Constructive lessons focus on students building connections between what they know and what they don't know. They rely on critical thinking skills and student problem solving more than the traditional method of teacher modeling or teacher indoctrination of knowledge. Though basic skills and tenets still need to be taught, they are taught as a toolbox that will be called upon for problem solving. Instead of a fourth grade student being taught to multiply a numerator and denominator by the same number to get an equivalent fraction, they use manipulatives to discover the relationship of objects that are the same size, but are divided into different numbers of pieces. The same exercise at the fifth grade level could lead to discovery of how we use the multiplicative identity property of the number 1 to create equivalent fractions.
In order to make these connections, our students have to analyze the problem and think about what they know that might help them solve it. They can gather clues from prior learning, from the current problem, even from how the question is being asked. This also encourages the development of their metacognitive thinking.
Authentic assessment mirrors constructive lessons in that they also rely more heavily on the student using their toolbox of skills to solve real-life problems. These assessments again rely more heavily on higher order thinking skills to analyze, apply and evaluate. In the fraction exercise above, in order to do an informal authentic assessment, you could have the fourth grade students play a game of "Pizza to Go". The object of the game is to be the first to make exactly two full pizzas. Using pizza pieces sized 1/2 to 1/12, and a spinner that indicates which size piece they can take from the pile, the students soon realize that they can take two pieces of 1/8 when the spinner says 1/4. Asking them to justify their turn, they may say something like "I can put two 1/8's together and it's the same size as a 1/4 piece, or they may say "1 times 2 is 2 for the numerator and 4 times 2 is 8 for the denominator".
Both a constructive lesson and authentic assessment provide student engagement and connection to the real world and previous knowledge. Both provide a depth of understanding beyond rote knowledge. That deep understanding of what we are being taught is the kernel that allows our future learning to explode.
