The Pain a Quadratic Can Bring...
Tears. So many tears.
It never ceases to amaze me how many tears can be shed over math homework. It was 9:15pm on a bitterly cold December night in Canada. My exhausted 11th-grade daughter, just home from dance class, decided this was the perfect time to tackle a tough set of quadratic equations — because she had a quiz the next day.
“Dad! I don’t get it! This is stupid!” she declared. Confused as to why the existence of math was my fault, I nonetheless proceeded to walk through the concepts with her. And despite peppering me with a nearly constant barrage of incinerating glares, she slowly started making progress. Fifteen minutes, a half box of Kleenex, and twenty deep breaths later, she felt ready for the quiz.
As I turned to leave, I casually mentioned, “You know, in America, kids learn this in 8th or 9th grade.” I didn’t know it was possible to display both intense horror and deep compassion in one expression, but I promise you, she somehow accomplished it. She couldn’t believe that those poor souls would be subjected to the evil of quadratic functions so young.
In the United States, students in Algebra 1 are expected to master quadratic functions in 9th grade and are even encouraged to take it in 8th grade if possible. By contrast, in Canada, such topics are spread out over the 10th and 11th grade and are reserved for students who choose to opt-in to a pre-calculus pathway in mathematics.
According to study after study, Algebra 1 is highlighted as the most failed course in America. Some estimates say that as many as 40% of students take the course more than once in order to achieve an acceptable grade. “It’s not surprising, then, that the most common remedial course for college students is — you guessed it — math. Something is amiss in America when it comes to Algebra 1. So much has been written on how to solve this issue:
- We need better trained teachers!
- Students are underprepared!
- We need more technology!
- We need less technology!
- It’s the “new math” that’s the problem!
- We need more “real world” applications!
So many opinions.
Yet with all of these diagnoses, there is one that I am shocked doesn’t get more air time in the conversation, and its absence is the reason I feel compelled to add one more voice to the discussion. What if the problem isn’t with the students, or the teachers, or the methods? What if the problem is the Algebra 1 course itself?
I want to propose that Algebra 1 is a broken course. I propose that it is too big and it is too hard. To add to that, I believe that the concepts it contains are taught too early for most students to understand. And call me a crazy Canuck, but I just can’t help but poke my head down below the 49th parallel, take a shot of maple syrup and say, “There’s a better way!!” In Canada, our students perform better on international math assessments than American students. We also have just as much success in the same AP Calculus course that high achievers aim for in both countries. And we accomplish both of those things by covering fewer topics and covering them later.
What if the problem is the Algebra 1 course itself?
So with all of that in mind, I want to suggest that maybe the solution to the Algebra 1 problem in America is this: Teach less content, teach it later, and thereby have the opportunity to teach it better.
Less. Later. Better.
Let’s examine each of those in turn.
I still remember the first time I was doing a comparison of the standards between my native British Columbia’s math standards and the arrangement of the Common Core standards that many states adopt for their high school mathematics pathways. I couldn’t believe my eyes. In shock and disbelief, I kept saying to my colleagues, and anyone who would listen, things like, “They cover all of that in 9th grade?” and, “Man, this is a big course.” and, “I don’t know how we can cover all of this in a reasonable time frame.” I would estimate that most American high school courses contain about 50% more required content than a typical equivalent course in Canada. This is a massive difference.
What is interesting is that from 9th grade to 12th grade, a typical student in British Columbia would need to take four courses to prepare for taking AP Calculus, just as a student in California might:
British Columbia
- Mathematics 9
- Foundations of Math & Pre-Calculus 10
- Pre-Calculus 11
- Pre-Calculus 12
- AP Calculus
California
- Algebra 1
- Geometry
- Algebra 2
- Pre-Calculus
- AP Calculus
Let’s examine these two particular jurisdictions because they are both fairly typical of the content covered across each country, and they both have easily accessible curriculum frameworks we can compare. On the surface, there appears to be a simple equivalence here. However, when you get into the details of the topics contained in each of those courses, you quickly notice that a student in British Columbia is covering significantly less content than a student in California.
British Columbia
Mathematics 9
- operations with rational numbers (addition, subtraction, multiplication, division, and order of operations)
- exponents and exponent laws with whole-number exponents
- operations with polynomials, of degree less than or equal to 2
- two-variable linear relations, using graphing, interpolation, and extrapolation
- multi-step one-variable linear equations
- spatial proportional reasoning
- statistics in society
- financial literacy — simple budgets and transactions
Foundations of Math & Pre-Calculus 10
- operations on powers with integral exponents
- prime factorization
- functions and relations: connecting data, graphs, and situations
- linear functions: slope and equations of lines
- arithmetic sequences
- systems of linear equations
- multiplication of polynomial expressions
- polynomial factoring
- primary trigonometric ratios
- financial literacy: gross and net pay
Pre-Calculus 11
- real number system
- powers with rational exponents
- radical operations and equations
- polynomial factoring
rational expressions and equations - quadratic functions and equations
- linear and quadratic inequalities
- trigonometry: non-right triangles and angles in standard position
- financial literacy: compound interest, investments, loans
Pre-Calculus 12
- real number system
- powers with rational exponents
- radical operations and equations
- polynomial factoring
rational expressions and equations - quadratic functions and equations
- linear and quadratic inequalities
- trigonometry: non-right triangles and angles in standard position
- financial literacy: compound interest, investments, loans
AP Calculus
- same standards as California
California
Algebra 1
- understand the concept of a function and use function notation
- interpret functions that arise in applications in terms of the context
- analyze functions using different representations.
Build a function that models a relationship between two quantities - build new functions from existing functions
- construct and compare linear, quadratic, and exponential models and solve problems
- interpret expressions for functions in terms of the situation they model
- extend the properties of exponents to rational exponents
- use properties of rational and irrational numbers
- reason quantitatively and use units to solve problems
- interpret the structure of expressions
- write expressions in equivalent forms to solve problems
- create equations that describe numbers or relationships
- understand solving equations as a process of reasoning and explain the reasoning
- solve equations and inequalities in one variable
- solve systems of equations
- represent and solve equations and inequalities graphically
- summarize, represent, and interpret data on a single count or measurement variable
- summarize, represent, and interpret data on two categorical and quantitative variables
- interpret linear models
Geometry
- experiment with transformations in the plane
- understand congruence in terms of rigid motions
- prove geometric theorems
- make geometric constructions
- understand similarity in terms of similarity transformations
- prove theorems involving similarity
- define trigonometric ratios and solve problems involving right triangles
- apply trigonometry to general triangles
- understand and apply theorems about circles
- find arc lengths and area of sectors of circles
- translate between the geometric description and the equation for a conic section
- use coordinates to prove simple geometric theorems algebraically
- explain volume formulas and use them to solve problems
- visualize relationships between two-dimensional and three-dimensional objects
- apply geometric concepts in modeling situations
- understand independence and conditional probability and use them to interpret data
- use the rules of probability to compute probabilities of compound events in a uniform probability model
- use probability to evaluate outcomes of decisions
Algebra 2
- perform arithmetic operations with complex numbers
- use complex numbers in polynomial identities and equations
- interpret the structure of expressions
- write expressions in equivalent forms to solve problems
- perform arithmetic operations on polynomials
- understand the relationship between zeros and factors of polynomials
- use polynomial identities to solve problems
- rewrite rational expressions
- create equations that describe numbers or relationships
- understand solving equations as a process of reasoning and explain the reasoning
- solve equations and inequalities in one variable
- represent and solve equations and inequalities graphically
- interpret functions that arise in applications in terms of the context
- analyze functions using different representations
- build a function that models a relationship between two quantities
- build new functions from existing functions
- construct and compare linear, quadratic, and exponential models and solve problems
- extend the domain of trigonometric functions using the unit circle
- model periodic phenomena with trigonometric functions
- prove and apply trigonometric identities
- translate between the geometric description and the equation for a conic section
- summarize, represent, and interpret data on a single count or measurement variable
- understand and evaluate random processes underlying statistical experiments
- make inferences and justify conclusions from sample surveys, experiments, and observational studies
- use probability to evaluate outcomes of decisions
Pre-Calculus
- perform arithmetic operations with complex numbers
- represent complex numbers and their operations on the complex plane
- represent and model with vector quantities.
Perform operations on vectors - perform operations on matrices and use matrices in applications
- interpret the structure of expressions
- rewrite rational expressions
- create equations that describe numbers or relationships
- solve systems of equations
- interpret functions that arise in applications in terms of the context
- build new functions from existing functions.
- extend the domain of trigonometric functions using the unit circle
- model periodic phenomena with trigonometric functions
- prove and apply trigonometric identities
- apply trigonometry to general triangles
- translate between the geometric description and the equation for a conic section
AP Calculus
- same standards as British Columbia
Now you might think that I am cherry-picking things or not comparing apples to apples here, but I included the links to both jurisdictions’ curriculum frameworks above so that you can take a look and judge for yourself. The expectations of the sheer amount of content that a California student is expected to encounter in their high school pathway to AP Calculus is monumental compared to a British Columbia student. In fact, in looking through the standards linked above, you might find that my initial guess of “50% larger” is even an underestimate of just how much more material is expected to be covered.
Perhaps one might argue that teaching more content gives students the advantage of being better prepared for the future due to having more math knowledge. Unfortunately, I don’t think this holds up to scrutiny for two reasons. First, students in both systems succeed at the same rates in AP Calculus, which is the same course at the end of the stream. Second, Canadian 15-year-olds consistently outperform American 15-year-olds in math.
Naturally, one might ask the question, what do you cut? For starters, how about some of Geometry? In British Columbia, we have a geometry course that covers most of what is covered in US Geometry, but it is a 12th-grade elective course. Spoiler: Hardly anyone takes that course. If we were to cut out most of the content from Geometry, we could spread the most important topics from Algebra 1, Algebra 2, and Pre-Calculus across four courses, and be left with a much more manageable scope and sequence that still prepares students for Calculus. And I wouldn’t stop there.
- Matrices and Vectors? Cut.
- Advanced Statistics? Cut.
- Conic Sections? Cut.
These topics can safely wait to be introduced at the college level, covered in separate elective classes, or included only in honors curriculum.
In addition to having too much content, I would argue that the content is too hard for the average 14-year-old. What if we simply waited to teach certain concepts until later? Think back to my daughter being horrified that students two years younger than her would be subjected to the evils of parabolas. What if we could just wait to cover these topics?
Another of my favorite stories to recount to my oldest daughter is the day she came home and let me know how she was learning about brain development. “Dad! Guess what I learned in science today! Did you know that my prefrontal cortex that helps me with abstract reasoning and non-emotional judgement ability is not yet fully developed?” That she was completely unaware of having just handed me a parenting weapon of incalculable power only reinforced her point. I promise I have (mostly) used this tool with restraint.
The core purpose of teaching kids Algebra is the ability to abstract ideas into numbers and variables that we can manipulate. This is why we emphasize the topics of “real-world” math and modeling. It takes a certain level of cognitive ability to do that effectively, and we might be expecting that too early for 13 and 14 year olds. Maybe it would be more effective to wait until most students’ ability to think abstractly has further developed.
For example, let’s go through the topics from Algebra 1, and see where they are covered in British Columbia:
Grade 9
- Interpret the structure of expressions.
- Create equations that describe numbers or relationships.
- Understand solving equations as a process of reasoning and explain the reasoning.
- Solve equations and inequalities in one variable.
Grade 10
- Understand the concept of a function and use function notation.
- Interpret functions that arise in applications in terms of the context.
- Build a function that models a relationship between two quantities.
- Interpret expressions for functions in terms of the situation they model.
- Reason quantitatively and use units to solve problems.
- Interpret the structure of expressions.
- Write expressions in equivalent forms to solve problems.
- Solve systems of equations.
- Represent and solve equations and inequalities graphically.
- Interpret linear models.
Grade 11
- Analyze functions using different representations.
- Construct and compare linear, quadratic, and exponential models and solve problems.
- Extend the properties of exponents to rational exponents.
- Use properties of rational and irrational numbers.
Grade 12
- Build new functions from existing functions.
- Construct and compare linear, quadratic, and exponential models and solve problems.
We can see that 9th-grade students in British Columbia are only expected to understand four of the 20 topics that a Californian student of the same age would be required to know, and many topics aren’t even covered until the 12th grade. Given the effectiveness of the British Columbian system, I (apologetically and kindly — typical Canadian) suggest that the Algebra 1 content is not only too much, it is too early. We need to teach much less, and we could teach it much later. Sorry, eh?
If we do take the less and later approach, then there is a wonderful corollary: We can teach it better. To be more specific, we can spend more time on the deeply important concepts, the truly foundational ideas. We can give students and teachers space to breathe.
Having this time is so important for mathematics specifically, because of its sequential nature. Not having a strong grasp of fractions will really make simplifying monomials tricky, which will in turn make polynomials seem unattainable, and so on. The concepts stack one on top of the other.
This is why, at StudyForge, we believe strongly in mastery-based learning, especially in mathematics. Mastery learning takes time, and if there is a constant pressure to cover more topics, the amount of time for the re-learning and extra practice required to be successful is squeezed out.
However, even if mastery is not your thing, consider how much more effectively your students might grasp the basics of the coordinate plane if you had twice as much time to introduce the concept. How much better of a foundation for relating two variables could be built if there wasn’t constant pressure to move on to the next thing? Doing less, and doing it later, makes it possible to do it better.
Perhaps you are starting to think that there just might be something to the “less, later, better” approach that I am advocating for. But, with the momentum that the American education system has, any action that you could take might feel a bit like spitting into the wind, so to speak. (We don’t recommend this, by the way.) As the publisher of what we believe to be the best Algebra 1 curriculum ever developed for online learning, we feel your pain. We know that our Algebra 1 course needs to cover the standards! But, given that constraint, we are obsessively passionate about doing whatever we can do to make it accessible for students.
So, here are some suggestions of actions that you can take, depending on the role that you have.
- Teachers
- Pick the most important standards and spend the most time on those. Aim for mastery there as your number one priority. Ensure students have enough time to deeply grasp the following:
- Basics of working with expressions
- Solving for a variable and keeping equations balanced
- The coordinate plane
- Functions, especially linear functions
- Relating “solutions” to equations and graphical representations
- If your students master these ideas, they have the core of “algebraic thinking” that you can continue to build upon for the rest of high school.
- This is the approach we took with our brand new Algebra 1 course. It covers all of the standards but prioritizes the most important ones.
- Schools
- Encourage your math teachers to collaborate on a scope and sequence to prioritize the most important things, and look at what tricky concepts can be delayed as late as possible.
- Districts
- Run a pilot project at a school with an integrated pathway that de-emphasizes geometry. Track AP Calculus results over five years.
- States
- Consider making Geometry an optional course for graduation.
- Change your standards to spread Algebra 1 and Algebra 2 over three years instead of two.
- Drop Matrices and Vectors from Pre-Calculus.
I cannot imagine how the story of my daughter fending off the evil quadratics would have turned out if she was two years younger. The amount of effort it took her to make it through her 11th grade course was monumental. Did I mention there were tears? I ultimately question if she would be able to do it. Would she have been one more victim of the most failed course in America?
Instead, I get to conclude by getting my “dad-brag” on and letting you know that my daughter worked extremely hard. She put in the time and effort to understand her Pre-Calculus 11 course, and she succeeded! She achieved an A and it was her highest mark in her first semester this year. She even decided to get a head start by taking Pre-Calculus 12 in her second semester to allow for the option to take Calculus next year. That is sweet, sweet music to her math-teacher dad’s ears.
Objection: Most students don’t take Calculus.
The focus of this article is preparing students to succeed in Calculus. This is crucial for most careers in STEM fields, but this does not need to be the path for every student, which would be a valid critique of the main arguments in the article. We agree. Other students would be better served by not taking Pre-Calculus, and substituting a Statistics elective instead, which could be done after finishing Algebra 2. Or, maybe the truly abstract students could take Geometry. Non-mathy students could take more practical courses instead of Algebra 2 that focus on more practical skills, such as math for trades and finances. Having these different math streams is the practice in most regions of Canada. All that said, the argument remains solid, that if you could teach 50% less content and still be prepared for Calculus, then there exists extraneous content being taught.
Objection: The real problem is that students are not prepared for high school math.
Objection: The amount of content is being overestimated because the standards repeat from course to course.
Objection: Spiraling curriculum is a beneficial way to scaffold learning across years.
Another related objection is that it can actually be effective to teach the curriculum in a “spiraling” manner. Saxon Math is a very successful homeschool curriculum that takes this approach. While this seems good in theory, it breaks down in practice, due to the overwhelming amount of content and the nature of different teachers. How often do you find that the Algebra 1, Algebra 2, and Pre-Calculus teachers are able to coordinate on the level of detail of one repeated standard among almost 100 that make up each course? If it is hard to imagine this happening within a single school. Now consider the reality that students might move from school to school, or teaching staff changes, or any number of other things that would disrupt the perfect synchronisation of a properly spiraled curriculum. And any such disruption wipes out any benefit that spiraling would have brought, and that time could have been better spent slowing down on other more core topics.
Objection: Geometry is inherently valuable!
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