While preparing to explore fractions with my Grade 6 students, some colleagues and I co-planned a lesson from the Pattern Block volume of the ETA Cuisinaire Supersource Series, an oldie but goodie when it comes to constructivism in math.
I had been introduced to pattern blocks and other manipulatives early in my teaching career, and found them helpful in teaching myself math concepts, never mind introducing these concepts to students!
We selected a problem that would require students to work in pairs to calculate the "value" of a design they had created using a baggie of pattern blocks (based on the wide range of needs in our classrooms, we decided to prepare baggies ahead of time, so that some students would have either fewer pattern blocks to work with, or no blue rhombi, or both fewer and less variety of blocks, in order to scaffold and make the task more accessible-- this turned out to be a very wise choice in my room!!) Depending on which block was worth "one whole", the design would have differing values.
The idea was to introduce improper fractions and mixed numbers through a constructivist stance, in a way that would appeal to both the hands-on learners, and those more math phobic students who enjoyed patterning in art.
And it worked! Students really engaged with the math, and using the pattern blocks as thinking tools allowed them to explore concepts that would have been foreign using merely standard notation in the abstract. Even with the blocks, many students had to be prodded to move beyond a superficial examination... but the blocks helped them to do so in a wide range of ways, as evidenced below...
Some of my students found apps or virtual pattern blocks online, and created screenshots of their annotated work to post on our Edmodo site. A few got so excited about the work that they even shared photos of their designs with students from our collaborative class in Australia!
One of my goals this year has been to integrate more equity and social justice issues into my math program, and while that endeavor has been going reasonably well, I worry that sometimes my focus on "real life application" comes at the cost of just engaging students in the beauty of mathematics.