Reflections From My Classroom: What I’ve

Learned About Supporting Student Learning

David Kukla

Sabino HS, Tucson, AZ

Editor's Note: David Kukla's name was given to me by someone who was obviously impressed with his teaching style and urged me

to have him write an article highlighting some of the "secrets" to his success. Although everyone realizes that there is no magic

formula and many good teachers have many "secrets" to success, it does not hurt to see some of our best practices being utilized in

another's classroom. Dave suggested that this article might be the first in a series of articles highlighting teachers' and students'

experiences so that we may all have a way of benefiting from each other’s best practices and re-examining our own classroom

environments. I think this is an excellent idea and I hope that readers will accept this invitation so that other's can value from their

experiences.

Preface: A dear friend asked me about being a teacher. He said that he had felt bored and disconnected in his high

school mathematics classes. He mentioned how the mathematics classes he took tended to be blank, nondescript affairs

reinforcing a prison-like feeling. I had the sense that what he was really asking was how I dealt with these feelings in my

classroom. We talked for quite a while and my responses form the basis for the following descriptive piece. As I began

to write it, in a traditionally neutral tone, I realized that that seemed to add to my friend’s concern about sterility. I also

began to think that some might take the piece as a proscriptive of what every teacher should do rather than a descriptive

piece about my room and what happens there. So I went with “I” in this piece as a suggestion of ownership rather than of

obligation. I imagined my audience being me, as a new teacher, walking into a blank classroom for the first time, keen to

think about how to use the space. I also imagined my audience being me now, having taught for 20 years, keen to know

what others do and what I might adopt to help my students to be more successful. And as I wrote the article, I began to

look at student experiences in the room with new eyes and had some new ideas to try out this year. Here are some

reflections of my classroom.

My classroom displays many things: the curriculum covered,

lessons learned, and experiences of the students. Its arrangement

suggests student learning and exploration. The walls are covered with

student projects and material to spark student thought. They are a

collage of colors and images: Word of the Day; Article of the Week;

Group Work Rules; Polya’s Problem Solving Strategies; money from

foreign lands; translations of “Where is my army of diabolical

squirrels” in many languages; student-made quilts, multi-dimensional

geometric origami, semi-Archimedean solids, and pyramid projects

from geometry; rotational volume projects from calculus; that Blue

Man Group Drumbone that some students made during lunch one

year; capes and expressive hats; extra credit problems for students to

explore; posters

of student

presentations; samples from artists like Basquiat and Escher as well

as student artwork; the perfect pitch wind chimes Johnny made using

a Vernier Labpro, a tuning fork, some metal tubing and a hacksaw;

and a giant geometric cereal box sculpture. [See the following page

for pictures of some of these.] These all serve the multiple purposes

of hiding the ugly paint underneath, showing off student mastery, and

making sure students have an enriching experience even if their

attention wanders from the lesson.

It’s a large room and because of this I can fit not only the

required 36 desks, but also three large tables. On these tables I have

pencils and paper and inexpensive cookies. I’m always impressed at

how access to these things shifts the attention of a student from thinking about leaving the room to go visit the snack

machine to focusing on the explorations at hand.

I also keep solution binders on these tables. These binders

contain my own step-by-step solutions to all the homework problems. I

write them for many reasons. I want to know how long the homework

assignments take and there is nothing like doing them yourself to get a

sense of that. I want to know if the problems I picked turn out to be

good examples of the concepts we’ve discussed in class. But most

importantly, I do them to provide a line-by-line way for students to

check their work. Students are allowed to check their work in these

binders anytime outside of their regular class period. This means that

they can use them an hour before school starts, at lunch, and during my

planning period. Students can also come in and use them while I’m

teaching a class other than their own. These binders help students find

new ways to think about problems that they were stuck on. From the

time-management perspective, they act as a filter, reducing the number of students who need to ask me simple questions

and allowing me more time to help kids who still don’t understand.

I know, you’re already saying, “Doesn’t he know that kids are just going to copy the answers?” Well, it’s true,

some kids do that. And I know it, because their work is line by line the same as mine. But generally speaking, these are

the kids who weren’t going to do it at all anyway. Kids have always copied from each other. Obviously, I would prefer

them not to copy at all, but if they must, I would just as soon they copied from me. Then, at least, I know that they are

copying the correct work.

I also have email listservs set up for my classes that help with this filtering process. These allow students to get

help in the evening and on weekends from me and from other students in the class. Students report that being able to get

help right away is much more useful than waiting until the next day (when they might forget they had a question in the

first place). Students who tend to quit when they get stuck are more likely to complete their assignments if they can get

help right away. Kids tend to tune out when someone asks a question they themselves didn’t have. Providing individual

assistance through the listserv and the solution binders leaves more time for class activities to engage all students.

My colleagues and I assign homework nightly. We generally assign the same problems and our expectation is

that it should take students less than 45 minutes to complete. But then I differ from my colleagues. Most of them only

give completeness grades. If it looks like the paper is “complete”, then the student gets full credit. Over the years

students have shown me “full credit” papers from other classes which had the correct problem numbers, but random

number answers filled in; or no problem numbers and just random numbers written; and in one case, a stream of

consciousness essay about how the teacher will give the student full credit even though they aren’t actually reading the

homework pages.

Students that do their homework deserve to know if they are doing their homework properly. Most spend a

significant amount of time on it. So, until my students can have online services that provide instant homework feedback,

such as WebAssign or MathXL, I pick 8-10 problems from each assignment and actually grade them. I don’t write

comments; I just mark of these that are wrong. A student aide in a higher-level class and a key can be a big help with the

grading. Students are then allowed to correct their mistakes and hand their work back in for re-grading. If a student has

the correct work and correct answer the second time, he/she gets full credit. If not, he/she is invited to try the problems

yet again and come in and get help. I find that the students who consistently correct their mistakes before tests and

quizzes tend to make fewer mistakes on those higher-stakes assessments.

Since I give cumulative semester finals, I allow kids to rework assignments until two weeks before the semester is

over. Some kids rework their homework daily, others weekly, some hand in a mountain before the deadline. I find that

this flexibility draws in many students who would otherwise opt out and stop trying. And while they are redoing their

homework, primarily to raise their grade, I know that they are also going through a process of review and mastery-

building that they wouldn’t otherwise.

I could have a more traditional homework policy, but I’ve thought about how homework can best support student

learning and implemented a policy to support that philosophy. This homework process takes time, for both the student and

the teacher. Some of my colleagues (and certainly my wife) think I’m a chump for putting this much time into providing

feedback on daily mastery. I feel that if a student is going to put in 30-60 minutes a day of his/her life on work for my

class, he/she has a right to know whether or not that work is correct. And when I take the time to give that feedback and

allow them to go back and revisit topics for full credit as well as for mastery, students feel that their efforts are valued and

respected.

Equally, how quickly students have access to this feedback is important. I always hand back their graded tests the

day after they’ve taken them. Homework grading is usually a three-day process from the time they hand it in until my

community service (a.k.a. detention) students are filing the graded assignments in student portfolios. It’s really frustrating

for students when they take the time to do their work and then “it takes forever” to get it back. Some students want to get

their work even sooner. I let them know where they can get the graded assignments that are awaiting filing. This is the

kind of thing that can provide students with a sense of control and help them feel more empowered with respect to their

educational needs.

To many students, school in general, and math in particular, often feels like a punishment. Anything I can do to

help mitigate those feelings while helping students master the material at hand is important, so I spend a lot of time

thinking about the experience of my students in our classroom. This extends to details like having cookies on the table and

cough drops in the closet. Cough drops and band aids and hair ties and safety pins are all little items that serve the

purpose of keeping kids in class and focused on learning, as well as letting them know that I care about them and that our

room is a place they can come to when they need help. I think this last is crucial. I want my kids to think of our room as a

place that they want to go to rather than a place they have to go to. When students feel safe and have their needs attended

to they can focus on their learning.

I can’t write an IEP for each of my 150 students. But all the same I think that it is critical that I come to know

each of my students as learners and work to support them as they confront their own challenges. This process starts with

learning and using their names when they come into class each day. When I greet my students, I try to remember that

they might not be excited about another challenging day in math class. I try to greet them not just by name, but greet

them with a exclamation mark. I want them to know that though the course may be hard, I am not the course, but rather

part of their trusty support system.

I had a conversation with a college professor not long ago and he asked me how I came to have so many kids

scoring so well on the AP Calculus AB exam (for the last 10 years we’ve had 85%+ getting a 3 or better, this year, 97%.)

I told him that I thought building a connection with students and creating a safe learning environment was key to this

success. Putting the material “out there” like he did in his lectures wasn’t enough. I need to “be there” beyond a lecture.

I need to know who students are as individuals beyond our classroom so that I can employ useful metaphors to help them

bridge gaps in understanding. Building rapport is a process that is continual but low key. I find it works well with a quick

question on their way in or out of class or while I’m walking around helping kids. I’ll ask about their weekend, or about

the play they are in, or about that pin or piece of jewelry they have on. I don’t make a big deal about it. I’m just letting

them know that I’m interested in them as a person.

One of the ways I support students individually is

through questioning. I ask a lot of questions. Whether one-

on-one, working with student groups, or working with the

class as a whole, I drive the narrative of the experience

forward through questioning. I think it’s Socratic and I

know it’s influenced by Bloom, but it’s not formalized. I

don’t have a script or lesson plan that says, “I’m going to

probe, process, and then ask them this application question.”

Rather, like a good comedian, I sense where the audience is

and use appropriate levels of questioning at different points

with different people. In this way I can guide the class in

the direction I need it to go and support as many students as

possible in the way that works best for them.

Some teachers say it’s antagonistic to call on

students who don’t raise their hand. I know it isn’t if I’m

setting them up for success with the question. For me

asking so many questions with respect, never with the intention to humiliate, is a tool for keeping the class on their toes,

focused, and involved. I want to have my students actively involved in the process of their learning. Questioning can be a

way to achieve that. I also don’t have many discipline issues, and I believe this questioning strategy is partly responsible

for that.

I want my students to think about their work, and to think about how they are thinking about it, rather than just

trying to keep a large number of recipes or algorithms memorized. When a student asks about a math problem, I try not to

give them the answer, but rather draw out steps, possibilities, options from them. I try to ask them about what they know

and about they are doing and what they are trying to achieve. With this consistent questioning, I want them to hear the

questions to ask themselves when they get stuck, be it on a homework problem, test question, or even when they are

looking to take out that car loan.

Having students think about their work is also facilitated by the student-centered nature of our day. I know some

days have to be all lecture, but I try to have as few of those as possible. In order for students to construct meaning and

lasting memory of the material, they need to be involved, physically involved, daily in math in our room. To that end I

employ many techniques and tools.

One of my students’ favorites is using mini-white

boards for a quick concept review. After introducing a new

topic, I can put a problem on my mini-white board and then

have the kids do it on theirs. Following a countdown, all hold

up their work. This gives me really quick feedback about who

has grasped which concepts. On review days of some topics,

like applying the rules for special right triangles or finding

trigonometric values for unit circle angles, we spend a lot of

time doing rapid-fire mini-white boards.

I think partner work is really useful too, even more so

than group work. It allows students to work one-on-one with

others. It’s much harder to wiggle out of actually doing work if

there are only two of them and, importantly, much easier to ask

questions when they don’t understand something. In the past

I’ve used partner work for discovery and exploration activities and for playing review games. Partners are really great for

homework checks, too. For this, I have partners go through and check to make sure their answers are the same and to

discuss any points where they are different. This is a good way for kids to get help from each other when they might not

normally ask for help or clarification.

Pairs of students can also tackle something I call “Two-minute Controversies.” I put up a couple of problems, or a

question to think about, or samples of work with common mistakes (for example: What’s the mistake here:

)?,

and I ask the partners to work together to resolve the issues at

hand. I have created lists of controversies for each day of each

class I teach, but I only use them at the beginning or end of a

period when it feels right.

In my AP Calculus classes, one of the ways I like to

review for the AP Exam is to have students work through sets of

three problems from previous years’ exams. Sometimes I merely

ask them to “score” the sample using the supplied grading rubric.

Sometimes I ask them to do them with a partner and discuss their

thinking about the problem. Sometimes I ask them to work the

problems alone and then with their partner compare answers.

Having the opportunity to experience these problems in so many

different ways not only allows for different learning modalities,

but also facilitates their thinking about the problems rather than

just doing the problems. This is a central part of my teaching

philosophy. I move students away from formulaic thinking and towards a more analytic approach.

I find I often reference the two posters in my class “Thinking is

good.” and “It’s ok to read, really.” On one hand, I think of our class as

being a preparation for an entrance exam or a mastery of an explicit set

of mathematical content, but I know that it might be even more about

preparing kids to be successful in their lives. Most of them will never

solve a problem with logarithms, use the quadratic formula, or find the

instantaneous rate of change of a function. But all of them will need to

solve problems that they have to think about and to deal with problems

where reading the details and reading for true understanding will be

terribly important. Learning to think about problems alone and with

partners is one of the most important skills I can help all my students

learn, not just those headed to the sciences or engineering..

Partner work isn’t the only student grouping I use. Before tests,

my students often do review stations, which allow for flexible groupings.

I organize the desks into groups of 4 and set up a station, a.k.a. copier

paper folded in half with problems printed so that people on either side

of the desks can see the question(s). Sometimes I put the solutions on

the back of these stations, sometimes I just put hints or clues, and

sometimes I just sit at the station waiting to lead students with probing

questions through the

experience, like Sybil

leading Aeneas through

the underworld. I like

to let students move at

their own pace from

station to station, sometimes traveling as a group of three or 4, sometimes

moving on alone, sometimes moving with a partner. This sort of

flexibility allows students to take on the stations at their own pace and

with as much support as they feel they need. By allowing this freedom to

move, students can spend more time on topics that they find personally

challenging. By being able to choose whom to travel with at any moment

they can ask others for help or even stay by themselves to puzzle concepts out. I usually put out 12 stations covering

samples from the major themes to be assessed. Based on observations I make during class, about 10 minutes before the

end of the period, I announce how many stations need to be complete for full, partial, or extra credit. And, as always,

students are invited to come in outside of class to continue working on their stations as part of their review for the up-

coming test or exam.

While partner and group work are important tools for students’ active learning, I also need to specifically support

individual active learning. One bridge to this individual experience is having partners go to the front of the room and

present their solutions to problems, or their ways of thinking, or the responses to our two-minute controversies. A

document camera allows my students to project their work, simply by placing it under the camera. They don’t have to

spend ten minutes copying something over while the rest of the class gets bored and drifts off. With this piece of

technology, I can have three groups present their thinking in the time it used to take one group – and not lose the rest of

the class.

With this approach, students have to stand up and explain their thinking, whether on their own or “encouraged”

by my questioning. I think it’s really important that students see diverse ways of thinking about or understanding a

problem rather than just the way I might think about it. I want them to know and see that there are often many paths to

understanding and resolution.

Homework is perhaps the most common form of individual active learning, and my students get daily practice and

support in that. I find that many of these tools I’ve mentioned also act as formative assessment devices, means by which I

can assess student learning prior to handing them a test. But sometimes I want to assess what they have learned in other

ways. As such, projects are a nice complement to traditional tests. In all my classes I have, minimally, a fall and a spring

project. I started with having my students’ research math topics or mathematicians. However, I realized that projects

would be even better if they promoted active engagement in authentic mathematics so I changed the format. Now my

students must present and defend solutions to open-ended problems like, “How many cows are needed to support our

city’s annual requirement for dairy products?” or, from John Allen Poulos’ Innumeracy, “How high would the walls need

to be if you poured all the blood from all the people on the planet on a football field?”

Now, due to the constraints of our high stakes state graduation

assessment, AIMS, I’ve changed yet again. In the fall, my students

partner up and create a math review game to be played during the

review period prior to the semester final exam. I try to keep the

requirements for these games minimal in order to allow students to

really take ownership of the development process. I ask only that the

game include fifty problems based on the math we’ve covered and

their worked out solutions to those problems. It needs to be a game

that can be played in the space of six desks and experienced in twenty

minutes and may not involve ping-pong balls. Students have produced

all manner of board games as well as games like math hopscotch,

calculus twister, and fishing for parabolas. I love this project because

it has the students reviewing by creating the game in the first place,

and then, instead of “another worksheet” review for the exam, they

spend a day playing each other’s games. Students are active learners

both in having to create the game and in playing someone else’s game.

I give these projects a grade that takes into account student feedback

on how much the game helped them prepare for the exam.

In the late

spring, I have all my

students write a story.

Students incorporate

four to six problems

from a list of problem

types I provide into a

story where the

solutions to the

problems drive the narrative forward. For example, “the troll won’t let

you over the bridge until you solve this quadratic equation” isn’t ok. An

acceptable example would be, “you meet a troll on a bridge whose wagon

wheel is broken and the troll needs to know how much material he needs for a new metal wheel band with a given radius.”

These stories are usually four to six typed, double-spaced pages. Writing these stories helps students review for the final

exam. Additionally it’s a wonderful project for that awkward time between standardized testing and the final exam. This

exercise provides students with a direct – and self-generated – answer to the question, “How does this math ever get used

outside of class?”

Our projects are not limited to these two I just described.

In Calculus, for example, my students build volumes, either

rotational or with cross-sectional shapes. They choose the material,

and I only ask for 32 layers and at least 6” in some direction.

Students need to find the volume through integration They also

build a spreadsheet to find the sum of the volumes of the individual

layers. For many students this project solidifies the concept of

volume in a way that doing the homework problems doesn’t.

Students can “see” these problems in a way they couldn’t simply by

drawing them on their paper. I do a similar project involving

pyramids with Geometry students, where they have to find the

volume of a

stepped,

polygonal-

based pyramid made from at least 24 layers and then compare this

volume with the volume of the sum of the layers. They are asked to

explain why the numbers aren’t the same and what reasons there

might be for this. Also in my geometry classes we make quilts

combining construction skills and Sketchpad transformations.

I want all my students to feel challenged, regardless of their

own level, but ALSO to feel like they are supported as they reach to

fulfill their potential. I want the classroom to be a room where

students feel comfortable taking risks and exploring new ideas. I

know in order to achieve this all our time together in it is precious

and I need to be very intentional about how we use it. Everything I

do is geared to promoting students to be actively involved in their

own learning.

It’s true that not all my students do well. I hope that by getting to know them as individuals, by providing a

variety of ways to approach problems, and by providing a safe place for them to do the best they could at the time, that

they come away from our classroom secure in the knowledge that, when they are next in this situation, they can be

successful. I hope that their experience in our classroom has enriched them, even if they didn’t succeed at the

mathematics.

The vast majority of students leave our time

together feeling like the door is still open both

actually and figuratively. They know that I’ll be here

to help, even when they graduate, and that, more

importantly, they are now armed with an array of

tools to be successful on their own.

Thinking is good!