"One of the most powerful moves a teacher or parent can make is in changing the messages they give about mistakes and wrong answers in mathematics." (p. 15)
"What is mathematics, really? And why do so many students either hate it or fear it - or both? Mathematics is different from other subjects, not because it is right or wrong, as many people would say, but because it is taught in ways that are not used by other subject teachers, and people hold beliefs about mathematics that they do not hold about other subjects. Students rarely think that they are in math classrooms to appreciate the beauty of mathematics, to ask deep questions, to explore the rich set of connections that make up the subject, or even to learn about the applicability of the subject; they think they are in math classrooms to perform." (p. 21)
"Teachers are the most important resource for students. They are the ones who can create exciting mathematics environments, give students the positive messages they need, and take any math task and make it one that piques students' curiosity and interest." (p. 57)
If educators enjoy and appreciate math, they will model this to children who will then think positively about their mathematical experiences as well. Educators can also use this new attitude to help change those of their students' families and the greater community.
Spending time creating an environment that is a community of learners who value each other as individuals who strengthen the group with their involvement and participation is essential. We learn better when we are together and it's okay to make mistakes because these are learning opportunities.
Many mentor texts helped us explore this idea with the children and consider what perseverance and resilience might look like in our classroom.
Boaler asks educators to consider the following reflective questions to help us move from product-driven, isolated, performance-based tasks to rich, collaborative, meaning mathematics for children. In my opinion these compliment the beliefs that many teachers also hold about developmentally appropriate, emergent, play-based practices leading them to be the perfect way to engage children in rich, robust math experiences in kindergarten. They naturally focus on process-based explorations and multiple ways of doing and knowing. They ask us to consider math from multiple perspectives and support and enrich content through meaningful ways of wondering about the world.
When considering math in the classroom Boaler specifically asks:
1. "Can you open the task to encourage multiple methods, pathways, and representations?" (p.77)
We naturally do this in an emergent, play-based classroom as we consider differentiated instruction and how to engage children in multi-faceted experiences that support preferred learning styles. Those influenced by Reggio Emilia think about the 'hundred languages of children' and how we can encourage and support children to explore a topic and also communicate their ideas and learning in new and innovative ways (e.g., drawing, painting, building, dancing). In our classroom we found that the loose parts, building, and art areas were rich with mathematical possibilities and because many children favoured these experiences, they also frequently explored and enjoyed math there as well.
2. "Can you make it an inquiry task?" (p. 78)
Inquiry is the heart of learning in the FDK, emergent curriculum classroom! When we honour children's questions and consider how to support their explorations we empower them as learners and members of an inquisitive community. There were many inquiries in our classroom this year, and reflecting back on each I can see the math potential and just how motivated children were to answer self-directed areas of wondering. Look for how to weave math naturally into an inquiry, instead of presenting artificial math situations in order to empower your learners and grow their mindset.
One of our richest inquiries was exploring how to help a local food bank that was in desperate need of items for the community. The rich math and language learning that emerged in this project is captured in the following video and article.
Kindergarteners Building Community One Can at a Time
3. "Can you ask the problem before teaching the method?" (p. 81)
Boaler suggests that the richest math teaching happens when there is a need to know how to solve the problem, instead of children being told math rules and procedures and then assigned practice questions out of context. In our room we are always open to using these 'teachable math moments'. We conduct daily number talks, usually beginning with our sign-in question, and the different ways children wonder about how to arrive at a solution helps us meaningfully embed strategies into the experience. Playing along with children at the learning centres during playtime is also a very important way to accomplish this. As we play with children, we discuss what we are doing and are able and available to support and scaffold rich math learning as problems in the play arise (e.g., how to build a tower that doesn't fall over).
4. "Can you add a visual component?" (p.81)
Because kindergarten children are emerging readers and writers, much of what we do in the classroom uses many other languages (e.g., drawing, painting, dancing, sculpting, building). Our play activities naturally differentiate content and provide multiple entry points for children to use when exploring and communicating their discoveries to others. Math happens everywhere, not just the 'math centre'.
5. "Can you make it low floor and high ceiling?" (p. 84)
This is one of the most interesting things I learned from the book. I strive to differentiate activities to best meet children's needs and support their emerging understandings, but I really liked how Boaler described tasks that had multiple entry points for children and endless potential as this. When I'm designing an activity to use during whole or small group time, or to offer as an interesting math activity or provocation during play, I will consider it a pedagogical success only if it's low floor high ceiling. Here's an example of one:
Children have a basket of subitizing cards (numbers 1 - 10) and are encouraged to match them on a number line with the numbers (10 - 20). They can do a straight match (10 on the 10 spot), or add the numbers in interesting combinations (2, 3, or even 4 numbers to total a sum as represented on the number line). They can even think in terms of multiplication (3 groups of 6 can go on the 18).
With an emphasis on exploring and representing math in different ways, it only makes sense that our assessment should reflect the unique interests and approaches of the children and honour and celebrate their stories. According to Boaler:
"The complex ways in which children understand mathematics are fascinating to me. Students ask questions, see ideas, draw representations, connect methods, justify, and reason in all sorts of different ways. But recent years have seen all of these different nuanced complexities of student understanding reduced to single numbers and letters that are used to judge students' worth. Teachers are encouraged to test and grade students, to a ridiculous and damaging degree; and students start to define themselves-and mathematics-in terms of letters and numbers. Such crude representations of understanding not only fail to adequately describe children's knowledge, in many cases they misrepresent it." (p.141)
We have tried to observe, document and honour children's experiences in a variety of ways in order to make these visible to others. We have also used this documentation to help our own emerging understandings of the children's interests, strengths and needs and help drive our instruction forward as we better support the children's questions.
We have used annotated photos and portfolios of children's work samples (art, drawings, writing, etc.). Each time we create an annotated photo story we send a copy home at the end of the day so that our families can be informed as to what has happened at school. This is especially important as some do not follow our twitter or blog and the portfolios are only sent home a few times a year. We want the rich, descriptive feedback to be ongoing and celebrated by families as well in order to continue to promote an interest in math and grow everyone's mathematical mindset.
Especially with bigger math explorations we like to keep a running record on our easel of our math thinking and colour code the text and pictures to help us make sense of our journey. We continually refer back to this documentation and add/remove ideas as needed.
We build the documentation directly into our centres - photos, artwork, writing, physical artifacts from the inquiries, and many other representations of our ideas are woven throughout the classroom.
Photos and writing on our walls show our journey and the progression of our ideas. We keep empty space available so we can continue to add to it as we progress on the journey.
Consolidating our understanding, sharing what we have done and learned with a wide audience, and connecting with others in order to learn from them is a major part of our documentation. The children like to blog and tweet together with me and we've been able to connect with many classrooms from all over the province. It's exciting to know that there are others who are on the same journey as us, and as Boaler states in her book, math is a collaborative and social experience and we learn best when we work together with other like-minded individuals!
The ending of Mathematical Mindsets was most powerful for me, and I'm going to revisit this paragraph many more times as I continue to expand my own mindset and continue to explore how to better implement meaningful math in an emergent program. I would love to continue to connect with other educators interested in the same!
"Teachers, parents and leaders have the opportunity to set students on a growth mindset mathematics pathway that will bring them greater accomplishment, happiness, and feelings of self-worth throughout their lives. We need to free our young people from the crippling idea that they must not fail, that they cannot mess up, that only some students can be good at math, and that success should be easy and not involve effort. We need to introduce them to creative, beautiful mathematics that allows them to ask questions that have not been asked, and to think of ideas that go beyond traditional and imaginary boundaries." (p.208)