Enrichment in the Math Classroom

Enrichment in the Math ClassroomColegio Bolivar Forum on Teaching and Learning, 2008Jeff Mahood and Matt StephensPLEASE TRY TO SIT AT A TABLE WITH PEOPLE WHO TEACH APPROXIMATELY THE SAME GRADE LEVEL AS YOU.ENRICHMENT IS ALL ABOUT DEPTHWhat is Enrichment, Anyway? What Is Enrichment? What Is It Not? Why Enrich?Brainstorm EnrichmentAt 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?Jeff4What 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.MattInstead of classroom time = working independently5What is it not?MandatoryJust a set of extra questionsWorksheets or rote workA time-killer

Jeff6Why Enrich?It may:Offer opportunities for further or deeper learning.Reinforce comprehension Expose students to aspects of math from beyond the curriculumAid in retention of fundamental skillsGet students excited about mathAllow students to use their skills and interests from outside of mathProvide opportunities to apply math in real-world situationsAllow students to practice higher-order thinking skillsAllows students to demonstrate creativity

Matt7Enrichment in Your ClassroomHow 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 Persons Way(Also known as Effective Time Management)

Have a look at the textbook or resource manual.Matt9How Do You Choose a Subject? Tangential SubjectsFrom personal knowledge or a little research, find a topic that is tangentially related to what youre doing in class.Broadening or deepening the ideas presented in the curriculum.JeffTangential: Knowledge extensions, or subjects from future classes are all possibilities. Then, design an activity to allow the student to discover that idea.10How Do You Choose a Subject? Mathematicians or Math HistoryYou 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.MattBiography: Of course, mathematicians are the roots of all our knowledge, and so theyre always relevant to what youre studying in class. Problems: Bridges of Konigsburg, Map-colouring, Fermats Last Theorem, even proving the Pythagorean Theorem11How Do You Choose a Subject? Recreational MathAspects 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 DevilJeffMany students like to think of math as solving equations and graphing functions, especially between grades 6-10, because thats what they tend to focus on there.

12How Do You Word An Essential Question? What Are The Learning Goals?The tasks learning goals allow us to decide how to focus the Essential Question

General Overview of Facts/ConceptsAnalysis of importanceTrivia knowledge for knowledges sake sparks interest!WHY does something work? HOW does something work?Relationships between ConceptsDeeper understanding of known conceptsEnjoyment of non-curriculum mathematics pattern recognition, logic and puzzle solving, etc.

Matt13How Do You Word An Essential Question? What Are They Handing In?PresentationPosterEssayComputer ProgramMobileTraditional Solution SetWhatever else you can think of...

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

I dont 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.Its not just giving them an hour and some glue to make a poster of mathematicians.Jeff14How 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?Matt15FOR 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 Eulers Formula. GOOD: Find a relationship between the vertices, faces, and edges of a polyhedron.Jeff16How 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 were 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.Matt17Examples: 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."Jeff18Examples - 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.Matt19Examples - 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?Jeff20Example - Number SenseTake integers from 13-24. Make a multiplication table of them, and identify the perfect squares. (Note: It wont just be along the diagonal)

Magic SquaresMatt21ACTIVITY PART ONEGroup by levelBrainstorm possible topics choose one from within the curriculum and one from outside. If youre 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 theyll be submitting at the end.

Jeff22Okay, 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.

Matt24How 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!)

Matt25Do 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: Standard bonus marksMarks to surpass a capNo grades.

MattExtra credit: you can always scale to reduce or augment importanceThree Options: Possibility of mixing in the grey areas between the three options.

26Do You Count It?Standard Bonus MarksProsConsEasyIncentiveHelps students improve their own grades through extra work.Encourages copying (short timeline corrects this) Often doesnt help the ones it needs to helpTakes focus away from the core curriculum.Matt27Do You Count It?Marks to Surpass a CapProsConsAllows 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 its impossible for an average student to get 100%.Jeff28Do You Count It?Not GradedProsConsOnly students who are actually interested do the workNo incentive for others to copyAllows a broader range of assignments outside the curriculum because not attached to a gradeSuper easy to implementDeadlines unnecessaryNo incentive for the student to try itDeadlines unnecessaryMatt29ACTIVITY 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

Jeff30Conclusion Session Evaluation Please also note down if there are any questions that you had about Enrichment that were not answered. Anything you expected but didnt 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.