Mathematics For Thought

Documenting an educator's love for all that teaching mathematics involves

August 13, 2015
by Mirela Ciobanu

Developing Visualization in Mathematics through the use of Singapore Model Drawing

The importance of visualization in mathematics education cannot be argued. By the time they reach middle school, Ontario students are exposed to a variety of visual tools such as, the number line, KWC – charts, tables or Venn diagrams, which are used to solve number sense problems.

In 2005 Ginsburg et al. published a comprehensive study in which they compared the educational system of the United States to that of Singapore. In terms of textbooks, the textbooks in the United States are very similar to those used in Ontario; often

Singapore Model Drawing Method

the publishers being the same. This study concluded that, though very visually appealing, most of the visualization tools in the United States Mathematics textbooks are introduced in context specific lessons and students practise using them mostly when solving routine, definition type of problems.

When students do not associate a context to a visual representation tool they usually use for that context or type of skill, and they are expected to create their own, they rely mostly on inefficient, concrete type of drawings. This is one of the many difficulties I believe our students encounter when asked to use visuals to help them solve word problems. Hence the need for a visual tool that can transcend the limitations of applicability to only familiar contexts related to a concept or set of skills.

When students solve problems, what information are they paying attention to? If they use a visual method, does the visual used represent some information in the problem or the entire problem? What is its role in the problem solving process? How efficient is it?

Duval (2006) emphasized the fact that students who have a hard time solving word (algebraic) problems rely too much on ‘shortcut” approaches, focusing mostly on algorithms, translating “key phrases” associated with computations, working only with symbolic representations and rushing to compute the numbers in the problems.

The mathematics education landscape in Ontario is experiencing a serious problem solving issue lately. In recent years (2012-2013; 2013-2014) the Education Quality and Accountability Office’s (EQAO) reported a decline in the Grade 3 (primary division) and Grade 6 (junior division) students’ ability to solve multi-step word problems in Number Sense, while they still demonstrate proficiency with computation skills. This data indicates clearly a difficulty to apply computations to the right context. Moreover, 26% of the entire cohort tested in 2014 achieved below the provincial standard in both grade 3 and Grade 6 and 19% met the standard in Grade 3 but not in Grade 6. That means that 45% of the students in our province are failing mathematics in elementary grades.

Additionally, reflecting on my own teaching experiences in the classroom I noticed a prevalence of students’ misinterpretation of the problem text and a rush to use arithmetic schemata to solve algebraic problems. When models are drawn they are mostly models of parts of the problem and not of the entire problem and they are often not used to solve it. This is the Model Drawing method or the Bar Method used in Singapore.

During the year 2014-2015 I conducted a TLLP (Teacher Learning and Leadership Program) project and implemented the use of Singapore Model Drawing with grade 6 students at my school. This model helps them understand a variety of problems, represent them in a way in which it becomes easy to understand the relationships between the known and unknown values in the problem translate them into correct algebraic equations, is the base of the solution plan and the core around which the student communicates his/her thinking.

I explored not only how students in Ontario develop and apply a visual model solve algebraic problems that is borrowed from a different culture but also what the causal conditions, the context and the intervening conditions are that would make the model method successful to them. Additionally, equally important is looking at factors that can prevent them from using the model to solve algebraic problems, such as conceptual, semantic, contextual and strategy knowledge.

Introduction – Why Singapore Model Drawing Method

The Primary Mathematics curriculum in Singapore is founded on the work of American Psychologist Jerome Bruner and on his the concrete-pictorial-abstract approach to learning. It was not the buzz surrounding recent TIMSS and PISA scores that lead me to the Singapore Primary Mathematics Curriculum but researching metacognition in mathematics while conducting a TLLP (Teacher Learning and Leadership Program) project in 2012-2013. In Singapore, metacognition is one of the core components of problem solving which is at the heart of their Mathematics curriculum. Exploring the way problem solving is taught in Singapore introduced me to the Mode Drawing method.

Model Drawing is the most unique feature of Singapore math. In 1983 Dr. Ted Hong Kho and his team introduced it in the Singapore Mathematics Curriculum. Initially, students were taught the model for the first time in grade 4 (Primary 4) and later this was changed to grade 3 (Primary 3). This method is believed to have helped Singaporean students bridge from arithmetic reasoning to algebraic reasoning when solving challenging word problems (Cheong, 2002; Ng, 2003). The model relies on drawing a visual diagram made up of rectangles that do not need to be scaled.

The model relies on the fundamental knowledge of part and whole relationship between numbers. A small rectangle is a part of a bigger rectangle representing the whole. The length is not important when the student represents an unknown value. Each rectangle is labeled with information from the problem, using either known or unknown values from the problem. The diagram allows students to see and draw meaningful relationships between the elements of the problem (both known and unknown) before they attempt to translate such relationships into abstract symbolic representations, the arithmetic or algebraic equations.

The model involves three stages: understanding the text of the problem, structural phase (drawing the model) and procedural-symbolic stage (translating the visual into arithmetic or/and algebraic equations). Therefore, the model drawing relies heavily on the process of understanding the situation presented in the problem before students attempt to solve it. It is also dependent on the use of metacognitive processes (Ng & Lee, 2009, p. 292). In Singapore, the model method is so prevalent in their mathematics textbooks that even parents are familiar with it (Yeap, 2013).

Theoretical Highlights

What is successful problem solving? This study’s understanding of successful problem solving does not equate with reaching an accurate answer but being able to understand the problem situation, represent it visually through a model that allows to work out the relationships between known and unknown and constitutes the basis of the solution stage (Bednarz & Janvier, 1996, Hegarty et al., 1992). Solving word problems has two major stages, the comprehension and the solution stages (Mayer and Hegarty, 1996).

What causes students to struggle with problem solving? When students struggle with the comprehension stage it might be because they have difficulties with translating the situation in the problem and representing it using various modes and switching back and forth between natural language and the other models of representation (Duval, 2006; Mayer, Lewis and Hegarty, 1992). Mathematical misunderstanding is caused, according to Mayer et al., by the “construction of a mental model of the situation that conflicts with the information in the problem statement (1992: 137)”. These researchers encourage the development of students’ qualitative reasoning as opposed to quantitative reasoning skills represented by performing an algorithm and focusing on an answer. The qualitative reasoning steps included in problem solving are translation (the problem’s text into a mental representation), integration (combining the relevant information into a mental representation), planning (the steps needed to solve the problem). The last step is execution (carrying out the planning) and uses quantitative reasoning. Mathematical misunderstandings are more likely to occur in the qualitative reasoning steps and not in the execution. Hence, students need more opportunities with the first three steps and not a focus on the last one. Obviously, the choice of problem-solving task becomes a condition here.

Why is the choice of problem solving tasks important? There are two types of problem solving tasks that need two types of reasoning: arithmetic and algebraic. An arithmetic problem is one where the start value is known and the unknown (or the result) values can be found by performing arithmetic computations to the known values. On the other hand the algebraic or “word” problem is one where the result value is known but not the start value. Solving direct computations to the known values will not solve the problem (Khng& Lee, 2009; Nathan & Koedinger, 2000). Mosely and Brenner (2009) argue that the two types of problems present different schemata that work together in order for the algebraic problem to be solved. However, when students apply only arithmetic schemata to solve algebraic problems they run into difficulties (Khng & Lee, 2009). As students attempt to use variables to express mathematical relationships other misconceptions may be revealed (Moseley and Brenner, 2009).

How can students make the transition between arithmetic and algebraic reasoning? The transition from arithmetic methods to algebraic ones is needed for students to be successful at solving algebraic word problems (Khng & Lee, 2009). Moseley and Brenner (2009) suggest (a) schema integration or the ability to successfully integrate variables into the process of solving arithmetic problems (b) using tables, graphics, diagrams that accompany the text so that the students translate a mathematical relationships presented in pictorial form, and (c) exposing students to non-routine problem solving situations which involves using variables to represent unknown. Kieran (2004) suggests a focus on numerical relationships and not only on the answer, a focus on inverses of operations, a focus on representing problems and not only solving them, a focus on the use of the equal sign and a focus on the use of both numbers and letters rather than only numbers to solve problems.

What constitutes an effective representation and how can students be helped in visualizing mathematical situations? Duval (2014) makes the distinction between visual representations and visualization. Visual representations are representations used in mathematics in mathematical treatment of concepts, heuristic exploration in problem solving and educational tools in the acquisition of math concepts (Duval, 2014, p.159). The model drawing approach creates a visual representation of the problem. Visualization is the recognition, spontaneous or not, of what is mathematically relevant in any visual representation produced (Duval, 2014, p.160). He argues that there is a gap between what teachers see and what students see and this is particularly one of the focuses of this study method. In what way, students’ visualization differs from that of the teacher? Figural units are individual elements that are recognized as significant or informative in a visual representation.


Bednarz, N. & Janvier, B. (1996): Emergence and Development of algebra as a problem-solving tool: Continuities and discontinuities with arithmetic. In Bednarz, N., Kieran, C. & Lee, L., (Eds.), Approaches to algebra. Perspectives for research and teaching, 115-136, Dordrecht, The Netherlands: Kluwer.

Cheong, Y.K. (2002): The model Method in Singapore, The Mathematics Educator, 6(2), 47-64

Duval, R. (2006): A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103-131

Duval, R. (2014): Commentary: Linking epistemology and semio-cognitive modeling in visualization. Zentralblatt für Didaktik der Mathematik (ZDM), 46, 159-170.

Fan, I. & Zhu, Y. (2007): From convergence to divergence: the development of mathematical problem solving in research, curriculum, and classroom practice in Singapore. Zentralblatt für Didaktik der Mathematik (ZDM), 39, 491-501

Ginsburg, A., Leinwand, S., Anstrom, T., Pollock, E. (2005): What the United States can learn from Singapore’s world-class Mathematics system (and what Singapore can learn from the United States): an Exploratory Study. American institutes for Research

Higgins, K.M. (1997): The effect of year-long instruction in mathematical problem solving on middle school students’ attitudes, beliefs, and abilities. The Journal of Experimental Education, 66(1), 5-27

Highlights of the Provincial Results Assessments of Reading, Writing and Mathematics, Primary Division (Grades 1–3) and Junior Division (Grades 4–6) English-Language Students, 2013–2014 Retrieved from

Highlights of the Provincial Results Assessments of Reading, Writing and Mathematics, Primary Division (Grades 1–3) and Junior Division (Grades 4–6) English-Language Students, 2012–2013, Retrieved from

Khng, K.H. & Lee, K. (2009): Inhibiting inference from prior knowledge: Arithmetic intrusions in algebra word problem solving. Learning and Individual Differences, 19, 262-268

Kieran, C. (2004): Algebraic thinking in the early grades: What is it? The Mathematics Educator, 8(1), 139-151

Lewis, A.B. (1989): Training students to represent arithmetic word problems. Journal of Educational Psychology, 81, 521-531

Mayer, R.E., Lewis, A.B., Hegarty, M. (1992): Mathematical Misunderstandings: Qualitative reasoning about quantitative problems. Advances in Psychology: The Nature and Origins of Mathematical Skills, 91, 137- 153

Moseley, B. & Brenner, M. A., (2009): A comparison of curricular effects on the integration of arithmetic and algebraic schemata in pre-algebra students. Instructional Science, 37(1), 1-20

Nathan M.J. & Koedinger, K. R. (2000): Teachers’ and researchers’ beliefs about the development of algebraic reasoning. The Journal for Research in Mathematics Education, 31(2), 168-190

Nathan M.J. & Koedinger, K. R. (2004): The real story behind story problems: effects of representations on quantitative reasoning. The Journal of the Learning Sciences, 13 (2) 129-164

Ng, S.F. & Lee, K. (2005): How primary five pupils use the model method to solve word problems. The Mathematics Educator. 9(1), 60-83

Ng, S.F. & Lee, K. (2009): The Model Method: Singapore Children’s Tool for Representing and solving algebraic Word Problems. Journal for Research in Mathematics Education, 40(3), 282-313

Yeap, B.H. (2013) The Model Method,, Retrieved on August 4, 15



April 6, 2013
by Mirela Ciobanu

Motivation: Predictor of Student Success in Math

Niños de Tilcara saliendo del cole
Photo Credit: (M) via Compfight


In December 2012 I have encountered an article published on the Time’s website (as well as published in the Scientific American) and many education sites that added a new direction to my learning. “Motivation, Not IQ, Matters Most for Learning New Math Skills.” It proved what we all probably knew for years. And this is the proof.

Post-doctoral researcher from UCLA, Kou Murayama, published this study in the journal Child Development. His study was conducted in Germany on 3,500 schools over a five year period. The findings: it is not how smart you are, it is how motivated you are to learn, your attitude toward learning is what matters most. He noticed that motivation is a greater predictor of success in mathematics than IQ. His study looked at how intellectually engaged students are in math, attitudes, making connections between concepts studied and use of strategies.

Do all of these ring a bell?

Let’s add Carol Dweck’s Growth Mindset theory into the mix. What motivates students is knowing and believing that the more they work the smarter and more capable they become to understand and do math. Cultivating this more into the classroom will increase students’ motivation.

So, how do we ensure that we create enough opportunities for students to see and monitor their successes and celebrate those as the result of their effort, their interest and perseverance?

Do we provide enough opportunities for the students to place themselves on a learning continuum and see themselves as progressing while learning? Do we give the students opportunities to equate this with the result of their hard work to progress?

Here are some of the things I do:


1. Individualized Learning Logs used by students to record and track success, set goals and control their learning in general.

2. Goal setting activities: “Today’s Lesson Personal Goal: I begin each lesson giving the students time to write a personal learning goal which they will revisit at the end of the lesson to tell me whether they have achieved it, approaching it or still working on it. They also give me a good measurement tool of how accurately they can assess their own progress.

3. Critical reflections: End of the unit reflections

4. Ongoing opportunities to compare and contrast solutions and ideas.

5. Ongoing opportunities to provide peer descriptive feedback.



I heard about building a relationship and a safe learning community very often.  I must say that I intuitively had taken care of this aspect in my classes. However, the more I read studies and research about factors that foster success and motivation, the more convinced I am of the importance of cultivating and fostering a learning environment that is conducive to success.

So, if the students’ voices are not heard or valued, do students feel like a sense of belonging? We all know that certain circles of people can intimidate and make us feel less worthy. Conversely, the ones that are supportive make us look bright, inspire us, raise us. This is in short what happens to students. Where do they feel like they can shine? Do they all have the chance to own the floor, not because we want everyone to have it, so we provide it for them, but because we actually discovered something they shared, the seed of a great idea and we use it in the consolidation of a new concept. Or maybe because they were attempting a unique way to solve the problem that might have been the most efficient one.

And so, do we listen to the student voice long enough? Attentively enough? Beyond what we would like them to say? Do we make them listen to each other patiently? To build on each others’ ideas? To acknowledge and clarify each other’s ideas? Do we give them a chance to analyze the process and not only the solution?

A few things I have tried so far: 

a. Students observing students solve problems ( with reflection, comparing and contrasting solutions)

b. Students listening to students’ pencasts ( using Livescribe pens) ( followed by providing peer descriptive feedback about problem-solving skills)

c. Students sharing their solutions during Bansho ( used in the consolidation part of the lesson)

d. Using to foster more opportunities to communicate, reflect and provide peer support

e. Student -student conferences about the pencasts

f. Student- student conferences about the problem-solving process they observed

One of the things anyone notices when they visit my class is how engaged all my students are. I still have a hard time naming what I do. It works and it does for years now. In fact, is a recipe with more ingredients than the ones I mentioned above. I strongly believe though that it takes a lot of hard work and persistence.

I see the ingredients mentioned above as the “Musts”. In my opinion, motivation in math cannot exist without having students set goals, “see”, “track”, monitor and celebrate their success. They should see that their hard work pays off.

To conclude with, it is my focus on researching metacognition in the math class that brought me close to realizing the important role of motivation and having a supportive community. I am happy with this encounter.


The link below will take you to the article about Kou Murayama’s study:

Read more:

And for the Scientific American link:


The partnership teacher-parents is build on the premise of trust. Usually parents want to know one thing, “What can I do to support my child’s learning”?

I created this tip sheet that can be shared with parents at the beginning of the school year during Meet The Teacher Night or Curriculum Night. If we want to send messages about growth mindset we need them to be reinforced at home and in the community. I think a good start is to talk to parents about what type of praise should be given to their children in order to support learning as a continuous process, perseverance and motivation.

Praising children



February 18, 2013
by Mirela Ciobanu

Searching for a Different Problem- Solving Model

maths in classroom
Creative Commons License Photo Credit: Charles Pieters via Compfight

     “If you can’t solve a problem, then there is an easier problem you can solve: find it.”

~ G. Polya

Why a “different” problem-solving model?

This desire came from my many attempts to make students adopt a model of problem solving that they can use independently and which can realistically describe the steps and the skills of a problem-solver. I have used the standard one introduced by Polya in 1945 in his book “How to Solve It”: understand the problem, make a plan, carry out the plan, judge the solution.

There is nothing wrong with this only that I felt that certain steps needed more detail and a lot of direct modelling. For instance: “Understand the problem.” Easily said but how hard is that? It is particularly because this is a reading comprehension skill and just like we use a variety of reading strategies: summarizing, making inferences, finding the main idea, visualizing, to name a few. Students need a lot of practice with this first step.

I don’t see the problem solving like a linear process but mostly like a circular model, almost like a spiral, if I have to use a metaphor to describe it. The more you understand it, the greater your ability to solve the problem. One has to revisit the context of the problem several times during the process, will re-read the problem even while carrying a strategy. We return to check the solution within the initial context. We adopt a plan and we test our understanding of the problem. More and more in mathematics I reach the same conclusion in my class: the students are solving a different problem. This is a very nice way of saying that they misunderstood the problem.

So, I started my quest. I dropped all the formulas I knew from before. All the KWC ( Know-Want to find-Conditions) and KWCD ( Know-Want to find-Conditions-Do) seem completely useless. Why? There is so much work that needs to be done by a teacher to show the students that KNOW means more than what is “given” or stated. “Know” includes one’s prior knowledge and the ability to infer . Ultimately, KNOW can be the knowledge about the context, a similar context or real life context that the problem reminds us of. I hope I am in agreement with whoever reads this blog entry that simply telling students to write what they know and expecting them to do a good job just because of a straightforward visual organizer won’t help. If you are like me, then surely you encountered the same problem. The students need to practice how to “KNOW”.


Luckily I discovered, like many others before me, Singaporean mathematics. Metacognition is part of their curriculum. Visualization is part of their problems-solving model. I initially used Dr. Lee Ngan Hoe’s problem solving model.

First, I felt  so vindicated because of the “circular” model he proposed (which he had borrowed and simplified from Richard Paul’s Reasoning Wheel) and secondly, I found out that he adopted this model as he felt dissatisfied with the results of usin Polya’s model as well.

The Problem-Solving Wheel includes the following steps:

1. The GIVEN ( main idea, stated and inferred information)

2. FIND ( what do I have to find out?)

3. PICTURE ( this helps students visualize the problem and understand it better as well as making the first connections betewen its components)


4. TOPIC ( identify the topics and strands that this problem might fit in and thus it provides students with opportunities to make connections/apply knowledge)

5. FORMULAE ( formulae that might have been used to solve problems that fit the topics)
Photo Credit: Kai Schreiber via Compfight

Now, this is a great model but as I teach in an Ontario school it needs more work on components number 3 and 5.

“PICTURES” because in Singapore, whose mathematics education is stellar and so well recognized internationally ( see PISA) as one of the strongest in the world, the “Model Drawing” is taught since kindergarten. Students are used to using rectangular diagrams to represent the contexts of problems and thus, their ability to generalize and use algebraic thinking is stronger from a younger age.

“FORMULAE” because of what I mentioned above. The connections between topics are so easy to make. Think: representing numbers as rectangles helps with understanding rational numbers, measurement, number lines and leads to algebraic generalizations.

These two are my current goals: teaching the model drawing and encouraging algebraic thinking ( which is based on making connections and generalizations, observing relationships between concepts, patterns) in order to make the above model of problem solving applicable in my Ontario mathematics classroom. I know from what I read and from my elementary schooling experience that this will lead to success.

February 6, 2013
by Mirela Ciobanu

NOV. 2012: Belonging – Community – Success

Learning is social and the feeling of belonging to a community affects performance” ( Annie Murphy Paul, New York Times, Sept 2012)

In November 2012 I launched this year’s online math community on Kidblog for my two grade 6 classes. I have taken this journey before and I have now reached a point where I know and can say that using kidblog as a medium of communication between my students have not only strengthened the relationships we built in the classroom but it helped the students develop their ability to communicate tremendously. This ability is strongly connected to their self-knowledge in math and self-regulation.

We all know that one cannot use language unless it is intentionally used and that language is a tool. This being said one  the language of mathematics cannot be activated with such proficiency without knowing when and how to use it or to communicate one’s thinking. Therefore, I noticed a direct and strong correlation between the level of knowledge and the ability to communicate knowledge, the depth and complexity of a student’s reflection and that student’s performance in mathematics, the depth of the mathematical reflection and the level of proficiency of the language used.


A beautiful thing happened on October 2, 2012 when I happened to check back on my last year’s blog and discovered that two students, who are currently in grade 7 ( in two different classes ) have returned to our last year’s class blog and posted a math question for whoever might have been “there”. This is their entry:

Well, I’m not too sure if anyone comes back here any more, but we were bored, so we just wanted to ask a question and see if anyone wants to solve it.

So, in this problem, there are 200 students and 200 lockers. ( heard of this problem before?)

The first student walks across the hallway and opens EVERY locker.
The second student walks across the hallway and opens every SECOND locker.
The third student walks across the hallway and opens every THIRD locker.

This continues all the way to the 200′th student opening just the last locker. 
Which of the lockers are open?
Is there any pattern to the lockers that are open? 
*Remember to show your work!*

BONUS: If you find all of the lockers that are open, draw a graph of the first 8 numbers that have their lockers open. Explain how this is related to “Exponential Growth”

(Remember we were just very bored.)

The replies came flying  from two other peers, two girls, who are also separated as they are in two other grade 7 classes. This is what they wrote:

“Well, we weren’t not too sure if anyone comes back here any more, but we were bored, so we just wanted to answer a question that was posted by (names of students), (our former classmates.) We hope that someone grades us.”

AND after I commented on their post:
One of them wrote:
“Thanks Ms.Ciobanu! Answering questions and getting feedback makes me feel like I am in 6D again. :D “
The other one’s reply: 
“Thank you, Ms.Ciobanu! I agree with (…), coming here again, I feel like I’m in 6D again. I miss the math discussions here and all the fun we had here doing math! 
True, for this problem many people when handed the problem had no idea how to solve it (me included) then we tried it out and usually the first attempt at solving it, we didn’t get it, but working into the problem we eventually figured out the pattern which led us to the answer.” 
Apart from the initial joy and overwhelming emotion I tried to reason as to why this “return” to our community happened?

Here’s what I found about the importance of having a learning community:

– Learning blossoms if done in a community that promotes the feelings of belonging – Uri Treisman( UC- Berkley), 1970

– Metacognition develops when math is done collaboratively ( idea repeatedly emphasized by Pina Tarricone who in her last book, her PHD thesis, The Taxonomy of Metacognition, reviews all the research about this “fuzzy” term in an attempt to clarify its usages. In most of the resources cited, solving challenging problems collaboratively is the one condition that leads to the development of metacognition.

Probably the most significant a-ha’s I had came from reading Annie Murphy Paul’s “A Brilliant Report” on January 18/13.

-In the issue entitled ” The Importance of Belonging” Annie summarized  important findings that linked the idea of feeling appreciated, loved and a sense of belonging to one’s academic success. She also quoted the work of Dr. Carol Dweck. The latter dedicated the last decades to the concept of “growth mindset”. In my previous entry I explained how I linked this concept to metacognigitive activities I use in my math classes. Here are some conclusions:

COMMUNITY: Learning done in a community will lead to improvement. Studying together leads to more success than studying alone.

TRANSITIONS: Changing a school, class, grades affects one’s performance in school. Students are still trying identify “cues” in a learning environment that will signal to them that they belong.

IMPOSSIBLE ROLE MODELS: When choosing motivational stories for students we should avoid the successful part but actually show the struggle that lead to success. The sole focus on success without the many fails that lead to success seem demotivating, particularly for girls.


1. My students worked together and commented on each others’ work, gave each other feedback and acted on it irrespective of their level of achievement.

2. The process of solving problems became more transparent and all students saw the peers that they thought are “born” with an ability to do well in math as problem solvers who attempt different strategies, reflect on their success and try to make sense of the intricacies of math collaboratively.

3. The blog reinforced Carol Dewck’s “growth mindset” concept : the more you work and the more determined you are to succeed, the more successful you will become. The students’ motivation increased. It was particularly evident in the struggling students and girls.

4. Hard work leads to success.

5. Transitions: the students who posted the question on October 2 were in an environment in which they did not feel at home yet: new colleagues, new teachers, new requirements. Their sense of belonging was in transition mode. They returned to what felt safe, to where they felt they belonged and appreciated.

6. Students need feedback and challenges.

And last, but not least:

7. Mathematics can be one of the most exciting topics to write about and try to make sense of together with your friends. The satisfaction that comes from working out a challenging problem is priceless. Or like one of my students tells me, “I feel the synapses charging. My brain is wrinkling!” – I AM LEARNING!

Currently the project Kidblog has expanded and involves partnering pre-service teachers from the Ontario Institute for Studies in Education of University of Toronto to grade 6 and 7 students. We are looking at observing closely how students employ metacognitive habits modelled and implemented in my classroom and the impact of descriptive feedback on students’ achievement and motivation.

To further build on Annie Murphy Paul’s article, I have now discovered that a community that students are devouring any feedback and require it. They want to talk to each other and learn from each other.

I had one separate class blog for each of my two grade 6 classes.

A beautiful thing happened: students from one class commented on the posts of students from another class and have collectively asked me for the permission to join and enlarge the math community. Al the pre-service students are impressed by the vocabulary used by my students and by their will and determination to grow in math – together.

Here are the links to the two classes’ blogs:

Note: For the Power Point Presentation used when I launched the project at OISE, please go to “Teacher Resources”

January 22, 2013
by Mirela Ciobanu

OCTOBER 2012 – The Student Learning Log and the “Growth Mindset”

Macro of a leaf unfurling in the cloud forest adjacent to our lodge Bosque de Paz
Photo Credit: Mike Baird via Compfight

October 2012 was dedicated entirely on providing my students with a tool that was “friendly” enough for them to be able to use independently to monitor and control their learning: “The learning Log”

The current Student Learning LogI am using had undergone a lot of metamorphoses from the wording, to the format and content. One thing I knew it has to have: it must indicate how the students see themselves progressing through each unit. I wanted them to be able at any given time to be able to describe what they know and can do well and identify a few goals that they are working towards achieving.

This tool was ultimately designed as an assessment as learning tool. In its first stage it was names “My Learning Path”. Given the many contexts of the word “path” mostly in professional learning communities I had to change its title to “My Learning Log”.

Initially the log included each expectation from the previous year and the current one. They were added in the log and did not make too much sense to the students. As well, to make their job less assiduous I provided them with easy symbols to denote: “progressing very well”, “progressing well”, “almost there” and “still working on it”. I found that most students did not understand how to use it (and that meant more modelling on my part) or they used it only when required and did not put much thought into using those symbols. Most importantly, they did not understand those goals that were already there, written  and got confused in the sea of expectations presented to them. Did I mention that they still had trouble understanding the purpose of this form. So, this was out and I knew I had more work to do modelling the use and explaining the purpose.

A NEW UNDERSTANDING: The forever truthful “less is more”.

Mathematics Teaching Translation:  “Let the students identify the big ideas and do the work”. It’s their learning.

I knew I had to get rid of those expectations and allow the students to go through the process of consciously deciding the following:

a. What they already know and can do – THE STRENGTHS

b. What they need to improve on ( what gaps to fill from the previous year(s) – THE LEARNING TARGETS THAT NEED TO BE ACHIEVED

c. What new learning they need to focus on during grade 6: THE LEARNING TARGETS CORRESPONDING TO THE CURRENT YEAR


I modelled this during two units and I must say that now it’s becoming less of a guided process and more independent. The students can do this at home or in class at the end of each instructional unit. This is another opportunity for them to develop self-awareness and get involved continuously into the learning process. Thus, they get to realize the amount of learning they get through.

The log is revised mid unit when we have peer-peer support carousel sessions based on student self-identified strengths and goals. Students get a chance to “openly” identify what strengths and targets they have in relation to every learning goal of the unit. They are partnered according to strengths and targets and work through a variety of challenges doing think aloud’s. These sessions are target based and they are lead by the students who need support not by the ones who is already strong.


I was asked on a few occasions whether I am not worried about asking students to publicly identify their strengths and learning targets. Am I not worried of embarrassing, poor self-confidence?

Here’s my take: First, I don’t start using these sessions in September. In September I talk to them about our learning community and I cultivate systematically the “growth mindset” I want my students to have. I think that given the experience I have with this so far teachers will be stunned to see that they only ones who are afraid of this are the teachers. We cultivate the idea that we all learn with and from each other. If you are an adept of Bansho, you will know what I am talking about. Many students who might be described as “struggling” surprise with solutions that make their way to the right side of the board. The same happens here: students who are very strong, will identify targets. Isn’t this what we want anyways? Students who are struggling will identify at least one strength a unit. It always happened to me. I started using these sessions in 2010 and I never had any surprises. The “worst” that can happen is they all have identified a learning goal as a target and so I get the immediate assessment data that I need to go back to that goal the following lesson.

Additionally, what motivates students is to see that math does not come easy for a few and hard for the same but rather that it takes time, effort and determination to grow into your strengths.

For more ideas on the how we can talk to students to cultivate Dr. Carol Dweck‘s “Growth Mindset” in the classroom please check the following link:

January 21, 2013
by Mirela Ciobanu

September 2012: instructional re-design of math units


For lack of a better word I will use the word, “design”. I knew early in September 2012 what type of class I want to have in June 2013: a community of learners of mathematics, fluent in the language of mathematics, comfortable with each other’s feedback and support, knowing at all times, where they can place themselves in the mathematics learning continuum, and who are permanently concerned with improvement, while enjoying math.


Each unit in mathematics is seen as a journey meant to make everyone meet success and being able to see the “change” at the end of it when a comparison with the entry stage is needed.

THE LEARNING CONTINUUM: Even before a unit starts I familiarize myself with the learning continuum as described in the Ontario Mathematics Curriculum. I look at the previous year’s expectations, this year’s and the following’s. I record the expectations in a simplified manner on an anchor chart making sure I do not leave out any terminology that I would need my students to know and use while working through the unit. I number the expectations as they correspond to sections in the student portfolios where students will insert their work addressing individual expectations.

THE STUDENT PORTFOLIO: The portfolio is the students’ tangible collection of evidence attesting to their progress through each unit. The first section is entitled: Entry Point. The last one: Exit point. The backbone or the internalizing and continuous monitoring and control of learning targets is done through the  Learning Log that students complete the  after each class.

a. The ENTRY – this is the diagnostic assessment tailored for the previous year’s expectations and directly rooted in the curriculum. Each diagnostic test is accompanied by a form on which the number of each question and the corresponding expectations are recorded. the students were taught and are always reminded to use that form as it describes their initial ‘strengths’ and the highlighted areas translate into their ” learning targets”.

b. The LEARNING LOG: the log is completed throughout the unit. It is work in progress and an “organic, fluid” document leant to record growth. Self-monitored and guided growth, ideally.  Students add new ones to the initial strengths as they become aware of them during the unit. To the previous learning targets, students add the new ones that match the grade 6 curriculum and are stemming from the previous ones. This way students can see how their knowledge develop on the foundation provided by previous ones and the focus is all on GROWTH.

c. THE WORK SAMPLES: I use three part lessons during every class. Bansho plays a great role in developing metacognition as it invites students to describe their strategies, compare them and build the new learning collectively. Of course it implies the fact students work collaboratively on tasks and that tasks are challenging enough and lend themselves to a variety of strategies to solve them. Research on metacognition has emphasized the fact that for critical knowledge and self-knowledge to develop the problem solving must be challenging. These work samples are: problem solutions, independent practice questions with success criteria checklists attached to provide descriptive feedback, investigations and reflections. They are sorted according to the section number describing the curriculum expectations addressed.


a. STUDENTS OBSERVING STUDENTS SOLVE PROBLEMS – two times a week ( with Assessment for learning sheet, metacognitive reflection and teacher -student conference)


C. MID-UNIT SELF ASSESSMENT AND PEER-PEER SUPPORT CAROUSEL ( students are paired according to their self- identified strengths and targets and rotate partners to help each other with difficult or unclear concepts)


D. STUDENTS LISTENING TO STUDENTS SOLVE PROBLEMS followed by metacognitive reflection on problem -solving

d. THE EXIT TEST: Just like the entry test, students are given the opportunity to see their end point and compare themselves to the entry point. This is the time in the unit when I ask them to reflect on their learning and become aware of the progress they made. I am a strong supporter the growth mindset theory. I also came to believe that students need to see and measure the extend of their progress. They need to see that they did not just “cover” the content but that they actually learned and grew. This is about their individual learning. This is also the moment, at the end of the unit when I evaluate them. Until this point in the unit there are absolutely no marks or levels communicated to students, only descriptive feedback: peer and teacher. I invited students to use the learning log and specifically self-evaluate their progress. I must say that I was stunned when I compared the first retrospective reflection they wrote during this year and I compared them to the exit point.

January 21, 2013
by Mirela Ciobanu

September 2012: Preliminary Diagnosis

Creative Commons License Photo Credit: trontnort via Compfight


Why Metacognition in mathematics? Where did my interest in metacognition? First of all, metacognition is essential to self-regulation and self-knowledge. In mathematics, students are required to use metacognition but it is not directly embedded in the mathematics curriculum but it is rather implied through math process expectations, such as communication, connection, problem-solving, reflection.

The problem solving model provided by the curriculum documents is the one developed by Polya in 1945 and presented in his book, “How to Solve It”. The steps for problem solving are: Understand the Problem, Devise a plan, Carry out the plan and Revise/Extend.

Like any teacher I have tried to create anchor charts explaining the steps, modelling them and expecting students to use the steps. This seems like a great recipe for success in problem solving if it were a linear process. Soon, I realized that there is a lot of work involved in the teaching of problem solving skills and that this is more related to teaching reading comprehension than doing math. The latter cannot be done without the former. This problem solving method works for students who know what to do to understand the problem.  It works for those who know what devising a plan means.  In fact, it means devising a few plans or strategies, and, through close monitoring and continuos assessment of their effectiveness in relationship with the problem, going back to the problem, review the understanding of this problem and try a new strategy. It is a redundant , circular process that good problem solvers do it unaware of the many times they re-read the problem, wondered whether a prelimiray result makes sense, whether they are on the right track.

Too many times I have to tell the students, “This would work if you were solving this type of problem.” Or: ” Given this information, you were actually solving a different problem.” It is because there is so much work involved in the understanding of the problem that depends on what one knows about mathematics ( math knowledge) about that content or similar ones ( ability to build on prior knowledge, apply or make connections), what one knows about himself as a problem solver( self- knowledge and critical self knowledge), the previous experiences ( failures or successes) with similar problems. In other words, a lot of thinking about themselves and a lot of skills that play a major part, I believe, in the students’ ability to do well in mathematics.


During the summer of 2012 I began my research for my project. I focused on ways to collect data at that time.  I discovered this Belgian research paper, published in the Journal of Research in Educational Psychology, that focused on assessing metacognition, Evaluating and Improving the Teaching-Learning Through Metacognition ( 2007). It contains two types of questionnaires used for metacognitive practices: one for children and one for the teacher’s observation habits. It also includes descriptors for “think aloud metacognitive behaviour”. I used this as my source for creating my questionnaire.

I administered the questionnaire to two grade 6 classes and the following behaviours scored consistently among the student population in both classes. Additionally, these answers were in line with my initial diagnostic assessment using the ONAP testing, journal entries, as well as observations taken in class during problem-solving sessions. The behaviours included in the questionnaire are prospective ( before ) and retrospective ( after) solving the problem.

1. Knowing in advance whether you will be successful or not

2. Underlining the relevant information needed to solve the problem

3. Checking the answer with the estimated result

4. Reflecting on how a task was solved

5. Thinking whether the solution or the answer would make sense in reality.

After analyzing the results and evidence, it was clear to me that this is an area of need and I needed my entire program to be designed in such a way that metacognitive behaviours are taught explicitly and reinforces continuously until they become habits.





January 13, 2013
by Mirela Ciobanu


I must admit that my journey with this project started before I received the TLLP approval for my project from the Ministry, in February 2012. I had already been involved in an inquiry in mathematics as part of the OAME’s initiative, launched in October 2011 through their Leadership Conference and whose focus was creating collaborative communities of practice in mathematics with a focus on assessment. I decided to conduct such an inquiry and share my learning at the annual OAME Conference in Kingston, Ontario in May 2012.

My presentation at the OAME conference was entitled “Students Observing Students” and it was co-presented with Caroline Rosenbloom, a coordinator and pre-service instructor from OISE/UT. Caroline was the strongest supporter of my work from the first day I met her. It was through her CLMT (Collaborative Learning for Teaching Mathematics) initiative, developed in partnership with TDSB that strengthened our bond. Her project had at its heart the intention of creating collaborative communities. TDSB  teachers and teacher-candidates co-planned, observed and discussed each others’ lessons, co-taught and collaborated during the project.

As a former math instructional leader with Toronto District School Board, Caroline knows that the best experiences and knowledge about math teac

hing are a result of teacher inquiry and action research. My inquiry and collaboration with her focused on the following formative assessment activities:

  1. –  students observing students solve problems in mathematics (developed and piloted during one of the teacher candidates placements in the classroom in spring 2010) and how this activity impacts their own problem-solving skills
  2. – students observers reflecting on their learning after observing the problem solving done by their peers in order to strengthen mathematical self-knowledge)
  3. – the use of  as an online collaborative “journal” and vehicle of strengthening our math community ( as well  self-knowledge in mathematics, self-confidence, communication and self-regulation)
  4. – the use of technology in the math classroom facilitate collaboration through the use of online whiteboards, such as Cosketch and Twiddla
  5. –  the use of livescribe pens ( learned from Caroline) to record the entire problem –solving process during a math class.


Caroline Rosenbloom and I co-presented at the 2012 Annual OAME conference in May and at the local TEAMS conference in October 2012. Both presentations received great feedback from the audience.

In May 2012 when the TLLP was launched I had a chance to share some of my artifacts with the teachers who had the same research focus a few of my findings related a blogging site for students: KIDBLOG.

We are currently collaborating on a project involving the use of KIDBLOG in mathematics. The progress has a new dimension: it involves another colleague of mine from school and the participation of OISE teacher candidates partnered with my students for an innovative learning and assessment initiative.

The following are a few of the questions that we intend to focus on during this project:

A. What happens when students collaborate outside school hours on solving problems, reflecting on mathematical learning, making connections and providing constructive feedback that is detailed and directly connected to the success criteria discussed in class.

B. What happens when teacher-candidates monitor the learning of a focus group of students and their collaboration as well as they provide additional descriptive feedback.

C. What are the implications for teachers pedagogical knowledge and student knowledge?

D. What impact does this collaboration have on student learning, and teacher knowledge of student performance (assessment knowledge)

E. What are the social implications for students?

F. How does this experience motivate students and builds their self-confidence, particularly when the learning experience becomes a less private  – a public experience?

For whoever reads this: Thank you for you interest and enjoy the ride!



December 29, 2012
by Mirela Ciobanu


I am a believer in the power of social networking for teachers. CollaborativeSociabilityDon’t get me wrong. I am not media savy and not even a tech junkie. But this new “hobby” can be very addictive if you are one of those teachers who want to keep up with the latest in their field from new teaching ideas, tools, virtual tools, apps, to organizational miracles meant to get rid of the paper clutter in which we can get trapped at times.

So, I thought I would share my new online discoveries and start a conversation about how they might apply to the math classroom:

Pintrest: Defined as an online pinboard, it allows people to create boards of “interests” and “pin” websites, ideas and images just as they would on a cork board. If you search “mathematics”  for instance you might feel like a kid in a candy store. You can be spending hours on end browsing from the ever increasing ocean of interests and picking the ones you have. As well, you get to add websites, pictures and start conversations around your interests with people who share them. You might also decide to follow them. Great opportunity to connect and network with other professionals.

How could this be used in a math class? Well, you can definitely use them as conversations starters during the “minds on”/warm up part of the lesson. With students old enough to get a Facebook account legally teachers might try to engage students into mathematical conversations around everything “mathematical” about our world. And what is NOT mathematical about it? Any other ideas?

KIDBLOG. I am a big fan of this site for blogging. It is absolutely safe for kids and teachers to use. I have used kidblog as a vehicle to engage students in math conversations with me and with each other for over two years now. I still use it in a very controlled way: that is I assign a mathematical question, usually a question that is rich enough that will allow for level 4 solutions ( yes, and a multitude of possibilities to answer it) and not only for the level 3 ones. I also prefer reflections or questions that invite students to generalize, conceptualize or hypothesize. Spontaneous blogging on an idea has been noticed in some students but I do invite and welcome question/problem posing and sharing of mathematical resources. During this “journey”Documenting Learning. Electronic Portfolios: Engaging Today's Students in Higher Education students are asked to post, read at least two posts and reply to them though a constructive, descriptive feedback. Consequently, the recipients must respond to the comments and try to act on the feedback.

This activity has beed incredibly successful for me and is welcome by students and parents alike. First, students love engaging in conversation with each other beyond school hours ( I know, nothing new here) but why not providing them with a forum and a channeled conversation to do so? The result? Pages of conversation on Pascal’s triangle or a graph that needed interpretation. Additionally, most students have become fluent in the language of “mathematics” and they try to increasingly refine the way in which they communicate their ideas to avoid ambiguity or misinterpretations. Communication is a tool in math. Last, but not least, it has strengthened the ties that existed between students. Learning is done through collaboration and support from peers and their teachers. This is an electronic math portfolio. I have meaningful conversations with parents who see the work done on kidblog and the improvement it leads to.

Parents enjoy seeing that the time spent by their children socializing online is dedicated to mathematical conversations and can monitor their children learning.


1. It requires a lot of time and commitment from teachers at least initially. Teachers must respond to every post and model the way students should do this. We want them to refer to the success criteria discussed in class, we must model this for them.

2. Online conduct and respect for intellectual property – also needs to be modelled. Maintaining a positive tone and one that is meant to construct not to ‘damage” is essential. If anyone seems out of line you can talk to them and definitely, if it is to late for them to edit the reply you may delete such comment.

Check my classes’ blogs:

and my last year’s blog:

Creative Commons License Photo Credit: Giulia Forsythe via Compfight studentsI have discovered that the


Creative Commons License Photo Credit: vaXzine via Compfight

December 12, 2012
by Mirela Ciobanu


4.321: You be the judge

 Students Observing Students (S O S) came into being two years ago when I was trying to find a solution to the problem of collecting enough and relevant qualitative data to track student progress. Best tool I had was the Assessment For Learning and Observation form. It allows teachers to letter code strategies they observe their students use during the lesson problem (“action” problem in a 3 part lesson) and number code their current understandings. It’s a very convenient and easy to use tool. However, I  was only able to observe snippets of each pair’s problem solving process during one class. I was missing a lot or did not get a chance to see every student’s solution or plan. I wanted a richer body of evidence so I can ensure conducting a strong consolidation during Bansho.

The idea: I looked at my class and realized that all I was doing was collecting data about their work without them ever having a chance to see their work through the perspective of an observer of the entire problem solving process and wondered what if they had this chance? We usually give them a chance to provide peer – feedback but this is about a product that is in a “finite” state. They are missing the most important aspect of problem-solving done collaboratively: the process, the understanding of the problem, the thinking and re-thinking of strategies and tools, the role the estimation or simply re-assessing the effectiveness of a strategy adopted. What about the dialogue, the negotiation and clarification of meaning that happens during the collaboration between the peers being observed? We can’t do think-alouds in math. It does not have the same efficiency as the real problem solving process taking place in front of their eyes.

So, I asked myself this question: What if the students had a chance to spend some time observing their peers solve math problems and compare the way they would have solved the problem to the solutions they were observing? What would be the benefits of such an activity? What if then, I would ask them to reflect on what they learned? This included mathematical strategies, tools, thinking process and judging the reasonableness of the answers/solutions reached. I gave my students the Assessment for Learning tool I had adapted for them (which also included reflection questions to track its benefits for the student observers as well),  and assigned them 1 pair of peers to observe as if they watched a video – no interaction, interference, just observation.

This is how Students Observing Students (SOS) came into being. 
Students Observing Students is an assessment as learning activity that I converted into a program that allows teachers not only to collect a richer body of evidence of their students’ problem solving skills and habits but it is a means of empowering students, of giving them leadership opportunities that will impact their learning. It provides them with great discussion points as they compare problem – solving strategies and reflect on their own multi- faceted learning in math.
This was the subject of an inquiry last school year , 2011-2012,  school year and this year it is part of an action research in my class as part of the TLLP  (Teacher Leadership and Learning Program) in Ontario, Canada.

My current findings  related to the benefits of this new assessment strategy impact the following broad areas:

– Development of mathematical concepts and understanding

– The process of problem solving

– Communication in mathematics ( accountable talk in math class)

– Self-regulation in mathematics

– Social interactions and collaboration in the math class

– “Consolidation” of concepts during a three part lesson ( whose voices are heard)

–  Students Listening to and Observing Students Solve Problems


Creative Commons License Photo Credit: Giulia Forsythe via Compfight

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