Week 2 Teaching - the Gentle Push Back

The second full week of school has been a very meaty one. The kids seemed very eager to learn after the first few unstructured socializing/cohort retreat days. And I am starting to see the various personalities starting to emerge, which is both wonderful and more challenging because now it is real teaching and real learning.

In my Algebra 2 classes, we had our first quiz, which challenged all the students in different ways. My vision for the start of Algebra 2 had been to lay down a solid tech foundation alongside review or re-teaching of linearity skills, so that kids realize that a corner stone of Algebra 2 has to be using technology flexibly in order to self-monitor accuracy. I told the kids that I don't want them to ask me, "Is this right?" but I would be very happy to hear them ask, "How can I check this using the calculator?" So, on the first quiz they needed to demonstrate this skill throughout the algebra problems, in order to earn full points on reflection. (I gave separate points for: Communication, Approach, Accuracy, and Reflection.) In the end, my two groups were challenged differently; the Grade 10 native-speaker class had holes in their algebra skills and couldn't complete all problems, and the Grade 9 largely non-native speaker class generally showed stronger algebra skills (even with less in-class whiteboarding practice), but struggled with using the calculator flexibly. But, some of the kids shared their effective studying strategies in class, and some of the other kids are planning to see me on Monday for a requiz, so I feel quite hopeful that this first quiz is just the start of a learning dialogue.

By the way, my relationship with the international kids is developing in an interesting way. Because I speak Chinese, I am able to help the kids in my class without watering down the level of tasks I am asking them to complete. But, at the same time I can carry some weight when I see them being off-task and I offer to call their parents in China to have the dialogue directly about their efforts, in Chinese. What an interesting situation for me and them. Interestingly, they are better-behaved for me, and they try hard to speak English in my class, except when they need to help each other translate something. I am curious how they are going to do on their first big writing assignment, which we will be doing next week...

In my Precalculus class, kids are wrapping up the rough drafts of their first big project on special (triangular and stellar) numbers. I had to be very explicit in helping them to format their writeup. (I had to say at one point, "Take out a sheet of paper. At the top, write, 'In this task, I was asked to...' Now, complete that thought in your own words. I give you 30 seconds to do that. ...Now, write down, 'In order to accomplish that, first we had to...' and go ahead and complete that thought, make sure you insert a diagram here." But, after about 5 minutes of modeling, I think they all got the idea and were able to continue the rest at home, because all the drafts that they brought back to me the next day looked pretty coherent. So, it has been a tough project for them, for sure, but I still think it had tremendous learning value. We also had a quiz, and the kids are doing fine with function identification, interpretation of f(3)=7, and writing both recursive and explicit equations for arithmetic sequences. Some of the more clever ones were able to write formulas for quadratic sequences already, based on their learning from the Special Numbers project. So, I am pretty happy so far. As we wrap up our project (meaning, as I read over their drafts), the kids are doing mixed lines review and checking all answers via their calculator. They use either the Table or Trace to check all equations that they write from given info, and they use [2nd][Math] to verify equivalent expressions after simplifying. So far, so good, because kids in this class seem to be quite independent.

In my Calculus class, I had one student come forward to say that he really enjoys the exploratory nature of our class, and two others who came to ask me to do more examples followed by practice. I thought over this carefully and decided that although I think it is awesome that kids are being advocates for their own learning, and I really wanted to acknowledge that and to encourage that, the issue is really much more complex than their individual learning styles. I ended up describing to the class two contrasting learning models, direct instruction and inquiry-based learning. I said that in most math classes they have had, they probably experienced the former (intro, example, guided example, individual practice, closure, and eval), and that that is fine. It is comfortable, you know what to expect when you come to class. But, that way only reaches the top half of the class, the half that is fortunate enough to maintain focused attention and to comprehend at the speed of material presentation. Then I showed them a diagram of inquiry-based learning, which is a cycle of asking questions, investigation, creation of model or new knowledge, discussion, reflection, and back to asking questions. I explained to them that what we do in class is NOT true inquiry, because true inquiry would be like me saying, "Go. Find out how much universal health care is going to cost our country, both in the short run and in the long run." The problem would be entirely open-ended, complex, and vast, and we would learn all the necessary math skills as we move along. I explained to them that what we do typically in our class is a smaller version of this; within the individual topics of Calculus, I try to think about ways to structure our class so that they can create their own understanding. I ended the class with showing a little clip from Sir Ken Robinson's tedX talk (the animated one), and saying that teaching creativity is hard, and that our traditional schools have been doing a good job killing creativity. I told the kids that, yes, I think even in math there is room for creativity in the classroom, and unfortunately we don't get that by me doing an example and then handing out 25 problems that look the same. So, although I will try to find a balance between direct instruction and exploratory learning, I want the kids to keep an open mind and to appreciate opportunities for creativity in any discipline.

After this talk, I heard from another student in this class who said that she really enjoys math this year, and that our chat helps her to understand and appreciate my philosophy even more. So, one point for being authentic with kids and treating them as intellectual equals.

Good second week. Our school is awesome, by the way. I love that kids clean the school three times a week, and I adore my colleagues!!!!!

Here's a Graph that Makes a Statement

Although I do realize that it's complicated, and that there are a lot of factors that go into this data, it's worthy of discussion in a class that cares about real-world statistical implications.

(from http://www.forbes.com/sites/danmunro/2012/12/30/2012-the-year-in-healthcare-charts/ ).


And, here's a follow up, to make the discussion even better:

 (from http://ucatlas.ucsc.edu/spend.php )

Week 1 Teaching

My year at school has begun, and as of today, we have had a full 5 days' worth of classes, even though lots of kids were missing class here and there for special retreat-type of activities. I feel quite settled, and I am starting to learn most of the kids' names despite having a terrible memory.


I am thrilled about how Precalculus is going! It's almost too good to be true -- the kids can build their own patterns from colored blocks; they can write recursive equations to represent those patterns; they can explain the limitations of knowing only the recursive formulas; they can write explicit equations that turn out to be linear; they can sum arithmetic elements and explain why that sum must be quadratic, both from a graphical standpoint and an algebraic standpoint. And they're ready to break down quadratic patterns into linear elements, in order to find the formula for the n-th quadratic element! ...Rock ON, 11th-graders!! They will start their first project very soon, either tomorrow (if they understand all preceding concepts) or next week. It is a project that I adopted from the IB, watered down quite a bit simply because I don't want the kids to be scared off right away.

My vision of tying linearity and quadratics neatly together for the kids with the concept of sequences is working out well so far; we'll see how they fare on the first quiz! :)

In Calculus, I discovered on Day 1 that the kids were strangers to the powers of the graphing calculator. I spent the next day walking them through basic features, and since then they've been flying through the conceptual material which I've presented through explorations. Unfortunately, this class has had the most amount of absences, since the Seniors and various special leadership folks had to miss class starting on Tuesday of this week. That has caused the pacing to be a bit chaotic. The kids who have been here everyday now have a solid grasp of the derivative: what it is, how to read it from the calculator, and how it relates to average rate. They can visualize derivative graphs when looking at an abstract original graph of f, and in general I think their calculator skills are slowly maturing. Yay!

My Algebra 2 classes are two diverse groups. One is full of international students, some of who had just stepped off the plane to arrive in America with very little English comprehension. This has made for a very interesting challenge, trying to teach those kids in the same class as the native speakers. I think it is great -- it helps to teach all kids the virtues of patience and kindness, because some of the non-native speakers are much stronger in math than some of their native-speaking peers, so in my mind at least, it's a bit of give-and-take for everyone. The other Algebra 2 class is full of bubbly tenth-graders, and most of them are my advisees. I simply love that class! They don't work as fast as the other group through the material, but they're very responsible, eager to learn, and nice. Kids from my Grade 10 class keep saying that that seems like the fastest-passing period, and they seem to genuinely enjoy their time in class even though we're still on the nitty gritties of basic algebra. We're working our way through some knowledge about how to use the graphing calculator flexibly to check our work alongside reviewing skills from Algebra 1 and wrapping our mind around visual linear patterns. The first big project will be coming up soon though (next week!), and it's the rubber chicken bungee jumping project (I think at most schools, they use barbies), which will involve a significant amount of writing and really show kids what I expect this year from their analyses. Whee! I'm excited!

How are your school years going??

2013 - 2014... ready to rock it!

I'm so-so-SO excited this year about teaching. I'm going to try to keep my collection of lessons running online at http://bit.ly/mimisMath2013-2014. My goal is to produced better-documented units this year -- ones that can be more easily shared with colleagues and tweeps, that have clearly defined goals and rationale.

Happy September!!

Grading by Criteria

This is an excerpt from the grading section of my course syllabi this year. My new school is very good about letting each teacher choose an appropriate system for their own class. Two of my math colleagues saw this and thought it was interesting, so I thought that maybe it'd be interesting for some of you to consider as well. For me, this is similar to the grading system used in the IB Middle Years Programme. It also just generally makes more sense to me to be grading by criteria instead of by the type of assignments, especially because as a department we are trying to foster some of those important qualities in our kids.

Grading: Your grades are a direct feedback to you. They will speak to your areas of excellence, versus areas where I believe that you can still grow and develop your skills. They should, therefore, reflect the values that we hold in our course. To this end, your class grade will break down into grading criteria as follows:

Grading Criterion                       Assessed using…
Communication in Math 30%  All collected written explanations (classwork and projects)
Mathematical Approach  40% All collected “algebra work” (quizzes, tests, and projects)        
Accuracy and Precision  20%  All collected “algebra work” (quizzes, tests, and projects)
Reflection on Results      10%  On-going observation (class participation and written projects)

This might be different from what you are used to. Do not worry! You can always request a re-test or a re-quiz if there is something that you wish you had done better on. Also, grading rubrics will be provided in advance of project due dates, in order to clarify how these criteria will be applied specifically to the project.

Even though I still have a lot of things to prepare before the year starts, I am very excited! I think it's going to be a great year.  

Precalc Breakdown

Continuing with my Precalc brainstorm, I went through my new school's textbook (Functions Modeling Change, Fourth Edition, from John Wiley & Sons, Inc.), and pulled out specific content strands to correspond to each of the core topics I will need to teach. The current Precalc teachers don't all teach sequences as a topic in Precalc, but when I took it out, I found that no matter how I sequenced the basic algebra topics, they always seemed somewhat random and disconnected. So, I will still start off with a sequences discussion, in order to tie everything together for my own students. I also brainstormed a project for each Precalc unit during the year. If I end up throwing them out, it's ok! But, hopefully this gives me ideas to fall back on throughout the year. If they don't have references, either I have that project lying around from my own creation before or I think it'll be fairly straight-forward to pull one together this year.

Thoughts??

Topic	Key Lessons	Project or Lab
From Arithmetic Sequences to Linear and Quadratic Functions	1.       Sum of arithmetic sequences 2.       Quadratic patterns and quadratic formulas from applying sequence formulas to sum 2nd differences Daily openers: Practice linear and quadratic skills	Triangular Numbers and Stellar Numbers (past IB Portfolio topic)
From Geometric Sequences to Exponential Functions	1.       Sum of geometric sequences 2.       Exponential formulas from applying sequence formulas to geometric sequences 3.       Recursive vs. explicit formulas Daily openers: Practice basic exponent and log skills	Retirement Project / writing letter to parents
Modeling and Visualizing Graphs of Basic Forms	1.       Interpretation of parameters within a formula, domain, range 2.       Compare and contrast linear vs. exponential word problem setups 3.       Using GCD to analyze word problems after proper setup Daily openers: Practice graph prediction from formula	Regression lab via RC-circuit
Transformations of Families of Functions	1.       Review function notation 2.       Use the studied functions to reinforce basic transformations knowledge Daily openers: Practice transformations, forwards and backwards to/from formulas	@Samjshah’s family of transformed functions art project
Mini Capstone Unit: Modeling with Functions	1.       Go over function forms and uses (including forms not yet studied in the course) 2.       Give students guidelines on modeling assignment 3.       Do one example during class and then provide time in class to complete individual assignments Daily openers: Adjust as perceiving student needs.	Use old IB portfolio topics
Trigonometry of Circles and Waves	1.       Review non-right triangle trig with project from @KFouss 2.       Motivate circular trig using rollercoaster problem 3.       Unit Circle 4.       Graphing and transforming waves in both modes 5.       Solving trig equations (graphical and algebraic) 6.       Trigonometric identities 7.       Tangent function and GCD solutions Daily openers: Unit circle memorization; from hand-drawn graphs to finding trig formulas (and thereafter, to finding GCD solutions for intersection points)	Non-right triangle trig project from @KFouss Building an animated rollercoaster in Geogebra
End Behaviors of Polynomial and Rational Functions	1.       Start with modeling problems to motivate the various forms of new functions 2.       Fully develop each function type completely, paying attention to Calculus terminology 3.       Graphical analysis of instantaneous rates along a graph Daily openers: Predictions of graphs based on formulas; verify with GCD	Stocks modeling project Field trip planning project / write proposal to principal
Logarithmic Functions	1.       Function inverses 2.       Motivate topic using exponential vs. log scales 3.       Mixed practice in interpreting log scale graphs Daily openers: Solving logarithm equations	Research/poster representing an exponential dataset using multiple forms of graphs, discussing visual tradeoffs
Non-functions and Modeling Using Different Coordinate Systems	1.       Circles and Ellipses 2.       Polar coordinate system 3.       Hyperbola and the idea of Locus Daily openers: From Cartesian to Polar Coordinates, vice versa	Art project via Desmos
Year-long, Ongoing Review		Create a Precalculus Magazine summarizing each unit that we have learned.

Precalculus Retirement Project

I came up with a pretty useful retirement project idea for my Precalculus class. I say this is useful because more and more, I find that a lot of our friends are stressing over what's going to happen when our parents retire. Are they going to have enough money to get by? For how long? It seems to be a somewhat complex problem to predict accurately, because each month both the interest compounds and the principal left in the bank decreases.

So, to that effect, I plan to do a retirement project at the end of studying sequences and recursive/explicit formulas, and the kids are going to write a letter to their parents to make recommendations on why planning for a continued source of income post-retirement is really essential. Hopefully, this will help their parents talk to them a bit about money and financial planning, which I find that in some families is not as open a topic as it ought to be.

Check it out: http://bit.ly/retirementProj

It's still early in my planning of the course, and I would love any suggestions you may have! Geoff recommended that I could teach the kids to cross-check the explicit formula results from their calculator tables against the results from Excel (which are essentially iterations of the recursion from row to row, when you drag the formula downwards).