Episode 230: Teaching Math Is Hard. Here’s Why… [Part 1]

Apr 24, 2023 | Podcast | 0 comments



Teaching mathematics in school is complex and often misunderstood.

This paper titled Why Are Learning And Teaching Mathematics So Difficult by Alan H. Schoenfeld does a great job shedding some light on why this is the case.

He argues that teaching and learning math is complex because of these three (3) big ideas:

  • Thinking Mathematically – What does it mean to do mathematics?
  • The Learning Environment – Structuring classroom experiences to allow students to think mathematically.
  • Cultural Surround – How surroundings help or deter students from experiencing mathematical thinking.

In this episode we’ll help shed some light here on the first of those three (3) big ideas: Thinking Mathematically. 

elements connected to Schonfeld’s first big idea – thinking mathematically – and how you can successfully integrate them into your program without adding more stress and time to your already jam-packed work load. 

You’ll Learn

  • Why teaching and learning math is difficult; 
  • Techniques you can implement to make the teaching of math less difficult for you and your students;
  • What you should focus on most in the classroom with students; and,
  • How to strike a balance between the specific content standards you have to teach and ensuring your students are thinking mathematically.


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Jon: Teaching math in school is complex and it's often misunderstood, and we came across a paper recently, not a super recent paper, but it's a paper that we found very intriguing. It's titled, Why are Learning and Teaching Mathematics So Difficult by Alan H. Schoenfeld. He does a great job of shedding some light on why this is the case.

Kyle: Yeah. He argues that teaching and learning math is complex because of actually three big ideas and they are actually massive as you're going to hear in this particular episode in some of the following episodes that we do. The first one he talks about is thinking mathematically, what does it mean to do mathematics? He then talks about the learning environment, structuring classroom experiences to allow students to actually think mathematically, and then the cultural surround. This is how surroundings can help, or actually deter students from experiencing mathematical thinking.

Jon: In this episode, we're going to shed some light here on the first of the three big ideas, which Sean Field calls thinking mathematically. In following episodes, we'll be diving into the other two pieces of his paper. Stick around and you'll learn four key elements connected to Schoenfeld's. First big idea, Thinking Mathematically, and how you can successfully integrate them into your math program without adding any more stress or time to your already jam-packed workload.

Speaker 3: singing.

Kyle: Welcome to the Making Math Moments That Matter podcast. I'm Kyle Pearce.

Jon: And I'm Jon Orr. We are from Makemathmoments.com.

Kyle: This is the only podcast that coaches you through a six step plan to grow your mathematics program, whether at the classroom level or at the district level.

Jon: And we do that by helping you cultivate and foster your mathematics program like a strong, healthy and balanced tree.

Kyle: If you master the six parts of an effective mathematics program, the impact of your math program will grow and reach far and wide.

Jon: Hey. Every week here you'll get the insight you need to stop feeling overwhelmed, gain back your confidence and get back to enjoying the planning and facilitating of your mathematics program for the students or for the educators you serve.

Kyle: All right there, Jon. I am super excited to chat about this particular topic. One piece of the three big ideas that were shared with us from Shoenfeld's paper, which as you mentioned, it was actually you that brought it to my attention. And we dug in, and we found that really his three big ideas are actually could be really easily integrated into those six parts of an effective math program that here at Make Math Moments, we've been advocating and chatting about most recently on the podcast, and in a lot of the presentations that we've been doing at conferences

Jon: As we kind of unpack Schoenfeld's kind of paper, especially in this dispersed kind of big idea, thinking mathematically and what it means to do mathematics, we'll point out where his ideas kind of intersect with our kind of framework for the six areas of an effective mathematics program. And so if we start with breaking it down is we did say in the intro that he outlines three areas why mathematics, teaching and learning is so hard as a difficult thing. I think you can agree that if you walk into any room and ask someone about math, there are people that shutter, there are people that kind of step back. They know that learning math is hard, and actually historical places it's been like if you were successful at math, it automatically meant you were like a smart person. You overcome this hard area. This is the hardest subject in school.
This is typically how people view mathematics. And so he kind of breaks down three big areas, thinking mathematically the learning environment, which is what happens in the classroom, "How do you structure experiences for students so that they can have what he calls a robust learning environment?" And then the third one is cultural surround, like, "What is it surrounding the classroom that deters or makes it really hard to teach math?" And we'll dive into those two elements, those two pieces in future episodes. So in this episode we're going to specifically talk about thinking mathematically and what it means and what he's saying means to do mathematics, or what it means to study mathematics, or teach mathematics in the classroom and what is most important, I think that's what we're going to focus on here is what is most important in teaching mathematics in the classroom?
There's a quote right at the beginning of the paper, which kind of stands out, it hits home. I think it kind of hurts a little bit, when you read it. And he says decades ago Hans Freudenthal used a phrase that said that, "School of mathematics is the fossilized remains of real reasoning processes, or real mathematical thinking." And I want you to stop there for a second, and think about that fossilized remains and the word, "Fossilized", what does it come to mind when I read that phrase to you? How do you see how Schoenfeld's highlighting Freudenthal's idea here?

Kyle: Honestly, when I read that I was like, "Oh my gosh. I got a little mathematical chill down my spine." Because you'll notice, and we say this all the time, that everything we talk about here at Make Math Moments often or I say often, really everything we do is taking all of the ideas, all of the research that's out there and bringing it together and trying to put it into a process that's easier for people to learn, but then also actually implement. And this piece for me, connects well to what we call the rush to the algorithm. And the algorithm itself is the fossilized remain. That's how I interpret it anywhere, how I interpreted it when I first heard that quote, or read that quote from this paper. Because when you think about it's like if math class is led, much like you and I have said it a billion times before, we led our classrooms where the goal was for kids to learn algorithms to learn steps, and procedures, and repeat those things over and over again.
What we were saying at that time are actions were saying that mathematics is taking these things that were created a long time ago. Anything that you find nowadays that you'd call it a fossil, right? Just like a dinosaur fossil, that you find it's there and it's all that's left. And that is what a lot of math classrooms, at least for the most part, involving students doing things, using what was created from real mathematical thinking and then basically getting them to just sort of regurgitate it. Or as Peter inaudible always says, "Mimic it", to mimic this thing. And really what we've missed is that we actually haven't necessarily engaged in any real mathematical thinking.
So for me, that was the biggest piece, this idea of teaching kids to memorize procedures, or just to calculate things over and over again, memorizing definitions. These those fossilized remains and this idea that they're packaged or canned into bite size chunks that really I think the goal was when people started to do this and add it to our education system, the goal was, "I'm going to make it as easy for everyone to do math as possible."
But the reality was is they weren't thinking about everyone out there who, some people have a hard time with memorization, some people have a hard time with doing things when there is no meaning or sense making there. And ultimately, at the end of the day, we have a lot of these things called computers that can kind of do these calculations, can use these fossilized remains to get us answers to problems, if that's really the end goal is just finding an answer. So I love that this paper kind of shines the light on one of the biggest pieces in our Make Math Moments three part framework, is that we don't want kids losing fossilized remains, but rather we want kids to be the creators.
We want these kids to be the creators of this thinking. And the beautiful part is when kids create and use the thinking in order to create similar strategies and similar algorithms and procedures, and then we can go, "Yeah. You know what you just did? You just did something similar to this mathematician hundreds of years ago, sometimes thousands of years ago. And look at you are a true mathematician. You are not just a mimicker of those fossilized remains."

Jon: Yeah. When I think about that in the paper, he goes on to point out, we came from a history that kind of tried to package math so that it was accessible. So when you think about that, you think about-

Kyle: That was the goal.

Jon: Right. We're coming from a history of people who are like, "I've learned mathematics, I've done mathematics, I was the thinker of mathematics." And for somewhere in that line of thinking, someone's like, "The procedures are the most important, I have to communicate that to somebody else. Let me show you how to do that." So you can see this history of teaching math to students has been about, "I want to communicate these ideas to you, I want you to be able to do them." But I think the focus has been like, "Let me show you how to be a calculator instead of, let me show you how to be a thinker."
And so that part of the history, and Shoenfeld goes into the paper ideas of how this came out to be, how we got to this place. And there's some elements that he's pointed out of why it is hard to teach math. And part of it is this belief of what goes into teaching, "What do you believe is real mathematics? Is it a calculator? Are we calculating, are we doing procedures or is it about thinking?" And that's where these math wars end up coming from. And he talks a little bit about the math wars, but another part that kind of has led us to this place of people thinking, "I got to package math so that everybody can do it, not just a select few." So that kind of addresses why we've kind of got to a place of procedural based mathematics. But another piece is he goes into thinking the history of where we got in terms of resources at our disposal.
He outlines that even the curriculums have been crafted, textbooks have been crafted to communicate ideas to us, instead of thought processes. And even the curriculums, they got to a granular level of saying, "We should be teaching all these ideas." And when the Common Core came in for American schools and from even for our curriculum here in Ontario, when they came in and added the math practices, and the elements of what goes into thinking about math, that was a step forward. We were like, "Hey. We're going to go away from always talking about content only", but we've got a system of people who have come through this content only system like you and me, Kyle, our teachers only thought about math as that content, procedural based math, and not about the actual practices of math because it wasn't in their curriculum. It wasn't even part of their discussion back then.
Maybe a select few thought, "That's important in mathematics." But we got to this place through a history of an army of teachers who've only gone through that system. And we're now realizing after this kind of change that we are switching to more a little bit practices in our mathematics. And that's where these math wars end up coming from, because we've got a whack of people who are still thinking math is this, but other people thinking math is actually this. And so that's where we're at in a sense of the history of why it's hard to teach math is because we got two different belief systems that kind of decide on, "No. We should focus here." Or, "No. We should focus here." And in his paper he says, "If you're going to really make a dent in mathematics, what we need to do is focus on the problem solving aspects of mathematics."

Kyle: I love it. I love it. It's interesting going all the way back to you had mentioned by getting students to sort of regurgitate what's already been done, it was almost a strategy, like the goal was for it to be more accessible to everyone. At least that was the thought. And it didn't turn out that way. It's taken us a really long time to realize that, "You know what? That actually wasn't the best way to go about it, especially if access and equity are actually on your mind." Now, something else to keep in mind is that when we talk about content knowledge, it's really hard. Not only that, I don't think we have enough educators out there that actually understand how to teach problem solving, right? We're not talking about just taking a framework, like a model and going, "Use this template." We've done it, districts have done it, schools have done it, different grade bands have tried it.
They're like, "If we just use this template, then all of a sudden all of our problem solving worries are going to go away." The reality is that it's a process. We actually have to understand the process. We can't just be trying to solve problems without any sort of strategies in there, and actually helping and mentoring students. So this brings up another problem. I'm going to be honest and say I wasn't a great problem solver as a student, and I think that if it was expected of me in the current or the way it was taught to me, mode of education, if I was truly being assessed on my problem solving skills, I actually wouldn't be a math teacher right now. Which is kind of like interesting-

Jon: inaudible think?

Kyle: ... but it also means we have to actually think about this. We want to shift towards problem solving and actually getting students to think mathematically.
We have a lot of work to do. It's not just a mindset piece. So for us, our six parts of the tree that we always talk about, that's like the soil, the water, and the sunlight. That is our mindset and beliefs part of the tree. Well, it's great if we shift our mindset to say, "Hey. Let's not just teach kids to play with math as fossilized remains, but rather get them to truly problem solve." Well, guess what? There's another part of our mathematical tree that we're going to have to build. And that is not only content knowledge around the mathematics itself, but content knowledge around, "What does it mean to think mathematically?" And to actually teach students how to develop problem solving skills because that piece is hard work. It isn't as easy as saying, 'We did Pythagorean theorem today. Or, "We did the area of a triangle today."
It is actually helping them to get better at the process. And lately, I've been bringing sports up, and having been on the ice with my son all year long for hockey, there's two aspects of hockey. We can teach them how to do a course around obstacles, and get them to do it as fast as possible, but that does not help them become a better goalscorer, because they have to understand the process of the game. They have to understand and they have to essentially build their problem solving skills. When I'm in this sport and these things happen, what do I do? And it's never the exact same thing. There's always a, "Yeah. But", scenario. So these are things that make teaching math as we're hearing in this paper incredibly challenging. And we're just talking about the first of those three big ideas so far. Yeah.

Jon: And when you talk about mathematical beliefs and understanding what it means to teach mathematics and what we should focus on that is what Schoenfeld goes on to kind of say. There are four areas that you should focus on if you're going to focus in your classroom on what matters most in terms of thinking mathematically. Remember that we're going to talk about the environment of teaching itself and cultural inaudible outside the classroom on later episodes. So in this episode we're focusing on just like what matters the most when thinking about learning mathematics itself. And you've said mathematical beliefs Kyle, is super important. That is one of the four areas Schoenfeld says is important, is making sure that not only we understand what it means to teach mathematics, but our students understand what it means to teach mathematics, and the belief that we have about ourselves in being capable of understanding mathematics, and doing mathematics.
So that's one of the four. The other one that I think most people think about when they think about learning mathematics, or understanding math, or doing mathematics is content and knowing our content knowledge that is super important, and having access to that content. But he goes on to specifically say that, "Not only the content is important to understand math and be able to think mathematically, but the processes that are involved in that." And so that is an element that I think we talk about here a lot on the podcast about really understanding the math, which is one of our key areas also in our six parts of effective math program, which is the roots of your tree, the content knowledge. And when we talk about the roots often here we are referring to the teacher's roots, roots of our own understanding of content knowledge at a conceptual level, and its connections to other areas of math like a math web.
If we can understand that at a strong level, then we're in a better position. We're more confident to show those same connections to our students and be able to tie the connections to other ideas with our students. So he says that, "Understanding mathematical processes or having access to resources that include the mathematical processes, and content, and practices is one of the elements that you need in your classroom so that you can think mathematically and focus on what's important in your math class." So I think we are all kind of in agreement there. That's not like a surprise, but maybe the surprise is again, to think about our conceptual understanding of math and then we'll talk about teaching pedagogy, which is that learning environment on a future episode. But Kyle, what else is important when we think about mathematics and think when we're helping kids think mathematically?

Kyle: Yeah. Jon. So another piece that they highlight is using different problem solving strategies and they even reference George Pólya, and he's sort of known as the godfather of problem solving, right? And many people may be familiar with his four step process that oftentimes got turned into these place mats I already referenced. So here's an example, "Step one is understanding the problem. Step two, devise a plan, step three, carry out the plan. Step four, look back, check and interpret." So most people are going like, "Yeah. I know, I have that hanging on my wall."

Jon: No brainer.

Kyle: Of course. Well, here's the thing though, just like everything in mathematics, there's so much more nuance to it, and I would argue that these four steps are something that once again, if you really think back to the first quote, these four steps are fossilized remains from his work. You can't just take these four steps and give them to kids and go, "This will help you think mathematically." He found through all of his work that it really gets boiled down to these four steps.
So now the question becomes, "How do we as educators help guide students to reveal that there are kind of these four big ideas in the problem solving process?" But it's not as simple as just saying, "Hey. You need to understand the problem. Do you understand the problem?" We as educators need to be working and listening, and this is why problem solving in our classrooms is so necessary is so that we can walk around and we need to be thinking about these four steps and we need to try to figure out, "Where in the journey are all of these students?" Like, oh my gosh, I have these students working at the whiteboard over there. Those students, they're still trying to understand the problem. That's where they are in the process. I'm not going to go up to them and be like, "Okay. You need to understand the problem and walk away."
I'm now thinking as the facilitator, "Hmm. What prompt can I offer this group of students to help them get a little further within that step of the process?" It's not as simple as just saying, "Look at the framework on the board and everything's going to go away." I need to really be able to look around the room and sort of go, "Where are they in the process? And then where are the next steps that they need to go in order to try to get them there, without me coming in and sort of just telling them the next step?" That's the hardest part, right? It's like I gave you a problem and then we as educators want to be helpful. So oftentimes we come in and then we do the next step for them and then we walk around, and then we come back, and then we do the next step for them.
That is such a instinctual thing for us to do because we want to be helpful. You see parents doing it to children all the time, right? Where a child's trying to butter a piece of bread and all of a sudden the mom comes and grabs the knife and does it for them, because they're going to do it better. But you just rob that child of the opportunity to of course make a mess in your house, but to also learn how to do that process. So really holding back on that and thinking about where in the problem solving process are the students and then at what point are students ready to sort of co-construct?
Are there different steps, or steps is almost the wrong word, it's almost different big ideas and problem solving that we need to work on. And then could we craft what would be helpful when we're devising a plan? That's where we ask questions, "What might be helpful? Is there any information that you want from me?" We are helping the students to devise a plan so that they can do problem solving. And the goal is that one day students will eventually feel comfortable and competent enough to sort of guide themselves through the process to figure out where they are, and then ask themselves the questions that they need in order to get to the next step.

Jon: And I think you've articulated Leah's ideas nicely and what we really need to do when we think about the Pólya type strategies, and I think also in Pólya's work in his book, each of these four steps are gone into very much detail and provide strategies. And I think the strategy pieces for each of these is what we need to help kids with. And that's us being a little bit more coach-like, like you've described, "How can we be more coach-like to help kids solve problems and thinking about these steps?" But it's tied into the last piece that we haven't mentioned yet from the paper, which is really important in teaching mathematics, which is kids metacognitive performance, and specifically the self-monitoring skills. So thinking about that and tying in with the problem solving process, or the problem solving steps, we want to help kids understand where they feel like they are on that problem solving process.
So these are the strategies that we want to help kids with. I'm in this problem, am I stuck? The fact that they're thinking about am I stuck is one of those metacognitive skills in thinking like, "Okay. What do I have to do? Okay. Maybe I'm going to use one of Pólya-like strategy", and kid obviously is not going to say that, but you as a teacher are that coach like going, "Okay. Well one of Pólya's strategies." Now, remember, you're not going to say that. But you're going to say, "One of the strategies he uses in his four step plan, which is not listed if you search it, but it's like, 'How can I break this problem down into smaller chunks so that I can understand it better?;" That's one of the strategies inside understanding the problem. If you can break this problem down into a smaller numbers, or a smaller scenario, sometimes we can understand that better, and then we can reapply it to the original problem.
Those skills are missing. They're missing in any of the resources that we look for or have access to. Think about professional development sessions. How many of the professional development sessions have you been in lately where they focused on the thinking about thinking skills, like the metacognitive skills? I'm in this problem and we're going to going down this route and I have to switch gears. I have to go, "You know what? I'm going to go over here." Helping kids realize that's okay and helping students think about, "What could I try next?" And what Shoenfeld's saying is more important, it is so important in teaching mathematics. \.
And this is why teaching math is so hard, because we're trying to blend how do kids can think about mathematics while integrating that with making sure there's content is covered, and making sure that we're following a problem solving process. So we want to make sure that these things come out. So one of the summary here that we've been thinking about is that, "How can we be more coach-like in helping kids solve problems first, and us kind of helping them think about how they're solving problems?" Because that skill is going to be translatable not only in just math class, but it's going to be translatable outside of math class. And so that is so important when teaching mathematics.

Kyle: I just want to highlight something too, because you had mentioned about just this idea of, "Where does this thinking about your thinking come up in curriculum and so forth?" And the reality is that it actually, it's in a lot of the curriculum out there, but it's usually in the part of the curriculum that isn't necessarily associated with the content. So for example, here in Ontario, the front matter as we call it, is really all the meat and potatoes of these things that we're talking about here. And to be honest, they're the things that matter, but yet we hyper-focus, because our standardized test hyper-focus on it because it's easier to measure. Let's be honest, I can tell whether you know how to complete the square a lot faster, or for me to be able to say how great or how well you can problem solve, right?
In general, usually what we do is we give them problems that are around the content and then we try to put them and make them thinking questions, or problems that are challenging to solve, because now that's problem solving. But then what we do is we sort of assign, even when we do standard space grading, which is definitely a huge step forward from more traditional approaches to grading, but when you address that, you typically are addressing their content knowledge of that idea, not necessarily how great they are at problem solving, right? They're sort of getting ranked on, "You can solve some of these problems, but some of the harder problems you don't." Or here in Ontario you used to explicitly call TACK or-

Jon: KIKA-

Kyle: You added KIKA. There you go, KIKA. Different districts had different acronyms for it, but TACK is breaking down thinking application, communication, and knowledge.
And those four things are still here in Ontario. They are still what we use to sort of organize our assessment and evaluation policies. But the reality is that when it comes to say the thinking area, which is where you'd probably lump problem solving, we're not very good at teaching it, or assessing and evaluating it yet. So it's something that people want and policymakers want these things, but what we get, we say we want this, but then our actions through standardized tests and the way we score, the way we assess and evaluate, typically are saying something very different. So if we truly want to break free from again, this fossilized remains idea, we're going to have to actually act on putting problem solving to the forefront, or thinking mathematically to the forefront, instead of content itself to the forefront.

Jon: In this paper, Schoenfeld brings some of that into the cultural surround when we get to that part of the paper, but he also kind of addresses this idea that you brought up that curriculums in lists of standards. So thinking about even Common Core, we have the practices that are listed at the front. When he says this in his paper, he is saying, "It's supposed to be embedded everywhere." So essentially it's everywhere. But the fact that when all of these standard lists, so here in Ontario we call it the Ontario curriculum, the Common Core, but when you go granular on a set of standards and get down to it, like you said, it's easier to measure, it's easier to track those things, but even the fact that you're just listing them, you're putting that emphasis on achieving those things and then by not listing specifically in those areas a standard that is attached to a practice, you're essentially saying, "All these practices are everywhere, but at nowhere."
They're everywhere but nowhere at the same time, because they're kind of like at the front or at the back or we're saying, "You can embed these everywhere, but we're not going to actually list them every time you need to think about it." So he addresses that as well in the cultural surrounding why that's the case when we've developed curriculums over the course of years and where this curriculum has come from. So if we go back here, Kyle, when we're thinking mathematically is one of the areas that we need to consider, and why teaching math is so hard and learning math is so hard. And inside that we've outlined his four key elements that go into how to help kids, and how to help yourself think mathematically. And the first one we talked about here was about what our mathematical beliefs and understanding what it means to learn mathematics for ourselves, also our identities around mathematics.
We talked about access to resources including mathematical processes, and content, and practices. So that is an important element of thinking mathematically, we talked, the third one here we talked about was access to a Pólya-like problem solving strategy or strategies. So helping kids understand how to actually think about problems and problem solve with those problems. We often omit those when we are teaching math. We need to bring them and focus on them more.
And then the last one here we talked about is kind of the metacognitive performance, and specifically our self-monitoring skills that we have to focus on with our students so that we can think mathematically at a deeper level. So if we focus on those four, we can help our students think mathematically, and eventually we can start to strengthen our students' abilities in mathematics, and have that more confidence in mathematics. Kyle, in future episodes here, we're going to take this a step further. We're going to look at the learning environment, which is structuring our classrooms to allow students to think mathematically, and have this robust learning environment. So we're going to dive there and then in another episode we're going to break down what Schoenfeld outlines as the cultural surround. Why math has historically been hard to learn and teach, and what outside factors are preventing us from learning this, or teaching this at a deeper level.

Kyle: I love it. I love it. And I think it's worth mentioning that what we're not saying is that, "None of the content is worth exploring", but I think it's like when you leave here today and you think about how important it is for us to help students develop true mathematical thinking, ability thinking skills. What I want you to think of your content as those standards that are specific to math concepts and skills that are in your curriculum, whether you're Common Core, whether you're here in Ontario like Jon and I, wherever you are, think of those as excuses ideas to inspire mathematical thinking in your classroom. And when you explore the Make Math Moments catalog of 60 plus problem-based units, what you're going to find is day one of every single unit, and typically day three as well, focus around a problem. And the problem is structured around one of those content pieces that you likely have to, "Cover", in your standards, but the purpose is to use that standard as an excuse to engage in the problem solving process.
So we've crafted these lessons with tasks so that students can truly problem solve using a low floor, having a high ceiling, and engaging in true mathematical thinking. On day two and day four, typically what we're doing is we're actually trying to build our conceptual knowledge and understanding of those big ideas from the content side of things, developing towards procedural fluency. So what we don't want you to hear is this idea that all we do is problem solving and there's never any sort of skill development, or any sort of procedural fluency being built. That is not what's being said here. All we're doing is we're trying to get you to sort of think differently about how we can use what you feel is these things you have to cover, and use them as a way to help you get to the problem solving. It's not an either/or.
It's how do we use one to promote the other? And that's how we do it here at Make Math Moments. Today as we summarize this one first big idea from that paper, and as Jon said, we'll be coming back with the two other big ideas in two future episodes. But in today's episode, I hope you heard us talking about how important math content, knowledge and proficiencies are. We call that the roots of our math program tree. We talked about mindset and beliefs. That's the sun, the water, and the soil of our tree. Those mindsets and beliefs are so key because again, your belief and your mindset around what math is going to dictate how that happens in your classroom. Whether we're just getting students to essentially leverage the use of fossilized remains to calculate, or whether we actually help them to think mathematically and create remains that they can leave behind when they leave your classroom.
And then finally, the third one that we dug into is the idea of the pedagogical content knowledge. Of course, again, as we mentioned, you can't just take a framework and go grab a place mat that says, "Problem solving framework at the top", and expect that that's going to be helpful. There are so many pedagogical moves that have to take place, and that is the part of our tree where we're exploring in the branches of the tree. The branches of the tree are sort of what gets us to the resources that we're actually going to use in our tree or our leaves that we're developing.
So keep that in mind. We hit three of them today. We will be hitting all six by the time we are done with his three big ideas. So I'm hoping that you are making connections based on all our conversations on how we can build an effective math program through our analogy of the tree, and how you see it relating to the research. Because everything we do here at Make Math Moments, it's not our research. We're taking research just like this particular paper, and we're trying to make it more consumable, more accessible for more educators around the world, so that you can actually put it into practice in your classroom.

Jon: And if you want to take a little bit of an assessment on your math classroom and the practices that you are currently using, or the resources you're using, or what your plans look like, you can get a full kind of assessment and see where you're strong, and where also maybe you can make some changes and make some growth in those areas. So you can take an assessment that we've built. And after taking that assessment, you will get a full report on those areas, and also some next steps on taking action. So you can do that right now over at makemathmoments.com/report. Whether you are a classroom teacher or a district leader, there's a little switch there for you to get to the right one. We have two different assessments depending on your role in education. So each one is going to help you with your current role.

Kyle: Awesome. If you haven't yet, do us a huge favor and leave us a rating and review over on Apple Podcasts. You don't know how much. It helps us reach more Math Moment Makers just like you. We're also on social media, so say, "Hi." Don't be a stranger. We're real people and we love chatting with the math moment maker community.

Jon: In order to ensure you don't miss out on new episodes as they come out each week, just hit the subscribe button on your podcast platform. And if you've already done that awesome job, welcome back. Show notes and links to resources. And a complete transcript for this episode can be found over at our show notes page, which is at makemathmoments.com/episode230. Makemathmoments.com/episode230.

Kyle: Hey, thanks Math moment makers for listening to the Making Math Moments that Matter podcast where we help you grow your mathematics program at the district level or in your classroom, like a strong, healthy, and balanced tree so you can impact more students and ensure that that tree reaches far and wide. Well, until next time, my math moment Maker friends. I'm Kyle Pearce.

Jon: And I'm John Orr.

Kyle: High fives for us.

Jon: And.

Kyle: Hi.

Jon: Five.

Kyle: For you.

Speaker 3: singing.

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