## Mathematics for Grades 4 and 8: International Benchmarks

Exhibit 1. Description of TIMSS international mathematics benchmarks, by grade: 2015
SOURCE: Mullis, I.V.S., Martin, M.O., Foy, P., and Hooper, M. (2016). TIMSS 2015 International Results in Mathematics. Retrieved from http://timssandpirls.bc.edu/timss2015/international-results/.
Advanced (625) Students can apply their understanding and knowledge in a variety of relatively complex situations and explain their reasoning. They can solve a variety of multistep word problems involving whole numbers. Students at this level show an increasing understanding of fractions and decimals. They can apply knowledge of a range of two and three-dimensional shapes in a variety of situations. They can interpret and represent data to solve multistep problems. Students can apply and reason in a variety of problem situations, solve linear equations, and make generalizations. They can solve a variety of fraction, proportion, and percent problems and justify their conclusions. Students can use their knowledge of geometric figures to solve a wide range of problems about area. They demonstrate understanding of the meaning of averages and can solve problems involving expected values.
High (550) Students can apply their knowledge and understanding to solve problems. They can solve word problems involving operations with whole numbers, simple fractions, and two-place decimals. Students demonstrate an understanding of geometric properties of shapes and angles that are less than or greater than a right angle. Students can interpret and use data in tables and a variety of graphs to solve problems. Students can apply their understanding and knowledge in a variety of relatively complex situations. They can use information to solve problems involving different types of numbers and operations. Students at this level show basic procedural knowledge related to algebraic expressions. They can solve a variety of problems with angles including those involving triangles, parallel lines, rectangles, and similar figures. Students can interpret data in a variety of graphs and solve simple problems involving outcomes and probabilities.
Intermediate (475) Students can apply basic mathematical knowledge in straightforward situations. They demonstrate an understanding of whole numbers and some understanding of fractions and decimals. Students can relate two- and three-dimensional shapes and identify and draw shapes with simple properties. They can read and interpret bar graphs and tables. Students can apply basic mathematics knowledge in straightforward situations. They can solve problems involving negative numbers, decimals, percentages, and proportions. Students have some knowledge of linear expressions and two- and three-dimensional shapes. They can read and interpret data in graphs and tables. They have some basic knowledge of chance.
Low (400) Students have some basic mathematical knowledge. They can add and subtract whole numbers, have some understanding of multiplication by one-digit numbers, and can solve simple word problems. They have some knowledge of simple fractions, geometric shapes, and measurement. Students can read and complete simple bar graphs and tables. Students have some knowledge of whole numbers and basic graphs. The few items at this level provide some evidence that students have an elementary understanding of whole numbers. They can match tables to bar graphs and pictographs.