First Grade Mathematics Class

This class is one month into the first grade.

The class starts with math questions:

Teacher:   What numbers are between 6 and 8?
Students (in chorus):   7
Teacher:   Is there a number between 8 and 9?
Students:   No, there isn't one.

The teacher then asks the class who would like to pose similar questions. Hands eagerly go up. A student is selected and he asks the class: What number is between 2 and 4? Other students volunteer to answer.

Teacher: Which is the smallest number?
Students:   1
Teacher:   Which is the largest l digit number?
Students:   9
Teacher:   Which is the largest 2 digit number?
Students:   99
Teacher:   It is useful to know numbers so that when you go to the theatre with your parents, you can find your seat.

The teacher points to some numbers on the board to illustrate relational symbols: =, <, >. The students had learned these symbols the day before.

Teacher: Would you like to write these symbols in your note book? Does everyone have a pen? Children all raise their pens enthusiastically.
Teacher: Let's do a finger stretching game. They all recite a little poem (which is something they learned in their reading class) while doing this finger exercise.

The teacher pushes aside a curtain which was hiding the blackboard and several rows of neatly written numbers appear:

The teacher tells them that whichever row [of students] is first at correctly filling in the numeric relational symbols for every number pair in his or her notebook will get a small flag. Students begin writing in their notebooks. The students copy the numbers from the blackboard and fill in the symbols (>,<,=). The teacher has put some classical music on, which plays softly in the background. The room is quiet and calm as the students busily do their work. She is quietly walking amongst the rows of students, and so as to maintain the solemnity and concentration of the moment, she speaks in a low, quiet voice, giving them positive reinforcement as they write. Some students are raising their hands showing that they have finished. After a certain time has gone by, she slowly turns off the classical music.

Teacher: Which rows [of students] are first? [soft competition implied]. [She hands out a flag to a student in the fastest row.] The slow rows didn't get a flag, because they wanted to take their time to write nicely and neatly. We all have different ways of doing the same things.

The music is off now. The teacher's voice is louder and her voice rhythm changes. The teacher now has students do some physical exercise. It is a type of Simon Says game where students squat and then stand up on the word "please." They finish the exercise by singing a nursery song.

She goes to the board and pushes back yet another curtain. On the board are artistically drawn butterflies and other animals. In the center of these drawings are some squares and circles, with numbers inside, connected by lines, e.g.:

She asks how many butterflies there are as well as how many of the other types of animals. She moves close to the children and touches some of them gently as they answer.

Teacher then starts a ball game. She tosses the ball to a student and asks what is 2 + 1 and the student responds, tossing the ball back to the teacher. She picks up the tempo of the questions. She is not tossing the ball now, just asking fast addition and subtraction problems. She goes to the board and points to some addition problems: (2 + 7 = , 9 + 6 = ) and says, "don't say the addition problem out loud, just give me an answer." During these exercises, she frequently asks, "Do you agree?", "Is it correct?", "Is it difficult?" "No, it is not," students respond.

Second Grade Math

This class is also one month into the (second) year.

The class begins with stories of rabbits and cabbages. If I have 6 rabbits, how many legs do they have? And if I have 8 rabbits? The teacher praises the studens a lot.

The blackboard has been divided into 3 or 4 sections containing many numbers, equations, shapes, drawings in different colored chalk. All of these are artistically done.

Teacher: I'm thinking of a number. Multiply it by 9 and I get 81. In which geometrical figure is it in?

Students:   In the rectangle.

Teacher: What math operation is needed for these to be correct? (The teacher is pointing to some equations on the board.)

Students come up and write the correct operations in the blue squares

Teacher: What numbers belong here (pointing to the empty squares below)?

After a brief period of silence, the students answer: 16, 24, 28, 36

The teacher now points to the formula: Y * 4 < 60

Teacher: Now which of these numbers above can be represented by the "Y."
Students: 4, 8, 12

Teacher: Which is the correct symbol for each of the following equations? >, < =

Teacher: In this next set, fill in the squares with the correct numbers and put the math operation you used in the circle.

At this point I had a brief conversation with Dr. Lozanov and missed some of the exercises. When I turned my attention back to the class, students were solving for the area of a rectangle (P) with one side of 36 meters and one side of 24 meters, providing answers like these:

P = (24 * 2) + (36 * 2)

P = (24 + 24) + (36 * 2)

P = (24 + 24) + (36 + 36)

P = (24 + 36) * 2 = 120

Here are some math word problems:

One tree has 36 apples. Another one has 74 more than the first one. How many apples altogether?

Apple production:

Day 1: 186 Kg.
Day 2: 254 Kg.
Day 3: ?
Total production: = 635 Kg.

The teacher tells the students that they can choose either A or B to solve the problem.

A: 186 + 254 + x = 635
B: 635 - (186 + 254)

On the board several kites are drawn in different colors with some simple math expressions inside the kites. Above, there are some numbers in circles. These numbers are the equivalents to the expressions in the kites. Students are to match the numbers in the circles with the appropriate kites. Two of the circles and two of the kites are left blank.

The teacher asks for a volunteer to put a number in the blank circle. One student comes up and writes in 16. Then she asks another student to come up and write up an expression in one of the empty kites so that it equals 16.

A girl comes up and writes in "4 * 4".

The teacher then tells the class that she knows they know there is another way to come up with 16. What is it? Hands go up. One student is chosen.

He stands up and says, "8 * 2".

The teacher continues: "There is still another way to come up with 16. Who can come up and show us? Hands go up again enthusiastically and a boy comes up and writes 1 * 16.

Teacher: So, you see, boys and girls, just as in mathematics we can come to the same answer using different equations, so too in life we have many ways of arriving at the same solutions.

On yet another part of the blackboard there is a beautiful tree drawn. There are circles also drawn on the tree, each with a number inside. The students are given autumn colored 'leaves' (cutouts) which have mathematical expressions on them, e.g. 56 / 7. The backs of these 'leaves' stick to the board. Students go up to the board and each matches his equation to its answer, which is one of the numbers in a circle.

At the end, the tree looks like a brightly colored tree in autumn. The teacher then tells the children a story about autumn.

At the end of the story she tells the class that they are now ready for big numbers. She dictates 3,000,194. One student volunteers to go to the board and he writes it correctly.

Summary Observations

Class instruction flows quickly, and with ease. I could not record all of the mathematical problems from the board because the children solved them quickly and moved on to other things. The teacher always seems to find a link between math skills and other school subjects or to life situations. Light physical exercises are integrated with the core subjects. These are never done as if 'we need a break,' but rather as a natural extension of the subject at hand. For example, in a mathematics class, the teacher has the students do some toe raises; she divides the class into two groups, and each group alternates with the other, counting out loud by threes. In the reading class, when the students do a yawning exercise (she asks them to 'roar like a lion'), she explains that this is important because it makes the vocal cords strong and "we need strong, solid voices when we speak to the entire class."

There is a sense that everything done in the class connects to a larger whole. Information is not presented as isolated data but rather as fascinating particles in a bigger picture. In her closing story about autumn (in the second grade mathematics class), the teacher ties in all the drawings of kites, butterflies, apple trees, math formulae, stories about rabbits and cabbabes and more.

According to Dr. Lozanov, neither of the classes I observed, nor any others in the experimental schools, were tracked by ability or in any other way.

I had always loved teaching. But after observing these classes I knew with certainty what I was seeking in my teaching.