In spring 2024, when over 3,000 math teachers were asked “How frequently have you used AI in your mathematics teaching this school year?”, about 82% responded, “Never.” (see Dan Meyer’s article). Does this mean that math teachers are not interested in learning about how AI can help them in the classroom? Does it mean that the professional learning about AI for math teachers is not very good? Does it mean that other areas of research are more advanced technologically than math education? I would have to say the answer to these questions is that “it’s complicated.” The ways in which other fields may be embracing AI may seem more exciting and invested, but math teachers are definitely talking about AI. However, the considerations that math teachers need to contemplate are not necessarily the same types of considerations that other professionals need to be thinking about. Most educators think very seriously about the effects that any change will have on their students’ learning. What are the ramifications of making this huge change in students’ learning process? Teachers can be very set in their ways, yes. What has worked for them for years can be very difficult to adapt to new developments in educational practice. However, I believe they do this out of a desire to do right by their students. Changing something too quickly can have huge affects down the line for learners. In my mind, math teachers and administrators are just waiting until some of their big questions are answered. What are these big questions? Well, here are some that come to mind.
One of the issues of using AI that comes to mind most for people is the question of “cheating.” Because learning mathematics has been limited in most people’s minds to the processes and procedures of algebraic manipulation, a lot of teachers find it difficult not to quickly scrutinize AI use as the AI “doing the math” for students. Alternatively, we could look at math learning as different from these procedures and think more about how AI could be even more helpful. In order to move in this direction, some things must change to allow math teachers to be creative with AI.
Curriculum Review
Although Common Core Math Standards were published only 14 years ago, it is clear that there needs to be a revisiting of those standards with AI in mind. A review of the current K-12 math curriculum in any state must occur, especially our algebra-heavy high school curriculum. We need to make it so that what students are being asked to learn is not how to factor, but why you might factor a polynomial. Instead of focusing on how to graph trigonometric functions, questions should focus on what affects the graph and why. AI can give students examples from which they can generalize about ideas instead of practicing dozens of examples that they should be mastering. There is so much that AI can do for students (and eventually employees) that can now be removed from the curriculum to focus on more critical thinking and higher-order learning skills.
Classroom Assessment
AI should not be used as a teaching tool until teachers can find ways to authentically assess these new skills – perhaps even with AI. We can no longer assess by giving a traditional test, where the goal is for students to show mastery by doing 5-6 problems that they have already shown that they can do in class. Is mastery really what we are looking for in the age of AI? This is not to say that students should not have number sense and be able to do some calculations in their head. AI will soon replace the basic calculator and be on everyone’s phone. There is room for both a need for basic understanding and skills on how best to use AI.
Culture of Standardized Testing
What has been on my mind recently, is the fact that math teachers are expected to find ways to teach with AI in the classroom, while many external standardized assessments have not changed. It is not possible for math teachers to consider what is being done in the classroom and ignore the expectations of a test on which much of their future rides. I am the last educator to tote the benefits of focusing on standardized tests or “teaching to the test,” however we cannot overlook the dependence of college admissions teams on these scores. Until the college admissions process faces the effects of AI, the trickle-down effects to math teachers are huge.
If you don’t come from an education background, the concept of a student-centered or discussion-based classroom can be confusing to you, especially when it comes to math. Your idea of a “good” or “rigorous” math class might be one that is traditionally teacher-centered and requires a passive stance from students. Even many educators haven’t been exposed to the latest educational research, recent pedagogical initiatives, or the need to support greater neurodiversity. Every school should prepare to have, and to revisit regularly, the conversations about what best suits their school’s idea of an innovative mathematics program. Those conversations will form the foundation of curricular commitments and changes in teaching practices, in support of our shared values about the subject of mathematics.
The Innovative Math Classroom of 2024
What does today’s innovative math classroom look like? Is it an abundance of technology use, including AI for student learning? Is it students seated in groups doing explorations of traditional mathematics topics?
I would say that it can be seen from this perspective, but the innovation that needs to occur is even more basic than this. As AI becomes more and more normalized in student work, it will be more important than ever in the classroom to filter out the parts of mathematics that are redundant or procedure-based. Instead of factoring complex polynomials in an Algebra II class, students should understand the reasoning behind what factoring tells you about the polynomial and why it’s important. An innovative classroom will value the parts of mathematics that allow for higher order thinking – deciding on strategies, comparing, conjecturing and critiquing different processes.
Understanding the processes and procedures of what most people consider “math” today will be superseded by critical thinking and overarching reasoning skills. Mathematics will be the topic for the study of writing algorithms and engineering prompts for AI. It will become the discipline of study for scholarly communication with artificial intelligence.
As AI becomes more and more normalized in student work, it will be more important than ever to filter out the parts of mathematics that are redundant or procedure-based.
Leading Towards an Innovative Math Program
How do schools decide to innovate in their mathematics program? How do they identify where the program is in the transition to an effective, innovative math classroom?
First, teachers must be aware and supportive of making change – changes to what texts they use, their familiarity with support services for students, and incorporating updated pedagogical practices that enhance and value specific higher-order student thinking skills. Students need to be taught how to become better mathematical communicators, and how to present mathematical ideas in a clear and engaging way for their peers. Instead of a classroom where students sit passively – asking questions of the teacher that only relate to their individual understanding of a problem –students should be more actively doing mathematics, conjecturing with peers and judging for themselves which problem-solving methods are optimal in different circumstances.
Next, a deep and rigorous curriculum review of the mathematics program needs to be complete for all grades. Here, we seek to discover whether there is alignment between curricula for lower grades, middle grades and upper grades. Are learning outcomes and prerequisites clearly outlined for each course in the sequence? Are teaching practices aligned so students have similar experiences in their courses across the program? These are the types of questions that a department chair or academic dean should consider before attempting to make changes in innovation.
Evolve Your Math Programming
All change comes with trepidation and consequences in the community. Teachers will need professional learning, and students will need help accepting different methods of instruction. However, reinventing a school’s mathematics program is essential to remaining relevant and innovative. These ideas are the future of mathematics education and should be explored to keep a school’s mathematics program modern and creative.
I have received some great inquiries since I wrote my last blogpost and I’ve spoken to many teachers who have ideas and questions about teaching online with PBL math. I thought I would share some of those ideas so that everyone can benefit from these great thoughts.
Collaboration
One of the aspects of the PBL classroom that can be missing or disappear altogether is the relational aspect of discussion or collaboration. It is difficult to have students interact, especially if your school has decided to not have synchronous classes (students in different time zones, allowing families to have their own schedules, etc.). Collaborating asynchronously is difficult, but can happen. Here are some tools where this is possible:
Voicethread: See my last post – I think Voicethread is the best app for asynchronous collaboration. It does take time to set up, but once it is, students can post the problem they have a question on and students can asynchronously post a video, audio, text or drawing question in response to each others’ comments. The thread keeps the order of the comments and the teachers can put their two cents in when necessary as well.
Flipgrid: Some teachers that I am working with are using Flipgrid to allow students to continue doing problem partial-solution presentations and then having other student post video responses. The responses are either in the form of feedback on the video or in the form of a question or clarification. Of course, this creates a great deal of work if you are going to watch all of these videos, so you must find a way to randomly watch videos every day so the students know that you will be looking at them.
Canvas Conferences: You might want to check whatever LMS you have, but I now that Canvas has a wonderful feature called “conferences” where you can set up a video of you chatting (it even has a whiteboard where you can write) and students can type questions. This can be recorded and posted on canvas for students that cannot be there synchronously. I guess is the same as using Zoom, but it is nice because there is communciation through the LMS automatically for the students the time and they don’t have to log onto another app.
Explain Everything Whiteboard: If you already have an account with Explain Everything, that allows you to make videos on your iPad (and I think on your laptop too), Explain Everything also has a web whiteboard where you can create an online lesson and invite people to watch as you talk. I have not used this before but iti’s kind of an interesting add-on.
Twiddla: This Free Online Public whiteboard is pretty cool, but it does have pop-up adds. You can make it private in the room settings and you don’t have to sign-up for an account. Of course their goal is for you to sign up for an account, so I’m sure there’s some thing in there about getting you to join, but if you just need something in a pinch, I think this is pretty great for synchronous collaboration. You can also leave anything that was written up and have kids go in later and see what was there. So it somewhat works for asynchronous as well.
Zoom Breakout Rooms: Boy, I wish I had bought stock in Zoom about six months ago. Zoom has become the most-used app, I believe for online and distance learning in the past three weeks. I hope you are all aware of the break-out rooms that you can do in Zoom, you can triage students by putting them in these break-out rooms in groups to work on topics or separate problems as well. Of course, this is also assuming you are using synchrous time for these classes, but it a really great way to have kids collaborating in smaller groups while you can jump in between the groups – sort of like in class.
“Lessons”
I know it seems weird to think about giving direct instruction during a PBL class, but I think at this time in a crisis, it is important to think about the stress, anxiety, isolation, and other emotions that these students are feeling. All of those feelings are exacerbated in a PBL classroom so whatever we can do to help ease any of those feelings without totally moving to a direct instruction or lecture based class I think is good for the students.
One thing I have done when I teach online, especially at the beginning of the year, is try to make small videos with an app like Educreations (which giving away free Pro Accounts right now) and Explain Everything, which I use a lot (and are also giving free accounts to closed schools). These are wonderful apps and I think easy to use. Here is a video below that is a quick tutorial on how to use Explain Everything on an iPad if it would help anyone.
When I make a video to help a student(s) move forward with a problem, I try to keep it the way I would motivate their thinking in class. If they are not seeing something that is holding them back from what is needed in the problem like visualizing moving from a cone to a sector as in this video:
Another thing that I think is important is to remind students of prior knowledge that they may not see and cannot benefit from the discussion and ability of other students in the class who may have that prior knowledge more accessible to them. This is a great advantage in the face-to-face classroom that many PBL teachers rely on, is that there is usually at least one student who can say “Oh, yeah, I remember doing that before” and can give a short recall of what that prior knowledge is. Online, it is more difficult for students to access that prior knowledge that might be needed. So another type of video I make for students is a recap of prior knowledge that they might need for problems that are coming up:
Now you might be saying, “I don’t have time to make all these videos, Carmel” and that may be true. You can feel free to search on YouTube, but what I’ve found is that many of the videos on YouTube are not what i want. They are either very much based on mneumonic devices or not the type of understanding I would want my students to have. So feel free to use any of mine or email me (cschettinophd@gmail.com and I will try to make a video for you if I have time and post it on my YouTube channel.
Assessment
So it is clear that giving tests with distance learning if very difficult. Parents are not really going to sit and proctor written tests or quizzes especially if they are working from home. The idea of test integrity is also quite difficult with different age groups that you are all working with. Many students are extremely trustworthy and some schools have honor codes which kids would probably adhere to even when at home. However, the stress of being online now with a new way of learning, would be enough to make any student crack under the pressure. So here are some of the ideas that you can use to be more formative in your assessment and then perhaps a bit summative as well.
LMS Quizzes: Many of the LMSs that schools use already have built in systems for making quick quizzes (Canvas, Schoology, Moodle, just to name a few). If one of your colleagues makes a quiz, it is very easy to share them among members of the same LMS. These quick quizzes can be open notebook and you can have 3 or 4 a week with only 3-4 questions, just to see how the students are keeping up with material. They could count towards a grade or not, but this is a wonderful way to check in. They can also be used as exit or entrance tickets.
Socrative App: When I was still in the classroom I used this app many times in order to have a quick quiz that was open notebook and gives students real-time feedback about whether they answered a question correctly or not. You can take the time to make your quizzes or you can use the many quizzes that are already in their data base at the Socrative Quiz Shop and use those.
Summary of Topic Arcs: I believe that writing is the best way for students to show their understanding of a topic, but writing online is very difficult. So it does make sense for students to make videos of themselves. I think a good summative assessment is for the teacher to assign to 2-3 students one of the topic arcs that has just been discussed in the recent 2-3 weeks. Students should work together synchronous or asynchronous (make a google doc perhaps) to :
review and research the problems that move through that arc
describe in the video how they understand those problems building up their understanding
each student needs to reflect on one or two problems each and talk about how that problem is connected to the overarching theme.
Use paper, a whiteboard or a virtual whiteboard to explain the problems and their understanding
The way these are graded should be up to the teacher, but what I would suggest is a collaborative grading system:
the 2-3 student group would get a grade based on a rubric for their own video
the whole class will get a single grade for the set of videos as a whole based on some basic requirements (deadlines, time limit, etc.)
individual students will get a grade for one feedback/question video posted for one other group’s summary video – again with a rubric so that it is clear what you are looking for
I will try to post some sample rubrics for this that I have used in the past
I do believe that a combination of these types of assessments will help students stress less about their grades before the end of the school year and will also help you see how much they have retained in terms of material. There is also daily contributions and completion of work that should be recorded, but I don’t think I need to tell teachers that!
I wish you all luck with this new venture and would love to hear any other ideas and your experiences with teaching PBL online.
About three years ago, I was invited to the great challenge of finding a way to teach Problem-Based Math online with a great school named Avenues: The World School, whose online school is called Avenues Online (AON). Little did I know then that COVID-19 would come in 2020 and the idea of online learning would be not only popular but necessary in this time of crisis.
Awhile ago, I wrote a blogpost about teaching with a relational pedagogy with technology, as it is something that has been on my mind for a long time. When we started the pilot of AON, one of my first challenges was to find a way for students to interact that somehow mimicked the classroom process and community of discourse and interaction that a Relational PBL classroom has. In the past, I had experimented with the wonderful app called Voicethread that allowed students to have threaded discussions about problems with video, audio, text and drawing. I had students try to have conversations before we came to class about some problems they were grappling with. Here is an example of one of those voicethreads:
Of course, this was always followed by a class the next day, so we could follow up on students’s questions and comments. After I did this small experiment, I did a survey of the students in my classes:
I think what I learned from this experiment was that using an app or interface that allowed for some type of interaction where students could hear, see and interact with each other helped them feel more heard, not judged, and more confident in their learning and discussion. They also did not feel like it was media overload, which was not surprising for this generation for sure.
This experiment helped inform the work I did with AON of course. There was no work that was F2F at all since this school is totally online. The coursework for other disciplines were split into World and STEAM, which are totally project-based (to learn more click the AON link above). But we kept math separate, in what we called the Math Inquiry Program, in order to make sure that the materials were consistently covered in a problem-based way.
The next question is, how do we get students in an online environment to have those relational interactions that were important to the students that used Voicethread? Clearly, it was seeing, hearing, interacting and being able to read each other’s work.
My ideas came to fruition with the amazing help of some exceptionally talented developers in our R&D team. We are lucky to have an interface now that allows for students to see, hear, write and watch each other write while discussing problems. The interface looks like this:
Each person in the class (cohort) as a “notebook” that shows up on a tab.
We have it to be very effective for interaction between students and teachers and to allow for collaborative problem solving. Is it perfect? Not quite, there is much that needs to be done, including teacher training and improvement in bandwidth needed for the video, audio and writing capabilities.
Here is a 5-minute video of a small cohort of 7th graders discussing a Mathalicious problem that you may be familiar with. It shows some of the ability of the interface and the ways in which students and teachers can interact in real-time.
The most important piece of teaching online with PBL is attempting to keep the relationships together – student-student, student-teacher and student-material. Keeping students involved in metcognitive activities – writing, making videos about their ideas, interacting with each other – this is what will pay off in the long run. It’s not something we are going to create in the next month or so, but if we keep in mind the relational aspects of learning, we may come out of this crisis with the strong connections we always had with our students.
When I was in elementary school, I was lucky enough to have a teacher named Mrs. Bayles who believed that what it meant to be “cool” was enjoy solving really interesting problems. I remember one time she gave everyone in class a piece of pie and asked us all “What’s the best way to start eating this piece of pie?” and everyone else in the classroom immediately took their fork and stabbed it right in that pointy corner, where, they argued, they would get the most of the juicy center of whatever type of pie they had. I was sitting with my group of friends (mostly girls) who were self-defined math “geeks” (although I think back in 1976, that’s not what they were called). We all kept thinking about that and eventually came to the conclusion that we wanted to start with the crust because thought saving the middle for last was a great idea.
Mrs. Bayles thought that was so awesome and asked the four us to come up to the front of the room, draw a diagram on the board and give evidence as to why eating pie from the back of the piece of pie was somehow better than eating it from the tip. We thought we were the Albert Einsteins of pie-eating. We just loved it. Even though the other kids in the class thought it was kind of weird, since we could justify our choice with a good argument, we stood together and most importantly, Mrs. Bayles respected our evidence and let us have our authority in our say.
One of the things in recent years that has become a passion of mine in the mathematics classroom, and more precisely, the mathematics probem-based learning classroom, is the idea of status and positioning of students in the discourse and learning that occurs. This has become such an important issue that I invited one of our keynote speakers this year at the PBL Math Teaching Summit to speak on this very subject. After many, many years of hearing teachers’ concerns about how to handle a student who tends to dominate a conversation, or who doesn’t speak enough, or what happens when kids get off on the wrong the track when discussing a problem – it is about time that this socio-emotional topic (which includes race, gender, privilege, equity, and all things relational that made me start studying this pedagogy in the first place years ago) be moved to forefront of the mathematics classroom once and for all. (Aside – huge thank you to Teresa Dunleavy who gave an awesome talk on this BTW!)
I have shared this specific story with so many people at this point in time, but I find it so important that I want to repeat it here for Sam Shah’s “How does your class move the needle on what your kids think about …. who can do math?” prompt for this “Virtual Conference on Mathematical Flavors.” I feel it is something I’ve worked on for almost twenty or so years and I still don’t have it down to a science, I just know that I can’t let it go anymore.
In my classroom, I allow students to use dynamic geometry apps or technology as much as they want to justify their answers or as evidence for their thinking, as long as it doesn’t go awry (and of course, as long as it is correct and they can describe their thinking). Two years ago, I had a student (for whom I will use a pseudonym here because I’ve used his real name in the story, but not on the Internet), I’ll call Ernie. Ernie was one of those kids who could do no wrong – very popular in his current class, very successful academically, which made him very outspoken in his ideas, very good-looking in our white, hetero-normative, social class acceptable way and to top it off – (what Dr. Dunleavy says is usually one of the highest privileges in white schooling) – an exceptional athlete. Mix all of these privileges together and what automatically came with him into this class? Mathematical status.
Mathematical status doesn’t mean that he was not a good math student, that’s for sure. Ernie worked very hard and had excellent intuition, as well as good retention from his past math courses (–hmm, I feel like I’m writing comments from the fall term here..) These were neither here nor there however to the rest of the class. When students bring their own thoughts and impressions of a student into the class with them, its the class itself that priveleges that other student (in this case Ernie) the high mathematical status that he had. There might have been other students in the class who should have had higher status but because they were not as outspoken, had different relationships with others, were messy or not as articulate about their ideas, asked “stupid” questions (you know that’s not what I mean) or whatever the behavior that was exhibited – the other students in the class would assign a low mathematical status to other students by the things they say, brushing aside questions or simply by just not listening.
So one day we were discussing a question about the congruence of two triangles that were in the different orientation, plotted with coordinates. Students were supposed to come up with some triangle congruence criteria (I believe it was supposed to be SSS) for why the two triangles were congruent – this was at the very beginning of the concept of Triangle Congruence. Ernie had plotted the triangles on GeoGebra and simply said, “These two triangles are not congruent because all of the correponding sides are not equal” stepped back, matter-of-factly with pride in a job well done.
There was thoughtful silence in the room as the class looked at this diagram up on the board projected from his laptop. There was no arguing with the fact that the sides of his triangles were not all the same. However, there was still confusion I could see in some of the students’ faces. Some kids asked him to find the lengths of the sides. “That’s what I got,” “Oh I see what I did wrong,” and “Thanks for clarifying” were some of the comments that Ernie received. Under her breath, I heard one girl just whipsering to herself, “That doesn’t make sense” and I tried to follow up on the comment, but she would have none of it. We spent maybe 5 more minutes of me trying to get anyone else to make a comment. It got to the point where I even got out my solutions because even I was doubting myself (the power of Ernie’s status) because I had sworn that those two triangles were supposed to come out congruent. I knew some of those kids knew it too. Why weren’t they all saying something? It was as simple as a misplaced point. Not a huge problem, why couldn’t anyone call him on it? I decided to do a little experiment. “OK, well let’s move on then, but I really think there’s a way to show that these two triangles are congruent.” Ernie was intrigued. He couldn’t be wrong so tried to start finding his error, but couldn’t. I said, “no, no, I want everyone to go home tonight and try to see if we can find a way to show that these triangles are congruent.”
Jump to the next day in class and kids are sharing their solutions from the previous nights struggle problems. Before we start discussing them, I say, “Did anyone think about the problem that Ernie presented yesterday?” Radio silence….I wasn’t sure that anyone would actually do it, so I had come prepared with a geogebra diagram of my own. I projected it on the whiteboard and asked if they noticed anything. Still no one said anything (outloud so everyone could hear, but I could tell that some students were at least talking to each other). Suddenly, Ernie says, “Oh my gosh, I plotted the wrong point! It was supposed to be (6,1).” There was this huge metaphorical sigh of relief from the whole class at this moment that could be felt by everyone. I just coudn’t understand it. Although no one was willing to speak up that they knew Ernie had been wrong, they were all relieved that that he realized his own error.
I expressed my concern with this dynamic in my classroom. Simply asking them why didn’t anyone help Ernie with the problem yesterday in class? or what kept anyone from speaking up when they thought the triangles were congruent? wasn’t getting us anywhere. So what I did was tried to let them know how much I wanted to hear their ideas – similar to what Mrs. Bayles did with the pie. If students can see and hear evidence that the teacher values all voices equally, not just those that the students have given high status, can truly make a difference in how they start placing their status beliefs.
What I saw change in the class slowly, wasn’t the status that the kids all gave Ernie. In fact, if anything he got even more from finding his own error – but what happened was that girl who had spoken under her breath, spoke a little more loudly. Students who presumed that Ernie was correct, asked an interesting question that Ernie had to justify. These other students were growing in the status that the others were giving them. I believe that it is very hard for us as teachers to control what the students come into the classroom believing about each other, but we can have an impact on what they believe is valuable and meaningful about what they do in the classroom.
As I write curriculum, I am constantly scouring the Internet for ideas and ways to improve my own work, as we all do. I was just on the NCTM resources page the other day at their “Reasoning and Sense Making Task Library” and found this description of a task called “As the Crow Flies”:
“The distance formula is often presented as a “rule” for students to memorize. This task is designed to help students develop an understanding of the meaning of the formula.”
OK, wait – shouldn’t this just say, we shouldn’t be presenting the distance formula as a rule for students to memorize? Instead we should be teaching it for understanding from the conceptual level and allowing students to realize the connection between the Pythagorean idea of distance and how it allows a student to find the distance between two points? Why should we have a specific task designed to create the understanding after learning the formula when the formula is actually secondary?
There is a series of questions in the problems I have written/edited that allow students to come to this realization on their own.
First, a few basic Pythagorean Theorem problems to practice the format, remind themselves of simplifying radicals, Pythagorean Triples, etc. Second, some coordinate plane review such as:
Given A=(5,-3) and B=(0.6). Find the coordinates of a point C that makes angle ACB a right angle.
This is really an interesting discussion question for many reasons. First many students have trouble understanding where the right angle is supposed to be. If they incorrectly read that the angle that should be right is ABC, then they are picturing a different right angle (and also doing a harder problem that we’ll get to soon!) but if they are reading ACB, it’s still an interesting question because there is more than one answer.
Students can sometimes visualize where the right angle can be (even both of the points) but may not be able to get the coordinates. This discussion is important however because in order to come up with the distance formula later in general (with the x’s, y’s and subscripts – whoa, way confusing!) they need to realize what’s so special about that vertex’s coordinates. So if there is a student who is confused I usually ask the student presenting this problem, “Can you describe the way you found the coordinates for C?” Their answer usually goes something like this: “You just take the x of the one it’s below and y of the one it’s next to.” and other kids are either totally on board, or totally confused. So then they need to make it a little more mathemtical so every else is on board. Other kids often chime in with words like horizontal and vertical, x-coordinate and y-coordinate. This is a really fun, useful and fruitful mathematical discussion in my experience.
We can then move to a problem like this:
2. Find the length of the hypotenuse of a right triangle ABC, where A = (1,2) and B = (5, 7). Give your answer is simplest radical form.
This is generally a problem that is given to students individually to grapple with for homework or in class in groups at the board. After doing the one discussed above, they at least are prepared to find the vertex of the right triangle and see where it should be.
It’s honestly rare that a student can’t even draw the diagram – especially if they can make the connection with the previous problem. (Connection is one of the four pillars of the PBL Classroom). One of the things that is often difficult for students is the idea of subtraction of the coordinates. The can easily count the units to get the sizes of the legs in order to do the Pythagorean Theorem, but in order to generalize, for a later purpose…sorry, don’t want to steal the thunder…subtraction would be an interesting alternate solution method if someone comes up with it – and they usually do.
At this point if someone does come up with it, I usually do ask why can you subtract the coordinates like that to get the lengths of the sides and (you guessed it) there was an earlier problem that had student finding distance on a number line, so, many kids make that connection.
So finally we get to this, maybe a couple of problems later:
Again, students are asked to use their prior knowledge and contemplate a way that they might be able to describe of finding a way to express the distance between two points in a plane. This is after discussing notation, discussing how to visualize that distance, discussing subscripts, and discussing the purposes (in other problems) of why we might actually need to find the distances between two points. Because the Pythagorean Theorem squares the lengths of the sides (BC and AC) I’ve never had a kid get all upset about the fact that we don’t put the absolute value signs around the difference for the sides – we’re gonna square it anyway, so who cares if it’s negative? Kids usually say, “if it’s negative, let’s just subtract the other way and it’s be positive.” We just get right to the point that all we are finding is the hypotenuse of a right triangle which has been the Pythagorean Theorem all along.
I generally have students write a journal entry about this amazing revelation for them at the beginning of the year and voila! It’s right there for them, in their journal for the whole year – no memorization needed. They understand the concept, know how to use it and actually love the idea because now they can just see a right triangle every time they need a distance. It’s how so many of my students say that have “never learned the distance formla” – they just use the Pythagorean theorem to find distance. I love it.
One issue that seems to arise after teachers have been teaching with PBL for some time is the question of how students can remain active learners while listening, taking notes, comparing solutions, being engaged in discussion, etc. All student-centered mathematics classrooms now have this issue don’t they? Can a student learn well when they are being active in their learning? How do you allow them to both have agency by being part of the construction of knowledge but also have ownership by taking responsibility for the active part of learning.
Here’s a scenario: Grade 8 class has a student at the board presenting a method of factoring that is obviously confusing everyone – you know, they learned “the box method” somewhere else and are presenting it like it’s just a given that you are supposed to know this. I’m observing this and I’m seeing at least 5-7 looks of confusion, maybe 1-2 students who are following the student and at least 3-4 who have checked out totally – maybe drawing a tree in their notebook.
How do you maximize this moment? It is imperative that the teacher move in and ask questions that get at the student presenter’s understanding, especially if the other students are not asking questions. There may be an air of “oh god, I’m supposed to understand what this kid is talking about” and others will not be asking good questions.
The teacher can ask questions like:
“OK, good work that you have a method that works for you. Can you back up and explain how this method is showing what the factors of the quadratic are?”
“Let’s slow down a minute and see if there are any questions.”
“Why don’t you explain why you chose the number and variable you did for each box and what those boxes represent?”
“Can everyone else write down a question for …. and then we’ll share out.” (this can include making up a problem for those who do understand and seeing the presenter do another example)
These will bring the moment back to the group, wake them up to the fact that it’s OK to have questions and maybe an alternate method as well. The kids who have checked out might feel validated in checking out. But at the same time, checking out shouldn’t be an option. . How can we teach students to remain connected even when they really feel like all is lost? In PBL, it is most important for student to have the tools in order to do this.
Today I saw this infographic tweeted by Brian Aspinall (@mraspinall) that does an excellent job of summing up ways to have students remain engaged when they want to check out.
So many of these relate to the expectations for students in the PBL classroom. Some of my favorites are
2. Relate it – not only to “something” you’ve experienced but another problem that you might have done that is connected to it!
3. Control your environment – How engaged you are is really your decision – How can students minimize their own distractions? Of course day-to-day this will vary, and is developmentally different from grade 8-12 of course, but students, when aware of being distracted, can often find ways to get back into the work.
4. Self-Assess – this is one of my favorites – it keeps them engaged, makes them think critically and thinking ahead.
Allowing kids to know that doing all of these different behaviors in the math classroom is not only OK, but expected and encouraged, is part of teaching with PBL and encouraging the “active” in active learning.
OK, so my online course for math teachers who are interested in learning about using meta cognitive journals is all ready to go. The official start day is next Friday Dec. 1, but if you register during this week prior, you get $50 off the full registration fee.
Here’s a description of the course if you are interested:
This course is an on-demand course geared towards middle and high school mathematics teachers who want to learn about journal writing in math classes. There are four main course lessons that range from the reasoning of using journals to how to assess them. Interaction can occur between participants in this course via the discussion forums with as much or as little time committment as participants desire.
Please share with anyone you might think is interested in learning about using journals in math class. Thanks so much!
