News
Entertainment
Science & Technology
Life
Culture & Art
Hobbies
News
Entertainment
Science & Technology
Culture & Art
Hobbies
Decimals and Place Value Mats – Educating Now
I was invited to come into Karen Fallan’s grade 4/5 classroom to kick off their decimal exploration, so I spent about an hour with this lovely bunch of students (and teachers) on Monday morning. Karen’s teaching partner, James, came to observe as he had some remedy time (I love it when teachers have the chance to watch each other- such a great opportunity to learn from each other!). Karen starts her day off each morning with a mathematical ‘check-in task’ and today it was a fantastic question that elicited some really interesting student insights and thoughts. Her question is pictured below and then students have their name on a magnet that they place under what they think is the correct answer: There was about the same number of kids who put their names under 230 and 300 and only one put it under 370. The discussions were so valuable because they showed us what they understood (not necessarily correctly understood) or assumed about the task and the math within it. This is such a fabulous way to start the school day -thinking, mathematically reasoning, communicating with each other, trying various strategies to understand others’ ideas. I encourage you to give it a try! Karen told me that at the beginning of the year some students would just put their name wherever without much thought but it didn’t take long before they started discussing and debating with each other and now they really dig into it and feel strongly about their ideas and so want to share. Next, I asked the students to discuss the following: “What is a decimal?” “What does it mean?” “When do we use decimals, other than in math class?” Again, the answers were varied and insightful. Many students used money and shopping as examples of where we use it and one even said weight! Only one student connected decimals to fractions and percentages and could explain clearly what they were. Most of the grade 4 students said they had no idea what they were or what they meant, which is expected because this is their first introduction to decimals. Then, we gave them their base 10 blocks and the Educating Now Place Value Mats and allowed 3 minutes of play-time (more on the Place Value Mats below). From here we followed the introduction to decimals lessons you’ll find on Educating Now. Essentially, we labeled the large flat block as 1 whole and then asked students to name the stick and the little cube and explain why. The place value mats really came in handy to connect the words with the size of the blocks and the symbols of decimals. There is a lot going on when we consider decimal numbers. We are now talking about numbers less than 1 whole but that are still in the base 10 number system. There is nothing about 0.1 that suggests it be called ‘one tenth’ AND one tenth is bigger than one hundredth! All of these can be very confusing for students so we found the mats along with the blocks very helpful for all of us. We also used contexts to help us understand decimals, like sharing an amount of money or food with 10 people versus sharing the same thing with 100 people and money (yes, we used pennies because they are still important). There were some ‘Ah-ha’ moments when we used contexts. I wrote some numbers on the board like 2.5 and 2.05 and students had to create them with their blocks on their place value mats and say it properly using place value language. “Two and five tenths” rather than “two point five”. We discussed why we need to use language and by the end of the class most of the students had a good grasp of it. I noticed at least 4 who were not there yet, however, they were engaged, trying, and working towards understanding. Two students continued to see them as whole numbers, completely ignoring the decimal but the mats and blocks helped us support them in understanding the numbers as decimals. Throughout the lesson I had students stand up to vote for things, or just to indicate that they’d discussed with their partner what I’d asked them to discuss. At one point a student said, “I really like that you keep getting us to stand up and move!”. Movement is so important for learning and it doesn’t have to be a lot of movement to be effective! After I left, Karen continued with teaching about decimals. She moved into benchmarking and rounding the decimals to the nearest whole or half (using the blocks and place value mats) and giving the students the numbers in words and asking students to build and write them numerically, for example, “three and sixty-five hundredths”. For the last 5 minutes of the lesson, we reflected on what helped us to understand decimals the most in that class. Most of the students said the mats and blocks and many also said the contexts of money that we used throughout the lesson. Some said working with their partners and a few said writing it down helped them. A few students wrote their own notes as they were trying to figure out how to say the numbers but I didn’t instruct them to write anything down for this lesson but hearing those reflections I would definitely do some writing next class. This is the power of reflections! They are fantastic formative assessments that will guide your practice so that you can meet your students’ learning needs best. Here are Karen’s own words about our lesson: “I feel so lucky to have been a part of many workshops put on by Nikki over the years. I am pretty certain my class wouldn't be loving math like they do now if it wasn't for the learning I have done from Nikki. I am constantly trying to bring in the lessons that I have learned from Nikki, into my classroom and I was overjoyed to have Nikki come into my classroom so that I could see her in action with my class. Here are my takeaways from that amazing experience: General Ideas: -Get the students standing when possible. When they have completed their task, either have them stand up or put their hand on their head, etc. -Use tools like a fair jar to call on students (instead of the same students volunteering). Students can say "I don't know yet", but they need to know that as a teacher, I will be checking in after to make sure they are trying to figure it out. -After one student has shown their work on the board, have another student explain what that student did -Do brain gym activities! Example from Nikki: Have your hands reach over to the other ear and touch that ear. Then swap. Count out loud, doing this 8-10 times. -Do partner talk as much as possible. Ideas connected to teaching numbers with decimals: -ALWAYS say a number with a decimal like it is written (ie 3 AND 40 hundredths) -Use place value mats and base 10 blocks for initial lessons on decimals. Students really buy in. If possible, also use the magnetic base 10's so that a student can show on the board what they have done. -Connect right away to concepts they know such as slices of pizza and money! Like my students, I look forward to math class daily. It is a great way to start our morning and I love seeing all the connections my students can make. I'm excited to keep adding to my math 'tool kit' of ideas as I keep working with Nikki and other colleagues to figure out the best way to support math learners!” Karen’s EA, Quincy, waited around to let me know that this is the first class he’s EVER worked in where when the teacher says; “Ok, it’s time to finish math” the class groans and when she says; “Ok, we can have another 10 minutes of math time” the class cheers! This is exactly why I’m so passionate about teaching math this way – it completely transforms a subject that has caused so much anxiety and fear into one that is exciting and fun and something that children actually want to do! Thank you to Karen and her class for allowing me to be a part of their learning community. It inspired me immensely in just 70 minutes! Place Value Mats Check out our new custom-made Place value mats! We just created a store on our site and are offering introductory pricing to get teachers using them and providing feedback. We have 3 different versions: I first used these mats during my summer camp and found them really useful in grades 4-8 (they reduce the noise of the blocks too, which is a great bonus!). The version on the far left is for primary students, I have not yet had a chance to pilot it in a primary class. The middle version for intermediate grades and the far right version for middle school grades.
PhD Series – Speaking Mathematically – Educating Now
Speaking Mathematically: Communication in Mathematics Classrooms – Chapter 4 (from this book) (1987), by David Pimm This is a blog post I did as part of my PhD studies that I am sharing here on Educating Now, it is more academic than my typical blog posts - Nikki Chapter Summary: In this chapter Pimm explores different linguistic aspects of mathematics; mathematics register, words and expressions that may cause confusion but mostly he focuses on metaphors including extra-mathematical metaphors and structural metaphors. Pimm builds on Halliday's explanation of mathematical register by summarizing that "registers have to do with the social usage of particular words and expressions, ways of talking but also ways of meaning" (p. 108). He discusses what mathematical register can include as well as provides examples of the different ways mathematical registers are created, including borrowing from everyday English (for example: face, degree, relation, power, etc.). he goes on to share examples of where ambiguity can create confusion and then further explores this idea by examining the effects of borrowing terms but using them in grammatically different ways. In this section he uses the example of diagonal and how it can be interpreted as a line that is not horizontal nor vertical versus the accepted definition of a diagonal of a polygon. Pimm further explores register confusion by more closely examining how and when misunderstandings arise in children and provides more examples that might be beneficial for teachers to know. Pimm argues that metaphor is "central to the development of the mathematics register" (p. 109) and that understanding of the processes will benefit math education. Below are the mathematical definitions of diagonal: https://www.chegg.com/homework-help/number-diagonals-diagonal-polygon-line-segment-connects-nona-chapter-1-problem-8t-solution-9781285454221-exc When asking students how many diagonals they wrote this because they interpreted diagonal to mean how many sides are slanted (neither horizontal nor vertical): This is an important reminder that we need to ensure students are understanding our intended meanings when we talk about math (visuals can help with this!). My reflections: 1) Metaphors in mathematical language - I feel embarrassed to admit that I hadn't 'crossed over' my own understanding of metaphor from literature to math. I also realized that I don't do a good job of articulating the fact that I'm using metaphors. The example that resonated with me was in thinking about an equation as a balance. I use this imagery, especially that of a teeter totter, often but have never gone further to accentuate that this is a metaphor and that an equation isn't literally a balance. Pimm suggests that we must be explicit in explaining that a metaphor is being used as a conceptual bridge. I appreciate that he suggests having students create their own metaphors for mathematical ideas as this would make it easier for students to see the metaphor as a tool. I also was thinking about English language learners and how our use of metaphors, similar to using more informal language, is culture-based and therefore not necessarily transferable to all/most students. Pimm does state that using our own (teacher's own) metaphor may not be understood by students but it can still benefit students by modelling the process of 'image-making' in mathematics. 2) I made many connections with this article and started creating a list in the margins of words and phrases that are confusing for students (improper fraction = there's something wrong with it, even numbers = whole numbers) as well as the metaphors I use (many had gone unexamined until reading this article). I feel that the use of metaphors is certainly powerful but am now encouraged to go the extra step in explaining that it is in fact a metaphor.
Incorporating the First Peoples Principles of Learning (FPPL) – Educating Now
In this blog I’m going to continue to explore some of the ways we can incorporate the First Peoples Principles of Learning (FPPL) into our daily math lessons. Last blog, we looked at incorporating experiential learning and multiple strategies and so I’d like to continue to unpack this principle:Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place). Holistic:My ethnomathematics experiences have taught me the value of holistic learning. I consider holistic learning to include learning in authentic contexts as well as learning that is centered on the wellbeing of the person and their communities. Learning in authentic contexts is integrated or cross-curricular learning (can also be problem-based or project-based learning). In terms of learning that is centered on the well-being of the learner, I think about the different worldviews and value systems as well as choice, being outdoors and moving while learning. Examples:a) This lesson centered on the Gorge Waterway incorporates different subject areas such as Social Studies, Science, Math, English Language Arts and focuses on the importance of the waterway for the Lekwungen peoples and all of us who live in Victoria. b) As a class discuss some problems that are specific to your community (classroom, whole school, surrounding community) and then students will work in groups to solve them or at least understand the problems in more complex ways. An example for teachers at Cedar Hill is to approach the owners or managers at Fairways grocery stores and discuss food waste while also looking at the statistics for how many folks in Victoria use food banks. Students can use different charts, maybe percentages and will likely perform many other operations as they determine the total food waste per day, week, month, year, etc. Ideally, they’ll be able to provide some ideas for reducing food waste and supporting those who don’t have enough to eat. Reflexive and Reflective:I’ve had a lot of experience with reflective learning and teaching but reflexive was a new term to me. I’ve found different definitions but have found the following ones useful: Reflexive:When we encourage students to be self-reflexive, we are asking them to understand what they are learning as they are learning. Additionally, self-reflexivity not only allows students to understand what they learned but why they learned it.(https://www.chronicle.com/blogs/profhacker/reflexive-pedagogy/22939) Reflective:Teachers who promote reflective classrooms ensure that students are fully engaged in the process of making meaning. They organize instruction so that students are the producers, not just the consumers, of knowledge. To best guide children in the habits of reflection, these teachers approach their role as that of "facilitator of meaning making."(http://www.ascd.org/publications/books/108008/chapters/Learning-Through-Reflection.aspx) -I consider both of these important aspects in developing student agency – helping students to be the drivers of their own education, rather than passive receivers of information. Examples:a) As students are problem solving stop them periodically to check-in. Ask them to reflect on what they are doing and have done so far: “what’s working?” “what’s not working?” “Do you understand why you are doing?” “Do you need to think about it differently?” “Could you think about it differently or solve it in a different way?” “What could help you in this moment (picture, discussion with others, tools, etc.)?”b) Students can keep a learning journal or keep their own notes to document their current understanding of math concepts and as they are doing this they can be asking themselves “if I read this in a month, will I know what I’m talking about?” They can use pictures, analogies, examples, or anything else that will help them to communicate their understanding.c) Use contexts often – this helps with the ‘why are we learning this’. When students solve problems rooted in contexts they see it as useful and meaningful.d) Ask “WHY?” all the time. Annoyingly often. We want our students to get into the habit of proving and justifying their ideas. You might also give sentence stems (on a poster) for students to use (especially helpful for ELL and FRIM learners):“My estimate is because ______”“I think the answer is ______ because ______”“I used (this method) to solve the problem because ______” In my experience it does take some time to get students into this habit and at first, they don’t like the “becauses”, which I understand – it’s challenging to explain our thinking at times but it’s also really importante) Do daily reflections at the end of every lesson. I recommend leaving 4-7 minutes for this. I usually do this orally for a few reasons: it’s faster, easier for most students and I like the students to hear each other’s reflections. I usually ask students to rate their level of understanding of the concept with their fingers (1-4) as well as a verbal reflection in answer to a question specific to HOW we learned such as “how did using the base 10 blocks affect your understanding of place value?” or more general questions like “what was the most challenging aspect of today’s lesson?” “What mistake helped your learning today (their own or someone else’s)?” etc. I also feel the need to reiterate the fundamental message from my first blog in this series: relational learning is key! If we truly want to use the FPPL then we must start by building respectful relationships with our students and also we MUST look for each student’s strengths and gifts, rather than viewing them through a deficit lens. I first heard the expression “bias of lowered expectations” used at a FNESC conference a few years ago and it hit me like a bus because I knew I was guilty of it. When we are looking at our students through our own set of values we may not even see their gifts because they are not what we might consider academic or ‘school related’ gifts. I wonder what would happen if we found their strengths and incorporated them into our teaching. Relational learning means that we learn from each other. There is a reciprocity of learning and teaching. This is also a way to share power with your students. Search for ways that you can learn from your students – I really think you might be surprised at what you learn and also how this relationship dynamic changes how students show up for learning. Lastly, I want to encourage you to really listen to your students’ mathematical thinking. It may seem wrong or illogical at first but be curious about their ideas, rather than being on the hunt for the ‘right answer’. If we can meet our students where they are at, in terms of their understanding, we can better support them in moving forward. I’ve often had to ask students to repeat their thoughts 2 or 3 times as I try to make sense of it and understand it and this act shows them that their ideas are worthy of talking about and thinking about.
My New Learning Journey: Why Culture Matters in Mathematics Education – Educating Now
As I’ve been working through my post graduate certificate in ethnomathematics, I’ve been deep diving into all sorts of new learning, especially how culture impacts not only mathematics but also the teaching and learning of mathematics. I still have so much to learn but I’d like to share my learning journey so far in the hopes that it may help you to meet more of your students’ needs. I’ll be writing some blogs over the next few months that will highlight some of my most relevant learning. Please also keep in mind that I feel like a fledgling at this point- still grappling with many BIG ideas and concepts that will take time for me to really absorb, make sense of and share in meaningful ways. I’m also making mistakes as I learn. It’s important to act anyways. Before I jump in, I’m going to share a story, as I’ll be using it as an analogy for this post. Before we started our ethnomathematics immersion week in Hawaii in July we were required to learn 2 Olis (the above picture is our practice round). An Oli is a song (in Hawaiian). I spent a crazy amount of time trying to learn the Olis because I struggle to memorize song lyrics, in English, and don’t speak Hawaiian. I wasn’t sure what Olis were or why I was learning them or if I’d have to perform them alone or in a group. The thought of doing it alone left me in cold sweats of anxiety. Once I arrived in Hawaii, I learned what the purpose and meaning of the Olis were and why they were important for us to learn and do. Essentially, they are a form of oral tradition – they share important information. In our case, one of our teachers, Kaipo Tam, created an Oli that represents the values and vision of the ethnomathematics program and seeks permission to learn with those we were singing to. During my very first official Oli to the crew of the Hokulea and their Oli back to us, I was moved to tears by the surprisingly powerful experience. Throughout the week we performed the Olis and I came to genuinely love performing them because it made me feel connected to my classmates and the program of ethnomathematics, which was a far cry from where I started. When I learned the importance of and actually performed the Olis (rather than just learning to sing them alone), I understood their purpose and power. I think this is true for most learning. We need to do more than just learn about different worldviews and value systems but also implement strategies that help us to better understand them in our practice of teaching. In case you are interested, this is the Mahalo Oli (the easier of the two), which is well known throughout Hawaii:http://kapalama.ksbe.edu/elementary/mele/oli_mahalo/olimahalo.php I’m going to start by discussing some differences in worldviews and value systems that different cultures hold. For context I’ll share a little bit about myself and my culture. I was born in Syilx/Okanagan territories, also known as Kelowna. We moved to the unceded territories of Skwxwú7mesh Úxwumixw, also known as Squamish where we lived from ages 6-11. I remember learning very little of their culture and ceremony through school. Then my family moved to the unceded territories of the Lekwungen and W̱SÁNEĆ, also known as Victoria. I grew up in Gordon Head and learned almost nothing about Lekwungen or W̱SÁNEĆ cultures. I didn’t have any Indigenous friends and my schools were predominantly white. This pattern continued throughout my university experience. Due to this, I recognize now that I had a very limited worldview and it wasn’t until this program that I started to understand how important worldviews are when it comes to teaching and learning. This was a major blind spot in my teaching practice. Different cultures hold very different values about learning and this impacts our students in ways that I was completely ignorant of. I’d love to share all my new learning with you but that would be a book, not a blog, so I’m going to pull out the pieces that I hope will help you as you grapple with these really big and important ideas.Let’s start by considering this chart from a book called ‘Why Culture Counts’: Look at the second and third rows and think about what you/we value as teachers. Do we value collaboration and interdependence or do we value and promote individual achievement? Look at the last row – how would you define the main purpose of our education system? Academics or social intelligence or an even balance? As a person who comes from an ‘Individualist’ family and community, this rocked my world and made me see my students and the way they approach their learning in a whole new way. Are we providing opportunities for both of these value systems in our daily lessons? How do we do this? I’ve also been learning a lot about the benefits of supporting each of our learners in feeling proud and grounded in their cultural roots as well as celebrating the diversity of cultures within our classrooms as a way to promote equity. Because I am of the dominant culture, this was another blind spot for me. How often do I refer to values and history that are Euro-centric? How often to I refer to values and history that are not? Just like me learning the Olis, knowing that there are different cultural values and that I need to try to incorporate this into my teaching feels immediately overwhelming and brings about thoughts like “how am I supposed to teach other cultures when I don’t know anything (or very little) about their cultures?” So, here are some practical ways that you can engage in learning more about and honouring different cultures and their values in your classrooms without spending a ton of time. I’ve experienced these first hand and just like singing our Olis together, I only understood their importance and power once I’d implemented them: 1) Build strong relationships with your students and their families. Some teachers might think that they don’t have time to do this, or maybe that their job is to teach math, not get to know their students better but in order to teach your students the math, building a strong relationship is key, especially for students who come from collectivist value systems. Examples:a) Ask students to share their culture/ancestry in a sharing circle. Keep track of the different cultures so that you can look for future opportunities to incorporate them into math lessons. Some students won’t know their ancestry or culture but will find out once they are asked (it seems to spark their curiosity) and so you can continue to ask if anyone has learned anything new about their ancestry that they’d like to share throughout the year. Connect with your school’s Aboriginal education teacher if your Aboriginal students are not sure of their nation. b) Invite students and family members to share about their culture. This can be in the form of story, song, sharing or making food, artifact sharing, etc. You can collaborate with your guests to ensure that they are sharing in a way that works for them and for your students. This puts the onus on them, not you, to teach about culture. c) Ask students to bring in an artifact or share about a mathematical contribution from their culture. This can be: a game, ways in which math concepts are used in art, building, weaving, etc., a mathematician, a historical fact, anything math related! You could even do a project where students explored math in a cultural context that is their own or one that they’re interested in learning more about. They could create a poster and/or an interactive display that others can explore. 2) Provide opportunities for students to work collaboratively, especially when learning a new concept or skill. Even students who come from an Individualist value system need to be able to work in a team as it’s the top employability skill. Examples:a) Use Complex Instruction (see collaborative math teams on Educating Now) to construct roles and norms for working in groups during math class. Here are some links for more on this: http://cgi.stanford.edu/group/pci/cgi-bin/site.cgihttps://complexinstruction.stanford.edu/b) Use ‘Thinking Classrooms’ to ensure students are working with different students each class. Here are some links for more on this:https://www.youtube.com/watch?v=HHTwr1jsB80http://peterliljedahl.com/wp-content/uploads/Building-Thinking-Classrooms-Feb-14-20151.pdfhttps://www.youtube.com/watch?v=hc0hp0d15-4c) Use ‘Think-Pair-Share’ often to allow for students to think quietly on their own as this is important for introverted students. Students then share, and have the opportunity to rehearse with a partner before sharing out whole class d) Consider how much you value students’ ability to work well with others – are you explicitly teaching what this looks like and co-creating criteria for group work?e) Allow for choice – your students will likely differ greatly and so allow choice for working together or alone at times as wellf) Consider how you are assessing and how much of it must come from individual versus group work. Are you using group assessments? I invite you to consider the idea of the differing value systems and how your own life experiences colour the way you teach, including what you value and what you think the purpose of education is. Do we teach our students how to be strong community members? We often see what we want to see and so I encourage you to take another look at our curriculum and see all of core and curricular competencies that include collectivist value systems and those which include individualist value systems. Like my Olis – if you don’t jump in and try new approaches and ways of thinking, the learning likely won’t be meaningful.
Culturally Responsive/Sustaining/Revitalizing Pedagogies Resources: – Educating Now
Books: Websites/articles: https://scenicregional.org/wp-content/uploads/2017/08/Mirrors-Windows-and-Sliding-Glass-Doors.pdf (Rudine Sims Bishop’s article)http://www.ascd.org/publications/educational-leadership/sept95/vol53/num01/A-Framework-for-Culturally-Responsive-Teaching.aspxhttps://www.tolerance.org/classroom-resourceshttp://www.teachingworks.org/images/files/TeachingWorks_Paris.pdf Webinars: https://www.youtube.com/watch?v=O2kzbH7ZWGghttps://www.tolerance.org/professional-development/webinarshttps://www.nelson.com/learningonline/pl.html (scroll down for culturally responsive math webinars with examples of lessons)https://www.angelanardozi.com/webinars https://www.youtube.com/watch?v=5m6c4aUYZow&t=43s Lesson plans: https://sites.google.com/site/ethnomathematics/lesson-planshttps://www.todos-math.org/index.php?option=com_dailyplanetblog&view=entry&year=2019&month=05&day=22&id=11:ethnomathematics-mathematics-de-todoshttps://coe.hawaii.edu/ethnomath/curricula/
Ethnomathematics – What It Is and Why It’s Important – Educating Now
I want to share some of what I’m learning through my ethnomathematics courses – we are currently focusing on how to teach culturally sustaining (this is the new term that replaces culturally relevant or culturally responsive pedagogy). To be honest, I’ve realized that I have had a huge blind spot in my own practice with this regard. It has been through my experiences in Hawaii that I realized that part of the problem was I didn’t necessarily feel a part of any culture and so it wasn’t on my radar. Now, I’m understanding culture in SO MANY different ways and am understanding how important it is to education. I’ll be sharing more about this as the year progresses and am planning some professional development opportunities in Victoria as well. This blog gives you a quick overview and how it is connected to our BC math curriculum (and all curricular areas) and some activities that would be a great way to start building a positive, inclusive math culture within your classroom. Day 1 in Hawaii we went to Diamond Head Beach to learn about intertidal species and marine debris Brief History: Ethnomathematics is far more complex than I originally thought and I’m still learning about it so certainly don’t feel like an expert in it yet, although I certainly feel a whole lot more confident creating ethnomathematics lessons now that I’ve been to Hawaii to actually experience lessons for 7 days and from all my readings and discussions with my classmates and professors. Ethnomathematics is a fairly new field of study – the term was coined in the late 70’s by Brazilian Educator Ubitarian D’Ambrosio (he was our guest lecturer last week! So inspirational). It is not yet been adopted by the Oxford Dictionary and is unknown to many, however, it also has its own global conference that occurs every 4 years around the world and there have been many scholars who have been studying it and many teachers who have been utilizing it, even before it was called ethnomathematics. What Is Ethnomathematics and Its Importance Ethnomathematics is the study of how cultures mathematize. Math was created by human need to solve problems. Unfortunately, most of our school math has been stripped of it’s story, context and history, leaving it meaningless for many. We tend to focus our math on the Greeks and other European cultures, while ignoring indigenous knowledge and contributions. The idea is to create holistic, integrated units of study that are relevant to students. This sounds like a very lofty goal (it is!) but we can start more simply, in our classrooms, by getting to know our students and their cultures. When I think about how ethnomathematics can be implemented into classes, I would say that in a nutshell, ethnomathematics focuses on culturally based math (this can include popular culture as it can be relevant to students) with the aim of social justice, land-based math, and environmental stewardship. I cannot honestly think of two more pressing issues in our world right now than the lack of respect for cultural diversity (this has been worsening in many respects, not improving, in the past couple of years) and the state of our environment. Living in a province that is on fire really makes this problem more relevant and immediate. There are several dimensions to ethnomathematics that I won’t go into detail here for the sake of brevity. How it Connects to BC Math Curriculum: Our curricular competencies include these: •The positive personal and cultural identity competency involves the awareness, understanding, and appreciation of all the facets that contribute to a healthy sense of oneself. It includes awareness and understanding of one’s family background, heritage(s), language(s), beliefs, and perspectives in a pluralistic society. • Engage in problem-solving experiences that are connected to place, story, cultural practices, and perspectives relevant to local First Peoples communities, the local community, and other cultures • Incorporate First Peoples worldviews and perspectives to make connections to mathematical concepts I’ve spent the last two years trying to do this authentically and certainly made some headway but ethnomathematics helped me to understand how to construct lessons for these competencies. If you are wondering how to incorporate these into your math program, ethnomathematics is the answer! Ethnomathematics seeks to make every student in the classroom feel valued and to feel connected to their roots so that they are more resistant to harassment and domination. The ultimate goal is to create and sustain world peace by imbedding into our education system the valuing and celebration of cultural diversity. Considering the vast diversity of students in each of our classrooms, this seems like a daunting task: to include the cultures of every student in my math class. So, here are a couple of ways that you can begin this journey with your class right away: 1) Start each lesson by standing in circle each morning. Sharing in circle is common to many indigenous cultures and it really does do magic as far as I’m concerned. It removes any hierarchy – we are all equal in circle and it does a wonderful job of building a solid learning community as students are all looking at each other and sharing. I used this every morning of my math camp and it worked really well to build a very respectful, inclusive learning community (and my camp was only 5 days!). We also did this every day in my program in Hawaii. When standing in circle you can have students do some or all of the following (I like to share with students WHY we are doing this so that they will be more motivated to participate fully): Say your name We stood in circle many times a day throughout my course 2) Artifact Sharing: We did this in our first day of my program and it was a fantastic way to get to know each other and our values/cultures. You can do this in any way you want but here are some ideas: • Ask students to bring in 2 objects that are important to them – one personal and one that represents where they are from (this can mean their family, ancestry, location, etc.) • Ask students to bring in a mathematical object from their home – this can be connected to their family, ancestry, childhood, etc. • Ask students to bring in 2-3 objects that represent who they are (their values, their beliefs, their family, their ancestry, their hopes and dreams etc.) Students have 4-5 minutes (keep time to honour the time of all students) to share these with the class over the first few weeks of school (I wouldn’t do them for a whole class but rather 4-5 students a day). 3) Math Biography: I’ve shared this before but it can be found here. I’ve been using this for years. In order to be able to better meet the needs of our students, we need to get to know their: likes, dislikes, attitudes and goals towards math. 4) Diagnostic Assessments: See my last blog post! Too often we have a binder or a program for teaching math to our students and we do the same things year after year despite the fact that our students change each year. If we use diagnostics and interviews to better understand what our students know and don’t know, we can build a responsive education program that will challenge them at the right level and fill in the gaps of missing knowledge. Someone once said that teachers are often over teaching or under teaching because we’re essentially flying blind if don’t find ways of knowing where our students are at and what strategies they have and use. 5) Go on math nature walk. September is often beautiful and math really is all around us…let’s encourage students to see this as well. You can do a number of things with this but here are some ideas: • Students can use some form of technology to take pictures of the math they see outside (you can use your school grounds or local community for this). They will present one of their pictures with a quick description of the math they noticed. • In groups, students hunt for things that have the following quantities and take pictures of them along with their reasoning (why they think what they chose fits). Things we’d count by 1’s, 10’s, 100’s, 1000’s, 10 000’s, etc. For older grades go all the way to 1 million. This will pull in estimating quantities, which is actually really challenging for most students but is really important for developing number sense. This could lead to some great debates and the use of math talk stems like: “I agree with ___________ because ______________” “I disagree with ___________ because _______________” “I’d like to add onto what ________________ said __________________” • Find items that are different shapes and take pictures: circles, rectangles, triangles, curved edges, all straight edges, quadrilaterals, etc. Once this is done as a class or in groups you can sort all the images into categories and define each category. • As you are on your nature walk or community walk, students brainstorm and write down ideas for problems that could be solved. For example: how long does it take to mow the lawn of the front school grounds? How fast is the lawn mower going? How big is the area? How many customers come to ________ store each day? Some of their questions might be good for further investigation (solving problems related to place) and so they are doing the work for you! AND how engaged would your students be knowing that THEY created the questions…..in general I’d love to see a lot more this in math classrooms.
Sequences For BC Math Curriculum – Educating Now
I get asked a lot for scope and sequence and so here are some ideas for each grade K-8. Please keep in mind the following when viewing the sequences for each grade level: [caption id="attachment_1260" align="alignnone" width="1200"] Math Scope and Sequence[/caption] Sequence: There are lots of right ways to organize the content and each sequence is one way you may choose to use. Combining topics that are applications of others or that naturally align will benefit students by giving them ways to implement their learning and by seeing the connections between concepts. The more connections students can make, the better! It also benefits you because it can reduce the amount of time you might spend on concepts if you taught them all as stand-alone concepts. Get creative – if you have ideas about how concepts can be connected, try it out! The content is NOT the most important aspect of the curriculum – the curricular competencies are the vehicle through which you will teach so they are to be used in order to teach the content. You will use multiple competencies in order to teach each content standard. Each sequence will start with review or the most basic content first so that we help our students get a strong foundation before attempting to build on it. However, I strongly advise using diagnostics (paper and pencil and/or interview) in order to know if you need to revisit previous years’ content areas. We need to meet our students where they are at and so charging ahead with the current grade level’s content may not be appropriate. Scope: See the elaborations in the curriculum for ideas for scope – they will guide you in a variety of activities to dig deeper into the content. Also use the Big Ideas as a guide. The goal is to teach students the content by using the competencies to get to the Big Ideas. If your students understand the big ideas, it’s time to move forward. If you aren’t sure what a content standard means or an elaboration means please comment on this post or email me so we can offer support! Sometimes the language used is really tough to understand (especially if you are not math trained). How long you spend on a given concept will depend entirely on the students in your class. Some years a fraction unit would take 2-3 weeks, while other times it took 6 weeks. The goal is for all students to achieve mastery at the most minimal level, with as many as you can achieving mastery at all levels. Students who understand the concept well conceptually and procedurally can be challenged with some problems related to the concept (nrich.maths.org has some great problems that are easy to give to students and also https://www.cemc.uwaterloo.ca/resources/potw.php has great problems for these students) These are the sequences I created for each grade level. Please share and give feedback! I’d love to hear what worked: Kindergarten Scope & Sequence Grade 1 Scope & Sequence Grade 2 Scope & Sequence Grade 3 Scope & Sequence Grade 4 Scope & Sequence Grade 5 Scope & Sequence Grade 6 Scope & Sequence Grade 7 Scope & Sequence Grade 8 Scope & Sequence
Using Collaborative Learning Teams – Educating Now
Happy Back to School! This September is a new beginning for me as I’m not only back (part time) to my position at Cedar Hill Middle School but I’m also a new PhD student at UVic so I’m back to TWO schools. I absolutely loved my post graduate certificate program in ethnomathematics and felt at the end of it that I still had so much more to learn so…. I’ll be delving even deeper into culturally sustaining math pedagogies and teaching for social justice during my PhD! This blog is going to focus on using collaborative learning teams for a number of reasons. I think they are the very best way to incorporate the core and curricular competencies (for all subject areas) and they are a culturally sustaining pedagogy (I’ll unpack what this means shortly). They are a great way to set the stage and build a safe learning culture within your classroom and finally, they save you, the teacher, so much time and energy and allows you the time to do great formative assessment and work with students who really need your support. Core and Curricular Competencies: When I look up the most important work/school skills each year, the list has been similar for a number of years. At the top of the list are things like: Communication, Collaboration, Problem Solving, etc. When I see these, I ask myself “how are my students ever going to become proficient at communicating, or working as a team unless they have regular practice doing these skills?”. Our students not only need to be experiencing these more often but also with specific feedback from us about how they are doing. In our curriculum, these important employability skills are found in the core and curricular competencies. These are not ‘one and done’ checklists of things to do, but rather they are the vehicles through which we teach the content. When I first started using the ‘teams’ approach a couple of years ago I was blown away at how much better of a pedagogical approach it was and the teachers that I worked with were similarly surprised and pleased (those that use it use the teams for more than math….it becomes how their classes operate most of the time). Since then, I’ve promoted them a lot but not many teachers were comfortable trying them. I’ve talked to quite a few teachers about this to better understand their hesitation and here is what I most often hear: • I’m still trying to teach the ‘new math’ using manipulatives, etc. and so I need to do one thing at a time • I’ve tried group work and it doesn’t work for a lot of my students • I’m not comfortable with the lack of control and noise • I just don’t know how to do it properly or even where to start – it feels overwhelming to me Can you identify with any of these? I can fully understand all of them and I think it’s tough because it’s such a big departure from the traditional approach AND (and this is just my own observation) I think we teachers are control freaks. I am one too. I get it. But I LOVE the collaborative approach. Here are just a few reasons why: Collectivist Worldview Approach There are two different main worldviews: Collectivist and Individualistic. As North Americans, we are generally an Individualistic society, however, considering our multi-cultural classrooms and the fact that Indigenous worldviews are collectivist, it is important for us to be aware of both so we can better meet all of our learners’ needs. Northern European and North American cultures are mostly Individualistic while countries like Japan, China, Korea, Taiwan, Venezuela, Guatemala, Indonesia, Ecuador, Argentina, Brazil, and India are collectivist. When we use collaborative teams we are equipping our students not only with those important employability skills and core competencies but we are also teaching using a method that can be more aligned with their collectivist cultures (this makes it a culturally sustaining pedagogy because it honours their cultural values). When I was in Juneau in June working with some Tlingit Elders, I learned that using the collaborative team approach (using Complex Instruction) is very aligned with how they have always educated their communities. One elder, David, explained to me that learning needs to be rooted in love, kindness and respect of all members as well as valuing the different contributions each member makes. In the First Peoples Principles of Learning it states that learning is relational, which means that we all learn from one another in relationship. This includes teachers learning from students and students from other students and working in properly structured groups is fantastic way to achieve this. Efficiency When I first piloted using teams I asked students to give anonymous feedback and I was really surprised by their feedback. Firstly, I imagined that they would find it somewhat annoying learning how to solve the same problem in different ways but this was actually something that the vast majority enjoyed the most! When I asked if they preferred the teams or the more traditional approach of working sometimes alone and sometimes with a partner, they said the teams. I literally had 1 student from each class that ticked both boxes (they valued the teams because it helped them learn but they still preferred to just work on their own). Here’s what else they shared: they got the help they needed way faster and more often from their group than they ever have from a teacher. On the flip side of this, YOU, the teacher, have way less running around to do, answering the EXACT SAME QUESTION six times (even though you already went over that question on the board in front of the whole class minutes before). It frees you up in ways that were incomprehensible to me before I experienced it. Agency The last benefit I’ll mention (even though there are so many more) is that students develop agency and confidence and their communication and social skills improve tremendously. How many times do you get asked “is this right?” in a math block? Students develop agency when we remove ourselves from being the ‘answer gods’ and allow them to collaboratively prove and/or disprove their ideas and theories. This then leads to an increased confidence and enjoyment of math. The team approach also requires some serious communication and social skills (our core competencies) and I can tell you that the classes of grades 4-8 students I’ve worked with do not naturally have these qualities. I was a little shocked to see how ill equipped they are to deal with simple communication, sharing, being patient with each other, and dealing with conflicts. How to Use the Teams: Using Complex Instruction (I feel this is misnomer as the process isn’t that complex) is a great way to avoid the common hazards of group work (students opting out, one student doing all the work, etc.) as it is very structured and every student has a role to play. I’ve developed a Collaborative Teams Course on www.educatingnow.com that has all of the how-to’s to get you started and overcoming common obstacles as you progress. I’m in the process now of updating it as I’ve learned a lot in the past 2 years since we launched it. I’ve most recently read the book ‘The Culture Code’, which was very helpful and interesting so I’ll be adding some of this new learning into the course. Because I would LOVE to see more teachers using this approach I am offering the course for $99 until October 31, 2019! Use the code TEAMME for a 60% discount and you will have lifetime access.
Indigenous Worldview’s And Math Education – Educating Now
In my last blog post we looked at Collectivist versus Individualist value systems and how they impact how and what we teach and asses. I also touched on how to incorporate culture into our math classrooms so that all of our students feel valued. In this post I’m going to explore Indigenous and Western worldviews in more detail as well as provide ways that we can incorporate more of the First Peoples Principles of Learning into our daily math teaching. There is no ONE Western or Indigenous worldview but the following are some commonalities that are seen in most: [caption id="attachment_1492" align="aligncenter" width="598"] Source: The marginalisation of Indigenous students within school mathematics and the math wars: seeking resolutions within ethical spaces. Mathematics Education Research Journal, 2013, Volume 25, Number 1, Page 109 Gale L. Russell, Egan J. Chernoff[/caption] These values are also seen in the First Peoples Principles of Learning (FPPL). Here are the first two principles of learning; see how they connect to the chart above: Learning ultimately supports the well-being of the self, the family, the community, the land, the spirits, and the ancestors. Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place). Again, there is so much to unpack here! I am going to unpack a lot of these ideas over these blogs but what I really want to highlight at this point is that very little of this has to do with content, but rather with HOW we are teaching math and what we are valuing in the teaching and learning of math. I believe that if we really want to improve the achievement gap between Aboriginal learners and non-Aboriginal learners, then we need to be having conversations about these different worldviews and value systems with our colleagues. Regularly. Starting with “Learning ultimately supports the well-being of the self, the family, the community, the land, the spirits, and the ancestors” I was listening to this podcast last week (it’s fantastic) and one piece of information really struck me as connected to the different value systems that this principle of learning refers to: for many First Nations, Metis and Inuit students, it is far more motivating and important to do well in school or improve their learning for the sake of their communities. Rather than “this will help you to be successful/make a good living”, instead “this will help you to serve/help your community better”. I’ve never thought about this and until I heard it, it hadn’t occurred to me that when we tell students how school benefits them, we are completely missing the mark with some of our learners as they are not individual focused. I’d also like to add that being numerate is a life skill that helps us make sense of the world around us and ensures that we are making informed decisions. I’m going to address a few of the parts of these principle next: Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place). Notice the overlap between the Collectivist value system, Indigenous worldviews and this (focus on relationships). Here are some practical ways to incorporate these principles of learning and Indigenous worldviews: 1. Include experiential learning often. This means using manipulatives, visuals, skits, nature, problem-based and project-based learning. Learning by doing can also be a powerful way to engage more learners and reduce behaviour problems. I always feel the need to reiterate here that manipulatives are NOT remedial, nor are they only for struggling learners. Manipulatives are tools that help all students to understand WHY the procedures work as they do or what is actually going with the operations and numbers they are using (this is conceptual understanding). Sometimes, using manipulatives is significantly more difficult/challenging that just using numbers because the concept is more challenging than the procedure (see Teacher Resource Bundle on Education Now for videos and lessons) An example of a concept that is more challenging to understand conceptually than procedurally using manipulatives: a.) 0.3 x 0.4 à the procedure is easy as long as you remember the rules but what does this actually mean? There are a lot of interesting things to examine here: firstly, the product (answer) is smaller than either of the factors you multiply…WHY? Take a look at this using base 10 blocks. The large flat is 1 whole, so the sticks are tenths and the small cubes are hundredths: 0.3 x 0.4 means three tenths of (a group of) 4 tenths. Here is ONE group of 0.4 : while here is 3 tenths of 0.4 : The product shrinks because we are NOT taking a full group of any number but rather we are taking less than HALF of a group of 0.4. See our video for more details: https://my.educatingnow.com/courses/97139/lectures/1414027 Although the manipulatives add a challenge, they are important for students to understand what is going on here and why a multiplication shrinks the numbers as this is new for most of them who have internalized a rule that ‘multiplication makes the answer bigger’. 2. Valuing and exploring multiple strategies and interpretations is another way to Indigenize your math classroom. When I first started using math teams and gave students feedback forms to fill out on the process I honestly expected them to complain about having to learn multiple representations as one of the group requirements was to explore all the different ways their group members thought about and solved the given problems. However, not one single student complained about this and in fact it was listed as one of the aspects they enjoyed most and found most helpful in the team approach. They found it useful and interesting to see all of the different ways they could solve the same problem. I found this also helped them to develop growth mindsets because they stopped viewing certain ways as better than others. These different strategies and methods occur when we give students problems and allow them the opportunity to solve them BEFORE teaching formal methods. Examples: a.) At the beginning of exploring larger number multiplication in a grade 4 class this problem was given: “there are 4 rows of 16 carrots in the garden. How many carrots is this?” Student 1 uses Cuisenaire Rods. She says “4 tens are 40 and 4 sixes are 24 so all together that’s 64 carrots” Student 2 says “4 groups of 8 is 32 and another 4 groups of 8 makes 4 groups of 16 which is 32+32 = 64”: Student 3 says, “16 + 16 + 16 + 16 = 64” So much amazing learning happens when students share out how they are thinking about the math. It benefits the one sharing as much, if not more than, the ones listening and working towards understanding all of the ways. It also sends a message that there are many right ways of doing math and some contexts and numbers are better suited to some strategies than others.
Race and Math Education – Educating Now
In this post I will share some thoughts about how we can reach towards equitable education for all of our students as well as work towards reconciliation. Difficult Conversations... I believe we need to educate ourselves on the true history of Canada and the systemic racism that is still embedded within our education system and other government systems. I know this is uncomfortable for many non-Indigenous people but without really understanding the impacts of colonialism – past and present – I’m not sure we’ll ever have an educational system that will be equitable. I’m in the rabbit hole with this concept and am not an expert at all but think it’s important to share these ideas because it’s all too easy for non-Indigenous folks to ignore what’s going on because it doesn’t affect us directly and because we can. However, BIPOC (Black, Indigenous, People of Colour) cannot ‘opt out’ because they face racism as a part of their lived experiences. Furthermore, if we don’t have a decent understanding of how our own worldviews differ from our students’ and how our lived experiences differ from theirs, then we are likely to have big blind spots in our teaching practices Until quite recently, I had been completely ignorant and naive about racism in our society. It wasn’t until I read “White Fragility” that I even understood that racism is a system, not an act, and that most people misunderstand what racism and white supremacy mean. Since reading that book (which I highly recommend), I’ve been following a lot of teachers on Twitter, who are actively addressing the systemic racism that lies within the education system. If you read the last few blog posts I’ve written then you’re now already aware that there are different worldviews and that those have a big impact on learners. Imagine you were educated in a system that promoted a totally different set of values to those of your culture? What would it feel like to be in a classroom, helping your friend understand a quiz so he can be successful and be called out for cheating, even though you’ve been taught from your family and community that the most important aspect of education is to help one another? How would it feel to be taught math using only symbols and abstractions when you’ve learned through experiential learning at home for your whole life and the symbols just don’t mean much to you? I’ve come to realize that we’ve been gearing our education system around Western worldview values. This feels to me like we’re continuing to colonize a group of people who have already been colonized almost to death (and many have been colonized to death). I’m well aware that we are living in a time right now that is so noisy with political strife and racial wars that it makes it difficult to figure out what feels right or wrong. As I read parts of the Indian Act, the Truth and Reconciliation report, ‘An Inconvenient Indian’, ‘Unsettling the Settler Within’ and listening to podcasts like “All My Relations”, I have come to understand that I really had no idea what was really going on within our political system and education system in terms of the treatment of Indigenous peoples. It’s been eye opening and sobering to say the least and I have so much more to learn. I feel it’s my responsibility as someone who some people listen to (that’s you!) to speak about these issues. Understanding the history of our country will better equip us with the tools we need to educate all of our students in ways that acknowledge and respect their cultures. This is what I believe ‘Truth and Reconciliation’ is. There are so many big issues like these that I’m grappling with and will continue to explore. If you’re anything like me, this may feel very overwhelming for you and you may be wondering “what can I do?” or “where do I start?”. I don’t have definitive answers for you as I’m just beginning my own learning but here are some suggestions: Read the books I listed above Share what other books, articles and podcasts we could read/listen to that will help educate us. I know we’re busy but summer break is coming up…maybe add a book or two into the rotation of your other ‘summer reads’. Follow people on Twitter who are from different cultures, races and places than you. Here are a few that I follow that have offered me some significant insights: @NativeApprops, @apihtawikosisan , @LBmathemagician , @DingleTeach , @ShanaVWhite , @ValeriaBrownEdu , @Mathgarden , @YehCathery ,@Pam_Palmater , @NicoleBridge1 , @beRealcoach Engage in these difficult conversations with others. I think we should be talking about this during staff meetings. I think we should be talking about racism with our students too but we need to be mindful and well enough educated to do so properly. Seek out professional development opportunities that are geared towards equity Read books written by Canadian Indigenous writers (I’ve really enjoyed the novels, memoirs and history books I’ve been reading and I feel like they give me a glimpse into a lived experience that is vastly different from my own, thus expanding my perspective, or the lens through which I see the world). Some that I’ve read recently are: ‘Monkey Beach’, ‘Johnny Appleseed’, ‘The Marrow Thieves’, ‘Indian Horse’ (I’ve loved all Richard Wagamese’s books). Here are some other suggestions: https://www.cbc.ca/books/108-indigenous-writers-to-read-as-recommended-by-you-1.4197475 Attend Indigenous community events (when public are welcome) – even when it feels scary being a minority as maybe experiencing this will be the best education we could ask for. We can’t magically learn all we need to in any short amount of time so I look at this as a life-long, continuous journey. I intersperse these books with my other reading and will continue to do so. I’ve got a stack of books that I’m diving into next that were recommended to me by colleagues: ‘A Fair Country’, ‘Indigenous Writes’, ‘Potlatch as Pedagogy’, and ‘Tilly and the Crazy Eights’ Thank you for taking the time to read about these important issues. Please comment and share to continue this much needed conversation! Note: I’ve given you some links to some of these books but many are also available through the public library, which is where I accessed some.