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TExES Math 235 Prep, Part 6: Mathematical Learning, Instruction and Assessment

Part 6, the final part of the TExES Mathematics 7-12 (235) refresher series (see Part 5: Mathematical Processes and Perspectives), covering how students learn mathematics, instructional design, and assessment (Competencies 020-021).

  • Describe the Van Hiele levels of geometric thinking and the Concrete-Representational-Abstract sequence.
  • Use prior knowledge and common misconceptions to inform instructional planning and differentiation.
  • Distinguish formative, summative, and diagnostic assessment.
  • Distinguish validity from reliability, and weigh selected-response against constructed-response items.

Mathematics Learning and Instruction

The Van Hiele levels describe how geometric thinking develops in order: visualization, analysis, informal deduction, formal deduction, and rigor. Instruction should match a student's current level rather than skip ahead. The Concrete-Representational-Abstract (CRA) sequence moves from manipulatives, to pictures or diagrams, to symbols, and is especially useful for students struggling with abstract notation.

Prior knowledge and common misconceptions should shape instructional planning, for example pre-assessing before a fractions unit to target known error patterns. Differentiation, through scaffolding, tiered tasks, and varied representations, reaches students at different readiness levels, including English learners and students with disabilities. Instruction should be TEKS-aligned and build toward grade-level standards through a logical progression from prior grades, known as vertical alignment.

Assessment

Formative assessment is ongoing and low-stakes, used to adjust instruction in real time (exit tickets, questioning, observation). Summative assessment evaluates learning at the end of a unit or course (unit tests, final exams) and is used for grading or reporting rather than adjusting instruction in the moment. Diagnostic assessment happens before instruction, to identify prior knowledge or misconceptions and inform planning.

Validity means an assessment measures what it claims to measure; reliability means it produces consistent results. Selected-response items (multiple choice) are efficient to score but may not reveal reasoning, while constructed-response and performance tasks reveal process and reasoning but take longer to score, so a balanced assessment plan uses both. Disaggregating results by item or standard, rather than just looking at an overall score, is what actually identifies specific reteaching needs.

Exercises

  1. A teacher gives a quick "thumbs up, thumbs down" check partway through a lesson to decide whether to reteach a concept before moving on. This is an example of: A) Summative assessment   B) Diagnostic assessment   C) Formative assessment   D) Standardized assessment
  2. Before starting a unit on solving equations, a teacher gives a short quiz to see which prerequisite skills students already have. This is: A) Formative assessment   B) Diagnostic assessment   C) Summative assessment   D) Peer assessment
  3. According to the Concrete-Representational-Abstract (CRA) framework, after students work with physical manipulatives, the next step should be: A) Move directly to abstract symbols   B) Use pictorial or visual representations before symbols   C) Skip to a timed test   D) Repeat the concrete stage indefinitely
  4. An assessment consistently gives similar results when the same student takes similar versions of it under similar conditions. This describes: A) Validity   B) Reliability   C) Differentiation   D) Formative feedback

  • 1. C. It's ongoing, happens in the moment, and is used to adjust instruction immediately.
  • 2. B. It's given before instruction to inform planning.
  • 3. B. Pictorial or visual representations come before abstract symbols in the CRA sequence.
  • 4. B. Consistency of results describes reliability, while validity describes whether the assessment measures what it's meant to measure.

That wraps this 6-part refresher series for the TExES Mathematics 7-12 (235) exam. If you worked through all six domains, from number concepts through mathematical learning and assessment, you've touched every competency on the official blueprint. Good luck on test day.

Want to review from the start? Back to Part 1: Number Concepts.

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