What does it mean to do cognitively guided instruction in math? A big concept in action

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.