September. Twenty-one first graders are sitting on a rug excitedly discussing their thought processes and math strategies, sometimes as a class, sometimes in pairs. Their teacher, Deb Arancibia, briefly shows them images with large black dots (counters) across the top row, some large blue dots in the bottom row and some empty spaces. The children are to raise their hands when they know how many counters were just shown. This is a lesson using Cognitively Guided Instruction (CGI) at La Ballona Elementary School.
Each student Arancibia calls on gives a numerical answer and tells the thought process that led to it. Perhaps the student saw a top row of ten big dots plus a group of five on the bottom plus two more or two groups of five counters on the left, another group on right plus two more. Or maybe the student counted each dot individually or saw the whole as a group of twenty with three blank spaces.
Listening to their fellow students exposes all the children to a variety of strategies, and helps the teacher know how each student’s mathematical understanding is developing so that she can guide them to full mastery of the math standards for their grade level (or beyond) by the end of the year.
“We’re hearing kids, even in kindergarten, having real discussions about mathematics,” says Atoosa Abascal, the CGI Math Mentor at Linwood E. Howe Elementary School, “explaining their thought processes to each other, asking each other about the different strategies they’re using and holding each other accountable.”
Next it is time for today’s story problem. “Story problems give numbers a context,” explains Tiana Rezac, the CGI Math Mentor at El Rincon Elementary School. “The kids learn to go back to the story to see if their answer makes sense in context.”
Today’s problem puts a boy whose birthday is this week at the center of the action. In the problem the birthday boy will invite his classmates, 12 boys and 8 girls, to his birthday party. How many children will be invited in all? Arancibia gives each student a sheet of paper with the problem and several number sets that can be substituted for the original 12 and 8. They spread out around the room. Some take out plastic boxes filled with manipulatives that represent single items or are marked to denote five or ten objects at a time. Individually they work their way through the problem.
At first glance this classroom seems completely open-ended. The teacher has not told anyone how to solve the problem. A casual observer might think the teacher isn’t teaching.
But look closer. CGI is a research-based pedagogical approach. The lesson, the problem and the questions have all been very carefully structured with a clear understanding of the cognitive developmental stage of each student, so that each child is challenged to move to the next level as they become ready.
One girl is clearly frustrated. Arancibia asks what’s wrong. The student has been showing her math strategies by drawing a representation of each child in the word problem, but now she has reached a number set with much higher numbers and she’s annoyed at having to draw so many things. It takes too long. Arancibia nudges her to consider other ways to account for the number of children in the problem. The little girl decides to draw bars, each one representing ten, only drawing individual items for those left over after she’s drawn all of the tens she can.
“When kids have that aha moment,” La Ballona Math Mentor Christie Uhe explains, “the learning sticks in their minds.” It’s a productive struggle, and the result is that the kids know math better and they enjoy it more.
The CGI approach to mathematics develops conceptual understanding and approaches to problem-solving before giving students ‘the rules.’ Abascal notes that, “With CGI mathematics we’ve had kids using multiplication and division skills in kindergarten. They weren’t writing the symbols, but they had the concepts.”
The first graders return to the rug in the front of the classroom to share their strategies for solving the word problem. When the first student called on explains her thinking it becomes clear that she has reversed the numbers, so that 12 girls and 8 boys were being invited to the party. This is not an uncommon mistake— girls often want to invite more girls and boys to invite more boys to their parties.
Other students have noted the error, but Arancibia doesn’t dwell on it, instead she asks, is the math the same either way? The children agree that it is, and what might have been a chance to call out someone for being wrong becomes a chance to learn something new. This is a safe, supportive space. The girl is smiling when she sits down.
Smiles and excitement are everywhere during this math lesson. Atoosa Abascal has seen this phenomenon at Lin Howe, too. “Kids are just loving math,” she says. “I’m hearing from parents stories like, ‘My kids are coming home and they’re having playdates and instead of playing with their toys they’re pulling out their white boards and doing CGI story problems together.’”
CCUSD had to put together a number of funding sources to bring this program to our district. As part of this effort the Culver City Education Foundation (CCEF) procured a grant from the California Community Foundation (CCF) that funded half the cost at one school over a two year period, enabling teachers at that school to receive extensive professional development through the Cotsen Foundation’s ‘The Art of Teaching Program.” The CCF grant also made it possible for a teacher to become a full-time mentor for her colleagues. This is a level of teacher support most teachers in other districts only dream of— and it has made a real difference.
CCEF raises funds for innovative programs, equipment and support that make the difference between a good education and an outstanding educational experience for our students. The State of California does not provide enough funding for our kids to have these opportunities. It is your contribution that makes them possible. Please go to www.ccef4schools.org and donate to CCEF today. Your contributions fuel everything we do.