Developing Primary Students’ Understanding of Mathematics through Mathematization: A Case of Teaching the Multiplication of Two Natural Numbers

<p style="text-align: justify;">Numeracy is one of the essential competencies that the objectives of teaching math to primary students should be towards. However, many research findings show that the problem of “innumeracy” frequently exists at primary schools. That means children still do not feel at home in the world of numbers and operations. Therefore, the paper aims to apply the realistic mathematics education (RME) approach to tackling the problem of innumeracy, in the case of teaching the multiplication of two natural numbers to primary students. We conducted a pedagogical experiment with 46 grade 2 students who have not studied the multiplication yet. The pedagogical experiment lasted in six lessons, included seven activities and nine worksheets which are designed according to fundamental principles of RME by researchers. This is mainly a qualitative study. Based on data obtained from classroom observations and students’ response on worksheets, under the perspective of RME, the article pointed out how mathematization processes took place throughout students' activities, their attitudes towards math learning, and their learning outcomes. The study results found that students were more interested in math learning and understood the concepts of multiplication of two natural numbers.</p>

A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function

The completed zeta function $\xi(s)$ is expanded in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) of quadratic factors by its complex conjugate zeros $\rho_i=\alpha_i +j\beta_i, \bar{\rho}_i=\alpha_i-j\beta_i, 0&lt;\alpha_i&lt;1, \beta_i\neq 0, i\in \mathbb{N}$ are natural numbers from 1 to infinity, $\rho_i$ are in order of increasing $|\rho_i|=\sqrt{\alpha_i^2+\beta_i^2}$, i.e., $|\rho_1|&lt;|\rho_2|\leq|\rho_3|\leq |\rho_4|, \cdots$, together with $\beta_1&lt;\beta_2\leq\beta_3\leq\beta_4, \cdots$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\xi(0)\prod_{i\in \mathbb{N}}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)} =\xi(0)\prod_{i\in \mathbb{N}}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(1-s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}$$ which, by Lemma 3, is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}, \text{from 1 to infinity.}$$ with only valid solution $\alpha_i= \frac{1}{2}$ (another solution $s=\frac{1}{2}$ is invalid due to obvious contradiction). Thus, a proof of the Riemann Hypothesis is achieved.

Short Note on the Riemann Hypothesis

Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)<e^{\gamma }\times n \times\log\log n$ holds for all natural numbers $n>5040$, where $\sigma(n)$ is the sum-of-divisors function of $n$ and $\gamma\approx0.57721$ is the Euler-Mascheroni constant. Let $q_{1}=2,q_{2}=3,\ldots,q_{m}$ denote the first $m$ consecutive primes, then an integer of the form $\prod_{i=1}^{m}q_{i}^{a_{i}}$ with $a_{1} \geq a_{2} \geq \cdots \geq a_{m} \geq 0$ is called an Hardy-Ramanujan integer. If the Riemann hypothesis is false, then there are infinitely many Hardy-Ramanujan integers $n>5040$ such that Robin inequality does not hold and we prove that $n^{\left(1-\frac{0.6253}{\log q_{m}}\right)}<N_{m}$, where $N_{m}=\prod_{i =1}^{m}q_{i}$ is the primorial number of order $m$ and $q_{m}$ is the largest prime divisor of $n$. In addition, we show that $q_{m}$ will not have an upper bound by some positive value for these counterexamples and therefore, the value of $q_{m}$ tends to infinity as $n$ goes to infinity.

Short Note on the Riemann Hypothesis

Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)<e^{\gamma }\times n \times\log\log n$ holds for all natural numbers $n>5040$, where $\sigma(n)$ is the sum-of-divisors function of $n$ and $\gamma\approx0.57721$ is the Euler-Mascheroni constant. Let $q_{1}=2,q_{2}=3,\ldots,q_{m}$ denote the first $m$ consecutive primes, then an integer of the form $\prod_{i=1}^{m}q_{i}^{a_{i}}$ with $a_{1} \geq a_{2} \geq \cdots \geq a_{m} \geq 0$ is called an Hardy-Ramanujan integer. If the Riemann hypothesis is false, then there are infinitely many Hardy-Ramanujan integers $n>5040$ such that Robin inequality does not hold and we prove that $n^{\left(1-\frac{0.6253}{\log q_{m}}\right)}<N_{m}$, where $N_{m}=\prod_{i =1}^{m}q_{i}$ is the primorial number of order $m$ and $q_{m}$ is the largest prime divisor of $n$. In addition, we show that $q_{m}$ will not have an upper bound by some positive value for these counterexamples and therefore, the value of $q_{m}$ tends to infinity as $n$ goes to infinity.

A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function

The completed zeta function $\xi(s)$ is expanded in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) of quadratic factors by its complex conjugate zeros $\alpha_i\pm j\beta_i, \beta_i\neq 0, i\in \mathbb{N}$ are natural numbers, from $1$ to infinity, $\mathbb{N}$ is the set of natural numbers. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)} =\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}$$ which, by Lemma 3 and Corollary 1, is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$$ with solution $\alpha_i= \frac{1}{2}, i\in \mathbb{N}$ (another solution $s=\frac{1}{2}$ is invalid due to obvious contradiction). Thus, a proof of the Riemann Hypothesis is achieved.

A Proof of the Riemann Hypothesis Based on A New Expression of the Completed Zeta Function

The completed zeta function $\xi(s)$ is expanded in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) of quadratic factors by its complex conjugate zeros $\alpha_i\pm j\beta_i, \beta_i\neq 0, i\in \mathbb{N}$ ($\mathbb{N}$ is the set of natural numbers, from $1$ to infinity). Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have $$\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)} =\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}$$ which, by Lemma 3 and Corollary 1, is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$$ with solution $\alpha_i= \frac{1}{2}, i\in \mathbb{N}$ (another solution $s=\frac{1}{2}$ is invalid due to obvious contradiction). Thus, a proof of the Riemann Hypothesis is achieved.

Proof of the Riemann Hypothesis

The Riemann hypothesis has been considered the most important unsolved problem in mathematics. Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function of $n$ and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We show that the Robin inequality is true for all natural numbers $n > 5040$ which are not divisible by the prime $3$. Moreover, we prove that the Robin inequality is true for all natural numbers $n > 5040$ which are divisible by the prime $3$. Consequently, the Robin inequality is true for all natural numbers $n > 5040$ and thus, the Riemann hypothesis is true.

Demonstration of a Technique to Construct a One-to-One Correspondence Between N and the infinite binary decimals in (0, 1).

In this paper we will see how by varying the initial conditions of the demonstration we can use Cantor’s method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the open interval (0, 1).

Arguments in Favor of the Riemann Hypothesis

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. This problem has remained unsolved for many years. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)<e^{\gamma } \times n \times \log\log n$ holds for all natural numbers $n>5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q\leq q_{n}}\frac{q}{q-1} >e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}>2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show some arguments in favor of the Riemann hypothesis is true.