School-Wide Math Culture: Model & Nurture

MODEL AND NURTURE POSITIVE ATTITUDES, SELF-EFFICACY AND ENGAGEMENT.

As educators gain the mathematical knowledge for teaching, they become more capable – and confident – in helping students extend and formalize their understanding of mathematical concepts. This can contribute to students’ development of positive attitudes toward mathematics and an increase in their sense of self-efficacy. Self-efficacy, which is an individual’s belief in whether he or she can succeed at a particular activity, plays an integral role in student success. Bruce and Ross discovered that “increases in teacher efficacy led to increases in student efficacy and outcome expectancy and to student achievement” (2010, p. 10).  In turn, strong student self-efficacy can contribute to greater enthusiasm and engagement in mathematics (Ross, 2007, p. 52). (Maximizing Mathematical Learning in the Early Years).

Goal:  Reach ALL Learners.

How?

Create a Math Talk Community by developing students reasoning along with the capacity to articulate and communicate.

• Turn & Talk - "The Worker is the Learner" - get students to restate their thinking & get students to comment on that thinking with the teacher being the lead learner/facilitator.

• Revoice - "Mathematization" - when the students are 'grappling' with an idea/concept and need clarity and then the teacher with 'highlight' the idea/concept.  The 'struggle' will stretch the mind and thus provide a foundation for "mathematization" to build on.

• Slow down - Who heard what?  We are not looking for 'right' answers but 'right' thinking.  The Lead Learners (i.e. teacher) needs to ask more questions when questions are asked, repeat/repeat/repeat and thus hold students accountable for their talk.

• Problem-solving - teach students from a problem-based perspective focusing on the Big Ideas.  Utilizing 3 Part Math as a way to structure our 60 minute learning block is a great starting point.

By incorporating these concepts in our planning & assessment practices, our entire school will see a dramatic change in the mathematical output of each of our students. 

Student-led Conferences

Student-led Conference

Student Self-Assessment

Student-led conferences offer authentic opportunities for students to share their learning with their parents/caregivers.  They engage and motivate both students and parents to participate enthusiastically in the teaching-learning process.  Student-led conferences strengthen the relationship between home and school and also bring students and parents closer together.  

Literacy & Numeracy Secretariat

Student-led conferences is the next step in our students evolution that will tie in and exemplify the power of learning goals, success criteria, anchor charts, etc...  The students will use their metacognitive skills to tap into their learning at an entirely new level. Moreover, as we move forward as a community of learners, the student success will reach new heights due to our consistent and collective strength.  

Our classrooms are set-up with the kids in mind, for the kids to utilize and thus there is meaning for the kids with everything in our rooms.  

When students articulate their learning, they consolidate their thinking. As they learn to express themselves, they are developing their communication skills and strengthening their understanding.

Literacy & Numeracy Secretariat

Follow the link to the Curriculum Services Canada website to watch a number of web casts that support Student-led conferencing.  

The Capacity Building Series (Literacy & Numeracy Secretariat) has a monograph on Student Self-Assessment.  This is a good resource.

School-Wide Math Culture: Mathematization

FACILITATE EXPERIENCES THAT ALLOW FOR MATHEMATIZATION OF EVERYDAY KNOWLEDGE.

Knowledgeable educators help students transform their everyday mathematics into a more formalized understanding that can be transferred and applied to other situations. Several researchers refer to this as “mathematization” which requires students to abstract, represent and elaborate on informal experiences and create models of their everyday activities (Clements & Sarama, 2009, p. 244). The educator can play an integral role by making meaningful connections between the mathematical strands, the real world and other disciplines, and most importantly, “between the intuitive informal mathematics that students have learned through their own experiences and the mathematics they are learning in school” (Ontario Ministry of Education, 2003, p. 14). For example, as a child naturally creates and extends a pattern while making a necklace, the educator can effectively pose questions that provoke the student not only to describe the pattern, but also to make predictions and generalizations. (from Maximizing Student Mathematical Learning in the Early Years)

Can U Beat These Mathematicians?

"Mathematization' can be an uphill battle but one worth fighting.

Honouring the Student Voice in the Mathematics Classroom is a webcast that highlights the goal of mathematization.  

The 'Student Voice' is exactly where your students are in their math journey.  It is important to listen to this 'Voice', support it and move it forward along the mathematical continuum (i.e. curriculum) through mathematization.

School-Wide Math Culture: Everyday Mathematics

IDENTIFY AND USE EVERYDAY MATHEMATICS KNOWLEDGE TO PLAN INSTRUCTION.

Knowledgeable educators begin planning by carefully observing children at play or engaged in other activities in order to identify their everyday mathematics. Next, they accurately interpret the mathematics underlying the behaviours and how it fits into the key mathematical concepts and curricula. Once identified, educators can create activities which allow assimilation of new concepts into the children’s prior knowledge (Ginsburg, 2008, p. 59). As educators observe student problem-solving, they can document what children say, do and represent in order to make both planned and “in-the-moment” decisions about how to respond, challenge and extend student thinking. (Maximizing Student Mathematical Learning in the Early Years)

Even though the above excerpt is taken from the Capacity Building Series: Maximizing Student Mathematical Learning in the Early Years it can be applied to all grades from a problem-solving perspective.  Consider the following thinking:

1. Understand your curriculum to the point you shape/form it into 'Big Ideas'
2. Find interesting ways to match the 'Big Ideas' to problems for your class to solve
3. Allow students to 'play' with the problem
4. Observe & Learn with your students
5. Intervene when necessary to 'mathematize' student thinking/work

Dan Meyer on Real-World Math

Many students struggle with math or have difficulty understanding the subject. Educator Dan Meyer has come up with various scenarios that can help students enjoy math and add some fun to learning it.

  

Dan Meyer blogs at http://blog.mrmeyer.com.

Challenge:

If you want to try something new and give it a 'fair' opportunity for success, then you must eliminate your current system (at least for a specified time period).  For example, one area to remove would be the reliance on 'class sets' of photocopies &/or textbook work.  If this is an area you currently use, then remove it for a period of time (i.e. 2-3 weeks).  See what happens....the 'withdrawal' symptoms are only temporary.

School-Wide Culture: Math Talk

(The concepts come from The Capacity Building Series: Maximizing Student Mathematical Learning in the Early Years)

The Starting Point...

Immerse yourself in the curriculum and supporting documents. 
Attain a better understanding of the expectations and the seven mathematical processes by reading about the explanations and rationale in the front matter of the Full-Day Early Learning Kindergarten Program and Grades 1 to 8 Mathematics curriculum. (p11-17)

Look before and beyond the grade you are teaching to see how concepts build upon each other. 
Utilize the document provided during our Divisional Meeting which shows the Math Curriculum on a continuum from grade to grade. 

There is a wealth of resources that can offer extra insight into the mathe- matics itself and can help to identify and connect the key mathematical concepts. 
Some of these include the Guides to Effective Instruction in Mathematics and the works of Dr. Marian Small, Catherine Twomey Fosnot and John A. Van de Walle. 

Your professional learning journey will be most effective when you delve into mathematical ideas with colleagues and together inquire about how your understanding impacts your related teaching.

Culture of Classroom Discourse

(Lucy West)

Teacher-facilitated “math talk”  significantly increases children’s growth in understanding of mathematical concepts.  Knowledgeable educators recognize that although children may have a beginning understanding of mathematical concepts they often lack the language to communicate their ideas.  By modelling and fostering math talk throughout the day and across various subject areas, educators can provide the math language that allows students to articulate their ideas.  

It is also important to encourage talk among students as they explain, question and discuss their strategies while co-operatively solving problems.  In order to facilitate mathematical thinking rather than direct it, knowledgeable educators recognize when student thinking is developing or stalled. If it is developing, the educator observes but leaves the students to work through their thinking (Sarama & Clements, 2009, p. 325). If it is stalled, probing questions can be asked that provoke thinking about alternate ways to perceive the problem.

After students have worked through solving a problem, educators facilitate consolidation time (either with individual students or with small groups or large groups) in order to allow students to talk about their thinking. This consolidation time is sometimes referred to as the third part of the three-part lesson in mathematics.  As educators value a variety of strategies and solutions, they guide students to make connections between them, to recognize how the thinking relates to the key mathematical concepts and to make further conjectures and generalizations.

Five productive talk moves ...to create meaningful mathematics discussions.

Gives the educator an opportunity to embed mathematics vocabulary

1. Revoicing – Repeating what students have said and then asking for clarification 

"So you’re saying it’s an odd number?"

2. Repeating – Asking students to restate someone else’s reasoning 

"Can you repeat what he just said in your own words?"

3. Reasoning – Asking students to apply their own reasoning to someone else’s reasoning     

"Do you agree or disagree and why?"

4. Adding on – Prompting students for further participation 

"Would someone like to add something more to that?"

5. Waiting – Using wait time   "Take your time ... We’ll Wait .."

(Chapin, O’Connor & Anderson, 2009, p.13)

Guidelines for Whole-Class Math-Talk

(taken from - Student Interaction in the Math Classroom: Stealing Ideas or Building Understanding)

Explain: “This is my solution/strategy ...” “I think _____ is saying that ...”

• Explain your thinking and show your thinking.

• Rephrase what another student has said.

Agree with reason: “I agree because ...”

• Agree with another student and describe your reason for agreeing.

• Agree with another student and provide an alternate explanation.

Disagree with reason: “I disagree because ...”

• Disagree with another student and explain or show how your thinking/ solution differs.

Build on: “I would like to build on that idea...”

• Build on the thinking of another student through explanation, example, or demonstration.

Go beyond: “This makes me think about ...” “Another way to think about this is ...”

• Extend the ideas of other students by generalizing or linking the idea to another concept.

Wait time:

• Wait to think about what is being said after someone speaks (try five seconds).

The Value of Student Interaction

In the math reform literature, learning math is viewed as a social endeavour.1,2 In this model, the math classroom functions as a community where thinking, talking, agreeing, and disagreeing are encouraged. The teacher provides students with powerful math problems to solve together and students are expected to justify and explain their solutions. The primary goal is to extend one’s own thinking as well as that of others.3

Powerful problems are problems that allow for a range of solutions, or a range of problem-solving strategies. Math problems are powerful when they take students beyond the singular goal of computational mastery into more complex math thinking. Research has firmly established that higher-order questions are correlated with increased student achievement, particularly for conceptual understanding.

Math PLC - Recap

Math PLC - Recap

Our collective strength 'trumps' individual strength each time.

A great week of Math PLC's!  We were introduced to our collective 'math culture' using the monographs listed below.  Our collective strength comes from having a consistent, school-wide & grade-by-grade approach to teaching mathematics.

Key concepts to create OUR collective math culture include the following: 

• Math-Talk
• Teacher as Facilitator
• Creating the Math Environment
• Mathematize
• An Open approach is a realistic approach

Professional Reading:

Asking Effective Questions - Provoking Student Thinking/Deepening Student Conceptual Understanding in the Mathematics Classroom
Excellent & Practical resource that is a major part of what will be our collective 'Math Culture'.  "In a constructivist classroom," Mariam Small writes, "students are recognized as the ones who are actively creating their own knowledge".  The teacher's role is to help the student in the following areas: (1)  identify thinking process, (2) see connections & (3) build new understanding.

Maximizing Student mathematical Learning in the Early Years
Excellent & Practical resource that is a major part of what will be our collective 'Math Culture'.  "Researchers have identified five common core characteristics of early learning environments that support effective mathematical pedagogy and foster positive attitudes and beliefs about mathematics."  

Student Interaction in the Math Classroom: Stealing Ideas or Building Understanding
Practical ideas to implement whole-class Math-Talk.

Word Problems: Connecting Language, Mathematics & Life
An open approach is a realistic approach.  Mathematize the student thinking.

Problem-Based Learning in Mathematics:  A Tool for Developing Students' Conceptual Knowledge
The teacher as the facilitator of conceptual knowledge.


We watched the webcast, Engaging Students in Mathematics during our PLC.  There are 'clips' that we didn't watch but might be helpful.  The Literacy & Numeracy Secretariat develops these webcasts (and other resources) and houses them within the Curriculum Services Canada website.  Everything is listed from the most recent publication.  The search bar is very good.

We will return in April with examples of student work, planning template, anchor charts, learning goals, success criteria & assessment criteria to share/discuss.

Creating A Mathematical School-Wide Culture

How to Apply OUR Knowledge of Mathematics for

Teaching to Improve Student Learning

A consistent, school-wide & grade-by-grade approach to mathematics will lead to improved student learning.  The information presented here comes from Asking Effective Questions & Maximizing Student Mathematical Learning in the Early Years.  

“Deepening of Teacher Understanding + Shifts in Instructional Practice = Impact on Student Learning” 

2010–2011 Early Primary Collaborative Inquiry

A Starting Point ...

Immerse yourself in the curriculum and supporting documents. Attain a better understanding of the expectations and the seven mathematical processes by reading about the explanations and rationale in the front matter of the Full-Day Early Learning Kindergarten Program and Grades 1 to 8 Mathematics curriculum. Look before and beyond the grade you are teaching to see how concepts build upon each other. There is a wealth of resources that can offer extra insight into the mathematics itself and can help to identify and connect the key mathematical concepts. Some of these include the Guides to Effective Instruction in Mathematics and the works of Dr. Marian Small, Catherine Twomey Fosnot and John A. Van de Walle. Our professional learning journey will be most effective when we delve into mathematical ideas with colleagues and together inquire about how our understanding impacts our related teaching.

As a school, we need to develop a collective understanding around the following concepts:

1. IDENTIFY AND USE EVERYDAY MATHEMATICS KNOWLEDGE TO PLAN INSTRUCTION
2. ENCOURAGE AND FOSTER “MATH TALK.”
3. FACILITATE EXPERIENCES THAT ALLOW FOR MATHEMATIZATION OF EVERYDAY KNOWLEDGE
4. MODEL AND NURTURE POSITIVE ATTITUDES, SELF-EFFICACY AND ENGAGEMENT...

Some Practical Tips for Creating a Mathematics-Rich Environment

Researchers have identified five common core characteristics of early learning environments that support effective mathematical pedagogy and foster positive attitudes and beliefs about mathematics 

(Clements & Sarama, 2009, p. 259).

Asking Effective Questions

    Provoking student thinking & deepening conceptual understanding in the mathematics classroom

Researchers support a problem-solving approach in the mathematics classroom because it engages students in inquiry, prompting them to build on and improve their current knowledge as they “construct” explanations that help them solve the task at hand. “In a constructivist classroom,” Marian Small writes, “students are recognized as the ones who are actively creating their own knowledge” (2008, p. 3).

The classroom becomes a workshop ...

“ ... as learners investigate together. It becomes a mini- society – a community of learners engaged in mathematical activity, discourse and reflection. Learners must be given the opportunity to act as mathematicians by allowing, supporting and challenging their ‘mathematizing’ of particular situations. The community provides an environment in which individual mathematical ideas can be expressed and tested against others’ ideas. ... This enables learners to become clearer and more confident about what they know and understand.”

(Fosnot, 2005. p. 10)

James Hiebert (2003) suggests three situations in which teachers might consider conveying information to students:

1. Students need conventional written notations. For example, how to represent a fraction, how to show that a quantity is greater than another and terms related to solving problems that require them. For example, “You’re saying this triangle has no equal sides. You’re describing a scalene triangle.” Students don’t always have the language to respond to open questions such as, “How do you know?” Modelling this language is important in building students’ sense of self-efficacy.

2. During consolidation, teachers may present alternative methods that have not been suggested by students. Teachers may choose to do this if a particular strategy would help students better understand the big idea underlying the problem. The strategy should be presented as just another alternative, and not as the preferred strategy.

3. Again during consolidation, teachers may highlight the mathematical ideas embedded in the students’ solutions. These can be made explicit by posing questions that focus the students’ attention on these ideas. The teacher may annotate the solutions to make these ideas visible, and add them to the chart of summary and highlights constructed by the class. For example, the teacher may record right on a student solution that the authors have used the strategy of making friendly numbers in order to solve the problem. Another teacher might highlight the way the students have shown how multiplication and division are related.

Questioning is a powerful instructional strategy. Open questions that are related to the big ideas embedded in the curriculum expectations and learning goals will excite student curiosity, provoke critical thinking, elicit reflection and help students construct their own meaning for the mathematics they are studying. Their responses will help the teacher assess what students know and what next instructional steps might be. Developing skills in questioning for understanding and content knowledge evolves over time and like anything else, requires practice. The payoff is significant in terms of students’ conceptual understanding.