Mastering Mathematics

One of the questions that sometimes surfaces from prospective families at Paideia is: “Do you teach Common Core math?” This question usually bubbles up from either a visceral loathing of Common Core or a nervous fear that students who don’t do Common Core will somehow be “behind.”

Before answering that question, I like to ask some questions of my own. “What does Common Core math mean to you? What do you not like about it?” Or alternatively, “Why do you think it is important?”

COMMON CORE: WHAT IS IT GOOD FOR?

Common Core math was created because educators noticed students having a lack in critical thinking skills in mathematics. Students appeared to just be memorizing ways to do problems in a “monkey see, monkey do” fashion, rather than understanding the why behind what they were doing. Common Core was an attempt to build number sense into mathematical learning, teaching students to recognize patterns and reason out steps on their own instead of simply memorizing processes.

But although creators of Common Core math correctly emphasized critical thinking skills, they incorrectly identified memorizing as a negative and as being at odds with comprehension. Over the last decade, addition, subtraction, multiplication, and division fact fluency has fallen by the wayside as the importance of memorizing basic arithmetic facts has been steadily deemphasized and denigrated. Students being able to reason out the process for the problem has become more important than students actually being able to complete the problem correctly. “Because…we all have calculators now, don’t you know?”

JOINING IMITATION WITH REASON

At Paideia, we believe that both number sense and fluency in arithmetic facts are important. In fact, the ability to do arithmetic fluently can actually lead to far greater number sense. The student who has no difficulty adding 5 and 16 in his head, will not have an extra barrier to contend with when understanding how to move numbers to opposite sides of the equation in Algebra. The student who is quick with her multiplication tables will be able to see at a glance how she can factor numbers out of the polynomial.

Memorizing math facts is important. Understanding place value, identifying patterns, decoding word problems, reasoning independently, and explaining your solutions are also important. As one of our favorite books, The Liberal Arts Tradition, puts it, imitation must be joined with reason before the art of arithmetic can be achieved. It is essential to realize that math fact fluency and critical thinking work together to build a strong math student.

BUILDING NUMBER SENSE…WE’RE HERE TO HELP!

The approach to math that we use at Paideia targets both of these skill sets. Our Math-U-See curriculum is consciously designed to foster number sense. Some of the ways of doing addition and multiplication in our Math-U-See curriculum can seem strange to parents at first. Students learn to estimate for approximate answers before solving for exact answers. Students expand the notation for their multiplication problems to understand what each number means according to its place value. “This isn’t how I learned to do math!,” some parents might think. “So how can I help my child?”

We are here to assist you. If you have questions about “the rule of four” or “the same-difference theorem” or any of the unfamiliar methods in our Math-U-See curriculum, don’t hesitate to ask. Most of our teachers would likely tell you that they understand how numbers work far better after learning the methods Math-U-See uses.

MEMORIZING MATH FACTS…EVEN WHEN IT’S HARD

Along with inculcating number sense, we are always working to build arithmetic fluency. At the beginning of each elementary class, students do a timed math facts page, mental Calisthenics to “warm up” for the practice that lies ahead.

In 1st-6th grade you will also see math facts on the at-home work checklist. It can be tempting to skip over these instead of making a daily effort to master them. (I know from personal experience that students who struggle the most with math facts are the most prone to shirk practicing them…) However, skipping math facts practice because they are hard for your child is counter-productive and ensures that your child will struggle with math all the way up through algebra.

Lean into the hard. Use strategies like breaking addition facts into small chunks and doing them over and over. Practice skip counting 8s first, and then time how fast you can do the 8s multiplication facts. Try to better your time the next day.

Think of arithmetic as any other skill that needs to be learned. When your daughter learns basketball, you don’t just teach her “the process of a layup”, you make sure she has the muscle memory to put the ball in the basket over and over again. When your son gets a new Chopin prelude, you don’t let him settle for “understanding how the chords work”, you have him play it over and over again until his hands transition fluently through each cadence. Math, like sports and music, is a skill that requires drill.

IN SUM

Giving our children the gift of mathematical reasoning and fluency is more than an attempt to make them successful in college or a bid to get them a well-paying career as an engineer. In a very real sense, it is an attempt to open their eyes to the order, design, and beauty of the world in which they live.

One of Isaac Newton’s most famous quotes is that “God created everything by number, weight, and measure.” Numbers are built into the fabric of our world, and the more we learn about the numerical patterns, processes, precision, and predictability of Creation, the more we learn about the nature of the Creator.

Proverbs 25:2 says “It is the glory of God to conceal a matter, but the glory of kings is to search out a matter.” Let us teach our children the glory of the search.

The Challenges of Writing a Curriculum Guide

“The question is not, — how much does the youth know? when he has finished his education — but how much does he care? and about how many orders of things does he care? In fact, how large is the room in which he finds his feet set? and, therefore, how full is the life he has before him?”
― Charlotte Mason, School Education: Developing A Curriculum

One of the drawbacks of a Curriculum Guide is that it, of necessity, produces division rather than integration. Classical education aims to show students the integration of all things in Christ, the One in Whom all things hold together. In some ways, the very structure of a Curriculum Guide is antithetical to this goal, for a Curriculum Guide—in order to categorize and communicate effectively—must divide and break down concepts into separate disciplines. The Curriculum Guide analyzes the animal of education in the same way that the scientist does—by cutting it apart, by tagging it, labeling it, defining it. It is the skilled teacher who can take these two-dimensional diagrams and parsed out pieces of pedagogy and pull them back together into a living, breathing animal that can run on all fours or take flight into the unknown.

Another challenge for a Curriculum Guide is conveying that the non-academic aspects of education are the most important. What does it profit a man to gain the whole world of Calculus and Physics and lose his soul? Since we believe that the foundation of education is piety, “the proper fear and love of God and man,” it is necessary that our Curriculum Guide should go beyond academic standards in some of its goals. It must advance from the realm of quantitative knowledge into the realm of wisdom and virtue. Those of us who have formulated Objectives and Key Results (OKRs) know that a key result must be measurable. And yet how can wisdom and virtue be measured by human means? The goal of education is “to love what is lovely,” but where is the yardstick to know whether that has been attained?

Because of these challenges, any Curriculum Guide will have limitations in conveying what it means to be a student. And yet, despite the limitations, the benefits of the Curriculum Guide still make it a profitable tool as we guide students along the way.

Who or What is the Curriculum Guide?

“Therefore we also, since we are surrounded by so great a cloud of witnesses, let us lay aside every weight, and the sin which so easily ensnares us, and let us run with endurance the race that is set before us, looking unto Jesus, the author and finisher of our faith, who for the joy that was set before Him endured the cross, despising the shame, and has sat down at the right hand of the throne of God.”
– Hebrews 12:1-2

THE COURSE AND THE GUIDE

The vision of Paideia is to train worshipers of Jesus Christ. We aim to assist parents in bringing up their children in the culture (paideia) and counsel of Jesus. This is the ultimate goal that informs all else in our specific Curriculum Guide.

The word curriculum comes from currere – “to run.” It is the course that students run as they grow in the paideia of the Lord. The author of Hebrews speaks of the whole Christian life as a course to be run. As teachers, we are guides along this course. We are part of the cloud of witnesses, mature Christians who have traveled farther along the course than they have. As human guides, we urge students to shake off any hindrances and point them forward to the True Guide: Jesus, the source and fulfillment of our faith.

The importance of the teacher as guide cannot be overstated. Luke 6:40 says: “A disciple is not above his teacher, but everyone who is perfectly trained will be like his teacher.” If students are to become like their teachers, the teacher must then be someone worthy of imitating. As Paul says, “Imitate me, just as I also imitate Christ” (1 Corinthians 11:1). In a sense, the person of the teacher is of primary importance in the classroom. The student goes to second grade, not merely to learn carrying and borrowing, but to learn “Mrs. True.” The student goes to sixth grade, not merely to learn sentence diagramming and Shakespeare, but to learn “Mrs. Barnard.” The best Curriculum Guide in the world will not make up for a foolish or ignorant teacher. An excellent teacher can make up for deficiencies in the Curriculum Guide.

Yet although the character and aptitude of the human guide is paramount, at the same time, a cohesive Curriculum Guide is necessary, especially in a school setting, so that students can gradually and incrementally work their way through the various disciplines under the tutelage of various teachers. The Curriculum Guide highlights the “must-see” sites along the way. The Curriculum Guide sets the wayposts and stopping points as students transfer from one guide to another. The Curriculum Guide makes sure that none of the journey is forgotten or overlooked. To attempt a journey of this magnitude without a map might mean that the Stage One scout is planning to hand off his wagon train in Kansas while the Stage Two scout is expecting the wagon train to make it all the way to Wyoming before the reins are given over to him.

But, like a roadmap, the Curriculum Guide requires its reader to have the skills to read it and implement it properly. Of itself, the map is only a pretty picture. In the right hands, it becomes a powerful tool. Teachers must realize that the Curriculum Guide is only effective insofar as they are effective guides who know how to hold the map right side up, assess the strength of this year’s pilgrim group, and plan for weekly stages and watering hole stops.