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Two and a Half Ways to Make Your Next Test More Readable

12 May

Math questions are hard to read. It’s easy to mix up numbers, mis-attribute modifiers, confuse powers with multipliers, etc. Unless you’re testing for reading skill, it’s important to put work into making questions as readable as possible. Here are two (and a half) ideas you can use on your next test.

One: Repeat the Question in an Answer Box

My test questions include space for work and a very specific location for answers: the Answer Box.

The answer box is the dark box you see in the lower-right corner of the question area. I pre-print that box on every test question I ever give so that, regardless of the sloppiness of a student’s work, I can see what they thought the final answer should be. I indicate the type of answer I expect right in the box. My hope is that even a student with low English-reading skills will be guided by that indicator to provide at least the right type of answer.

Two: Provide a graphical representation with your text, where possible.

In this problem I wanted to test students’ ability to transform parabolas. I accompany the text with a picture of what I mean.

Without the picture, “compressed by a factor of 2″ is ambiguous. When a student answers $y=\frac 1 2 (x+2)^2 +1$ I don’t want to have to guess if he doesn’t know what to do or if he just got right and left mixed up. In the previous example, the MS clipart of a die reminds ELL students what a die is (not obvious from the word itself!).

Two and a Half: Easy Stuff

Actually, more like five tenths of a big idea:

Use large fonts (obviously easier to read)
Include grids on your graphs (the grids above are more obvious in print)
Use the same format every time (students know what to expect even when they don’t command the language)
Number and name questions with the standard they are testing (make double-sure ELL students know what you’re asking)
Leave room between things (not only room for work. Spaces help us separate ideas)

If you don’t make your questions crystal clear, you can’t be sure you’re testing what you think you’re testing. Reading is hard, everyone messes it up occasionally, and students who have only been learning English for a couple years need more help than you think!

Of course, I’m just a math teacher, not a design expert or language coach. What have you used to make your print materials more readable?

4 Comments

Posted in assessment, bag o' tricks

Tools in Geogebra

04 Apr

My blog has gotten a little lofty lately, and it’s been a while since I just posted some plain old good ideas you could use tomorrow. Here’s one if you have access to a class set of laptops or a computer lab: have your students make tools in geogebra. I’m not going to try to frame this in a lesson plan – it’s just a tutorial for you. Open geogebra to follow along.

We’ll make a midpoint tool¹ in two different ways today. The first way will be geometrically, via construction. The second way will use an algebraic formula. This might be a fun way to connect geometry and algebra! If you already know both of these methods, skip down to the “Toolify” section. If you already know how to make tools, skip down to “The Point” section at the bottom!

Midpoint via Construction

To make tools in geogebra, you first do what you want the tool to do, and then tell geogebra about it. So to make our midpoint tool via construction, we first have to actually do the construction.

Start with two points.
Use the circle tool to draw a circle with center A and perimeter point B. Draw a second one with center B and perimeter point A.
The circles intersect at two points. Use the line tool to draw the line between them. Also draw the line from A to B.
Of course, these two lines intersect at the midpoint between A and B. Use the point tool to give it its own name.
A crucial step: test your construction by moving the points A and B. The entire construction should move, but E should still be the midpoint. Do not move on if your construction does not pass this “wiggle test.”When your construction passes the wiggle test, go to the “Toolifying” section below.

Midpoint via Algebra

We’ll do this construction entirely from the input bar. Text in bold is text you can type directly into the input bar.

A = (2, 4)
B = (5, 6)
Typing these commands creates two points, A and B, at the specified coordinates.
x_A = x(A)
y_A = y(A)
x_B = x(B)
y_B = y(B)
These commands create variables with which you can access the coordinates of points A and B. The thing on the left of the equal sign is the NAME of the variable. The thing on the right of the equal sign is the VALUE of the variable.
Do the wiggle test on your variables. When you wiggle points A and B, all four of the variables from step 2 should change. You can move point A by dragging it with the mouse, or by redefining it with something like A = (1, 4). Do not move on until your variables have passed the wiggle test.
E = ( (x_A + x_B) / 2 , (y_A + y_B) / 2 )
This command creates the point $( \frac{x_A + x_B} {2}, \frac{y_A + y_B}{2} )$ . If all has gone well, you should see the midpoint appear.

When this point E passes the wiggle test, move on to “Toolifying” below.

Toolifying

Regardless of HOW your construction was made – via algebra, geometry, or even calculus² – if it passes the wiggle test, you can make it into a tool.

From the “Tools” menu, choose “Create New Tool.” You’ll be presented with a dialog like the one below.
The most crucial part here is to identify to geogebra your output object. What is the RESULT of your tool supposed to be? In our case, we were trying to make a tool that finds the midpoint of two points. The output is that midpoint. We called it point E. Select that object from the list. You could also click on that object from the graph view.
Go to the “Input Objects” tab. On this tab you will select the objects that your tool needs to work. Our tool is supposed to create the midpoint from two starting points, so those two starting points must be listed as input objects.

So far, in my experience, Geogebra has always guessed the necessary input objects for me. Point A and B are already listed because geogebra knows that they are at the root of your construction. This will save a lot of confusion with your students.
Head to the “Name & Icon” tab to personalize your tool. The “Tool Name” is what will appear on the tool bar. The “Command name” is what you would type on the input bar to use your command. The “Tool help” will appear in the toolbar when your tool is selected.
After you click Finish, your tool is created and ready to use. Let’s test it! First, make two new points.
Then, choose your tool from the toolbar and click those two points, one after the other. The order you click is important in some tools: the first object you click becomes the first input object, the second you click becomes the second input object, etc. If everything is working, you should see the midpoint appear between your two new input points!
Remember to try the wiggle test by pressing escape and dragging F and G around! You will not be able to drag H around – geogebra cannot (yet?) run tools backwards like that. Note that geogebra does not create all the intermediate objects needed for the construction. If you want those objects to appear, include them in the list of output objects when you create the tool.

Your students will enjoy this and feel a sense of ownership of the math. It’s fun to use the tool on its own output, for multiple nested midpoints and things like that.

You can even create other objects with the output of your tools for extra fun. Below is an actual geogebra applet – drag the blue points around for fun!

{{Sorry, the GeoGebra Applet could not be started. If you're seeing this in a reader application click here to see the post live.. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)}}

Riley Lark, Created with GeoGebra

The point (heh)

When your students can program geogebra to perform a mathematical feat:

They will feel (and be) smart and empowered.
They will learn about programming computers – an invaluable tool not just for “the future” but for exploring mathematical concepts later in your class.
They will necessarily have mastered the concept at least one time with enough specificity that a computer can understand what they mean
They will have fun!

Yes, geogebra does have a built-in midpoint tool. ↩
define an f(x) in geogebra, and then type f’(x). Geogebra automatically calculates the derivative! ↩

1 Comment

Posted in bag o' tricks, teaching through computers

Teaching note-taking with notes templates

15 Feb

One of the questions I struggle with is “what is my class for?” This has been a question forever, of course: is my most important lesson about Algebra, about being ready for college, about independent thinking, about forming a community, about the connection between me and my students? It’s easy to get bogged down in this but it’s also easy to see that the answer is not Algebra alone.

One of the other things I try to teach my students is the skill of taking notes, and my approach revolves around templates for notes for each class. At the beginning of the year, their templates will be almost completely filled out, and colorfully organized. I use the trick of leaving everything but the f____ l____ of some words blank to give kids some ownership of the notes but to simultaneously make sure that kids have really important points written down. There are spaces at the top and bottom of each template for a title (a super-short summary) and a summary of the class (a couple of sentences max).

A complex notes template (click to download pdf)

As the semester progresses, we talk occasionally about what from the class would be important to remember, and how the students could have guessed at what material to write on their own notes template. On a couple of days in the second or third month I give them the notes template at the end of class, and we take a little time for them to realize what they missed, and what they included on their own notes that I didn’t. And the templates get simpler.

A simpler template (click to download PDF)

Creating these templates for class every day is a fair amount of work, but I find that focusing around the template can actually help me focus my lesson. And, in those halcyon days of my imagined future self, in which I am following the same sequence in two consecutive years and sipping white russians and reading Robertson Davies instead of working frantically until 10 PM and still going to bed unsatisfied, these notes templates can be reused.

I don’t have data for you about how well these work. I don’t grade the notes and I don’t control any part of the experiment. Some students say they like them, and some don’t. I can tell you this:

When a student comes to my office hours and asks a question, I can ask him to pull out his notes on the subject. If he can’t do that, I’ve immediately identified a problem in his work habits. Better still, it’s a problem that we can manage and check in on the next week. If he can do that, he is empowered to help himself, and good study habits are fortified.
I am showing kids what it means to take good notes. They can compare their own notes to mine. I am no longer just expecting them to figure this skill out and getting frustrated when they don’t.
The kids who keep all of the templates in order have a binder they can really be proud of. It looks nice.
I don’t have to give kids a ton of time to accurately copy complex diagrams or equations. I just put those things on the template.
You can really supplement a powerpoint presentation with the right stuff on a complementary handout. And there are some fun classroom activities to be had by subtly changing a few of the handouts in ways that students won’t notice right away

I’d love to hear of any other ideas for teaching note taking, or any responses to mine. My method is based on the Cornell notes system.

3 Comments

Posted in POI, bag o' tricks

Polydoku!

08 Dec

I’m trying out CPM’s Algebra 2 book and so far it’s pretty much fantastic. I don’t want to give away many details of the lesson plan I used today, since they asked me not to reproduce it. But I have to tell you about the basic mechanic: polydokus.

A complete polydoku has 4 main sections – one for each of two polynomial factors, one for the product of said factors, and another area for the work. You can figure it out from this already-solved puzzle:

polydoku_solved

Of course, much of this data is redundant. Try your luck at this unsolved puzzle:

polydoku_unsolved These puzzles are fun and satisfying. I tried it in class today and students had the perfect amount of difficulty with them. I explained the puzzles only as much as I explained them here, and the students seemed to enjoy figuring them out.

The payoff comes in when the puzzle looks like this:

polydoku_division

Have your kids solve this polydoku, and then ask them, “Hey, by the way, what’s ?”

The CPM lesson went on to have the kids discover remainders, and even connected it to the factor theorem for finding roots of an equation, but I’ll let you ask CPM about that.

You know who’s a polydoku convert?

[pointing emphatically at myself with my thumbs] This guy.

5 Comments

Posted in Lessons, bag o' tricks

Bag of Tricks #1 – Index Cards

06 Dec

In “Bag o’ Tricks” posts, I’ll give activities that require almost zero prep, but inject a shot of fun, practice, activity, assessment, remediation, or whatever in a small amount of class time.

This post’s focus is index cards. My students like them – I think they are just nicer objects than sheets of paper. These are perhaps my favorite no-prep activities.

Memory (20 minutes)

Each student gets two index cards.
On one index card, each student writes an expression of a given type (e.g. an anonymous differentiable function like “2x+sin(x)”). Every student must use a pencil.
On the other index card, each student writes a corresponding expression after a given operation (e.g. differentiation – “2 + cos(x)”). After this step each student has two cards that are connected by the given operation, but not by name or any other property.
In pairs, students swap cards and check each other’s work.
Each student gets another two index cards and repeats the process. Each student now has a total of four cards, two pairs of linked cards.
Students form groups of four, shuffle their combined sixteen cards together, and lay them out upside down. The cards are (hopefully) indistinguishable.
The students play memory (in teams of two, or not). A team flips over one card, and then another. If they match through the operation, they keep the pair, get a point, and go again. If the cards don’t match, the next team is up.

This activity is great, after you figure out how to make sure students write problems of the appropriate difficulty. They need to be pretty easy. Memory is hard when its just pictures of barnyard animals, you know? I use it to have students practice derivatives over and over again. Every time they see, for example, “2x,” they have to think “what is the derivative of 2x, and what might have 2x as a derivative?” You need a problem that’s easy, but takes lots of practice. Distributing polynomials, finding logarithms, solving linear equations, etc. The first time I used this activity I put, like, physics word problems on one card and answers on another. Let’s just leave it at “don’t do that” and move on, please .

Benefits of memory:

A bunch of practice
It’s reasonably fun
Kids write their own problems and solve them
Each student gets the advantage of knowing 2 of the 8 answers right away. This almost guarantees some success for every student – everyone can feel engaged, even if their skill level is lower than the others’.

Write and Swap (5-7 minutes)

Each student gets an index card and creates an example problem.
Students swap cards at their table (I have tables of two) and confirm that the problems are in the proper form, etc. Any questions about problem creation are resolved.
The teacher moves quickly and energetically around the room, picking cards swiftly out of kids’ hands and giving them replacement cards from other kids. This works elegantly – the teacher can move in any pattern, so as soon as problems are written they can be swapped out, but students who need more time may take it as the teacher is passing out cards.
After this step each student has a new card in front of them, and they don’t know exactly where it came from.
Each student solves the problem on his or her card.
Students swap cards at their table and confirm solutions. Any questions about problem solution are resolved.

Benefits of write and swap:

Each student gets practice writing a problem, which may involve critical thinking about what is important to include.
Each student thinks about four different problems in a row, but a physical interaction between each problem keeps attentions focused.
Student responsibility is diffused. Limited responsibility can help students feel safe, which can be important (though students should be fully responsible for at least some work every day).
A peppy teacher can infuse the activity with energy on a slow day by zipping around the classroom in the big card swap. Carry around a funny container instead of just holding the cards in your hand if you want.

Write and Swap is great for those times when you just want students to practice something kind of boring a few times. It’s not great for longer problems because the phases get unsynchronized.

Most confusing part (5-7 minutes)

I got this from Science Formative Assessments, by Page Keeley.

Each student gets one index card near the end of the period.
Each student writes, anonymously, the thing about the class that was most confusing, least fun, whatever.
The cards all go in a box and are redistributed, one card per kid. Page Keeley recommends having the kids literally throw the cards around, but I admit to not being brave enough to try this yet. It might make this activity really fun… or just add two minutes to its execution.
Kids read their new cards aloud to the class.

The first time I tried this, I wasn’t that impressed with the results, but like any new technique I’ve gotten better at making it succinct and useful. This activity is mostly to get a quick sense of how your lesson went, if you didn’t have any better way to do it built in.

Benefits:

If a theme emerges, you know, that’s a great piece of information for the teacher. Write that down on your lesson plan!
You get to hear from every kid in a very low-pressure way.
I imagine that kids who are embarrassed by a lack of understanding are heartened when they (inevitably(!)) hear that someone else had the same problem.