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Elements of Effective Lessons and
Lesson Planning

There are many ways to structure an effective math lesson.  Our lesson plan template outlines one way to structure a standard math lesson intended to deliver, investigate, and practice new math content.  This is intended as a good starting point for new teachers, but it is important to remember that as we learn more strategies, use more rich tasks, practice using games and activities, and learn more about varied math instruction, this standard template will vary.  Regardless of the template, pacing, content, and order of instruction, it is important to remember some elements of an effective lesson that should be considered:

  • Connection to prior learning and experience (often in a Warm-Up)

  • Connection to real life

  • Collaboration with others (often a cooperative learning task)

  • Opportunity to practice problems independently

  • Closure (often in an Exit Slip)

  • Assessment (formal and/or informal)

In the sections below, you will read about some important elements of the standard lesson plan.

Warm-Up

Getting Ready to Learn

The warm-up is at the beginning of class (often on the board as students walk into the classroom).  It is intended to help students get ready to learn.  Warm-ups are typically 5 - 8 minutes long and may

  • review prior skills and understanding

  • activate prior knowledge

  • make connections between prior learning and new learning

  • Practice essential math fluency

  • Play with, make sense of numbers

Warm-Ups should be designed so that students are able to do them independently.

It is not necessary (nor is it a good investment of time) to "go over" warm-up problems daily.  Instead, it is helpful to walk around assessing student understanding (and on-task behavior) and address a common error or misunderstanding before moving on.

 

Click the button below to access an article with some great warm-up ideas! 

Also called hooks, bridges, or attention-getters, the anticipatory set engages your students in the learning that is about to happen. It sets the tone for the lesson and makes students hungry to learn more. Think of the anticipatory set as a movie trailer. The trailer doesn’t tell everything about the movie but provides enough glimpses to leave you wanting more.

An anticipatory set might include a:

  • Short Video

  • Picture or icon

  • KWL chart

  • Cartoon

  • Personal story or experience

  • Discussion about movie, book, current event that pertains to topic

  • What do you notice?  What do you wonder?

You're not going to be able to create an anticipatory set every day, but you'll be inspired to try -- at least at the begininig of a standard, a content area, or a unit, when you see the engagement fostered by a good anticipatory set!

Anticipatory Set

Set the Stage for Learning

Direct Instruction

Learning or Extending Content

Direct instruction is a teaching model that includes the teacher's explanation of students' concepts or skills, often in introducing new material and concepts. In this part of a lesson, the teacher provides a combination of facts and concepts in connecting prior knowledge to new learning.

The instruction provided usually revolves around small chunks of information where the teacher makes connections, provides background, or introduces a set of procedures that align with the learning intentions. 

In the standard lesson plan, direct instruction occurs at the beginning of the lesson, leading students to practice the new procedures, problems, and concepts collaboratively and independently.  Students will extend learning through guided practice, problem solving, and inquiry.

Direct instruction should take up no more than 12 minutes of class time.

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In some lessons, students will be asked to conduct inquiry and investigation before returning to direct instruction where the teacher will discuss students' thoughts, generalizations, and discoveries, and solidify into practice.

When material is complex, teachers may chunk direct instruction allowing students to digest and practice smaller parts, and then returning to direct instruction of another "chunk" (which students will then digest and practice).

For instance, instruction about properties of triangles might go something like this:

1) Inquiry/investigation: Triangle Sum Theorem

Students work in groups to construct 3 triangles (maybe one acute, one obtuse, and one other of their choice), or you can provide groups with cut-outs of different triangles.  Then they will use a protractor to find the measure of each angle in the triangle.  What do they notice?  (If they don't notice anything, you may prompt them to find the sum of the angles).

2) Direct Instruction ChunkI

Consolidate discoveries and generalizations to introduce the Triangle Sum Theorem.  Guide students through a problem or two, and give them a problem or two to solve in their groups

3)  Inquiry/investigation: Exterior Angle Theorem

Students work in groups to find the missing angle in several triangles (using the Triangle Sum Theorem (you can go back to the problems above), Then they will use a protractor to find the measure of one external angle in each triangle.  What do they notice?  (If they don't notice anything, you may prompt them to compare the exterior to the interior angles).

4) Direct Instruction Chunk II

Consolidate discoveries and generalizations to introduce the External Angle Theorem.  Guide students through a problem or two, and give them a problem or two to solve in their groups (some include both theorems above).

This is probably enough for students to really dig into and understand in one lesson.  The following day, you might review these two properties before continuing in the same manner for two or three new properties.

Direct instruction may followed by Guided Practice.  This requires asking students to test their understanding by practicing  under the guidance of the teacher.  In Guided Practice, students might:

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  • Practice skills under direction of teacher

  • Think aloud

  • Solve problems on the board or whiteboards

  • Practice in groups while teacher circulates

  • Students practice/teacher questions

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Guided practice should be limited to a problem or two under teacher guidance before asking students to practice in groups and/or independently.

In some cases, Guided Practice may be combined with DI, when students are called on to contribute to the instruction and practice (rather than the teacher doing all the "telling").

Guided Practice

Practicing Content

with Support

Collaboration and Cooperative Learning

Working together to increase understanding

Cooperative learning is more than just "students working together". Good collaboration involves pairs or small groups of learners working together as a team to solve a problem, complete a task, or accomplish a common goal.  Research has show that collaboration in secondary math is important because:

  • Students are more willing to solve challenging problems as a group.

  • Students are often able to explain things to each other in ways that make more sense than the teacher’s original explanation.

  • Students are more willing to ask questions and take risks in small groups.

  • Students learn more when they invest in math discourse.

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Students do best in cooperative learning when problems are well chosen, students are assigned roles, specific meaningful goals are set, and students must demonstrate both individual responsibiliy and positive interdependence in the task(s).

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Throughout our school year we will investigate different ways to incorporate cooperative learning in the classroom, but even without formal cooperative learning structures we must understand that it is important for students to work together, contribute to group problem solving, and engage in math discourse in every lesson.

It is very important for students to practice math concepts independently.  Independent practice helps engage students formatively in their own learning. It also provides vital assessment information to the teacher regarding student understanding, mastery, and misunderstandings that may inform future instruction.

As an analogy, let's think about learning to play a musical instrument. During a piano lesson for instance, the teacher provides the student opportunities to practice playing scales, notes, and chords through guided practice. The teacher makes corrections and suggestions in real time, helping the student make improvement in practice, leading to the ability to eventually play more complicated pieces of music.

The student then practices independently outside the lesson, without the teacher.  In fact, the instructor plans the next learning session under the assumption the student is practicing regularly between lessons. (Solution Tree, 2018).  If students are only playing the piano during piano lessons, their ability to retain knowledge, build upon prior knowledge, and apply their knowledge to more complex pieces is compromised.

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In the same way, in the math classroom, if students are only doing the math alongside the teacher or with group members, they will have more trouble achieving mastery of concepts, retaining knowledge of procedures, and applying understanding to novel problems, It is also impossible for the teacher to assess individual understanding and mastery.

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Sometimes, independent practice will include homework (we'll talk about homework in later posts), but homework poses some problems.  Teachers cannot rely on informative assessment through homework for several reasons:

  • Many students do not complete homework

  • Cheating/copying is always a problem with homework

  • Assistance at home is inequitable for our students.  

  • Technology can "do the problems" for students, making it impossible to know what they can do themselves.

Therefore, it is vital to include some independent practice on a regular, frequent basis in the math classroom.

Independent Practice

What can our students do on their own?

Closure

Wrapping up the Lesson

Closure in a lesson often involves an Exit Ticket or Ticket-out-the-Door. The purpose of closure in a lesson is that it helps solidify the understandings and skills learned. It benefit the students by reviewing and extending their understanding, and it also benefits the teacher to provide assessment and information in a quick, easy-to-access format.

Good closure should be done by students and might include:

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  • A review or consolidation of practice, procedure, and knowledge learned in the lesson

  • A summarization of concepts learned

  • An opportunity to ask questions about the lesson

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Student should not be ending your class (and moving to the next class) standing at the door chatting.  They should be wrapping up their learning, making connections, and thinking about the lesson and what might be coming next.

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Take a look at the following resources for ways to vary your Exit Slips or Closure tasks:

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