Teaching through the Mathematical Processes Session 5: Assessing with the Mathematical Processes Retention Timeline 0 2 4 6 years Pink: time students retain knowledge through procedures Yellow: time students retain knowledge through concepts Blue: time students retain knowledge through mathematical processes 8 10 What are we teaching? • Reflect on the relevance of the video to retention and life-long learning. • Share your ideas with a partner. Assessing the Processes Assessment - Evaluation - Reporting What is Classroom Assessment? One possible answer: Classroom Assessment refers to the collection of information teachers use to monitor students’ learning, provide feedback and to make appropriate adjustments to instruction. Exploring Classroom Assessment in Math NCTM 1998 Name That Process… • Identify the Mathematical Process that can be assessed for the given criteria. • Share your choice with a partner. Mathematics Processes Rubric Think →Tools Pair → ShareStrategies Selecting Reasoning Problem Communicating and Representing Connecting Reflecting Computational and Solving Proving Criteria Criteria Criteria Criteria Criteria Criteria Criteria Creates a model to represent the problem Uses clear language to make presentations, and to explain Formulates and defends a hypothesis or conjecture Makes connections among mathematical concepts and Selects, sequences and applies mathematical processes Uses metacognitive skills to determine which mathematical (e.g., numerical, algebraic, graphical, physical, or scale and justify solutions when reporting for various purposes procedures appropriate touses the processes to revisit in order to reach the model, by hand or task using technology) Selects and tools and strategies to goal solve problem and different audiences Makes inferences, draws conclusions and givesajustifications Relates mathematical ideastotosolve situations drawn from other Uses critical thinking skills a problem Reflects on the reasonableness of answers Makes connections between numeric, graphical and Uses mathematical symbols, labels, units and conventions contexts Interpretsrepresentations mathematical language, charts, and graphs algebraic correctly Uses mathematical appropriately Translates from onevocabulary representation to another as appropriate to the problem Generic Processes Rubric BLM5.1 Thinking Problem Solving Criteria Below Level 1 Level 1 Level 2 Level 3 Level 4 Specific Feedback Selects, sequences and applies mathematical processes appropriate to the task Selects, sequences and applies mathematical processes to the assigned task with significant prompting Selects, sequences and applies mathematical processes to the assigned task with minimal prompting Selects, sequences and applies mathematical processes to the assigned task independently Selects, sequences and applies mathematical processes to the assigned task independently with a broader view of the task Uses critical thinking skills to solve a problem Uses minimal logic and precision in mathematical reasoning to solve problems Uses logic to solve problems but lacks precision in mathematical reasoning Solves problems logically and with precision in mathematical reasoning Demonstrates a sophisticated level of mathematical reasoning and precision in solving problems Reasoning and Proving Criteria Below Level 1 Specific Feedback Level 1 Level 2 Level 3 Level 4 Formulates and defends a hypothesis or conjecture Forms a hypothesis or conjecture that connects few aspects of the problem Forms a hypothesis or conjecture that connects some of the pertinent aspects of the problem Forms a hypothesis or conjecture that connects pertinent aspects of the problem Forms a hypothesis or conjecture that connects aspects of the problem with a broader view of the problem Makes inferences, draws conclusions and gives justifications Makes limited connections to the problem-solving process and models presented when justifying answers Makes some connections to the problem-solving process and models presented when justifying answers Makes direct connections to the problem-solving process and models presented when justifying answers Makes direct and insightful connections to the problemsolving process and models presented when justifying answers Interprets mathematical language, charts, and graphs Misinterprets a critical element of the information, but makes some reasonable statements Misinterprets part of the information, but makes some reasonable statements Interprets the information correctly and makes reasonable statements Interprets the information correctly, and makes insightful statements Opposite Sides Agree / Disagree / Don’t Know! The distance around a tennis ball can is less than the height of the can. Justify your answer with your group. Those who ‘don’t know’ explore further. = 3 diameters h = 3 balls C = diameters The distance around a tennis ball can is less than the height of the can. Tennis Ball Can Problem • What Mathematical Processes were used in solving the problem? • Which criteria could be used to assess the Mathematical Processes with this problem? How do we effectively teach Mathematical Processes? • Teach to the curriculum expectations. • Do investigations and solve rich problems. • Ask questions and provide feedback related to the Mathematical Processes. • Explore the TIPS lessons that demonstrate how lessons can be adjusted to focus on particular Mathematical Processes. Students learn content and problem solving by solving problems and sharing solutions Problem Selection … Choose one of the following problems to form working groups. Volume of Three Dimensional Shapes Develop, through investigation (e.g., using concrete materials) the formulas for the volume of a pyramid, a cone, and a sphere. Painted Cube Problem A 3 x 3 x 3 cube made up of small cubes is dipped into a bucket of red paint and removed. a) How many small cubes will have 3 faces painted? b) How many small cubes will have 2 faces painted? c) How many small cubes will have 1 face painted? d) How many small cubes will have 0 faces painted? e) Generalize your results for an n x n x n cube. Temperature Problem The inhabitants of Xenor use two scales for measuring temperature. On the A scale, water freezes at 0° and boils at 80°, whereas on the B scale, water freezes at -20° and boils at 120°. What is the equivalent on the A scale of a temperature of 15° on the B scale? The Math Forum@Drexel All rights Reserved. Dart Board Problem This dart board is designed with a square inside a circle and a square outside the same circle. Assign numerical values of 2, 5, and 8 to the three coloured regions on the dart board such that regions with smaller areas are assigned higher scores. Justify your solution. Minds On: Decking Problem Deck Problem You have been tohired build a adeck attached Youhired have been to build deck attached to the to secondof flooraofcottage a cottage using exactlyexactly 30 m of deck30m railing. the second floor using (Note: the entire outside edge ofoutside the deck willedge have railing.) of deck railing (note: the entire will Determine the dimensions that will maximize the area of have railing).the deck with the configuration below. І http://www.beachside-bb.nf.ca/Accomdations.htm DECK = = І Determine the dimensions of the deck that follow the specifications in the diagram and maximize the area of the deck. COTTAGE Combination of Functions Card Game Examples The initial graph of sin(x) and 2x can be combined to produce the graphs shown below it. Determine what operations are used to combine them and explain the reasoning. sin x and 2x Problem Selection … Consider your group problem to answer the question: “Which Mathematical Process(es) could be assessed?” Assessing the Mathematical Fishbowl Processes Strategy • Half of the group members “solve” the problem. • The remaining members observe and assess the Mathematical Processes, using the given rubric. • The observing members identify other Mathematical Processes that become apparent during the problem solving. • Discuss the solution strategy and observations. • Group members switch roles to solve the same or a different problem. Debrief • How did solving this problem provide opportunities for you to apply Mathematical Processes • How did the rubric facilitate assessing the Mathematical Processes specifically? • What other observations did you make? Home Activity Journal Reflection: What role will the Generic Rubric for Mathematical Processes play in my teaching and assessment practices?