Jamie S. Pyper, PhD, OCT

...thinking about.... the learning and teaching of mathematics... (and on Twitter @314_per)

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What is teacher’s pedagogy to EQAO? #teaching #secondary #mathematics

Using EQAO, which makes standardized content tests for students, for content tests and pedagogical tests for preservice secondary school mathematics teachers suggests to me that there are some questions that need to be answered.

1. To focus on the mathematics content:

A) p. 3 from EQAO technical reports (2017). “It should be noted that not all expectations can be measured in a largescale assessment. Measurable curriculum expectations are clustered by topic, and items are then mapped to these clusters. Not all of the measurable expectations in a cluster are measured in any one assessment; however, over a five-year cycle, all measurable expectations in a cluster are assessed.”

This means student achievement is an incomplete picture for each year because NOT all the curriculum expectations are tested each year. And, in my review and analysis of 5 years of sample tests, there is often NOT an equitable weighting of test items across all strands. What will be emphasized (read: particular nugget of math someone thinks is important) each year for the ‘one-off’ test for graduation for preservice teachers?

For other expectations of the curriculum, such as how to learn and do mathematics, this also means that the curriculum document “Program Planning” (e.g., Gr 9-10, p. 23) statements, and any curriculum expectation that states “by investigation” “with manipulatives” “with technology” are not reflected in the EQAO standardized test. 

For other expectations of the curriculum, such as the mathematics processes (e.g., p. 29, Gr 9-10) of problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing, and communicating, these are also, NOT, being assessed by EQAO, but are expected to be taught and practiced by students. To me, these key features of the mathematics curriculum, which do not appear in the EQAO test for which students’ achievement is being judged, indicates a problem with the intent of such a standardized test and its alignment with the curriculum it is purportedly measuring student achievement.

  B) EQAO report p. 11, “Scorers are trained to refer constantly to the anchors to ensure consistent scoring. The rubric codes are related to, but do not correspond to, the levels of achievement outlined in the achievement charts in the Ministry of Education curriculum documents.”

This means how teachers are to teach and assess math according to Growing Success and the curriculum document is NOT how the curriculum is being assessed or items designed for the EQAO test. The reported EQAO student achievement levels may not match the stated achievement levels detailed in the curriculum documents. For me this calls into question the ability of teachers and schools to be able to make changes to learning (which is seamlessly linked to assessment) if there are different assessment perspectives in play with teaching and with standardized testing. 

These problems and issues, 1A and 1B above, will be equally present for preservice teachers’ measurements of mathematics knowledge, as they are for students’ measurements of mathematics achievement. And for preservice teachers, they will be five (5) or more years AWAY from the last time they were learning secondary school grade 11 mathematics content... and what if the preservice teacher learned mathematics in a non-Ontario or non-Canadian secondary school!? (n.b. teacher preparation has mathematics content pre-requisites, and the program itself is about pedagogical knowledge learning and pedagogical-content knowledge learning (Shulman, 1986, 1987) NOT about learning mathematics content.)

2. To focus on pedagogy:

Since preservice teacher learning is at the university level, there is NOT a common curriculum, nor a common pedagogical strategy espoused by any governing body. The OCT and Ministry of Education conduct program certifications, but at no time has anyone watched me teach, interviewed me about my pedagogical approach, or judged my pedagogy against the artifacts and course syllabi I submit for certification evidence. And, I have never had a pedagogical discussion, workshop, conference with EQAO in the 16 years I have been a teacher educator, nor the 12 years I was an Associate Teacher with preservice teachers in my classroom on practica.

This means… I am not sure what pedagogy EQAO thinks should be practiced in teacher education programs, and what pedagogy teacher education programs should be teaching preservice teachers to use in their classrooms. How can a test be created and be valid and reliable, each year, given 11+ teacher education programs across Ontario, and the many teacher education programs that have taught non-Ontario preservice teachers who come to Ontario to teach?

 If improved learning is truly desired by society, then let the experts do their work. Move the barriers, such as ineffective and/or misaligned testing, away. Stop looking back at the ‘good old times’ and trying to replicate them for the new generations today, because one’s present success—while an outcome of that history—is not relevant to the lives the new generations are living right now.

There are experts in education as a human endeavour, the teaching and learning of mathematics, the creation and implementation of educational policy, the organization and structure of curriculum, who are up to date with this century’s learning needs. Politicians, elected by the people, should be responsible to the people by recognizing their own knowledge limitations and taking advantage of others’ expert knowledge.

There is a risk of oversimplifying the issue of standardized tests and teacher testing that I put up front here; I welcome evidence- and literature-based discussion (and even correction) because I am really just another fallible human being trying to do the best for our shared human condition.

Reference: the Technical report, at the least the last available report from EQAO about test design, evaluation, and reporting, found online at http://www.eqao.com/en/assessments/Pages/Technical-Reports.aspx

Curriculum documents at http://www.edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf http://www.edu.gov.on.ca/eng/curriculum/secondary/math1112currb.pdf http://www.edu.gov.on.ca/eng/policyfunding/growSuccess.pdf http://www.edu.gov.on.ca/eng/literacynumeracy/index.html

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.  Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22.

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Parallel or not!? Why do you say that?

at Federal Triangle Metro stop in Washington DC, June 2019.

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mathematical arts as elementary school curriculum

I think about elementary preservice teachers – many arrive in teacher preparation with little mathematics learning after grade 11, many with varying degrees of anxiety for mathematics and teaching mathematics, and with a small appreciation for the beauty, aesthetic, relevance, and structures and syntactics of mathematics. But, many preservice elementary school teachers arrive with degrees and experience in the arts, languages, literature, etc. and so have great experience with playing with language. (I am thinking about what is still often heard in society with a laugh, “oh, I can’t do math!” but not “oh, I can’t read or write”).

What if elementary school curriculum was less about what is perceived as the formal (read algebraic and algorithmic here) presentation (read teaching) of mathematics and more about the exploration and expression of thinking and communication through a mathematical lens? Channeling ideas of

a) definitions of literacy which involve reading, writing, thinking, and problem solving for societal involvement,

b) the English language curriculum that is often called ‘language arts’ as they play with various forms of language expression,

c) elementary school teachers’ more often seen abilities in the arts and languages, and

d) what we know about young children’s play-based learning, learning through the arts, and learning because of creative opportunities and exploration,

can we consider a shift in the elementary curriculum towards a sense of mathematical arts? For example, play with manipulatives; play with digital media; play together; allow and encourage creative expression; make mistakes and be ready to just try it again; learn about various ways of thinking mathematically – geometrically, numerically, statistically, graphically; problem solve because the complexity of the situation suggests it; be interdisciplinary because those kinds of situations/problems/activities can be more engaging…

What if the elementary school curriculum was shifted to be about “mathematical arts” and offer more opportunities for teachers, students, and parents to engage in the learning because it IS engaging according to their ways of thinking. Elementary school mathematical arts could relieve a lot of everyone’s anxiety, and potentially groom the paths through the forest and trees of mathematics so that everyone could enjoy mathematical experiences, and when they are ready, choose different paths into more rigorous areas of mathematical thinking or more creative expressions of life that naturally use mathematical thought.

....

In 2014, at the CMESG/GCEDM annual meeting, I led an ad-hoc conversation about reconceptualizing elementary school mathematics, and I think the issue and my motivation for thinking this way is still relevant today given the conversation about STEM and STEAM.

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@LisaThompson @fordnation @ONeducation #OntEd class-size details in the rest of those quoted articles... in case you are interested...

Referring to the evidence presented by the SUN news article by [email protected] and promoted by @LisaThompsonMPP @fordnation about class sizes: It is important to read the whole article and not (what some call) “cherry-pick” sentences from abstracts and conclusions. The details of the participants under study, and in a quantitative study such as those studies quoted, the variables NOT included or controlled for are very important to appreciating how generalizable the particular study is to other educational jurisdictions.

Researchers know that context matters. Context gives the possibility for generalizability of a study’s result to the greater population, but then it also inhibits the generalizability and transferability to OTHER populations. Context matters – the evidence Ontario uses for its decisions should be coming from Canadian (or at least North American) schools if it is being used for local effect.

For example, Project Star is a large scale experiment (that means “Gold Standard” research according to the USA government) in Tennessee schools “students in smaller classes have significantly higher achievement that students in larger classes” (Shen & Konstantopoulos, 2017, p. 99).

“Project Star is considered an exceptional study and perhaps the most impressive and powerful field experiment about class size effects in education (Mosteller, 1995)” (Shen & Konstantopoulos, 2017, p. 99).

This is important because it is a study about North American schools, and can be informative when thinking about North American schools. Context matters – the evidence should be coming from Canadian or at least North American schools if it is being used for local effect.

Cho, Glewwe, & Whitler (2012) completed a study with Minnesota’s elementary schools. This research on class size in Minnesota suggests that reducing class sizes by ten students in elementary schools in Minnesota increases students’ test scores by 0.04-0.05 standard deviations.” But other variables such as family wealth have an impact, “most students in Minnesota (and Connecticut) may be too wealthy to benefit from small classes” (p. 93). Thus, the North American context matters.

In Kentucky schools, Barrett & Toma (2013) stated that the discussion about class sizes must take into account the local practices of principals, “in particular, principals place more students in the classroom of more effective teachers, controlling fo all other factors” (p. 51). “But if estimates from past models have been biased because class size is endogenous [that means affected by external influences], there is little policy advice that can be made based on prior estimates. The efficiency of public resources devoted to class size reduction should be further addressed” (p. 51).

Now, concerning the Shen & Konstantopoulos (2017) and Li and Konstantopoulos (2017) articles, these findings represent the schooling in eight to 14 European countries: Austria, Croatia, Czech Republic, Denmark, Germany, Hungary, Italy, Lithuania, Malta, Portugal, Romania, Slovak Republic, Slovenia, and Spain. I hazard a guess and suggest the cultural, economic, racial, etc. diversity of these European countries does not align with North American educational jurisdictions. We can learn some great things from these studies, but our job is then to think about how those findings are relevant and transferable to our jurisdictions. It is not a straight one-to-one match.

Some points from within those Shen & Konstantopoulos (2017) and Li and Konstantopoulos (2017) articles. Shen & Konstantopoulos (2017): “Variables such as family wealth, parental involvement in school or parental pressure regarding student assignment to classes, teacher quality, motivation, or commitment as well as relationships between principals, teachers, and parent may be critical factors regarding assigning students and teachers to smaller classes” (p. 99).

“The assignment of students and teachers to classrooms within grades and schools is rarely random” (p. 99).

Two of the eight countries [that’s 25% or one quarter of the countries in the study] involved in the Shen & Konstantopoulos European study showed smaller class size effects with student achievement (Romania and Germany) (see p. 109). So why do the other six of eight, given all the school systems in the world, make for a ringing endorsement for change in Ontario schools?

“These differences may indicate as a whole that class size effects are country and context specific. Unfortunately, classroom data (e.g., observations or videos) were not available and thus we had no knowledge of actual classroom practices” p. 109. The researchers themselves said be careful how one interprets their findings -- class size effects are country and context specific.

A limitation to their study: “it is unclear that the [data] instrument was unrelated to unobserved variables” p. 112, meaning, context matters and context from unmeasured variables might likely change the results.

Li and Konstantopoulos (2017): “TIMSS does not provide data about classroom dynamics, instruction, and practices, it is difficult to know exactly why class-size effects were non-significant” (p. 308). “Perhaps teachers did not modify their instruction accordingly” for class size differences” (p. 308). “This kind of information was not available to us” (p. 309). This means, context matters, the variables not measured likely would change the results.

The most important thing I have learned as an education researcher, is that the context of the study matters. Generalizability is wonderful, but not always possible, and it is definitely not necessarily possible into other jurisdictions. Transferability of findings is wonderful, but for findings to be transferable to other jurisdictions, a lot of critical thinking and even more reading of relevant literature from research is necessary before making a research decision – and that is not even considering a social policy decision that has enormous impact on society and people directly. The other important thing I have appreciated – bias must be reduced. One cannot only look for the details that support one way of thinking. Combined with the first point, attention to bias and context are critical to evidence-based decision making.

References

Barrett, N, & Toma, E. F. (2013). Reward or punishment? Class size and teacher quality. Economics of Education review, 35, 41-52.

Cho, H., Glewwe, P. & Whitler, M. (2012). Do reductions in class size raise students’ test scores? Evidence from population variation in Minnesota’s elementary school. Economics of Education Review, 31, 77-95.

Li, W. & Konstantopoulos, S. (2017. Does class-size reduction close the achievement gap? Evidence from TIMSS 2011. School Effectiveness and School Improvement, 28(2), 292-313.

Shen, T. & Konstantopoulos, S. (2017). Class size effects on reading achievement in Europe: Evidence from Pirls. Studies in Educational Evaluation, 53, 98-114.

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engaging students in a concept

I am continually surprised by my students, they are so unique and individual, they find ways to surprise me about when and how they learn – often when I least expect it, and most often when I don’t expect it. They are inconsistent (in a good way) because if I think I can categorize them as using a particular approach to engaging in a topic, they show me some other motivation that works for them with another topic. This can be challenging as a teacher with a room of diverse-thinking and -learning students. For example, if I rely on words I may miss engaging the student who needs graphics to approach the topic. There are many ways a student might engage in a topic.

The above does not represent an exhaustive list of ways to engage, it is just a quick way, at the moment, that I am thinking about student engagement. I used self-efficacy as an inspiration. Students’ self-efficacy is related to their achievement – the higher their self-efficacy (their beliefs in their capabilities to perform a particular task) the greater chance they will persevere, take risks, be resilient, problem-solve, etc. The cycle above is inspired by the four sources to self-efficacy: verbal persuasion, vicarious experience, mastery experiences, and physiological/affective experiences.

Students care about what they are learning – they ‘tell’ me when they pay attention and when they don’t pay attention. Relevance to their thinking and the way they feel they fit into the world around them that day seems to be a key predictor of student engagement.

I wanted students to understand (not calculate, but understand) the concept of intersections of lines. So I put three notes up on the board, and asked them to read the one that appealed to them the most.

Then we talked about the importance of the intersection point of lines.

  They talked about what they cared about – which was spending their money wisely. We then talked about how details can be made in to lines, and the point where those lines intersect could be helpful in our decision-making.

(p.s. I used this in a grade 10 applied class. I didn’t catch everyone, but more were engaged than last year when I started with note 1, then went to note 2, and then gave them a ‘problem’ with note 3. This is anecdotal evidence – but based on research evidence: working towards an evidence-based practice. For details about self-efficacy, see Bandura, A. (1997). Self-Efficacy: The exercise of control. New York, NY: W. H. Freeman.)

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Looking at one thing, and seeing another thing.

(Chichen Itza, Mexico,  Mar2019)

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as a respondent, who was given an elicitation tool, from the researcher, ...a struggle to let you write me...

…a struggle to let you write me…

do I trust1 you? and you, me? do I trust you to do the right thing and inspire confidence? or do I trust you to do the wrong thing and inspire expectation?

your story is persuasive2, more than if you had given me a chronology of events, neatly ordered and numbered and coded with the appropriate ‘isms’ and multi-syllable words.

it sounds like it could be mine! and that makes me feel very vulnerable3. I am tempted top resist4 the emotions your story evokes rather than look deeper into myself to my own story. my identity is constructed5 from the stories of my life my identity is tangled in the multiple issues of my gender, my race, my class6 and how they bump up against another’s issues. my construction is shaped by who has authority, and personally what living means to me.

I won’t use your story to write mine. as I may tend to reinvent7 my past, to fit yours, to give meaning to my present according to my need to conform to you, to peers, to my public image rather than my authentic self.

I am not you.

can I honestly revise my story to find a focus and so find the meaning of the past to explain the meaning of today? it will demand cooperation8 between you and me, to listen to my story for the meaning in it for me, rather than in you for me.

there has been security9 in my silence, it has been only my story and no one else has had influence or could subjugate my meaning in my life. my meanings have served me well, are they to serve you too? is letting go of my silence and my story becoming vulnerable to your insight affording me the opportunity to know me, more?

can I realize, do I realize that the individual needs are restrictive10, the social needs are enlightening, from the expression of my story? I’ll have to rely11 on you to help me, so that I can rescue and be rescued12, from the vicious circle of self-avoidance and complacency into a re-building of my story and a re-construction of its meaning for today,

from trust, in you, in me.

…

Notes

1.     Baier 2.     Ritchie, p.19 3.     Ritchie, p.27 4.     Ritchie 5.     Ritchie p. 13 6.     Ritchie, p. 12 7.     Ritchie 8.     Baier, p. 606 9.     Hooks, p. 156 10.   Ritchie, p. 11 11.   Baier, p. 608 12.   Hooks, p.159

References

Baier, A. (1997). Trust and antitrust. In T. J., Meyers (ed.). Feminist social      thought: A reader (pp. 604-629). New York, NY: Routledge. Hooks, b. (1994). Talking Back (pp. 155-159). New York, NY: Routledge.

Ritchie, J. S., & Wilson,. D. E. (2000). Teacher narratives as critical inquiry (pp.      1-28). New York, NY: Teachers College Press.

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“Telling math stories: Math makes me feel happy!”

            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.”

                              Oliver Sacks                               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.

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#PBLmathEd #preservice #secondary #mathematics #PBL #constructivefeedback

PBL 4 – day 3, team- and self-assessment: today was a bit different because the presentations won’t happen until classes in April when everyone returns from a classroom practicum block. But the consolidation session and team- and self-assessment session occurred today.  We turned the peer- and team-assessment discussion into a self-assessment discussion, because it can often be easier to talk about one-self rather than another.

Being able to talk in a critical and improvement-focused manner with peers is often challenging and difficult. And this can also be the experience when trying to talk about oneself. Even if one has something to say, it is not always easy to start to speak. This is partly due to our (gentle, sometimes more intense) fear of being vulnerable, or showing a weakness and thinking we will be considered ‘less than’ for ever afterwards, and sometimes a lack of practice at risk-taking. All very human, natural, and normal feelings and thinking. 

One strategy might be to think of this scenario: in an interview, the interviewer will often ask for you to talk about a weakness. This is a very challenging question to answer and not feel you are leaving the interviewer thinking you are quite unfit for the position! But the “Mac sandwich” (from the McMaster Med School PBL process) of a) tell something positive about their contribution, b) provide constructive feedback on something to improve (this is the ‘weakness’ part), and c) provide a positive example of how they can use this feedback going forward (tell the interviewer how you have learned and implemented improvement with respect to your ‘weakness’) can be a conversational way to finesse the question, still be honest and forthright, and ultimately be able to talk.

What math gets you excited! into Rich Learning Tasks (RLTs).

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#PBLmathEd #preservice #secondary #mathematics #PBL #RLTs #habitsofmind

PBL 4 – day 1, Everyone has some mathematics that excited them, that made them think, wow! That is cool! And that mathematics might have been the inspiration for continuing learning math, and maybe to become a secondary school math teacher. This is their chance to show us the math that got them, and still gets them excited, and then find a connection, a relevance to the expectations of the mathematics curriculum, and turn their mathematics into a rich learning task (RLT) (see Flewelling & Higginson’s 2002 A handbook on rich learning tasks).

There is also something different with this PBL task, can you see the difference? Yes, it is shorter, and, it also does not list Learning Goals. Perhaps making this task a bit more open-ended, and maybe appearing less like a ‘school/course’ learning activity. This is the fourth PBL cycle, and I am also wondering if the PBL protocol we have been using for that past three cycles and 13 meeting days has taken root in the preservice teachers’ thinking when facing a problem of practice… has it started to become a professional practice ‘habit’…

What math gets you excited! --> Rich Learning Tasks (RLTs).

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#PBLmathEd #preservice #secondary #mathematics #PBL #planning #growthmindsets

PBL 3 – day 4, presentations, and self-assessment. The presentations were awesome! Five distinctly different approaches to course outlines, and all so very detailed from the perspective of student success first. Spiraling expectations, Backwards Design (UbD) approaches, summative tasks and exams, and coherency of a course from day 1 to day n, were the major learning points.

Self-assessment and peer-assessment: First, it can be very courageous to make yourself vulnerable with peers. But the benefit of a carefully structured approach to help navigate the action and the experience can be incredible for personal and professional growth. Second, it can require a lot of practice. It is not easy to tell others about something you think you could do to contribute more, to be a better team-member, and to take action and responsibility. The McMaster “mac sandwich” structure of i. say something positive, ii. Indicate an improvement, iii. Clarify how that improvement will have an effect on future contribution, professionalism, collaboration, productivity, is one of those carefully structured approaches. As stated and appreciated in the room, a growth mindset is not just for our secondary school students, but for us as teachers too.

Course Planning for secondary school mathematics -- curriculum development and implementation.

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#PBLmathEd #preservice #secondary #mathematics #PBL #planning #differentiation

PBL 3 – day 3, differentiation in higher ed! It sounds like every team was able to find their own direction and every person who wanted to was able to participate and produce what they felt was important. Very nice! Consolidating thinking around curriculum content (order, purpose, approach), and Achievement Categories, technology use, and formative assessment and summative assessment issues! How does it all fit together!? Just like the “real world” when teaching a new course means the course plan needs to be done before you have ever taught the course – normal! And even with teaching experience changes appear: new pedagogical approaches, a new curriculum, a new initiative from the education ministry/department or school board, means revising a course plan without having taught it yet – very normal! And frequent! ;-). Can’t wait to see what the teams learned, created, and decided upon in the final day presentations.

Course Planning for secondary school mathematics -- curriculum development and implementation.

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PBL 3 – day 2, what are the big ideas? what will student be able to do/show at the end?  how do we get students there through the course or unit?

Course Planning for secondary school mathematics -- curriculum development and implementation.

Sounds like backward design! If curriculum documents were written in an ordered, conceptually appropriate manner, showing the development of concepts from the beginning of the course to the end of the course, then we might not need course design strategies. But curriculum documents are not written that way. And textbooks are not written showing the same order of curriculum expectations in their chapters as we see in the curriculum documents. AND, teachers bring a LOT to the classroom – like their own ways of knowing mathematics, their own ways of listening to and attending to students’ learning needs and their knowledge of students’ engagement issues, or the use of rich learning tasks in optimal places in the course, and how formative assessment can inform instructional practice decisions, and how the context of the surrounding community and the school population matters when implementing context in problem-solving activities… and perhaps, just using the curriculum as a vehicle for building learning skills and a safe learning community in the classroom. All of this knowledge and ways of knowing (yes, just like Indigenous ways of knowing, or mathematical thinking ways of knowing, or the scientific method) means some kind of a strategy needs to be in place.

How many times has it been said by a teacher, “I didn’t get through the curriculum”, or to another teacher at the end of the course, “don’t put that on the exam because I didn’t teach it”, or “students did fine during the tests but failed the final exam”, or students say “but we used calculators and technology for the lessons, why NOT for the text or exam”, or “when will we ever use this?”, “why is this necessary?”…

If 1. The big ideas, or the essential questions of the course, or the main topics are understood, then 2. A final, summative, evaluation tool can be created that maintains a curriculum standard, a beacon and goal for learning for everyone in the course. Then 3. once we know where we are going, we can design the daily lessons, activities, tasks, assessments, and learning to build towards that ultimate goal, and we all get there.

There are lots of ways to think about this: archery clearly knows where the target is before shooting the arrow; baseball clearly requires knowing where to throw the ball before letting go; religion demands one think of the end (e.g., heaven or hell) before one acts; Stephen Covey’s “7 habits of Highly Effective People”… keep the end in mind; when traveling, we know the destination and what it is all about before we pack (e.g., for the beach or for the ski slopes); even a good guess is better than nothing – research requires a hypothesis before the experiment is started; learning goals in Ontario teaching practice; and Jim Collins’ “Good to Great”… put the right people in the right seats on the bus (know the curriculum), then decide where you are going (the goal/destination/final), and start moving to get yourself there (the daily activities).

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#PBLmathEd #preservice #secondary #mathematics #PBL #planning

PBL 3 – day 1, a new cycle – course/unit planning

Course Planning for secondary school mathematics --  curriculum development and implementation.

A new PBL cycle. This one is four days in length, and with a product at the end; an open-middled kind of task. However, even though everyone is working in their own way towards the same kind of final product, many differences will be visible across all the final products. PBL teams are working in a school department scenario developing a course outline for grade 9 applied and academic courses, or grade 10 applied and academic courses.

Yes, trust the textbooks and materials created for the curriculum, but trust in one’s own creativity, thinking, problem solving, and mathematical knowledge to develop a course outline that maintains the curriculum standard, and reflects the teacher’s pedagogical style...that can then also be responsive to students’ learning needs.

First on the list of action items for self-study: What curriculum development strategies exist? (e.g., like Understanding by Design or Backwards Design.) What is spiraling a curriculum all about? If the curriculum document is just a list of expectations, can the expectations be organized in a meaningful manner? (e.g., looking for the Big Ideas that anchor days of learning together into a cohesive unit). And how does assessment and evaluation work with, or drive, or respond to instructional strategies? (e.g., looking at Growing Success for insight.)

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#PBLmathEd #preservice #secondary #mathematics #PBL #technology

PBL 2 – day 4, what was learned….

Technology for teaching & technology for learning secondary school mathematics.

The five PBL teams focused either on grades 9 and 10, or grades 11 and 12: and the relevance of the Ontario curriculum’s mathematical processes with the use of technology for the teaching and learning of mathematics, the use of technology for instruction and for assessment, and student collaboration in the process of their learning of mathematics were prominent themes. Interestingly too, from the five PBL teams came five different pieces of technology and five different perspectives on the use of technology for the teaching and learning of secondary school mathematics. Technology explored --> Minecraft, IXL, Geogebra, EquatIO and Desmos, and the use of ipads with free apps.

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#PBLmathEd #preservice #secondary #mathematics #PBL

PBL 2 – day 3, specific technologies selected and application in curriculum expectations explored today.

Technology for teaching & technology for learning secondary school mathematics.

Today was about consolidation and consensus. Large group problem solving usually means individuals do lots of work, and sometimes none of their effort appears in the concrete results… BUT, has definitely had an impact in the collective thinking that moved the group to its final outcome and product! Schools, departments, divisions, are all large group professional spaces where it requires many to do work that may not appear ‘on paper’ but does appear in changing mindsets, attitudes, and collective thinking. Ultimately, as teachers, our collective work is for our students’ benefit and their learning.

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PBL 2 – day 2, curriculum reasons for technology! And problem solving…

Technology for teaching & technology for learning secondary school mathematics.

Lots of great discussions today! Issues of software and student accessibility – perhaps hardware concerns? definitely learning ability/disability concerns. Knowing about ‘mathematical processes’ which are present in the curriculum…but are they really understood? And are they present in our thinking and decision-making about technology? Following the PBL steps and reviewing the Learning Goals for the PBL task were key features to conversations today, and to re-shaping and forming arguments and directions for inquiry over the next few days as team members get ready to do their ‘homework’ so the team can move on and be productive in the next team meeting. (actually, very much a ‘flipped classroom’ kind of experience I think)

A question has been rumbling around in my head lately, especially today, is mathematical thinking for secondary school students the same as the mathematical thinking I believe I am hearing in the PBL groups from preservice mathematics teachers? If so, does the same definition apply or should there be a subtle shift in the articulation? This feels like it could be a similar kind of argument about problem solving as a definition – are the “steps to problem solving” the same as “mathematical thinking as problem solving”? (In my opinion, they are different… and pedagogically necessarily so…

“Pólya is likely the most familiar person associated with describing the steps of problem solving. In mathematics curricula, for example, the various National Council of Teachers of Mathematics (NCTM) principles and standards publications, and the Ontario Ministry of Education curriculum (OME, 2005), problem solving is described as a mathematics process or performance skill, a way of ‘doing’ some (but not all) mathematics, where “[i]t is considered an essential process through which students are able to achieve the expectations in mathematics” (OME, 2005, p. 12).

Problem solving is more than just a set of steps to successfully find a solution to a mathematics problem; it is also how one thinks as a problem solver. Some have explored ways one can be a problem solver, and explain what knowing mathematics is all about, for example, “knowing-to” from Mason and Spence (1999), the Harvard verbs of mathematical inquiry (Harvard, 1995), and Cuoco, Goldenberg, and Mark’s (1996) mathematical habits of mind.

It appears problem solving has two meanings, a) a noun: the way some mathematics is done, and b) a verb: the thinking and reasoning mathematically through a problem to come to a successful conclusion or solution. People who are proficient at mathematics may be unaware of how problem solving can exist as a noun and as a verb, and that this distinction can make a difference in their thinking and actions, for example, preservice teachers who are teaching others to learn mathematics and mathematics problem solving. If preservice teachers are not crystal clear that teaching mathematics as problem solving is different from teaching mathematics through problem-solving, what learning success can we expect from students?” (from Pyper, J. S. & MacGregor, S. (2018). Problem-solving: How preservice teachers understand it during their preservice learning. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.). Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 19-26). Umeå, Sweden: PME.)

References from above text.

Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. Journal of Mathematical Behaviour, 15, 375-402.

Harvard. (1995). Assessing mathematical understanding and skills effectively. An interim report of the Harvard Group, Balanced Assessment Project. Harvard Graduate School of Education. Retrieved from http://hgse.balancedassessment.org/amuse.html

Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment. Educational Studies in Mathematics, 38(1/3), 135-161.

OME. (2005 (revised)). The Ontario Curriculum: Grades 9 and 10 Mathematics. Toronto, Ontario: Queen's Printer for Ontario Retrieved from http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf.