All Grades: 'Novice', 'Apprentice', and 'Expert' Tasks

Balancing assessment of content standards and practices

The goal of mathematics instruction is to equip students with the tools of the trade (mathematical practices, content, and productive dispositions) so that they can successfully engage with complex problems. Thus the goals for curricula, and assessment, should be to have students develop such skills and understandings, and to asses those competencies, on rich problems. However, the pathways to such competencies and their assessments are not linear. Some skills may be best motivated, or developed, in the context of working rich problems; the idea of not engaging students in rich mathematics until they have "mastered" all the relevant skills can be intellectually problematic and and deadly in curricular terms. For purposes of assessment, however, it is often useful to judge students' understandings in expanding levels of complexity. Thus, mathematical skills and practices might be assessed partly in isolation, partly under scaffolded conditions, and partly when students face substantial problems without scaffolded support.

In the Mathematics Assessment Project we have classified tasks that assess these three different types of performance novice, apprentice, and expert tasks respectively. A balanced test should combine all three types of task.


Short items focused on specific content or skills.

Novice tasks are short items, each focused on a specific concept or skill, as set out in the Common Core State Standards. They involve only two of the CCSS mathematical practices (MP2 – reason abstractly and quantitatively; MP6 – attend to precision), and do so only at the comparatively low level that short items allow.

Apprentice: Substantial tasks, structured to ensure that all students have access to the problem.

Apprentice tasks are substantial, often involving several aspect of mathematics, and structured so as to ensure that all students have access to the problem. Students are guided through a “ramp” of increasing challenge to enable them to show the levels of performance they have achieved. While any of the CCSS mathematical practices may be required, these tasks especially feature MP2, MP6 and two others (MP3 – construct viable arguments and critique the reasoning of others; MP7 – look for and make use of structure). Because the structure guides the students, the mathematical practices involved are at a comparatively modest level.

Expert: Rich, less structured tasks requiring strategic problem-solving skills as well as content knowledge.

Expert tasks are rich tasks, each presented in a form in which it might naturally arise in applications. They require the effective use of problem solving strategies, as well as concepts and skills. Performance on these tasks indicates how well a person will be able to do and to use mathematics beyond the mathematics classroom. They demand the full range of mathematical practices, as described in the Common Core State Standards, including: MP1 – make sense of problems and persist in solving them; MP4 – model with mathematics; MP5 – use appropriate tools strategically; MP8 – look for and express regularity in repeated reasoning.