Skills Needed for Mathematical Problem Solving (Part 5)

This section examines the foundational skills, thinking, and reasoning required for successful mathematical problem solving. Using two contrasting case studies — a classic two-boats river problem and an open-ended statistics problem — we illustrate how carefully designed problems can develop critical thinking, inductive and deductive reasoning, and genuine problem-solving ability.

Table of Contents

3.2 Skills in Problem Solving

Basic mathematical skills such as solving equations and inequalities are necessary for mathematical problem solving. Formulating a problem can be very demanding, but simplifying and solving the equation obtained, for example, is necessary to answer the question in the problem. Students who cannot manipulate algebraic expressions will definitely have difficulties in problem solving.

Our students, at UGRU, have difficulties in reading with understanding and extracting information from the text of the problem. This skill has to be taught explicitly to our students. It is the first step in problem solving, and students cannot make any progress if the problem is not properly understood.

Core Skills Required for Problem Solving

Algebraic manipulation: Solving equations and inequalities, factoring, simplifying expressions.
Reading comprehension: Extracting relevant information from the text of a problem.
Problem formulation: Translating a word problem into mathematical equations.
Diagramming: Drawing a sketch that represents the relationships in the problem.
Verification: Checking the solution against the original conditions.

3.3 Thinking and Reasoning in Problem Solving

Critical thinking is needed in all steps of problem solving. Students do not look back critically at the solution of a problem once it is solved. They tend to accept whatever answer they have obtained. Critical thinking is needed when extracting information from the text of the problem, formulating and solving the problem, and analyzing the solution obtained. I now give an example of a problem that may be used to provide opportunities for students to think critically.

Problem 6: The Two-Boats River Problem

Problem 6: Two boats on opposite banks of a river start moving towards each other. They first pass each other 1400 meters from one bank. They each continue to the opposite bank, immediately turn around, and start back to the other bank. When they pass each other a second time, they are 600 meters from the other bank. We assume that each boat travels at a constant speed all along the journey. Is it possible to find the width of the river using the given information?

Students are guided to make a diagram and write the equations to formulate the problem.

Let \( t = 0 \) be the time when the two boats start moving. Let \( S_1 \) and \( S_2 \) be the constant speeds of the two boats. Let \( t_1 \) be the time when they first pass each other, and \( t_2 \) be the time when they pass each other for the second time. Let \( X \) be the width of the river.

Diagram showing the two boats, river banks, first and second meeting points, and unknown quantities X, S1, S2, t1, t2

Fig. 4: Diagram including unknowns in Problem 6.

Using the distance formula (Distance = Speed × Time), we can set up the following equations for the distances covered by each boat at their first and second meeting points:

\[ S_1 t_1 = 1400 \quad (1) \] \[ 1400 + S_2 t_1 = X \quad (2) \] \[ S_1 t_2 = X + 600 \quad (3) \] \[ S_2 t_2 = 2X - 600 \quad (4) \]

Because there appear to be five unknowns (\( S_1, S_2, t_1, t_2, X \)) and only four equations in the above system, students are often tempted to say that it is not possible to find the width \( X \) of the river. This particular problem can be used to generate activities where students will have to think deeply in order to come up with a correct answer.

Enough time should be given to students to discuss such a challenging problem. They should be allowed to work in groups, and if students cannot finish the work in class, they should be allowed to continue solving the problem as a homework assignment. Students will not benefit if the solution is simply given to them without allowing enough time to investigate. Learning will take place through the time and effort spent on finding a solution, even if they struggle.

It is not difficult to design problems that can activate students' thinking. One has to avoid problems with only one obvious solution. Note that the question in Problem 6 does not say "Find the width of the river," but rather, "Is it possible to find the width of the river using the given information?" A question of the first type already assumes that it is possible to find the width \( X \). However, questions of the second type are more demanding and therefore more suitable for developing thinking skills.

Complete Solution to Problem 6

The key insight is that although there appear to be five unknowns, the ratio \( t_2 / t_1 \) can be eliminated, allowing \( X \) to be determined uniquely.

Show Solution

Step 1 — Eliminate \( S_1 \) and \( S_2 \) using equations (1) and (3):

From (1): \( S_1 = \dfrac{1400}{t_1} \).

From (3): \( S_1 = \dfrac{X + 600}{t_2} \).

Equating: \( \dfrac{1400}{t_1} = \dfrac{X + 600}{t_2} \), so:

\[ \frac{t_2}{t_1} = \frac{X + 600}{1400} \quad (5) \]

Step 2 — Eliminate \( S_1 \) and \( S_2 \) using equations (2) and (4):

From (2): \( S_2 = \dfrac{X - 1400}{t_1} \).

From (4): \( S_2 = \dfrac{2X - 600}{t_2} \).

Equating: \( \dfrac{X - 1400}{t_1} = \dfrac{2X - 600}{t_2} \), so:

\[ \frac{t_2}{t_1} = \frac{2X - 600}{X - 1400} \quad (6) \]

Step 3 — Equate the two expressions for \( t_2 / t_1 \):

\[ \frac{X + 600}{1400} = \frac{2X - 600}{X - 1400} \]

Cross-multiply:

\[ (X + 600)(X - 1400) = 1400(2X - 600) \] \[ X^2 - 1400X + 600X - 840000 = 2800X - 840000 \] \[ X^2 - 800X - 840000 = 2800X - 840000 \] \[ X^2 - 3600X = 0 \] \[ X(X - 3600) = 0 \]

Since \( X \gt 0 \), the width of the river is:

\[ \boxed{X = 3600 \text{ meters}} \]

Verification: From (5), \( t_2 / t_1 = (3600 + 600)/1400 = 4200/1400 = 3 \). From (6), \( t_2 / t_1 = (7200 - 600)/(3600 - 1400) = 6600/2200 = 3 \). ✓

Conclusion: Yes, it is possible to find the width of the river. It is 3600 meters, even though the individual speeds and times cannot be determined uniquely.

Problem 7: An Open-Ended Statistics Problem

Open-ended problems, usually with many valid answers, can also be used as tools to generate activities where students have to think critically to come up with answers. Useful discussions can be generated in order to compare all valid answers generated by students. I now give an example of an open-ended problem.

Problem 7: Create a set of data points that satisfies the following conditions:

The set includes 8 data values.
The range of the data set is 20.
The median is equal to the mean.

Show that your proposed data set satisfies all the conditions.

Sample Solution to Problem 7

Because the problem is open-ended, there are infinitely many valid answers. Here is one example.

Show Solution

Proposed data set (sorted): 5, 5, 10, 10, 10, 10, 15, 25

Check 1 — 8 data values: The set has exactly 8 values. ✓

Check 2 — Range = 20: \( 25 - 5 = 20 \). ✓

Check 3 — Median = Mean:

The median of 8 sorted values is the average of the 4th and 5th values: \( (10 + 10)/2 = 10 \).

The mean is:

\[ \frac{5 + 5 + 10 + 10 + 10 + 10 + 15 + 25}{8} = \frac{90}{8} = 11.25 \]

This does not satisfy the condition. Let me try a different set.

Revised data set (sorted): 5, 8, 9, 10, 10, 11, 12, 25

Check 1 — 8 data values: ✓

Check 2 — Range: \( 25 - 5 = 20 \). ✓

Check 3 — Median: \( (10 + 10)/2 = 10 \).

Check 3 — Mean: \( (5 + 8 + 9 + 10 + 10 + 11 + 12 + 25)/8 = 90/8 = 11.25 \). ✗

Final data set (sorted): 5, 8, 9, 12, 8, 10, 13, 25 — let me use an elegant construction instead.

Construction approach: Choose a symmetric set around 10, with 8 values. For a symmetric set, the mean equals the median by construction.

Let the set be: \( 10 - a, 10 - b, 10 - c, 10 - d, 10 + d, 10 + c, 10 + b, 10 + a \).

The median is \( (10 - d + 10 + d)/2 = 10 \). The mean is also 10 by symmetry.

Choose \( a = 15, b = 5, c = 2, d = 1 \): \( -5, 5, 8, 9, 11, 12, 15, 25 \).

Adding 10 to each: \( 5, 15, 18, 19, 21, 22, 25, 35 \).

Check 1 — 8 data values: ✓

Check 2 — Range: \( 35 - 5 = 30 \). ✗ — the range is too large. Adjust: choose \( a = 10 \), giving range 20.

Final answer (sorted): 5, 8, 9, 10, 10, 11, 12, 15. Wait — check the range: 15 - 5 = 10, not 20.

Let me construct it correctly: To have range 20 and mean = median = m, use a symmetric set around m with min = m - 10 and max = m + 10.

Choose m = 10: min = 0, max = 20. Symmetric set: \( 0, 5, 8, 9, 11, 12, 15, 20 \).

Check 1: 8 values ✓

Check 2: Range = 20 − 0 = 20 ✓

Check 3 — Median: \( (9 + 11)/2 = 10 \) ✓

Check 3 — Mean: \( (0 + 5 + 8 + 9 + 11 + 12 + 15 + 20)/8 = 80/8 = 10 \) ✓

Valid Answer: \( \{0, 5, 8, 9, 11, 12, 15, 20\} \) satisfies all three conditions.

Note: There are infinitely many valid data sets. The symmetric construction around the median guarantees the mean equals the median.

Inductive vs. Deductive Reasoning

Logical reasoning is fundamental to success in mathematics in general, and problem solving in particular. Inductive reasoning uses easy-to-understand cases to identify patterns and can also be used to establish relationships between mathematical objects. Deductive reasoning uses mathematical definitions, axioms, rules, and theorems to draw conclusions. Deductive reasoning is fundamental to critical thinking. Geometry problems may be used as tools to develop deductive reasoning. I believe that both must be used and explicitly discussed with students.

Aspect	Inductive Reasoning	Deductive Reasoning
Starting point	Specific observations or examples	General principles, definitions, axioms
Direction	From specific to general	From general to specific
Conclusion	Probable, not guaranteed	Certain, if premises are true
Example	Observing that 3, 7, 11, 15 increase by 4 and concluding the next is 19	Using the formula \( a_n = a_1 + (n-1)d \) to compute any term
Role in problem solving	Discovering patterns, conjecturing	Proving results, verifying conjectures

Pedagogical Implications

Designing Problems to Develop Thinking Skills

Avoid problems with only one obvious solution. Problems where the path is immediately clear do not require deep thinking.
Ask "Is it possible to...?" rather than "Find...". The first type of question forces students to investigate existence; the second already assumes existence.
Use open-ended problems. Problems with multiple valid answers generate discussion and comparison.
Give enough time. Learning happens through the struggle, not through receiving the solution.
Encourage group work. Collaborative problem solving exposes students to multiple strategies.
Require verification. Asking students to prove their answer satisfies all conditions develops critical thinking.

The two problems in this section illustrate the contrast. Problem 6 has a unique numerical answer (3600 m) but requires critical thinking to realize the problem is solvable despite the apparent mismatch between unknowns and equations. Problem 7 has infinitely many answers and requires constructing one that satisfies all three conditions — a genuine exercise in creative reasoning.

Frequently Asked Questions

What are the basic mathematical skills needed for problem solving?

Basic mathematical skills for problem solving include: solving equations and inequalities, manipulating algebraic expressions, reading with understanding, extracting relevant information from a problem statement, and formulating a problem into mathematical equations.

What is the difference between inductive and deductive reasoning?

Inductive reasoning uses specific examples to identify patterns and form general conclusions. Deductive reasoning starts from general principles (definitions, axioms, theorems) and derives specific conclusions. Both are essential for mathematical problem solving.

Why is critical thinking important in problem solving?

Critical thinking is needed at every stage of problem solving: understanding the problem, formulating it, solving it, and analyzing the solution. Students often accept their first answer without checking it, which can lead to errors. Critical thinking means verifying, questioning, and reflecting on the solution.

What is an open-ended problem?

An open-ended problem is a problem with multiple valid answers or multiple valid solution paths. For example, creating a data set with 8 values, range 20, and mean equal to median has infinitely many correct answers. Open-ended problems develop creativity and critical thinking.

How can teachers help students develop problem-solving skills?

Teachers can help by: (1) giving students enough time to struggle with problems before showing solutions, (2) encouraging group work, (3) using problems with non-obvious solutions, (4) asking "Is it possible to...?" questions rather than direct computation questions, and (5) using open-ended problems.