The Genesis of Our Algebra 1 Course, part 1: Foundations
Mar 31, 2021
In the United States, Algebra 1 is a core math course typically completed in 7th, 8th, or 9th grade. It marks the first time students study functions and equations to describe relationships between various entities. Often seen as a "gatekeeper" to higher education, the course can significantly impact a student's future academic success. Our aim is to transform Algebra 1 from a barrier to a gateway to college.
Problems with mathematics education are widespread on both sides of the Atlantic and they already show up in elementary school. Nevertheless, there are additional problems arising from the mathematical content that Algebra I introduces. The topics contain families of functions, linear equations, systems of linear equations, inequalities, basic statistics, polynomials, exponential functions, quadratic functions, quadratic equations, etc....
Abstractions and "aha" moments
Before Algebra 1, math courses focus on numbers and simple geometric shapes that are more tangible to students. A simple example would be measuring the weight of different coins. In pre-algebra courses, a typical question would be “Weigh the 10¢, 25¢, 50¢, and 1$ coins! How much does each coin weigh?” In Algebra I, the focus shifts to relations. It is no longer about 1 unit of measurement but about 2 units of measurement and their relationship to each other. Extending the previous example to include the monetary value would look like this: "Is the weight w(c) of a coin c correlated to its value v(c) for the 10¢, 25¢, 50¢, and 1$ coins? If yes, describe the relation between w(c) and v(c)."
Even though there are examples of these mathematically comprehensible relationships in everyday life, they are not obviously recognizable. A higher degree of abstraction is required of the students as they move to a higher mathematical level. Many students completely lose their orientation and thus their confidence in their mathematical abilities. One of the students we interviewed expressed her frustration with this as follows: "What are letters doing in mathematics? No thanks, not with me!"
One should not overlook the fact that before Algebra 1, during the development of the number concept, many abstraction achievements must be furnished which are far from obvious. Even though many children develop a sense of how to deal with these basic concepts since they are omnipresent in life, many students fail to reflect on how they do it and this leads to problems at the next level of abstraction. The numbers themselves are already a cumulation of several abstraction steps, starting with the very abstract concept of a set, i.e the freedom to treat things as elements of something, and to conceptualize totally different things as equal with regard to their quantity. There are several examples of languages that use different counting words for different types of objects. And getting back to the coins, have you noticed that young children who learn to pay with money initially equate 3 monetary units like Dollars or Euros with 3 coins? For them, the number of coins alone reflects their value. But they first have to learn that different coins are worth different amounts. Through a lot of practice, they gain experience and at some point, they suddenly go "Aha". They pay correctly and have understood the abstract meaning of the coin’s value. They have achieved a feat of abstraction.
How to deal with a lack of foundational knowledge?
It is a fact that many students do not succeed in building resilient foundations in pre-algebra courses. And it is the most natural desire of every idealist (which is inherent in many dedicated teachers) to say: Stop! If the foundations are missing, it would be best to go back and lay secure foundations step-by-step first, before we try to build another level on top. But the pragmatist (which is a characteristic of experienced teachers) knows: unfortunately, there is not nearly enough time for this in the classroom. Moreover, classes are often very heterogeneous, and what would the students do who have already acquired these foundations?
This is indeed a contradiction that many dedicated and experienced teachers face in their daily lives and one that we also found in our teacher interviews. “The biggest challenge for me is trying to always move forward with the curriculum while knowing that there are a lot of gaps in core math literacy skills.”
We conducted five deep interviews with five Title 1 school algebra 1 teachers. They named the following as the most crucial problems:
Not seeing the subjects’ relevance: “It is a big challenge to motivate kids when they realize that they are not gonna use a lot of algebra when getting out of high school.”
Serious knowledge gaps: “Trying to always move forward with the curriculum while knowing that there are a lot of gaps in core math literacy skills.”
Serious real-life difficulties: “It is tough for them to practice outside of school with other matters to take care of such as jobs or taking care of siblings.”
Lack of motivation: “I think the immediacy of the digital world is having an impact on how much effort kids are willing to put into that. If you can just type it in google [and now in ChatGPT] and get an answer, why would you put your brain power into it?”
No STEM-related positive role models: “Most students don’t have a role model at all because a lot of kids have parents out making money.”
This is where Mastory comes in. Because we are one thing above all: innovators who are motivated by precisely what others give up on in frustration. We build a story that bridges the many levels of abstraction between students’ everyday life and the topics of the Algebra 1 course to overcome the contradiction of step-by-step approaches.
Actually, it is a very basic and uniquely human ability that we build on: humans’ incredible mind-reading skills and the ability to link our minds together. “This allows us to take advantage of others' experiences, reflections, and imaginings to prudently guide our own behavior.”- says Thomas Suddendorf, an evolutionary psychologist at the University of Queensland in Australia in a BBC article.
We usually know mathematics as a school subject and mathematics teaching as a teacher explaining to students some abstract methods through rational argumentations and/or some visual illustrations. Normally, the teacher wants students to be quiet to focus on the rational explanation. Normally, the mathematics class ends without having become personal. Normally, the only reason students participate is the fact that it is obligatory and the most important subject regarding their future possibilities. However, students do not get “real” guidance in mathematics. They do not experience any natural connection between mathematics and their lives, and they do not see how the teacher would behave if he/she was in their situation: that is if he/she encountered new problems which he/she did not immediately know how to solve.
We introduce characters who are in this situation and who can serve directly as role models for students. They provide a great deal of support when students encounter new problems. Not only do they show how to treat difficult problems with faith in future success, how to break them down into smaller subproblems, and how to collaborate as peers, but they also show their own associations and emotions while dealing with math: doubts, surprise, disbelief, pride, philosophical thoughts, etc. By integrating our story into everyday situations where math becomes essential for solving problems or satisfying curiosity, we create a natural connection between mathematics and students' lives. In a way, we shift mathematics into the life of (fictional) people who are ready to take on mathematical challenges and show how to do it - without instructions on what to do.
What kind of a story should it be?
When creating a new course, Mastory always starts with the question: What story could bring the given set of mathematical topics to life in such a way that it remains interesting from beginning to end, while incorporating all the given mathematics into its dramaturgy?
This question is the prelude to a long iterative process that addresses several aspects: adequate translation of mathematics into story elements, dramaturgy of the story, dramaturgy of student activities, dramaturgy of the used technologies and devices, psychological needs of students, and of course, the moral of the story.
The Mastory story is thus deeply interwoven with all aspects of mathematics teaching and can unfold freely but only within the space predetermined by mathematics and the purpose of teaching. So first of all, we investigate the mathematical topics to explore the space in which our story is meant to come alive.
A basic decision required is what genre to select. Mastory has stuck with one genre so far, which is science fiction. Part of the reason why the science fiction genre, for over 100 years, has remained popular, is its playfulness with real scientific, or at least, apparently true data and research. Through its use of what our species can do or is on the road to achieving, it leaves speculation to the mind on whether what we see can someday become a reality. Therefore, we create a story that uses science in a fun manner and leaves our users intrigued and in constant engagement with its characters. We could also say that instead of looking for the algebra in the day-to-day lives of average people - which leads easily to too difficult, too boring, or too artificial examples - we create fictional lives in a fictional world that is designed to provide the most exciting algebra examples.
In the next blog post series, we are going to report to you about the co-creation process of the Algebra 1 interactive storyline game. We have just started with our first concept test series involving six Black and LatinX students from Title 1 schools in the Chicago area.
To learn more about the concept and practicalities of Mastory, please stay tuned for our next blog entries and read the ones we have already posted. Be a part of our journey!
This blog post is part of a series where we introduce the various components of our upcoming Algebra 1 course. If you have missed out on the first part of this series, make sure to have a look!