Reverse Intervention
Below are some less than organized ideas about "math intervention," the concept that certain students need to brush up on material from previous grades to access grade-level math.
Thoughts on Intervention
Progression is inherent to modern mathematics education. The Massachusetts Frameworks for Mathematics Education prescribe the expectations for that progression, and encourage "intervention" for students who may be behind, as if they are experiencing an emergency akin to addiction.
Many schools have free blocks that serve as times for student services without loss of classroom time. When most students get free choice, but Suzy gets to go to math intervention, Suzy knows there's something "wrong" with her. Instead of building and learning on who she already is, Suzy must first fix the inherent wrongness requiring intervention.
Long before I was introduced to Paul Lockhart's Ladder Myth of Mathematics, the myth that math has some type of inherent order, I was experimenting with what I was calling "reverse intervention." I knew math didn't go in order. 7th graders were working with proportional relationships, and I was determined to show that students who did not know multiplication facts (those in need of intervention) could understand proportionality and manipulate values with the same proficiency as their times-table confident peers. Even my student who was without the ability to add whole numbers without a calculator could work with proportionality. I proved it when my students began to succeed in the general education classroom. With newfound success and confidence--and an understanding of proportionality--my students began to fill in their gaps. Instead of needing to fill those gaps first to then access grade-level math, grade-level math unlocked abilities to look backwards once students were ready. Hence "reverse intervention."
The reason students could interact with proportional relationships without knowledge of simple math facts is that math is relational. Most things in this world are. Every math concept is deeply related with others to form a diverse ecosystem of contextual ideas. The relationships between concepts may strengthen or weaken depending on a combination of factors, like interest or repeated practice. Learning is not linear, and forgetting does not mean learning never happened, and connections were never forged.
Traditionally (and according to state frameworks) students build understanding of proportionality from understanding of multiplication, which they build from understanding of addition. This process can easily go in a different order, it just requires a little creativity in the current educational climate.
I don't love the idea of reverse intervention now, because I don't believe these students need intervention. Having gaps of knowledge does not make anyone less worthy of education among peers. I tell middle schoolers all the time that despite their struggles with percentages, I have yet to meet a single adult that can't pull out their iPhone and calculate a tip. At some point, concepts click into place. It's my job as a teacher to provide entry points to those concepts, not to control the timing of the click.
Worked example: Proportional Relationships
I understand it might be hard to believe my students in the special education learning center could succeed in grade-level proportions without understanding addition. Here is a more specific description of what worked.
Looking at an architect's scale
My mother was an architect back before computers, so I was lucky enough to own a few of these types of rulers to bring in. We simply measured the classroom with a tape, and then drew it to different scales using the ruler. No math needed, just measuring. We each chose a different scale to use and compared our drawings. All the same shape! And the smaller the scale, the smaller the drawing. Makes sense. A 1ft:2in ratio will make a bigger drawing than a 1ft:1in ratio. We introduced the idea of scale here as something that can change, but once you choose a scale, the image is what it is. It's constant.
Next, we took out our normal rulers and measured how many feet fit in one inch. For a larger drawing, that number was smaller. Students could make the connection that more feet fit in less space. The 1in:1ft drawing was smaller than the 1in:0.5ft drawing.
I introduced students to the necessary vocabulary, constant of proportionality, and showed them the magic that each ratio's reciprocal made the other constant. I showed students that we can treat the ratio as a division relationship to see the magic easier with a calculator, since our number system is decimal and it's often hard to see decimal values with ratios. I explained we could dig into why division works later, but for now it's enough to trust me that the numbers have a division relationship.
Enhance! Enhance!
I opened my iPhone camera and took a photo. I then asked a student to help me zoom in and out. I took a photo zoomed in, and the number on the screen read something like 3x. I took a photo zoomed out and the number read something like 0.5x. We talked about how everything got smaller in the 0.5x, but more stuff fit in the frame. Similarly, everything got bigger with 3x but less stuff fit in the frame. Same idea as the architect scale, but now with the idea that anything larger than 1 makes something expand and anything smaller than 1 makes it shrink.
We then watched this incredible montage clip that mashes up moments in TV and film where a crime is solved by zooming in on some sort of surveillance footage. There's other great math vocabulary in there--75 degree rotation around the z axis, eigenvalue, algorithm--that all sounds real smart and math-y, especially to a middle schooler. Every time the characters zoom in, the details get bigger, and less things fill the frame. Zooming out, details get smaller, and more fills the frame. From then on, proportions became "Enhance! Enhance!" and the constant of proportionality became, "Freeze! And Enhance!" We all had a good laugh.
That's it.
That was it. Two lessons and students could easily manipulate proportions and identify the constant of proportionality by using values as a division relationship in their calculators. We practiced plugging them into the calculator over and over.
Our two learning center lessons took shorter than the general education's curriculum, so students felt pride in being able to breeze through as quickly as their peers with their secret knowledge and access to a calculator. This confidence made fractions and multiplication more accessible, and we worked relationally from proportion knowledge to "fill in gaps."
Comments