Episode #29: Concept-Based Mathematics: An interview With Jennifer Chang Wathall
In this episode, we’ll dive into a great conversation about shifting mathematics teaching from what many of us remember from our K-12 learning journey to a more progressive, inquiry approach to teaching mathematics conceptually. In particular, we’ll discuss with Jennifer Chang-Wathall how to be brave in your classroom to make necessary changes to promote student understanding, unpacking what teaching conceptually really means, and how we can ensure that students still learn the necessary facts and skills alongside conceptual understanding.
- How to be brave in your classroom to make changes that are needed for your students understanding.
- What does teaching conceptually really mean?
- Why we need to have facts and skill AND conceptual understanding. It’s not either or.
- How we can build confidence in our own math learning.
- Why knowing where students will create misconceptions will make you a better teacher.
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Jennifer Chang: I probably developed my teaching craft in terms of teaching for understanding and inquiry based learning approaches later. I probably said… towards the 15th to 20th year. I taught traditionally for a good 15 years I would say. This is not binary. I don’t want to say traditional teaching is wrong or right, but there’s no such wrong. There’s nothing wrong in any kind of teaching strategy. I want to get that clear, that point across, because as teaching is this complex, exciting, dynamic craft and we have this toolbox of different pedagogical strategies and there are times where we have to use direct instructional- instruction, which I think is what we term as traditional teaching. But for me, traditional teaching is that you only use that particular pedagogical strategy of direct instruction. I think we really need to expose our students to a whole variety of different approaches to learning mathematics and that may include some direct teaching when necessary… [crosstalk 00:01:05]
Jon Orr: You’re listening to Jennifer Chang-Wathall. Jennifer is an independent educational consultant, the author of the book Concept Based Mathematics and part time instructor for the University of Hong Kong. With over 25 years experience in the educational field, Jennifer has worked in several international school including South Island School, Hong Kong and the United Nations International school, New York and Island school in Hong Kong.
Kyle Pearce: In this episode, we’ll dive into a great conversation about shifting mathematics teaching from what many of us remember from our own K-12 learning journey to a more progressive, inquiry approach to teaching mathematics conceptually. In particular, we’ll discuss how to be brave in your classroom to make necessary changes to promote student understanding. Unpacking what teaching conceptually really means, and how we can insure that student’s still learn the necessary facts and skill alongside the conceptual understandings.
Jon Orr: Join Kyle and I from here in Windsor, Ontario, Canada and Jennifer all the way from Hong Kong in this awesome episode. Let’s go.
Kyle Pearce: Welcome to the Making Math Moments That Matter podcast. I’m Kyle Pearce.
Jon Orr: And I’m Jon Orr. We are two match teachers who, together…
Kyle Pearce: With you, the community of math educators worldwide who want to build and deliver math lessons that spark engagement…
Jon Orr: … fuel learning…
Kyle Pearce: … and ignite teacher action. Jon, are you ready for episode 29?
Jon Orr: Yes, sir. I can’t wait to share this great conversation with the Math Moment Maker community.
Kyle Pearce: Before we begin, the Making Math Moments That Matter podcast is excited to bring you the Math Moments with Corwin Mathematics book giveaway. That’s right, we’ll be giving away 10 books from Corwin Mathematics, including Jennifer’s book Concept Based Mathematics, Teaching for Deep Understanding in Secondary Classrooms. Plus, you’ll receive Corwin discounts and digital downloads just for entering the draw. Get in on the giveaway by visiting MakeMathMoments.com/giveaway by Wednesday, July 31st, 2019.
Jon Orr: Are you listening after July 31st, 2019? No sweat. We are always actively running giveaways, so check out MakeMathMoments.com/giveaway by Wednesday, July 31st to get in on this Corwin math book giveaway. But, if you’ve already missed it, no worries. Go to that same link, MakeMathMoments.com/giveaway, and you can get in on the new giveaway.
Kyle Pearce: Awesome. Don’t miss out. Dive in at MakeMathMoments.com/giveaway.
Jon Orr: That’s MakeMathMoments.com/giveaway.
Kyle Pearce: We want to give a short Math Moment Maker shout out to Jim Murray who left us an awesome review on iTunes. Jim wrote, “Jon Orr and Kyle Pearce offer practical ideas that can be used in the classroom immediately. Their suggestions will increase student engagement and help teachers improve the atmosphere for learning in their classrooms. These guys know what they’re talking about and they inspire teachers to grow professionally. Thank you, Jon and Kyle, for sharing your knowledge, experiences, and ideas.”
Kyle Pearce: We will give another shout out to a Math Moment Maker on another upcoming episode and maybe this time it will be you. Just hit that review button and tell us how listening to the podcast has helped you. Thanks so much, Jim, we really appreciate you taking the time to give us some feedback and also to help spread the podcast to more listeners.
Jon Orr: Without further ado, here’s our chat with Jennifer Chang-Wathall.
Kyle Pearce: Hey there, Jennifer. Welcome to the Making Math Moments That Matter podcast. We are so excited to have you on the show today. How are things over on the other side of the world?
Jennifer Chang: I want to say greetings from Hong Kong! So, we’re actually I think 12 hours ahead. I’m really happy to be on the show, so thank you so much for having me.
Kyle Pearce: Awesome stuff, awesome stuff. We are getting ready to shut her down for the night and you are just waking up. So cool to be able to talk across the world like this.
Kyle Pearce: Jennifer, would you mind helping our listeners understand just a little bit about yourself. Tell us a little bit about you backstory. What’s your role in mathematics education and how’d you get into this gig?
Jennifer Chang: So, thank you for asking that very interesting question. Hi, everyone. I’m an independent educational consultant and I run professional development courses and coach schools and school partnerships in a few different areas, actually. So, I do help schools transition towards concept based curriculum and instruction for all subject areas from K to 12. I also look at concepts and inquiry but really, my specialty area is concept based mathematics. So, I’m very fortunate to be able to collaborate and coach educators around the world to transition towards teaching for more conceptual understanding and really reframing content and building curriculum that focuses, I think, on the deep and meaningful concepts of different disciplinarians, but more so in mathematics.
Jennifer Chang: I’m also a part-time instructor at the University of Hong Kong, so I have the honor of supporting masters of education students, in particular mathematics teachers, and it’s probably one of the only master’s programs that has a practicum. So, I’m really lucky to be able to go into the classroom and support teaching and learning with my students in university.
Jennifer Chang: But my backstory really is that I enjoyed a 27 year teaching career in the classroom. So, my roots really were about being a teacher and I feel like I was born to be a teacher. I love those light bulb moments that you see when your students really understand and they’re engaged and there’s joy in what they’re doing and, for me, that’s the biggest reward. So, it was actually very difficult for me to leave the classroom. I only left the classroom two years ago, I think it was, and it was a difficult decision because I loved my students, I loved being in the classroom and I said previously that I was born to be a teacher.
Jennifer Chang: When I was 12 years old, my mother took me to a fortune teller in the mountains of Taiwan, that’s where she was from. And this fortune teller looked into my face and she said, “You’re going to be a math educator for the rest of your life.” And who knew that I would actually make this vocation, that she was going to be correct. But she also looked into my brother’s face and she said, “You’re going to be a doctor.” And my brother is a surgeon in Australia.
Kyle Pearce: Wow.
Jon Orr: Wow.
Jennifer Chang: Yeah. People always ask, “Can I please have the details of this fortune teller so that..”
Jon Orr: Send them over my way, please.
Jennifer Chang: And so, whether she was right or not, and it really did kind of, I think, prepare me for a very long and very enjoyable career in the classroom. So, when I decided two years ago to leave the classroom, it was difficult to leave my students, first of all, but I thought, “Well, I just want to try to have a bigger impact on mathematics was taught.” How can I have more impact on more students worldwide rather than just the 30 students or the 20 students that were in front of me.
Jennifer Chang: So, I made that difficult decision, but now I haven’t looked back because I really do love what I do. I think we’re one of the few people and I think you, Jon, and Kyle really love what you do. We actually are able to be in the position where we share our passion for mathematics learning, I think, and our goal is really to try and move away from this very traditional procedural approach and help students see the creativity, the beauty of the subject. And so I haven’t looked back and I just absolutely love what I do. In fact, I’ve just come back from the Hague in the Netherlands and I was working with a group in Brussels in Belgium. So, a lot of my work kind of tends to be either in Europe, I do a little bit of work in the States, Australasia, all around Asia. But I am looking forward to, some day, coming to Canada and collaborating with some of the educators there where you are.
Kyle Pearce: We will welcome you with open arms. I’m still blown away by the fortune…
Jon Orr: You’re still stuck on that, huh?
Kyle Pearce: Yeah. I’m thinking, I’m like, “That sounds like an amazing probability problem.” Like what’s the chances for not just for one but for both of you and brother to be able to… If that wasn’t “real”, then she’s awfully lucky, so that’s a pretty amazing story.
Kyle Pearce: I want to back up just for a second. So you’ve been out of the classroom for just a couple years now. 27 years in the classroom and I’m staring at your book sitting on my desk right now and… So I know that you have this book called Concept Based Mathematics: Teaching For Deep Understanding In Secondary Classrooms. So you are coming from secondary, I just want to clarify. Did you teach certain grade levels or were you across the board or, actually, were you teaching like K-12 and just the book tended to focus more on secondary?
Jennifer Chang: Well, first of all, I want to say yes, I enjoyed 27 years. I started when I was 12 years old in the classroom. Not at all, I wish. But, in terms of my background, I taught 10 year olds to 18 year olds and I’ve really made it my focus, I think as well, to professionally develop myself in terms of the K-6 fields. And I think working with elementary teachers and especially early years teachers has really been one of the most rewarding and fulfilling, I think, parts of my role as a coach. I think elementary teachers are multi-talented. If you can teach half a dozen to 10 different subjects to your students and engage the same students day in, day out for a year, you’re extremely talented. And I’ve actually learned quite a lot about, I would say, K-6 education so that I can really support and coach elementary teachers and, fortunately, I do still get to go into classrooms. So sometimes in my school partnerships, we actually co-create lessons and design lessons and then we co-teach in the classroom, so that could be a variety of different ages and different schools, yeah.
Kyle Pearce: I can definitely relate myself, now, in a role, a K-12 math consultant for my district. I just, three years ago, came out of the classroom as well, so I’m feeling that same longing to be back in the classroom like you’ve expressed here. But, for me, I’m just fascinated. I’ve said it before on the podcast, the younger the children, the more fascinated I am by just how mathematics develops and just gaining my own understanding of concepts that I had been just using in my secondary classroom without even really thinking about how complex and like what is actually going on under the hood. So, just like yourself, I think it’s fascinating. I love, love, love hanging out in elementary classrooms now. And I think it’s more because I want to fully understand it before I go back to secondary world to start piecing all of that together, too. Just seeing how these big ideas, how early they start to develop and how easy it is for students to have gaps and maybe have fallen off the wagon because they weren’t really given an great opportunity to learn outside of the procedures, right?
Jennifer Chang: Yes, I agree, Kyle. I think at the same time that we have a lot to learn from early years teachers. I really have to shout out to early years teachers because everything at that age is conceptual. So they actually teach conceptually because there is no skill and drill, there is no flow order facts to memorize when you’re that age. And so, normally, in my workshops when we have that mix of teachers that are K-12, it is the early or pre-K, it’s the pre-K teachers that are actually showing us and helping us to understand how to teach conceptually better because everything is play based, which I think it should be all the way through. Everything is about either starting with concrete manipulatives, which I think is lacking in secondary schools as well. And so, normally, they are the rock stars in my workshops. So, I think we have a lot to learn, yeah.
Jon Orr: For sure.
Kyle Pearce: Abso…
Jon Orr: For sure. And you know what? I think we can unpack a lot of that statement and some of the statements you’ve just made as we dive into the rest of this chat. But before we go too much further, could you fill us in on a couple more details? Could you fill us in on what would be your most memorable math moment from math class as a student or even as a teacher? Like when we say math class, what sticks out in your mind as something that has stuck with you for the 27 years as an educator or those years as a student. A lot of us have those student memories.
Jennifer Chang: Wow. I’ve got so many, but I’m going to just focus on one that I actually do write about in my book and she’s under the alias of May. So I had a student, May. She was in secondary school and she’d experienced five years of very traditional teaching. And she came into my class, this is grade 11, so this is the last two years before entering university. And, for the first couple of weeks, I noticed she was very quiet. She was a hard working, dedicated student that was used to, I think, achieving excellent results through this traditional approach.
Jennifer Chang: At the time at the school, we were doing IGCEs, which is a kind of grade 10 qualification in the UK and it was an international qualification. So, she had got top scores in the previous year, she was in this class, and I noticed that she was really quiet and she came up to me a couple of weeks into term and she said, “Mrs. Wathall, I really do not understand a single thing that’s going on in our lessons in maths.” And I was devastated because I’ve made it my career, my vocation, this is my passion. I’ve put all my energies into really trying to help students have that light bulb moment when they’re learning mathematics and to be engaged and to really enjoy it. And here is this lovely student who is really high performing but had been exposed to the traditional approach for the last five years and she said to me, “Can you just tell me the formula? Show me three examples and I will work from my textbook.”
Jennifer Chang: And my lessons, we didn’t use a textbook in the lessons because I feel like if there is a textbook, use it at home. There is a place, yes, there is a purpose for the textbook, but it’s not during our time together where it’s a social endeavor and where adopting a constructivist approach to really make meaning and understand what we’re doing. And when she asked me questions, I wouldn’t provide any answers because I see my role as not the answer machine in the classroom. My role is not sage on the stage, it’s not even guide on the side. I’m the meddler in the middle, so I’m going to be really pushing your thinking by asking you questions, not answering your questions but maybe scaffolding and providing prompts to help you. Especially in the initial stages of that, when students form their understanding of particular mathematical concepts. I really believe in the let go for a period of time, but with scaffolding and prompt if necessary.
Jennifer Chang: So, she was not used to this. The beginning of our lesson, I’d walk in, we’d have this lovely prompt that would be high ceiling, low threshold. She would have no idea where to go, how to answer this, and she wasn’t used to being independent, resilient, and being able to embrace that productive struggle. For her, mathematics was about being shown what the formulas… I’m sure you’ve heard this a thousand times as well. Be shown what the formula is, show me three examples, and I will just go and practice.
Jon Orr: Yeah. Most of our high achieving students are exactly like that. They’ve had a lot of success as students in class in those previous years going in and maybe from listening to the podcast or we’ve talked about that idea a lot is they might’ve been successful in that system, but how much problem solving or how much actual math learning is what’s going on here. And I think it comes down to the teachers who are teaching in that way, what are they defining as math learning and what are we defining as math learning, because some of those teachers are saying math learning is I show you how to do it, you do it, hey, you’re doing math. Like that’s what it is, whereas, like you, we tend to differ on that and we’d rather our students be problems solvers and think critically…
Jennifer Chang: And engaged.
Jon Orr: … and all that stuff. Yeah.
Jennifer Chang: Yeah. It’s not monkey see, monkey do. That’s not our philosophy at all. It really is about trying to engage our students intellect and I think there’s a lack of… Well, we’ve talked about the procedural approaches and [inaudible 00:19:11] emphasis on that, but I see in a lot of classrooms the lack of showcasing the creativity and beauty of mathematics in itself, the conceptual language of mathematics.
Jennifer Chang: There is a conclusion to the story of May, by the way. So it doesn’t end there.
Kyle Pearce: Happy ending?
Jennifer Chang: Well, okay. This is going to be a cliffhanger for a little while while I explain what happened. So, after a month, she did an assessment for me. She did very well, of course, because she’s a hardworking student and I said, “May, what happened? You’ve done very well.” She said, “Well, I had to study so much by myself because I didn’t understand what was going on in classes. I had to the get the textbook every night.” And she said… And this is the IB diploma course, so it was an SL, a standard level of class, which… So in IB, in the diploma, there’s high level courses and standard level courses and you pick three standard level, three high level. So this was not one of her higher level courses that she had to focus a lot more time on.
Jennifer Chang: And she said, “Look, this is the standard level course. I don’t have time and energy to do this every time we have an assessment.” And I said, “May, just give me some more time.” This is a month in to the semester and I said, “Just give me some more time.” I said, “Look. If you do have questions in class, do you want to see me outside of class and then during lessons…” I said I’d check in with her regularly. And I said, “May, please just give me a little bit more time. Just give me some more time.” And in my head, I’m saying, “There’s method to my madness.”
Jennifer Chang: You have to really believe in what you think is right when it comes to mathematics learning, especially when you have students that come to you and say, “I don’t understand anything that you’re doing in your lesson.” There’s a big self reflection that happens there. You’ve got to be so confident with what you’re doing and lucky that was probably my 25th year of teaching, so if she would’ve come up to me in my second year of teaching, I probably would’ve wobbled and succumbed back to just telling her the problem. You’ve got to really have confidence to believe that what you’re doing for students is the right thing in terms of their learning.
Jennifer Chang: So I just kind of stuck with it and so I think two months, we’re two to three months now into the semester and we had another assessment and she did extremely well. But I said, “May, what happened?” And she said, “Well, I didn’t actually study. It was just all there.” And I said, “Well, what do you mean?” And she said, “Well, everything was just all interconnected and I seemed to just really deeply understand and I kind of understand why you’re not answering my questions now.”
Kyle Pearce: I love that.
Jennifer Chang: And then the next lesson, we were I think just stating binomial theorem, but we were staring our binomial theorem with a game. And the game was… I think I called it Sam the Taxi Driver. You start off at home, you can only go to the right, you can only go down in a grid to get to the school or something like that and it’s a lovely…
Jon Orr: Pascal’s Triangle.
Jennifer Chang: … a lovely pattern. Exactly. So, introduction and as she was doing this, she’s like, “Oh, this is really fun. Oh, I want more.” And I just felt elated because I think a few weeks later she actually wrote me a Christmas card because this was so… We start in September, this is December now. And she said, “Thank you for not giving up on me and I really do understand.” I nearly had tears in my eyes and I have her card, laminated, posted on my wall as a reminder of just to keep persisting and to not give up in terms of confidence and what we believe in is the right thing to do by students.
Jennifer Chang: And I think the other lesson that I got from may was that students, like teachers, if they’re learning something new or trying to transition and trying to progress their practice in some way, they are going to go through a dip. And I call it the Fallen Dip, it’s by Michael Fallen. Or you might have heard from the UK, there’s another guy, that calls it the learning pit. It’s going to be very natural and normal for teachers and students to go through some kind of dip, but that dip, I think, we have to embrace, we have to accept, and we have to say that productive struggle in the dip is part of the learning process. And if you don’t experience it, you’re not learning. So, if I give you 10 questions and you know the answers to those 10 questions, have you learned anything at all? I would say no, you haven’t learned anything, actually, because you knew the answers to them already.
Kyle Pearce: Yeah, exactly.
Jon Orr: I always think back to the beginning of my career. It was exactly that. I planned lessons hoping that I did such a good job preparing and planning that no one would have questions at the end. And once in a while, you get that lesson where that would happen, but you look back on it and you go, “Maybe they just didn’t care or maybe they already knew all the answers and I actually didn’t actually teach them anything new.” And that’s a huge challenge.
Jon Orr: I’m happy you articulated that after that story was in year 25, let’s say, and you have that confidence, you knew what it was you were trying to achieve and you had the belief that this was the right thing to do. Like you said, if you went back and it was a year two teacher and you’re there… I was just thinking to myself, I know when I started to shift my teaching because my teaching wasn’t always the way that we’re describing right now. My teaching was very procedural initially and that’s what I thought it was supposed to be. And when I started changing, the traditional high achievers, and I’m using bunny ears because like Jon said, it was like they were good at the procedural game, but they didn’t actually understand the math. They’re pretty unforgiving.
Jon Orr: When you change the game on them, especially I find like later in the game, when you get into high school and now kids are starting to think about post secondary and it sounds like you were teaching in the IB program. So, these are going to be students that are not going to be so understanding when things aren’t working out right and, lucky for your student there, that something in her… or maybe she could just tell that you knew what you were doing, like the confidence you had. I’m wondering what advice could you give to a teacher who’s listening to this going like, “I wish I could be that brave, but I’m just not.” What might that look like or sound like?
Jennifer Chang: Well there a great question because I actually had this question last week from teacher. So, a teacher said, “I have these students that don’t understand the value of what I’m doing because they think tradition… A maths lesson looks like me standing at the front of the board, giving the formula, and then students practicing from the textbook.” And I think one of the things that the reason why May and I actually came to an understanding and she experienced success was that we built this relationship.
Jennifer Chang: I think that we, as teachers, will never ever be replaced by robots or any type of AI because of that human element that we bring to the classroom and the human relationships that we bring. We’re not going to be replaced by textbooks or videos either. And I think it’s all the studies and research, especially from the World Economic Forum, that said teachers in all the industries that will require all those human connections and human relationships and human skills are not going to disappear at all. So I think building a relationship, and this is what I said to this teacher. I said, “You have to be prepared for your students to go through this transition and this dip and if you go back to your classroom and you teach completely different to how you taught yesterday, then that’s going to be a very difficult, very hard sell.” You’ve got to really hold their hand and I really believe it’s baby steps. You as a teacher, if you are actually evolving in terms of your own practice and different pedagogical strategies, you’re not going to come into your classroom the next day and be 180 degrees from where you were. I think it takes baby steps.
Jennifer Chang: So I said, “Start just introducing small things.” Another strategy I recommended to her was start of with some collaborative essential protocols or agreements about what the students want from their maths lessons. Do this at the beginning of the year. Let’s co-create some guidelines and protocols in what students expect to see and you’ll be surprised at what students will say. Students will not say stand at the front of the board and make me practice out of my textbook. And so I suggested that to her. I said, “Set up some protocols, have some common understandings with the students of what they expect maths lesson should look like, but then baby steps. So start off with just a few things. Don’t change 180 degrees overnight because everybody’s just going to be in shock and not ready.”
Jennifer Chang: And then I said, “In terms of stakeholders in the school community, it’s not just students. You know, we have parents because it’s parents that start complaining to the school board or to the administration about this teacher’s not teaching.” They’re not standing at the front, they’re not lecturing, they’re not teaching my kids properly. Their scores may initially go down because they’re floundering and they’re not understanding the purpose, the bigger purpose. And so in a lot of my travels, I actually give parent talks to really help parents… And I’m a parent myself, I’m a parent of two adult children, I can’t believe it. They’re 18. I have twin boys, so that’s why I’m saying plural.
Jennifer Chang: So I’m a parent myself and I went through traditional schooling, so I know exactly what other parents are thinking when it comes to, in their minds, this progressive fluffy type of teaching that’s not very rigorous. That’s one of the myths, right, of concept based curriculum and instruction and teaching for understanding and inquiry that there is no rigor and challenge. And after the parent talk, I’ve given quite a few at different levels of parent talks, but after the parent talk I give parents the visible thinking routine I used to think, now I think. And there’s huge shifts. Parents, they want the best for their children and they just need to be, I think, shown and they need to be educated about what really is effective learning and what’s going to really enhance their children’s education in the long run.
Jon Orr: Kyle and I were both just at the NCTM conference in San Diego and we were both… had the pleasure of seeing Sarah Vanderwerf present and she told an interesting anecdote. When she travels, she always asks the cab drivers, or the people that she meets, what do you think about math and math class. And they always have these memories that aren’t so glamorous. And she says like two thirds of the people are like that, they give those stories about math class. And then she said it’s the same two thirds that are parents that say, “But I still want you teach math the same way I was taught.”
Jon Orr: It’s like those parents still have… What your saying is about… Educating the parents that it’s going to be changing and you want to present that to them. Like did you really enjoy math class when you were young? And most of them are saying, no, but you still are angry that math is coming home and it’s not the math that you learned, or it’s not the way you learned it. And the frustration is they’re probably frustrated in a sense that they’re having trouble helping their sons or daughters or their children with that math and it’s coming out that way. But I always think about that we do need to educate the parents and I think that’s something that we need to do more of for sure.
Jennifer Chang: Yeah, definitely I think. Once you have parents onboard… You have students onboard first and foremost and then once you have students onboard it’s just a matter of just helping our parents understand what 21st century mathematics learning looks like.
Jon Orr: It’s very clear that you are so passionate about teaching math conceptually, as shown in your book Concept Based Mathematics. Did you always believe that math should be taught this way? You’ve kid of alluded to it earlier. I’m sure this is not exactly how you were taught, but when did this shift happen for you? Early in your career, late in your career? Can you give us a little bit of timeline there? I know that you’ve mentioned a little bit about baby steps for teachers, that’s your recommendation, but what’s an actual good timeline for teachers? Maybe they want to model it off what you did. Could you give us a little insight in there?
Jennifer Chang: I’m a bit older, I would say. So, for me, I probably developed my teaching craft in terms of teaching for understanding and inquiry based learning approaches later. I’d probably say towards the 15 to 20th year. I taught traditionally for a good 15 years, I would say. This is not binary. I don’t want to say traditional teaching is wrong or right, there’s no such wrong. There’s nothing wrong in any kind of teaching strategy. I want to get that clear, that point across, because as teaching is this complex, exciting dynamic craft and we have this toolbox of different pedagogical strategies. And there are times where we have to use direct instruction on some explicit instruction, which I think is what we term as traditional teaching.
Jennifer Chang: But for me, traditional teaching is that you only use that particular pedagogical strategy of direct instruction. I think we really need to expose our students to a whole variety of different approaches to learning mathematics and that may include some direct instruction when necessary of very low order thinking skills and facts. I think there’s nothing wrong with direct instruction when it comes to the low order things that we want our students… But when it comes to the high order thinking, that’s something we want to draw out from our students through learning experiences.
Jennifer Chang: But in terms of my own career, I’m hoping that people now, because I’m of an older generation. I was in a generation where there was still corporal punishment in school, so I don’t want anyone to follow my career or my pathway. But, nowadays, I know that there’s this huge movement of back to basics, I’ve heard. In the UK it’s the same thing. There’s this, I think, misconception that you either only teach facts and skills and you go back to basics or you teach conceptual understandings and that’s a complete false dichotomy. In fact, you need both. You need both facts and skills that go hand in hand with conceptual understandings. The facts and skills are the low levels of thinking. The conceptual understandings are the higher levels of thinking. We want to be able to connect those two levels by creating some kind of synergy between those two levels of thinking. We achieve that through using inquiry based learning in a mathematic, so your inquiry will actually connect those facts and skills with those conceptual understandings. In other words, you need to be able to know when to do something in order to understand. You can’t understand by itself.
Jennifer Chang: I’m hoping that… I know that there’s huge movements at the moment and pockets around the world with go back to basics and focusing on the skills and facts, but I’m hoping that there are more pockets around the world that believe, well, let’s try to evolve our practice or let’s try to adapt what mathematics learning looks like now so that we actually do prepare our students for this unknown, very exciting world because of the exponential growth in technology. We know many studies, Oxford University did one that basically said 47% of jobs are no longer going to exist in the next two decades because of automation. The World Economic Forum actually just released something and they said in 2022, they think that… Many companies think 50% are no longer going to exist because of automation. So, what are we preparing our students for? So I’m hoping there are pockets. Don’t follow my path necessarily, but maybe follow my path from the 15th year. Really think about how are we going to bring about that joy into the classroom? How am I going to engage my students but give them these necessary, I think, not just skills and facts but understandings that they can transfer and apply to different situations? How can I develop their intellect? Because, basically, information without intellect is absolutely meaningless.
Kyle Pearce: I’m so happy that you helped the listeners hear that it wasn’t like you came out of your pre-service and you’re not like had it all figured out. I think that’s always… Like people give a sigh relief, like, “Oh. It’s okay that maybe I’m feeling like I’m not where I want to be in terms of my math instruction,” for example. And I think by you sharing that… It’s been a long journey, I think that helps people see that… Like, “Hey, wait a second, we can all do this and it does take time.” And I appreciate you saying to take those small moves.
Kyle Pearce: Something else I really appreciated as well, I jotted a note down here, was just this idea that direct instruction, there is always going to be a place there and I’m going to argue that it’s really… Like I think one of the mistakes I made early on… And I’m saying it’s a mistake, I mean you learn from your mistakes, so I don’t… it’s not regret, but what I did early on was I sort of went and I let the pendulum swing all the way to the other side and my class, it was tons of inquiry going on all the time, nonstop. But on thing, it was like I felt afraid to actually use any sort of direct instruction to actually consolidate the learning and that is something that’s so important because if I’m doing an investigation and I’m too afraid to actually use any form of explicit instruction, then half of the kids may completely miss the learning objective of that lesson.
Kyle Pearce: And this is something that I hope people listening when we talk about it… It’s like we talk a lot about inquiry and sparking curiosity and fueling sense making and all of these things on this podcast, but, at the end of the day, the reason we talk about those things is because they’re kind of harder and they’re not something we all feel 100% comfortable with. But it doesn’t mean that we want to shift all the way over and I think you did a great job of articulating that.
Jennifer Chang: Thank you, Kyle. I also want to bring up the levels of inquiry as well. And I know that Trevor MacKenzie talks about this. Our dear friend who connected us, thank you Trevor. Shout out to you.
Kyle Pearce: Another shout out. This is like podcast, what, three or four where Trevor gets shout outs there Jon.
Jennifer Chang: But also the work of Andrew Blair, I’m not sure if you’re familiar with him. He is from the UK and I’ve been following him for years. His website is InquiryMaths, with an S, .com. And he talks about the levels of inquiry, too. So, whether you believe there’s three or four, actually doesn’t matter, but there are levels. There’s highly structured levels on inquiry for those students and teachers that are just transitioning and they’re not ready to like really let go yet, but here are lots of prompts to help you and here’s lots of scaffolding to really help you, but we’re still not going to tell you what you should understand from these lessons or from this lesson. But at least you have more confidence or more opportunities to build your confidence.
Jennifer Chang: And then there’s the guided approach, which is less prompting and there, of course, we have the open approach. There’s not that many opportunities, I see, for authentic open inquiry, but there are and it’s not something that you would do all the time in your lessons and so I think to help teachers really try and use more inquiry in their lessons, think about those levels and start off highly structured with lots of prompts, but just don’t tell students that… The key here is not to tell students what those conceptual understandings are because, for me, that would be like robbing them of that experience and that a-ha moment. Imagine if like I gave you a present and I told you what that present was before you opened it. Or if I… Instead of showing the trailer to a movie, I tell you the spoiler to the movie.
Jennifer Chang: So I want to set the stage for learning. I want to get you excited. I want to, at least, tell you certain things about it, but you’re going to find out what that present is at the end of the lesson or at the end of the series of lessons.
Kyle Pearce: I think about it in a math classroom as well and I tend to be somewhere in the middle of those different levels of inquiry where it’s very well thought out as to the learning objective and I think the part I really like about math… And it’s an opportunity… It’s something that Cathy Fosnot speaks a lot about in her resources including Context for Learning Kits is just this idea of like not trying to fix the mathematician, she says. And we’re letting the kids work through and see their solution through, but like, at the end of the day, we’ve put them in this scenario. We know where… We can anticipate where students are likely going to go, at least these different types of strategies and, of course, they surprise us every now and again. That’s the part that I think is just so awesome and in my classroom, early on in my career, that just never happened. It was just me telling and there really wasn’t any sort of investigation, any sort of inquiry going on.
Kyle Pearce: One thing I want to come back to and I’m worried… I don’t want to move too far away from it because I wanted to ask this a little earlier. My district, we’ve done a lot of work around shifting beliefs and we focus a lot on the mathematical proficiencies through the National Research Council and the adding it up document. And after years of work in and around our district around the five proficiencies, one of those proficiencies being conceptual understanding, I feel like teachers in our district have a much better view of what it means to be mathematically proficient, but yet, through conversations, we still hear teachers with different conceptions themself of what it means to teach conceptually. So, I’m wondering what does it mean to you? Like if someone’s at home and they’re going, “Yeah!” I feel like a lot of teachers are like, “Of course I teach conceptually.” But when you actually observe a lesson, often times what you find is it’s less conceptual and more like, “Well, I’m going to teach you why this works but it’s based on another set of rules where we’re not really sure where they came from either.”
Kyle Pearce: So, what does that mean to you and I don’t know if you have like a layman’s term sort of way to describe it, but I’m curious and I’m sure there’s some people at home going, “Yeah, I think it’s this but maybe I’m off the mark.”
Jennifer Chang: Yeah. I liked how you said, Kyle, about the why. So teaching conceptually really helps students to understand the purpose of what they’re doing and the reasons why. And I’ll give you an example, because I think it’s always good to have concrete examples. One of the tenets of concept based curriculum is that we use an inductive, or we try to, encourage an inductive teaching approach by looking at concrete examples first and then helping students to generalize for themselves.
Jennifer Chang: I’ll give you an example. I had a grade 8 class one year and I said to them… At the beginning of the year, we’re about to start trigonometry and I said, “Have you learned about those ratios and corresponding sides and the right angle triangles?” And they had no clue of what I was talking about.
Kyle Pearce: Surprise, surprise.
Jennifer Chang: Yeah. And then as soon as I said Sohcahtoa, they got their calculators out and they went, “Yeah!” They’re formuling on a calculator. And they really did not understand…
Jon Orr: Right. They’re just black boxes.
Jennifer Chang: … what those ratios were. They were just like boxes, substitute or, I hate the word, plug. Substitute numbers in, get an answer out, no idea what this means that they’re really about similar triangles and those ratios of different corresponding sides. And so, for me to teach conceptually means that you think about what is the purpose or what is why or what is the really deep understanding of Sohcahtoa, those trig ratios. So the deep understanding is that you really want students to understand the similarity of right angle triangles and different corresponding ratios, corresponding sides and their ratios.
Jennifer Chang: So that would drive learning then. So what does learning look like in a conceptual classroom? Well, you would have different tables perhaps with different similar triangles. So maybe one table has the 45/45 similar triangles and they’re all different sizes and they would actually either measure and they can use, I think, any online graphing software as well, so they don’t have to measure. But sometimes I like to ask students to pick up their rulers and measure things, especially when it comes to introductions to calculous, I think, before we move into graphing software.
Jennifer Chang: Anyway, getting back to the trig ratios.
Kyle Pearce: No, that’s definitely important. Not jumping to the technology too quickly.
Jennifer Chang: Exactly. I love technology but there’s a time and a place. You would ask them to measure and then they would look at different sides like opposite over adjacent or opposite of hypotenuse and see what they notice. And then you would ask them to communicate, to articulate what is the understanding that they are coming up from that learning experience. And I probably wouldn’t even mention Sohcahtoa, because the minute that you mention Sohcahtoa, all the conceptual ideas then become reduced down to this formula as procedural approaches without really focusing on that beauty of well, how come all those ratios of corresponding sides and similar right angle triangles always the same. It’s like pi, the same thing. Why? How come all the ratios of circumference to diameter is the same? So that really is beautiful and very creative and we need to bring that a out a little bit more.
Jennifer Chang: So that’s an example of teaching conceptually, so it’s really understanding your purpose and your goal and that purpose and goal should be articulated by teachers first so that we design learning engagements around that goal and that purpose and we call them generalizations. So they’re statements of conceptual understanding. Now when I know what that statement is, I can design a lesson or a series of lessons around that and then I can use particular questions to draw the understanding from my students. I can use a combination of factual and conceptual questions to then draw their understanding so I give my students the opportunity and the agency to really own what they understand through that learning experience.
Kyle Pearce: Well that’s an awesome example. I really appreciate you clarifying that for folks who are listening. And what I’m hearing is… Well, I’m hearing a few things, quite a bit actually, but one that’s popping into my mind is just this idea that when you’re teaching conceptually, to me, it sounds like it’s not just about answer getting. It’s about something so much more. It’s… When you mentioned Sohcahtoa and the kids are, “Hey, all right, we’re good, we’re good to go. Let’s fire this into the calculator.” You said plug into the calculator right? And I almost think that the word plug is appropriate in that scenario because there is no mathematical language or thinking there. We’re just aimlessly firing things in and when I go into a classroom…
Kyle Pearce: I’ve done some traveling. I was part of a project called the NORCAN project and we had an opportunity to go Alberta, province here in Canada, and then go to Norway and go into classrooms and I would always seek out classrooms… I was always curious about trigonometry at the time, this was a couple years ago. And something about trigonometry, it was like no matter where I went, the students didn’t have any conceptual understanding of what it was they were doing. And so it was exactly what you had said, it was like you were reading my mind. And what I realized eventually was this idea that even if we back it all the way up just to proportional relationships in general, students don’t fully understand what’s going on there anyway and that’s, to me, where… I was thinking, “How could I help students with trigonometry?” I’m like, “Oh boy, I think we need to back up even earlier.” And it introduces this other challenge that is a huge concern for Jon and I is this idea of even just our own content knowledge as teachers can be really difficult for me to teach conceptually if I am not feeling like I have a conceptual command of the topic.
Kyle Pearce: I always say I got a degree in procedural fluency, I did not get a degree in conceptual understanding. And that is like a big, big hole to fill. I don’t know if you can speak to that at all. Like how do you suggest teachers go about trying to figure out how the curriculum that they have to teach… How do they begin unpacking the concepts?
Jennifer Chang: I mentioned one of the tenets as being the inductive teaching approach and when I learned about this probably, I would say, seven years ago. So, quite recent in terms of my career. It really reframed everything I did in the classroom. So if you look at some of the websites such as Ed Rich from Cambridge University, if you look at James Tanton’s lovely stuff, my dear friend James and explaining dotso. I was listening to…
Kyle Pearce: Everybody loves James. We all love him.
Jennifer Chang: Yeah. Big fan. It’s all very inductive, which means that you start off with concrete examples first. So, when you’re designing, let’s say, learning engagement, it could be highly structured to begin with in terms of the level of inquiry and you want to give students the opportunity to generalize by giving them a few concrete examples, very carefully chosen, thought out examples. So I think that helps in terms of that journey. Confidence with teachers is probably one of the biggest issues that I really am focusing on in my coaching role because if you don’t have confidence, then you can’t do anything, right? And it really is sometimes about confidence in that you’re ready to admit that you don’t know something. That takes quite… It’s a big deal. And I do this a lot in my workshops to model a learner. To model that… I’m a lifelong learner, I don’t know everything. I’m going to make mistakes and I’m going to say when I don’t know and when I’m struggling, I’m going to let everybody know that I’m struggling. And so it’s very important that you model that as a teacher.
Jennifer Chang: Now if you model that and you believe that you’re a lifelong learner and that you’re supposed to go through this, I think then it’s a matter of before I even teach a lesson, I still do the lesson myself. I’ve just co-authored these reference books for the new diploma, IB diploma mathematics courses and I’m working through those investigations myself. After 27 years. There was a school a month ago and the day before my visit, they said, “Oh, can you come and teach come lessons?” And I actually said, “I’m really sorry. I’ve got to plan it. After 27 years, I still plan my lesson. I’m not going to have time, I’m going to be on a plane. I won’t have time.”
Jennifer Chang: So I think it’s important that we do mathematics as mathematics teachers and I still do this. I have a degree in implied and pure mathematics but I still make sure that I do the activity so that I can see misconceptions, so that I can, I think, address my own misconceptions and try to guide my own thinking in terms of what I think is important. Because, at the end of the day, everything in mathematics is very, very important. So we then have to make that decision, well, what is the most important things on a conceptual level that we want to decide to invest time in in our lessons. And that’s the decision that we have to make. What do you think this… So I decided to design those lessons around trigonometry because even you, Kyle and Jon, has probably faced that. The same misunderstanding and misconceptions from students from different places about that. So, we have to decide, well, what are those areas that students are going to not understand fully on a conceptual level and let’s design learning experiences to really help them understand that.
Jennifer Chang: I think another one is the ambiguous case of the sin rule. It’s so important. That’s something that’s so confusing for a lot of students and I think even just looking at circular functions and generating those trigonometric functions. Yes, you can have an applet that will show you something like that in five seconds, but I invest 80 minutes in an activity called Spaghetti and Sin Curves where we get spaghetti… You probably heard of that one. It’s in my book. You get sticks of spaghetti and you generate the curve using a unit circle and we spend 80 minutes instead of five seconds looking at an applet. So we, as teachers, have to decide what do we think, conceptually, is important to actually invest and design learning experiences for our students. What do we value.
Jon Orr: For sure. And you’re talking about challenges that teachers have when they’re transitioning to this more progressive inquiry approach. You brought up confidence. We need to have this confidence and it’s something that… It is a struggle for some teachers to have that confidence to go forward. You talked about knowing where the misconceptions were, you talked about knowing what, conceptually, you want to bring out. These are all kind of struggles to transition. Are there any more struggles that you want to bring up and then maybe strategies to help teachers overcome a couple of these struggles?
Jennifer Chang: Probably just to reiterate that confidence with the actual mathematics and sometimes it’s not confidence, sometimes it is that maybe a teacher doesn’t have a degree in mathematics and they love the subject, they love teaching, but it’s not necessarily their background. And so I would still encourage math teachers to do mathematics regardless of their age, regardless of their experience. I still enjoy doing mathematics and I think just being able to model that lifelong learner that I’m still learning about mathematics and different areas of mathematics in these new diploma math courses. We put in some new topics in mathematics and so I’m relearning those from university so that I can be confident in teaching them and creating learning engagements that will help my students understand or teachers understand those different topics in mathematics. My advice would be just do the mathematics. Go onto InquiryMaths.com, go into Enrich by Cambridge, Illuminations is wonderful, but they’re subscription now.
Kyle Pearce: I get E-mails from my district all the time, “Do we have a subscription?” And I’m like, “We don’t right now.”
Jennifer Chang: But they’re wonderful, too. And just do it. Sit down and think about the next unit that you’ve got to teach your students, look for some resources, and do it and improve and modify and add your own flavor.
Kyle Pearce: Right, right. I love how you mentioned just about us doing the mathematics because, at the end of the day, if we’re not doing it, we’re not going to learn anything new about it. It’ll be very difficult for me to be able to draw out what’s important if I’m not too sure what’s important. I think now, I’ve been playing a lot, like I said, with elementary math and looking at the implications of partitive and quotative division when we’re working in proportional relationships. Like Jon and I actually just released a new course inside of our academy, the Math Moments Academy, and that’s something that we’re really focusing in on for the middle school classrooms is really trying to get a handle on proportional relationships and you would be just shocked at how many different… What would seem like fairly simple problems, we’ve been unpacking and a colleague of mine, Vet Leamen from my district. We’re constantly going back and forth about some of these things that seems pretty basic but is not, it’s actually quite complex, so as you move up that ladder into trigonometry, you get close to calculus and things just explode. So I love that recommendation.
Kyle Pearce: Well, Jennifer, listen. We are looking here and Jon and I had a long list of more ideas that we wanted to pull out here. Including some ideas from your book, including the structure of knowledge versus process. We wanted to get into this idea of like planning units of study to draw out conceptual understanding. So we’re looking at the time and we don’t want to take up too much more of your day. It is our evening but your day. Is it possible that maybe in the future, we’ll be able to get you back on the podcast so we can dive even deeper into your book…
Jennifer Chang: I would…
Kyle Pearce: … Constant Piece Mathematics.
Jennifer Chang: I would love that, Kyle, that would be fantastic. Yeah.
Kyle Pearce: Awesome.
Jon Orr: Aw, that would be fabulous.
Jennifer Chang: It’s so wonderful to be able to have a conversation with like minded people.
Jon Orr: Jennifer where could our listeners find more about you? Where could they go soon as they hit the pause button?
Jennifer Chang: Yeah, so my website. It’s actually JenniferChangWathall.com. Maybe you can just share the links.
Jon Orr: Yes, we’ll put that in the show notes for sure.
Jennifer Chang: Yeah. I’ve also got the domain ConceptBasedMathematics.com as well, which I ran online courses. And please find me on Twitter, too. It’s great professional learning for me. So that’s… my handle is @JenniferWathall.
Jon Orr: Perfect. Awesome stuff. Thank you, Jennifer, we will put all that in the show notes and I’m sure people right now are clicking those and heading out to see what else they can learn from you.
Jennifer Chang: Thank you. Thank you so much for having me.
Kyle Pearce: Thanks for taking the time, Jennifer. It’s all our pleasure and we hope you have an amazing day as Jon and I look towards ending ours. So enjoy the day and we can’t wait to have you back on the podcast sometime soon.
Jennifer Chang: Thank you so much. Enjoy your evening. Looking forward to future chats.
Kyle Pearce: We want to thank Jennifer for our wonderful conversation. We know that you and the Math Moment Maker community received answers to questions you had or now have new questions that you’ll be seeking answers to to dig and think deeper on.
Kyle Pearce: The Making Math Moments That Matter podcast is excited to bring you the Math Moments with Corwin Mathematics book giveaway. That’s right. We’ll be giving away 10 books from Corwin Mathematics, including Jennifer’s book Concept Based Mathematics: Teaching For Deep Understanding in Secondary Classrooms. Plus you’ll receive Corwin discounts and digital downloads just for entering the draw. You can get in on the giveaway by visiting MakeMathMoments.com/giveaway by Wednesday July 31st, 2019.
Jon Orr: Are you listening after July 31st, 2019? No sweat. We are always actively running giveaways, so check out MakeMathMoments.com/giveaway by Wednesday July 31st to get on this Corwin Math book giveaway. But if you’ve already missed it, no worries, go to that same link, MakeMathMoments.com/giveaway, and you can get in on the new giveaway.
Kyle Pearce: Awesome. Don’t miss out. Dive in at MakeMathMoments.com/giveaway.
Jon Orr: That’s MakeMathMoments.com/giveaway. If you haven’t checked out JenniferWathallChang.com yet to learn more about her book, Concept Based Mathematics, as well as her other professional learning resources, be sure to do so now. Kyle, what was your big takeaway from our conversation?
Kyle Pearce: My biggest takeaway is just how far Jennifer has come in her own mathematics teaching journey. When you hear a mathematics consultant speaking at a conference or you read a book and you think about how we can do this better, I think it’s easy for each of us to jump to the conclusion that they must’ve just always known this stuff or they had the secret sauce to teaching mathematics. It’s like our irrational brain throws these thoughts in our heads to make us feel inadequate, when in reality, Jennifer openly shared with us how long it took before she started to shift her teaching practice. This was just another great example of how it is never too late to continue learning and trying to do the best we possibly can for our students. I really appreciated that. How about you, Jon? What resonated with you?
Jon Orr: That definitely resonated with me when she talked about this being… We’re all on a journey and we’re here just to do better and it’s not the same for everyone. We all progress at different speeds just like our students do in math class. I really appreciated when she called herself the meddler in the middle, like she’s not the teacher at the front giving the advice and step-by-step instructions, she’s in the middle with the students helping them overcome their productive struggles and giving them the feedback they need when they need it. I really appreciated that with her.
Jon Orr: I also appreciated when we asked her about what struggles teachers have in changing their practice. She not only let us know what struggles those are so we can watch out for them, but she gave us tips and strategies on how to overcome those struggles, which is great for you to know as a teacher so that you can be aware of those things, but also, if you’re in a coaching role or consulting role, you now have some better tools to go into those conversations with your teachers and know what to watch out for. So I appreciated all of those things that she shared in this conversation.
Kyle Pearce: Awesome. Yeah, definitely some new tools for our tool belt. So how about you at home? What’s your big takeaway from this episode? Share it with a friend, a colleague, or send us a message on social media. @MakeMathMoments on Twitter, Instagram, and Facebook.
Jon Orr: In order ensure you don’t miss out on new episodes as they come out each week, be sure to subscribe on iTunes or your favorite podcast platform.
Kyle Pearce: Also if you’re liking what you’re hearing, please share the podcast with a colleague! And help us reach an even wider audience by leaving us a review on iTunes. And tweeting us @MakeMathMoments on Twitter. Show notes and links to resources from this episode can be found at MakeMathMoments.com/epsiode29. That’s again MakeMathMoments.com/episode29.
Jon Orr: We release a new episode every Monday morning. Keep an eye out for our next episode.
Kyle Pearce: Well, until next time, I’m Kyle Pearce.
Jon Orr: And I’m Jon Orr.
Kyle Pearce: High fives for us.
Jon Orr: And high fives for you.
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