I teach Pre-Calculus and here are some assumptions that I had that were wrong!
1) Students can order numbers when they are given in different forms, like this.
What I found was that students didn't realize they needed to convert all the numbers to the same form (decimal) in order to compare. They thought they should know just by looking at them, and of course felt stupid when they couldn't. Some students didn't know how to convert the numbers to decimals even with a calculator.
2) If students can factor, then they can simplify a rational expression like this.
In problems like this, students are looking to cancel individual terms rather than factor then cancel.
3) In my Algebra 1 class: If I teach students how to find slope given two ordered pairs, then they can find slope given a graph.
Again, I discovered that students aren't comfortable with converting forms without instruction. To take a graph and, on their own, find ordered pairs, then find the slope, was something they didn't think was "allowed".
This year I decided to teach graphing lines to my students a little differently based on these incorrect assumptions that I've been making year after year.
First, I can't teach topics in isolation, I need to help students make connections between the methods of graphing. I always assumed that students were making these connections because they were obvious to me, so they should be obvious to the students.
I started teaching graphing of linear equations with tables. To me, that seemed to be the easiest entry point into this topic to help the students feel comfortable with graphing. Once students were proficient with this I asked them to start picking out patterns in their tables. I encouraged them to keep their tables organized. We discussed the change in y (delta y). We discussed the change in x (delta x). We looked for these numbers in the equation. Low and behold, the students noticed that delta y over delta x was the fraction next to x in the equation (after we solved for y).
Next, we went over how to graph by finding the intercepts. We noticed that this was very similar to graphing with tables, except that we filled in two zeros in the table immediately (one for x and one for y). Discussions on why to use zero were completed.
Now we're back to slope. We know that we can find slope from a table and from an equation when it's in slope-intercept form. What about other forms? What if I give you two ordered pairs? One student suggests putting those numbers in a table **Angels start singing**. We discovered the slope formula. Next I give them a graph and ask how we can find the slope. Another student says, find two ordered pairs and use the slope formula **An entire choir of angels sing**. We discover rise over run.
Helping students make connections is time consuming, but oh so worth it.