Students Can Demonstrate Understanding with a Shorter Assessment Piece
I've been working on changing my assessments. Marian Small has been a huge influence on this ever since I attended her session on the High School Math Institute. Now, every time that I am creating a new assessment, I search "Marian Small" +topic in google to help generate some ideas rather than always recreating my own. I always try to change the question so that I'm not just stealing her ideas, especially since I share so many of the things I create with the teachers around Central Alberta during Assessment sessions. However, I certainly use her framework as it makes the questions so much more interesting and checks for a deeper understanding.
When I present a session on assessment, one of the first slides I show asks
What Does...
on a test tell us about the student? 
The inevitable response is that "They can operate a calculator". It terrifies me that when I first started teaching, I would strongly populate my exams with questions like this...ones that tested their ability to operate a calculator. Of course, I would have to have many questions like this to test whether they could do it in multiple settings. So, my students were answering a bunch of repetitive questions and I was stuck marking a bunch of repetitive questions. Of course, it was easy because a simple answer key worked but what did it tell me about my students' understanding? Pretty much nothing.
Once I began my assessment journey, I realized how easy it would be to change these questions to better get at their true understanding. Yes, it makes my answer key pretty much useless. Yes, it might take me extra time to mark the question. However, I was then able to cut down 2050 questions to just a few deep questions so it balances out in the end.
Giving up a little bit of control during the assessment process allows students to demonstrate their knowledge more fully.
I wanted to know if students truly understood that 3^{4} is really 3x3x3x3. Why not let them pick the numbers they used? By wording the question carefully, I could prevent "easy outs". This is how I changed the first question: (The bold portions would not be bolded on the test. They're bolded so you can see how I tried to ensure I didn't get situations like 1^{1}
. The others are samples of how I would assess the other outcomes.
Question 1:
Choose a base greater than one: ____________
Choose an exponent greater than the base: ______________
Create a visual representation for the power you produced.
Question 2:
You simplified an expression. The result was (3/4)^{1/2}. What could the original expression have been if you were using the following operations?
(a^{m})(a^{n}) = _____________________
a^{m}/a^{n} = _______________________
(a^{m})^{n} = _______________________
Question 3:
Choose one of the following operations: (ab)^{m} OR (a/b)^{n }
Choose your own bases and exponents, however "m" and "n" can NOT be a whole number.
Explain TWO ways to determine the result without using any "rules".
Question 4: (From Marian Small)
Write an expression that you would likely use three power laws to simplify. Use a variety of exponents.
Simplify it.
Question 5:
Jamie states that a^{0} is a. Pat says that it is 1. Jessie says that it is 0. Casey says that none of them are correct. Who is right and how would you convince everyone else without saying "The rule is..."
Like I said, when I first started teaching, I would overkill on questions. I believed an assessment was supposed to take all class which meant I would have to put a LOT of questions on it. Then, of course, when you have a class of 1630 students, you had to MARK all those questions. I would now use this assessment to check students understanding in Math 9:
Side Note #1: These types of questions are excellent discussion starters, review questions, exit slips, "homework checks" and formative and summative assessment questions.
Side Note #2: Students should NOT see this style of question for the first time on a final assessment! They should experience it ahead of time. I would worry that they would feel like they were being "tricked" otherwise.
I've considered including the outcomes on the assessment to give students and parents a better idea of students' success within each outcome. I'm trying to decide between two formats:
Option 1:
I could attach the table included below to their paper after the assessment. I would highlight the questions they were successful (or not successful) at.
Number


1
Q1 Q5 Q4 
Demonstrate an understanding of
powers with integral bases (excluding base 0) and whole number exponents by

2
Q2 Q2 Q2 Q3 Q3 Q4 All 
Demonstrate an understanding of
operations on powers with integral bases (excluding base 0) and whole number
exponents

OR, I could add a column to the assessment (something like below) but then I would have to make sure to blank out the pieces that give answers.
Exponent Laws Assessment
Question 1: Choose a base greater than one: ____________ Choose an exponent greater than the base: ______________ Create a visual representation for the power you produced.  Demonstrate an understanding of powers with integral bases (excluding base 0) and whole number exponents  
Question 2: You simplified an expression. The result was (3/4)^{1/2}. What could the original expression have been if you were using the following operations? (a^{m})(a^{n}) = _______________________ a^{m}/a^{n} = _______________________ (a^{m})^{n} = _______________________ 
Demonstrate an understanding of
operations on powers with integral bases (excluding base 0) and whole number
exponents
 
Question 3: Choose one of the following operations: (ab)^{m} OR (a/b)^{n } Choose your own bases and exponents, however "m" and "n" can NOT be a whole number. Explain TWO ways to determine the result without using any "rules 
Demonstrate an understanding of
operations on powers with integral bases (excluding base 0) and whole number
exponents
 
Question 4: (From Marian Small) Write an expression that you would likely use three power laws to simplify. Use a variety of exponents. Simplify it.  Solving problems involving powers  
Question 5: Jamie states that a^{0} is a. Pat says that it is 1. Jessie says that it is 0. Casey says that none of them are correct. Who is right and how would you convince everyone else without saying "The rule is...  Show that a power with an exponent of zero is equal to___  
So there it is. A short assessment piece I could use to check my students' understanding of the power laws. What do you do in your classroom to assess students' understanding of the power laws? I would love to hear ideas for a performance assessment piece, other assessment questions (ie. a fabulous word problem), etc. How would you share the connections between their responses and the outcomes (like I was attempting in option 1 and 2)?
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Sandi