What does it take for parents to connect their children to a seamless and rigorous K-12 curriculum?
By “rigorous”, I mean “presenting a challenge that is appropriate for families with high educational aspirations and that has students with the ability to fulfill them”. For some, the solution has been to get one’s child into a rigorous private school. Yet, private school admission is not a guarantee. And what about the parents of students in public school? So far, my exploration has only turned up partial answers.
What I’ve learned in general:
- It is up to parents to cobble together the best experiences they can find in the short run while researching and advocating for future opportunities, mainly because:
- The authority to build a seamless curricula in public schools is diffused among many entities— district level administrators, state legislators, state departments of education, school boards, individual sectors (high school, middle school, primary and elementary school), specialized programs, and individuals schools; and
- In private schools, if a student’s performance in early grades is bumpy, that could set them on a trajectory for the rest of their middle and high school experience that could be considered less than rigorous.
This means that the original idea I had that I could get someone to give me a map of a rigorous curriculum, (with perhaps variations or alternate routes) which I could in turn pass along to you, was profoundly flawed. So then I began refining the question to see how much I could sketch out and still have something worth sharing.
The first refinement was to focus on Prince George’s County, Maryland and ask the question—“What does a parent of a public school-bound student in Prince George’s need to do to access a rigorous and seamless curriculum?” Given its predominantly African American school population, perhaps it could provide insight into other diverse school districts. It turns out that Prince George’s parents can find options that offer “rigor” or “appropriate challenges”. For example, advanced math courses like Calculus, other advanced placement courses, and the international baccalaureate programs are offered in high school. However, in order to get students into the classes that will best prepare them for those greater challenges in high school, parents have to be flexible, creative, and determined.
Special programs that can provide academic preparation like language immersion, talented and gifted, STEM (science, technology, engineering and math), and charter schools abound. But access to most of these programs is determined by lottery. On one hand, lotteries nullify bias. On the other, when demand exceeds supply, parents will need to either shake the doors of multiple programs to gain entry into the one that opens first or go outside of the system for supplementary preparation.
“Going out of the system” to get supplementary programming or to get appropriate preparation means using websites like Kahn Academy (mentioned in a previous post), hiring a tudor, enrolling in summer programs like the Johns Hopkins Center for Talented Youth, or enrolling in programs offered by private online course providers. Each of these options has something to offer, but they don’t illuminate potential pathways for parents to follow.
So I then refined the question again. “What would it take for parents to prepare their children for calculus by the 12th grade?”. A rigorous curriculum would include calculus as an option. Pursuing this question helped build ideas about where a pathway could begin.
Getting to a potential beginning meant “mapping backward” from the target. In 2019, researchers at the Charles Dana center at the University of Texas published a report called, “Re-envisioning the Pathway to Calculus”. In it, they pointed to four overarching concepts that students needed to solidify in order to be ready for calculus:
- A deep understanding of the processes and functions of early math.
Students need to understand the process of mathematical operations in order to apply them to any set of numbers. For example, a young student may be able to repeat that 3 + 3 is 6, when said by an adult, but do they understand the operation of adding 3 units to another 3 units? Do they understand that “addition” or “subtraction” is a function? - The ability to follow changes In two sets of numbers.
Students need to be able to observe and operate on two sets of quantities simultaneously—seeing how those sets of numbers are related, and how they might differ when various processes are applied to them. - Fluency in mathematical notation.
Students need to be able to describe all of the functions they’ve had to use: addition, subtraction, division, fractions, decimals, etc. They also need to understand the symbols that represent them. - The ability to think algebraically.
Students need to be able to use algebraic expressions to find unknown information based on the information that’s been given. “Algebraic thinking” also requires using multiple steps to solve a problem.
The last overarching concept, “algebraic thinking” also provides a milestone to use to map backward again. If knowing algebra is one of the keys to success in calculus, what needs to be known first to succeed in algebra? According to Hung-Hsi Wu, a professor of Math at the University of California, Berkeley, the short answer is “understanding fractions”.
“We have children who are completely lost as to what a fraction is, and educators who publicly bemoan students’ failure to grasp the concept. Yet, strangely enough, no clear definition of a fraction is ever offered.”
In the rest of his article titled, “How to Prepare Students for Algebra”, he also echoes some of what the researchers at UT found as fundamental to succeeding calculus. Wu suggests that the comprehension of algebra primarily requires two foundational kinds of thinking: generality and abstraction. General thinking means understanding how a process (or set of processes) that’s applied to one set of numbers— such as the adding of two numbers, can be used in the adding of all numbers. By “abstraction”, Wu means understanding the ways that certain sets of numbers or results may occur in patterns. Given a set of numbers like 3,6,9,12,15, and 18, one needs to be able to explain the underlying pattern.
Understanding fractions requires both kinds of thinking. When seeing 2/5— thinking about that fraction generally means understanding that any number to the left of the slash is being divided by any number to the right of the slash. The process could also be described as 2 ÷ 5.
Seeing pairs of numbers like 10,5; 4,2; 20,10 should trigger the thought that the second number is being divided by 2 to get the second number – which could also be explained as the second number being 1/2 of the first number. Exhibiting that kind of insight is abstraction.
Wu’s critical point deeper in the article is that students are rarely given a precise definition of fractions. “We have children who are completely lost as to what a fraction is, and educators who publicly bemoan students’ failure to grasp the concept. Yet, strangely enough, no clear definition of a fraction is ever offered.”
According to Wu, student inability to define fractions starts with the way fractions are often taught. First, he claims that children are taught that a fraction like 3/5ths means three pieces of a pie that was evenly sliced into five pieces. The problem with that explanation is that it does not help a young student comprehend what 3 divided by 5 means—which is what the expression—”3/5ths” also requires.
The second problem with how fractions are taught comes when a student moves from adding whole numbers to adding fractions. The relatively straight-forward mental process of adding small whole numbers with one’s fingers becomes irrelevant once one is taught to search for common denominators as a part of string of other steps used to add fractions. The result leaves the false impression that fractions and whole numbers are not the same kind of mathematical entity. Yet, when a whole number like 6 is seen as 6/1 (as a whole number divided by one is that same number), it can become clearer that fractions and whole numbers are not all that different after all.
Avoiding mistakes in how fractions are taught, learning how to think about numbers in a general and abstract manner, learning the computations involved in adding and subtracting fractions, and recognizing the connections between whole numbers and fractions are critical to learning algebra.
So now we can start forming a sequence of milestones, having mapped backwards:
calculus –> algebra –> fractions –> elementary operations like addition, subtraction, division and multiplication –> and generalized and abstract thinking.
Let’s then use generalized and abstract thinking as the last milestone. “What does it take to prepare young students for this kind of thinking?”
This question was also pursued by Robert Moses, the civil rights activist (he just passed away in July of this year) and author of the book, Radical Equations. In his book, he shares his revelation that the teaching of algebra to young African American students was like ensuring the right of their parents to vote, particularly in states like Mississippi. He considered both to be acts of liberation.
That revelation followed him from the combination of protests and teaching he engaged in throughout the deep south to teaching in Tanzania to Cambridge, MA, where he eventually settled with his family. As the Moses family enrolled their daughter in the local public school, they also began advocating for his daughter’s school to make algebra available to her in the eighth grade. When the school said they lacked the capacity to meet his request he wound up volunteering to teach it himself.
Soon thereafter, he was encouraged by the school to take on more students. While he experienced success early on, he eventually encountered students who struggled. By reflecting on his exchanges with one struggling student in particular, Mr. Moses realized that the student’s real challenge was abstract thinking. A key example of that challenge was the student not fully understanding the meaning of negative signs when they are associated with numbers.
A negative number, like -5, is an early abstract concept. A student who misses this also might not understand that adding +4 and -5 is -1, not 9. So once Mr. Moses detected the student’s challenge, he sought to ground the abstract concept in a concrete example. He brought the students to the train system in Boston called, “The T”. By using the movement of trains through stations along metro lines, he could then teach the number line—the tool used to explain positive and negative numbers.
From Moses’ example we have final keys to integrate within our potential pathway:
- Look for ways to ground abstract concepts in concrete examples; and
- Consider taking a role in teaching challenging concepts for your children — especially early on, to make sure they are ready for more difficult concepts later. Even if the best you can do is to teach the number line, your relationship with your child may reveal what they are missing better than others outside the home.
Those keys or milestones can now be added to the others that we deduced from the previous questions.
3. Understand patterns of numbers and how those patterns might be explained.
4. Understand the function of operations (same link) like addition, subtraction, division, and multiplication. More than filling in blanks in homework packets, make sure students know what the processes mean.
5. Use a number line (while they are easy to draw, here’s a digital tool) to help students understand the abstract concept of negative numbers, and how addition and subtraction that includes negative numbers works.
6. Thoroughly understand fractions, not just as portions of a pie, but also as a concept that requires generalized and abstract thinking to comprehend, and a set of processes to add, subtract, divide, and multiply them.
7. Understand all the rules of algebra like the commutative and distributive laws.
8. For each step along the math pathway, use quizzes and exams (those given by the school, and perhaps those you can obtain on your own such as IXL) to determine what your child understands, and which areas need more work. Use parent/teacher conferences to ask more questions about what your teachers observe and the approaches they are taking to help reinforce key concepts.
With these high level milestones, you now have some initial considerations when plotting out a math pathway for your child. More details concerning your own children can be worked out with trusted teachers and advisors in your school, the county level school administration, or with friends and family who have had successful experiences.
Moreover, there are many underlying skills at each level of math that could be considered–too many to cover here. But in Prince George’s County, for example, their talented and gifted office could potentially offer advice to county residents about how one can get what they need for their child, even if it can’t be obtained through one’s school.
And while it is true that success in calculus is one measure used to determine whether your child is prepared for a competitive college and used by gatekeepers to STEM fields in college, I’m not declaring that every child must take it. The intent here is to suggest that the preparation for any challenging course in high school begins with seeds planted as early as pre-school. Calculus is just one example of a high aspiration.
In honor of the life and work of Robert Moses, next month I will look deeper into areas where parents’ advocacy can potentially provide better opportunity for students.
