--Originally published at FLN – matthewtmoore
Even in a highly structured content area like math non-linear organization provides choice and helps students make connections.
Math people love lines. High school and middle school teachers love straight lines. y=mx+b …. y-y=m(x-x) …. Ax+By=C… I am getting excited already!!! It is little wonder then that when we approach teaching we also approach the artistic and performance part of our job in a linear fashion. While this might be our primary organization pattern, it may not be the most beneficial for some students. In fact, limiting mathematical concepts to a linear pattern minimizes the interconnected nature of many topics in mathematics. higher order thinking in math is not best viewed as a straight line, but more as a constellation and collection of related ideas. Incorporating the interconnections of ideas in our planning and presentation can provide a better picture for students and reintroduce choice and freedom.
“But math builds on itself!”
To be fair, there are some areas of math in which one topic sits squarely on the shoulders of a previous skill or idea but just a quick look at a curriculum reveals opportunities for choice. Take a look below at graphing linear equations.
In the graphic you can see a 
screenshot from my LMS I have used in my flipped classroom. The topics are clearly arranged from top to bottom with each module including an instructional video, notes, some form of quiz, and a list of practice problems. This is better than 45 minute lectures but there is no choice as to content order and the unspoken message is if you get stuck on one topic there is no moving on.
“Now for something completely different”
Let’s take a look at how I reorganized the exact same digital content, paying close attention to first impressions and implied messages.
Click here to access the live document then just point and click http://bit.ly/GLENLBLOG
I have clearly defined a starting point to introduce key terms and the basic overarching concept. I have even included a “Pre-Skills” component for students who need the assistance. In this case, graphing points and understanding slope are pre-requisites for success and therefore supports are provided. Another subtle message is moving generally from left to right from pre-skill to mastery demonstration. I tell the students in class about this subtle pattern but clarify for them that they may attempt any piece in any order until finished. After years of teachers working linearly students feel the need for permission to deviate from the linear pattern. There are cases where a student chooses a topic and finds they need information from another topic somewhere else in the constellation. Once students find that they are free to jump across the circle to any topic to get what they need there is a new sense of freedom and control.
Choice lends itself to mastery
I must admit that because I used existing video instruction and notes that there is likely a “best” path where students are presented information in a linear fashion. I justify recycling because there is value in collecting information from a variety of sources much like most problems in life where information gathering is part of the process (and because I don’t wanna make new videos for this). Because students will work on different topics at different times, I also included a tech exploration, and redefined success by having the students create their own assessment demonstration problems increasing ownership and choice. Because not all students are doing the tech exploration at the same time I can troubleshoot individuals without dragging down the whole class. I can also assist students individually as learning issues arise rather than addressing the whole class. Now my mastery learning friends respond with “duh”, but non-linear organization of units or topics allows a more mastery style space within a school and community with very traditional expectations.
A final note about choice, ownership, and mastery. Moving to a non-linear lesson plan fundamentally changes the group space as well. It becomes nearly impossible to provide whole group instruction via lecture as it is now an interruption of the learning process rather than the basis of the learning process. If a need arises for whole group instruction it is because there is significant need for clarification voiced by a significant portion of the students, in which case whole group instruction is the correct medium. This is a type of learning ownership that both teachers and students must be trained and conditioned to accept but is an ownership that only choice and the rigors of mastery can provide.
Addendum:
It might be tempting to think that non-linear organization and its associated changes can only be accomplished with more academically talented students. My experience is with standard high school Algebra 1 students, which in my district represent third string math students at best. We pull the best and brightest out for Algebra 1 in 7th grade, and the next most talented group out for algebra 1 in 8th grade, leaving freshmen in Algebra 1 as the third group. This does not even address the many sophomores, juniors, and seniors that find themselves in Algebra 1. My argument for choice in learning for these students comes from the benefits of giving students who have traditionally had very little choice or ownership in learning the opportunity to have a stake in their own learning.
