How to teach math is a big controversy in educational
circles. In the United States, math has
often been taught by showing kids how to do a particular operation, then having
them practice it until they can do it independently. This would be called an algorithm driven
approach, or doing “math basics”.
However, when worldwide annual math and science testing began in the 1995,
with the Timms testing (timss.bc.edu/), people were shocked to see that American students did not do as well as their
counterparts in places like Japan and Singapore. Some people thought that kids in these places
simply did more “drill and kill” practice of math problems, but when the
situation was analyzed, it turned out that the countries that did best were
ones which spent more of their math teaching time on understanding the number system and solving problems rather than on practicing operations.
Many people wonder what a math teaching approach based on
teaching for understanding would look like.
At Peregrine School, we use Singapore Math, which is considered a
Cognitively Guided approach to teaching math.
In this approach, a lot of time is spent on discussion, dialogue, and
problem solving. The following lesson is
an example.
We are in Marcia Margoniner-Reilly’s first-second grade
classroom. Children are sitting at three
tables, facing each other and able to see a white board in front. A large box of ones, tens, and hundreds
squares are on a demonstration table in front.
(M stands for Marcia, and S1,2,3… for student speakers):
Children are given number squares of 1 and 10.
M: How many white squares fit in a 10 strip? Children count by lining them up and answer:
10.
Now how many fit in a big square?
S: 100
M: Prove it to me.
S1: Uses 10 strips to cover the 100 square. Then shows that each of these is 10, so 10 x
10 make 100.
M: Are you sure that 10 x 10 make 100?
S2: I can prove it to you!
He puts all 100 small squares on the big one.
M puts a number chart as follows on the board. 125=
Hundreds tens ones
|
1
|
2
|
5
|
M writes the number 125 on the board. Kids are challenged to make it with the ones,
tens, and hundreds squares they have.
They all agree it is as shown above.
They try some other numbers, first building them with the blocks, then
putting them in the chart. They do this until everyone can do it easily.
M: Now I challenge you to think of a number you create for
other people. You will make it with the
squares, and we will decide as a group how to put it on the chart. She makes the big box of squares available. Children get out lots of squares.
(This is an example of how children can differentiate their
own learning in this kind of open-ended setting. One child, who is advanced at math, gets out
a lot of hundreds. Others get out
simpler numbers.)
M: (Referring the child with big numbers) Look, S2 is coming up with a big number. He wants to scare us! Are you making your number, S3?
S3: (watching S2) I am trying to figure out what he is
doing. Then she begins to build a number,
saying “S2, you won’t be able to guess this one!”
S2: I’ll bet I can!
(In this case, the student who created the challenging
number has created a challenge for his peers.)
S2: (to S3) Oh, you’ll never guess mine!
S3: 50,000?
S2: No, not at all.
I’m not allowed to go into thousands.
M: Is that true? I
didn’t say that.
(Question: Students want to get more than 10 of the 100’s,
but in this lesson they have not yet gotten into using cubes which represent
1000- so they do not have a correct way to represent more than one
thousand. These are the kinds of questions
that teachers deal with when doing this kind of lesson. Should they make game rules that keep student
responses within certain bounds, covered by the concept of the time (standards
for first grade are to work with numbers up to 1000), or should they let things
go beyond these limits and then challenge the students to explain what
happened? The answer to this kind of
teaching question will vary with each group of students. If the students barely understand the concept
at hand, going beyond might be confusing.
On the other hand, some students may have the ability to go beyond and
benefit from it.)
M goes from student to student, asking them to place numbers
on the chart on the board to represent the squares they see built on the
students’ desks. When they get to S2’s
number, which has more than 10 hundreds, M points out that it raises an
interesting question: what to do with numbers over 10 in any category?
As M moves through the children, several have more than 10
squares in a category, causing them to turn in the squares and get 10’s or 100’s
to replace them. This is the beginning
of learning to carry numbers. Students
have practiced this with smaller numbers (up to 100), but not previously up to
1000.
Numbers evolve as shown below:
|
Hundreds
|
Tens
|
Ones
|
|
8
|
11
|
15
|
|
|
11 + 1=12
|
5
|
|
8 + 1= 9
|
2
|
5
|
|
9
|
2
|
5
|
Number= 925
S1’s number is ten one hundreds, ten tens, and ten
ones. This challenges the kids as they
think about how to represent it.
|
Hundreds
|
Tens
|
Ones
|
|
10
|
10
|
10
|
|
10
|
11
|
0
|
|
11
|
1
|
0
|
M begins by discussing what we do when we have ten
ones. One kid knows that
5+ 5=10, but this equation throws things off, since it is
not relevant.
M: What should I do, put 10 in the ones column? How many tens do we have?
S3: We have one ten, because 5 + 5 = 10.
M: But how many ones are there in ten?
S1: Ten! So we need to make another ten in the next row.
M: Great idea! (See
red numbers above) Now how many ones do
we have?
Students agree that is hard.
One comes up with the idea that no ones are left.
M: Great. But now we have a new problem! What about the tens? Can we have eleven tens?
S2: No. Now we have
to move a ten to the 100’s.
M: Right, let’s try it.
(See green numbers) But now we
have a big problem! We have 11
hundreds. Can we have that?
This leads to a discussion of thousands, which is the point
of the lesson, to understand how the number system works up to thousands. (Students have up to now been working with
numbers only up to hundreds). They found
that if you have more than 10 hundreds, you need to add another column to the
chart, and another column to the numbers we write. The thousands column.
Students are now given a work sheet in which they have to
transfer numbers back and forth from drawn squares of ones, tens, and hundreds
to numbers written as usual. They use
their cubes to help do these conversions, and either draw the squares on their
paper or write the numbers that represent the squares. This is the independent work that follows and
reinforces the discussion which took place.
This math lesson illustrates how the strategy from my last
blog, the collaborative conversation,
is an essential tool in teaching math. This
conversation consists of a small group and a teacher discussing an issue or
concept together.
The essential parts of a cognitively guided math lesson are:
1.
The introduction of a question (How to represent
big numbers with cubes?)
2.
A mini-lesson which demonstrates a skill or concept (how numbers look as cubes and on the number
chart)
3.
A challenge, presented to individuals or small
groups, to solve a problem their own way (make your own big number with cubes)
4.
Group sharing and analysis of people’s solutions
(how to write the numbers students created on the number chart).
5.
Individual practice of concepts learned (writing
your own numbers back and forth from drawn cubes to written numbers)
Cognitively guided math is challenging to teach because
student solutions can take different directions that can lead to new cognitive
challenges. The teacher constantly
defines the boundaries of the lesson, so that the lesson ends up taking a clear
direction which illustrates the point s/he is trying to make. At the same time, students are encouraged to
come up with novel solutions and even to try out ideas which they know but
which might not prove relevant, in order to learn from their own process. Balancing keeping a clear direction with
letting students experiment is a constant challenge. There is no one way to do it.
This kind of math lesson illustrates how teaching is an art,
in which new challenges constantly arise and require the teacher to improvise
solutions. Marcia illustrates a skillful
approach to meeting these challenges.
Yet for each teacher, there are different directions which discussions
can take, and a constant set of new decisions to be made. The exciting part, for teachers and students,
is that the challenges created by novel solutions shared by different students
always expands the lesson in unpredictable ways, and in the end, causes everyone
to see that the number system makes sense and yet provides constant new
explorations.
