Enrichment in the Math Classroom

Colegio Bolivar Forum on Teaching and Learning, 2008

Jeff Mahood and Matt Stephens

PLEASE TRY TO SIT AT A TABLE WITH PEOPLE WHO TEACH APPROXIMATELY THE SAME GRADE LEVEL AS YOU.

ENRICHMENT IS ALL ABOUT DEPTH

What is Enrichment, Anyway?• What Is Enrichment?• What Is It Not?• Why Enrich?

Brainstorm “Enrichment”

At your tables, take five minutes to discuss the concept of “Enrichment.” Use the following question to guide your discussion:

What does “Enrichment” mean to you as a teacher, to your classroom, and to your students?

What is Enrichment?

Us: “Providing extra opportunities for learning. It may take place either outside classroom time, instead of classroom time, or to keep fast-working students on-task during classroom time once they have finished the required work.”

What is it not?MandatoryJust a set of extra questionsWorksheets or rote workA time-killer

Why Enrich?It may:• Offer opportunities for further or deeper learning.• Reinforce comprehension • Expose students to aspects of math from beyond the

curriculum• Aid in retention of fundamental skills• Get students excited about math• Allow students to use their skills and interests from

outside of math• Provide opportunities to apply math in real-world

situations• Allow students to practice higher-order thinking skills• Allows students to demonstrate creativity

Enrichment in Your Classroom•How Do You Choose a Subject?•How Do You Word the Essential Question?•How Do You Evaluate It?•Examples

•ACTIVITY – PART ONE

How Do You Choose a Subject? –The Lazy Person’s Way(Also known as “Effective Time Management”)

Have a look at the textbook or resource manual.

How Do You Choose a Subject? – Tangential SubjectsFrom personal knowledge or a little

research, find a topic that is tangentially related to what you’re doing in class.

Broadening or deepening the ideas presented in the curriculum.

How Do You Choose a Subject? – Mathematicians or Math History

You can design an option to get students to write a short biography on said mathematician.

There are loads of famous historical problems that can be given to students to explore.

How Do You Choose a Subject? –Recreational Math

Aspects of math such as Graph Theory, Matrices, Logic Puzzles, Infinity, The Golden Ratio, Pi, and even convergent and divergent series are part of what we call “Recreational Math” and can be given as research topics.

Number Devil

How Do You Word An Essential Question? – What Are The Learning Goals?The task’s learning goals allow us to decide how to focus the

Essential Question

General Overview of Facts/Concepts Analysis of importance Trivia – knowledge for knowledge’s sake – sparks interest! WHY does something work? HOW does something work? Relationships between Concepts Deeper understanding of known concepts Enjoyment of non-curriculum mathematics – pattern

recognition, logic and puzzle solving, etc.

How Do You Word An Essential Question? – What Are They Handing In?

Presentation Poster Essay Computer Program Mobile Traditional Solution Set Whatever else you can think of...

The question must also address the depth of the task. Some quotes from Matt Stephens:

“I don’t want a timeline with just dates and events – I want a demonstration of the connections between when the events happened and why they happened in that order.”

“It’s not just giving them an hour and some glue to make a poster of mathematicians.”

How Do You Word An Essential Question? – Can You Be General?

If the question is a traditional challenge problem, the question can be as vague as you want it to be.

For example:

How many blades of grass are in the area bordered by H-Block, I-Block, the main path, and the science building?

FOR EXAMPLE...BAD (too unfocussed): Write a biography

of Blaise Pascal. GOOD: Research and report on the contributions that Blaise Pascal made to number patterns.

BAD (too much direction): Research and describe Euler’s Formula. GOOD: Find a relationship between the vertices, faces, and edges of a polyhedron.

How Do You Evaluate it?Because of the varied nature of the projects,

most of them need to be evaluated by rubrics. Depth of questions means that we’re not just asking “right/wrong” answers, but rather expecting students to give a more in-depth type of response. This precludes traditional “mark out of 20” grading.

Examples: Math JokesExplain why the following joke is funny:

A physicist, biologist, and mathematician were sitting on a park bench across from a house. They watched two people go into the house, and an hour later, they watched three people come out.

The physicist said, "The original measurement was inaccurate."

The biologist said, "They reproduced while they were inside."

The mathematician said, "If one more person goes inside, the house will be empty."

Examples - Coordinate GeometryShow them how to approximate a parabola or

hyperbola by drawing straight lines on a coordinate plane.

Ask them to create another curve by drawing straight lines, and describe the process.

Examples - ProbabilityHow many people do you have to have in a

room before the odds of two of them sharing a birthday are more than 1:1?

Example - Number SenseTake integers from 13-24. Make a

multiplication table of them, and identify the perfect squares.

(Note: It won’t just be along the diagonal)

Magic Squares

ACTIVITY – PART ONEGroup by levelBrainstorm possible topics – choose one from

within the curriculum and one from outside. If you’re having trouble with the latter, talk to the group one step up from you.

Develop an Essential Question for that one of your topics. Make sure to consider both what you want them to learn, as well as what they’ll be submitting at the end.

Okay, now how do You use it?• Should You Target Students?• How Do Students Get It?• Do You Count It?

Should I Target Students?In our opinion:It depends on the kind of enrichment being

done. Work-ahead needs to be carefully chosen, where extra assignments could be offered to anyone.

Students could benefit from individual targetting.

How Do Students Get It?Available in-class for those who take initiative to get it.

(Folder at the back of the room, PoW on the board, etc.)Direct handout for students who finish earlyA bonus question at the bottom of each assignment that

extend/generalize the conceptsCompletely separate assignments Separate agreements with studentsAfter-school clubs. (Yay math club!)

Do You Count It? It can be hard to justify “recreational math” – math outside

the curriculum – for grades.“Extra credit is worth as many points as you want it to be

worth in their final grade.”

Three options: 1. Standard bonus marks2. Marks to surpass a ‘cap’3. No grades.

Do You Count It?Standard Bonus MarksPros Cons

EasyIncentiveHelps students

improve their own grades through extra work.

Encourages copying (short timeline corrects this)

Often doesn’t help the ones it needs to help

Takes focus away from the core curriculum.

Do You Count It?Marks to Surpass a CapPros Cons

Allows talented students to demonstrate rewards for their talents and extra effort.

clearly shows who is putting in the extra effort.

Hard to justify to administrators, parents, and students (or even yourself!) why it’s impossible for an “average” student to get 100%.

Do You Count It?Not GradedPros ConsOnly students who are

actually interested do the work

No incentive for others to copy

Allows a broader range of assignments outside the curriculum because not attached to a grade

Super easy to implementDeadlines unnecessary

No incentive for the student to try it

Deadlines unnecessary

ACTIVITY – PART TWOUsing your Essential Question, develop a set

of instructions for your studentsDetermine your expectations of the students’

submissionsDevelop a complete rubric for your task

Conclusion• Session Evaluation• Please also note down if there are any questions that you had about Enrichment that were not answered. Anything you expected but didn’t get? Any way we can improve for another presentation?• www.mathninja.com/forum - whatever you finish with today will be typed up and posted for everyone to use. (Credit will, of course, be given.) Also, there are several webpages and books linked from there that might help.