Notes from Grade 3 - 5 Math Teaching and Assessing for Mastery Workshop
Bethany Center, Diocese of St. Petersburg and Catapult Learning
October 16, 2014
Skill Fluency and Deep Understanding - balance between these through coherence (conceptual velcro)
"Understanding is not just about coverage of knowledge...but about 'uncoverage' - being introduced to new ideas and being asked to think more deeply and more carefully about facts, ideas, experiences, and theories previously encountered and learned" (Grant Wiggins).
Approaching 13 x 7 with deep understanding that recognizes place value, multiplication as repeated addition, multiplying 7 x 10, not 7 x 1 in the "tens column"
Multiplying is totaling equal groups, a number's digits represent different place values
-Key steps: multiply via place value
Teach with fidelity for deeper understandings
What does deep understanding really mean?
-ability to justify why a particular mathematical statement is true or where a mathematical rule comes from (from CCSS)
What happens to students without deep understanding?
-they memorize more than they need to
-students can't apply
-"jumping through hoops" to get the answer / receive the grade
-"fed" steps as opposed to discover how to do things themselves
-use approaches in situations where they don't actually apply
-lose sight of the beauty of mathematics
Deep Understanding can lead to:
-appreciation of beauty in the math world
-transfer to applications
-support for later learning
Conceptual velcro - making connections between and among knowledge and skills to deepen understandings
-instruction must support the connection of ideas and rooting knowledge to other pieces of information
Coherence - connect related ideas, connect facts and skills to key concepts, memorize for application
Direct Instruction (I do), Guided Practice (We do), Independent Work (You do)
Mathematical practices - by CCSS math lessons should have as many of the mathematical practices as possible:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
-These Mathematical Practices describe what mathematicians do
-Get students to see themselves as mathematicians
-See ourselves in this way, too
-These MP standards usually "live" together and are intertwined
-Connection to Literacy standards about reading and writing (writers / authors)
-Valuable skills in math and life
Resource for Mathematical Practices Posters:
http://elemmath.jordandistrict.org/mathematical-practices-by-standard/
Take time throughout lessons to allow students to reflect on their learning
-slow down and afford them time to reflect on their understandings / misunderstandings / cloudy spots
-give students time to self-assess
How do you assess depth of understanding?
-it does not need to be a formal test
-Webb's Depth of Knowledge components (DOK)
1. Recollection and reproduction
*recollection of a fact, term, principle, concept OR performance of a routine or procedure that can be learned through practice
2. Basic application of skills / concepts
*Using information or concepts to choose a procedure, make a decision, solve a routine problem, organize and interpret information, or make a simple decision
3. Strategic thinking
*Use deeper understanding and analysis in abstract, complex, non-routine contexts to reason, draw conclusions, develop arguments, make decisions or develop a plan
4. Extended application and thinking
*An investigation or application of deep understanding requiring time to research, integrate knowledge, process multiple conditions, solve problems, draw conclusions
Webb's Depth of Knowledge does not correlate with Bloom's Taxonomy of Learning
-Webb's DOK have "Bloom" verbs in most of the four components of DOK - it is richer, more rigorous
-DOK is more concerned with the challenge behind an assessment, not just the verb being used - i.e. a DOK level 4 can use the verb explain or draw depending on the depth at which you are challenging students to demonstrate their understanding
-In DOK, even those application problems, if repeated with students enough, can become level 1 tasks
When using manipulatives, allow students time to "play" before asking them to use them to learn
-also ensure that policies and routines are clearly established and enforced
Florida has moved from FCAT to AIR testing to address CCSS (DOSP has not yet moved)
-Site for details about the new testing: http://www.fsassessments.org/resources/?section=1-students-and-parents
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