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What role does explicit instruction play in Mathematics? Sometimes we feel we are being told to avoid direct instruction, or that direct instruction is less preferred than small group instruction, PODS, technology based learning, math games, centres, journals, etc. We may even feel that direct instruction is viewed negatively, evidenced by coined terms such as “chalk and talk”. We know its better to be less of a “Sage on the Stage” and more of a “Guide on the Side”. But in fact, there is still a role for direct instruction in Math.

Hattie’s research (2012) found “direct instruction” to be a useful instructional practice, with an effect size of 0.59, and rated “teacher clarity” even higher, at 0 .75. The National Mathematics Advisory Panel recommends that struggling students need instruction that is explicit and systematic, including verbalizing the thinking process, guided practice, and feedback.

“This kind of instruction should not comprise all the mathematics instruction these students receive. However, it does seem essential for building proficiency in both computation and the translation of word problems into appropriate mathematical equations and solutions. Some of this time should be dedicated to ensuring that students possess the foundational skills and conceptual knowledge necessary for understanding the mathematics they are learning at their grade level.” (Final Report of the Mathematics Advisory Panel, 2008, p. 49)

We may need to teach full group lesson before students are able to make meaningful use of technology or practice stations. Explicit instruction is our chance to model mathematics, and our chance to model “think-alouds” as we verbalize our thinking through the work. A large portion of our math students are predominantly visual learners: They need a chance to see the mathematics and watch the procedures in order to internalize them. Logic and systematic work are an important component of mathematics. We need to lay out clear, logical, sequential, neat, orderly, well-planned and well-explained lessons. Furthermore, by middle years, students should be expected to maintain notebooks. Of course we wouldn’t expect students to “copy notes”!…but they could be asked to keep some notes (Marzanno lists Summarizing and Note-taking as second highest effect in his list of 9 most effective instructional strategies), examples of work, explanations, instructions, and journal responses. We need to model the logical and sequential order of recording math processes, with appropriate terminology, models and symbols in order to immerse students in mathematical language, logic, and sense-making.

Does explicit instruction fly in the face of constructivist teaching?** Not at all! Appropriate explicit instruction is not “chalk and talk”.** **It is not “stand and deliver”. **Good direct instruction involves as much or more student talk than teacher talk!!

- We provide opportunities to introduce the topic and engage thinking: “What are some ways we use fraction at home?” Or, “What are some graphs we find in everyday life?” “How do we know how many numbers we need in our phone numbers in Saskatchewan. Why does that matter? How do we figure it out?”
- We communicate the learning target clearly.
- We introduce concepts, then within the lesson we offer opportunities for students to make connections: “Do you recognize the pattern of this arithmetic sequence? What if we graph it?”
- We allow opportunity for conjecture and exploration: “What do you think happens we….. “, “How might we simplify…” “Try this at your desk…”, “Experiment with these values and see if there is a pattern.”
- We build in opportunities for communication, because we know that learning is socially constructed. “Check with your elbow partner”. “Share your results”. “Can you write a rule for this relationship?”
- We break for moments of deliberate, guided practice with immediate feedback: “Now try this one at your desk. Check with your partner to see if your work meets the criteria from the example”. “Do the work on a white board and show me”. “Share your diagram on the document camera or with your ipad”.
- We create opportunities to share strategies and representations.
- Good instruction allows opportunity for open questions, forming and debating opinions, and wrestling with cognitive dissonance. We choose questions carefully to challenge thinking and allow multiple points of entry. We can direct conversation to pairs, groups, or full class discussion.
- Direct instruction allows us to model appropriate symbols and vocabulary: “
*We write*å*and we say*‘Find the sum of’ “. - We can give clear directions, we can chunk information, we can provide a solid summary of the days’ learning.
- Direct instruction can help model learning. Students need to be explicitly taught how to learn, how to self-regulate their thinking processes (Hattie, 2012).

We know the person in the room doing the most talking is the person doing the most learning. It mustn’t be the teacher! A balanced approach to math means using *all* teaching methods *where appropriate*, including small group work, centres, independent practice, technology, experimentation, dialogue, debate, discovery and even direct instruction. All things in moderation.

There is no “one” correct way to teach. We think carefully about what methods are most likely to help lead students to deep understanding of the concept, ownership of the learning, flexible reasoning, application and transfer, and proficiency. More importantly, to attend to all our learners, we should have students engage in learning in a variety of contexts. But I hope we can “rebrand” quality whole-class teaching to include practices we know are effective, and to not think of it narrowly as “stand and deliver” teaching. It has its place.

References:

Hattie, J. (2012). *Visible Learning for Teachers: Maximizing Impact on Teaching and *

*Learning*. New York, NY: Routledge.

Marzanno, Pickering, and Pollock. (2001). Classroom Instruction that Works: Research-

based Practices for Improving Student Achievement. Alexandria, VA: ASCD

National Center for Education Evaluation and Regional Assistance (2009), U.S. Dept of

Education, Institute of Education Sciences

National Mathematics Advisory Panel. *Foundations for Success: The*

*Final Report of the National Mathematics Advisory Panel*, U.S. Department of

Education: Washington, DC, 2008.

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__Vertical Math Teams in your School__

We’ve often created opportunities for math teachers to converse with grade alike partners, but to facilitate setting curricular priorities and supporting strong math achievement in our schools, we need math teachers to have conversations with the entire math team, up and down the grades, in order to get a sense of the flow of outcomes up through the years. In truth, this team already exists, and we are all players, whether we’ve recognized it or not.

Understanding where an outcome is going, or what future learning depends on its mastery, helps teachers prioritize topics and also to make those connections to past and future learning. Those **connections** to math concepts already taught or coming in the future are pivotal to advancing a strong mathematics program. Similarly, when introducing a new topic, it is extremely beneficial for a teacher to understand where the students are in that progression of understanding, what **vocabulary** and skill has been taught to date, what **models** have been used, and what to expect of the level of understanding of students.

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We know that mathematical understanding is logical and sequential; ideas are built on concepts that are already learned. We know that the development math understanding is fluid and connected as a student studies math from K-12. But do our students know it? Or do they feel that each grade is new learning and discreet content? Sometimes we have to help them remember the connection.

Middle years and elementary teachers can help us understand how math concepts develop. How are these understandings built? High School teachers can explain the secondary goal of the strand. What will students ultimately need to be able to do with this skill/concept?

The centre of this conversation is not *teachers* but *curriculum*. What are students learning and when? What are the gaps in curriculum and how can we address them? What does it look like at each grade level? How do I support my colleagues that teach the grades before me and the grades above? Recognize that each grade has its own challenges, children are at different stages developmentally, and our system imposes artificial “grade level” expectations on what is simply a continuum of learning and development that evolves differently for every child.

“By promoting communication among educators about curriculum[outcomes]and instructional strategies, Vertical Teams foster the development of an educational community committed to improving student performance. This community then creates acontinuum of learning among classes and across grade levels, and also reduces the redundancy and reteaching of content across grade levels and courses. This process can enable teachers to encourage students to apply previously gained skills and knowledge to new and more challenging material. Increased coordination can also help counselors, administrators, teachers, and students to develop a clearer vision of how the curriculum unfolds, enhancing their ability to understand its objectives.”

-The College Board, (2006). New York

Vertical teams of teachers look through the continuum of learning of math outcomes and decide what the non-negotiable learnings are. What are the things I need the teacher of the grade prior to really drive home to students? What are the promises I make to the next teacher? Does this mean we don’t teach all outcomes? Well, it might mean we don’t teach them all with the same depth (this is where UbD planning comes in). We will identify those **concepts that are essential** to our program and work collectively toward proficiency there.

We know there is no more time for regular meetings! But a couple collaborative opportunities to start, and some ongoing understanding and practices (maybe a common area to post things) will make a huge difference to our learners.

**Vertical Teams: **

- Take a Big Picture view of curriculum
- Focus on student understanding across curriculum
- Focus on assessment
- Focus on difficult concepts to teach or to learn
- Reveal and deal with student misconceptions
- Enhance vocabulary coordination across curriculum
- Enhance notation consistency

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**Departmental Exam Prototypes** up on blackboard. These exams are** interactive**; you can create a class of students, log them in and let them practice online. They will receive a grade and feedback. Thankyou Sask Math Leaders and Ministry of Ed!!!!

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Click the link above to go to the word document with working links

**Summer Math Institute and Math Mentors **Every summer 20-30 K-8 teachers are invited to take part in a 2 day math instruction workshop. Topics include problem solving, inquiry, assessment, manipulatives, technology for teaching, journaling, and have included guest presenters such as Mike Fulton and Trevor Brown. Attendees at this workshop become *Math Mentors*, and are called back for two days during the school year for follow up training and support, and once in the following year. They also join an online community to share expertise and ideas. Our digital learning coaches support teachers in integrating technology in instruction as well as housing a collection of virtual manipulatives and ipad apps. *Math Mentors* are asked to become instructional leaders in their school, and in this way we hope to build capacity in our schools to improve student achievement.

**Grade Alike “Let’s Talk Math” Days**

Teachers are invited to attend a one-day workshop with other teachers of the same grade/course. While there is some presentation by math and technology coaches most of the workshop is developed and delivered by teachers. There is always an assessment component where teachers share formative and summative assessment strategies, and also a time of open sharing where teachers bring their favourite resources, web sites, lessons, classroom management tips, manipulatives, activities, catalogs, and any other math support items. There is round-table sharing and digital collection of resources. This has been a very successful PD model; it allows us to capitalize on capacity we have built through years of the *Math Mentor* program, and teachers give very positive feedback about being able to learn from their colleagues and take away ideas and resources they can use in the classroom the very next day. High school sessions have focused on collaborative learning activities, curriculum, inquiry, technology, assessment and instruction, and teachers have brought lesson, activity, and project ideas and samples to share. We’ve been fortunate to have Lisa Eberharter share her expertise with us at some these days.

**Math Pods**

Good Spirit school division supports teachers in establishing flexible group instruction, such as Guided Math. PODS (Pedogogy that is Outcome focussed, Differentiated, and Student centred) has students working through stations that are all focussed on the same outcome. Stations generally involve a technology station (online instruction, practice, creation/documentation of learning, or analysis), manipulatives, paper-pencil practice, reading, journaling, collaborative creating, games, guided practice, exploration and a teacher table for assessment, support, or small group instruction. To learn more, watch our video.

**Math Coach**

Cindy Smith is our division Math Coach. She works with teachers to support math instruction, assessment, and RTI. Areas that are commonly addressed are word walls and vocabulary programs, technology for teaching, manipulatives, curriculum and content support, collaborative learning, formative assessment, resources, and effective instructional practices. Cindy also supports teachers as part of GSSD’s “PAALS” program. PAALs stands for Positive, Accountable, Active Learning. Each year a number of schools are designated as PAALS schools, and in that year they receive ongoing focused support for Administration, Student Services and Instruction. Each teacher chooses a Professional Growth Goal which will deepen their understanding in any curricular areas and each goal is supported by a division coach. Math goals may be technology, assessment, FNMI content, RTI, manipulatives, resources, lesson or unit planning, goal setting and student engagement. The coach and teacher plan some strategies to try, research, practice, and reflect together. You can check out Cindy’s blog, as well as math newsletters.

**Benchmarking**

Once a year, every student in Grades K-8 complete an Exemplars word problem, with emphasis on communicating and representing mathematical reasoning. These problems are scored collaboratively on an analytic rubric by teams of teachers. Benchmarking supports math instruction through mathematical processes; problem solving, reasoning and proof, communicating, making mathematical connections, and making mathematical representations. Collaborative assessment is a powerful professional development opportunity, and there are many critical conversations that take place at these full-day scoring events (one per grade). This data has been collected for six years in our division, and is part of division data and schools’ learning improvement plans.

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Providing specific, non-graded feedback as formative assessment, on the other hand, is very effective. We now understand that ongoing formative assessment is essential to student achievement, and using feedback is one of the most effective methods of informing students about their learning. When students are able to make an attempt, receive specific advice from the teacher, and then be allowed to redo the work, the visible improvement is extremely motivating.

Feedback is essential if we are asking students to self-monitor and set goals. Stiggins wrote that students can hit any target they can see and that doesn’t move. Providing timely feedback shows students the way to the target.

Some tips for providing effective feedback:

It must be specific, not just “good job” or “well done”, but instead it must state specifically what needs to improve to demonstrate improvement. The feedback must be in student friendly language: make sure the learner knows what you mean! Do they know exactly what they did, and what they need to do to improve? When giving verbal feedback, some people advocate for “sandwiching” their comments: say something positive about the work, then some constru ctive advice, then finish with something positive again. This is also a great model for training students to peer tutor.

Rubrics are often used to make criteria visible, and as discussion tools when giving feedback. Be sure the rubric is clear and understandable, and appropriate to the age of the learner. Using concrete examples of work like anchors and exemplars is also very effective ways of making criteria clear to students. While we may understand what we want went we ask students to “show appropriate steps” or “explain your reasoning clearly”, but students might not know what we mean. By showing examples of good student work, or samples of student work along a spectrum of achievement, we can show students exactly what “fair” “good” and “excellent” mean. (Tomlinson and McTighe)

Not only can we advance learning, we can demonstrate to students that they can improve, a key factor in establishing a growth mindset. Being shown how to work to improve and seeing their own progress is empowering to students, and contributes to a sense of self efficacy, which is a major predictor of success in mathematics. This is truly assessment “as” learning! Helping students understand that they can improve with effort will help them become lifelong learners.

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