Motivational constructs and students' engagements have great impacts on students' mathematics achievements, yet they have not been theoretically investigated using international large-scale assessment data. This study utilized the mathematics data of the Trends in International Mathematics and Science Study 2011 to conduct a comparative and empirical study on exploring: (1) the changes of motivational constructs from 4th grade to 8th grade; and (2) the effects of motivational constructs from the expectancy-value model and students' engagements on mathematics achievements. The countries investigated include Chinese Taipei, Singapore, and the USA. The results showed that: (1) students' motivations deteriorate over school years; (2) self-confidence in mathematics (SCM) has the strongest relationship with mathematics achievements. Furthermore, it is evident that Singapore has the most effective schools in students' mathematics education. More findings of this comparative study are subsequently discussed.