Jana, I see the work we’ve been a part of together through MEC landing solidly in the top-left-green-inquiry-based-tough-to-name circle and share your curiosity of how it most powerfully intersects with your other two areas of focus. The resources you’re currently immersed in are exciting and I’m appreciative of the opportunity to broaden my understanding of how the work we do is positioned in the landscape you’re navigating. The book layout draft you shared sounds very interesting to me. My question always in reading books that share multiple strategies, tasks, anything is, do they also give me a sense of how these pieces might fit together or how I might begin to integrate them into my current classroom? I imagine this might already be on your mind (I feel like we’ve discussed having shared this experience!). Again, thank you for bringing us all along on this ride!
Debbie, I have been thinking about your question. You are right about how the first thing we much consider is how this will work for me. I am shifting a bit to think about “tools” that teachers can use so that students can choose the problem solving “tools” they use. I am not sure, but that may be a useful way of thinking about all of this.
Yes, for sure Maurine. I have a contact in the south of the Netherlands who teaches special education with a problem-solving emphasis. I wonder what overlaps you see between multi-lingual education and special education.
Jana, I’m looking forward to learning more about the ideas outlined in your graphic and how they will intersect. I see clear connections to some of our most glaring student needs. Like Debbie, my mind went right to our MEC sessions as I read through the summary of van den Boer’s work: the value of student voice and importance of representing student thinking vs. teacher thinking, reminders of the hazards of presenting work as “easy,” and the importance of structuring learning to make misconceptions transparent. Thanks for sharing your thinking.
Rosalyn, yes, I saw those connections too. It is so interesting to see so much that is familiar in such a different context. And I still so curious about how to fine-tune what we know for all of our learners and convince other educators of its efficacy for EVERYONE! Imagine MEC practices being common-place in special education classrooms and intervention classrooms.
Thank you so much for the Jo Boaler reference. In addition to visiting classrooms, I am auditing a course on crafting messages for a popular audience. I am thinking about how an important part of this work is in persuading people that culture and language should be no barrier to students engaging in powerful math learning. A CERME colleague suggested that I look into the work of Peter Liljedahl. He writes about A-ha moments as purely emotional experiences that are possible in some but not other conditions. It looks interesting. http://www.peterliljedahl.com/wp-content/uploads/Thesis.pdf
Pamela, Thank you for your question. I know that Inquiry Based Math Education is not the right label, nor is RME, quite. RME includes inquiry and it is also more, for example, Math in Context supports number strings much like those you provide in your work. The strings are written to be introduced with a context however and the materials talk of about models of thinking and models for thinking. A model for thinking is a problem-solving tool, while a model of thinking is a means of communication a process, if that makes sense. I am visualizing classroom communities characterized by students doing mathematical reasoning, so that means all of the following: inquiry; problem-solving; mathematizing with both real world and imagined contexts; modeling; multiple representations; defending reasoning; working in groups; working alone; selecting tools for problem-solving and more. I know these communities are not communities in which teachers have answers and students try to figure out what teachers are thinking. They are communities in which, to quote Ruth Parker, “Every mathematical idea is worth having.” And to quote Patty Lofgren, students and teacher strive “to understand each learner’s thinking for themselves.” The goal of this kind of community is for students to have their own A-Ha moments.
Dear Jana
Thank you for the opportunity to think deeply about mathematics instruction with you. 🙂 I am wondering about the graphic and the label of bilingual education. Are the students taught in their native language also?
Much Love,
Linda
Linda, Thanks for your question. I think it depends on the setting. At a school I will be visiting next Thursday, students are Dutch and learning in English for half the day in a bilingual school. Next Friday I will be visiting a school for refugees. While students have math classes in that school, the goal is for them to learn Dutch and they come from all over the world. There, I expect them to be taught primarily in Dutch. There is strong support here for refugees to gain proficiency in Dutch, including grants for adult learners to attend classes. The idea is that language will help them become more able to contribute to society. In order to become a citizen all immigrants must pass a language exam.
After reflecting on your post, I was drawn into your comments about van den Boer’s discourse rich classroom. As an ELL teacher, I am always awed by classrooms that enable deep and thoughtful thinking about a topic that includes all learners. I have found students need supports like sentence frames to support them to compose their ideas in a more formal way or other scaffolds to use as their stretch themselves. I have used a book called, Common Core Standards in Diverse Classrooms by Zwiers, O’Hara and Pritchard. It has excellent ideas in it that may be useful to you.
I also identified with not saying, “that’s easy.” I often tell someone who says that to stop because what is easy for one is hard for another. It builds empathy. I find your learning journey to be facinating. Thank you for sharing.
Thank you Sheila for that reference. I will look it up. I will be visiting classes who use a protocol called CLIL (Content Learning in Language) in the coming weeks. Does that sound familiar to you? It has been fascinating to watch for myself how I am making sense of situations with my small but growing grasp of Dutch. Until soon!
I, too, “just plain think your book is good idea.” I’m am curious what 6-9 practices you find most important.
I am also very curious about the research on discussion-based for EL students, as I have been a proponent of that in my my workshops and presentations.
What is the American equivalent of the Freudenthal Institute?
Hi Chris, I am glad you like the idea. I am not sure there is an equivalent of FI in the United States, although there is a Freudenthal Real Mathematics Education effort at CU Boulder: https://outreach.colorado.edu/programs/details/id/147. There is also Math in the City in New York. The link is in my post. Other than that, I don’t know. FI is completely devoted to research in math and science education and supports masters and doctoral students.
Hi Jana … There’s lots going on in your post that draws me in, the book outline, the conceptual model, the resources and your exciting itinerary of classroom connections! I’m interested in all of them. For a start, could you sort out what you mean by “practice”? The list you shared include dispositions and specific words. Are you thinking about specific teaching “moves” (e.g. asking students to revoice other students reasoning) or ways of engaging students in the mathematics (e.g. presenting worked examples of mathematical contradictions)? It may be useful to develop a list of half-dozen or so “routines” that promote engagement by each and every learner. The development of “routines for learning” was the topic of Robert Berry’s NCTM President’s Message from December, are central to the work of Amy Lucenta and Grace Kalamanik and may include several teaching moves. Here’s an example of some teaching routines:
• Structuring students’ math-talk
• Selecting and sequencing students’ ideas in a whole-class discussion
• Eliciting visual representations
• Working with stuck points and disequilibrium
• Researching students’ thinking
• Co-constructing a public record
Keep making sense, tot ziens!
Thank you for your thoughts Fred. As I keep thinking about this and attend CERME this week, I am starting to think that the project may be centered on Mathematical Reasoning and having the kind of class or community in which students know how to choose tools for problem-solving. The most simplistic illustration of not choosing the right tool is the story of the shepherd who has 3 dogs and 48 sheep. When you ask students how old the shepherd is, they immediately say 51, grabbing whatever tool is at hand, or whichever tool they think the teacher wants them to grab. I have had students set up the ratio 3:48 and respond that the shepherd must be 16 because we are in a ratio unit. These students are not reasoning mathematics. So if I think in terms of tools rather than practices the question becomes, what tools can a teacher choose to ensure that all students regardless of culture and language can choose tools from their toolkits for problem solving in ways that make sense. But to answer your question, a practice is a routine that teachers and students engage in within the community’s practice of doing mathematics. I like your list very much. It makes me ask, within those routines, how can we enact them in ways that increase equity and access?
Jana, I finally found time to get involved with your journey on a snow day! 22+ inches here! Close to home for you. On your original post, you listed saying “this is easy” as a taboo phrase. I’d like to add what I’ve heard many teachers say lately: “This is tricky.” I’ve heard this several times in my coaching efforts and find it unintentional for teachers. They intend to encourage students to pay close attention, but instead they may mislead students into believing they will succeed if they learn the “tricks”. Or worse, students may believe that teachers or texts are trying to trick them, supporting a fear of mathematics. We want students to know that math is sensible. Our common background with MEC has certainly taught us the importance of making sense of math with students and teachers. Luckily, when I’ve called this phrase to the attention of teachers involved , they’ve all responded positively and changed their practice, with honest enthusiasm. Thank you for sharing your adventure with mathematics. We have much to learn from each other.
A few questions and wonderings come to mind. I love the idea for a book focusing on instructional practices that provide access for all students to engage deeply in mathematical reasoning and discourse. As we all know, this is not an easy transition for teachers – knowing the 3 years of continual growth for those in the MEC project. I would love to learn more about how Dutch educators support shifts in instruction – do their teachers come much better prepared from their Universities? What opportunities and committments to PD is there in the Dutch educational system? My wondering: in what ways can your book provide guidance to support teachers who teach in a traditional format and knowing that for someone to continually growth in their teaching practices, takes someone willing to risk and wiling to try, fail, and learn from those mistakes (just like our students!). I think about my work in my district (any my past lens across the state) and how challenging it is to work with every teacher (many are in small rural areas) and support growth and learning. How could the book provide teachers engaging in some of their own routines (not really sure what kind of routines) that support their continual growth- perhaps self reflection, discourse with others about their own learning…….. as you know I process “aloud” – so there you have it!
The intersection of these three ideas is an interesting concept! As I was thinking about the difficult to label section, I was wondering if the 8 teaching practices identified in NCTM’s Principles to Actions somehow apply to this area. I see them as teaching practices that would be present in the type of instruction this area describes. Hopefully that’s helpful! Interesting work!
Jana,
You are several weeks into your quest now. I hope you are learning a ton and having rich experiences. I have a few thoughts about your February 5 blog, but first, I love the pictures you are posting on Facebook. But at risk of exposing my own lack of some cultural awareness or having missed some clue, what are you counting? In asking this, I wonder if I have exposed some lack of problem (or puzzle) solving on my part. It’s hard for me not to see parallels to our students here, right? To quote from your blog as you were discussing C.J.E.M. van den Boer, “She found that minority students and their teachers often didn’t know what they didn’t know. In other words, neither teachers nor students were aware of gaps in conceptual understanding during the learning process and only became aware of problems after students failed final assessments and entrance exams. Students and teachers were not understanding each other.” Don’t many of us experience this when put in new learning environments, or new cultural environments where we aren’t quite “in the loop”. I certainly have been as I pursue new experiences in retirement.
I was puzzled at first by your Venn Diagram, as I believe from our many conversations that the IBME is the core of your beliefs around teaching and learning math. The diagram makes it appear that all three parts are equal, whereas I think of how bilingual and multicultural education can enhance the IBME experience for students. Reading your words, I don’t think that’s what you intended, rather that you are using it to illustrate your focus on the three intersection of the three ideas.
I am especially intrigued lately by the idea of how important language (academic and conversational) is to learning, for bilingual and native speaking students, and for cultural knowledge. I would be interested in how you find that the strategies that work for students trying to learn in cultures and/or languages that are not their first would also be beneficial for students who are struggling even as they are learning in their first language and culture.
Well, Jana, it has taken me a few days to get to this next step, thanks to your sharing the Thesis on “AHA Moments” by Peter Liljedahl. It is a long piece, but so worth the read!!! I absolutely could not agree more about the power of those moments and how invaluable they are in developing a sense of agency and mathematical power in students, teachers, and the population in general. Regardless of the content we are teaching, it is giving participants opportunities to have those moments that seems critical to growth. As teachers, it requires that we become comfortable with our students experiencing confusion/cognitive dissonance/disequilibrium, but the bank for the buck is huge!!!
It continues to surprise me when I see how often teachers still believe their job is to put confusion to rest. I would like to think about ways to make use of this article with project participants if you think that is OK…
Jana, I see the work we’ve been a part of together through MEC landing solidly in the top-left-green-inquiry-based-tough-to-name circle and share your curiosity of how it most powerfully intersects with your other two areas of focus. The resources you’re currently immersed in are exciting and I’m appreciative of the opportunity to broaden my understanding of how the work we do is positioned in the landscape you’re navigating. The book layout draft you shared sounds very interesting to me. My question always in reading books that share multiple strategies, tasks, anything is, do they also give me a sense of how these pieces might fit together or how I might begin to integrate them into my current classroom? I imagine this might already be on your mind (I feel like we’ve discussed having shared this experience!). Again, thank you for bringing us all along on this ride!
Debbie, I have been thinking about your question. You are right about how the first thing we much consider is how this will work for me. I am shifting a bit to think about “tools” that teachers can use so that students can choose the problem solving “tools” they use. I am not sure, but that may be a useful way of thinking about all of this.
Jana,
I didn’t see anything about students with IEPs. Will that be addressed?
Yes, for sure Maurine. I have a contact in the south of the Netherlands who teaches special education with a problem-solving emphasis. I wonder what overlaps you see between multi-lingual education and special education.
Jana, I’m looking forward to learning more about the ideas outlined in your graphic and how they will intersect. I see clear connections to some of our most glaring student needs. Like Debbie, my mind went right to our MEC sessions as I read through the summary of van den Boer’s work: the value of student voice and importance of representing student thinking vs. teacher thinking, reminders of the hazards of presenting work as “easy,” and the importance of structuring learning to make misconceptions transparent. Thanks for sharing your thinking.
Rosalyn, yes, I saw those connections too. It is so interesting to see so much that is familiar in such a different context. And I still so curious about how to fine-tune what we know for all of our learners and convince other educators of its efficacy for EVERYONE! Imagine MEC practices being common-place in special education classrooms and intervention classrooms.
I don’t know if there’s a formal name for Inquiry based math instruction. I usually think about it as student centered learning where the students’ thoughts and questions move the learning forward. Jo Boaler just put out a blog post that speaks to the fact that math should be accessible for all students. Everyone Can Learn Mathematics to High Levels: The Evidence from Neuroscience that Should Change our Teaching https://blogs.ams.org/matheducation/2019/02/01/everyone-can-learn-mathematics-to-high-levels-the-evidence-from-neuroscience-that-should-change-our-teaching/?fbclid=IwAR3kXCK9v9YXf3HLPenQs2xO6q48FRJ6IkcZN-hMIqACkPb9MpB796B7ad8
Thank you so much for the Jo Boaler reference. In addition to visiting classrooms, I am auditing a course on crafting messages for a popular audience. I am thinking about how an important part of this work is in persuading people that culture and language should be no barrier to students engaging in powerful math learning. A CERME colleague suggested that I look into the work of Peter Liljedahl. He writes about A-ha moments as purely emotional experiences that are possible in some but not other conditions. It looks interesting. http://www.peterliljedahl.com/wp-content/uploads/Thesis.pdf
I wonder how your “inquiry-based math education” intersects with RME (Realistic Math Education)? Same, different?
Pamela, Thank you for your question. I know that Inquiry Based Math Education is not the right label, nor is RME, quite. RME includes inquiry and it is also more, for example, Math in Context supports number strings much like those you provide in your work. The strings are written to be introduced with a context however and the materials talk of about models of thinking and models for thinking. A model for thinking is a problem-solving tool, while a model of thinking is a means of communication a process, if that makes sense. I am visualizing classroom communities characterized by students doing mathematical reasoning, so that means all of the following: inquiry; problem-solving; mathematizing with both real world and imagined contexts; modeling; multiple representations; defending reasoning; working in groups; working alone; selecting tools for problem-solving and more. I know these communities are not communities in which teachers have answers and students try to figure out what teachers are thinking. They are communities in which, to quote Ruth Parker, “Every mathematical idea is worth having.” And to quote Patty Lofgren, students and teacher strive “to understand each learner’s thinking for themselves.” The goal of this kind of community is for students to have their own A-Ha moments.
Dear Jana
Thank you for the opportunity to think deeply about mathematics instruction with you. 🙂 I am wondering about the graphic and the label of bilingual education. Are the students taught in their native language also?
Much Love,
Linda
Linda, Thanks for your question. I think it depends on the setting. At a school I will be visiting next Thursday, students are Dutch and learning in English for half the day in a bilingual school. Next Friday I will be visiting a school for refugees. While students have math classes in that school, the goal is for them to learn Dutch and they come from all over the world. There, I expect them to be taught primarily in Dutch. There is strong support here for refugees to gain proficiency in Dutch, including grants for adult learners to attend classes. The idea is that language will help them become more able to contribute to society. In order to become a citizen all immigrants must pass a language exam.
After reflecting on your post, I was drawn into your comments about van den Boer’s discourse rich classroom. As an ELL teacher, I am always awed by classrooms that enable deep and thoughtful thinking about a topic that includes all learners. I have found students need supports like sentence frames to support them to compose their ideas in a more formal way or other scaffolds to use as their stretch themselves. I have used a book called, Common Core Standards in Diverse Classrooms by Zwiers, O’Hara and Pritchard. It has excellent ideas in it that may be useful to you.
I also identified with not saying, “that’s easy.” I often tell someone who says that to stop because what is easy for one is hard for another. It builds empathy. I find your learning journey to be facinating. Thank you for sharing.
Thank you Sheila for that reference. I will look it up. I will be visiting classes who use a protocol called CLIL (Content Learning in Language) in the coming weeks. Does that sound familiar to you? It has been fascinating to watch for myself how I am making sense of situations with my small but growing grasp of Dutch. Until soon!
I, too, “just plain think your book is good idea.” I’m am curious what 6-9 practices you find most important.
I am also very curious about the research on discussion-based for EL students, as I have been a proponent of that in my my workshops and presentations.
What is the American equivalent of the Freudenthal Institute?
Hi Chris, I am glad you like the idea. I am not sure there is an equivalent of FI in the United States, although there is a Freudenthal Real Mathematics Education effort at CU Boulder: https://outreach.colorado.edu/programs/details/id/147. There is also Math in the City in New York. The link is in my post. Other than that, I don’t know. FI is completely devoted to research in math and science education and supports masters and doctoral students.
Hi Jana … There’s lots going on in your post that draws me in, the book outline, the conceptual model, the resources and your exciting itinerary of classroom connections! I’m interested in all of them. For a start, could you sort out what you mean by “practice”? The list you shared include dispositions and specific words. Are you thinking about specific teaching “moves” (e.g. asking students to revoice other students reasoning) or ways of engaging students in the mathematics (e.g. presenting worked examples of mathematical contradictions)? It may be useful to develop a list of half-dozen or so “routines” that promote engagement by each and every learner. The development of “routines for learning” was the topic of Robert Berry’s NCTM President’s Message from December, are central to the work of Amy Lucenta and Grace Kalamanik and may include several teaching moves. Here’s an example of some teaching routines:
• Structuring students’ math-talk
• Selecting and sequencing students’ ideas in a whole-class discussion
• Eliciting visual representations
• Working with stuck points and disequilibrium
• Researching students’ thinking
• Co-constructing a public record
Keep making sense, tot ziens!
Thank you for your thoughts Fred. As I keep thinking about this and attend CERME this week, I am starting to think that the project may be centered on Mathematical Reasoning and having the kind of class or community in which students know how to choose tools for problem-solving. The most simplistic illustration of not choosing the right tool is the story of the shepherd who has 3 dogs and 48 sheep. When you ask students how old the shepherd is, they immediately say 51, grabbing whatever tool is at hand, or whichever tool they think the teacher wants them to grab. I have had students set up the ratio 3:48 and respond that the shepherd must be 16 because we are in a ratio unit. These students are not reasoning mathematics. So if I think in terms of tools rather than practices the question becomes, what tools can a teacher choose to ensure that all students regardless of culture and language can choose tools from their toolkits for problem solving in ways that make sense. But to answer your question, a practice is a routine that teachers and students engage in within the community’s practice of doing mathematics. I like your list very much. It makes me ask, within those routines, how can we enact them in ways that increase equity and access?
Hi Fred!
Anne Gallagher from PPS!- it is wonderful to see your name and reconnect through Jana – who knew!!
Hi Anne … What a small blogsphere! Great to “meet up” again. Thanks for reaching out. §FR
Jana, I finally found time to get involved with your journey on a snow day! 22+ inches here! Close to home for you. On your original post, you listed saying “this is easy” as a taboo phrase. I’d like to add what I’ve heard many teachers say lately: “This is tricky.” I’ve heard this several times in my coaching efforts and find it unintentional for teachers. They intend to encourage students to pay close attention, but instead they may mislead students into believing they will succeed if they learn the “tricks”. Or worse, students may believe that teachers or texts are trying to trick them, supporting a fear of mathematics. We want students to know that math is sensible. Our common background with MEC has certainly taught us the importance of making sense of math with students and teachers. Luckily, when I’ve called this phrase to the attention of teachers involved , they’ve all responded positively and changed their practice, with honest enthusiasm. Thank you for sharing your adventure with mathematics. We have much to learn from each other.
Jana,
A few questions and wonderings come to mind. I love the idea for a book focusing on instructional practices that provide access for all students to engage deeply in mathematical reasoning and discourse. As we all know, this is not an easy transition for teachers – knowing the 3 years of continual growth for those in the MEC project. I would love to learn more about how Dutch educators support shifts in instruction – do their teachers come much better prepared from their Universities? What opportunities and committments to PD is there in the Dutch educational system? My wondering: in what ways can your book provide guidance to support teachers who teach in a traditional format and knowing that for someone to continually growth in their teaching practices, takes someone willing to risk and wiling to try, fail, and learn from those mistakes (just like our students!). I think about my work in my district (any my past lens across the state) and how challenging it is to work with every teacher (many are in small rural areas) and support growth and learning. How could the book provide teachers engaging in some of their own routines (not really sure what kind of routines) that support their continual growth- perhaps self reflection, discourse with others about their own learning…….. as you know I process “aloud” – so there you have it!
The intersection of these three ideas is an interesting concept! As I was thinking about the difficult to label section, I was wondering if the 8 teaching practices identified in NCTM’s Principles to Actions somehow apply to this area. I see them as teaching practices that would be present in the type of instruction this area describes. Hopefully that’s helpful! Interesting work!
Jana,
You are several weeks into your quest now. I hope you are learning a ton and having rich experiences. I have a few thoughts about your February 5 blog, but first, I love the pictures you are posting on Facebook. But at risk of exposing my own lack of some cultural awareness or having missed some clue, what are you counting? In asking this, I wonder if I have exposed some lack of problem (or puzzle) solving on my part. It’s hard for me not to see parallels to our students here, right? To quote from your blog as you were discussing C.J.E.M. van den Boer, “She found that minority students and their teachers often didn’t know what they didn’t know. In other words, neither teachers nor students were aware of gaps in conceptual understanding during the learning process and only became aware of problems after students failed final assessments and entrance exams. Students and teachers were not understanding each other.” Don’t many of us experience this when put in new learning environments, or new cultural environments where we aren’t quite “in the loop”. I certainly have been as I pursue new experiences in retirement.
I was puzzled at first by your Venn Diagram, as I believe from our many conversations that the IBME is the core of your beliefs around teaching and learning math. The diagram makes it appear that all three parts are equal, whereas I think of how bilingual and multicultural education can enhance the IBME experience for students. Reading your words, I don’t think that’s what you intended, rather that you are using it to illustrate your focus on the three intersection of the three ideas.
I am especially intrigued lately by the idea of how important language (academic and conversational) is to learning, for bilingual and native speaking students, and for cultural knowledge. I would be interested in how you find that the strategies that work for students trying to learn in cultures and/or languages that are not their first would also be beneficial for students who are struggling even as they are learning in their first language and culture.
Well, Jana, it has taken me a few days to get to this next step, thanks to your sharing the Thesis on “AHA Moments” by Peter Liljedahl. It is a long piece, but so worth the read!!! I absolutely could not agree more about the power of those moments and how invaluable they are in developing a sense of agency and mathematical power in students, teachers, and the population in general. Regardless of the content we are teaching, it is giving participants opportunities to have those moments that seems critical to growth. As teachers, it requires that we become comfortable with our students experiencing confusion/cognitive dissonance/disequilibrium, but the bank for the buck is huge!!!
It continues to surprise me when I see how often teachers still believe their job is to put confusion to rest. I would like to think about ways to make use of this article with project participants if you think that is OK…