Student guiding his own learning

Below is an example of how a parent allowed his child to guide the learning.  Good job Mr. Hansen!!

My 3rd grade son is deep into multiplication right now and our discussion about groups of things and multiplication eventually led to factors. And while he was pondering how many numbers can be multiplied together to get 12 or 16, he asked a very good question. He asked me "How come there isn't a division table?" and he went on to say "We could make a division table and sell it!"

I explained to him that division tables do not exist because once you put the numbers down the sides, the majority of the inside is empty because most numbers are not (evenly) divisible by the other numbers. I told him that when we do division mentally we actually use the multiplication facts in reverse, as we did with addition. But I could see in his eyes that my explanation didn't phase him much and he was still dreaming of selling millions of division tables to other 3rd graders. So I said "Let's make a division table!" Big smile.

So I opened a spreadsheet and we started making our table. I said to him "let's just go up to 20 for now and we can expand it later", otherwise we would be there for hours if we went all the way to 144 on both edges. So I numbered the rows and columns from 1 to 20 and then like a game of battleship I started calling off the rows and columns and waiting on him for the quotient, or as is the case in the majority of cells, his response "it doesn't divide".

It took less time than I imagined before he saw the light as to why it is difficult to make a good division table. Maybe it was all the dead ends with him responding "it doesn't divide", but he realized that most of the cells are empty and when you are talking about "whole" division you are actually talking about knowing which combinations of numbers are divisible in the first place which is essentially the multiplication table. But something neat happened. He noticed the diagonal of 1's (you can't really miss that) but then he spotted the 2's, so we studied that for a bit and I started coloring the 1's, then the 2's and then the 3's and so forth so that he could see how they each repeat and form a line. And then I pointed out that there were some numbers with no quotients at all, except for 1 and themselves. I colored those red and pointed out that every now and then those patterns of 2's and 3's and 4's line up in such a fashion that they skip a number entirely, we call those primes. They aren't divisible by any number, except 1 and themselves.

In the end (actually there is no end to this) I told him that what we are doing is turning the multiplication table inside out and if we go in and count all of the filled in cells there will be 144 of them (assuming we did the whole table), one for every entry in the multiplication table. You will find all of the 2's and 3's and 4's and so forth but spread out in a 144x144 table, instead of a 12x12 table.

Our division table is here...

http://dl.dropbox.com/u/39455389/DivisionTable.pdf

It was a very productive evening to say the least

Differentiated Assessment

I have had a great enlightening summer and would like to start things off with a bang!  Differentiated Instruction (DI) and Differentiated Assessment (DA).

Most teachers speak of DI as a common practice in their class and truly teach to the needs of EVERY student.  However, some teachers require these same students to jump through a common hoop of assessment.  This assessment can take forms of unit exam, a worksheet, a quiz, or any assignment which is the same for ALL. 

If we speak of DI so commonly, where is DA?

Rick Wormelli in Fair Isn't Always Equal: Assessing and Grading in the Differentiated Classroom states that "Assessment informs practice, and we take action".

DI MUST lead to DA in a classroom!  In a truly differentiated class, students can work towards learning outcomes at different paces, using different strategies, and mastering outcomes in different order.  As a result, teachers will need to assess these strategies differently and accommodate the assortment of learning styles while still measuring the learning outcomes.  Assessment and instruction does not have to be different but can occur simultaneously and appear different for each individual student. 

Alberta Education states, "The goal is not to have an individualized assessment plan for each student, but to have a manageable class assessment plan that is flexible enough to accommodate a range of student needs."

Some say assessment OF learning has to be common (or standardized).  Here are some examples of how we can have DA in assessment OF learning. (From Alberta Education)

Assessment of learning (sometimes called summative assessment) is the process of collecting and interpreting information to judge student achievement against predetermined criteria for the purposes of grading and reporting. Assessment of learning occurs at benchmark points in learning, such as the end of a unit or chunk

of learning. Consider the following examples of differentiating assessment of learning.

• Some students in a class choose to demonstrate their learning by writing a report, while others choose to create a poster, and still others choose an oral presentation.

• A teacher provides text-to-speech software and a digital version of the test to a student who has significant difficulty reading the questions in a social studies test.

• A teacher discards some marks collected early in the semester for a student

It can be done, and some our already doing it.  Assessment should not be a democratic process but an individual process.  Nor should assessment be done TO students, but actually WITH students.  Always remember you are not teaching statistics, data points, trends, or even groups, but actually living students with heartbeats, emotions, interests, and passions.

Math motivation through creative assignments

How do you motivate students to show their best without the use of marks?  The answer to this is very simple; Allow them to complete an assignment in a way such that their creativity, passion, and interests can shine.  Instead of marking this assignment, let them show it off.

Back in elementary school I had “show and tell”.  When it was my day I was allowed to stand in front of the class and show off who I was and tell my class who is the real David Martin (to this day I am still unsure though who I really am ;) ).  My teacher never graded me on this, and I was allowed to bring in anything from home.  How can this activity and true intrinsic motivation be mimicked in other grades?

I wanted to see if “show and tell” could be used in a gr. 12 classroom.  In my calculus class, we were studying functions and how to graph them.  Instead of asking my students to graph a function that had no meaning to them, I asked them to create a drawing (using multiple and piecewise functions) that represented who they truly were.  I was amazed by what I got, and here are some of the pictures:

Before I hear criticism and the stereotype that only calculus students are high achievers, I also gave an assignment about the Alberta winter games to my math 30 applied class.  In the assignment, students were allowed to represent the math in a creative way.   A student used an actual hockey jersey to represent their work.

Why did this work?

Students were asked to create something they can truly call their own and then to present their work to the class.  They were told to inform the class on WHY they drew or created the project they did.  Not only did we learn about math throughout this assignment, but the class also learned a little more about each other.  Students want to be treated as individuals and our assignments should allow this individualization to shine. 

Differentiated instruction first, differentiated assessment second.

I teach my students in many different ways, but I grade all using the same process.  There is a major problem occurring here.  Every teacher, I would hope, in Alberta knows what differentiated instruction is, but do they know what differentiated assessment is?  Educators know that every student learns in a different way, but do we know that every student can demonstrate their learning in a different way?

In our province, students write standardized exams in grades 3, 6, 9, and 12.  When our school is evaluated for “students’ achievement”, we are assessed on standardized exam participation, acceptable standard (over 50%), and standard of excellence (over 80%).  I find this very contradictory!  Our province is forcing every student to demonstrate learning in the same way, on the same exam and on the same questions.  Where is the differentiated assessment?

Currently, due to a mandate of my department, I am administering common exams to all my students.   Every test day I shake my head as I use differentiated instruction in all my courses, but then grade all my students the same. 

Next semester, I will be changing my grading process.  I will keep differentiated instruction, but I will be implementing differentiated assessment.  Students will inform me when, during the term, they want their outcomes assessed.  No longer will I grade based on my progress through the course, but actually grade the students on their own progress through the outcomes.  Students will also be allowed to demonstrate any outcome as often as they would like.

To truly be teaching for the students, we need to realize that differentiated instruction is no longer enough, we need to start implementing differentiated assessment as well.