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.

The Importance of Talking

I walked back to my apartment with a student last week, discussing with her the TOEFL (Test of English as a Foreign Language) exam. For foreign students, the TOEFL is #1 priority for getting into US universities, as "good" scores are required before the application is even considered.

I asked her, "What can I do in math class to help you get ready for TOEFL?" Her response was not quite what I expected. I knew that having them present problems, as required in a discussion-based classroom, was important, as were my efforts to correct their English. What I didn't realize was the importance of my speaking.

Harkness/Discussed-based classrooms put a premium on student speaking, and I can certainly see how important this is, in any learning situation. However, second-language learners also need to hear the language spoken properly to pick up on the many nuances, particularly in English. My efforts to get them to speak and listen to one another are certainly worthwhile; however, my situation is unique in that they need to hear me a fair amount, also.

The goal remains the same - do what is the best for your students. So, while their discussion (primarily through presentation) remains at the top of the list, my speaking has moved up the list to further help them in an area of need.

Interestingly, in using Exeter's math materials, for which I don't have the answers, my presentations are often prefaced with, "Now remember, I don't have the answers. So you may well need to correct me as you know how I make mistakes." Correcting my mistakes, or challenging my methods (and often showing me another, usually easier, way), is equally valuable. Just because I'm speaking doesn't mean they're not thinking and learning.

To Answer, or Not To Answer

Something struck me over these past couple of weeks - How much answering of questions should I be doing prior to the discussion?

Like many/most (all?) teachers, I've told my students that I'm here for them, that I'm available for questions. "Don't understand something? Ask." As I've said, I can't see inside their heads (and, even if I could, I can't read Chinese! :-D).

However, what I've found myself doing is helping them to answer homework/discussion questions BEFORE the discussion. Eager to capitalize on their interest in the problem and their willingness to come to me to seek assistance, I'm only too happy to help, often leading them to the answer. This is, after all, what I've always done with homework since it was always about practicing something that had been taught.

But, this is a different way of teaching. Learning through problem-solving and the ensuing discussion. How much should I be involved BEFORE both the individual attempts at the problem and the class in which the problem is discussed?

My sense is that, in some (many?) cases, I'm doing too much. Word/Meaning clarification? Yes. (English is, after all, their second language.) Ask questions to help them clarify their thinking? Yes. Lead them to the answer, my answer? No, at least not nearly as much as I've done.

My background as a teacher has always been "dispenser of knowledge." Leading students to the knowledge - through questioning - is so much more difficult.

Step back and allow the struggle to take place. THAT'S when the real learning happens.

Asking Questions

I've yet to fully crack the how-to-get-students-to-ask-when-they-don't-understand nut.

What I really like about a discussion-based (Harkness, Socratic) classroom is students talking and sharing. It's well-documented that real understanding is demonstrated when you can teach someone else, and the discussion-based method approaches this ideal far better than most.

That said, some students simply won't ask questions when they don't understand something, despite:

1) my emphasizing daily that mistakes are welcome and how we learn

2) my making mistakes and letting everyone know that I made a mistake

3) my asking questions when I don't understand something

4) thanking each student who asks a question

5) thanking each student who shares a mistake

6) providing (or attempting to provide) a classroom environment in which everyone is supportive

The struggle I have is between this being OK and how frustrated I feel when a quiz/test grade is low, as I want all of my students to succeed. I mean, part of learning is learning how and when to seek assistance. And part of learning, a BIG part, is failing.

On the plus side, no one is able to remain silent in my class. Each day, at least four students stand in front of the class and speak/present and I rotate through all of the students. But, how to get questions when something's not understood...

Learning the Language of Mathematics

My Chinese tutor (I'm taking Chinese lessons) spoke recently about the sequence of learning a new language - you first listen, then learn to speak, then read, and finally write. (I recall quite clearly this being the case with my niece.) Each builds upon the former.

In the Chinese education system, dominated by teacher lecture and next to no student involvement or interaction, the emphasis is on the latter two building blocks, reading and writing, at the expense of the first two building blocks, listening and speaking. According to her (and the students I asked about this), this results in a very surface level of understanding, one that more quickly disappears once out of school.

(I've had students come up to me who have taken 10+ years of English and tell me that speaking with me is the first time they engaged in an English conversation!)

As many of us know, math is very much like a language - there are unique symbols and either unique words or words that have mathematical meanings. Given this, it's very important that students are engaged in parts one and two of language acquisition - listening and speaking.

I wish I had a nickel for every time a student said to me, "I know what it means but I can't explain it." If you can't explain it, you don't know what it means. If you don't know what it means, chances are pretty good you won't remember it.

Listening and speaking are the foundation of the Harkness Method. Students are required to listen to one another and to present their solutions orally (and, in my case, in writing on the board). This is taking some getting used to (my not speaking), but it has become very clear how important it is for their learning.

Oh The Things You Can Learn

Easily one of my favorite things about teaching is learning from the students.

Prior to taking this position in China, I spent 5+ years tutoring SAT math. I used the College Board's "Big, Blue Book" which meant that I saw each problem (particularly those in the first three tests) many times.

What amazed me was the variety of ways that students solved the same problem. Here I was, the teacher ("I know how to do this, and this is the way"), learning different ways to solve the same problem, in some cases, from the 100th student. To say this thrilled me is an understatement. I so enjoyed being taught a new way that I wrote the student's name next to the problem so that I could reference her/him in the future.

(One student commented, "Really? I made the book?!")

The crux of the Harkness Method is students sharing their solutions. Combine this with several ways to solve a particular problem and learning something new is almost guaranteed. This has happened numerous times in the first four weeks (The Exeter materials are SO rich. I read somewhere that their problem sets were 8 years in the making.), and it brings a smile to my face each time.

A couple of notes:

1) I've gone back to one group in each of my classes. (Remember, my largest class size is 13.) I wanted to hear more and focus more and wasn't able to do this with two groups, particularly given English being their second language.

2) My chalkboard is divided into four panels, so I'm starting off the class picking four students to display the first four problems. This is giving me the opportunity to speak with the other students individually, checking homework and answering questions.

Harkness Good... or Bad

As you know, this is my first experience with the Harkness Method. As is usual for me with such things, I research and tweak, reach out and learn. In my experience, things are usually far more grey than black or white.

With regards to Harkness Math, I came across the following two articles, both speaking to Harkness Math at Exeter:

Harkness - Cons/Negative

Harkness - Pros/Positive

As I've said, I'm quite impressed with the methodology; however, as one who has read the book "Quiet," I can certainly identify with the "Cons" article author. How do you effectively involve extroverts and introverts so that all are able to maximize their learning?

No matter the methodology, one of the struggles I've always found is getting students to ask questions when something is not clear, or not understood. I'm trying so hard to welcome (and appreciate) mistakes, but that doesn't mean that there's an increased willingness to make them publicly. The quiet students can easily slip under the radar and that's frustrating.

The basics of the Harkness Method for math, as I currently understand them, are:

1) working with math at a deeper, more applicable level.

2) owning the struggle

3) recognizing that, more often than not, learning comes from making mistakes

4) presenting your efforts in a coherent way

All of these things are VERY valuable, and the Harkness Method emphasizes all of them. What's important for the teacher is to keep the instruction on topic and to step in when necessary to answer questions, thus meeting the needs of those students looking for a bit more structure and direction than that provided only by the discussion. It's a fine line to walk.

Students (and the Teacher!) Learning From Students

I had a wonderful experience on Wednesday.

The PreCalculus question had to do with whether or not a larger area was possible. One group (each class has two groups made up of either 6 or 7 students each) disagreed as to the answer and I watched as an animated discussion took place, going back and forth with drawings on the chalkboard, IN ENGLISH! It was math, it was in English, it was problem-solving, it was learning from one another.

It was wonderful.

One of the things I love most about teaching is learning from the students. (During my five plus years of SAT math prep, I was always learning new ways to solve the same problem. It was just awesome.) In my case here in China, teaching at one of the best schools in this "medium-size" city of 10-12 million, I have some very bright students (especially true in a STEM subject). Allowing them to discuss ways of solving problems offers a FAR richer learning opportunity than I could ever do as "sage on the stage."

I read somewhere that teaching is listening (to which I'll add and asking good questions); learning is talking. Thus, if my job is to teach, my job is to listen and ask questions. This is difficult for me; it doesn't fit the standard definition of "teaching" as I know it. That said, the standard definition of teaching (I talk, you listen) is rapidly being shown to be ineffective, as learners need to talk, struggle, and make mistakes.

The Harkness Method has taken me out of my comfort zone (and I have a long way to go), but it has, so far, been SO rewarding. Seeing students learning from one another through discussion is incredibly satisfying.

A couple of notes:

1) Following in the footsteps of Johnothon Sauer, I took my two PreCalculus classes of 12 and 13 students and broke them into two groups each. I wanted more opportunity for discussion, which he was also looking for when he experimented and found that groups of 6-8 were pretty ideal. (This is particularly true in my situation with students for whom English is not their first language.)

2) In my classes, discussion will make up 10% of their grade. I'm not sure how'd I handle this in a class in the US (English speakers), but I wanted to emphasize the importance of speaking as a big goal for us is to develop their English.

3) I created a "Discussion Log" and each group has a Leader who keeps track of the presenter and who speaks (asks a question or presents another method) for each problem.

4) So far, we're going through one Exeter math page every two days. I'm hopeful that I'll be able to speed this up a bit as the English and presentation skills improve; however, I'm also teaching technology (using the TI-84 calculator) - brand new to these students - so that will take some time as well. Time well spent.

To Harkness or Not to Harkness. To Harkness.

A roller-coaster week.

On Tuesday (we didn't have school on Monday)...

1) I was asked, "When are we going to start using the textbook" (which they purchased).

2) I was struggling to equate the textbook topics with those in the Exeter materials.

3) I was, quite frankly, struggling to solve some of the initial problems in the Math 3 materials. (For example, it's been some time since I've done anything with vectors.)

4) I was wondering how I was going to complete a required curriculum map using Exeter's discovery and spiraling methodology.

5) I questioned how I was going to get through the questions in a timely fashion using Harkness/discussion given the language issues, particularly in the Algebra 2 class.

My first time teaching since 2006 (I've been tutoring high school math and SAT and ACT math for the past 5+ years) and I was going to take on high school math (most of my teaching experience is in middle school math), teaching in another country, and an entirely new way of doing things!?! It was all too much. Time to simplify. Follow the book.

For the next day-and-a-half, my gut ate at me. I wasn't here, I hadn't gotten back into teaching, to do things the same. I wanted to incorporate a curriculum of problem-solving and critical thinking. I wanted the students to think, make mistakes, struggle, grow. The students need to talk, not me. A comment I had read regarding Harkness - "Teaching is listening, talking is learning" - just kept resonating with me.

So... To Harkness.

Decisions I've Made:

1) Two groups of 6-7 students each in my PreCalculus classes. Johnothon Sauer talked about how he felt 6-8 was an optimal size as it allows for more discussion opportunities. To improve their English - a primary goal - these students need to talk.

2) A discussion leader for each group with a sheet to track who presents each question, who contributes to the discussion, and whether or not everyone understands the question.

3) A guidelines sheet (modeled after the one I got from Johnothon) for the discussion leader, the presenter, and the participants. Talking/Discussing is new to them - they certainly don't do it in their Chinese classes - so these guidelines will help.

4) Trust the Exeter materials. There's simply no way Exeter's Math 3 materials (for PreCalculus) aren't plenty robust for any required curriculum. I'll record the main ideas of the questions as the topics I'm covering.

5) Quit worrying about "Am I getting through a textbook or an as-of-yet defined curriculum?" I want to teach depth and problem-solving. The greatest thing I can do for these students is develop their critical thinking skills and hone their English skills and vocabulary. (They do, after all, hope to attend US universities.) And, let's not forget, they're getting curriculum coverage in their Chinese math classes.

What would I tell my best friend if he were in the same situation? Take the risk. You know what the right thing to do is, you care, so follow your gut.

I have a lot of learning to do, but I'm looking forward to the challenge.

Week One - What Have I Gotten Myself Into?

First things first, as I embark on this adventure (implementing the Harkness Method in my PreCalculus and Algebra 2 classes here in Wuhan, China), I have to thank Phillips Exeter, Johnothon Sauer, and Glenn Waddell. I'm humbled by all three.


Phillips Exeter - Pioneering a discussion/problem-solving method of teaching, creating all of the materials, and exporting their expertise via their website (http://www.exeter.edu/academics/72_6539.aspx) and summer institutes. Incredible!

Johnothon Sauer - Blogging/sharing his experiences, and materials, with implementing this in his public school classes via his blog (http://harknessforthirty.blogspot.com), teacher website (http://www.edline.net/pages/Mason_High_School/Classes/1415_2431_1_1), and email. Thank you!

Glenn Waddell - Blogging/sharing his experiences via his blog (http://blog.mrwaddell.net) and email. Thank you, as well!
Like those above (and others, I'm sure), I'm acutely interested in honing my "teaching math" craft and more than willing to assist those in doing the same. This blog serves as an opportunity to do both.

Background on my situation:

As the math teacher (Algebra II and PreCalculus), my situation is rather unique as the students will be taking two math classes each day - their usual Chinese class and mine. Given that their Chinese math class is traditional (40+ students/class, lecture-driven, very little student/teacher in-class interaction), my goal is to provide a problem-solving, student-driven math class. Internet searches on how best to do this (more fully than I’ve done in the past) led me to the Harkness Method.


I’m the high school math teacher at St. Mary’s Wuhan No. 2 School (http://www.smschool.us/page.cfm?p=561). Our international school is a partnership between Knowledge Link (http://kleducation.org/KnowledgeLink/index.php), St. Mary’s School in Medford OR, and Wuhan No. 2 High School. My school (and, currently, six others) was developed to provide a creative, critical-thinking education, in English, to area students interested in pursuing studies at both China and U.S. colleges and universities.

My classes are enviably small - two PreCalculus classes of 12 and 13, and one Algebra 2 class of 6. (Some recruitment struggles this past summer.) The main difficulty will be generating discussion as the culture generally teaches the students to not speak in class. Throw in the second language of English (not to mention the language of math) and a fear of making mistakes and this method will prove to be both challenging and necessary for future success for those students pursuing college in the USA (the goal for most of them).

From what I’m learning, their math skills are already strong and/or developed/developing in their Chinese classes. What they need from me is discussion/English skills and creative/critical//problem-solving skills. Thus, I’m far less focused on covering some pre-defined, US curriculum than I am on getting them to think, challenge, discuss, make mistakes, etc. Exeter’s materials and the Harkness method really seems to fit the bill.

Week One:

Delightful students (discipline won't be a problem)

Lots of silence; too much talking by me (will I be able to out-wait them?)

An intense need for all of the students to SPEAK UP (this will be a challenge)

Wide range of English ability ("What is an 'estimate'?")

Introducing a template for solving each problem (thanks, Johnothon!)

No Harkness Table so desks currently arranged in an oval

Limited chalkboard space

Smartboard and worksheet projector (use projector for student problem display?)

Working to align Exeter materials to my classes

How am I going to grade? (discussion and HW completion need to factor in somehow)

The Harkness Method is brand new to me and them, so we'll all be learning and growing this year. I have to focus my efforts on getting them to talk (loudly!) in class, discussing and learning from one another. This will be a challenge. That said, teaching via the Harkness Method is pretty much exactly what they need if they are going to be successful in future US math classes.