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Early algebraic reasoning can be viewed as developing a bridge between arithmetic and algebra. Accordingly, this research examines how middle-grades students' arithmetic reasoning, classified by their number sequences, can be used to model their algebraic reasoning as it pertains to generalizing, writing, and solving linear equations and systems of equations. In the quantitative phase of research, 326 students in grades six through nine completed a survey to assess their number sequence construction. In the qualitative phase, 18 students participated in clinical interviews, the purpose of which was to elicit their algebraic reasoning. Results show that the numbers of students who had constructed the two least sophisticated number sequences did not change significantly across grades six through nine. In contrast, the numbers of students who had constructed the three most sophisticated number sequences did change significantly from grades six and seven to grades eight and nine. Furthermore, students did not consistently reason algebraically unless they had constructed at least the fourth number sequence. Thus, it is concluded that students with the two least sophisticated number sequences are no more prepared to reason algebraically in ninth grade than they were in sixth. / Doctor of Philosophy / Early algebraic reasoning can be viewed as developing a bridge between arithmetic and algebra. This study examines how students in grades six through nine reason about numbers, and whether their reasoning about numbers can be used to explain how they reason on algebra tasks. Particularly, the students were asked to extend numerical patterns by writing algebraic expressions, and were asked to read contextualized word problems and write algebraic equations and systems of equations to represent the problems. In the first phase of research, 326 students completed a survey to assess their understanding of numbers and their ability to reason about numbers. In the second phase, 18 students participated in interviews, the purpose of which was to elicit their algebraic reasoning. Results show that the numbers of students who had constructed a more sophisticated understanding of number did not change significantly across grades six through nine. In contrast, the numbers of students who had constructed a less sophisticated understanding of number did change significantly from grades six and seven to grades eight and nine. Furthermore, students were not consistently successful on algebraic tasks unless they had constructed a more sophisticated understanding of number. Thus, it is concluded that students with an unsophisticated understanding of number are no more prepared to reason algebraically in ninth grade than they were in sixth.

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Doctor of Philosophy / Department of Mathematics / Todd E. Cochrane / Let $p$ be an odd prime and $\gamma(k,p^n)$ be the smallest positive integer $s$ such that every integer is a sum of $s$ $k$-th powers $\pmod {p^n}$. We establish $\gamma(k,p^n) \le [k/2]+2$ and $\gamma(k,p^n) \ll \sqrt{k}$ provided that $k$ is not divisible by $(p-1)/2$. Next, let $t=(p-1)/(p-1,k)$, and $q$ be any positive integer. We show that if $\phi(t) \ge q$ then $\gamma(k,p^n) \le c(q) k^{1/q}$ for some constant $c(q)$. These results generalize results known for the case of prime moduli. Next we generalize these results to a number field setting. Let $F$ be a number field, $R$ it's ring of integers and $\mathcal{P}$ a prime ideal in $R$ that lies over a rational prime $p$ with ramification index $e$, degree of inertia $f$ and put $t=(p^f-1)/(p-1,k)$. Let $k=p^rk_1$ with $p\nmid k_1$ and $\gamma(k,\mathcal{P}^n)$ be the smallest integer $s$ such that every algebraic integer in $F$ that can be expressed as a sum of $k$-th powers $\pmod{\mathcal{P}^n}$ is expressible as a sum of $s$ $k$-th powers $\pmod {\mathcal{P}^n}$. We prove for instance that when $p>e+1$ then $\gamma(k,\mathcal{P}^n) \le c(t) p^{nf/ \phi(t)}$. Moreover, if $p>e+1$ we obtain the upper bounds $\ds{\gamma(k,\mathcal{P}^n) \le 2313 \left(\frac{k}{k_1}\right)^{8.44/\log p}+\frac{1}{2}}$ if $f=2$ or $3,$ and $\ds{\gamma(k,\mathcal{P}^n)\le 129 \left(\frac{k}{k_1}\right)^{5.55/ \log p}+\frac{1}{2}}$ if $f\ge4$. We also show that if $\mathcal{P}$ does not ramify then $\ds{\gamma(k,\mathcal{P}^n) \le \frac{17}{2} \left(\frac{k}{k_1}\right)^{2.83/ \log p}+\frac{1}{2}}$ if $f\ge 2$ and $k_1\le p^{f/2}$, and $\ds{\gamma(k,\mathcal{P}^n)\le\left(\frac{f}{p^{f/2-1}}\right)k}$ if $f> 2$ and $k_1> p^{f/2}$.

published_or_final_version / Computer Science / Master / Master of Philosophy