This is a ‘throw-back Thursday’ story – one that I have remembered as I scroll through peoples’ stories of their math exploits, experiences, challenges, and learning on social media. It is a strong enough memory for me to dig through old backup drives to find the pictures! I think it is helpful to write and read every-day stories as well as the ‘official’ literature of journal articles and books by authors we hold as knowledge keepers and role models. I enjoy reading others’ stories, and I know I learn from them. Conle (1996) recognized the connections that can be made by “seeing one experience in terms of another” as resonance (p. 299). The experiences of others that I gain by reading their stories are not generalizable, but transferable to me because their stories resonate with me in some way. Resonance can be an inspiring feeling; almost a drive to do something in response to that feeling. I feel I know what the other person’s story was trying to say to me and I am motivated to add their story to my repertoire of thinking and let it influence my future decisions and actions.
In the field of teaching, Clandinin and Connelly (1995) have led me to the “professional knowledge landscape”, a textured and contextual expression-space of my values, beliefs, experiences, and practice that has shaped who I am today, and continues to shape how I think, and act tomorrow. They said personal and practical knowledge held tacitly, that comes out in stories, potentially shapes others’ knowledge (1988). I think of stories as a form of personal and professional reflection-on-action (Schon, 1983) that helps me unpack, think through, and learn from a particularly memorable experience, especially if I write the story down. As with narrative inquiry writing, the reader is expected to make their own interpretation as well as read my interpretation, or not. This is my every-day story, it resonates with me today as strongly as I remember feeling and thinking when these pictures were made.
It was a winter Saturday afternoon, and it had just started raining. While I was reading in the living room, my seven-year-old daughter, Emma, wandered in looking for something to do. Having come in from playing outside and carrying her hot chocolate I had just made for her, she wanted an activity to do that would keep me nearby. I suggested she colour a picture that I could add to the others already on the wall in my office. Emma thought this was a good idea! I already have wonderful, colourful, energetic landscapes pictorially drowning out the usual office texture of books, texts, papers, and computer screens, and I expected this afternoon would provide me with yet one more addition to the gallery.
After several ‘snow-scapes’ lay on the floor, I was asked “what can I draw now?” Always the teacher I guess, I took advantage of the opportunity to ask her to creatively express what she thinks about school – and ultimately about learning math in particular. I said, “at school, you do reading, gym, art centres, news-books… Can you draw what reading is to you?” Emma got right on the task. This is the result.
It is Emma’s picture of “what reading is” to her. I thought we were on a roll, so then I asked her to draw “what math is” to her. This caused Emma to pause with a quizzical look, but within a few moments she started. This is what math is to her.
I noticed some differences between the two pictures and some concern rose in my ‘math-teacher’ heart. But wait… let me point out what I now understand, from my position of ‘being there’ and watching the process, and knowing my daughter. And as an aside, so you are not side-tracked by possible issues of math anxiety or poor achievement induced attitudes of resistance to learning math; Emma’s report cards consistently maintain level 3 or 4 achievements (75% to 100%) in both language arts and mathematics. In fact, all subject areas are performed well, and she participates willingly in the classroom learning environment. My daughter is seven years of age, in grade two, and of school she says… “weekends are fun, but they get in the way of school!”
Here is my interpretation of her drawing: the ‘reading’ picture shows a person engaged in the story with two hands gripping a book held immediately in front of the eyes. The book envelops the reader; both reader and book are dominant objects in the picture; and the title of the picture is left unfinished. According to Emma: the picture says it all. The ‘math’ picture shows a page of addition facts neatly arranged on the page, correct, and no mistakes shown. Someone’s hand – we do not get to see a face – is poised over the page. The person appears to be distanced from the math – so far so, that they are out of the picture. This picture required a title, “Math is working hard, because it is difficult.”
With my heart in my throat I sought to soften the blow to my pride and ease the rising panic… with all the work we have done in education – reforms, programmes, books, resources, manipulatives, professional development, testing – how is it possible that math, right now, today!, is still seen as distant and difficult by our children? Why is math shown to be just number facts? Where are the manipulative experiences, the games, the fun?!? They do not appear in this picture of ‘what math is’ to Emma.
Tentatively, I asked… “so… how do you feel about math?” Emma replied without hesitation. “Math makes me feel happy because sometimes I get it and that makes me feel happy, and sometimes I don’t, but that’s okay because then we just start again and that’s what you do.”
Yahoo! My faith is restored!
When I looked at the drawing again, I realized that the picture showed math that would be a very small part of what she was to be learning at this time. In Emma’s experiences numbers have only been used in the contexts of games and manipulatives – dice games, snakes & ladders, yahtzee – where addition and subtraction are concepts and mental actions, done as simply as reading a birthday card. When I asked Emma about manipulatives, she easily listed a number of manipulatives used regularly in class. However, the most significant impression is the more formal look of math. It is a formal process – like addition and other operations on numbers.
Also, for Emma, working hard is natural and expected. She sees me work hard at teaching, and her older brother work hard on his school projects and studying for tests. The fact that math is difficult does not mean she cannot do it or that she should respond with avoidance, resentment, or frustration. Em knows that some things are easy, and some things are hard. It is the process of doing that makes a difference. “Sometimes I don’t get it, but that’s okay because we just start again and that’s what you do.” The matter of fact tone, the subtext that this is such an obvious explanation and I am silly for asking clarification, the simplicity of her approach to my (hidden from her) concern, represented a truly mature and balanced attitude, and gave me more relief than a package of Rolaids.
It also made me sad…
In my experience, I see children, who are just starting their career in learning, have this cavalier, risk-taking, attitude to problems – like ‘hard math’ – and I see adults such as myself (math teachers, math educators, professors, and others generally interested in mathematical endeavours) at a later part of their career in learning have a similar attitude of perseverance and an enjoyment of the challenge and fun with finding a solution when stumped. However, many learners appear to lose their way after some years. They seem to be stuck in a ‘middle’ period of their learning career. Some elementary school preservice teachers call it ‘stalling’. Many of them ‘stalled’ somewhere between grades 4 and 12, and from adolescence onward math is not viewed as an enjoyable activity. What happens in this ‘middle’ period? Why do so many children, who then grow into adult-hood, so often express feelings of fear, frustration, and resentment towards math and take great care to avoid it? This question feels very important to me and has become a catalyst for future thought and action. For many children, there is a wonderful learning attitude that encompasses all subjects, including math, at the beginning of our learning trajectory.
Perhaps beliefs may be the key. My daughter reminded me that learning math is so much more than numbers, operations, mechanical drill, and working with various tools (like pattern blocks)—it is the belief one has when learning math. If it is belief – we seem to have it, we lose it, and sometimes we get it back.
In her infinite grade 2 wisdom, Emma reminds me to question what I am doing as a math educator and how I can help shift our understanding of what it means to ‘do math’. What can I do so that it does not get lost in the first place? I have three thought bubbles floating around in my mind right now, about teachers, about students, and about curriculum. I am going to talk about teachers in this next part. Personally, I can share stories, work to understand what my stories mean for me, and then share that thinking, and maybe that will resonate with someone else and we can work together to “be the change.” Professionally, I can work to help preservice and inservice teachers understand their beliefs and help them change their beliefs. My professional focus is on how mathematical problem-solving and teacher efficacy support problem-based learning as a professional collaborative learning model for preservice teachers. Other mathematics education research is ongoing into the issues of elementary teachers’ math anxiety, math teaching anxiety, math teaching efficacy (and often these constructs measure out on the low end of the scale). For example, read Brady & Bowd (2005), Bekdemir (2010), Burleigh (2010), Dowker, Sarkar, & Looi (2016), Gresham (2017), and watch for J. Bosica’s upcoming work.
Now, I am going to talk about students in this part… There might be a disconnect, and entirely our fault. When we talk about learning mathematics, we may not be making a distinction between three important subsets of our collective humanity and society: i., those students who will go on to mathematics, the sciences, and engineering etc. ii., those students who will use mathematics in an integral way in their jobs or careers, and iii., those students who embody mathematics everyday but it may be invisible to them – they live life. Arguments about what should be in the curriculum does not seem to distinguish who is the learner – which of the three subsets? What is good for one subset of learners is not necessarily good for another subset. For example, if one is a university mathematics professor, then the mathematics that is important for students to learn is the mathematics students will see in their courses… not the mathematics someone needs to read a particular set of directions as a trades-person might in the ‘Machinery’s Handbook’ of formulas and tables to reference for a particular job site task (a friend who is a plumber with a gas license showed me his handbook of formulas he uses – he says “no way can I remember all these formulas, but I know the dimensions of the installation, and I know where to find the formula to use, substitute everything together and I fix it right the first time”), … and not the mathematical literacy someone needs to be a part of society, to understand marketing graphs can mislead, be critical of political poll results in the news, (recognize the fake news), and know when a mortgage term is not reasonable for their personal finances, etc.
And this part is about the curriculum… These are all DIFFERENT mathematics thinking needs – mathematics for mathematics sake, mathematics in application, and mathematics for life. I completed a quick investigation on the Stats Canada website https://www.statcan.gc.ca, (and all misinterpretations are mine alone, and apologized for now). Under the Educational attainment of the population aged 25 to 64, by age group and sex, OECD, Canada, provinces and territories data set, the approximate ratio of attaining less than high school to high school to tertiary looks like it is 9 to 34 to 57. So approximately 57% of Canadians go to post-secondary schooling. Of that 57%, the Postsecondary enrolments, by program type, credential type, Classification of Instructional Programs, Primary Grouping (CIP_PG), registration status and sex data set looks like it states the approximate ratio of students in mathematics, sciences, and engineering etc. programs to students in all other programs is 21 to 100. This means approximately 57% of Canadians go to post-secondary education, and approximately 21% of those go to a program that requires mathematics in a rigorous way… 0.57 x 0.21 = 0.1197 (or approximately 12%). And I return to the elementary school and secondary school years:
So, what mathematics do this (approximately) 12% of Canadians need?
And what mathematics do the other (approximately) 88% of Canadians need?
. “To restore the human subject at the centre—the suffering, afflicted, a fighting human subject—we must deepen a case history to a narrative or tale; only then do we have a ‘who’ as well as a ‘what’, a real person.”
The Man Who Mistook His Wife for a Hat
I will leave my thinking somewhat unfinished… There are a few ideas and thoughts to unpack and consider, and I will take a bit of time to do so. Three closing remarks though, for clarification which I feel might be helpful, i., the section on students is connected to the section on teachers because teachers were once students, and these sections are connected to the curriculum, because the curriculum is experienced by students who can become teachers… of another generation of students… a social knowledge propagation cycle, and ii., ‘need’ in the two sentences above encompasses epistemological as well as ontological considerations/aspects, and iii., the point about being careful who the subject of our discussion is, is vitally important.
Selected References (and I added a few…)
Barone, T. & Eisner, E. (1997). Touching eternity: The enduring outcomes of teaching. New York, NY: Teachers College Press.
Ben-Perez, M. (1995). Learning from Experience: Memory and the teacher’s account of teaching. Albany, NY: State University of New York Press.
Ben-Perez, M. (1991). Scenes from the past: Retired teachers remember. Paper presented at the Fifth ISATT Conference, University of Surrey, Guilford, England.
Brady, P., & Bowd, A. (2005). Mathematics anxiety, prior experience and confidence to teach mathematics among pre-service education students. Teachers and Teaching: Theory and Practice, 11(1), 37-46.
Bekdemir, M. (2010). The pre-service teachers’ mathematics anxiety related to depth of negative experiences in mathematics classroom while they were students. Educational Studies in Mathematics, 75(3), 311-328.
Burleigh, C. (2017). Exploring early childhood preservice teachers’ mathematics anxiety and mathematics efficacy beliefs: A multiple case study. Available from ProQuest Dissertations & Theses Global. Retrieved from https://search-proquest-com.proxy.queensu.ca/docview/1925933744?accountid=6180
Carter, K. (1993). The place of story in the study of teaching and teacher education. Educational Researcher, 22(1), pp. 5-12.
Chafe, W. (1990). Some things that narrative tells us about the mind. In B.K. Britton & A.D. Pellegrini (EDs.) Narrative thought and narrative language, (pp. 79-98). Hillsdale, NJ: Erlbaum.
Connelly M. & Clandinin, D.J. (1999). Shaping a Professional Identity: Stories of Educational Practice, London ON: Althouse Press.
Connelly M. & Clandinin, D.J. (1995). Teachers Professional Knowledge Landscapes, New York, NY: Teachers College Press.
Connelly M. & Clandinin, D.J. (1988). Teachers as Curriculum Planners: Narratives of experience, New York, NY: Teachers College Press.
Conle, C. (2003). An Anatomy of Narrative Curricula. Educational Researcher, 32(3), 3-15.
Conle, C. (2000a) Narrative inquiry: Research tool and medium for professional. European Journal of Teacher Education, 23(1): 49-63.
Conle, C. (1999). Why Narrative? Which Narrative? Struggling with Time and Place in Life and Research. Curriculum Inquiry, 29(1).
Conle, C. (1997). Community, Reflection and the Shared Governance of Schools. Teaching and Teacher Education, 13(2), pp. 137-152.
Conle, C. (1996). Resonance in preservice teacher inquiry. American Educational Research Journal, 33(2), pp. 297-325.
Dowker, A., Sarkar, A., & Looi, C. Y. (2016). Mathematics anxiety: What have we learned in 60 years. Frontiers in Psychology, 7,1–16.
Gadamer, H-G. (1979). The Problem of Historical Consciousness. In P. Rabinow and W.M. Sullivan (Eds): Interpretive Social Science: A Second Look. Berkley, CA : University of California Press.
Gadanidis, G, Hoogland, C., Sedig, K. (Eds.), (2004). Mathematics as Story, A
Symposium on Mathematics Through the Lenses of Art & Technology. Toronto, ON: The Fields Institute for Research in Mathematics.
Gresham, G. (2017). Preservice to inservice: Does mathematics anxiety change with teaching experience? Journal of Teacher Education, 1–18.
Hannula, M. S. (2002). Attitude towards mathematics: Emotions, expectations and values. Educational Studies in Mathematics, 49(1), 25-46.
Stats Canada. Educational attainment of the population aged 25 to 64, by age group and sex, OECD, Canada, provinces and territories data set, retrieved from https://www150.statcan.gc.ca/t1/tbl1/en/tv.action?pid=3710013001
Stats Canada. Postsecondary enrollments, by program type, credential type, Classification of Instructional Programs, Primary Grouping (CIP_PG), registration status and sex data set, retrieved from https://www150.statcan.gc.ca/t1/tbl1/en/tv.action?pid=3710001101
Schon, D. (1983). The reflective practitioner: How professionals think in action. USA: Basic Books.