Sal put 5 apples on the kitchen counter. Mel put 4 more apples on the kitchen counter. How many apples are on the kitchen counter? Use pictures, words or numbers to show how you solved the problem.
- Student 1 draws this on his paper: OOOOO OOOO “9”
- Student 2 writes: OOOOO “5”, “6-7-8-9”
- Student 3 writes: “5 + 4 = 9”
Obviously, all three students have some representation of the number nine as an answer to the question, “How many apples are on the kitchen counter?”. But, how does this help you, as the teacher, know what each of these students understands or doesn’t understand?
Analyze.
Student 1 uses a pictorial representation to demonstrate his knowledge. You know that he can represent 5 by drawing 5 circles and he can represent 4 by drawing 4 circles. He even separates the two sets with some space, indicating that he knows he is dealing with two groups of apples. You do not know from the drawing what strategies for addition this student knows, so to dig a little deeper, you call the student over and ask him to explain his picture. Here is his response as he counts the circles:
- OOOOO “12345”; OOOO “1234”;
- then, OOOOOOOOO “123456789;
- Nine apples are on the table.”
Now you know what the student knows:
- he has one-to-one correspondence when counting
- he can count to nine,
- he can distinguish that he has two groups of apples, and
- he knows that he is joining the two groups apples to find out how
many in all
he
is using a strategy called “counting all”. That means he counted out the 5
apples, then counted out the 4 apples, then began over at one and counted all
of the apples to get a total.
You also know what he doesn’t know:
You also know what he doesn’t know:
- he was not
able to conserve the number in the first group (5 apples) and then count
on the other four apples to get the total.
Being
the perceptive teacher you are, you recognize this and proceed to explain and
demonstrate how he can count-on from 5 instead of starting over and counting
from one.
Score! You have taught this student a new counting strategy!
However, 8 more of your students used the same strategy that Student 1 used. Are you going to meet with each of them, one-on-one, and give the same explanation? That would be ideal, wouldn’t it? Giving individual attention to each student? Perhaps, but now think about you giving the same explanation eight more times.
Using the process of Cognitively Guided Instruction (Teaching Children Mathematics: Cognitively Guided Instruction, Carpenter, et al) to teach math, not only will you not have to give the explanation nine different times, or teach to some students while other students who already know this skill sit and listen, but you will be engaging every student in active learning and the students are the ones giving the explanations when they share their strategies for solving problems.
You will circulate among the several small groups of students as they work on word problems. Students have been instructed to solve the problem using any strategy. They are required to show their work on their papers and explain how they use the strategy to solve the problem. As you walk around, you are paying attention to the discussions at each group, gauging levels of understanding and skill, and asking clarifying questions to guide student thinking.
Score! You have taught this student a new counting strategy!
However, 8 more of your students used the same strategy that Student 1 used. Are you going to meet with each of them, one-on-one, and give the same explanation? That would be ideal, wouldn’t it? Giving individual attention to each student? Perhaps, but now think about you giving the same explanation eight more times.
Using the process of Cognitively Guided Instruction (Teaching Children Mathematics: Cognitively Guided Instruction, Carpenter, et al) to teach math, not only will you not have to give the explanation nine different times, or teach to some students while other students who already know this skill sit and listen, but you will be engaging every student in active learning and the students are the ones giving the explanations when they share their strategies for solving problems.
You will circulate among the several small groups of students as they work on word problems. Students have been instructed to solve the problem using any strategy. They are required to show their work on their papers and explain how they use the strategy to solve the problem. As you walk around, you are paying attention to the discussions at each group, gauging levels of understanding and skill, and asking clarifying questions to guide student thinking.
After students have had ample time to solve the problem, you will call on 3 preselected groups to share their work and explain their strategies with the class. The secret is that you have purposefully sequenced the groups according to the strategies that were used as you observed while circulating. The magic is that every student will have an entry point. Have the group with the least sophisticated explanation, such as a pictorial representation, to share first. You have just placed value on the thinking of these students-the ones who probably never share in a traditional setting.
Score for the boost of
confidence for the lower-achieving students!
So, how does this benefit
those students who are writing equations? Think about this. Have you ever had
that student who gets a correct answer but has no idea how he got it? Sometimes
those students need to see pictorial representations or models to help clarify
their own thinking and to help them put their thoughts into words.
You will have sequenced the groups to present from the least sophisticated strategy to the most sophisticated. Therefore, in the example above, Student 2’s strategy would be shared next. This student has demonstrated that he can conserve a number and begins at that number to count-on. (This is the strategy that you explained 9 times in the traditional scenario above, remember.) So now Group 2 has shared the counting-on strategy to the class. Group 1, who presented their strategy first, has now learned how to count-on to get a sum and you didn’t have to explain it 9 times.
But, Group 2 still has room to grow. They can conserve numbers and they can count-on, so they are ready to learn a more abstract strategy. Bring on Group 3. They wrote an equation to solve the problem. They explain to the class how the equation works to solve the problem. Group 1 may not be ready to use equations to solve problems, but at least now they have been exposed to the notion. Moving from counting-on to writing an expression is a natural “next step” for Group 2. Again, you still haven’t had to “stand and deliver” or repeat yourself 8 times.
But how will the members of Group 3 grow if they’ve already presented the most sophisticated strategy? You can now pose some “what-if” situations for this group. For example, “What if there were 4 apples on the table first, then someone put 5 more apples on the table?” (Commutative Property of Addition) . “What if there were 9 apples already on the counter, but Sal took 5 away to give to his friends for a snack. How many apples are left on the table?” (Inverse Operations; Fact Families). Everyone in the class gets the benefit of the discussion generated from the “what-if” questions.
