Teach 180: Snow Days (Day 78 & 79)

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.

Teach 180: Connecting Representations (Day 77)

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. 

Teach 180: The Beauty of Math (Day 76)

 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.