“Our need to have things explained is as strong an impulse in our kids, and in us, as being hungry and thirsty,” Aungst said. “The problem with how we usually teach math is that we take all that wondering away.”
Educators usually teach math by laying out the facts, showing them processes, and asking students to practice until they achieve “mechanical perfection”–students have nothing to wonder about.
“One element of conjecture is being able to provoke that sense of wonder in kids, and allowing them to look for explanations and let that drive keep them engaged,” Aungst said.
But it goes deeper than that, he said.
“It’s about students not just solving problems–it’s about them looking for problems, too,” he added. “Innovators are looking for problems and they try to solve them before anyone even realizes the problem exists. We need innovators. Math class is a great place to start doing that.”
Educators should strive to avoid ending with the answer. Instead, they should ask students why they think the answer is what it is, how they arrived at the answer, if other answers are possible, if other methods of solving are possible, if students encountered difficulty, and if so, how they overcame it.
Digital tools to support conjecture include:
When students are able to explain their thought processes and understanding, their own knowledge increases.
One way to promote better math learning is to think of math as if it were a foreign language.
“If all we’re doing is teaching students how to move the symbols around and get an answer out of it, without embedding meaning into that, then the meaning behind the math is completely lost,” he said. “Learning how to do math is like learning how to read a foreign language.”
Students should be able to explain, in their own words, what numbers and symbols mean and represent.
Instead of asking students to show their work, ask them to convince mathematical experts that their solution is a good one–students understand what they do, but communicating it to someone else is a challenge.
Digital tools for communication include:
Infographics such as http://piktochart.com and http://infogr.am
Social media (speaking to others about the math students are doing)
YouTube and Vine
“Problem solving in the real world is nearly always collaborative,” Aungst said. “In fact, competition might even serve to dampen innovation. We want to get our kids working together.”
Working together inspires students to consider other points of view and other approaches to problems. This, in turn, informs, and may change, their thinking.
Educators could begin with a “You, Y’all, We” approach: present the problem first, and let students work on that problem individually. They’ll struggle, Aungst said, but that’s OK. Then, move to small-group discussion, before involving the whole class in the discussion or in solving the problem.
Aungst also recommends the “three before me” strategy, in which students consult three other resources or people before bringing an “unsolvable” problem to their teacher.
Digital tools for collaboration and building classroom teams include:
Wikis and Google Sites
Skype and Google Hangouts
As odd as it seems, chaos promotes learning and discovery, Aungst said.
“What it really is about is the fact that problem solving is messy–it’s not a linear step-by-step,” he said. “Real world problem solving is a messy thing.”
Students should struggle in productive ways, and if they’re not, instruction isn’t particularly effective. In short, they need “cognitive sweat,” Aungst said.
Digital tools to support chaos include:
Educators should celebrate students’ growth, successes, “and even their failures, and what you can learn from their failures,” Aungst said.
Sometimes, a “catch me if you can” strategy works well. Educators tell students they plan to make mistakes, and students must try to identify those mistakes. This makes it safe for students to point out errors.
“It’s really important that you validate effort, and not answers,” he said. “It’s really important that we recognize that the students who start out as the smartest at the beginning of the year may not be the smartest at the end of the year.”