Does everybody count?

In this article Nel Nodding raises the problem of the purpose of learning mathematics at school. Her main idea is that school mathematics isn’t such as essential subject as we are told every time. Namely, there are many talks about high correlation between math achievements and further success at work, and therefore these beliefs emphasise a significant role of mathematics in overcoming inequalities in our society. Moreover, our efforts to teach mathematics in school led to enhancement of national economics.    

In her turn Nel doubts in causality between mathematics and further success in life, and argues that school math rather plays a role of artificial barrier to success than accounts for high-paid job.  As soon as math doesn’t play a crucial role in our life, we should stop to oblige everybody to learn mathematics, at least in high school.  According to this paper it is better to allow students to decide for themselves whether they need to study math or not.  In other words we should provide student an opportunity to choose their own way in life (not only in mathematics) and respect their choice.

It is should be noted that the author is not arguing about mathematics itself, but states that students could have different interests and talents in multiple spheres of life. And instead of focusing just on teaching mathematics, it is better for school to support development of everyone's interests.  For this purpose Nel suggests to enrich math curriculum with references music, art, philosophy, theology, etc.

Generally I agree with Nel’s argumentation and I am sure that conscious choice is always better. Moreover, it always better to help students to understand their interests, to work with their anxiety and beliefs regardless of school subject. However, I find the idea of interdisciplinary connections very attractive, I also doubt that such serious changes could happen in the nearest future.

I was also wondering about role of mathematics in our life. In my research I asked several math teachers about it. Surprisingly, most of them define threefold role of school of mathematics:

1.  Mathematics is the main science among others because only mathematics could teach students to think and solve different problems; it improves abstract reasoning
2. Mathematics plays a crucial role in professional and life success since many students are aimed to go to the universities and become economists, IT-specialists or lawyers  in future   
3. Mathematics helps us to solve our everyday problems. Namely, we need know math in order to buy something or to understand what is going on with our credits  

Honestly, I favour the first role of mathematics more than others. But I guess it is hard to define is it true or not, and which part of math curriculum accounts for t, and by what time the formation of reasoning skills is completed.

Have you been forced to study math at school or university? If yes, how could your life have been changed without that forcing?   

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)

Future directions and perspectives for problem solving research and curriculum development

The topic of the present research belongs to the educational studies of problem solving. The general goal of the current article is to suggest new perspective to teach mathematical problem solving through reviewing history of research on that topic and revealing several existing limitations.

However history of research on mathematical problem solving begun 60 years ago, according to the literature review there were revealed fairly ambiguous results about optimal teaching practices towards problem solving. From one point of view it is better to teach specific heuristics or strategies to solve problems, and from another point of view it is more rational to develop the whole concept understanding (or develop metacognitive strategies). These two traditions of studies are called descriptive and prescriptive, respectively. And the problem is the first approach of teaching focuses on strategies without understanding the whole concept and the second approach of teaching suggests apply too general and for this reason unclear rules to problem solving.  In other words it is still unclear whether the concept development or the development of competencies of problems solving is the best way to learn problem solving. 

Taking into account limitations of previous studies authors suggest Models&Modelling perspective as an alternative way to teach problem solving. Instead of describing the external characteristic of problem solvers’ behavior, authors propose to look deep inside the process and try to understand the nature of this process. In other words they are more interesting in such steps of modelling process as mathematizing problems, quantifying a situation, interpreting results, etc. Moreover, according to their concept the M&M approach helps learners consider the problem as a whole process rather than parsing a problem in separate pieces.

From my opinion the Models&Modeling is the most optimal approach to learn problem solving, however there are still remained many unanswered question about teaching practices. For example, whether we should start with analyzing examples of solved problems, or it is better to start solve problem without knowing how to do it, or it is better to teach solve problem step by step according steps in modelling process, etc. Moreover, it is possible that students could have different problems with acquiring process of modelling. Some of them will struggle at mathematizing step while others will stick at the step of interpreting results.