Education ∪ Math ∪ Technology

Month: August 2014 (page 1 of 1)

Formative Assessment Responses

Formative assessment means more than just giving a quiz or an exit ticket. An assessment is only formative if the teacher (or her students) respond to the information gathered.

However coming up with an appropriate response is typically hard to do. After all, the most common finding in formative assessment is that a significant, but perhaps minority, group of students still do not understand a concept, after the teacher gave her best shot at helping students understand. No teachers save their best strategy for teaching a topic until later.

I’m working on a menu of possible responses teachers could come up with. Some of these responses depend on the nature of the formative assessment gathered, but most of them can be applied in many different contexts.

If you have other possible strategies teachers can try, please feel free to add them here or comment below.



Two different approaches to teaching

Here’s an approach to teaching about the relationships between the different forms of the equation of a line that is based on constructivism.

  1. Be clear with students what the objective for the lesson is today.
  2. Give students a tool for looking at the relationships between the different forms of the equation of a line.
  3. Ask them to write down what they notice and what they wonder after they have played around with the tool themselves. Circulate around the room to see what they write down so you can better make use of it in the whole group discussion.
  4. Next, as a group, discuss what students notice and what students wonder about the graph, or use a different talk protocol.
  5. As a group summarize what appear to be the most important observations made, and everyone (including the teacher) writes them down. If important points about these representations of a line are not brought up, either ask questions to prompt students to realize these points themselves or make a note to yourself to structure a task that will help draw out these points next time.
  6. Come back to this concept multiple times in a variety of different ways over the next few weeks, and even find ways to connect to this topic through-out the rest of the year.


Here’s an approach to this same lesson based on cognitive load theory.

  1. Be clear with students what the objective of the lesson is for today.
  2. Give them notes on the different forms of the equation of a line and some worked examples.
  3. Give students twenty or so practice problems to do themselves. Circulate around the room while students do this and give them timely, efficient, and useful feedback.
  4. Quiz students on their understanding of the concept. Use this to guide your planning in subsequent lessons.
  5. Review this concept with students at spaced intervals over the course of the rest of the year.


My recommendation is that if you are unclear on how either of these approaches is helpful for students, then you should try it at least once. I have in fact tried both of these approaches myself, and I’m quite clear on which approach helps students make better connections to other areas of mathematics, and which approach did not work.

Which approach is more likely to lead to an instrumental understanding of mathematics? Which approach is more likely to lead to a relational understanding of mathematics? For reference, here’s a comparison of instrumental to relational understanding.


This Kind of Teaching

I read an important passage in Elizabeth Green’s book “How to Build a Better Teacher” and decided that I had missed something important when reading about Magdalene Lampert’s teaching. Below is a summary of some of the important features of her teaching as I see them. The first tweet is the thing that I had misinterpreted as students talking to justify their reasoning, which is similar but not exactly the same as students talking to prove their reasoning is correct.

A catalog of mathematics education resources

I’m tired of having to search all over the place to find the-link-to-that-mathematics-resource-I-really-want-now-and-bookmarked-a-year-ago and so I created a spreadsheet to keep track of the various mathematics resources as I learn about them. Yes, I know I could have done this with Diigo or Delicious, but I prefer the portability and simplicity of a spreadsheet.

I’ve opened the spreadsheet up for editing and will curate it to ensure it only includes what I consider to be the highest quality resources. The spreadsheet contains the categories of websites, mobile apps, software, books, research, and organizations. If you have a suggestion for a resource you think I should include, please offer it below, or add it to the spreadsheet. Note that I have currently protected the first sheet (the list of websites) so that you can comment on it, but not edit it directly.

This is by no means complete and will continue to evolve over time as I learn about new resources.