One way to understand this is to start at the end and work backward, so let's do that. Let's start at advanced high school mathematics.

Since high school advanced math courses are electives, students have either mastered the prerequisite skills to take the course, or they have

*almost*mastered the prerequisites and convinced the powers that be they can handle the rigor and rise to the occasion. Some struggle in advanced math , but some for whatever reason, don't struggle as much. It could be they have developed a love for complex problem solving, or it could be they have had stellar mathematics instruction through the years, or it could be something else. Many students, however, don't get the chance to fail or succeed in advanced math courses, because they don't take them.

We know that a significant number of students in high school don't take calculus or pre-calculus before graduation. Some high schools don't even offer calculus. Research says students who take advanced math in high school are more likely to enter into higher education, so why is it that so many do not qualify for or opt out of advanced math in high school?

Moving one step further back, we look at regular high school math, that is, algebra and geometry. Students who don't take advanced math, or fail to earn prerequisite credit for advanced math, either struggled with algebra and/or geometry, or they did not have a positive enough experience to want to take the step to advanced math. Why is that?

High school teachers report that many 9th graders show up for their algebra class without a solid understanding of fractions and division. When it comes time to operate with fractions, they get confused and start sweating. They wonder: Do I invert and multiply? HOW do I invert and multiply? WHY and WHEN do I invert and multiply? Which one is the numerator? When is it that denominators have to be common, and how do I get them that way? What is a

*rational number?*What's the difference between a

*ratio*and a fraction? I'm not even sure I know how to do long division, so how on earth am I supposed to be able to

*divide fractions*?!

These questions indicate a lack of confidence in understanding and operating with fractions. So let's take the next step back and dive into middle school math, which tackles operating with rational numbers and integers (part of what is often called "pre-algebra") as well as basic linear algebra concepts. The basis of linear algebra on a coordinate plane, dependent and independent variables and relationships, "rise over run"and slope, and y-intercept all require students to not only understand fractions, but to

*use*them and multiply with them in equations. A linear equation in slope-intercept form is a two-step equation that involves multiplication and division to solve. Students will struggle with the basics of linear algebra in 7th and 8th grade if they do not have a functional understanding of fractions and mastery of basic and long division.

The next step back brings us to fifth and sixth grade and application of multiplication and division, as well as fractions, into solving problems relating to money and probability. Without an understanding of the relationship fractions have to decimals, students will likely struggle with everything from finding unit rates to determine likelihood of an outcome in simple probability. Adding, subtracting, multiplying and dividing fractions and mixed numbers becomes a tedious task of memorizing steps rather than a process of estimating, applying understanding, and choosing an efficient strategy.

Back still more to 4th grade. Here the relationship between division and fractions is front and center, unless it isn't. For students who fail to grasp that relationship, conceptual understanding of fractions, (beyond unit fractions and equal parts, is likely to be weak. Students are still developing their fluency in division of whole numbers, exploring division algorithms, and understanding remainders. Students who do not make the connection between remainders and fractions will struggle to understand and operate with mixed numbers.

3rd grade. Multiplication strategies and introduction to fractions are a big focus at this point. This is where we expect students to become both

*fluent*in multiplication (able to understand

*how*to multiply, what multiplying is, and to choose from multiple strategies to solve multiplication problems) and to achieve

*automaticity*in multiplication facts (the ability to

*know*in an instant that a multiplication fact is accurate, having already understood multiplication as repeated addition, and used a variety of models like arrays and areas of rectangles to visualize and deconstruct multiplication problems). If students are not both

*fluent*and

*automatic*with their multiplication facts, division will likely not make a lot of sense, and bigger multiplication problems will be a more arduous and time consuming cognitive process.

Alas, 2nd grade, where students are mastering subtraction strategies and gaining understanding of place value. Subtracting is far less intuitive than addition, and presents a linguistic challenge. Students used to finding out "how many are in all" with addition are now expected to understand,

*How many are left? How many more? How many less? What is the difference? Take away, subtract, minus,*and

*how many in all NOW?*Subtraction presents the first real algebraic thinking students encounter, as the

*difference*represents an unknown addend in an addition problem. Simply memorizing subtraction facts is not enough for students to truly master and become both fluent

*and*automatic with subtraction. It is essential students understand exactly what it is they are doing when they subtract. Without a complete understanding and automaticity of this operation, students will likely struggle with every algorithm for division.

We could go further back... The building blocks of subtraction and place value are assembled starting in kindergarten and first grade, but I singled out subtraction in 2nd grade and multiplication in 3rd grade because these are the first real significant hurdles that many students have trouble clearing. Students who do not become fluent

*and*automatic in subtraction

*and*multiplication

*will*struggle when they attempt to take on division and fractions. The algorithms they use will not be as efficient as intended, and division and fractions will become a roadblock to understanding, to engagement, to finding joy in mathematics and mathematical problem solving.

To conclude, we need to plan our units in ways that emphasize opportunities to explore subtraction and multiplication concepts in the early grades, and division and fractions in the upper grades, and follow that up with consistent practice for fluency and automaticity. It's not easy, but I want to be a resource for making that happen, and utilizing all that EM has to offer, in any way I can.

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