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# What does differentiation of instruction mean?

Differentiation of instruction means that the teacher is able to engage ALL students within THE SAME lesson. The lesson being taught should incorporate specific tasks that allows for multiple entry points.

Jo Boaler in her book, Mathematical Mindsets refers to these tasks as being low floor and high ceiling tasks. If you are looking for a rich educational resource to read this is one I would most definitely recommend.

# EXAMPLE OF LOW CEILING/HIGH FLOOR PROBLEM

You went to Walmart with your mom and saw a game you would like to buy for \$20.00. Mom said you had to earn the money by helping out with the dog.

Mom's deal was this: each time you help with the dog she will pay you 2 quarters. You told dad when he came home from work and he told you he would match the two quarters with 5 dimes.

• When you have earned \$20.00 how many quarters and how many dimes would you have?
• How many chores do you need to do?
• You must work in small groups and represent your answer in different ways.

## You may ask how is it this a low floor and high ceiling task?

• Some students may take out paper or plastic coins and lay out ALL coins and COUNT ALL until they reach a sum of \$20.00. Then they will count all the repetitions of the pattern core/unit to determine the number of chores to be done to purchase their game.
• Other students might show the unit/core that is 2 quarters and 5 dimes and know that is a total of \$1.00 so therefore 20 chores would have to be done to earn \$20.00 because 20 groups of 1 is 20 or 1 x 20 is 20
• Other students might show/draw 10 repetitions in sequence to \$10.00 and just realize 10 and 10 is 20 so they just need to double it.

Can you think of another solution/pathway a student may take depending if they are at a surface, deep or transfer level of learning with patterns, money, counting by 25’s, 10’s, 50’s, as well as their overall number sense?

You may ask where the terminology of surface, deep and transfer learning comes from. This is from a book I read, Visible Learning for Mathematics, by John Hattie, Douglas Fisher and Nancy Frey. This is another excellent resource to get your hands on!

## Another way to differentiate is to provide a question and then build on it by asking, if you found that solution, what if…  or are you able to start at and….

Example 1:

Your class is discussing number patterns or divisibility and you ask your students to skip count. For example, we are skip counting by 2. I would say your count can start a 0, 20, 78, 134, 546. (record these choices on the board)

Once students start their work, I walk around because sometimes students will make the choice that is too difficult or too easy and I will quietly encourage them to try another.

Example 2 - Problem:

My natural path doctor prescribed a special capsule for me to take. The recommended dosage on the bottle is to take 2 every morning with breakfast. I opened the bottle this morning and took 2. What date would I take my last two capsules or how many days would it take until the bottle is empty?

Again I walk around, engaging with my students, and if some students are finishing  and others need more time or are needing my guidance, I will extend this problem with the following question:

However, my doctor said if I wanted to give my body a good kick start, I could take 3 capsules daily. If I did take 3 this morning what date would I finish the bottle of 30 and would it be more or less days than if I took 2 daily and why?

I could further extend this problem by asking,

I was recommended to take these capsules for 3 months. If I took 2 a day, how many bottles would I need? If I took 3 a day how many bottles would I need?

If the cost of one bottle is \$15.75 including GST, what would the total cost be if I took 2 a day? 3 a day? Would the cost be greater or less if I took 2 a day or 3 a day? Why?

THIS REQUIRES YOU TO BE ON YOUR TOES. IT IS QUICK AND EASY TO INCORPORATE BUT YOU NEED TO BE THINKING. I EITHER JUST WRITE KEY WORDS ON THE BOARD OR YOU CAN HAVE THIS TYPED UP.

I MAY STOP BEFORE ALL SOLUTIONS ARE FOUND, BUT IT GIVES TIME FOR ME TO WATCH AND LISTEN TO MY STUDENTS.

IT ALSO GIVES A LITTLE EXTRA TIME FOR THOSE WHO MAY ONLY FINISH THE FIRST QUESTION AND THIS IS PERFECTLY OKAY. ALL STUDENTS ARE ENGAGED AND SOME OF MY STUDENTS EVEN COME UP WITH MORE QUESTIONS OR THOUGHTS/CONNECTIONS RELATING TO WHAT WAS ASKED!!

Throughout my Math For Success teacher guide books, at every grade level K-6, you will find these types of problems!

Day By Day Math Guide

# How does/should a teacher approach this in his/her classroom?

I am going to begin by saying how this SHOULD NOT be approached and that is by homogeneous or ability grouping students. I am so very disheartened to hear of teachers and/or schools (districts) making the choice to group students according to ability.

The talk is that the reasoning behind this decision is based on the RTI model in which students with challenges are to be provided targeted instruction.

I have read the book, Simplifying Response to Intervention, Four Essential Guiding Principles by Austin Buffum, Mike Mattos and Chris Weber,  and I will differ in opinion that the founders of RTI are moving schools in the direction of ability grouping with insurmountable research that proves the negative effects of it.

Is it perhaps being interpreted to meet the ideas of specific schools and individual teachers? No matter what the thinking is and I am confident that these decisions are being made in the best interest of our students, BUT it still comes down to grouping students into low, medium and high levels of ability.

In my Grade 5 and 6 years of school,  I was ability grouped for L.Arts. There was the A, B and C groups and I was placed into the B group. Behind backs and in corners students talked and everyone knew, despite the positive talk of the adults, who was in which group.

That stigma stayed with me right through to college. I failed my first English writing assignment in College and was absolutely devastated. I went to the Learning Centre to help improve my writing skills and without any challenges, I was successful. I completed my Masters in Elementary Mathematics and throughout my Masters program I was writing at an academia level and achieving scores of A and A+. With that said, I will put forth the following question, was I really a ‘B’ level student or could have I excelled if provided rich instructional tasks in a heterogeneous class?

Again, if you were to read Jo Boaler’s book, Mathematical Mindsets, which I highly recommend, she cites countless research from various countries, in which ALL students who were homogeneously grouped according to level of ability in math scored lower on testing than students who had rich instruction in heterogeneous classes. She also said even the students in the the so called ‘high level’ classes held negative attitudes towards math and didn’t go further into math studies after high school or if they did move on they dropped math after their first year of college or university.

I have taught at a grade 3 level for quite some time and it is so sad to count how many students over the 25 years have come into my class telling me they hate math or they are not good at math. HOW CAN THIS BE AT AGE 7 AND 8?

We want to help ALL STUDENTS develop a mathematical mindset through RICH, HIGH LEVEL INSTRUCTION; SENDING THE MESSAGE WE BELIEVE IN THEM; AND CELEBRATING THEIR MISTAKES!

Mathematical ability is NOT hereditary, nor is it for ONLY those with a math brain. ALL our students can learn mathematics and that is the mindset we as teachers need to have as we teach to our class that includes children of all ability levels.