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Strategies for Successful Mathematics Instruction and Assessments
Presenter: Amanda [email protected]
https://sites.google.com/isd31.net/amandamix/home218-333-3100 Ext. 44211
Introduction:1. “Whole group instruction directed from the front of the room for an entire
lesson needs to disappear” (pg. 1).
2. Mathematics needs to be an active process (learning by doing model).
3. Each student needs to build his/her mathematical knowledge from “personal experiences, coupled with feedback from peers, teachers and other adults, and themselves” (NCTM).
4. Students must move beyond “the practice of procedures and skills to engagement in answering meaningful questions (high-cognitive demand)” (pg. 2).
Clearing up Misconceptions between MN Standards and Common Core:
● The MN Math Standards and the Common Core Math Standards come from the same two foundational resources: Principals and Standards for School Mathematics (NCTM, 2000, originated in 1989) and the National Research Council’s Adding it Up: Helping Children Learn Mathematics (Kilpatrick, 2001)
● The NCTM expanded and extended the original standards (from the 1980’s) to not only include content standards but also process standards (problem solving, reasoning and proof, communications, connections, and representations).
● The definition of mathematical proficiency, defined by Kilpatrick et al., in 2001, includes five strands.
Definition of Mathematical Proficiency :1. Conceptual Understanding-comprehension of mathematics concepts, operations,
and relations
2. Procedural Fluency-skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
3. Strategic Competence-ability to formulate, represent, and solve mathematical problems
4. Adaptive Reasoning-capacity for logical thought, reflection, explanation, and justification
5. Productive Disposition-habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.
What should I be modeling in order to
promote number sense in the classroom?
Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the 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 and repeated reasoning (reasonableness).
Principals to Action (NCTM, 2014)1. Engage with challenging tasks that involve active meaning making and support
meaningful learning.
2. Connect new learning with prior knowledge and informal reasoning and, in the process, address preconceptions and misconceptions.
3. Acquire conceptual knowledge as well as procedural knowledge, so that they can meaningfully organize their knowledge, acquire new knowledge, and transfer and apply knowledge to new situations (the students).
4. Construct knowledge socially, through discourse, activity, and interaction related to meaningful problems.
Principles to Action (NCTM, 2014, cont’d.)5. Receive descriptive and timely feedback so that they can reflect on and revise their
own work, thinking, and understanding.
6. Develop metacognitive awareness of themselves as learners, thinkers, and problem solvers, and learn to monitor their learning and performance.
These action statements are what the NCTM wants students to be able to do.
The MCTM (The Minnesota Council of Teachers of Mathematics) is an affiliate of the NCTM (National Council of Teachers of Mathematics) and promotes the same
Principles to Action.
In the end, what are the differences?1. Both the Common Core Standards, and MN Math Standards/Benchmarks come from the
same two original sources.
2. Both the Common Core Standards, and MN Math Standards/Benchmarks focus on students understanding math at higher cognitive levels.
3. Both the Common Core Standards, and MN Math Standards/Benchmarks focus on active guidance from the teacher to help promote number sense, reasoning, and problem-solving in the classroom.
4. The differences are: the name of the sub-strands (domains) and some of the content (what, when it is taught).
5. Friendly reminder: MN adopted the CCSS in language arts because the CCSS increased the rigor (cognitive levels) at a higher level than the MN ELA standards.
What to Look for in Any Math Lesson:1. Who is doing the mathematical thinking-teachers, students, or both?
2. What is the goal of the instruction-understanding mathematics or simply getting answers?
3. What is the cognitive demand of the tasks students are being asked to do? What happens to the demand as teachers introduce the task, and as students work on the task?
4. To what extent are all students engaged in mathematics learning?
How should I change my math instruction? What should my math
class look like?
Math Classroom: Math Instruction:1. Math centers and classroom activities
focusing on: inquiry, collaboration, and hands-on learning (NCTM).
2. Balanced mathematics program (promotes conceptual understanding, competence with procedures, and effective/efficient problem solving (pg. 3).
3. Discussion-based environment with peers, teachers, and class.
4. Standard-based, with formative assessments that guide and change instruction (not event based).
5. Models real-world application6. Is not worksheet focused.
1. Integrates rigorous (higher quality, not harder), engaging instructional strategies.
2. Promotes practical teaching strategies (to meet diverse needs) (pg. 3).
3. Applies: Number Talks, Mathematical Mindsets, and promotes the Growth Mindset in our students.
4. Focuses on instruction that is meaningful, significant, and relevant to OUR students (this will be ever-changing from year to year). The pronoun is WE, not US/THEM.
5. Models mathematics and embrace mistakes!6. Embraces the positivity mindset with
student learning (Eric Jensen).
What does proficient mathematical learning
involve?
Math Proficiency Includes Five Attributes:1. Conceptual understanding-comprehension of mathematical concepts,
operations, and relations.2. Procedural fluency-skills in carrying out procedures, flexibility, fluently, and
appropriately.3. Strategic competence-ability to formulate, represent, and solve mathematical
problems.4. Adaptive reasoning-capacity for logical thought, reflection, explanation, and
justification.5. Productive disposition-habitual inclination to see mathematics as sensible,
useful, and worthwhile with a belief in diligence and one’s own learning efficacy (Kilpatrick, Adding It Up).
Visual Model:
Conceptual Understanding
Procedural Competency
Application of Mathematics to
Real-World Problems
Frequently Asked Questions: What is number sense? How can I integrate it?1. Number sense pertains to number “flexibility.” 2. Jo Boaler, author of Mathematical Mindsets, provides an example:
https://www.youcubed.org/what-is-number-sense/
Ways to integrate (Examples):
1. Number Talks2. Mathematical Mindsets3. Modeling (in the classroom and during an intervention)
Examples of Highly Effective Strategies for Interventions (pg. 119):
1. Deliver direct, explicit, and systematic instruction that models problem solving, metacognition, and step-by-step procedures.
2. Provide students with immediate and specific correct feedback in areas of need.3. Provide structured opportunities for students to talk with one another about
mathematics.4. Provide scaffolded opportunities for students to verbalize their understanding
(crucial).5. Visually represent and model mathematics problems.6. Carefully select problems that address areas of need.7. Build automaticity with computation (not just flash cards or practice drills).
Examples of Highly Effective Strategies for Motivating Students (pg. 119):
1. Include motivational strategies with instruction, providing support that builds student confidence and competence with skills that are prerequisites of the target topic.
2. Involve students in assessing their own work, goal setting, goal identification, and monitoring.
3. Use ongoing formative data and feedback, combined with targeted reteaching, to immediately address gaps in understanding.
4. Provide feedback to students on the impact of their effort and growth (compliment this).
5. Embrace and encourage students to celebrate mistakes (growth mindset)!
Assessments for Interventions
(Linda Snowberg, National Viewpoint Conference, 2017)
What Learning, Teaching,
Assessing, and Interventions
Involve(s)
High-Quality Assessment Practices:● Motivate and engage students● Communicate strengths in terms of learning● Provide intentional opportunities to learn from mistakes and failures● Generate confidence and success
“If we gather information but never stop to analyze it or use it in the way it was intended, we can create a lot of additional work for students and teachers.”
- Nicole Dimich Vagle
(Vagle, 2015)
“So, what is the difference between formative and
summative assessments?”
Formative vs. Summative AssessmentsFormative Assessments (for learning):
When: Simultaneously while teaching
Examples: Informal teacher questions, informal observations, rough drafts, student reflective statements, practice student checklists or rubrics, rehearsals, setting up of projects, discussions, visual representations (graphic organizers), exit slips, individual whiteboards, think-pair-share, learning/response logs (in process)
Summative Assessments (of learning):
When: At the end of the selected time period
Examples: Final copy of any student work, final checklists, final rubrics, teacher-made summarizing assessments, high-stakes standardized test, end of unit compilation, portfolios, conferences
(Bailey, 2012)
“Okay. So what is the difference between formative
assessments and common formative assessments?”
Formative vs. Common Formative AssessmentsFormative Assessments (individual):
1. Assessments given by the classroom teacher only.
2. Reflecting/Individual learning3. The classroom teacher determines
the instructional planning for the students.
4. The classroom teacher determines adaptations/modification of assessments and curriculum.
Common Formative Assessments (group):
1. Assessments given by two or more teachers with the intention of collaboratively examining data results.
2. Shared learning3. Shared modifications4. Collaboratively determines “take-
aways” to put into current and future lessons.
(Bailey, 2012)
When Working with Collaborative Teams:1. What do we want all students to know and be able to do?2. How will we know if they know it?3. How will we respond if they don’t know it?4. How will we respond if they do know it?
“Planning can be done in many ways, but the most powerful is when teachers work together to develop plans, develop common understandings of what is worth teaching,
collaborate on understanding their beliefs of challenge and progress, and work together to evaluate the impact of their planning on student outcomes.” - John Hattie
(Schuhl, 2016)
Effect Size Information:Based off of
John Hattie’s
model.
(Link)
(Hattie, 2015)
What Does This Look Like?
Start (Pre-Instruction) End (Summative)
Are we going at the necessary pace in the right direction?Data/
AssessmentData/
Assessment
Data From Formative Assessments (Ongoing-During Instruction)
Not Event-Based
All in All...Collaborate, Teach, Question, Tweak, and Repeat!1. What’s important to teach?2. How will we know that our students learned it?3. What will we decide to do if our students don’t learn it?4. If they already know it?5. How can we promote the growth mindset in our students? (Model)
“Do it...study it...tweak it and keep the cycle going” - Kim Bailey.
“Alone we can do so little; together we can do so much.”-Helen Keller
(Bailey, 2012)
Clarifications about C.F.A.’s:1. Bigger (longer) is not beneficial (If 50% or more need help=C.W. or group
intervention). Does not have to be 30 minutes/30 days for one intervention.2. Constant assessing is not effective (Enough to gain useful information, but
not so much that it overwhelms.)3. You don’t need to reinvent the wheel, but you do need to make sure the
wheel is working (efficient, reliable, and valid).4. Assessments and tests are not synonymous (review collaboratively before
using, and exercise professional judgement).5. “If you don’t use common formative assessments to make a difference in
student learning, the assessments are summative” (Kim Bailey, 2012).6. When working with Power Standards/ELO’s, don’t lose the rigor (quality) of
the standard (verb). There can be consequences to this!!!
“What should I remember when writing common
formative assessments?”
Ten Tips for Writing Common Formative Assessments:1. The purpose of CFA’s is to get information efficiently and easily (to collect it,
and act on it quickly).2. Collaboration is Key! (Write your assessments/rubrics together.) Even if that
means delivering fewer common assessments.3. Assess ONLY the identified essential learning targets.4. Ask a minimum of 3 questions per learning target that you are assessing.5. When determining mastery, assess no more than 3 or 4 learning targets per
assessment.6. Clearly connect every single question to an essential learning target.
(Ferriter, 2014)
Ten Tips for Writing CFA’s (Continued):7. Choose appropriate assessment types for the content/skills that you are measuring.
8. If you decide to write multiple choice questions, use wrong answer choices to highlight common student misconceptions (to learn more about your students).
9. When writing constructed response questions, provide students with context so that they are able to answer the question.
10. If you include higher level questions, make sure the questions ask students to apply knowledge and/or skills in new situations.
(Ferriter, 2014)
“But, how do I connect C.F.A.’s (provide
interventions) to the state standards?”
Designing Assessments from Standards:Phase One:
1. Choose the standards.2. Plan engagement (It answers the question, “Why do we have to know this?”)
Phase Two (on next slide):
1. Analyze the standards (for clarity).2. Sketch out the learning goals (ladder).
(Vagle, 2015)
When Clarifying the Standards or Benchmarks (Phase 2):1. Circle to verbs (determines cognitive levels).2. Underline key concepts, vocabulary words, terms.3. Star what is familiar or an “ah-ha” moment (moment of realization).4. Place question marks by standards or words that need clarification.5. Label standards that have not been taught yet or assessed.
“Standards are meaningless until you define how you will assess them.” -Bambrick-Santoyo
(Vagle, 2015)
Different Levels of Cognition (Verbs-Standards/Benchmarks):● Simple: Define, recognize, describe, explain, recall, identify, relate, label,
review, memorize, state, name, summarize, compute● Complex: Apply, analyze, connect, examine, appraise, categorize, distinguish,
compare, infer, classify, interpret, contrast, predict, recommend, differentiate, support, defend, formulate, judge, evaluate, criticize, synthesize, investigate
● More Complex Verbs that Involve Doing: Operate, make, calculate, match, compose, measure, construct, plan, demonstrate, practice, show, use
● More Complex Verbs that Involve Creating: Produce, create, write, design, develop
(Vagle, 2015)
Designing Assessments from Standards (Continued):Phase Three:
1. Identify the learning goals for the assessment (simple and complex).2. Choose the method of the assessment.3. Determine the weight/number of items for each learning goal (1 to 4 scale).
Phase Four:
1. Create or revise assessment items (rubrics, etc.) and tasks for each learning goal (rigor and relevancy).
2. Develop student documents and gather necessary materials (the why of learning).
(Vagle, 2015)
Designing Assessments from Standards (Last Step):Phase 5:
1. Create a scoring scheme that reflects the learning (organize grades by learning goal, not by assessment method).
2. Choose strategies to foster student investment (I can statements, feedback).
“Assessment in collaboration has great potential to empower educators to help them grow in their own practice as well as contribute to the overall success of a school,
team, and classroom. It is one of the most significant professional learning practices in which educators can engage” - Nicole Dimich Vagle
(Vagle, 2015)
“What are some resources, or examples, that can help
me start this process?”(Provided)
Questions?
Other Things to Keep in Mind...
Why, on average, do we feel more
comfortable teaching reading than math?
The Reasons (To Promote a Solution):1. It is not uncommon or unacceptable for adults, including educators, to say, “I
never liked mathematics as a student” or “I’m not really good at math” (pg. 1).2. An educator’s levels of confidence in math impacts his/her students
mathematical mindset (sad spiral of negative influence).3. Educators feel “less professionally satisfied with the mathematics instruction in
their classrooms” (pg. 1).4. We reflect on the way WE learned mathematics, which is not the way we
SHOULD have learned mathematics (performance based, when it should be process based).
5. “Teachers have not received sufficient professional support in instructional practices” (pg. 2).
6. “Assessment(s) and intervention strategies for mathematics are not as accessible to many teachers” (pg. 2) Until recently, (last 4 to 5 years) less research available.
What else should I know to make my math
instruction more engaging?
Four Phases of Application-Based Mathematics Learning (pgs. 93-102):1. Explore and Inquire (Direct instruction is not provided.)2. Teach and Explain (based on observations and assessments)3. Practice (Does not imply pages of problems.)4. Demonstrate Ownership (through assessments/tasks)
Questions to Use When Choosing Tasks to Support Problem Solving and Making Sense of Problems:
1. Is the problem interesting to students?
2. Does the problem involve meaningful mathematics?
3. Does the problem provide an opportunity for students to apply and extend mathematics?
4. Is the problem challenging (not synonymous with hard) for students?
5. Does the problems support the use of multiple strategies?
6. Will students’ interactions with the problem provide real information about students’ mathematics understanding? (pg. 16)
Questions to Ask, if Unsure:1. What is the problem asking?
2. How will I solve it?
3. Is this strategy working?
4. Does my answer make sense? (pg. 32)
Five Areas of the Classroom Climate that Contribute to Students’ Ability to Learn:
1. Knowing others: “I know these people and they know me.”
2. Greeting, acknowledging, listening, responding, and affirming: “I feel accepted and included.”
3. Group identity, responsibility, and interdependence: “I am a member of this group…we want each other to succeed.”
4. Cooperative learning, social skills, group meetings, and group dynamics: “I can help others and they will help me.”
5. Problem solving and conflict resolution: “We can solve problems that arise between us” (pg. 68).
How Teachers Can Encourage Active Listening:1. Make sure multiple students respond to each question before asking another.
2. Ask questions that have multiple possible answers (Why might someone else represent it differently?).
3. Encourage students to reflect back when they are hearing or offer alternate ways to say the same thing, as well as asking “Am I understanding you correctly to mean this?”
4. Listen to students’ ideas, don’t listen for correct thinking (pg. 69).
Action Steps to Promote Effective Discussions:1. Anticipating likely student responses to challenging mathematical tasks.
2. Monitoring students’ actual responses to tasks (while students work on the tasks in pairs or small groups).
3. Selecting particular students to present their mathematical work during the whole-class discussion.
4. Sequencing the student responses that will be displayed in a specific order.
5. Connecting different students’ responses and connecting the responses to key mathematical ideas (pg. 84).
“Too many students are unaware that they can apply mathematics outside school” (pg. 97). NCTM suggests:
1. Students should use multiple forms of representations to make sense of and understand mathematics.
2. Students should describe and justify their mathematical understanding and reasoning with drawings, diagrams, and other representations.
3. Students should contextualize mathematical ideas by connecting them to real-world situations.
4. Students should consider the advantage of suitability of using various representations when solving problems (K-12) (pg. 97).
Teachers must re-evaluate a math tool’s effectiveness. Ask yourself these questions:
1. Will the tool deepen student understanding?
2. Will the tool allow for student verification of ideas?
3. Will the tool provide opportunities for students to explore mathematical concepts and rules?
4. Will the tool allow students an entry point into solving higher-level cognitive tasks?
5. Will the tool help students make sense of abstract ideas?
The goal is that students should ask themselves the following questions:
1. How will this tool help me understand the task?
2. How can this tool help me solve the task?
3. What other tools could we use to solve the task?
4. Is this the most effective and efficient tool to use? (pg. 138)
Six-Step Process for Teaching Academic Vocabulary:1. Provide a description, explanation, or example of the new term (A non-linguistic
representation as well).
2. Ask students to restate the description, explanation, or example in their own words.
3. Ask students to construct a picture, symbol, or graphic representing the word.
4. Engage students periodically in activities that help them add to their knowledge of the terms in their notebooks.
5. Periodically ask students to discuss the terms with one another.
6. Involve students in games that allow them to play with terms (pgs. 155-156).
Helping Students Make Connections Between Tasks:1. How did you solve the previous task? Will that strategy work to solve this task?
2. What ideas in this tasks are similar to the previous task?
3. What in this task is different or new? How will you deal with that in your solution?
4. What skills and strategies have you learned that might help you solve this task?
5. What do you wonder about in this task?
What to ask students with giving complex tasks to solve:
1. What is the task asking you to find?
2. What do you know? What do you need to know?
3. Can you draw a picture to help you?
4. What is one thing you think you need to do to solve this task?
5. How does this task resemble something we have been working on in class?
What to ask students when making predictions about patterns:
1. Does your answer make sense?
2. What is repeating?
3. What pattern are you noticing?
4. Why might this be true?
5. Could there be a different pattern?
Your input, please!Write your name, a +, and a ? on your post-it note.
I will e-mail you and answer any questions you may have.
Thank you very much!
PLEASE WRITE YOUR NAME
+ Write down one thing you liked or learned about this presentation.
? Write down one question you would like answered.
Resources:1. Bill Ferriter, “Ten Tips for Writing Common Formative Assessments,”
https://www.teachingquality.org/content/blogs/bill-ferriter/ten-tips-writing-common-formative-assessments (2014).
2. Common Formative Assessment: A Toolkit for PLCs at Work by Kim Bailey and Chris Jakicic (2011)
3. Design in 5: Essential Phases to Create Engaging Assessment Practice by Nicole Dimich Vagle (2015)
4. Engage in the Mathematical Practices: Strategies to Build Numeracy and Literacy with K-5 Learners by Norris and Schuhl (2016)
5. John Hattie Ranking: Visible Learning http://visible-learning.org/nvd3/visualize/hattie-ranking-interactive-2009-2011-2015.html (2015).
6. Effect Size Graphic: https://image.slidesharecdn.com/vlfeedbackwholeed-141124064757-conversion-gate01/95/5th-annual-conf-feedback-that-makes-learning-visible-8-638.jpg?cb=1416811714
7. Cake Graphic: Linda Snowberg, 2017 National Viewpoint Conference
Continued Resources:8. Collaborative Common Assessments: Teamwork. Instruction. Results. (Practical Steps for Teacher Teams to Examine Assessment Data) by Cassandra Erkens (2015)
9. Grading from the Inside Out Bringing Accuracy to Student Assessment Through a Standards-Based Mindset (How to Give Students Full Credit for Their Knowledge) by Tom Schimmer (2016)
10. Balanced Assessment from Formative to Summative by Kay Burke (2010)
11. Mindful Assessment: The 6 Essential Fluencies of Innovative Learning by Lee Wanatabe Crockett and Andrew Churches (2016)
12. Simplifying Common Assessment by Kim Bailey and Chris Jakicic (2016)
13. Professional Support From: Jake Seuntjens from Northwest Service Cooperative
14. Strategies for Mathematics Instruction and Intervention by Chris Weber and Darlene Crane (2016)
15. Making Number Talks Matter by Cathy Humphreys and Ruth Parker (2015)