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.

Thursday, December 7, 2017

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.

Tuesday, December 5, 2017

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.

Monday, December 4, 2017

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.

Saturday, December 2, 2017

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%.

Thursday, November 30, 2017

Teach 180: Would It Be Possible To... (Day 60)

I love it when my students challenge me.  My favorite types of questions from students are the "What if..." or "Would it be possible to..." questions.  And then if the answer is yes, the natural follow-up is "Why does that happen?"

This line of questioning happened today in Calculus class.  The actual question was "Would it be possible for a function to have both a sharp point and a smooth part?" I should have asked the student to be more precise and reword his question.  But I knew the question he was really asking was: "Is it possible for a function to have both a relative max. or min. with an undefined derivative and a relative max. or min. with a defined derivative?"  Although I could have (and maybe should have) thrown the question back to my entire class, in a matter of about 15 seconds, I produced the following graph in Desmos and I asked "Why do we have cusps on this graph?"  The student recognized that it was due to the absolute value being used in the function.


Next, the student asked if it would be possible to have a cusp in the middle of the graph.  Within another few seconds, I produced the following graph.  To which I heard a student whisper, "that's cool".  Thanks again to Desmos for making it easier for me to keep me and my students curious.


(After class ended, I spent about 15 more minutes playing around with functions in desmos and have the beginnings of some pretty cool ideas for Christmas designs!  I can't wait to share my finished creation with my students.)

Wednesday, November 29, 2017

Teach 180: Starting to See Connections (Day 59)

One of the things I love about Desmos (besides the fact that it has better resolution than a graphing calculator and it is simple easy to use) is that it makes it easier for kids to see connections.  Today in Calculus we considered at the function f(x) = (x - 2)2/3 + 1.  Prior to actually graphing the function in Desmos, we calculated the derivative and determined that there would be a critical point at x = 1, because the derivative was undefined at that point.   After we did a quick sketch of the graph by hand, we looked at both the function and its derivative in desmos.  The two graphs are shown on the same axes below.



We could easily see when the derivative was negative and when it was positive and how that corresponded to the left and right sides of the graph.  We could also see that the derivative had a vertical asymptote at x = 1 and that made sense since the derivative of the function was undefined at x = 1.   What was more interesting however was that the derivative had large positive values immediately to the right of x = 1, but then the derivative had smaller positive values as x got larger.  It was right around that moment that I could see the synapses in some students' brains firing as they were getting a better understanding - a visual understanding - of the derivative and its relationship to the graph of a function.  Thank you, Desmos!


Tuesday, November 28, 2017

Teach 180: Informal before Formal (Day 58)

Two years ago I created my very first Desmos Activity Builder lesson.  It was called "What is the Derivative, Anyway?"  This year I am teaching Calculus again and I was able to pull out this Desmos AB lesson to use in class today.  As I was reviewing it last night, I noticed several things that I think make this a good introductory AB lesson.

First, students are set up to be successful.  They have the foundation needed to do the lesson based on what they learned in our previous chapter.  And since we just had Thanksgiving break, it was a gentle way to get them thinking about math again.  Plus, I could easily use the teacher dashboard to identify student pairs who had "off" answers.  (One pair said "No" to a question that was not a "Yes/No" question and I was able to visit them for a quick discussion.)

Second, the screens build off of each other.  For example,  students are asked about intervals where the tangent lines have positive slopes and then are asked about intervals where the tangent lines have negative slopes.   

Third, it introduces vocabulary after students have had a chance to informally describe a concept for themselves.  On the screen below, students are asked "or is something else going on"?  Some students simply answered this with one word "constant" or horizontal".  Jack and Grayson said "there is a horizontal tangent, therefore the function is at a peak or valley".  To encourage more of a description of what is going on, I'm going to modify the directions on this slide slightly for future use. The informal idea of a peak or valley lead to formally talking about relative maximums and relative minimums. 


After we get through much of this current unit, I am hoping to create a Talkers and Drawers activity, like what is described by Job Orr in his blog post "Three New Desmos Activities: Talkers and Drawers".   I'll probably use it as a way to review concepts before our test in a few weeks, but it might also be a good activity to do as we ease back into things after Christmas break.

Monday, November 27, 2017

Teach 180: Working with Colleagues (Day 57)

I didn't teach until 1:15 today due to my schedule for the day.  However, I was still plenty busy.  I met with one my my colleagues to talk about students who were struggling and revise a test we are giving in PreCalculus tomorrow.  I also met with the Upper School director for about 20 minutes to discuss several items, including our new bell schedule and the challenges it is presenting with teaching in the math department.  The Upper School Director and I brainstormed some ideas for how we could deal with some of the difficulties.

https://pixabay.com/en/cooperate-collaborate-teamwork-2924261/
Next, I met with another colleague who had just come back from maternity leave.  I showed her how to create some formulas in Excel for her gradebook.  She had been using Rediker last year and we no longer use that system.  We also brainstormed some ideas relative to the schedule and course sequencing, which may lessen some of the problems in future years.  One of the best things about working with the people in my department is that they are willing to share ideas and if one of my ideas is a bad one, they let me know about it.  If they think of a better way to do something, they willingly share it. 

In a previous school (this was about 18 years ago), I wanted to work with students during study hall with their math.  They weren't my students, but I wanted to help them succeed and they were willing to work with me.  Unfortunately, a colleague at that school saw my help as a personal affront to his ability to teach.  At no point did I say to the student that their teacher didn't know what they were doing.  I simply offered alternative strategies to solving some problems.  Rather than working together with the teacher to come to an understanding, I was told that I could no longer help the students with their math.  I am very grateful that this is not the case at my current school.  The best outcomes in education are achieved with cooperation, negotiation and being willing to listen with empathy to the perspectives of our colleagues.  

Tuesday, November 21, 2017

Teach 180: Grandparents' Day Part 2 (Day 56)

(Note: Today (the day before Thanksgiving break) is a 1/2 day of school and many students choose to take the day off.  Rather than talking about that for Day 54, I will be talking about Grandparents' Day for both Day 53 and Day 54 of my blog.)

For Grandparents' Day in calculus students constructed boxes out of paper.  The task: cut squares of equal size out of the corners of an 8.5" x 11" piece of paper and fold up the sides to make a box without a lid.  The goal: make a box with the maximum volume.  You can see the variety of boxes here.



Next, we used calculus to answer the question.  What should the length, x, be for the side of the square to maximize the volume?  We created a function and then took the derivative.  Students knew that the maximum value would occur where the tangent was horizontal.  

I love that this question doesn't lead to an easily factorable quadratic.  For a purely calculus and algebraic approach, students actually needed to use the quadratic formula!  And there was one solution that we had to throw out.  It wasn't the negative solution, because both solutions were positive!  Why couldn't x be 4.915? Since the one side of the paper was 8.5 inches, cutting in 4.915 inches from both sides would mean there would be nothing to fold up.  The domain for x was 0 < x < 4.25.  Finally, we looked at the graph in desmos to confirm our solution.  




Who had the box that was closest to being the one with the maximum volume?  Caroline S. with squares that were 1.5 inches in length and the volume of her box was 66 cubic inches. 

Teach 180: Grandparents' Day Part 1 (Day 55)

(Note: The day before Thanksgiving break is a 1/2 day of school and many students choose to take the day off.  Rather than talking about that for Day 54, I will be talking about Grandparents' Day for both Day 53 and Day 54 of my blog.)

The Monday prior to Thanksgiving, we have an event called Grandparents' Day.  Students get to bring their Grandparents to two classes, attend chapel with them and have lunch with them.  Today I not only had the opportunity to teach my daughter in Probability and Statistics, but her three grandparents - my mother, my father and my mother-in-law.

In that class, we did the following activity involving probability and simulation.  I had 4 index cards; one card had a wave, one card had a circle, one card had a plus and once card had a star.  I asked for a student volunteer to guess what card I was holding to my head.  We did 10 trials and the student got 9 right!  Amazing!!!  Since the average number correct would be 2.5 in 10 attempts, getting 9 right seemed very unusual.  But just how unusual?  Was I giving Neo signals to increase the probability of getting it right?  Using dice and cards, the students worked with their grandparents to conduct a simulation to see how many they would get right in 10 trials if the probability of success was 1/4.  You can see the results from our dotplot below.


Next we used Fathom to run the simulation and we also calculated the binomial probability of getting 9 right by guessing.  The simulated results show that in 100 trials none resulted in 9 correct matches.  This makes sense since the theoretical probability is 2.86 x 10-5.  Our Fathom results are shown below.



Friday, November 17, 2017

Teach 180: A Personal Day (Day 54)

I am not playing hookie today.  I took my second personal day to visit a college with my daughter, who is a senior.  (How did that time go by so quickly?) It has been an eventful trip. Our flight leaving Newark was delayed by 2+ hours, there was a scuffle that required a police escort of two men off our plane when we landed, and our rental car was reserved at the other Dallas airport.  We finally got into our hotel at 3 AM CST (meaning 4 AM in Easton, PA).  Despite the travel troubles, the visit has been informative for both my daughter and me.

What are my students doing while I am away? Today is E day in the 8 day rotation of our schedule and that means I would be teaching period B and F.  So, my sub is just teaching the first period of the day (8AM - 9 AM) and the last period of the day (2:15 PM - 3:15 PM).  The first period is Calculus and the students took at test and then watched the video "How Big is Infinity?".  After a 40 - 45 minute test, students are pretty spent and it doesn't make sense to try to move onto the next unit.  But I don't want them to sit around shooting the breeze either.  So, we either do a hands-on activity to introduce a topic or we watch a video like this one.

Period F was Probability and Statistics.  Because I am very concerned about falling behind with content due to our new schedule this year, I had the sub do the lesson I had planned for they day.  It included reviewing problems related to probability and having students work on a graded problem set.  Luckily, we have a math teacher that retired from my school a few years ago and having her as a math sub is a great asset. I know I can count on her to teach almost any content and she will do it well.

It looks like the weather for Sunday may cause us more travel headaches when we head home.  I hope not.  Monday is Grandparents Day and I have some special lessons and activities planned to have students interacting with their grandparents.

Teach 180: Rethinking Giftedness (Day 53)

A general perspective: I just finished watching Jo Boaler's film Rethinking Giftedness and although there are some valid things that the students say, I feel that the film is very one sided.  All of the students have a similar opinion - that being labeled as gifted was bad for them.  At no point did they mention a benefit to being labeled gifted.  Does this mean that we shouldn't label students as gifted?  What about labeling students as learning disabled?  The students labeled gifted saw it as a reason for why they couldn't learn something - "the gift running out".  Would those that are labeled learning disabled use it as an excuse for why they can't do well?  I would contend that we should label students (gifted, learning disabled, etc.) not to harm them or make them feel badly about themselves, but to make sure they get the help they need to maximize their potential.

Image from: jcoulter1992.wordpress.com/including-all-learners-2/gifted-and-talented/

My personal perspective: As a child I was labeled as gifted and I do recall the special test that was given to me at age 7 that the students in the film reference.  My parents didn't praise me for this label or expect more of me, because of it.  In fact to this day, I don't actually know the specific test scores or results. They did what all good parents do - encouraged me to learn a musical instrument, read for enjoyment, ride my bike and play with friends.  Because they saw my giftedness as just a part of who I was, I saw it as a part of me and it wasn't something that made me special or different.  It just made me, well, me.

After I was labeled gifted, I was put into a pull-out class with some of my other classmates.  If it wasn't for this class, I would have not learned the BASIC programming language (on a Radio Shack computer).  I would not have written and published a short story (after numerous rejection letters).  I would not have been exposed to logic puzzles and "thinking outside the box".  And I would not have experienced frustration that can sometimes happen in learning with supportive teachers to guide me through that frustration.  Perhaps if the students in the video had experienced their label in a similar way to the way I had experienced it, they would have seen their "giftedness" label differently.


Wednesday, November 15, 2017

Teach 180: Thoughts about Tracking in Math (Day 52)

Today at our department meeting we had a discussion about pre-requisites for certain courses, specifically Honors courses.  To remain in the Honors track, students need to maintain a B+ grade in the prior course.  Even with a grade that high, there are students who struggle when they are in Geometry Honors.  To lower the grade to a B would mean that we would have more students struggling in that course.  Alternatively, I would be forced to slow the pace of the course down for those students.  Although you may think that teaching material at a pace that is appropriate for the weakest student in the class is a good idea, it actually does a disservice to those students who catch onto concepts easily.  They become bored. Or they get into trouble. Or they do what my best friend and I did in fifth and sixth grade - complete their math work quickly so they can continue to silently read their favorite book.  To remove all Honors math courses completely or lower our current standards would be a mistake.  I don't have research or data to back my claim, but I do have twenty-five years of experience at six different schools in four different states and my experiences with tracking was the same at each school.  It was necessary to help those students who were struggling and important to keep top students challenged.

In my next blog, I'll critique "What Tracking Is and How to Start Dismantling It".  It is an article I found while composing this blog entry and I am curious as to what it says.  Plus I am interested in the short film by Jo Boaler called Rethinking GiftednessHer film is not so much about tracking, but more about labeling kids and the damage that can come with a label.  We wouldn't think labeling a student with a learning disability would be negative. So, why is "gifted" a negative label to use?

Tuesday, November 14, 2017

Teach 180: Math League (Day 51)

One of the things I wish I had more time to do during my day is to solve math problems.  I am not talking about writing answer keys, but problems that I do not initially know how to solve.  Problems that are novel and interesting.  Today I was given one of those opportunities.

Each month a handful of students get to solve challenging problems with the PA Math League contest.  It is a contest that consists of 6 questions and students have 30 minutes to solve those questions.  They span a variety of areas of mathematics including number theory, probability, geometry, algebra and trigonometry.  It is very challenging to get a perfect score on this contest.  When I took this contest in high school as a student, I would typically get 2 or 3 questions correct.

After the contest today, I had one senior who wanted to work with me at the whiteboard after the contest was over.  Since I had not been working on the contest when the students were taking it, (I was helping a student from one of my classes) I was initially stumped by the problem.  We discussed ratios of area of similar figures and between the two of us we generated some equations that eventually led us to the correct answer.  Our lively chatter about the solution brightened my day and made me wish for more moments like this.

As I was driving home in the car tonight after an 8th grade parent night, I was struck with another idea about how to solve the problem!  Since not all students have taken this contest across the country yet, I'll hold off on posting the problem and my solution for now.  In addition, I have some ideas that I want to try out in an effort to generalize a solution.  Generalizing solutions is the pinnacle of solving a math problem. As Fermat once said with his generalization, "I have discovered a truly marvelous proof of this, which however the margin is not large enough to contain."

Monday, November 13, 2017

Teach 180: Pseudo-Context Problems are Dull (Day 50)

I'll admit it.  I sometimes pull things from my file and say to myself, "This is what I did last year and this is what I will do this year."  And then afterwards I say to myself, "What was I thinking last year? That was awful."

Well, today was one of those days.  In PreCalculus we are studying function composition.  Here is the problem I pulled last year from the textbook to model a problem that students would see in the homework. 

There really is no context here.  Why 1000?  Where did 5 come from?  What is it that we are selling?  The problem itself involves a bunch of algebraic manipulation and there is no real reason for doing it.  There is no question to solve.  At one point, I may have even yawned while going over this question.   No, I am pretty sure I yawned.   Why did I drag my students through this pseudo-context?  Actually, I am not sure I can even call it pseudo-context.


Tomorrow is another day and I have made notes for next year to scrap this problem and replace it with a problem grounded in real context and not pseudo-context. 


Thursday, November 9, 2017

Teach 180: Connecting Inverses (Day 49)


Anytime you can help students to connect new learning to previous learning, I believe that the understanding of the new concept is stronger. Today I set out to purposefully connect the concept of function inverse to the familiar concept of inverse operations.

I began by having each table find f(g(x)) and g(f(x)) for the two cards at their table. Then, I had each table report out on their results. (You can see the pairs of functions at the left.)

Me: "What????? Wait a minute. You mean each of you got that f(g(x)) = x
and g(f(x)) = x. I wonder if that always happens. Hmmmm. Let's switch g(x) cards and see if that will happen again."

So, I had the student groups swap their g(x) functions and calculate f(g(x)) and g(f(x)) again. This time no groups got a result of x.

Me: "Well, that's thoroughly disappointing. I was hoping we would get x again. I wonder why we didn't. OK, let's look at the initial pairs we had."

Next, we wrote the original pairs of functions on the board and I asked students what they noticed about each pair of functions. One student quickly realized that one pair of functions had squaring and square rooting and she described those operations as "opposites". After that other students noted the other inverse operations they saw, like multiplying by 2 and dividing by 2, and adding 3 and subtracting 3.

I pointed out that not only did each pair of functions have inverse operations, but that they were done in an inverse or opposite order.  For example, in the first pair of cards for f(x) we would add 3 and then square the result.  For the inverse function g(x), we would do the inverse of those operations in the opposite order.  This means add 3, then square becomes square root, then subtract 3.

Our next class will meet on Tuesday (there is no school on Friday due to Parent-Teacher conferences and no class on Monday due to the new bell schedule).  For now, I am going to ask students to create inverse functions by using inverse operations.  We'll graph our functions and inverses in desmos to see that the composition of a function and its' inverse is the identity function.

We will also note that if a and b are numbers that are defined for both functions that f(a) = b and g(b) = a.  This will also be easy to show on desmos.

I will give my students "easy" inverses to create initially, but then they will be given ones that are more challenging, such as find the inverse of f(x) = (1 + x)/x.  This will lead to a need for the formal way to find an inverse - replacing f(x) with y, switching x and y, and solving for y.











Wednesday, November 8, 2017

Teach 180: The Calculus Toolbox (Day 48)

One of the things I love about teaching at a small school is that students find it easier to take risks and ask questions during class.  Today, we were working on learning the Chain Rule in Calculus.  One of the problems we reviewed involved using both the Chain Rule and the Quotient Rule.  However, one of the students in my class raised her hand and asked if we could rewrite the quotient as a product and use the product rule instead.  At this point, I told my students that they had a "Toolbox of Calculus Tools" at their disposal and that they could choose to do many of the problems in a variety of ways.  In fact, one of the things that I love about math is that there are often a variety of approaches to solving a specific problem. We ended up working through the problem using both the Quotient Rule and the Product Rule and saw it resulted in equivalent forms of the solution.

I am curious as to which tools students would prefer to use when given the option.  So, for our class opener on Monday (that's our next day of class - 5 days from today), I plan on giving students a few problems that lend themselves to using different tools in their Calculus Toolbox.  It will be interesting to see which tools they choose. 

Tuesday, November 7, 2017

Teach 180: Bad Test Grades (Day 47)

(Warning: This blog entry is being written after spending almost 10 hours at school and working 3 hours at home.  There may be some ramblings in this particular entry.  You have been fairly warned.)



I just finished grading a Pre-Calculus test.  Sometimes students do poorly on a test.  It happens. It could be for a variety of reasons or combination of reasons, such as:

1) The student didn't study.
2) The student wasn't paying attention in class.
3) The student stayed up late the previous night.
4) The student just broke up with his/her girlfriend or boyfriend.
5) The student didn't have a good foundation from previous classes.
6) The student has test anxiety.
7) The student has senioritis and doesn't feel like learning.
8) The student is waiting to turn 18 for his trust fund to kick in. (This was true one year.)
9) The student didn't meet with the teacher to review prior to the test.
10) The student didn't understand the material and/or had major misconceptions.

And there are probably many others that I didn't even consider that could be on this list.

A test is designed to assess student understanding of concepts (Reason #10) and not other things that can impact a student's performance (Reasons #1 - 9).  I know many of my math teacher colleagues at other schools allow for re-testing.  I even re-test from time to time.  But at some point students need to be held accountable for what they learn.  And if they don't learn it, they re-take the course in high school or take a remedial course in college.

If I constantly allow re-testing and it is because of reasons like #1, #2, or #3, then I am just reinforcing bad habits.  If it is because of Reason #6, then the student should get help from the learning specialist.  If it is a reason like #7 or #8, then at what point am I the fool for wasting my time and energy?  Time and energy are finite resources and I would rather give them to a student who is willing to work and learn.

Teach 180: Always Plot Your Data (Day 46)

Today in my Probability and Statistics class we did one of my favorite activities.  We used the Anscombe quartet to learn the lesson that summary statistics are only part of data analysis, and the fact that it is very important to always plot your data!  The Anscombe quartet has 4 sets of bivariate data.  You can see the data below.

Each of the four data sets have the same correlation coefficient of about 0.816.  They also have the same least-squares regression line.  But is a linear model appropriate in each situation?  What do the graphs tell you?

You could see the initial disbelief on my students faces when we looked at the results of each group on the board.  First, they couldn't believe that there could be data sets that could have such a high correlation coefficient that were clearly not linearly related.  Second, they realized that the only way they could see the relationship was non-linear was to look at the graphs.

For more on Anscombe's quartet, I invite you to read this interesting blog post.

Sunday, November 5, 2017

Teach 180: Irony and Chocolate Consumption (Day 45)

We finally came to a place in my schedule with my Probability and Statistics classes where I could actually teach the students the same content on the same day! (This year I teach 2 sections of Prob/Stat, 1 section of Calculus and 1 section of PreCalculus.)  I was so excited to have a day where I would only had to prep for teaching 3 different classes, instead of 4!

However, after teaching my first class of Probability and Statistics, I realized that there would be no way to get the new content taught in just 20 minutes.  (Students who needed extended time on the quiz in my first class could use part of the lunch period to finish the quiz.  There was no lunch period in my afternoon class and this led to 10 minutes less instructional time.)  Plus, the three big ideas to be taught with my second class were to be taught between 2:55 PM and 3:15 PM on a Friday afternoon.  Would my students remember these ideas when we had class again on Tuesday? Unlikely.

So...did I plow through the content?  Did I plan for this and "flip my classroom", having all students watch video explanations of the content after the quiz? I did neither of these things.  Instead, I had students read an article from The New England Journal of Medicine called "Chocolate Consumption, Cognitive Function and Nobel Laureates".  It is a brief article that shows that there is a strong positive linear correlation between chocolate consumption (in kg/yr/capita) and the number of Nobel Laureates per 10 million people.  The article suggests three possible reasons for the association, but clearly we cannot assume that creating a  law that requires people to eat more chocolate will increase the number of Nobel Laureates within a given country.  The brief, but important, lesson students learned on a Friday afternoon is that correlation does not imply causation.









Thursday, November 2, 2017

Teach 180: The Document Camera (Day 44)

Today we did more work on Pre-Calculus with transforming functions.  When several transformations are involved at once, it can be quite challenging for students to draw the image from the preimage.  Today, I decided it was best to use my HoverCam document camera to work through several problems with students. This allowed students to better see how each of the points was being transformed.  A screenshot of a completed problem is shown below.  The original function is in black and the transformed function is in purple.  The red annotation was added using the HoverCam software to show a pair of corresponding points.  I also had several students come to the front of the room and they also used the document camera to demonstrate how to transform various functions.


Depending on the assignment, I have found the HoverCam to be a helpful way for showcasing and critiquing student work.

Wednesday, November 1, 2017

Teach 180: Celebrating Students (Day 43)

                                                                                                                                          

One of the aspects of my school that I really like is that as a whole school we celebrate student successes.  We aren't just talking sports, but also successes in the arts and academics.  Today, I was able to announce in front of the entire student body that our Math Madness team won its first round of bracket competition with a score of 21-15. Plus, we have a student that is ranked #72 out over 20,000 students nationwide in Math Madness for the fall season.  The students cheered and clapped just as enthusiastically for this as they do for winning a district sports championship.

And speaking of district sports, good luck to the field hockey team and boy's and girl's soccer teams on their district games today and tomorrow. Go MA Lions!

Tuesday, October 31, 2017

Teach 180: A Little Competition Can Be a Good Thing (Day 42)

Today was our first day in Probability and Statistics for analyzing bivariate data.  One of the concepts that students sometime struggle with is estimating the strength of a linear relationship from a scatterplot.  They think there is no association when there is a weak, negative linear association.  Or they think there is a strong, positive linear association when it is more of a moderate, positive linear association.

Today, we used the Rossman-Chance applet called Guess the Correlation to improve our estimation skills with correlation.  Student were arranged in a bracket-style competition.  Each student was shown a randomly produced scatterplot with 25 dots and the student had to guess the correlation coefficient.  The student in the pair that was closest to the correct value moved on to the next round of competition.  An example is shown below.  You can see that I did fairly well. But not as well as the winner of the tournament, who was within 0.005 with his estimate!


Teach 180: Sometimes I Don't Listen (Day 41)

One of the things I love about working with the other math teachers at my school is the fact that we enjoy collaborating.  If we had a common office space, I am guessing it would be a challenge for us to get anything accomplished individually, because we would be sharing ideas all the time.

Even though we share ideas frequently, I don't always listen.  Marilyn told me that this one activity took her longer than she planned and I didn't listen.  In my plans, I had even made a note that I thought the activity would only take about 15 minutes.  Then, 20 minutes elapsed, then 25 minutes and finally the last group completed the activity in about 30 minutes.  Students were to match a function with its graph, domain & range and characteristics. You can see a grouping of four such cards below.


It was a valuable activity and students did really well working together.  Why did it take so long?  Part of it was the fact that there were 40 cards in front of them and I gave them no guidance in what might be easiest to match first.  In addition, I let students debate which cards were to be matched together and did not step in when there was a disagreement.  Usually, the student with the right answer prevailed and convinced the other students at the table as to why the answer was correct.

So, what would I do differently?  I might do eight groups instead of ten and I might model the thinking process for completing one match.  This would allow us to review key ideas, like open and closed intervals, prior to having them work on the activity in their groups.

Sunday, October 29, 2017

Teach 180: The German Tank Problem (Day 40)

I spent about two hours on Friday finalizing what I needed to do to prep for a talk that I was giving on Saturday.  The talk was given at the Association of Math Teachers of Philadelphia and Vicinity (ATMoPAV) fall conference.  The title of my talk was "The German Tank Problem: Simulating a Statistic".  For those of you who teach AP Statistics, this is an activity you likely do to introduce sampling distributions.  When I first did this activity, it bombed, because I wasn't comfortable with using Fathom to create a simulation of a sampling distribution.

Then in 2009, I took an online course to learn how to use Fathom.  I still wasn't comfortable with using it entirely and spent about a year screencasting demos to show Fathom in class.  Then in 2010, I started to use Fathom with my students.  It is a great visualization tool and it is great for helping students understand the difference between the distribution of a population, the distribution of a sample and the distribution of a sampling statistic. (Note: Fathom has a free trial version and a 1-year subscription is $5.95.)

For the German Tank problem, students are given a bag with N "tanks" numbered in their bag consecutively from 1 to N.  They randomly select 7 tanks and use the numbers on the tanks to create a statistic to estimate the total number of tanks. The eight participants at my workshop on Saturday worked in pairs and spent a good fifteen minutes to create their statistics.  At the right you can see the common statistics students often create.  The Partition Method is the one that was used by the mathematicians in WW II to estimate the number of tanks.  At the workshop on Saturday, participants created several of these common statistics.

I had been doing this problem for several years and last year I had a pair of students create a statistic that was good as (or perhaps better than) the Partition statistic.  I named the statistic after the two students, Cecily Redfern and Neelam Ferrari.  It's called the Redfern-Ferrari statistic.  Two hundred trials of the simulation can be seen on the screenshot at the left.  Note that 344 is the true number of tanks in the bag and that both Partition and Redfern-Ferrari are centered near 344 with similar variability.

What actually is the Redfern-Ferrari statistic?  The formula for the statistic is Max. + Range/6.  The students found the average difference between two consecutive numbers by calculating Range/6. Then, they added it to the maximum number they drew to get the approximation for the value of N.

If you have tried this activity and abandoned it, I would encourage you try it again. If you are interested in the Fathom files to do this activity and/or the handout I give to my students to do the activity, send an email to me at mathteacher@ptd.net or lnataro@mamail.net.

Thursday, October 26, 2017

Teach 180: Marbleslide Fun! (Day 39)


Today I filled in for one class of Algebra 2 and I was asked to lead a Marbleslides activity on lines .  Students primarily used slope intercept form and worked on restricting the domain of the function.  Although the teacher I was subbing for had shown students graphs in Desmos, the students did not have much experience using Desmos for themselves.  So, I used the teacher pacing feature to lead the students though the first few slides of the activity.  The few screenshots seen here show what the teacher pacing looks like on the teacher dashboard.

When you select the Pacing icon on the left side of the screen, you see this on the dashboard.


When you click on a slide, it highlights it and then you click on the orange "Restrict to Screen X" button to place all students on the chosen screen.


Once on that screen you can advance or go back one screen at a time by moving left and right with the < and > buttons.  But did you notice the plus sign?  Do you wonder what happens when you hit that?


What?!?! You can select multiple screens!  Now I can assign certain tables to one problem, if I wish and then we can have a breakout discussion after a few minutes.


The students were fully engaged and working with a partner for about 45 minutes.  Two girls did not want to work together and I allowed them to work alone.  I soon noticed that both of them struggled and did not persevere for as long as those working in pairs did.  In addition, I spent most of my time working with the two of them instead of checking in with the other students in the class.  This definitely showed me the value of a 2:1 ratio of students to computers.  In the future, I'll use my "random assignment by playing card" method to pair all students up and require students to work together.