With my year 10 "literacy focus" mathematics class we have been looking at the language of measurement, specifically area and perimeter. The idea has been to get students to clearly understand the importance of units and how to interpret what questions are asking.
When thinking about perimeter we are only thinking about the boundary outside a shape. We then looked at some real world shapes and measured their boundaries, followed by drawing some scale diagrams of shapes and measuring their boundaries.
We then tried to describe how much space was in the shapes. I tried to drive them to describing dimensions, but some students had enough prior knowledge to say calculate the answer by multiplying lengths.
The main issues came as we arrived at composite shapes where composite shapes had to be split into basic geometric shapes. Deriving missing measurements really brought home how much trouble low literacy students can have with basic mathematical concepts such as subtraction. Assuming that a student can see how to find missing sides is in many cases overestimating their ability.
Even at year 10, the majority of low literacy students fail to see f- c = a whereas they are more likely to be able to see a+ c = f.
Because of this to use a scale drawing to assist these students see subtraction in action requires some thought. If a student drew the above diagram with the following measurements they would
not use subtraction to find the missing side on the top rectangle. They would count the squares (if on grid paper) or measure the vertical gap and put the measurement of one on the page.
To encourage students to use subtraction I needed to encourage students to first look at the diagram and do a number of examples with them without using scale diagrams, explicitly doing subtraction sums. We checked our examples with scale diagrams, rather than finding the solution using scale diagrams.
Though a subtle difference, this was far more successful.
Another successful strategy used during these lessons was using formal layout during the early stages with simple examples. By doing this, students were able to see the connection between diagrams, algebraic substitution and the usage of formulae.
For instance:
By learning this and ensuring each example was completed thus, when triangles and circles were introduced it was a trivial case of changing the formula and adding a line to the bottom
Area(Shape) = Area(S1) + Area(S2)
It was interesting to note that students at this stage had now forgotten that perimeter was the outer boundary and were including the S1 - S2 boundary in their perimeter calculations. It had to be explicitly explained that
Perimeter(Shape) != Perimeter(S1) + Perimeter(S2)
and that the intersecting boundary needed to be subtracted. For me, this is learning real mathematical literacy. Students are becoming able to exactly describe their intent on the path to a solution.