These are some of my thoughts about teaching mathematics. The purpose of this blog is to help me reflect and become my best teaching self. #MTBoS #iteachmath
Today we began a unit on exponential functions in PreCalculus. Each group of students was given a function and asked to do the following: create a table of values, graph the function, find any asymptotes, determine the domain and range and find any intercepts. Then, they put their work on large sheets of chart paper, which we posted in the back of the classroom. You can see two samples of student work here.
Once they were posted I invited students to stand in front of the posted graphs and I asked them the following questions. How do they compare? How are they alike? How are they different? These are the types of questions we need to ask our students. Answering these questions help students see connections between ideas and it allows them to reconstruct their understanding of a concept long after the exam is over. The more we ask these questions of our students, the more comfortable they get with making observations for themselves and the fact that in math there can be more than one approach or one right answer to a question. Slowly, I am getting my students to think like mathematicians.
At the beginning of the year, I jumped into #teach180 blogging. I had heard of other teachers doing it
and wondered "How? Where do they find the time?" and "Why? What is the point?" Well, today is Day 95 for me and I don't have an really have an answer to the first question. I blog during school
http://canacopegdl.com/images/why/why-15.jpg
(sometimes), after school (sometimes), on the
weekend (sometimes), and late at night (sometimes). Basically, I blog when I have a moment to sit still and it happens, because I have made it a priority. (An alternative explanation is that I have found a way to make days have more than 24 hours.)
The answer to the second question is I blog because it has given me a more formal way to reflect on my teaching. Scribbled notes in the margins of my planning book will still be a thing, and I will never want to give up on the conversations about teaching with my colleagues. But blogging is different. Because of my blog, I often think ahead about what I am teaching and what I might include in my blog. My blogging has led me to try new things and publicly share my successes and failures. Ultimately, I blog because it makes me a better teacher. Here's to 85 days more!!!
In AP Statistics today, I used an activity that Bob Lochel introduced at a PASTA (Philadelphia Area Statistics Teacher Association) meeting about two weeks ago. The AP Free Response questions have model solutions that are posted by the College Board and it is sometimes hard to know exactly how to use those solutions with our students. If the students simply read the solution, they are lulled into the sense of thinking they know how to explain a concept when they really don't. Bob proposed a better way to use the model solution as a teaching tool. This blog entry describes Bob's suggestion.
Today we were reviewing for the Chapter 7 test on Sampling Distributions. We used the Question #3 from the 2007 AP exam. I had students initially work independently to solve the problem. We then discussed the answers to parts a and b. Finally, they worked in pairs to organize the following strips of paper into a model solution. Among the 15 strips of paper, there where 5 "red herrings" or "distractors". Although students had difficulty with the ordering of some of the strips, they were able to identify the extra slips of paper pretty easily. If you are an AP Stat teacher, I encourage you to try it out for yourselves.
I enjoy going into other teachers classrooms to see how they teach and learn from them. On Friday I was fortunate to have some time to visit our new Algebra 2 teacher's classroom. At the previous day's faculty meeting, she told me that she was planning on using the Desmos Activity Builder "Polynomial Equation Challenge" to introduce the concept of polynomial functions and their zeros. The activity description says "In this activity, students will create polynomial equations (of degree 2, 3 and 4) to match given zeros and points. Students will explore how the factored form of the equations relates to the zeros and the order of those zeros."
As I watched the lesson unfold, I saw her use the teacher dashboard very effectively. First, she limited the students to about three to four screens to create a good combination of individual time for students to work and whole class discussion around the ideas the students were discovering.
Second, she stayed at her computer to monitor the dashboard as students worked. This is counterintuitive to how I teach. Normally, I circulate around the room to check on what students are doing and I have continued to do this during desmos activities. But staying glued to her seat, she had a better sense of what was going on in the moment and the students were engaged without her walking around the room! (Note to self: Use this teacher move more when using Desmos AB. It works!)
Third, she showed sample solutions on some of the slides to encourage precision in communicating. One student had said "when you solve them" and she turned it back to the class asking what was meant by "them".
Finally, rather than answering some of the student questions directly, she had students figure it out for themselves. One student asked, "What happens if you do a negative?" and she replied, "I don't know. Try it." She knew the answer, but was encouraging the student to use Desmos to investigate what happens.
If you are a math teacher (or any kind of teacher, really), I highly recommend that you visit the classroom of a colleague. I guarantee you will learn something about teaching and about your teaching self.
Today it was like my PreCalculus students were reading my mind. We just finished reviewing domain, vertical asymptotes and horizontal asymptotes for rational functions using some individual practice with deltamath. A.K. raises her hand and asks, "Can there be asymptotes other than vertical and horizontal, like slanty ones?" She asked this about 10 seconds before I was going to introduce the concept! I replied, "Let's look at one of those in Desmos."
The students could see the slant asymptote in the graph, but how did I know what it was? I told students to rewrite f(x) by doing the division. At that point the students could easily see the equation for the slant asymptote was the quotient.
In previous years, most students would be ok with that and not think anything of the remainder. But G.L. raises his hand and asks, "But what about the remainder?" More music to my ears! Noticing something and wondering about it! Unfortunately, I erred at that point and took the wondering out of the students hands. (Face-palm.) I explained that as x got bigger the remainder divided by the divisor would get smaller, meaning it was approaching 0. Why didn't I turn that back on the class? I could have had them enter 6/(x-1) in Desmos in a table to investigate what happens as x gets larger (or smaller). It's great that they are noticing and wondering, but now I need to get out of the way and give them a chance to figure it out.
As much as possible, I think it is important for students to discover math for themselves. Depending on the topic, it can be challenging to think of ways to do that. Today I combined the technologies of StatKey and Desmos to help students see that the standard error of the sampling distribution of a sample mean is the standard deviation of the population divided by the square root of the sample size. You know. . . this formula.
First, I assigned students a value for n. We looked at nine values that between n = 2 to n = 50. Each student created 400 samples based on their value of n. The population distribution was the percent of people with internet in various countries. You can see screenshots for the original distribution and for sampling distributions based on sample sizes of n = 4 and n = 16.
On the board each student listed their sample size, mean and standard deviation for their simulated statistic. It was very easy to see that each simulated distribution was centered at the population mean, and that as the sample size increased, the variability decreased. Was there a formula for calculating the variability or standard deviation from the sample size? To see what the relationship might be, we plotted the following in desmos and did a regression.
As a class we noticed that a is close to the standard deviation of the population, 29.259, and b is about 0.5. This also allowed us to review regression and the fact that this equation is based on sample data from our simulations. If we ran the simulations again, we would get slightly different data and as a result, a slightly different equation. One thing to make this lesson better would be to have students discover this for themselves individually or in groups. However, crowdsourcing this discovery as a class worked fairly well.
When one of my students was taking his PreCalculus exam yesterday, he tried to use his graphing calculator to solve a rational inequality. During the exam, he brought his graphing calculator up to me and asked me why his calculator was showing what it did. On the left, he has the graph and on the right is what he entered. Can you tell what is going on?
It took a few seconds for me to understand what it was graphing. What were those horizontal segments of y = 0 and y = 1? If you are having trouble figuring it out, look at the graph without the inequality graphed on top of the one produced by my student. Can you see it now?
When the function is below the x-axis, we know the y-coordinate is less than 0. When the function is above the x-axis, we know the y-coordinate is greater than 0. So, if the result of (x+2)(x-1)/(x+5) is negative, we have a true statement to the inequality the student entered and a y-value of 1 is plotted. If the result of (x+2)(x-1)/(x+5) is positive, we have a false statement to the inequality the student entered and a y-value of 0 is plotted.
If we look at the original graph, we can more easily see the solution to the inequality (x+2)(x-1) < 0.
(x+5)
It happens when the horizontal pieces are at y=1. In interval notation, the solution is
(-∞, -5) U [-2, 1]. Note that the student still needs to recognize that -5 is not in the domain of the original function.
A special shout out and thanks to H.B. for making me think about boolean algebra, graphing style.
Today was a 2-hour math midterm followed by three one hour classes. Teaching new content after a two hour exam?!? If I were to continue on with teaching the curriculum, this is what I would have taught. The Central Limit Theorem in AP Statistics - one of the most fundamental ideas to the entire course and vertical and horizontal asymptotes for rational functions in PreCalculus - a challenging topic when seen for the first time. If I did that, here is what my student's brains would have looked like.
Instead I taught a concept that only relied on counting and identifying numbers as even or odd. I thought that my students' brains could handle these second grade concepts after a two-hour comprehensive test. I combined my AP Stat class with a Calculus class and a PreCalculus class with an Algebra 1 class and we set out to discover what the conditions needed to make Euler paths and Euler circuits. I encouraged other members of my department to do this lesson and several of them did. One of my colleagues noted that she was surprised how uncomfortable many of her students were with an open-ended question with no immediate solution method.
Here is the outline I sent to my department.
1) Pose bridges of Konigsburg problem. Have kids
determine if a walking tour of the bridges is possible or not possible and
explain why.
2) Ask if we could remove or add a bridge to make this
possible and describe the location of the bridge.
3) Show how the bridge problem could be modeled with line
segments (edges) and dots (vertices).
4) Have them work in groups of 3-4 at the boards to create
a set of edges and vertices that can be traced where you start and end at
different vertices. (Could give restriction like, you must have at least 5
vertices and 5 edges.)
5) Ask students to count the valence or degree of each
vertex. (This is the number of edges going into a vertex.) Have each
group report out on the number of even and odd vertices. Amazingly
enough, each group should have exactly two odd vertices! Try to have a
student or two articulate why this makes sense.
6) Now have students create a new graph where they would
start and end at the same vertex. Go through a similar process to have
each group report on the number of even and odd vertices. Amazingly
enough, each group should have all even vertices! Again, call on a few students
to try to explain why this makes sense.
7) Have students come up with a class generalization about
Euler Circuits and Euler Paths. For example, to start and end at the same
vertex, all vertices must be even. This allows us to have an Euler circuit.
To start and end at different vertices, there must be exactly two odd
vertices. This allows us to have an Euler path.
8) Finally, give each group one of the problems from the
attached materials to do. They do their work on the board together and then
each group shares out the solution to their problem with those at their seats
critiquing the solution.
Below you can see some of the graphs made by my students.
We only had 10 minutes of the 60 minutes of class time left, which was not enough time to complete item number 8. Instead, I beat a few volunteers in several games of Nim. There was a game show atmosphere reminiscent of the Price is Right, as students called out suggestions for the next move in the game. Were the goals of the lesson achieved? If the goal was to expose students to new ideas in mathematics that they have never seen and would probably not see in a traditional curriculum, the answer is yes. If the goal was to have students work together to solve a non-routine problem, the answer is yes. If the goal was to have students learn about one of the most prolific mathematicians of all time, the answer is yes. Teaching after a Two Hour Exam? It can be done and the sizzling of brains was avoided.
We are currently in the middle of exam week and I although I did not teach on any of these days, I did do some teaching related things.
Day 86 - It was a snow day! But to the south, the roads were clear. So at 3 PM, I drove a little over an hour to Germantown Academy to attend a PASTA (Philadelphia Area Statistics Teachers Association) meeting. At this 1.5 hour meeting we heard a talk from a statistician who works at a local pharmaceutical company and two teachers shared ideas related to teaching.
One idea was presented by Beth Benzing about students doing experiments with paper airplanes. She split students into two large groups of 10 students each. One did a matched pairs design and the other did a completely randomized design to test the flying distance of their planes. (The students chose the experimental design and Beth did not realize until later that the groups had chosen different designs.) After students gathered the data, they analyzed their data using Fathom and StatKey and drew conclusions based on hypothesis tests and confidence intervals. They entered all this information onto a google slides presentation AND gave the presentation to the class. Sounds like an activity that could take several class periods. Right? No - one 85 minute class period was all it took! Amazing!!!
Day 87 - Today I taught a lesson to students at Kent Place School in Summit, New Jersey. It is always challenging to teach lessons to students you have never met before and jump in during the middle of a unit. However, I had fun teaching the lesson and using ideas from an NCTM 2017 session presented by Jonathan Osters (@callmejosters).
Day 88 - Today I am working on putting together a lesson to use on Monday after our math exams. We have classes after the students take their math exam, but not all students will be there due to make-up exams. Plus their brains will probably be a bit fried from taking the 2 hour math exam. Rather than trying to force new material in to an tired brain, I am suggesting that math teachers teach a lesson on Euler paths and Euler circuits. I'll describe how that lesson went in my next blog. I will be joining my PreCalculus class with an Algebra 1 class for this lesson. I am hoping they will find the topic as fun and interesting as I do!
Today was the first day of midterm exams and a grading day. In the morning, I proctored a midterm exam in English and then I worked on grading quizzes, homework and poster projects for Probability and Statistics. I wrote a little about these projects in my Day 82 blog entry. For about 3 hours today, I spent about 10 minutes or so reading each poster and writing detailed comments to give the students feedback. In addition, I read all of the comments students gave each other and realized that there are some areas where I need to improve when I give this project as an assessment next year.
Here are some specific things I need to do differently:
1) Require students to use a larger font size and suggest a word limit. Students are comfortable with writing more, but they also need to learn to communicate the same ideas while writing less.
2) Revisit my directions to this project to make sure it is clear that students need to include certain statistics directly on their poster.
3) Not all students had a clear understanding of how the results of their simulation either supported a biased result or failed to support a bias result. Although we have done simulations with dice and cards, I need to spend part of a day reviewing how simulations work in Fathom. Next year we will review the video called Bias Project Simulation Results during class, instead of having students do this outside of class.
The reason I became a math teacher was because I truly love my subject and was told by one of my math teachers that I was good at working with students. She said that I would ask questions that the students should be asking themselves, modeling the thinking process the students would need when working by themselves. At that point, I thought that is all that teaching was. Working with students on math and helping them to understand math. In fact, that is what I thought it was for the first several years of my teaching career.
Well, teaching is more than just teaching a subject. At the beginning of my career, I would have said that I am a math teacher. Now, I say I am a teacher who just happens to teach math to high school students. I say this because there is so much more that teachers teach their students than just a specific subject area. We teach by the examples we set and we teach based on the stories we tell from our lives. In advisory today, we were talking about the need to Speak Up! if we hear talk that is racist or stereotypical. (The link takes you to the program that is being used within our school this year.) We looked at several scenarios and because my advisees were seniors, they had much to say on this topic. There was also an alumnus who happened to be visiting the school and I invited him into my classroom to participate. He gave some suggestions based on his experiences at college and even though I could have offered those same suggestions, the alumnus' words were probably seen as more valid than if I had said them.
There is no school on Monday, but I will be busy preparing a chapel talk, grading, finishing a consulting project and getting ready for the second semester.
It is around this time of year that I get bogged down in piles of administrative stuff. Today I only taught two classes, which meant my teaching day was done at 10:05 AM. However, my afternoon was filled with the following:
1) Working with a teacher on a new course proposal for an Introduction to Computer Programming course.
2) Working with that same teacher to get her NCTM membership renewed.
3) Coordinating midterm exams for 12 students who would be missing their math midterm due to County Chorus next Friday.
4) Asking for teacher input for students who should take the AMC contest in February and printing off contest information to prep for the contest.
5) Checking with five different teachers to see if a classroom would be open to administer the contest in February.
6) Organizing my midterm exams for next week.
7) Organizing the midterm exams for the extended time room.
8) Posting results from the latest Math League contest.
With all that you may think that I have finished everything that I need to do. Nope. I still have the following on my "To Do" list for tonight:
1) Grade Stat homework (only 2 remain)
2) Grade PreCalc projects (only 4 remain)
3) Grade Calc quizzes (16 to grade)
4) Create a 12 question Kahoot to review for PreCalculus
5) Create a "speed dating" review for Calculus
6) Finalize grades for all classes.
"Would you eat this pizza?" Sure. It looks delicious. "Would you eat this vegan pizza?" Maybe not. But it's the same pizza!?! Did introducing the word "vegan" in the question bias the results? This is an example of two questions that were asked as part of an experiment for the bias project.
In my Prob/Stats class, I have students create an experiment where they purposefully see if they can bias the results of a question. After gathering data, students used a simulation in Fathom to see if the results based on the biased question are likely to happen by chance alone. Students put their results on posters and I hang the posters around the room. Then, half the class stands by their posters and they present to the few students that are in front of them. This takes about five minutes. Then, the students move on to another poster and pair of presenters. Students get to give their presentation about three or four times. No powerpoints, no notecards, less nervousness and similar to informal presentations they might need to make at some point in their lives in the future.
I told my husband about this project this morning and he asked if my students peer review the posters as part of assigning a grade. When I asked him how I would go about this, he said to create a rubric. I am ok with creating rubrics, but I often feel that they can be challenging to have students (and adults) use the rubrics in the same way. By this I mean that it is hard to have evaluators get similar results with a rubric unless there is extensive training. (AP Statistics teachers who have been to the AP Stat reading can attest to this.) So, instead of a rubric I had students give qualitative feedback. For each presentation they saw, they had to write a "Kudos" and an "Improvement". As I read through the responses this afternoon, I felt like the students gave honest and helpful feedback to each other. Although I am a math and numbers person, qualitative feedback can often be more powerful than a single number.
A few weeks ago, I gave my students a project called the "Birthday Polynomial". This is a project I found on Twitter when I first started using Twitter a few years ago. Students create a polynomial based on their birthday and that means each student has a different solution to the assessment. They used word and copied and pasted images from Desmos into their work. It was not as easy for students to do this as I thought. If only I had waited until now to give this assignment.
Why? Today I got an email today with the following title "Desmos graphing now fully integrated into EquatIO". What is this? I read the email more closely.
Writing math in google docs and inserting desmos graphs in those docs easily? No more screenshots? When did this happen??? I clicked on the link that took me to a blog by texthelp that was dated January 8th. So, a recent development. But then I read further. "Back in September, we announced a partnership with Desmos..." This partnership happened 4 months ago!!! That's a lifetime in the world of technology.
But on a closer read I see, "Today, Desmos graphing has been FULLY integrated into EquatIO across all platforms - EquatIO for Google, Windows, and Mac, and the web app, EquatIO mathspace." So cool! In fact, I told my students this news and they equated it with the draft of a valuable football player. A powerhouse had been born, they said. Their next question for me was "Now can we use desmos on tests?"
Looks like I have some work to do. I sense more open-ended questions, like graph two lines that are perpendicular and explain how you know from the slopes that they are perpendicular. Students can confirm their answers are right with the graphs, but still need to explain their thinking. We have exams coming up next week and I'll have some time to think about how to begin to integrate this new tool.
My day started at 7:25 AM by meeting with a student. The student was in PreCalculus and we were reviewing the following problem. Given the polynomial and its factor, determine all other zeros and factor the polynomial.
The student knew that +2i was another solution, but didn't know how to use the two solutions to answer the question. Although the student knew how to do polynomial long division from a previous lesson, we first looked at a graph of the function to see if this function had 4 complex zeros or 2 complex zeros and 2 real zeros. From the graph, we can see that there are real zeros at -3 and 5.
Next, we divided (x - 2i)(x + 2i) = x2 + 4 into P(x) and found our quotient to be x2 - 2x - 15. This quadratic factors into (x + 3)(x - 5), which agrees with what we saw in the graph.
"When am I ever going to use this?" the student asked me. I told him truthfully that he would probably never personally use this outside of another math class. Then he asked a different question, "Where are complex numbers used?" I told him in engineering - working with electronics and with fractals. Then I referenced the Genesis planet in Star Trek and computer graphics as a use for fractals. (After posting this blog, I'll be sending the student the video link.)
Special Note: Much of the math I teach, other than statistics, will never be used by many of my students. So, why teach it? Of course I want students to learn math concepts, but more importantly I want them to notice, wonder and ask questions. Why does this polynomial go up on the left and the right and the other one goes down on the left and up on the right? What about the numbers in the polynomial make them do that? Helping students make connections between ideas and noticing relationships. Having them make and test conjectures. Having them say "What if?" or "Does this always happen?" This type of thinking is what I want my students to do beyond my classroom. The mathematics will be important to some of them later. But for all my students, I hope the math we discover together encourages them to be curious and to think deeply.
No school yesterday and no school today. You may think that meant that I celebrated by doing something special for myself. In fact, I kind of did. I went to the doctor with abdominal pain and had a CT scan and blood work done. Nothing unusual was found and luckily, my abdominal pain has subsided with appendicitis being ruled out.
So, what am I doing today? First, I sent emails to all of my students letting them know how their assignment calendars would change over the next two weeks. Next, I ate breakfast and created a video of the lesson my Prob/Stat students would have in class today. The lesson is on Sampling Distribution for a Population Proportion and I posted it in my statistics playlist at my YouTube channel, mathteacher24.
One of these polynomials only has real zeros and the other one only has complex zeros. Which is which and how do you know?
F(x) = x2 + 4 G(x) = x2 - 6x + 8
I asked this question of my PreCalculus students today. I was curious if they would focus on an algebraic approach or a graphical approach. My bet was on an algebraic approach. They took a minute or two to discuss this at their tables and all groups reached the same conclusion based on, no shock, an algebraic approach. They set each function equal to zero and solved the resulting equation.
Because I think it is important for students to connect various representations, we graphed the two functions in Desmos. F(x) is the red parabola and G(x) is the blue parabola.
I asked, "Why does the red parabola have complex zeros? How could you tell that from the graph?" One student answered, "It has complex zeros, because the vertex is on the y-axis." This was not what I was expecting. But rather than throwing that back at the class to see what they would do with it, either confirm or refute it, I entered y = x2 - 1 into Desmos to show a parabola with a vertex on the y-axis and two real zeros. What I should have done was had the students use Desmos themselves to either prove or disprove the student's statement. Clearly, I understood the connection between the algebraic and graphical representations, but did my students? I have some thoughts about how I will assess this when I see my students on Friday. That will be for another blog entry.
This is where I was over Christmas Break - in Guatemala. We went to visit our exchange student and her family. I was mostly relaxing with my daughter and missing home and my husband, but I did take the time to re-read "A Mathematician's Lament" by Paul Lockhart.
On top of Casa Encantada in Antigua
Volcán San Pedro on Lake Atitlán
As I read the book, I asked myself, "What should be part of the math curriculum at my school?" I agree with the author, Paul Lockhart, that many attempts done by textbook authors to make math "real world" often fall short and are extremely contrived. If you don't think so, take a minute (well, 30 minutes) to watch Dan Meyer's TED talk "Math Class Needs a Makeover".
Contrived pseudo-context aside, can we teach math, as Lockhart suggests, just for the sake of the beauty of the subject itself? Would students still do well enough on the SAT and ACT to get into top-tier colleges? Would they have the math skills needed to do well on the AP Calculus exam or in their college chemistry class? Reports from alumni at my school are that the math they have learned has prepared them well for what they are doing now. Should I rock the boat and scrap the entire math curriculum as suggested by Lockhart? If the cart isn't broken at my school, should I be fixing it? I think the true answer lies in modifying the cart. Right now it is useful and getting the students at my school where they need to go, but it isn't a very aesthetically pleasing cart. If most students see math as something they must endure to get them to their goal of "The College of My Choice", I have fallen short as a teacher of mathematics.
What first drew me to math at the age of seven was the relationship between numbers. I recall having difficulty with memorizing my addition and subtraction facts and getting extremely frustrated in the process to the point of tears. However, I soon learned that if I knew one fact, I could easily figure others. I also noticed patterns. For example, a "teen number minus nine" was one more than the ones digit of the teen number. Consider 13 - 9. Then answer is 4 and 4 is one more than 3. What about 17 - 9? The answer is 8 and 8 is one more than 7. The only thing missing at the time was an understanding of why this "teen number minus nine" thing always works. [Notice it is simply regrouping. Think of 17 - 9 as (10 + 7) - 9 and rearrange to be (10 - 9) + 7 = 8.]
This curiosity about the patterns in math and why they work is what makes math beautiful and interesting. Making and testing conjectures. Discovering relationships between ideas. Finding generalizations and proving they always work (or not). If this sort of thinking and play is not at the heart of a math curriculum, the math being taught will be seen as a set of cold and unforgiving rules to be followed. As I begin teaching in 2018, I hope that I can help more of my students to see the beauty and creativity that can be found in mathematics.
