See also my presentation for ESP, “Why your math textbook sucks” https://dl.dropboxusercontent.com/u/27883775/wiki/files/Why%20your%20math%20textbook%20SUCKS.pptx
Here are some thoughts on writing curriculum I’ve had while working at Gliya (an educational startup).
First, read Winston’s advice on writing (What I learned from Patrick Winston). Excerpt:
Begin each chapter (and begin each lesson in a classroom) with an empowerment promise - what you will be able to do after the lesson - and show that you have delivered upon that promise at the end of the chapter. For an example of a chapter opening, see http://courses.csail.mit.edu/6.803/chapteropenings.html.
Break up into large number of sections/subsections for easy reading. (This limits the amount of time you have to read before you get to a place when you can pause, a kind of “local goal” you can keep aiming towards.) Lots of titles together act as a road map, and point out the salient points of each section.
Making section titles complete sentences helps the reader summarize the main point achieved from reading the section.
Parallel structure is good organization. (Ex. between congruence and similarity)
Always, always maintain an encouraging tone. When you give the student instructions, make it seem like the student is ACTIVELY doing something CREATIVE, rather than just following instructions.
- For instance, rather than “In the following problems you will use proportions to find side lengths” say “You’ll put to use the proportions you get from similar triangles to compute side lengths.”
- Rather than say “The diagram is as follows” say “We first draw the diagram:”
- “Besides just measuring shadows, there are many other ways to find the heights and lengths of objects you can’t measure directly. Below you’ll discover several ways to do so, all using AA similarity.”
- Now ‘’you’re ready’’ for some more complex problems. Use a bit more thought to find the correct pair of similar triangles!
- Thales’s theorems remain the earliest attributed theorems in mathematics. In this section you’ll follow in Thales’s footsteps yet again as you rediscover an important theorem on similar triangles for yourself.
- Using “let’s” in solutions.
- We can create new information by [drawing a parallel line, extending a segment,…]
If followed consistently this makes a big impact on the tone of the whole lesson.
At the beginning of each chapter, have a summary and visual organizer of the material to motivate the reader. The summary should + give an empowerment promise: what will the reader have gained from learning these cells? What will (s)he be able to do that (s)he couldn’t before? Aim for concrete examples (ex. measure the height of a tree, rather than measure the height of tall objects). + give a quick summary of the different cells, how they relate to each other logically. Give a visual road map (“advance organizer”). + show how the lessons fit into the bigger picture – how they extend what the reader has already learned, and how they will enable the reader to learn future lessons.
Good organization will make students feel like they’re making concrete progress, and organize in their minds how all the different lessons they learn are related.
The 5 S’s, applied to textbook writing
Include all the following:
- Slogan: A repeated phrase that sticks in readers’ minds.
- Symbol: A repeated picture that sticks in readers’ minds.
- Salient: Have one idea that sticks out. (If you put “too many” good ideas in, then none of them sticks out and the reader won’t know what to make of the piece.) (Warning: this page does not follow this principle.) Crystallize the core ideas in concept boxes.
- Surprise: Something unexpected.
- Story: Have a narrative.
For instance, here’s how I implemented the 5 S’s in the Similar Triangles.
- Slogan: “Same shape but not the same size.” “Too big to measure.”
- Symbol: A picture of measuring the height of a tree using similar triangles. I put this in the intro, in the actual lesson, and in the summary.
- Salient: In several places. First, the concept boxes. Second, presenting the 3 common similar triangle situations (2 with parallel lines, one with right triangles) over and over again. Using similar triangles to measure the height of things you can’t measure directly. This is in the main narrative, and there are 3 exercises in the AA section (all with the story attached) on this.
- Surprise: The fact that there are so many different ways to measure lengths you can’t measure directly is rather surprising.
- Story: Two main storylines. First, Eric (the proportions guy) and Connie (the congruent triangles girl) coming together to find the height of a tree. Second, Thales of Miletos used similar triangles to find the distance of a warship from shore 2600 year ago. (History is a good source of stories!)
The 5 elements of a story are:
- Characters: Each character should have a distinct personality. Think through the character’s heritage (parents), dialogue (what makes it unique?), worldview, and daily life.
- Conflict: This can be an outside problem the characters have to use their brains to solve, or just some friendly rivalry (which always works… for instance, Connie and Karen competing to see who can come up with more congruence criteria).
- Theme: What’s the overall message? (Ex. you can use similar triangles to measure things you can’t measure directly)
A good curriculum should contain a narrative, that satisfies the following.
- Relevant/natural: It seems like a exploration that could actually happen in real life (ex. using fractions in recipes, vs. shooting fractions dropping from alien invaders). The characters should build upon information that they have. Occasionally you can have a corny/far-fetched story IF it is…
- Memorable: Have a key idea or bit of story (salient) that stands out. Even if students don’t remember much of the chapter, they’ll remember this, and hence a key idea of the chapter.
- Worldbuilding: Develop the characters and the setting, and keep building upon the same set of characters and settings in future lessons.
- Identification: Readers need to identify with the characters. Make them think, “Oh, that sounds like a problem I could encounter!” or “Oh, I could have come up with that too!”
Details matter! Try to be as specific as possible. Rather than “model airplane” for instance, write “model Wright Flyer (the Wright brothers’ plane).”
Try to include a few problems with stories attached (stories that use characters in the “world” you developed, in situations they would encounter – NOT random characters taken off the street like John Doe). Start with these problems if possible.
Organize similar problems together, with guiding text! (For instance, in problem solving with similar triangles: Multi-step problems, computation problems, How does it all fit together?)
Don’t just put a random progression. Each problem should have some important key idea, that makes a student remember the problem.
Reason before fact!
Use concept boxes as salients. Keep the text in the boxes short, punchy, and memorable.
Encourage students to ask questions, so put questions in the solutions at difficult points where an intellectual leap is necessary.
Diagrams, diagrams, diagrams! Using different colors, bold lines, and hatching helps to emphasize different parts.