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What is algebra? Why is it important? Why is it so difficult for students?
I had never really thought about these questions until they were presented to me by my professor during the first day of a Teaching Algebra course. The idea that I had never sat down to define algebra for myself-and that I didn't have an immediate and clear answer-bothered me and I am now on a mission to answer these questions for myself and the Kid's Math Talk community.
My initial thoughts about algebra include knowing that it is embedded within the common core state standards, that it is important, and it involves letters, representations, and proportional thinking. But why is it important? And what kind of definition would I provide to teachers during a professional learning session, or to elementary students I am teaching? I have started to unpack these ideas for myself in order to have a clear understanding and thought process when entering elementary classrooms as a mathematics coach.
WHAT IS ALGEBRA?
After getting home from that initial class, I immediately turned to one of my most reliable sources- the National Council of Teacher's of Mathematics (NCTM) website-with its many position statements and collections of research-based articles and texts. According to this organization, "algebra is not confined to a course or set of courses in the school curriculum; rather, it is a strand that unfolds across a pre-K–12 curriculum" (NCTM). My search on this website also took me to various books that focus specifically on the algebraic concept development in the elementary grades K-5. What I found is that NCTM's research-driven perspective is in direct contrast to what many pre-service and practicing teachers believe about teaching of algebra. Researchers of elementary mathematics education are explicitly stating that:
[Algebra] does not wait until an isolated grade 9 course that emphasiz[es] rote memorization, symbol manipulation, and artificial applications. It is now widely accepted that this traditional treatment of algebra must be replaced by an across-the-grades focus on algebraic thinking in which reasoning and sense making are front and center.
(Stephens, A. and Blanton, S., p. 71)
NCTM believes that all students are capable of learning algebra and deserve to be a part of rich and engaging tasks that involve the exploration of algebraic concepts. It doesn't matter whether a students is "on the algebra track" or not - in fact, NCTM is of the position that schools should de-track mathematics courses in the hopes of shifting the view that only some students are capable of higher level thinking. If a student is not performing at a level at which a teacher thinks they should be, instead of sending them off with low level work, it is the responsibility of the teacher to know what rich task would be within that student's zone of proximal development and to be able to differentiate within the classroom.
I have realized that there are key understandings within the domain that were never completely unpacked for me during my time in my undergraduate teacher program and I am venturing to say that most elementary teachers around the country never had this unpacking opportunity either based on the fact that despite the research stating otherwise, American teachers, curriculums, and most textbooks are often conditioned to "view arithmetic and algebra as distinct and different" which pushes algebra outside of the realm of elementary mathematics (Bamberger, H. et al, p.56).
I am now in the process of developing a blog series that address each of these key understandings from the standpoint of the elementary classroom.
These Algebraic key understandings are:
•Equality, expressions, equations, and inequalities
•Generalized arithmetic
•Functional Thinking
This detailed process has also allowed me to create my own conception of algebra, which I share with you below.
WHY IS ALGEBRA IMPORTANT?
One goal of mathematics education is to have students grow into self-confident, logical, fact-assessing, fully-human persons who are able to have a sense of joy and wonder about their world."
-Singh, S. and Brownell, C. (2019)
Whether teachers realize this or not, algebra begins before students ever step foot in a classroom. Children are naturally curious and are looking for and making use of structure, patterns, and numbers when they play. We have learned from texts such as Math Recess (Singh, S. and Brownell, C., 2019) that play is extremely important in the conceptual development of a child, and leads to "improvisational thinking...the same kind of thinking that creates multiple strategies in solving a math problem" (p.xxxi).
Along with this natural sense of play and curiosity is the fact that humans are story tellers. Mathematics, and the domain of algebra in particular, helps teachers meet students "where they are not only intellectually, but socially and emotionally as well" and helps students to better understand the story around them through the use of tables, graphs, equations (p.118). It is the responsibility of the teacher to "help students and their families better understand that algebra is not just about factoring and solving equations...[it] is a 'tool for understanding and describing relationships' in a variety of settings and being able to represent these relationships and others in many ways" while ensuring that curiosity, meaningful contexts, and rich story telling remains in the classroom (Bamberger, H. et al, p.50.).
Algebra is also important for all students, not just those geared toward a career that involves mathematics, because it teaches students perseverance and logical thinking skills. Singh and Brownell define perseverance as "a willingness to sit with discomfort and wrestle with an idea until your mind brings it into submission"(p.xxxix). The challenges of life bring people to these places of discomfort and we as educators can position students to receive a strong mathematics education in order to best prepare them for how to approach and handle these challenges. Therefore, teaching algebra by first having a deep conceptual understanding of algebra yourself, provides students "with the facility and confidence necessary to view their world with eyes wide open with both wonder and analysis" (p.62).
WHY IS ALGEBRA SO DIFFICULT FOR STUDENTS?
Without extensive mathematics methods courses available in most elementary teacher preparation programs, new service teachers are forced to rely on their own understanding and interpretation of the common core standards and various mathematical concepts. This comes with excellent intent. Unfortunately, the ultimate impact of teachers delivering instruction with a fragile understanding of the content themselves is that their students are at risk of developing the same misconceptions and fragile understandings as their teachers. This leads to countless cognitive obstacles for students to overcome in middle and high school - with them being forced to unlearn previously taught tricks and rules that separate math and context from problems such as- bigger bottom better borrow, and add a zero, to name a few (see the NCTM article 13 Rules that Expire for more information).
What you practice first usually becomes what you accept as the truth, and so it's no wonder that students are confused and turned off by deeper level mathematics learning such as integers and ratios.
Unlike the subject of reading, if a student is not on "grade level" in math, many pre-service and first year teachers are at a loss for what that student needs instead. And while the research shows that "provided the appropriate instruction—children are capable of engaging successfully with a broad and diverse set of big algebraic ideas" (Blanton et al. 2015), the scenario too often becomes that of offering rote "below grade level" to students via computer work and worksheets instead.
Elementary teachers are unknowingly hindering their students by providing them with these false supports- "Struggle is a twenty-first-century idea linked to humanization" and we are robbing students of this when we remove all of the complex thinking from their instruction (Sunil, S. and Brownell, C., p. xxx ).
Thus, my hope is that the resources with the Kid's Math Talk -Algebra section help to break this cycle of fragility in mathematics where algebra is concerned. Kid's Math Talk is here to empower teachers by increasing background knowledge, implementation tips, and access to additional readings.
Use these pages to shift your practice and to help positively impact the success of your students. Click on the picture below to start your journey.
Sources and further reading:
Bamberger, H., Oberdor, C. and Schultz-Ferrell, K.2010. Math misconceptions: from
misunderstanding to deep understanding.Heinemann:Portsmouth, NH.
Blair, L. 2003. It's Elementary: Introducing Algebraic Thinking Before High School.
American Institutes for Research. SEDL Letter Volume XV, (1).
Sigh, S. and Brownell, C. 2019. Math Recess: Playful learning in an age of disruption.
IMPress: Lexington, KY.