Journal of Science and Mathematics Education Vol.
2 No.
March 2026, pp.
E-ISSN 3090-0336
Strengthening Logical Foundation in Mathematics: A Pathway to Enhance Reasoning Skills Hitesh Choudhury1* 1 Department of Computer Science.
Dimoria College (Autonomou.
Assam.
India.
* Corresponding author : Chansiri23@gmail.
ARTICLE INFO
Article history a.
Received : January 19, 2025
Revised : February 25, 2026
Accepted : March 20, 2026
Published: March 28, 2026
Keywords Mathematical Reasoning Inference Rule
Reconstruction Model
Step-Reason Mapping Logic-Integrated Mathematical Expression
ABSTRACT
Logical thinking serves as the backbone of understanding mathematics which helps learners to connect concepts, justify methods, and construct meaningful explanation.
Many students acquire procedural competence to solve a mathematical problem but they remain unable to clearly explain the reasoning that justifies each mathematical step.
This gap between performing mathematical problem and explaining solutions undermines conceptual understanding and hinders the development of higher-order reasoning This research paper addresses this issue by presenting a structured pathway for rebuilding a clear and structured logical framework for mathematics learning.
This proposed framework emphasizes on deliberately integration of symbolic logic and formal inference rules into regular mathematical pedagogy.
The explicit incorporation of logic within problem-solving practices helps learners to move beyond intuitive thinking toward well-justified reasoning.
In support of this objective, this study presents three interrelated approaches: Reasoning Reconstruction Model (RRM).
StepAeReason Mapping (SRM) and Logic-Integrated Mathematical Expression (LIME).
Together, these approaches support students in structuring their thinking, aligning each solution step to logical rules and rebuilding coherent mathematical arguments.
License by CC-BY-SA Copyright A 2026.
The Author.
How to cite: Choudhury.
Strengthening Logical Foundation in Mathematics: A Pathway to Enhance Reasoning Skills, 2.
https://doi.
org/10.
70716/josme.
INTRODUCTION
Logical reasoning serves the foundation of mathematical understanding as it enables learners to justify their methods, connects concepts and develop valid arguments.
Even gaining procedural proficiency, many learners are unable to explain the logical reasoning that justifies each step of the solution.
Hanna & de Villers .
state that proof is fundamental for fostering learnerAos explanatory understanding instead of focusing only on procedural skills.
Similarly.
Stylianides et.
emphasized that inclusion of reasoning and proof into everyday classroom teaching is necessary for developing deep conceptual understanding.
However, research vindicate that many learners tend to emphasize procedural execution while paying less attention to logical justification (Kunth, 2.
Researchers have pointed out that explicit engagement with formal logic improves students reasoning abilities and enhances their mathematical thinking (Simpson & Ingli.
The integration of logical frameworks into mathematics education enhances students ability in analyzing and justifying mathematical claims (Weber, 2.
In addition, developing reasoning skills assists students to build clear logical arguments and improves better understanding of mathematical proofs (Selden et.
, 2.
According to Harel .
, meaningful mathematical understanding requires students to understand the logic behind procedures instead of simply memorizing rules.
Recent studies have highlighted the value of systematic reasoning to support learnerAos mathematical For instance.
Lithner .
made a distinction between imitative and creative reasoning, emphasizing the need to promoting learning approaches based on reasoning.
Similarly.
Maher & Muller .
showed that structured reasoning activities improve students ability in mathematical communication and logical understanding.
Stylianou et al.
also showed that mathematical argumentation is crucial for building students conceptual understanding.
Furthermore, educational research indicates that students frequently struggle to relate symbolic manipulation with proper logical justification (Bieda et al.
, 2.
According to Inglis & Mejia-Ramos .
, the integration of logic into mathematics education enhances clear Journal of Science and Mathematics Education Vol.
No.
March 2026, pp.
reasoning and strengthen students proof skills.
Borwein and Bailey .
stated that mathematical thinking fundamentally based on logical structure and logical organization.
In addition, current educational reforms stress reasoning-focused instruction for foster higher-order cognitive skills (National Council of Teachers Mathematics, 2.
Recent research suggests that teaching reasoning helps learners to develop logically sound solutions and enhances their mathematical confidence (Weber et al.
Inglis & Mejia-ramos, 2.
spite of these advancements, many learners continue to lack structured and systematic support for organizing their reasoning logically.
To address this gap, this paper present three interrelated structured logical framework namely Reasoning Reconstruction Model.
StepAeReason Mapping and Logic-Integrated Mathematical Expression aimed at to strengthen learners coherent mathematical reasoning and logical justification.
Preliminaries This section briefly reviews the fundamental logical and mathematical ideas that provide the theoretical foundation of the proposed framework.
These preliminaries lay the logical foundation needed for incorporating of formal reasoning into mathematical problem solving processes.
1 Propositional Logic A proposition is a declarative statement which has a truth value.
The truth value may be either true (T) or false (F) but not both.
Propositions are generally represented by symbols like ycy, yc, yc, yc.
U U, which are connected using logical connectives such as O .
O .
O .
Ie .
and Ii .
i-conditiona.
Numerous mathematical assertions are formulated as implications which make propositional logic as a powerful tool for identifying assumptions and conclusions within mathematical reasoning.
2 Predicate Logic Predicate logic builds upon propositional logic through the inclusion of predicates, quantifiers and variables in which predicates describes the characteristics of objects while the quantifiers such as universal quantifier (OA) and existential quantifier (OE) define the scope of applicability of statements.
Mathematical definitions and results are formulated using predicate logic.
For example, an integer ycu is even if there exists an integer yco such that ycu = 2yco.
In predicates it can be expressed as OAycu OO ycs .
cu yceycyceycu Ie ycu = 2yco, yceycuyc ycycuycoyce yco OO yc.
This approach allows mathematical assumptions and outcomes to be stated with precision.
Logical Equivalences Two propositions are said to be logically equivalent if they have same truth value under all Some common equivalence are give in the following table:
Table 1: Some Standard Laws of logical equivalences Equivalance ycy O ycN O ycy and ycy O ya O ycy ycyOycN OycN ycyOya Oya ycy O ycy O ycy and ycy O ycy O ycy .
O ycy ycy O yc O yc O ycy and ycy O yc O yc O ycy ycy O .
c O y.
O .
cy O y.
O yc ycy O .
c O y.
O .
cy O y.
O yc ycy O yc O y.
O .
cy O y.
O .
cy O yc ) ycy O .
c O y.
O .
cy O y.
O .
cy O yc ) .
cy O y.
O ycy O yc .
cy O y.
O ycy O yc ycy O .
cy O y.
O ycy ycy O .
cy O y.
O ycy Name of Laws Identity Laws Tautology Law Contradiction Law Idempotent Laws Double Negation Law Communicative Laws Associative Laws Distributive Laws De MorganAos Laws Absorption Laws Choudhury (Strengthening Logical Foundation in Mathematics A) Journal of Science and Mathematics Education Vol.
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March 2026, pp.
ycy O ycy O ya ycaycuycc ycy O ycy O ycN Negation Laws These logical equivalences are often used to rewrite mathematical statements while keeping logical Understanding these equivalences is essential in refining arguments and simplifying logical expression in proofs.
Rules of Inference The Rules of inferences provide the justification of the steps to show that a conclusion follows logically from a set of hypotheses.
Some popular types of Rules of inference in propositional logic are Modus Ponens.
Modus Tollens.
Hypothetical Syllogism.
Universal Instantiation, and Existential which are given in the table 2:
Table 2: Some Standard Rule of Inference Inference Rule .
Modus Ponens Symbolic form Universal Instantiation Premises: p Ie q, p Conclusion: q Premises: p Ie q, q Conclusion: p Premises: p Ie q, q Ier Conclusion: p Ie r Premises: OAycu ycE.
Conclusion: ycE.
Existential Instantiation Premises: OEycu ycE.
Conclusion: ycE.
Modus Tollens Hypothetical Syllogism Explanation From Auif p then qAy and AupAy we infer AuqAy From Auif p then qAy and Aunot pAy we infer Aunot qAy From Auif p then qAy and Auif q then rAy we infer Auif p then rAy From AuP.
is true for all xAy we infer AuP.
is true for a particular element cAy From Authere exists an element x such that P.
is trueAy we infer AuP.
is true for some particular element cAy In mathematical proofs, these inference rules validate that each conclusion logically follows from preceding statements rather then relying on intuition alone.
RESEARCH METHODOLOGY
In this paper, we introduce three new approaches of pedagogical structures to rebuild mathematical learnerAos logical reasoning.
Reasoning Reconstruction Model (RRM) The Reasoning Reconstruction Model can be considered as a mathematical pedagogical framework that can be used to identify, analyze, and rebuild flawed or incomplete mathematical reasoning.
Rather than focusing on the final answer of a mathematical problem.
RRM focuses how a learner thinks and reconstructs correct reasoning step-by-step to solve the problem.
It can encourages learners to articulate their thought processes, thereby can promoting their clarity, precision, and coherence in mathematical In this approach, the whole RRM framework is structured into the following three stages.
Stage 1: Decompose (D): In Decompose stage, the given mathematical problem or statement breaks into its basic logical components.
During this stage, the mathematical problem or statement is reformulated formal logical expression through either propositional logic or predicate logic.
This stage enables learnerAos to clearly identify the underlying assumptions, given conditions and conclusions embedded within the problem.
For example, if we consider a problem:
Choudhury (Strengthening Logical Foundation in Mathematics A) Journal of Science and Mathematics Education Vol.
No.
March 2026, pp.
ycnyce AycuA ycnyc yccycnycycnycycnycaycoyce ycayc 6, ycEayceycu ycu ycnyc yceycyceycu.
Decompose this problem into propositional logic as: Let ycy: ycu ycnyc yccycnycycnycycnycaycoyce ycayc 6 yc: ycu ycnyc yccycnycycnycycnycaycoyce ycayc 2 ycNEayce ycoycuyciycnycaycayco yceycuycyco ycuyce ycEayce ycycycaycyceycoyceycuyc ycnyc ycEayceycyceyceycuycyce, ycyIeyc At Decompose stage, learnerAos can clearly identify the given assumption .
hat is give.
, the required conclusion .
hat is to be prove.
and the logical relationship connecting the propositions .
ow the propositions are logically connecte.
Stage 2: Align (A): The main purpose of Alignment stage is on mapping each step of mathematical solution to a corresponding inference principle or logical rule.
This process ensures that each transformation in the mathematical solution is logically is logically grounded rather than intuitively During this stage, mathematical learners explicitly integrate mathematical operations such as algebraic manipulations, formal definitions like evenness or divisibility, and logical inference rules like Modus Ponens (MP) or Universal Instantiation (UI) etc.
Such alignment makes the reasoning process transparent and logically sound.
As a result learners come to understand not only what sequence of steps are performed in the solution but also into the logical justification behind each step.
The alignment of each step in above problem is shown in the following table:
Mathematical Logical Expression Rule Applied Step Definition of divisibility ycu = 6yco, yco OO ya ycy Universal Instantiation/Algebraic ycu = 2.
yc Even Implication Modus Ponens (MP) Stage 3: Rebuild (R): The Rebuild stage involves reconstructing the entire argument into a wellorganized and logically valid explanation using a combination of mathematical and logical language.
this stage, the output of the Decompose as well as Alignment stages is integrated to form a clear, structured and comprehensive explanation.
The reconstructed explanation emphasizes maintaining a smooth logical flow, ensuring mathematical correctness and providing explicit justification of conclusion.
This step helps mathematics learners to express mathematical reasoning in clearly and rigorously, which is essential for constructing proofs and engaging higher-level mathematical thinking.
Example of Rebuild Explanation : Since n is divisible by 6, then there exists an integer yco such that ycu = 6yco.
This can be written as ycu = 2.
Which demonstrates that n is even.
Therefore by systematically applying algebraic decomposition and logical inference, we can conclude that ycu is even.
Alternatively, since the implication ycy Ie yc is obtained by algebraic decomposition and logical inference, so ycu must be The overall framework of Reasoning Reconstruction Model can be consciously expressed by the formula:
Rebuild Explanation= ya ya ycI Where, ya.
breaks the problem into its logical components, ya involves verifying each steps through logical rules, and ycI(Rebuil.
focuses on combining these steps to a clear and coherent StepAeReason Mapping (SRM) Step-Reason Mapping works as a core component within the Rebuilding Logical Foundation in Mathematics and align strongly with the Reasoning Reconstruction Model(RRM).
The key concept of SRM is that every mathematical step must be accompanied by an explicit logical justification.
Thus each solution steps are systematically paired with the logical rule, definition, or inference rule that confirms its validity.
According to SRM, every mathematical solution must be presented as a sequence of collection of stepreason pairs as (SRM Forma.
UycI1 , ycI1 U.
UycI2 , ycI2 U.
U U U U .
UycIycu , ycIycu U Where, ycIycn represents the ycn-th step in the solution process, while ycIycn denotes the logical justification or rule for that step.
Such a mapping guarantees that all solution steps of a mathematical problem are Choudhury (Strengthening Logical Foundation in Mathematics A) Journal of Science and Mathematics Education Vol.
No.
March 2026, pp.
firmly based on valid logical rule such as rule of inference, definitions, axioms, equivalence transformations or previously proven results.
For example, solving a mathematical statement using SRM:
Statement "ycnyce yca ycuycycoycayceyc ycu ycnyc yceycyceycu ycEayceycu ycu2 ycnyc yceycyceycu" Step-Mapping Reason table:
Steps .
cIycn ) Mathematical Action Reason.
cIycn ) & Logical Rule ycI1 Given, ycu ycnyc yceycyceycu Given assumption ycu 2yco, yceycuyc ycycuycoyce yco ycI2 Definition of even numbers ycI3 ycu2 = .
2 Substitution rule ycI4 ycu = 4yco Algebraic simplification ycI5 ycu2 = 2.
yco 2 ) Factorization ycI6 Definition of even numbers ycu ycnyc yceycyceycu ycI7 Conclusion holds Logical inference (MP) SRM makes a clear distinction by separating assumptions, definitions, transformations and conclusions thereby helps to eliminate the common misconception that algebraic operations alone amount to proof.
Rather it emphasizes the idea that a proofs are built in a structured sequence of steps supported by logical Logic-Integrated Mathematical Expression (LIME) LIME is a formal reasoning approach aimed at rebuilds the logical foundation of mathematics through the integration of logical inference rules at each step of mathematical reasoning.
In conventional mathematical practices, the justification of transitions between expressions is often left unstated, which can obscure the logical flow of proofs and promote procedural thinking rather than conceptual LIME method mitigates this issue by framing mathematical reasoning as an explicitly structured sequence in which every transaction is validated by a named logical rule, definition or inference A Logic-Integrated Mathematical expression can be formally expresses as a sequence like:
ya1 ya2 yaycu ycA1 Ne ycA2 Ne U U U U Ne ycAycu 1 yaycn Where ycAycn are mathematical statements and the notation Ne denotes that the transition from ycAycn to ycAycn 1 is logically justified by the rule yaycn .
For example, to show that if "ycuA = 25, then x = 5 or x = Oe5 ycycycycaycyce ycycuycuyc ycoycayc ycuA = 25 Ne OOeIntroduction ycu = A5 Ne .
cu = .
O .
cu = Oe.
Through this structured representation, students grasp not only the mathematical result but also the logical principles that justify every transition, strengthening them both mathematical understanding and logical reasoning skills simultaneously.
RESULTS AND DISCUSSION
Illustrative Example Using the New Frameworks Let us consider the following problem AuShow that if an integer n is divisible by 4 and n is odd, a contradiction occurs.
Ay 1 Reasoning Reconstruction Model (RRM) RRM focuses on the systematic construction og reasoning rather than merely checking the correctness of final result.
In this approach the solution is developed through three organized stages namely Decompose.
Align, and Rebuild.
Choudhury (Strengthening Logical Foundation in Mathematics A) Journal of Science and Mathematics Education Vol.
No.
March 2026, pp.
Stage 1: Decompose: Let ycy: "ycu ycnyc yccycnycycnycycnycaycoyce ycayc 4" yc: "ycu ycnyc ycuyccycc" Using standard definitions:
From ycy: ycnyce ycu ycnyc yccycnycycnycycnycaycoyce ycayc 4 ycEayceycu ycu = 4yco, yceycuyc ycycuycoyce ycnycuycyceyciyceyc yco.
From yc: ycnyce ycu ycnyc ycuyccycc ycEayceycu ycu = 2yco 1, yceycuyc ycycuycoyce ycnycuycyceyciyceyc yco.
To examine the conjunction ycy O yc and show that it leads to a conclusion.
ycy O yc NeOu .
Stage 2: Align: During this stage, every mathematical transformations is clearly justified by an apporiate logical rule, definitions, or principle of inference.
Mathematical Step ycu = 4yco, yco OO ya ycu = 2.
ycu is even ycu = 2yco 1 ycu is even and odd Contradiction A .
cy O y.
Logical Expression Rule Applied ycy Oe Oe yc Even O odd Ou Oe Definition of divisibility by 4 Algebraic manipulation Definition of even numbers Definition of odd numbers Conjunction Mutual exclusivity of even and odd Negation introduction Stage 3: Rebuild: In this stage, the decomposed components and their corresponding aligned steps are assembled into a clear and logically valid argument.
Rebuild Explanation: Assume that integer ycu is divisible by 4.
Then ycu = 4yco, for some integer yco, which can be expressed as ycu = 2.
which establishes that ycu is even .
y definitio.
Simultaneously assume that, ycu is odd.
Thus, ycu is simultaneously both even and odd, which is impossible.
Thus, the assumption that ycu is simultaneously both even and odd leads to a contradiction, consequently, the conjunction ycy O yc cannot hold.
Step-Reason Mapping (SRM) By distinctly separating assumptions, algebraic manipulations, definitions, and logical inferences, the SRM framework emphasizes that proof is a logically justified steps rather than computational Step-Mapping Reason table:
Steps .
cIycn ) ycI1 ycI2 ycI3 ycI4 ycI5 ycI6 ycI7 ycI8 ycI9 Mathematical Action Given, ycu ycnyc yccycnycycnycycnycaycoyce ycayc 4 ycu = 4yco, yceycuyc ycycuycoyce ycnycuycyceyciyceyc yco ycu = 2.
ycu ycnyc yceycyceycu yaycnycyceycu ycu ycnyc ycuyccycc ycu = 2yco 1 ycu ycnyc yceycyceycu ycaycuycc ycuyccycc A .
cy O y.
Reason.
cIycn ) & Logical Rule Given assumption .
Definition of divisibility Algebraic transformation Definition of even numbers Given assumption Definition of odd numbers Logical conjunction Even and odd are mutually exclusive Logical inference Logic-Integrated Mathematical Expression (LIME) LIME represents the reasoning process as a connected series of mathematically stated transitions, where every transitions is labeled by an explicit logical justification.
Choudhury (Strengthening Logical Foundation in Mathematics A) Journal of Science and Mathematics Education Vol.
No.
March 2026, pp.
yaycoyciyceycaycyca ycu = 4yco Ne yayceyceycnycuycnycycnycuycu ycu = 2.
Ne yayceyceycnycuycnycycnycuycu ycu = 2yco 1 Ne yaycuyciycnycaycayco ycoycayc Even O odd Ne ycu ycnyc yceycyceycu ycu ycnyc ycuyccycc Ou .
The final logical conclusion becomes:
ycy O yc NeOu and hence A .
cy O y.
Discussion The proposed framework (RRM.
SRM.
LIME) collectively contribute in rebuilding logical foundation in They make the implicit reasoning processes explicit that are often hidden in conventional mathematical practices.
By merging symbolic logic with mathematical expression, these three frameworks help learners to establish clear connections between mathematical computation with formal reasoning.
They offer structured guidance to learners through a systematically and well-organized reasoning process.
As a result this structured approach reduces learners overreliance on memorized techniques as well as mechanical methods and encourages meaningful understanding.
Furthermore, these frameworks provide learners with clear representational tools to articulate and communicate mathematical justification with clarity and precision.
CONCLUSION
Rebuilding logical foundation in mathematics learners is fundamental for strengthening their strong reasoning skills.
Structured teaching methods grounded in logical rules and symbolic expressions helps learners to develop clear and more consistent thinking.
The models like RRM.
SRM and LIME strengthens this structured reasoning process.
These approaches enables learners to understand not just what steps to take in mathematical procedure, but why these steps are logically valid.
As a result, teachers gain a powerful and systematic framework for integrating formal logical reasoning into regular mathematics teaching.
REFERENCES