﻿ It's All About Practice: Free, Unlimited, Randomly-generated ... and Beautiful!

[First of three talks as a Featured Speaker at CAMT 2015; updated August 2022.]

# It's All About Practice: Free, Unlimited, Randomly-Generated—and Beautiful!

Since 1999, Dr. Carol JVF Burns has spent more than 15,000 hours creating 400+ free online math lessons ranging from arithmetic to calculus. Every lesson offers unlimited, randomly-generated exercises and worksheets for both online and offline practice and effortless assessments. Carol gives an overview of her materials and offers suggestions for use in both lecture-style and flipped classrooms.

To pique your interest in Carol's two subsequent sessions, she will briefly address Algebra Pinball (a fun way to master Algebra I skills) and her free online book One Mathematical Cat, Please! that empowers people to teach themselves mathematics.

## 60 Minutes, 60 Morsels

This is designed as a self-guided talk. Just click to open each morsel and then follow the suggested links. Feel free to email me with any questions or comments. Enjoy!

1. Welcome, everyone!
2. I'm excited to share with you over 15,000 hours of work since 1999.
3. I'm thankful to the conference organizers for inviting me; I guess someone stumbled on my math web site and thought it might be a useful resource for all of you.
1. I'll present 60 morsels in 60 minutes.
2. My entire talk is online, so there's no need to take notes. If you're interested in (say) morsels 27 and 53, just jot those numbers next to this talk in your program guide.
1. My waaayyy too long name is Dr. Carol JVF Burns. If you're curious:
• ‘J’ is for Jane, my Dad's birth-mother
• ‘V’ is for Vreeland, my maiden name
• ‘F’ is for Fisher, from my first marriage
2. If you Google (just) Carol Burns, you may not get me! (I'm not the Australian actress.)
3. Here's my online vita. I have a Doctor of Arts in Mathematics, which is a doctoral-level degree designed for effective teaching.
4. I've taught mathematics for about three decades at both the university and high school levels.
1. Raise your hand (or make some noise) if you have any association with each place I've taught
(if you taught or went there yourself, or if you know someone who teaches or goes there):
1. University of Massachusetts (Amherst): Bachelor of Science degree Magna Cum Laude in Civil Engineering
2. University of Oklahoma (Norman): Master of Arts in Mathematics
3. Idaho State University (Pocatello): Doctor of Arts in Mathematics
4. Miss Hall's School (private all-girl's day and boarding school, Pittsfield, Massachusetts):
My web site got started to give my girls extra practice. Then, it took on a life of its own!
5. Lenox Memorial High School (public high school, Lenox, Massachusetts):
I was Chair of the Mathematics Department for a year, before I left to pursue math-on-the-web fulltime.
6. Northern Arizona University (Flagstaff):
I moved closer to my daughter, who is pursuing her doctorate in neuro-linguistics at the University of Arizona. Then, I met my husband Ray through dancing, and moved to Tucson.
2. NAU required a teaching philosophy for my application. It's a great summary of my views.
1. Here's a quick digression. This is great if your class is ever bored and you have the internet available!
2. Go to WolframAlpha and type in (say) ‘Carol’.
3. Scroll down to the graph titled ‘History for US Births’. The popularity of my name peaked in about 1945; I was born in 1958.
1. You can type ‘Carol Burns math’ or ‘math cat Burns’ in most any search engine.
2. Take the ‘One Mathematical Cat, Please!’ entry to get to my homepage.
1. Everything I offer online is completely free and immediately available. No logins. No downloads. No pop-ups.
2. I have 400+ sequenced lessons from basic arithmetic to calculus. I certainly can't show you all of them in this talk, but I'll give you a good sampling.
3. Know this, though: Every lesson is the result of decades of refining and tweaking to maximize understanding. I begin creating each lesson with notes that I've used for many, many years in my teaching.
Let's use Divisibility to look at features common to every single one of my 400+ lessons:
1. Many of my users find me from a search engine, seeking information on a particular topic. But, every lesson is actually part of a sequence, so they can easily be used for entire courses.
• Sequencing Info:
At the top: Which course? What position?
Take a look at the entire Table of Contents for Algebra I (164+ lessons; some optional, for scheduling flexibility). It can be used for a year-long high school course, or one semester in college.
2. As I'm sure you've discovered, there are lots of sites that offer math content. But, only a few offer practice problems. Of those that do, it's usually only a few questions.

My lessons offer unlimited, randomly-generated exercises:
There are always links to jump right to the exercises.
• Exercises:
If you're working on a screen in front of a classroom, use browser features to enlarge the font-size, as needed.
3. Every lesson has the ability to create randomly-generated worksheets. These can be used for:
• practice away from the computer
• quizzes
• homework
Try one, with (say) the Divisibility lesson. (Enlarge the font-size if needed; scroll down to the solutions.) Change the number of problems desired. Try another!
4. As you'll see, many of my lessons have two different types of exercises: skill and concept.

Skill exercises, like the one we just did, can be ‘graded’ by the computer—they have exactly one correct answer, with little or no variability in how you type in that answer. Skill exercises are designed to master essential foundational skills—usually, things you want to be able to do without calculator or paper-and-pencil.

Concept questions have more flexibility in their answers. These are tightly linked to the lesson on the page. For concept questions, you think about your answer, and then click to check yourself.

5. (In Divisibility, scroll down to the concept questions exercise.) Get a new problem, check your answer, then ‘click here for the same type of problem’.

Sometimes, there's just a (particularly important) single static question. Usually, however, there's variability in each concept question. Some problems just vary numbers, some use different wordings—the randomness changes from question to question. However, each question addresses a particular aspect of a concept. Try a few!

6. Each type of exercise (skill and concept) has its own randomly-generated worksheets. For the Divisibility concept exercise, there are $\,12\,$ different types of questions.

On worksheets, you're always guaranteed to get distinct problem types. (Click to get a worksheet. Scroll down to see $\,12\,$ different types of questions.) Instead of using the in-page exercise form, some prefer to project a worksheet in class (enlarging font-size as needed), to be sure to get an even sampling of all problem types.

Of course, every time you click you get a different worksheet. (Click several times, to produce different worksheets.) You can effortlessly produce different quizzes for different classes, sections, meeting times, and so on.

Students often assess their readiness for a quiz by printing out a worksheet, setting aside the solutions page, taking the quiz, and then self-grading.

7. There is no ‘print’ button on worksheets. A worksheet is a regular web page, so just use your browser features to print as you would any web page. In many browsers, you just go File-Print.

When printed, the solutions always begin on a separate page, so it's easy to separate the questions from the answers.

8. I've talked about the exercises before talking about the lesson. (Hmmm... how many students do you know who skip the lesson and jump right to the homework?) In a perfect world, you should always study the lesson before practicing with the exercises!

The lesson is always in the white section. Easy to read. Short sentences. Lots of white space. Key results are boxed for easy identification.

I always write the lesson first (typically 2–4 days), and then create the exercises (typically 2–4 more days). I take every result, example, notation, terminology, and important sentence from the lesson and turn it into a question. So, the lesson and exercises are very closely connected.

9. At the bottom of the white (lesson) section, there's always a link to the next lesson in the course sequence. (Go on to the next lesson, then use your browser ‘back’ feature to return.) Of course, you're reminded to master the ideas from the current lesson first by practicing the exercises at the bottom of the page.

10. My site has Fun Facts about me. I believe it's more fun to learn from someone you ‘know’!

• Any derivative of my materials must clearly display a link to my homepage.
• Use of my materials must be non-profit, unless written permission is obtained from me.
12. On every page (say, the Divisibility page) there's always a search box for my entire website (many hundreds of pages).

Looking for an exercise on adding fractions? Type ‘adding fractions’ in the box and press  Enter  or the Search button (try it).

Even though I've created my entire site, and should theoretically know where everything is, I find myself frequently using this ‘Search’ feature to quickly locate a desired section.

Here are more common features:
1. My exercises are designed to have fast and easy keyboard entry. After your initial click of the ‘new problem’ button, you never have to click on another button. You just:

• press  ENTER  to check your answer
• press  ENTER  again to get a new problem
• repeat as desired

Try Mixed Basic Add/Subtract Multiply/Divide Practice (jump right to the exercises). If desired, check the ‘no variables’ box.

• Click the ‘new problem’ button (this is your initial click).
• Type in your answer; press ENTER to check.
• Press ENTER a second time to get a new problem.

When all your answers are correct, it's just:   type, ENTER, ENTER (repeat).

2. But what if you type in a wrong answer? Get a few wrong, on purpose, on the Mixed Basic Add/Subtract Multiply/Divide Practice.

At first, it says ‘Sorry. Please try again.’ Notice that the wrong answer is highlighted, so you get a second chance to get it right. Type in the new answer and press ENTER.

If you have two successive wrong answers, then you're told the correct answer (try it). (By the way, if you now try to type in that correct answer, you're just reminded to go on to another problem.)

3. All my skill exercises (about $\,93\,$ of them) have a timing feature:

• Teachers can require a desired level of mastery of basic skills. For example, a timing sheet that shows achievement of a required level of mastery can be passed in for homework or a quiz.
• Students can easily track their progress.
• The timing feature is the basis for Algebra Pinball, a fun math competition (more on this later).

Try Addition of Signed Numbers (jump right to the exercises).

• Click in the ‘Want to time yourself?’ box and type your name.
• Ready? Click the  ‘start timing’  button, then press  ENTER  to get your first problem.
• After each correct problem, you're updated on number correct and average speed.
• Click the  ‘end timing’  button when done.

(Don't close the timing sheet report yet; we'll talk about it in the next morsel.)

4. The report that is generated when you click  ‘end timing’  includes:

• name, date, and time
• the exercise title
• # problems attempted
• # problems correct
• average seconds per correct problem
• different levels of praise, depending on how well you do

(Still don't get rid of the timing sheet report! More in the next morsel...)

5. The Timing Sheet Report is a regular web page, so just use browser features to print as you would any web page. In many browsers, it's just:  File-Print.

When students are trying to get a really good time, they may go through many trials. Don't print a hard-copy of the sheet until the desired time is achieved! Instead, save the report as a temporary PDF file; keep over-writing it with better times. Only print a hard-copy for the best time! [Close the Timing Sheet Report when you're done.]

6. Possible levels of praise on the timing sheets are:

• Good job!
• Incredible job!
• YOU BROKE THE ALL-TIME RECORD!
UNBELIEVABLE!!

I've had many years of many students working on these exercises, and have recorded the fastest times achieved. The praise cutoffs are determined by comparison with these fastest times (sometimes adjusted, if it's just too blazing fast).

7. The fastest times are recorded in Algebra Pinball:

• In the Algebra Pinball chart, scroll down to the first timed exercise: ‘Expressions Versus Sentences’.
• It says the ‘Best Score at Miss Hall's School’ is 0.7 seconds per correct problem.
• All the fastest times, for every timed exercise, are recorded in this chart.
8. There's a link to the ‘Algebra Pinball’ page on each web page that contains a timed exercise:

9. [update August 2022: Fastest time is no longer available from source code]

10. Some students work hard at ‘cheating the system’. In all my lessons, I've worked hard to make it difficult to show ‘fake mastery’.

For example, try Identifying Place Values (jump right to the exercises). Notice that the radio-button place values don't always appear in the same order!

11. There are no ads on my site. I don't want you to be distracted from the math.

1. [Update August 2022: It happens so fast on most computers, you probably can't see it.]

That ‘dissolving green’ is the mathematics being processed.

I use MathJax to get the beautiful mathematics. Its initial release was in 2010. It's the gold standard for mathematics on the web, and it's free!

2. Several years ago, I created an interactive slide show on the history and philosophy of my web site. If you're interested in seeing the whole thing (at some future time), it starts here.

Here are a few slides from the show:
HTML Can't Do Math
MathML (Math Markup Language)
MathML Is Really Verbose

MathJax uses MathML ‘under its hood’, but it ‘compensates’ in environments where MathML just doesn't yet work.

MathJax solves the ‘unwieldly source code’ problem by allowing authors to write in $\,\TeX\,$ (pronounced TEK) syntax, which is concise and intuitive. $\TeX$ is a typesetting system for mathematics. To get the fraction $\displaystyle\,\frac{x^2}{3}\,$ using MathJax, here's all I had to type:

\frac{x^2}{3}

MUCH EASIER!

3. If you decide to use MathJax, this resource may be useful: TeX Commands Available in MathJax. I created it to thoroughly familiarize myself with the $\,\TeX\,$ commands that are available in MathJax, and to provide a resource for MathJax users.

It's a BIG page, because I made the decision to put all the commands on the same page. This way, it's easier to search for a command that you might not know. You can see the MathJax-loading progress in the lower-left corner—it might take a couple minutes to finish.

Davide Cervone, the lead developer of MathJax, most generously provided extensive edits.

4. I use JSXGraph for all my graphing and graphics needs. (Follow the link, scroll down a bit, and play with the ‘packing’ graphic, as an example of what JSXGraph can do!)

• It's free!
• It works really well.
• The source code is compact so it doesn't slow down page loads.
• It works in all major browsers and platforms.

I created a document for JSXGraph, similar to the one I created for MathJax. If you end up playing with JSXGraph, you may find it useful: JSXGraph Commands and Examples. It doesn't cover everything that JSXGraph can do, but it does cover most of the features that I use on a regular basis.

The next few morsels show how I use JSXGraph on my site.

5. Almost every image on my site is created with JSXGraph. If I only need a static image, then I take a snapshot of the JSXGraph output and post the picture. Here are examples:

• Explore the difference between precision and accuracy in: Significant Figures and Related Concepts.
[Scroll down to the exploration.
Choose a random TRUE VALUE.
Click ‘precise, not accurate’ several times.
Click ‘precise and accurate’ several times.
Click ‘not precise, not accurate’ several times.]
• Explore base/height pairs in a triangle: Area Formulas: Triangle, Parallelograms, Trapezoid. [Scroll down to the exploration. Go through the sequence with several different triangles.]
• See randomly-generated examples of probability tree diagrams: Probability Tree Diagrams. [Scroll down to the exploration. Click for several sample probability tree diagrams. Then, scroll slowly through the rest of the page to see MathJax being used inside JSXGraph.]
• Graphs can be annotated—even dynamically: The Testpoint Method for Sentences Like ‘$\,f(x) > g(x)\,$’. [Scroll slowly through the lesson, pausing to look at the MathJax inside JSXGraph. Then, try exercise type 4 several times.]
• Properly-positioned arrows, text, and math can be combined to help explain concepts: Fundamental Trigonometric Identities. [Search for ‘cosine has this property’, look at those pictures, and then scroll down a bit.]
6. I'm particularly proud of this lesson: Long Division of Polynomials.
• Search for ‘the thought process’; the hover-highlighting helps clarify the process.
• Continue to ‘Step-by-Step Long Division Practice’; click through one entire problem. To check their understanding, students should think ‘What do I do next?’ before clicking for the next step.
7. I have a ‘mastery’ feature for all my concept exercises. Try it out on (say) Logarithm Summary: Properties, Formulas, Laws:

• Jump right to the exercises.
• Click through a few—the ‘in progress’ case is shown in black.
• Click on problem type #1; get the same type of problem. Students can explore the amount of variability within each problem type. Then, click that you've mastered type #1; notice that it turns green.
• Students preparing for quizzes can have complete confidence they've mastered every problem type!
• Teachers can easily have students focus on particular problems.
8. One of my users, who works with people with disabilities, asked me to explore a feature where people could type answers into the worksheets before printing them. Check it out here:   Signs of All the Trigonometric Functions.

• Quickly scroll through the lesson, to the worksheet.
• Check the ‘ Want textboxes to type in your answers? ’ box.
• Click to create a randomly-generated worksheet and answers.
• Type in some fake answers. When the worksheet is printed, these answers will appear.
• Any mathematics in the answers will be hard to represent properly, since you're limited to text.

Right now, this feature only appears in a few of my exercises. Let me know if it is highly desirable to you!

Let's take a look at my online courses:
1. Here's a key philosophy of my web site:

I don't just want to teach people a bunch of different math topics.

I want to teach people how to teach themselves mathematics.

I want to teach the language that all mathematics is expressed in.

Flash back to the late 1990s. I wrote a tiny book, called One Mathematical Cat, Please! that explores the language of mathematics. The entire book is online.

2. Brooks-Cole Publishers loved it. They said:

It is wonderfully written and crafted with a care you rarely see.

If you can find the right place to put it, this book will do a great service to the mathematical educational world.

You can read their review here.

3. However, ‘One Mathematical Cat, Please!’ is a supplement, not a primary text, and (after a year of deliberations) the conclusion was that supplements don't sell. Brooks-Cole told me to turn it into an entire course, and then get back to them. (By the time I did get back, however, they were working with someone else.)

But it was okay—by then I had discovered a new way to reach my audience: the World Wide Web.

4. The ‘One Mathematical Cat, Please!’ supplement (morsel #44) was filled out to an entire Algebra I course.

• Scroll through the Algebra I Table of Contents. There are 164+ lessons: a year-long high school course, or one semester in college. There are optional sections, for flexibility.
• If you go through this Algebra I course, you get the entire ‘One Mathematical Cat, Please!’ supplement, plus a lot more!
5. If the web-based lesson isn't enough, you can get a more traditional text book that I wrote (in pdf form). Check it out with Introduction To Sets. [Click ‘read the text’. Scroll through a few pages. Notice the fun alternative page numbers in the lower left corners!]

6. For almost two decades I've scoured the web for the best (mostly mathematical) FUN STUFF: games, puzzles, optical illusions, more. Here are a couple of my all-time favorites:

• Number Gossip
Type in your favorite number and find out lots of cool things about it!
• Mudd Math Fun Facts
Perfect for a five-minute digression when class interest is fading, or as an end-of-class reward for a focused day. Take a look at ‘Pizza Slices’. Your students may never slice pizza in a traditional way again!
7. Work Problems was an experiment with a totally different style. Search for ‘choose your own names’ and put in a few. Then, play the ‘Guessing Game’. When I ‘tested’ this lesson on my family, there were loads of laughs as familiar names came up!

• logic and mathematical language
• proof
• geometric concepts needed for success in Precalculus and Calculus (that aren't already covered in my other courses)
9. There's of course traditional geometry content (triangles, congruence, parallel lines, etc.), but I'm just going to show you some of the logic and proof lessons:

1. If... Then... Sentences
How many ways are there to say ‘If $A$, then $B$’?
2. Contrapositive and Converse
Don't confuse them, because they aren't equivalent!
3. Equivalence Relations
Want to partition your class? Use an equivalence relation!
4. Parallelograms and Negating Sentences
The emphasis here is on negating AND, OR, FOR ALL, and THERE EXIST sentences. A theorem characterizing parallelograms provides fantastic practice!
10. Here are some lessons exploring proof techniques:

1. Proof Techniques [Print a worksheet. Look at the three questions/answers that help students understand the difference between a direct proof, an indirect proof, and proof by contraposition. There are MANY ‘fake words’ used here!]
2. Introduction to the Two-Column Proof [Print a worksheet. Look at one of the proof problems. By using familiar algebraic statements, students can focus on proof techniques and not get bogged down in ‘Which geometric results am I allowed to use?’]

• Recursion and Sequences
• Graphical Understanding of Sentences like:
• $\,f(x) = 0\,,$   $\,f(x) > 0$
• $\,f(x) = g(x)\,,$   $\,f(x) \le g(x)$
• Probability and Statistics
• Systems of Equations; Matrices
• Graphical Transformations (moving up/down/left/right, stretching/shrinking)
• Logarithmic and Exponential Functions
12. Some of my FUN STUFF links come from the National Laboratory of Virtual Manipulatives (NLVM). If you haven't yet discovered them, you're in for a treat!
Here are a couple examples:

13. I sometimes use WolframAlpha ‘widgets’ to focus attention on specialized results. (The WolframAlpha default results can be a bit overwhelming!)

For example, search for ‘widget’ in Solving Polynomial Equations, and click the ‘Submit’ button.

Widgets are:

• free
• easy to create
• easy to embed in web pages

Check it out, if interested: WolframAlpha Widget Builder.

14. My Precalculus course can be used for a full year in high school, or one semester in college. A couple samples:

15. Here is my Calculus course. Some highlights:

16. My Scholastic Aptitude Test Prep Course has all the materials for an 8-week course, one hour each week. There are printable (pdf) handouts to copy for your students. The short course covers critical concepts and strategies, common problem types, and lots of practice problems (with answers).

17. Here's how you might use my materials in lecture-style or flipped classrooms:

• LEARN:   Project online lesson to class; go through content. Note: If students (and/or parents) want to re-visit the lesson—it's available online!
• IN-CLASS PRACTICE:
• Class game (see Talk 2)
• Teacher-led: use a randomly-generated worksheet, which is guaranteed to have one of every problem type
• HOMEWORK:   a (different) randomly-generated worksheet
• Hand out copies of a single worksheet to the entire class; OR
• Have each student print their own at home; self-grade; pass-in or check off
• SHORT QUIZ (next day):   a (different) randomly-generated worksheet (possibly very few questions, depending on available time)

## FLIPPED CLASSROOMS:

In a ‘flipped’ classroom, instruction is delivered online, outside of class, freeing class time for homework and enrichment.

• LEARN (homework): Assign online lesson for students. Explain that a vital component of learning is lots of (online) practice!

Optional: Have students write up an index card that summarizes the key concepts.

Optional: Use social media for teacher/student collaboration and questions outside of class.
• IN-CLASS PRACTICE:
• Small groups: discuss, clarify, re-teach (teacher circulates); and/or
• Class game (see Talk 2)
• SHORT QUIZ (end of class): a randomly-generated worksheet (possibly very few questions, depending on available time)