School days, school days
Dear old golden rule days
Reading and ‘riting and ‘rithmetic . . .
I learned this song in elementary school and sang it every day as I walked to school. I wasn’t entirely sure what “‘rithmetic” was until years later as I watched my college algebra students struggle because they lacked these ‘rithmetic skills—or, simply put the math that precedes algebra.
Pre-algebra, the bridge between arithmetic and algebra
Pre-algebra is the bridge connecting arithmetic to algebra, and algebra is the foundation for all future STEM courses. If we were to build a house of mathematical knowledge, arithmetic would be the piers of the foundation, pre-algebra would be the beams, and algebra would be the joists. If any part of the foundation begins to falter, the entire building is in jeopardy. The same is true for students’ mathematics education. If their foundation is weak, their ability to learn higher levels of math or science is in serious jeopardy.
Nearly every college major requires that students pass a college algebra class, because those skills serve as a foundation for upper-level courses. It’s disheartening to watch calculus students struggle because of insufficient algebra skills and algebra students struggle because of an insufficient background in arithmetic. The piers of arithmetic aren’t missing completely—they were simply spaced too far apart. This caused the beams to also be spaced too far apart, which resulted in the joists having longer (and thus weaker) spans, which in turn made learning higher levels of math, business, and science a shaky proposition. The foundation began to falter. Faulty foundations can be fixed—more piers and beams can be added to give the joists more support—but the best course of action is to build a solid foundation from the beginning.
How to build a strong foundation in math
I don’t believe there is a one-size-fits-all solution. However, I do believe that there is a set of principles that can lead to the establishment of a strong mathematics foundation for your students.
- Invest sufficient time for understanding. Students spend the majority of their day reading: literature, science, history, Bible, etc. In turn, students tend to score better in subjects based on reading. What would happen if students spent more time working on math? As you are planning each school day, be sure to invest sufficient time both in math instruction and in having your students independently work through math problems.
- Keep a nice, steady pace. Don’t rush through your math curriculum. Remember, you are building a foundation. Moving too quickly is akin to “cram and forget” methods of study. Retention—establishing the concepts in students’ long-term memory—requires a more sustained effort that continues even after short-term mastery has been achieved. Assign daily homework that is cumulative and ask students to rework problems until they are all correct.
- Ban multiple-choice assignments. Something happens when students are given multiple-choice assignments. They abandon the math skills being taught in favor of test-taking strategies that help them determine the correct answer by process of elimination. They ace assignments without ever using the math concepts being assessed; thus, the purpose of the assignment was never truly fulfilled. Multiple-choice assignments also rob students of the opportunity to develop their ability to communicate mathematically. Students need to practice showing their work in steps that are easy for someone else to follow. This habit will develop their communication abilities and help them track their thoughts and monitor their work for mistakes.
- Teach the thought process, not the example. When students watch you work through example problems, everything seems clear and understandable. But when it’s their turn, they aren’t sure how to start. Teach the thought process, not just the example. Examples will fail them as soon as the problem is presented in a slightly different manner. However, understanding the required thought process (the questions they should ask, why certain steps are useful at certain times, etc.) will help guide them even when the format of the question changes.
- Embrace the challenge. Math isn’t easy. That’s why I love it! Embrace the challenge. Help students build perseverance by allowing them to struggle with a problem. If they are stuck, assist them by having a conversation about why they are stuck and what they don’t understand, then guide them through the general thought process. Don’t tell them they are wrong. Ask them to explain how they got the answer. Respond with questions, not answers. Otherwise, your students may become dependent on your bailing them out of tough problems and they won’t learn to think their way through them independently. Often, students give up before putting in a real effort. They feel they should be able to work the problem immediately, without the need to stop and contemplate. Help them understand that sometimes contemplation is required.
- Challenge them. Math that is too easy for students encourages the development of poor problem-solving habits. If they are completing many of the steps in their head or finishing every problem in under three minutes, then the math is too easy for them. Don’t go faster—go deeper. Students should periodically encounter problems that take them longer than ten minutes to figure out.
- Include logic and problem-solving. I once left my abstract algebra textbook at my grandma’s house. She called me and said, “You left a book at my house. It says algebra, but when I open it up, all I see is Greek.” At that moment, I realized my math books hadn’t used actual numbers for a while. This is because math is about more than just It is also about developing logical and analytical thinking. Incorporate critical-thinking activities and problem-solving in your math course. Problem-solving presents students with problems that have no clear approach and a non-exact solution. Students may never use synthetic division after completing an algebra course, but they will use the analytical thinking you helped them to develop while they learned synthetic division.
- Incorporate writing. The mathematics classroom shouldn’t be devoid of writing. Periodically, students should be required to explain their answer, concept, or method in writing. Additionally, math journals are a great tool to enhance learning. Journals should be unique to each student but should, at a minimum, include a quick summary of what the student has learned, written in his or her own words.
Why the struggle is worth it
Building a strong foundation in mathematics takes work, and at times it will be a struggle. However, as your students progress through their education and their foundation doesn’t falter, it will be worth it. High school and college math, calculus, chemistry, business, engineering, physics, and economics courses, even ultimately working in corporate America—these areas are all a lot easier for students with strong algebra skills, built on strong pre-algebra skills, resting on a solid foundation of good old “’rithmetic.”
Is your student ready for pre-algebra?
Readiness for pre-algebra is less a matter of age than of fluency. A student who is ready can add, subtract, multiply, and divide whole numbers without hesitation, and can do the same with fractions, decimals, and percents, moving between the three forms with ease. He or she understands place value well enough to estimate an answer before working it out, follows the order of operations without being reminded, and can read a multi-step word problem and decide what it is asking. Just as important is the habit of showing work, writing each step in order so that someone else could follow the reasoning, a discipline explored in Justify Your Answer.
Gaps in any of these areas are common and are no cause for alarm, but they are a reason to pause. Another season spent with fractions and decimals usually costs far less than a year of algebra built on sand. Parents who are unsure where their student stands can request a pre-algebra placement exam from Scholé Academy, which helps a family decide whether the Pre-Algebra course is the right next step or whether another year of arithmetic would serve the student better.
Frequently asked questions
What is a pre-algebra readiness test?
A pre-algebra readiness test, sometimes called a placement test, is a short assessment of the arithmetic a student needs before beginning pre-algebra: operations with whole numbers, fractions, decimals, and percents, along with place value, order of operations, and simple word problems. Its purpose is placement rather than a grade. It shows whether a student should move ahead or spend more time on arithmetic first.
What grade is pre-algebra?
Pre-algebra is most often taken in seventh or eighth grade, but the range is wide. Some students are ready in sixth grade, and others do better to wait a year. Because the course rests on arithmetic rather than on age, a student’s fluency with fractions, decimals, and percents is a better guide than the grade printed on a transcript.
What if my child is not ready for pre-algebra?
Spend another season on arithmetic, with particular attention to fractions, decimals, percents, and the order of operations, and ask for written work on every problem. The time is well spent; it is the work of setting the piers and beams described above. A student who enters pre-algebra fluent in arithmetic tends to move through algebra with far less struggle than one who was rushed.
