When you're faced with multiplying 24 x 15 and you figure out that you can break the problem down to something you might figure out all in your head—say, 20 x 15 plus 4 x 15—you're doing something that Darien educators increasingly want every student to learn.
If you've figured out this multiplication shortcut on your own (and it's the type of thing that many students and adults already do), then you're practicing a useful, real-life mathematical skill.
You're also doing essentially what mathematicians do—using an algebra concept called the "distributive property, said Cathy Fosnot, a math educator who has been helping to train Darien elementary school teachers in a different way of teaching math.
If you've figured out this and similar mathematics shortcuts in elementary school, you're playing around with numbers in a way that could lead you to better math grades in high school and maybe even a lucrative career after college, Fosnot said last week at a meeting for parents of Darien elementary school students.
Later in the week, Darien Mathematics Coordinator Stephanie Furman and Judith Pandolfo, assistant superintendent for elementary education, presented the Board of Education with a description of how mathematics teaching is changing for Darien elementary school students. The process is expected to change in Middlesex Middle School somewhat later.
These changes in the math curriculum started several years ago—Fosnot, for example, first started working with Darien teachers six years ago, and parts of her "Contexts for Learning Mathematics" program have already been used to teach Darien elementary school students.
A different way of teaching and learning math
The Darien public school system and other school systems in the state and nation are de-emphasizing rote drilling of math exercises and emphasizing ways to guide students into figuring out some concepts on their own, playing with the numbers, presenting their answers to other students for critique and discussion, and critiquing each other's work.
The way math is taught in Darien public schools, particularly this year and particularly in the earlier elementary school grades, is moving toward more discussion of how students worked out solutions to problems—getting students to question each other and explain their thinking—and farther from having teachers explain how to use particular equations.
"Math isn't about trying to transmit strategies," Furman said. "What teachers are being taught is how to develop kids' strategies—helping the kid construct a strategy by showing particular examples, not giving out the strategy."
For instance, rather than a straightforward lecture from a teacher about a particular equation, or even about the principles behind the equation, a teacher might present several similar problems to students and ask students to solve them.
Then the teacher may ask a student who solves them how she did it. The student may then come up with a math concept like the distributive property, getting a deeper understanding of it by having used it repeatedly.
The teacher may also break the class up into many teams who then discuss among themselves how to solve particular problems. Then some teams are chosen to present their answers to the class, which then sits as a "math congress" as various teams explain how they arrived at their solutions.
The other students then may question the team's answers, all the while becoming better at discussing math and describing what's being done. Later, all the teams may post their answers on the classroom walls as other students do a "gallery walk," examining the posted explanations, asking questions and making comments about the work done.
Where and how the change is taking place
This kind of math teaching is beginning to be done across the nation, pushed on by new standards in math education coming from the National Council for Teachers of Mathematics and nearly all state governments, which have subscribed to Common Core State Standards in math.
Darien and other school districts in Connecticut are being prodded on in part by the Connecticut Department of Education, which (in the spring of 2015) will replace the current statewide Connecticut Mastery Test (CMT) with a new test for mathematics that emphasizes understanding of concepts and being able to explain how math problems have been solved.
Some homework will be a lot different, too
This fall, elementary school students are starting to use a software program called DreamBox. The program asks users to solve certain math problems and closely monitors how well each problem is solved.
The program is so sensitive, Fosnot told parents, that it will recognize when a cursor is hovering over a particular answer before a child clicks on a multiple-choice option. When students demonstrate that they understand a particular concept, the program records that, then leads them to harder concepts.
For students, DreamBox may seem a bit like a computer game, but teachers can then use DreamBox to assess how well a student is doing—and also decide whether or not the student needs to go back over certain topics in order to understand them.
Eventually, Pandolfo expects to have DreamBox become part of homework for Darien students, but not before the program makes initial assessments about how well each child understands various concepts.
Making DreamBox part of homework is being delayed partly because Pandolfo fears some parents may help a child too much with this program, to the point where the computer will record that the student understands particular topics when what's really being recorded is how well the too-helpful parent understands the topic—defeating much of the purpose of the program.
Aside from DreamBox, other homework might include assignments with math problems that a student has never seen before, where "the teacher says, 'Try to figure out how to solve it,'" Fosnot said.
In that case, parents have to be especially careful not to help so much with homework that the parents are essentially doing it rather than the children, she said.
With all the changes in the way math is being taught, Pandolfo and Furman said the general content of what's being taught remains generally the same.
"The question isn't, 'Do they need to know the basic facts?'" Fosnot said. "Everybody agrees with that. The question is, 'How do you get there?'"