Professional Literature Review: Capacity Building Series: Maximizing Student Mathematical Learning in the Early Years

Source:

http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/cbs_maximize_math_learning.pdf

Summary:
This monograph explores how educators can take advantage of students’ mathematical knowledge and experience which they bring to the classroom (p. 1). From the research, in order to establish an environment with rich mathematical opportunities, educators must understand the early mathematics learner and have a good understanding of the mathematics of teaching (p. 2).
Some key characteristics of early mathematics learners:
- students enter school with and abundance of mathematical understandings
- play is “the gateway to engaging in mathematical inquiry... and need not complete for time in the classroom [with other subjects as it should be embedded into the play itself].
-- play that involves mathematics & play with mathematics itself (p. 2)
- manipulatives are great tools to help students build and explain their understanding
- teachers need to allow time for students to reflect and discuss their play in order to make connections between their mathematical ideas and how they are represented in play.

Teachers’ mathematical knowledge is actually linked to student achievement.

“Teachers need to be able to reason through and justify why certain procedures and properties hold true, to talk about how mathematical language is used, to see the connections between mathematical ideas and to understand how they build upon one another” (p. 3).  Even early math concepts of number sense and numeration are actually very complex and it is very important to understand these complexities in order to be able to teach the mathematical skills.

This monograph then identifies some general suggestions and practical tips which I have included at the end of this review. The key ideas are that math is embedded into the day and across the curriculum subjects, that students are encouraged to talk and learn the language to communicate their mathematical thinking, and teachers should be promoting positive attitudes towards math.

Ontario Mathematics Kindergarten Curriculum Connections:
The Ontario Full-Day Kindergarten Curriculum section on Mathematics links directly to the idea that early learners bring prior math skills which need to be further developed in every-day life activities at their level: “Young children use mathematics intuitively and develop their understanding of mathematics through their individual approaches to learning, as well as through their prior experience of their linguistic, family, cultural, and community backgrounds. It is therefore important that children’s existing conceptual understanding of mathematics be valued and that children be introduced to mathematical concepts in an appropriate manner and at an appropriate time in their development... Rich mathematical problems involve important mathematical ideas and arise out of real-life situations, and can be approached in a variety of ways so that all children can be involved in exploring solutions. Solving such mathematical problems requires persistence, since they do not have one easy-to-find correct answer. Through active participation in mathematics investigations, including problem solving and discussions, children develop their ability to use mathematics as a way of making sense out of their daily experiences.” (p. 92 ELKP)

Another aspect in the monograph is that math needs to be integrated across subjects and each concept should not be compartmentalized (p. 7). “When developing their Full-Day Early Learning–Kindergarten mathematics program from this document, Early Learning–Kindergarten (EL–K) teams* are expected to weave together the mathematical processes and related expectations from the five mathematics categories, as well as relevant expectations from other areas of learning (e.g., science and technology, language, the arts). It is important that the study of various aspects of everyday life should permeate young children’s mathematical experiences... Children demonstrate their understanding of these counting concepts in all five areas of mathematics – for example, a child might demonstrate his or her understanding of one-to-one correspondence while analysing data on a graph made by the class.” (p. 93 ELKP).

Personal Reflection:

• What resonated with you?
  Teachers’ mathematical knowledge is actually linked to student achievement. I did not find this surprising at all. I find, in fact, this can be said for any subject matter. If a teacher is knowledgeable and confident, students will ultimately be more engaged and thus learn more. And of course it is difficult to teach something if one is lacking the knowledge/skill/confidence.

“As educators observe student problem solving, they can document what children say, do and represent in order to make both planned and “in-the-moment” decisions about how to respond, challenge and extend student thinking” (p. 4). This notion of observing students and guiding their learning by posing appropriate questions and bringing in the mathematical learning and language at the same time, this is exactly what we are doing in full-day kindergarten play-based programming. And by using pedagogical documentation, I am becoming very aware of my own thought processing and the language and questions that I am using to help guide students in the moment and make both their and my learning visible.

The suggestion to model and encourage “Math Talk” also resonated with me since my colleagues and I are conducting a Number Talks for the first time and inquiring into their application and influence on students. Language and communication is extremely important and therefore I fully agree that fostering an environment and encouraging students to discuss their mathematical thinking is imperative if they are going to be able to successfully solve problems in the real-world. Consolidation time must be integrated into the day to regularly give students a chance to articulate their mathematical thinking and assist them in making connections within their strategies and solutions and make more generalizations (p.3). “Suzanne Chapin proposes five effective talk moves which help create meaningful mathematics discussions: revoicing, repeating, reasoning, adding on, and waiting” (p. 4 & 5). These are also excellent practical tips to use in “Math Talk”.

• What challenged you?
  I found it very interesting that “early mathematics skills were more powerful predictors of later academic achievement in both mathematics and reading than attentional, socioemotional or reading skills” (p. 1).  I always thought attentional skills had a high impact on student success. I am not surprised that early math skills are indicative of future achievement but I would not have thought they are more powerful predictors than reading level. However, upon reflection, it makes sense that early math skills are powerful because math is about problem-solving and this permeates all subjects both in the world of academia and in life. Thus, this fact should really influence our teaching and place a high priority on the importance of guiding students to further develop their mathematical knowledge, understanding and communication skills related to solving everyday problems tailored to their interests and developmental level.

I wasn’t surprised to hear that “while technology was being used in many classrooms, its potential to promote student thinking was being greatly underutilized” (p.2). According to Yelland and Kilderry, many computer tasks in primary classrooms had the equivalent complexity of “pencil and paper mathematics tasks, were teacher-directed and anticipated correct answers with narrow solution strategies” (p. 3). There are so many computer games that focus on one particular question with only one possible answer and seem to be the drill type of questions we are supposed to be moving away from. This type of activity is mindless and rote and without any real-life context.
It is recommended that educators should offer activities that are “multidimensional mathematical tasks ensuring both student input into the direction of their learning and supporting more varied learning outcomes (p.3). These types of activities are certainly ideal but most definitely challenging to invent and unfortunately, this monograph does not offer any examples. I must admit, even many of the technology resources that myself and my classmates discovered were mostly drill type of games.
I read an article on the SAMR model in the last year, which I roughly recall but it commented on the stages of using technology to innovate our education. Providing students with the opportunity to use technology to complete a task in an alternative way was on level of using technology but to actually invent a task that could not otherwise exist without technology is truly the most innovative way and I would love to know how to do this in the kindergarten classroom and link it to ‘playing with math’. I find we tend to resort to these simple, one-solution type of games for children since it is fast and easy for students to use. It is much more time-consuming to actually teach early primary learners how to use the technology in order to show their thinking. It think this would be the necessary step in order to provide students with opportunities to use technology to represent their mathematical thinking. Perhaps this might be a good inquiry for myself: to teach students at least one means of using technology to represent their mathematical thinking. I could present it to a small group and see how it goes.

• How might this text be helpful/useful in our teaching practice?
  This text gives many important suggestions and tips to get started such as immersing yourself in the curriculum and reading up on the grades below and above your current grade. It is also highly recommended to study these mathematical ideas with colleagues and “together inquire about how your understanding impacts your related teaching” (p. 4). Hence, why there are so many collaborative inquiries being conducted and funded by ministry programs.

Four main suggestions in order to improve student learning and achievement:

1. Identify and use everyday mathematics knowledge to plan instruction
2. Encourage and foster “math talk”
3. Facilitate experiences that allow for mathematization of everyday knowledge
4. Model and nurture positive attitudes, self-efficacy and engagement.

Five practical tips for creating a mathematics-rich environment:

1. Use problems that have meaning for children (both practical and mathematical).
2. Expect that children will invent, explain and offer critiques of their own solution strategies within a social context.
3. Provide opportunities for both creative invention and practice.
4. Encourage and support children by providing carefully scaffolded opportunities which allow them to deepen their understanding and use meaningful and elegant solution strategies and confidently engage in the mathematics.
5. Help children see connections between various types of knowledge and topics, with the goal of having each child build a well-structured coherent knowledge of mathematics.

Professional Literature Review: Capacity Building Series (K-2) : Pedagogical Documentation

Source:
http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_Pedagogical.pdf

Summary:
Pedagogical documentation is explored in this piece of literature as a means of finding a deeper connection to students’ thinking and deepen the assessment feedback loop. It draws on inquiries from the Early Primary Collaborative inquiry (EPCI), an Ontario Ministry Program. Even though it is a primary-age based inquiry, the suggestions are useful for the junior grades as well.

The monograph suggests give main benefits of integrating pedagogical documentation into pedagogy:

1. Create shared understanding
2. Celebrate rights of individual learners
3. Recognize student ownership of knowledge
4. Actualize shared responsibility
5. Provide voice in learning for everyone

It also suggests tips for getting started:

1. Use a collaborative inquiry approach
2. Establish your primary purpose
3. Think About your audience
4. Decide on your methods for collecting information
5. Remember - There is no one right way to do it!

Other key ideas:

• Pedagogical documentation is an ongoing, cyclical process (assessment for and as learning, vs. of)
• It “needs to become a habit of classroom practice... device... for documenting... need to be available at all times to capture learning when and where it happens” (p. 5)
• Embrace it as an attitude of knowing versus a technical process/product
• Can transform education in the early years and beyond

Ontario Mathematics Kindergarten Curriculum Connections:
“Children bring with them an intuitive knowledge of mathematics, which they have developed through curiosity about their physical world and through real-life experiences.” (p. 20 Ontario Full-Day Early Learning Kindergarten Program-ELKP) Pedagogical documentation allows educators to uncover the prior mathematics knowledge that students bring with them and continue to encourage this natural curiosity about their surroundings as time goes by, not only with their interests in mind but also with their skill-level. By setting up the classroom as a play-based setting to further propel their curiosity, teachers reflect on their observations of students and continue to uncover their interests as they change, in order to plan for future materials that will engage students’ curiosity and naturally compel them to practice certain skills. “The Early Learning–Kindergarten team plays a critical role in fostering a positive attitude towards mathematics by valuing a child’s early attempts at problem solving, by sharing and celebrating the child’s learning, and by encouraging in each child a love of mathematics.” (p. 20 ELKP). By documenting students’ mathematical learning and sharing it with other students, students take pride in their mathematical abilities at an early age and develop a positive mind-set. “They should also be encouraged to begin to represent their mathematical understandings in ways that are meaningful to them” (p. 21 ELKP) By documenting students’ thinking in various ways, educators allow for different representations of learning to occur within the classroom and display all different kinds of student voice. Pedagogical documentation is significantly important in the Kindergarten classroom across several domains and as learning is encouraged in an integrated fashion across domains, mathematics can be drawn into any play area through educators’ guidance.

Personal Reflection:

• What resonated with you?

My “ah-ha” moments while reading this monograph were more like “oh yeah, that’s so true” moments. I found what I was reading was not anything particularly remarkable, but rather clarified and reinforced what I already knew about pedagogical documentation.

“In school teams, learning communities and ministry programs, such as...the Collaborative Inquiry for Learning Mathematics (CILM)... they are experimenting with innovative ways to observe student learning, study student work and gain greater insight into effective instruction”.

This phrase resonated with me since I am working with a school team on two different mathematical inquiries: one for this class and the other as part of the ministry program. This clearly outlines our main purpose for the inquiry and makes me re-evaluate the focus of my math inquiry with my other team. I believe that perhaps it is the exact same goal: to gain insight into effective instruction of mathematics by experimenting with Number Talks.

This phrase really struck me: “It is not only our students’ thinking that becomes visible through pedagogical documentation but our thinking as educators because visible as well...” (p. 2). I was so focused on students’ thinking that I forgot that as I document my process of guiding them, I am also showing my teaching practices and thinking processes.

A few other important points addressed in the monograph that I strongly agree with:
“Through documentation, the potential for engagement and learning increases because students reflect on their learning throughout the process rather than at the end... Students... become partners in the process of determining how best to express their ideas” (p. 4). When students are given the opportunity to be part of their learning process, engagement increases. As noted later besides ownership over their learning, documentation provides voice in learning for everyone: “Students and teachers alike are demonstrating ownership of and engaging in teaching and learning... bridges understanding of children and adults” (p. 4).

Giving students the chance to also play the role of the teacher is very powerful. Recently I wanted to assess how high students could rote count in French. I decided to just go up and ask them to count for me, at any time during play / table top activities. As soon as one student was counting and saw me giving praise, others were observing and were very eager to be next to show me how high they could count. Then, as students complete activities and continue to practice their counting, I am able to partner up accordingly those that strong and can play the teacher role for those students that are weaker at a certain skill.

• What challenged you?

“Pedagogical documentation stops the train of standardized expectations and slows down our thinking processes to consider some topic with exquisite care” (Wien, Guyevskey & Berdoussis, 2011) (p. 2). Clearly, it is a time-consuming process to write learning stories and it cannot be done for all students all the time. The part that I find challenging, which is not addressed in this monograph is how to conduct this documentation for all students and how to decide which students and topics to focus on. It is suggested to become an attitude/habit. But how are we to do this in a busy classroom with only 5 hours a day? How do we use pedagogical documentation and reach every student? The only suggestion to solving the overwhelming feeling that educators face when it comes to documenting is that it be a collaborative approach and that teachers should be working in teams. “...it is the team’s collective reflection on and analysis of the evidence which deepens understanding” (p. 6). Yes, working on a team is helpful and helps us find a focus/purpose and decide which students to observe and how to analyze student-work. But what about all the other students? What is the best system/approach to reach them all? Is it possible to use pedagogical documentation all the time for all topics? I highly doubt it is. I already found it so daunting that I was only able to have a few students complete our chosen task for our collaborative math inquiry. It is certainly very interesting and very engaging and worthwhile but I am still in search of how to do so efficiently and how to do it systematically in the classroom and make it a habit.

• How might this text be helpful/useful in our teaching practice?

One of the purposes of the inquiry is to develop effective instruction and this is done in pedagogical documentation by creating a shared understanding between educators, students, and parents. Through pedagogical documentation we are able to provide parents will more precise feedback on their child’s learning which will ultimately help the parents better understand and engage in their children’s learning. “Documentation encourages educators to step back to listen and allow the child to take the lead in the learning, inviting students into the learning process” (p. 3). This documentation is also tremendously helpful for planning purposes as educators not only gain insight into student interests but also better understand students’ development needs (p. 3).

Number Talks - week 3

After reflecting on our first couple weeks of Number Talks, this week we decided to make some changes. First, we thought it would be helpful to try doing them in small groups during table tops to help those that had trouble focusing. This time, I made sure to only briefly show the cards as my teaching partner was already doing so. To refocus on numbers versus pictures we also changed the question from “What do you see?” to “How many dots?”.  Since we started them in November we decided to review the first four strings of cards for the number three. We recorded their answers, and finally, we discussed them.  The results were very interesting.

Since I am the French teacher I was asking them in French "Combien de points?" I recorded their answers in the charts below, noting if they replied in English. We started the Number Talk sitting at a table but I found it distracting and moved the group onto the carpet. Unfortunately I can't recall if we moved for string 2 or string 3.

String 1 – Number 3
Student	Card A	Card B	Card C	Card D
A	1	2	3 (English) I encouraged in French "1, 2, 3"	1, 2, 3 (Counted but card wasn`t in front of her)
B	1	No answer (didn`t see card?)	3 (English)	3 (English)
C	5 (English)	2 (English)	5 (English)	12 (English)
D	No Answer	No answer	(showed 3 fingers)	(showed 3 fingers)

String 2 – Number 3
Student	Card A	Card B	Card C	Card D
A	2	3 fingers	3 (English)	3 (English)
B	2 fingers	3 fingers	3 fingers (encouraged to speak… 3 English)	3
C	2	10 (English)	8 (English)	8 (English)
D	2	No answer	No answer	No answer

I recorded their answers for strings 1 and 2 but found it unnecessary to continue as that was enough data to give us a sense of their thinking. Both Students A & B seem to have a strong sense of the number of dots but still must count from one in French.  found they qucly became unnterested for strngs 3 and 4 as t was not challengng enough so  ased them to show dfferent ways wth ther fngers. Student D was very hesitant and shy to answer but began to hold up three fingers. She may have been inimidated working with the others or speaking in French. I would like to redo the Number Talk on-one-on with Student D. Student C gave random answers and will clearly need extra assistance and practice to develop his number sense. This student`s thinking is revealed below during our discusson.

Here is an account of our conversation after completing the Number Talk:

Teacher: What do you notice about the cards?
Student A: They`re all the same.
Student C: No.
Teacher: How are they the same?
Student A: They`re all sets of 3.
Teacher: Does everyone agree with student A?
Students B & D: Nodded.
Student C: They`re not all the same to me.
Teacher: (Spread out the cards) Look at the cards. How are they different?
Student B: They`re in different ways.
Student C: Different ways.
Teacher: How are they different to you (student D)?
Student C: Mumbled.
Teacher: Asked to repeat but he was shy and needed encouragement.
Studen C: One is down and one is up.

Possible Next Steps:

• Further discuss with English Partner on Monday to see her results.
• Continue with small group Number Talks to assess students and make groupings for future learning
• Read through Kindergarten chapter on Number Talks for more ideas
• Some possible suggestions from my English partner who was concerned they are just expectng the same answer over and over:
• move onto number 4
• Mix the cards up from 1-4