Live and Learn

In a nutshell... The change to two groups - one faster - didn't work.

As I alluded to in my last post, I have a range of math abilities in each of my precalculus classes. I have 2-3 students who could easily be successful in an advanced calculus class this year alongside students who would be better served by a normal precalculus class. (Groups in-between could include an advanced precalculus class and a normal calculus class.)

The idea was to see if I couldn't serve two groups' needs in one class, and I discovered that it just didn't work. I found that, within a 40-minute class, there wasn't enough time for me to be available for both, and I felt that my attempt to split time (quite unevenly) resulted in less quality time with both.

Live and learn.

Things have settled down quite a bit this semester, however. I've discovered that homework of 6 problems/night is close to ideal for my students and the time we have, and, thanks to some construction changes that took place over our break, I now have plenty of chalkboard space on which to display problems. The students know that they'll be presenting every other day, so they come to class prepared to get started.



Quite a learning process for me. I'm lucky to have such great students.

Mid-Year Thoughts

After a long Spring Festival break, second semester begins on Monday. Thus far, it has been quite an adventure, both teaching in China and teaching in a Harkness/discussion-based format. Time to reflect:

1) I've definitely become quite a fan of the Harkness Method of teaching. It is student-centered, problem-centered, and understanding-centered which I absolutely love. In addition, it really brings forth the different ways math problems can be solved, which deepens learning.

My experience immediately prior to coming to China was as an SAT (and ACT) math tutor and what I found was that students didn't really know how to problem solve and didn't have a good grasp on the concepts they were working with. This leads me to my second thought...

2) The Exeter materials are pretty amazing. I've written previously that they were 8 years in the making and you can tell. The problems are so rich, the spiraling is impressive, and the concept development is really well thought out. I'm humbled that they've done all this work and made it available for free.

3) The teacher's role as facilitator/guide is quite different from that of "sage on the stage," and mastering it will take some time. As I've mentioned, I need to get better at asking questions: questions to clarity, questions to draw out, questions to expand upon. No doubt, this is part of the art of teaching.

4) Allowing for different ways of solving problems means you, as the teacher, really need to know your stuff and be paying attention. Things are definitely easier when "there's only one way, the way I taught you;" however, this is, to me, incredibly anti-student and anti-learning. Suffice it to say that I've learned A LOT thus far this year and I have relished all of it.

5) It's difficult having an extended discussion in a second language. I applaud what my students are doing (taking classes in English), but I've come to realize that I need to temper my expectations as to  discussions. They are presenting in English and asking questions of me in English; however, sometimes, when other students have questions, I let them switch to Chinese. There's no shortage of discussion, questions or answers, when they speak in their native language.

Decided to make a few changes for this semester, principally dividing the students (precalculus) into a "faster" group and a "normal" group. One thing Exeter does is have their incoming students take placement tests, and I can see why - some students can simply fly, while others need more time. Curious to see how this is going to work.

Student Thoughts on Harkness

Our first term ended last week. As I look to make some changes after Spring Festival (Chinese New Year), I asked students to provide me with some feedback as to how the class has been thus far.

A couple of background comments:

1) Our (international) classes are MUCH smaller than typical Chinese classes. My largest class size is 13, most of the Chinese classes have between 40 and 50 students. My students are therefore having a very different experience to begin with.

2) As one who is taking Chinese lessons, I'm amazed at how tiring it is trying to think and speak in another language. My students have generally had around 8-10 years of English; however, I remain humbled by their ability to take 6 high school classes each day in English.

First Question: "How has the math class been for you thus far? Are you enjoying it?"

• I really like this way of learning math.
• Good. Fine. OK. No I'm not, I hate writing things down.
• I think most of the things are OK. And the way that students present the questions on the blackboard is good. I am enjoying studying the math. I think I can understand most of the problems very well.
• Good. I enjoy the math class because I learn things little by little.
• Not that enjoy. Some of the questions that (are) in the last (recent) pages are kind of complicated. I hate complicated questions.
• I think it's kind of hard for me, and I got (have) to spend much more time than others to solve the math problems in English.
• I think it's pretty good. I really enjoy it. The questions we (have) done were not too hard.
• I like the method. I feel better and more comfortable to speak in front of people.
• I enjoy it. Some problems are hard for me and I can't figure it out by myself. But listening to others explanation is pleasant to me.
• I think the math class is really good. I like the method we used that (for) learning new knowledge through new question(s). I enjoy it.
• It's good. It's exciting.
• Yes, I enjoy this class. I practice my English in this class. I had a good time in the math class.
• I think I feel good about it now. Maybe I feel a little bit uncomfortable at the beginning of the semester, but I gradually felt much better.
• I like math class. Everyone has opportunities to show their method during the class. I learnt (learned) many different ways to solve same questions. And also, when I did question on blackboard, other people could check my answers.
• I enjoy this class because was can participate in the class. We share the ways we solve the problems and discuss. It's helpful.
• I enjoy the math class so far. The teacher gives plenty of time for me to practice both solving problems and speaking English. I learn some mathematical terms during the class so that I will demonstrate more accurately next time.
• I like the way of our math class. I enjoy it because everyone can share different opinions in the class and we have more chance to practice our spoken English.
• I'm not that enjoying the math class because I'm not good at math all the time. I think it's hard for me to improve my math skill.
• Great. I am enjoying the discussion because this process let me know more way to solve questions and fix my mistakes.
• Good. I enjoyed it.  I like to see using different ways to solve problems because sometimes others' way is quicker and easier than mine. And we have the chance to correct our quizzes, that's so helpful for us to learn.
• I think it is very good. I enjoy it because I think we have different kinds of questions for a day, so it is not very boring to do homework.
• Great! I really enjoy it. I love different ways doing the problems and people showing their own ways.
• Yes, as discuss questions together, we know more methods.
• Yes, I enjoy this class because in this class I feel very free. I can discuss with class and I can find lots of different answers. That's great.

Second Question: Would you like to see anything done differently? If so, what?

• I wish I can do more difficult problem.
• I prefer more geometry questions. (My note: comment from precalculus student.)
• I don't really want to do solve many problems. I actually want to study the book. (My note: they purchased a textbook. I've using Phillip Exeter's "book" instead.)
• To be honest, sometimes I did not listen to the discussion. Maybe the discussion time could be shorter. In this case, we can learn more things. I always do my homework during the discussion, except there is something important.
• I hope that next semester, we could do new question that teach us new math such as (logs). I don't want to do many easy questions again and again. I want the same amount of homework but it could be harder.
• I think the math class now is good.
• I like the present condition.
• I don't know why don't we use our textbook instead of so many paper. Maybe we could use textbook next semester. (My note: The Exeter materials have been copied onto paper for each student. I'm putting it all together in a "book" for next semester as all the paper is a bit of a pain.)
• Yes. More questions with graphs and less words.
• I think it is fine now.
• Everything is great.
• I would like to see something different.
• I have one suggestion. For problems like exponents and simplification, we can just go over the answers, and leave much for other problems, like parametric equation. I don't study this in my Chinese math class, so it's new to me. You can introduce the math skill to me starting with some easy problems not based on the problem we have on the paper. That will help me understand the parametric equation more deeply and clearly.
• More and harder problems on geometry; less problems on three-dimensional. (My note: from a precalculus student)
• I think the math class we have now is pretty good.
• Yes, I'd like to have more geometric questions because graph is more interesting that words.
• I think the Harkness Method is great.
• Yes, I would like to see more geometry questions. (My note: from a precalculus student)
• No, it's good.
• It is good now.
• I'd like to have more time to do homework in the class, because we have much homework and TOEFL class. So we are a little busy. That is why I want more time to do homework at class. If we find the homework we don't know, we can ask questions in time.

In general, a lot of positive comments and some good suggestions. I'll be making some changes for sure.

Feedback Matters

Via Scott MacClintic's weekly blog, Pelican Ponderings (http://smacclintic.edublogs.org), I came across this 2012 piece by Grant Wiggins - http://inservice.ascd.org/less-teaching-and-more-feedback/#.VLleImBWNSM.twitter.

As he states, "You don't need any 'teaching.' You only need a good feedback system."

This has been resonating me recently as I've utilized Wikipedia, Khan Academy, and the like to further explore math topics. Students today have SO many resources at their disposal that the role of the teacher can no longer be that of "dispenser of knowledge." After all, the knowledge is available to anyone at any time.

My purpose, as I see it, in a discussion-based classroom is to provide immediate feedback (hopefully in the form of good follow-up questions) and to enable feedback to be provided by others, as applicable, so that learning takes place. Try. Get feedback. Question. Get more feedback. Fail. Get feedback. This isn't easy, but it's necessary.

Interestingly, I experienced some frustration with an online class recently due exactly to the lack of this - timely, focused feedback. I wanted to talk and discuss what I was thinking and all that was available was asynchronous communication. I found this frustrating.

Immediate, applicable feedback is the foundation of a discussion-based classroom, and it's necessary for learning to take place.

Effective Instruction

In a recent blog update, Grant Wiggins (through whom I learned of Phillips Exeter's Harkness Method) shared the results of a large UCLA study that found a lack of creativity in classroom teaching. A few excerpts:

- Students ''scarcely ever speculate on meanings'' or discuss ''alternative interpretations.''

- a ''sameness'' of instruction, with teachers doing virtually all the talking ... on the average, only seven of 150 minutes of instruction in the course of a school day involved teachers' responses to individual students. Feedback and guidance to help students understand and correct their mistakes were often ''almost nonexistent.''

- for most students school is a place in which to listen, to respond occasionally when called on, to read short sections of texbooks and write short responses to questions on quizzes. Students rarely, the study found, read or wrote anything of some length. Most of the time, they listened or worked alone.

A few days ago, I finished the book "Building a Better Teacher," one of the highly-rated reads from 2014. A few excerpts:

- It was crucial, for instance, to make sure that students did not talk just to (the teacher), but to the entire class.

- Watching (them) teach, (the teacher) had grasped the importance of getting her students to talk.

- "How do you know that?" "What do other people think about that?"

- The key to moving a discussion forward was to listen to students' questions, figure out what they needed to understand, and construct a response to pull them there.

- moving from the usual ask-tell ping-pong to something looser. "Facilitating."

As I read these, I thought about what I'm trying to do in my classroom. I want a discussion-based, facilitated classroom, with the student at the center, in charge of her/his learning. I will readily say that I (will most definitely always) have a lot of learning to do, but I am finding that the Harkness Method is proving to be a very effective way to provide high-quality instruction.

Who's Doing The Learning? (Therefore, Who Should Be Doing the Work?)

Like many teachers, my first years of teaching were exhausting! It's a tough profession, and I could see why it had such high, within-the-first-five-years, drop-out rate.

At the beginning of, I believe, my third year, I enrolled in a Fred Jones professional development course (Positive Classroom Management). He has a number of "Fredisms" to describe education and the bedrock of his system, one of which is really resonating with me this year - 

"There is no reason a teacher should work him/herself to death while the students sit back and watch. Effective teachers work the students to death while they sit back and watch."

Now, in no way am I advocating anyone "working to death;" however, think about it. I know the math, they don't. They're doing the learning, so they are the ones who should be doing the work (and getting tired :-). This year, teaching via the Harkness Method, this is exactly what is happening.

In my classroom, students are:

1) solving math problems (sometimes struggling)

2) presenting their solutions on the board

3) sometimes leading the discussions (made more difficult due to English being their second language)

4) answering my questions (and other students') as to how to do something

In other words, they are often the ones doing the doing; I'm the one "sitting back and watching." Learning requires involvement, the crux of the Harkness Method. In my classroom, the students are involved each and every day.

Is my job "easy?" No way. I'm constantly engaged/focused during class to understand all their different methods. (I'm learning the math so much more deeply.) As I mentioned in my last post, I'm having to relearn my profession, to ask high quality questions instead of providing immediate answers (and I have a long way to go). My job is different, focused so much more on their learning, and, given my experience with the Harkness Method thus far, I wouldn't have it any other way.

Questions. It's All About the Questions

The basis of a discussion-based classroom, is, of course, discussion. This requires good questions in two areas: 1) questions to be solved and 2) questions to be asked to complement, facilitate, and address difficulties.

After three months of using Phillips Exeter's math materials, I will readily admit that I am quite impressed with the level, the depth, of their questions. The vast majority of what I read when it comes to improving our math education is the need for more problem solving, more discovery learning, a greater focus on the richness of mathematics as a whole. Exeter's materials do all of these. I'm so appreciating questions that don't have whole number solutions (why do textbooks insist on always have "perfect" answers?), that spiral through topics, that lead students to derive formulas, and that often require knowing more than one mathematical idea to solve. This is, after all, what mathematics is like in the real world.

(I read somewhere that it took the Exeter faculty eight years to create their textbook! I'm humbled by such an effort.)

So, I have good questions to be solved. Now, as the teacher/guide/facilitator, the issue is asking good questions. This is definitely an area in which I need work. I'm so used to wanting to answer questions to keep things moving that I'm not nearly as good at what's really important - asking questions to guide in decision making, further challenge, or deepen learning. Students need to think and I need to help make that happen.

Area 1, good. Area 2, in process.