Saturday, 26 May 2012

Experiments in Grade Configuration in Math

We experimented this year with a new grade configuration in math in the middle school. We had a great in service PD in the beginning of the year which made the compelling case for certain assessment practices and how they would benefit math learning statistically through externalized (MAP) scores. Up to that point, we had a grade configuration which had a mix of percentages such as 40% on tests, 30% on projects, 20% on quizzes and 10% on homework. Digging around the internet, I found many middle schools have a similar policy (give or take 10%) but I could never find any research supporting while this was infinitely superior to one configuration or another.  One of our school goals this year was to use data to make informed decisions about teaching and student learning, and I decided to pilot the grading policies recommended in my grade 7 classroom.

In short, the premise was that by putting 100% on summative assessments, students will achieve more. I was skeptical at first and this sounded to me like "high stakes grading" but the argument was that if you give students enough practice and have deliberate, spiraling teaching, by the time students reached their final unit test, they were most prepared to demonstrated their understanding. Key data points that supported this were:
  1. Our adopted California curriculum standards clearly states that "Summative Assessments are the most valid and reliable indicator of in-depth understanding." (California Curriculum Framework)
  2. Statistics supported that through enough practice such as assignments, quizzes to check for foundational knowledge, projects and practice tests, students had the highest level of understanding in summative tests. 
  3. To grade everything puts undue stress on students and does not allow them to benefit from mistakes early in the learning process. A teacher must know that some work is "practice" on the road to understanding. 
All of these were valid points. I had just finished reading a policy on California grading standards that showed that "yet to date, there are no national, evidence based studies addressing grade-span configuration issues, and few studies actually use empirical data" (National Forum), but I decided to jump in feet first. We had problems in the past with grade inflation giving the illusion of higher understanding which often contradicted external test scores. For instance, little Johnny had a "C" or even a "B" through retesting or resubmission of work in class, although our MAP test data might put him in the 40% percentile which made for awkward recommendations on IEP's or math placement for the next academic year. We wanted to design a grading policy that was accurate and honest.

Our principal was very supportive of our efforts, although there were a number of parents in the community that did not like it. Typically, they were parents of middle and low achieving students as you can imagine. I did survey the students anonymously at the end of the year to get their opinion on the policy and most of them had softened to the notion. Many of them were tired of classmates not doing their share of the work in group based projects or investigations, and this inequity reared its ugly head to me as well. Students that "demonstrated mastery" through an project-based assessment or authentic assessment did poorly on a summative task on their own. Were they lucky? Or did someone else like a classmate, parent or tutor do the work for them?

Some of my key findings in action research after the school year were:
  1.  Although MAP test data did improve significantly in median score, mean scores were little changed. 
  2. Students did complete learning tasks that were not graded, as they were rungs on the ladder towards master, but there was almost an obsessive nature that some students had in test preparation. 
  3. For end of the year portfolios, I had students reflect on their "best learning experiences" on their blogs and told them not to focus so much on the grade, but rather the experience. Many students did reflect on their learning of topics, projects, but many of them chose a particular test that they did well on.
There were a lot of other conclusions that we made, but as they were more speculative than data based, so I shan't share them minus one-our students already consistently exceeded external test scores of typical 7th graders in the United States. Our seventh grade students usually test at the 10th grade level and as students math proficiencies increase, so does their tendency for growth. So, as the "smarter" and older your classes get, you can expect smaller gains from them (NWEA). Finally, if a hypothesis is not supported by the data, than it's unsupported by the data. Perhaps the data presented to us earlier was before and after a huge curriculum change or integration of differentiated instruction. We already had been differentiating instruction and our curriculum is fairly sound, so perhaps we were overly optimistic in expecting the same gains.
I want students to learn and appreciate math on it's own terms, and not create a culture of "math test takers." This seems to be the bottom line in American education these days and although test preparation is a necessary evil for high school and college, to create a classroom of test taking machines seems like a scene out of Pink Floyd's "The Wall". I can just hear the music playing-"We don't need no education!"

The challenge for me now is how to balance the academic rigor in a way that allows students to make mistakes and learn from them, connect math to real-world learning experiences, flesh out 21st century learning in problem solving and differentiate instruction for each an every student in my classroom so that every student learns. Oh, not to mention using assessment that is accurate and effectively indicates to students and parents areas for remediation. Through this whole experience, I am reminded that teachers do not merely teach, we are psychologists, scientists of our craft and politicians as we communicate with parents and the larger community.

I reminded of something Bill Powell told me in a workshop: "Good teaching is hard. If you want to do something easy, become a doctor."

  • Northwest Evaluation Association (NWEA) RIT Scale Norms 2011-Table 5.3 page 46
  • Mathematics Framework for California Public Schools Kindergarten to Grade 12. page 240
  • Policy Statement on Grade Configuration-Issue 5, July 2008 "The National Forum to Accelerate Middle Grades Reform"

Friday, 18 May 2012

Teaching Creativity in Mathematics

It's a shame that the arts do not have the same priority as do core subjects such as reading, writing, math and science in public schools. When the fat is in the fire, and budgets need to be cut, they're the first to go. There's even a hierarchy in the arts as music and art are generally given a greater priority than dance and drama. I feel very fortunate working in a school where they are all valued and students are given equal opportunity to develop these interests.

I've always been inspired by the arts and love the creativity that they instill in students. It makes so much sense that to become good at mathematics, students cannot resign themselves to learning the "one right way to do it" in order to derive an answer, but to cultivate multiply ways of approaching a problem and solving it through a variety of ways. I've undergone a metamorphic change in my career as a math teacher by shying away from multiple choice assessments to use assessments that require open-ended answers that require justification and analysis. When students are able to use these skills rather then just fill in the correct "bubble" with a 25% of a correct answer, we really start to see the depth of math education and cognitive skills grow by leaps and bounds. Some things that I like to do generally follow this pattern:

Beginning of the Year
During the first two weeks of class, I share a number of problem solving strategies that I have students repeatedly practice and apply over and over. We have a large poster of these strategies in the classroom and we refer to them throughout the school year through large group or whole class investigations. They are:
  • Doing a similar problem
  • Trial and Error
  • Constructing a table
  • Making an educated guess
  • Working backwards
  • Looking for mathematical language
  • Drawing a picture
  • Looking for a pattern
  • Making a list
Research generally supports that when students are well acquainted with a number of problem solving strategies, it reduces their anxiety, gives them a greater tendency to attempt problems and not to give up prematurely.

During the Year
We have some great math standards that are under the umbrella of "Mathematical Reasoning". The strand is intended to integrate into other strands such as Algebra, Computation, Geometry, etc., but our math program does not exactly dictate where these standards of mathematical reasoning will fall. The standards typically read like this:
  • Break a problem into similar parts
  • Eliminate unnecessary information
  • Develop a conjecture about this problem
I think these strategies are so applicable in life and take themselves out of the box of merely mathematics and use the creative skills. I've been learning how to take this to the next level by integrating problems like this into summative assessments. For example instead of merely having a linear equation such as y = 2x -3, and having students solve for it, you could ask them:

"Write a linear equation with "y" and "x" where the solution for "y" will always be 1 whatever is substituted in for "x". Explain and support your equation with mathematical reasoning, notation or other supporting evidence.".

Problems like this really demonstrate not only a students ability to solve a problem but to understand the complexity and underlying relationships inherent in math. Answers for this will be varied and many explanations can explain this relationship. How can we expect this creativity to be measured by computers on a multiple choice test?

Game Based Learning

Since I've finished my yearly math curriculum with two weeks to spare I've decided to enter my students in a gamed based video game mystery called "The Lure of the Labrynth". It was developed by some researchers at MIT who wanted to see if video gamed based learning infused with mathematics would improve students math skills. I learned about it through my math learning community through "Edumodo". Check out "Lure of the Labrynth" here.

A teacher registers their class and puts them into groups of 4-6 students and then each student builds an avatar in an underground virtual world and sets out to save "pets" through a number of math challenges. The more rooms one unlocks and more pets one saves, the higher score a student gets. There is a story line much in the medium of a comic book where the reader receives clues surrounding the larger mystery.

While students play the game, data is collected an analyzed by researchers at MIT to determine the effectiveness of this means of teaching math. Many of the math applications are subtle and require critical thinking and multiple problem solving strategies such as trial and error, reflecting on similar problems an so on.

I'm curious to what the data will support. My students are chronically addicted to video games and comic books too, so hopefully they'll respond favorably to the challenge. I'm sure the working hypothesis is whether bringing math to settings students enjoy will bring a heightened sense of understanding as students will see the immediate application and "reward" afterwards. I'll comment later when our class is finished!