Skills Needed for Mathematical Problem Solving (Part 6)

This section examines two critical but often overlooked dimensions of mathematical problem solving: attitude (perseverance, tolerance for failure) and metacognition (awareness and control of one's own thinking). Using two classic problems — Eratosthenes' measurement of the Earth's circumference and the average-speed round-trip paradox — we show how attitude and metacognition shape problem-solving success.

Table of Contents

3.4 Attitude and Problem Solving

Learners must persevere and take the risk of failing when solving a given problem. They must also understand that learning takes place even when they are not able to solve a problem. What counts is the time and effort spent looking for a solution. Students do not learn much from problems that they can easily solve; they learn more from challenging problems where they have to persevere.

However, students left alone with challenging problems to solve may get frustrated and develop a negative attitude towards mathematical problem solving. The instructor's task is not easy. He must not give too much information so that students will not have anything to think about. But also, he should give enough help so that students will not be frustrated.

The Instructor's Balance

Effective problem-solving instruction requires a careful balance:

Too much help: Students have nothing left to think about; no learning occurs.
Too little help: Students become frustrated and develop a negative attitude.
The right amount: Students struggle productively, make progress, and learn through the effort.

I found that historical facts about mathematics and real-life mathematical problems motivate students. I use both to highlight the power of mathematics and its applications. Below is a real-life problem related to trigonometry about measuring the circumference of the Earth.

Problem 8: Eratosthenes and the Earth's Circumference

Problem 8: More than two thousand years ago Eratosthenes of Cyrene (276 BC – 194 BC), a Greek mathematician, used the rays from the sun and the shadow of a stick to measure an angle \( \alpha = 7.2^\circ \) (representing exactly \( \frac{1}{50} \) of a full circle). Assume that the ray from the sun falls vertically at Syene and the stick used at Alexandria is also vertical. Assume also that the Earth is circular and find the circumference of the Earth.

Diagram showing the Sun's rays falling vertically at Syene and casting a shadow at Alexandria, with angle alpha = 7.2 degrees at the Earth's center

Fig. 5: Rays from the Sun and the shadow used by Eratosthenes to measure angle \( \alpha \).

Students are more interested in real-life problems because they make sense and also give practical reasons to study mathematics.

Complete Solution to Problem 8

Show Solution

Geometric principle: The angle \( \alpha = 7.2^\circ \) at the Earth's center is equal to the angle of the shadow cast by the stick at Alexandria. This is because the Sun's rays are (approximately) parallel, so the angle between the vertical at Alexandria and the direction to the Sun equals the central angle between Syene and Alexandria.

Key relationship: The arc length \( s \) between Syene and Alexandria corresponds to the central angle \( \alpha \). If \( C \) is the circumference of the Earth, then:

\[ \frac{\alpha}{360^\circ} = \frac{s}{C} \]

Eratosthenes knew the distance between Syene and Alexandria was approximately 5000 stadia. Since \( \alpha = 7.2^\circ = \frac{360^\circ}{50} \):

\[ \frac{7.2^\circ}{360^\circ} = \frac{1}{50} = \frac{5000}{C} \]

Solving for \( C \):

\[ C = 50 \times 5000 = 250{,}000 \text{ stadia} \]

Modern equivalence: Depending on which stadium unit is used, Eratosthenes' result corresponds to between 39,000 and 46,000 km. The modern value of Earth's circumference is approximately 40,075 km at the equator. His estimate was remarkably accurate — within about 1–15% of the true value, depending on the stadium conversion used.

Why this problem matters: It shows students that mathematics is not just abstract — it is a powerful tool that has been used for over two millennia to measure the world around us. This historical context increases motivation and demonstrates the practical power of mathematical reasoning.

3.5 Metacognition and Problem Solving

The term metacognition refers to a learner's knowledge about his or her process of cognition and the ability to control and monitor those processes as a function of the feedback the learner receives via the outcomes of learning. Learners must control and reflect on their cognitive processes in order to solve challenging problems. It is thought that the best way to help students become aware of their own thinking is to create opportunities where they have to explain their thinking explicitly. As an example, let us look at the following problem.

Problem 9: The Average Speed Paradox

Problem 9: Ahmed walked at a constant speed of 6 km/hour along a straight line from point A to B, then walked back along the same line from B to A at a constant speed of 4 km/hour. What is the average speed over the entire trip?

(Definition: Average speed = Total Distance / Total Time)

When the above problem was given to students in Level 2 advanced mathematics, nearly all students incorrectly answered the question as follows:

\[ \text{Average Speed} = \frac{6 + 4}{2} = 5 \text{ km/hour} \]

When asked to explain their answer, they said that any average is found by adding all data values and dividing the sum obtained by the number of data values. They simply ignored the definition given to them.

I next guided them to work in groups, look back at the problem, and strictly use the mathematical definition to answer the question. They had to justify the steps of their calculations and their thinking within their group.

When the correct answer based on the definition was obtained, I asked students to discuss within their groups what happened and why all students initially answered the question without taking into account the formal definition. I explicitly asked them to think about the way they solved the problem and why.

Metacognitive skills help students to analyze both the question and the solution developed. They also help students to look back at the problem and restart if necessary.

Complete Solution to Problem 9

Show Solution

Approach: Let \( d \) be the distance from A to B (in km).

Time going (A → B):

\[ t_1 = \frac{d}{6} \text{ hours} \]

Time returning (B → A):

\[ t_2 = \frac{d}{4} \text{ hours} \]

Total distance: \( 2d \) km (going and returning).

Total time:

\[ t_1 + t_2 = \frac{d}{6} + \frac{d}{4} = \frac{2d + 3d}{12} = \frac{5d}{12} \text{ hours} \]

Average speed (by definition):

\[ v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{5d/12} = \frac{24d}{5d} = \frac{24}{5} = 4.8 \text{ km/hour} \]

\[ \boxed{v_{\text{avg}} = 4.8 \text{ km/hour}} \]

Verification with a concrete distance: Suppose \( d = 12 \) km.

Going: \( 12/6 = 2 \) hours
Returning: \( 12/4 = 3 \) hours
Total distance = 24 km, Total time = 5 hours
Average speed = \( 24/5 = 4.8 \) km/h ✓

Note: The average speed is not the arithmetic mean of 6 and 4. The harmonic mean applies here: \( v_{\text{avg}} = \frac{2v_1 v_2}{v_1 + v_2} = \frac{2 \cdot 6 \cdot 4}{6 + 4} = \frac{48}{10} = 4.8 \) km/h.

Why Students Answer 5 km/h — A Metacognitive Analysis

The error of answering \( \frac{6+4}{2} = 5 \) km/h reveals several metacognitive failures:

Ignoring the definition. The problem explicitly states "Average speed = Total Distance / Total Time," yet students used a different procedure.
Overgeneralizing "average." Students applied the arithmetic mean rule from statistics without checking whether it applied to speed.
Not verifying the answer. A quick sanity check (does the answer make sense given more time is spent at the slower speed?) would have revealed the error.
Not monitoring comprehension. Students did not ask themselves, "Does my answer match what the problem asked?"

This is precisely why metacognitive training matters: students must be taught to monitor their reasoning, check their answers against the given definitions, and reflect on why a particular strategy was chosen.

Pedagogical Implications

Designing Activities That Develop Attitude and Metacognition

Use historical problems. Eratosthenes' measurement of the Earth engages students by showing mathematics as a living, applied discipline.
Use real-life problems. Problems connected to daily experience increase motivation and make the mathematics meaningful.
Teach metacognition explicitly. Metacognitive skills do not develop automatically; they must be modeled, discussed, and practiced.
Use genuine problems. Only problems students have not seen before help them develop metacognitive skills.
Require explanations. Ask students to explain their thinking to peers and to the teacher — articulation reveals gaps in understanding.
Model your own thinking. Show students how an expert solves a problem, including the false starts and revisions.
Encourage productive struggle. Allow enough time for students to wrestle with problems before revealing the solution.

As the paper states, metacognitive skills help students analyze both the question and the solution, and look back at the problem to restart if necessary. This makes them essential components of mathematical problem-solving competence.

Frequently Asked Questions

Why is attitude important in mathematical problem solving?

Attitude matters because students must persevere and take the risk of failing when solving challenging problems. Learning takes place even when a problem is not fully solved — what matters is the time and effort spent searching for a solution. Students learn more from problems they initially cannot solve than from easy ones.

What is metacognition in mathematics?

Metacognition refers to a learner's knowledge about their own cognitive processes and their ability to monitor and control those processes. In mathematics, it includes analyzing the question, planning a solution strategy, monitoring progress, looking back at the solution, and restarting if necessary.

Why do students incorrectly compute average speed as (v₁ + v₂)/2?

Students often confuse the arithmetic mean of two numbers with the average speed over a round trip. The correct average speed is defined as Total Distance / Total Time. For a trip at 6 km/h and back at 4 km/h, the average speed is 4.8 km/h, not 5 km/h, because more time is spent at the slower speed.

How did Eratosthenes measure the circumference of the Earth?

Eratosthenes measured the angle of the Sun's rays at Alexandria at noon on the summer solstice and found it to be about 7.2°, while at Syene (modern Aswan) the Sun was directly overhead. Since 7.2° is 1/50 of 360°, and the distance between the two cities was known to be about 5000 stadia, he estimated the Earth's circumference as 50 × 5000 = 250,000 stadia.

How can teachers help students develop metacognitive skills?

Teachers can help by: (1) teaching metacognitive skills explicitly, (2) using genuine problems students have not solved before, (3) asking students to explain their thinking to peers and to the teacher, (4) modeling their own thinking when solving problems, and (5) providing opportunities for group discussion and reflection after solving.