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.
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)
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.
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:
- Identify and use everyday mathematics knowledge to plan instruction
- Encourage and foster “math talk”
- Facilitate experiences that allow for mathematization of everyday knowledge
- Model and nurture positive attitudes, self-efficacy and engagement.
Five practical tips for creating a mathematics-rich environment:
- Use problems that have meaning for children (both practical and mathematical).
- Expect that children will invent, explain and offer critiques of their own solution strategies within a social context.
- Provide opportunities for both creative invention and practice.
- 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.
- 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.
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