Re-Imagine Algebra

Art Bardige with Peter Mili

 What if Math

We haven’t thought very creatively about education…We are in the midst of a once in a century technological revolution, and the last time we did this, in the 20th century, hand-in-hand with the industrial revolution went a revolution in education. We haven’t had near the revolution in K-12 commensurate with the digital revolution we are in the midst of. So I don’t think we are thinking ambitiously enough or investing enough in the young…

The Economist editor-in-chief Zanny Minton Beddoes 12.9.20

We have to make room for “revolutionary” math. Business has become dependent on data science and statistics, financial reasoning and coding, and certainly STEM problem-solving. To prepare our students for the world they will inherit, we have to also ensure that they are spreadsheet fluent, think functionally to build models, and represent concepts and data visually because images now dominate communication and collaboration. To make space in the math curriculum for these critical topics, it is not enough to take out a concept here or remove a chapter or two there. We need to be much more ambitious.

What if we eliminate Algebra I? Our math curriculum is jammed full. The Common Core’s emphasis on rigor, coupled with the importance of AP Calculus for college admission, have compressed and elaborated K-12 math substantially increasing the demand on students and teachers. The results are clear. Teachers tell us that students are not engaged, they ask, “Why do I have to learn this!” NAEP scores remain flatlined despite increased per student expenditures and demanding testing. Colleges and businesses shout, “K-12 has failed math.”

Technology makes Algebra I computation obsolete. “Ask Google” is now the answer to Algebra I exercises. With topics ranging from solving linear equations to the quadratic formula, Algebra I (The Algebra of Equations) is saturated with paper algorithms practice, now totally unnecessary because students can type “solve 3x-5=7” into the search box. Solve, factor, graph most any equation is now built-into your phone. Try it!

Algebra I is the point of failure. If Algebra I were easy for all students, if it could be learned in a month, then perhaps we can make an argument for keeping it. But Algebra I is hard. A year-long course at best, it is the critical test of math competency, required by states for a high school diploma and placement into college math courses. Algebra I is not a benign problem, just a waste of time. It is a destroyer of dreams, the barrier to college degrees. Over a third of all college entrants must take it again as a non-credit “developmental” course. Half will fail these “killer courses” each time they try. And these are the students who passed it in high school!

Algebra II concepts repeat Algebra I. Nothing is lost by dropping Algebra I and going straight to Algebra II (The Algebra of Functions). It covers the same topics using functions instead of equations. Do your students ask, “Why is y now called f(x)?” “Why is x now called a variable not an unknown,” and “What’s the difference between an equation and a function.” Do they see the power, the importance, and the simplicity of functions?

Algebra II (The Algebra of Functions) is valuable. Functions are the building blocks for STEM models, the basis for spreadsheets, the heart of data science, and central to coding. Algebra II is the foundation for College Algebra/Precalculus. Statistics, and Calculus. 

Spreadsheets are function machines. But, you may ask, don’t we need Algebra I as a stepping stone to Algebra II, to support students as they move from unknown to variable, from equation to function? Perhaps, just perhaps, it was required in the past, but spreadsheets make it unnecessary today. Spreadsheets represent functions concretely. Variables are tables, and functions, entered with an = sign, are rules that extend tables with each pair of cells showing the function as numbers.

Spreadsheets are creative. Spreadsheets enable and encourage students to experiment, to build models out of functions, and to use Parameter Tables to change constants to ask “What if…” not “What is ___? By automatically linking tables and graphs, spreadsheets represent functions with formulas, numbers and visualizations. And by giving students the power to manipulate constants, variables, and rules, spreadsheets enable students to compare models with real data, change models and build ones.

Spreadsheets enable real problem-solving. Math educators have long sought to make problem-solving the core of the math curriculum. But the difficulty all too many students have with paper algorithmic calculation and the complexity inherent in working with real data have limited so-called problem solving to simplistic artificial exercises. Spreadsheets, the essential quantitative problem-solving tool in the world of business, solves this problem. By handling repetitive complex calculations, they enable students and teachers to attend to concepts. And with the power of spreadsheets to handle rich data, problems can reflect the real world, gaining the relevance that excites and energizes learning. 

“Why do I have to learn this?” Imagine what it will mean to have the tools to enable the problem-solving we dream of. Imagine a 6-12 grade math curriculum rich in a wide variety of problem-solving explorations. Imagine students exploring and solving problems of interest to them and their design teammates while learning 21st century skills including data science, coding, and financial reasoning. Imagine students graduating high school with spreadsheet skills and problem-solving capability. Imagine students finding math relevant, not irrelevant.

Spreadsheets bring equity to math education. Today, far too many students enter school with limited numbersense practice, start out behind and fall further back as the ladder of math skills builds upon previous math skills. The demanding bloated curriculum leaves them little chance to catch up thwarting a school’s remedial efforts. Spreadsheets avoid paper calculation weakness, support visual thinking and concrete representations to offer every student a fresh star. They have a new chance to see math as a subject they can understand and use, an opportunity to believe in a career that requires math or functional thinking.

Learning math as a laboratory science. How do we re-imagine our math curriculum? We built the conceptual foundation with over 125 What if Math Spreadsheet Labs as a platform on which to build a new problem-solving curriculum. The Labs develop functional thinking, standard student use of parameter tables, and treat functions as tables, providing student lessons in the key concepts of the Algebra of Functions to access as needed. Students can work individually or in teams. Available free on the Web, they support spreadsheet problem-solving modules.

Explorations. We are building a digital age library of multi-week, multi-disciplinary problem-solving modules for students across a wide range of interests. Explorations are rich in visualizations, Web-links, shared activities, and data sets. They begin with what Einstein called “Proper Questions” as the secret to problem solving. Divided into a series of Inquiries, they use spreadsheets, links, and functional thinking to solve interesting real-world problems that are STEM cross-disciplinary engaging students in financial reasoning and coding.

A curriculum promoting choice and engagement. Explorations fill a matrix of Interests by Algebra of Functions concepts. We will be supporting teachers and others to build-out this curriculum, Exploration by Exploration, around topics that excite them and their students. We suggest that some be required modules for common class experiences and others open to student choice. Spreadsheets and Explorations offer every student math that will prepare them for their future.

A house of cards. Once we admit that our students no longer need to learn Algebra I, the whole structure of the math curriculum, so long an immutable staircase, collapses. If we don’t have to prepare students for the calculation and concepts of the Algebra of Equations, why, we must ask, should they have to master the paper algorithms of whole numbers, decimals, and particularly fractions? Why must we teach the operations in the canonical order (addition, subtraction, multiplication, division) marching up through grades 1 through 4, when the concepts don’t require it? Why would we withhold spreadsheets and internet technology from students who have already made it a critical part of their lives outside of school?

ELA as a model. We ditched the artificial sequence of grade dependent learning in reading and writing more than 20 years ago. English Language Arts no longer follows a defined path, a set of cumulative stairs to knowledge and competence. In the primary grades it focuses on developing reading and writing skills and from then on it has students use those skills in a wide variety of contexts and interests to practice and steadily strengthen those skills. Why shouldn’t math follow a similar pattern? Why should our students and teachers be bound by a rigid structure they will never use? 

Reimagining and Reinventing Education: The canonical math sequence we all know so well is no longer relevant, no longer needed. The sequence that has been used to “track” and classify students, way before they are ever ready, is gone. The courses and tests that have for more than a half century been “cheap” ways of evaluating students no longer have meaning. Even the step-by-step ancient “grade” structure of our schools is brought into question. We are free to not only re-imagine math education, we are free to reinvent all of education.

Re-Imagining Algebra Presentation.