Once again, Nrich.Maths.org is offering twenty-four math activities chosen to encourage your students’ mathematical creativity, one for each day in the run-up to Christmas. Click either image to visit the appropriate Advent Math Calendar page at Nrich.
Educational psychologist Richard Skemp popularized the terms instrumental understanding and relational understanding to describe these two ways of looking at mathematics. It is almost as if there were two unrelated subjects, both called “math” but as different from each other as American football is from the game the rest of the world calls football.
Which of the following sounds the most like your experience of school math? And which type of math are your children learning?
Instrumental Understanding: Math as a Tool
Every mathematical procedure we learn is an instrument or tool for solving a certain kind of problem. To understand math means to know which tool we are supposed to use for each type of problem and how to use that tool — how to categorize the problem, remember the formula, plug in the numbers, and do the calculation. To be fluent in math means we can produce correct answers with minimal effort.
Primary goal: to get the right answer. In math, answers are either right or wrong, and wrong answers are useless. Key question: “What?” What do we know? What can we do? What is the answer? Values: speed and accuracy. Method: memorization. Memorize math facts. Memorize definitions and rules. Memorize procedures and when to use them. Use manipulatives and mnemonics to aid memorization. Primary benefit of this approach: testability.
Instrumental instruction focuses on the standard algorithms (the pencil-and-paper steps for doing a calculation) or other step-by-step procedures. This produces quick results because students can follow the teacher’s directions and crank out a page of correct answers. Students like completing their assignments with minimal struggle, parents are pleased by their children’s high grades, and the teacher is happy to make steady progress through the curriculum.
Unfortunately, the focus on rules can lead children to conclude that math is arbitrary and authoritarian. Also, rote knowledge tends to be fragile, and the steps are easy to confuse or forget. Thus those who see math instrumentally must include continual review of old topics and provide frequent, repetitive practice.
Relational Understanding: Math as a Connected System
Each mathematical concept is part of a web of interrelated ideas. To understand mathematics means to see at least some of this web and to use the connections we see to make sense of new ideas. Giving a correct answer without justification (explaining how we know it is right) is mere accounting, not mathematics. To be fluent in math means we can think of more than one way to solve a problem.
Primary goal: to see the building blocks of each topic and how that topic relates to other concepts. Key questions: “How?” and “Why?” How can we figure that out? Why do we think this is true? Values: logic and justification. Method: conversation. Talk about the links between ideas, definitions, and rules. Explain why you used a certain procedure, and explore alternative approaches. Use manipulatives to investigate the logic behind a technique. Primary benefit of this approach: flexibility.
Relational instruction focuses on children’s thinking and expands on their ideas. This builds the students’ ability to reason logically and to approach new problems with confidence. Mistakes are not a mark of failure, but a sign that points out something we haven’t yet mastered, a chance to reexamine the mathematical web. Students look forward to the “Aha!” feeling when they figure out a new concept. Such an attitude establishes a secure foundation for future learning.
Unfortunately, this approach takes time and requires extensive personal interaction: discussing problems, comparing thoughts, searching for alternate solutions, and hashing out ideas. Those who see math relationally must plan on covering fewer new topics each year, so they can spend the necessary time to draw out and explore these connections. Relational understanding is also much more difficult to assess with a standardized test.
What constitutes mathematics is not the subject matter, but a particular kind of knowledge about it.
The subject matter of relational and instrumental mathematics may be the same: cars travelling at uniform speeds between two towns, towers whose heights are to be found, bodies falling freely under gravity, etc. etc.
But the two kinds of knowledge are so different that I think that there is a strong case for regarding them as different kinds of mathematics.
This is the second post in my Understanding Math series, adapted from the expanded paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…
Welcome to the 92nd edition of the Math Teachers At Play math education blog carnival—a monthly smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college.
Let the mathematical fun begin!
By tradition, we start the carnival with a couple of puzzles in honor of our 92nd edition…
What is the maximum number of queens that can be placed on an chessboard such that no two attack one another?
Spoiler: Don’t peek! But the answer is here—and the cool thing is that there are 92 different ways to do it.
Table Of Contents
And now, on to the main attraction: the blog posts. Many articles were submitted by their authors; others were drawn from the immense backlog in my rss reader. If you’d like to skip directly to your area of interest, click one of these links.
Joshua Greene (@JoshuaGreene19) offers some great ways to tweak an already-wonderful multiplication game in Times square variations. “It was really interesting to see the different strategies that the students took to determining what would go on their boards.”
Tina Cardone (@crstn85) experiments with Bar Models in Algebra to help her students think about linear equations. “I did not require students to draw a model, but I refused to discuss an incorrect equation with them until they had a model. Kids would tell me ‘I don’t know how to do fractions or percents’ but when I told them to draw a bar, and then draw 4/5, they could do that without assistance…”
How can we get a peek at how our children are thinking? Kristin Gray (@mathminds) starts with a typical set of 1st Grade Story Problems and tweaks them into a lively Notice/Wonder Lesson. “When I told them they would get to choose how many students were at each stop, they were so excited! I gave them a paper with the sentence at the top, let them choose a partner and sent them on their way…”
Tracy Zager (@tracyzager) talks about her own mathematical journey in The Steep Part of the Learning Curve: “The more math I learn, the better math teacher I am. I keep growing as a learner; I know more about where my kids are headed; and I understand more about what building is going on top of the foundation we construct in elementary school.”
And that rounds up this edition of the Math Teachers at Play carnival. I hope you enjoyed the ride.
The December 2015 installment of our carnival will open sometime during the week of December 21-25 at Math Misery? blog. If you would like to contribute, please use this handy submission form. Posts must be relevant to students or teachers of preK-12 mathematics. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.
We need more volunteers. Classroom teachers, homeschoolers, unschoolers, or anyone who likes to play around with math (even if the only person you “teach” is yourself)—if you would like to take a turn hosting the Math Teachers at Play blog carnival, please speak up!
All parents and teachers have one thing in common: we want our children to understand and be able to use math. Counting, multiplication, fractions, geometry — these topics are older than the pyramids.
So why is mathematical mastery so elusive?
The root problem is that we’re all graduates of the same system. The vast majority of us, including those with the power to shape reform, believe that if we can compute the answer, then we understand the concept; and if we can solve routine problems, then we have developed problem-solving skills.
In the last part of the twentieth century, Reform Math focused on problem solving, discovery learning, and student-centered methods.
But Reform Math brought calculators into elementary classrooms and de-emphasized pencil-and-paper arithmetic, setting off a “Math War” with those who argued for a more traditional approach.
Now, policymakers in the U.S. are debating the Common Core State Standards initiative. These guidelines attempt to blend the best parts of reform and traditional mathematics, balancing emphasis on conceptual knowledge with development of procedural fluency.
The “Standards for Mathematical Practice” encourage us to make sense of math problems and persevere in solving them, to give explanations for our answers, and to listen to the reasoning of others—all of which are important aspects of mathematical understanding.
Through all the math education fads, however, one thing remains consistent: even before they reach the schoolhouse door, students are convinced that math is all about memorizing and following arbitrary rules.
Understanding math, according to popular culture—according to movie actors, TV comedians, politicians pushing “accountability,” and the aunt who quizzes you on your times tables at a family gathering—means knowing which procedures to apply so you can get the correct answers.
But when mathematicians talk about understanding math, they have something different in mind. To them, mathematics is all about ideas and the relationships between them, and understanding math means seeing the patterns in these relationships: how things are connected, how they work together, and how a single change can send ripples through the system.
Mathematics is the science of patterns. The mathematician seeks patterns in number, in space, in science, in computers, and in imagination. Theories emerge as patterns of patterns, and significance is measured by the degree to which patterns in one area link to patterns in other areas.
This is the first post in my Understanding Math series, adapted from the expanded paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…
Have you noticed a new math blogger on your block that you’d like to introduce to the rest of us? Feel free to submit another blogger’s post in addition to your own. Beginning bloggers are often shy about sharing, but like all of us, they love finding new readers.
Don’t procrastinate:The deadline for entries is this Friday, November 20. The carnival will be posted next week right here at Let’s Play Math.
Need an Idea-Starter?
If you haven’t written anything about math lately, here are some ideas to get your creative juices flowing…
Elementary Concepts: As Liping Ma showed, there is more to understanding and teaching elementary mathematics than we often realize. Do you have a game, activity, or anecdote about teaching math to young students? Please share!
Arithmetic/Pre-Algebra: This section is for arithmetic lessons and number theory puzzles at the middle-school-and-beyond level. We would love to hear your favorite math club games, numerical investigations, or contest-preparation tips.
Beginning Algebra and Geometry: Can you explain why we never divide by zero, how to bisect an angle, or what is wrong with distributing the square in the expression ? Struggling students need your help! Share your wisdom about basic algebra and geometry topics here.
Advanced Math: Like most adults, I have forgotten enough math to fill several textbooks. I’m eager to learn again, but math books can be so-o-o tedious. Can you make upper-level math topics come alive, so they will stick in my (or a student’s) mind?
Mathematical Recreations: What kind of math do you do, just for the fun of it?
About Teaching Math: Other teachers’ blogs are an important factor in my continuing education. The more I read about the theory and practice of teaching math, the more I realize how much I have yet to learn. So please, fellow teachers, don’t be shy — share your insights!
Would You Like to Host the Carnival? Please?!
Hosting the blog carnival is fun because you get to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival often brings a nice little spike in traffic to your blog. If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.
Explore the Other Math Carnivals
While you’re waiting for next week’s Math Teachers at Play carnival, you may enjoy:
You know you’re a math teacher when you see a penny in the parking lot, and your first thought is, “Cool! A free math manipulative.”
My homeschool co-op math students love doing math with pennies. They’re rather heavy to carry to class, but worth it for the student buy-in.
This month, I’m finishing up the nearly 150 new illustrations for the upcoming paperback edition of my Let’s Play Math book. I’m no artist, and it’s been a long slog. But a couple of the graphics involved pennies—so when I saw that penny on the ground, it made me think of my book.
And thinking of my book made me think it would be fun to share a sneak peek at coming attractions…
The Penny Square: An Example of Real Mathematics
Real mathematics is intriguing and full of wonder, an exploration of patterns and mysterious connections. It rewards us with the joy of the “Aha!” feeling. Workbook math, on the other hand, is several pages of long division by hand followed by a rousing chorus of the fraction song: “Ours is not to reason why, just invert and multiply.”
Real math is the surprising fact that the odd numbers add up to perfect squares (1, 1 + 3, 1 + 3 + 5, etc.) and the satisfaction of seeing why it must be so.
Did your algebra teacher ever explain to you that a square number is literally a number that can be arranged to make a square? Try it for yourself:
Gather a bunch of pennies—or any small items that will not roll away when you set them out in rows—and place one of them in front of you on the table. Imagine drawing a frame around it: one penny makes a (very small) square. One row, with one item in each row.
Now, put out three more pennies. How will you add them to the first one in order to form a new, bigger square? Arrange them in a small L-shape around the original penny to make two rows with two pennies in each row.
Set out five additional pennies. Without moving the current four pennies, how can you place these five to form the next square? Three rows of three.
Then how many will you have to add to make four rows of four?
Each new set of pennies must add an extra row and column to the current square, plus a corner penny where the new row and column meet. The row and column match exactly, making an even number, and then the extra penny at the corner makes it odd.
Can you see that the “next odd number” pattern will continue as long as there are pennies to add, and that it could keep going forever in your imagination?
The point of the penny square is not to memorize the square numbers or to get any particular “right answer,” but to see numbers in a new way—to understand that numbers are related to each other and that we can show such relationships with diagrams or physical models. The more relationships like this our children explore, the more they see numbers as familiar friends.
The Penny Birthday Challenge: Exponential Growth
A large jar of assorted coins makes a wonderful math toy. Children love to play with, count, and sort coins.
Add a dollar bill to the jar, so you can play the Dollar Game: Take turns throwing a pair of dice, gathering that many pennies and trading up to bigger coins. Five pennies trade for a nickel, two nickels for a dime, etc. Whoever is the first to claim the dollar wins the game.
Or take the Penny Birthday Challenge to learn about exponential growth: Print out a calendar for your child’s birthday month. Put one penny on the first day of the month, two pennies on the second day, four pennies on the third day, etc. If you continued doubling the pennies each day until you reach your child’s birthday, how much money would you need?
Warning: Beware the Penny Birthday Challenge! Those pennies will add up to dollars much faster than most people expect. Do not promise to give the money to your child unless the birthday comes near the beginning of the month.
A Penny Holiday Challenge
The first time I did pennies on a calendar with my homeschool co-op class was during December, so we called it the Penny Christmas Challenge:
How many pennies would you need to cover all the days up to the 25th?
I told the kids that if their grandparents asked what gift they wanted for Christmas, they could say, “Not much. Just a few pennies…”
The Penny Square, Dollar Game, and Penny Birthday Challenge are just three of the myriad math tips and activity ideas in the paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…
There is a huge elephant standing in most math classrooms, it is the idea that only some students can do well in math. Students believe it, parents believe and teachers believe it. The myth that math is a gift that some students have and some do not, is one of the most damaging ideas that pervades education in the US and that stands in the way of students’ math achievement.