Context, reality and ambiguity

Real-life math problems are supposed to teach students to apply their knowledge in their lives beyond classrooms. But is it true? In chapter “Context, reality and ambiguity” authors discus the role of context in math problems and conclude that context of tasks is rarely relevant to real-life situations. In other words, the actual purpose of these tasks is to recognize school math in the problem and then find the right (expected) solution, rather than to force students to solve this problem as if it exists in real-life.

However, an ambiguity, which could arise due to real-life context, can enrich typical math problem, can force us to emerge beyond the simple answer and to look for further interpretations and explanations. Eventually ambiguity helps students to learn mathematics.

Here are four steps to exploiting ambiguity in math problem:

1.     Restoring reality/ identifying the variables.

Here you are supposed to imagine the situation to work out it. For example, you could reveal new details of the situation or new possibilities to solve the problem.  

2.     Playing with the mathematics

Here you can work out the mathematics in the problem. You could play with numbers and with approach to solve this problem. (For example, you could look through different grades or different levels of learners)

3.     Handling the variables/ imposing some constraints

Now you have to decide what kind of task you want to create. What is the purpose of this task? To work with ratios? On which level? What mathematics do you want to focus on?

4.     Defining some tasks.  

Finally, write down your renewed math problems! Now they look fairly different (and the purpose of this task could change completely), but they have true realistic context at least.

Let’s try to exploit ambiguity in three math problems from Russian textbook.

How many ways can the agricultural company sow four different kinds of herbs in four plowed fields? There are seeds of rye, weat, corn and barley.

Step 1:

-  Why does the agricultural company need these calculations?

-  Who is the person in that company? What kind of responsibility does he have?

-  Does this company need crop rotation?

-  What is the shape of these fields? Is it 2x2 or 1x4?

-  Is it any special approach of choosing seeds for crop rotation? For instance, it could be better to sow corn at first in order to enrich soil with nutrients.

-  Can we sow one kind of seeds in all of plowed fields? Or do we need to place different kinds of seed  in every field? And is it any special order to place these seeds?

Step 2:

Solution of this problem: P_n=n!. 4!=24 and so we have 24 ways to sow 4 different kinds in 4 plowed fields.

If we have special order to sow seeds, what formula do we need to choose?

What if we need to sow rye next to barley and wheat next to corn? How does this fact change our formula?

Step 3:

         Variables: number of different seeds, shape of field, constraints to sow in definite order, necessity to sow all of seeds.

Step 4:

a.   The agricultural company needs enrich soil in their fields by crop rotation. Currently they have 4 plowed fields (2x2) and 4 kinds of seeds, which are perfect for enriching soil with variety of nutrients: rye, wheat, corn and barley. How many ways can the company sow all kind of their seeds in all of their fields? What would happen if the field has a 1x4 shape?

b.   The agricultural company needs enrich soil in their fields by crop rotation. Currently they have 4 plowed fields (1x4) and 4 kinds of seeds, which are perfect for enriching soil with variety of nutrients: rye, wheat, corn and barley. But the problem is they can’t sow rye next to wheat and corn next to barley. In how many ways can the company sow all of their seeds in all of their fields? What would happen if they need to sow corn seeds only at the nearest fields? (Because corns are the heaviest ones among other seeds and the company want to reduce transportation costs)  

                    

What is an angle (in degrees) between the minute and hour hands at 4 PM?

Step 1:

• Which angle do we need find - the small or the big?
• Why does anybody need know it?
• Does this angle depend on the type of clock? (for example, special furniture, shape or mechanisms)
• What if we have an inverted clockwise?
• What if we have clockwise without numeric marks?

Step 2:

What if we have another size of the angle? Where is our 'starting point'?

Step 3:

Variables: features of watch, characteristics of clockwise, angle size.

Step 4:

A forensic expert found an ancient wristwatch on the crime scene. The minute and hour hands show a time when the mechanism stopped (and perhaps that was a time of victim's death), but the wristwatch was old insomuch that a clockwise lost all of their numeric marks. Luckily the angle between these hands was right. Can you say in what time the mechanism stopped? How many possible solutions do you have? How next details could change you solution:

• What if there is only one mark on this clockwise, for example, 12? 
• What if the angle between hands is 120 degrees? 
• What if that wristwatch was found without a wrist belt?  

(Yes, the initial purpose of the task was changed completely)