Thank you for taking the time to write these posts and delve into the research. All of this is so important. Discussing the different types of mathematical learning and approaches that work in each case is extremely practical. In my 10th grade math class I use a wide variety of instructional approaches depending on the content, where we are in a unit, student mastery data, and the nature of the mathematics involved. Most of the time students are constructing meaning by working through problems and discussing them in a guided format. In my support block at the end of the school day where I work with 10-15 students who are struggling to access the curriculum, I lean heavily on explicit instruction. Teachers need to know how to leverage a variety of instructional approaches and when each one is appropriate but that will never happen if there are entrenched viewpoints that only one type of instruction is “right”.
Kristen, this is music to my ears. What you're describing is a thoughtful effort to choose the right tool for the job, exactly the thinking I'm looking for.
“Teachers need to know how to leverage a variety of instructional approaches and when each one is appropriate but that will never happen if there are entrenched viewpoints that only one type of instruction is “right”.” <— agreed! However, with certain ages and populations of students, I think we can say that some approaches are likely to be more beneficial as a default. I teach in a Title I elementary school with a high percentage of students experiencing poverty.
Most current curriculum mandate problem-solving first constructivist approaches. The problem is that these approaches assume a certain amount of background knowledge that many of our students don’t possess. The subsequent lack of systematic practice (the guided inquiry takes up the bulk of the lessons) means many of our students never reach automaticity or secure conceptual understanding. The kids who thrive are those who get extra support and practice outside of school. In my context, the research supports explicit instruction being the default, with inquiry added on as students progress and show mastery.
Explicit instruction, for most of the students I teach, is the path that leads to equitable inquiry later on.
I agree, Daniel, that many curriculum materials are lacking sufficient explicit instruction and practice resources. In my current school 50% of our 9th grade was level 1 on SBAC, and before this I taught 6th grade in a Title I school. What I have found works best is to actually front load the explicit instruction and practice of root skills in class (I use a progression of Do Now then Skill Builder for fluency followed by explicit instruction of any content skills that are needed for the inquiry task) that means by the time students get to engage with the inquiry task they have had 30 minutes of explicit practice and skill building first. It also ensures that they get that explicit practice because it happens at the beginning. I’ve found this also makes the inquiry task move more quickly because students can access it better. The key is that I don’t explicitly teach the concepts that students arrive at through the inquiry task — I only teach the skills they need to access it. Most curriculums don’t do enough to point out to teachers what the root skills are that are needed for access, and it has taken many years of work for me to modify my curriculum in this way.
That sounds like a solid approach, and I’m glad you found something that works. I love the idea of front loading the practice. Do you have any thoughts about K-2? I’ve been thinking a lot about this age and inquiry. Two things make me wary about curriculum that relies on problem-solving first approaches at this age. The first is that students working memory is still developing, so their working memory is easily overwhelmed. The second is that their executive functioning is also still developing. When you combine these two factors, I’ve seen that by the time many students reach 2nd grade, they are already years behind. When I taught 3rd grade, this meant only a portion of the class had enough secure knowledge to profit from all the rich tasks I was using.
I keep thinking about this, because if we get K-2 right, I think it opens a lot of doors for different approaches later. If we don’t, the task of catching them up can feel herculean.
In the real world what I see happening is this: problem-solving first curriculum is used K-2. This separates out kids who have support at home. Kids who don’t have this privilege never build the foundations necessary for inquiry in later grades. The rich get richer and so many kids get left behind.
I don’t have any experience with elementary and I do believe it is different. What you are describing sounds like a genuine problem. My thoughts based on what I’ve seen work in 6th grade would be again to front load explicit instruction and practice and then use the inquiry tasks to build conceptual understanding when there is already fluency with facts. I wonder if your school has any built-in support time? In middle school I have found that to be game-changing when you can pull a select group of students who need additional support with fluency outside of class time so that in class they can have a more level playing field.
Thank you for this—this is exactly the kind of concrete, experience-based perspective I was hoping to hear. And I think the research backs you up in important ways. The Gersten IES Practice Guide, which I cited in the post, recommends explicit and systematic instruction specifically for students who are struggling, with the strongest evidence rating of any recommendation in that guide.
Where I'd love to hear more from you is about what explicit instruction looks like in your classroom on a good day. When it's working for your students, what's happening? How do you sequence modeling, feedback, discussion, and so on?
Thanks for the question, Bill. It's a good one to reflect on. I especially like the focus on a "good day" because this is such a work in progress for me.
For context, I teach in a district where only 30% of students pass 5th-grade end-of-year tests. By eighth grade, only 17% are on grade level, and fully 50% of our current 8th-grade cohort is two or more grade levels behind. COVID plays a role, but the crisis is deeper than that.
More context: I've been a die-hard believer in inquiry-based teaching for most of my career. My critique of our curriculum was always that the inquiry tasks weren't sufficiently low-floor, high-ceiling. I used to substitute IM's material even though it's not our adopted curriculum because it was so much more teacher- and student-friendly.
But over the past year and a half, my professional practice and beliefs have shifted. What changed was watching my daughter, who was struggling to learn to read, finally start to make rapid progress as soon as she started receiving systematic instruction in foundational skills as a result of our district embracing the science of reading. The sudden growth and success that followed led me to interrogate my beliefs when it came to math. This all led to a deep dive into cognitive load theory, the instructional hierarchy, desirable difficulties, and a hard look at the evidence for the specific population I teach.
As a result, I shifted. Now I think of my teaching as a trajectory, with inquiry as the destination, not the starting point.
To answer your question more directly, I can share what I aim for based upon what we’re working on now. It's the end of 4th grade and my students are learning about decimals. I anchored the unit with a story about Olympic sprinters. I used Usain Bolt as an example and had students predict how many seconds it took him to run the 100-meter dash. We watched a clock tick upward, looked at a photo of a track, and estimated. From there, we discussed the problem with measuring a runner in whole seconds. I showed them a stopwatch, drew their attention to the hundredths place, and taught them directly and interactively why decimals are needed. Students made updated predictions, then watched the world record sprint to see how close they were. Lots of storytelling, explicit vocabulary, turn and talks, and conceptual engagement up front.
The next phase was more direct. I explicitly taught the procedures for working with decimals, such as converting tenths to hundredths. We practiced with paired problems, scaffolding, and frequent checks for understanding. Kids used mini-whiteboards constantly. Critically, I didn’t embed any of this learning in story problems or rich tasks at this point. I wanted students to become secure in their knowledge and vocabulary first. Once they show enough success, I will move on to a generalization/adaptation level inquiry task.
Am I pulling this off perfectly? Not even close! Some days I wing it. I have three young kids at home and am teaching all subjects. The struggle is real. The curriculum is imperfect. But following this approach, even on rough days, I see students experiencing far more success than I have in the past. And when I employ inquiry tasks now, they are more joyous and productive for everyone. It also has alleviated a lot of anxiety for everyone.
When I meet with families at the beginning of the year, the number one thing parents mention is how anxious their children are about math. I now directly attribute this to an overuse of inquiry. Our curriculum uses inquiry every day for nearly every lesson, which leaves many students engaging unproductively on a daily basis. By giving them more explicit instruction, with lots of opportunities to respond successfully, I've watched their attitudes toward math transform. They are more willing to engage and more confident. This isn't brought about through productive struggle. It's engineered through productive success. And once students experience that success and build confidence, I see them naturally desire more challenges and engage in deeper thinking.
Something I think gets overlooked in a lot of these conversations about what approaches are best is that in high-achieving districts, many students are getting explicit instruction outside of school, through family support, tutoring, and structured practice at home. What those students experience in the classroom looks like successful inquiry, but the foundation was often built somewhere else. My daughter is in the 99th percentile, but I’ve been teaching her explicitly since she was 3. Most of the math she’s learned has been from me, at home, and not in school. So the question I keep coming back to is: how do we build that foundation inside the school day for students who don't have that privilege?
It's telling that when I engage online with teachers experiencing strong success with highly inquiry-focused programs in elementary schools, and I look into their context, they almost invariably teach in a high-SES setting. My hunch is that those students would do well regardless of approach, because so much support exists outside of school.
I want to be careful here, because I know how easily this gets pigeonholed. I am not saying explicit instruction is for poor kids and inquiry is for rich kids. I want to be emphatic that this is not my argument. What I am saying is that I want all students to be able to access rich tasks, and I think there is sometimes a real blindness about what it takes to get there and about the outside influences that make inquiry appear to work seamlessly in certain settings.
I've also been able to collect a piece of data I think is pretty rare in these conversations. I got my union to survey all of the teachers in our district on our math curriculum. Overwhelmingly, teachers say there is not enough practice. The curriculum moves too fast, too many strategies are introduced at once, and students aren't retaining what they've learned. Roughly 9 out 10 of teachers say our curriculum is failing students who are below grade level or receiving special education services. I think if you were to ask teachers across Title I schools who are asked to teach with inquiry-first curriculum, you’d find similar responses across the nation.
So I’ve shifted. I used to believe in productive struggle as it’s popularly conceptualized as the best way to teach young students mathematics. I no longer think productive struggle, in the sense of having students grapple with novel problems they haven't been taught how to approach, is appropriate as a daily default for our youngest learners, particularly outside of highly privileged contexts. It eats up too much instructional time and leaves too many kids behind.
Two other shifts have made a difference. First, I now teach math facts explicitly, daily, and systematically using Facts on Fire, which is structured and incremental timed practice. In three months, I went from 1 out of 23 to 18 out of 23 students fully fluent in multiplication facts. Before, I had internalized that all timed practice was harmful and anxiety-producing. I actually see the opposite now. Once students achieve automaticity, everything else becomes so much easier for them. Second, I use SpringMath for high-quality fluency practice in the core knowledge and concepts students need to access on-level mathematics. Together, these two tools are building the foundation that makes everything else possible.
After all that, my clearest answer is that on a good day, explicit instruction looks like hooking them with a connection to lived experience, teaching a procedure or concept with lots of checks for understanding and interaction, some discussion, and finally independent practice. But it doesn't always follow that pattern. It depends on how much knowledge the majority of students have and their level of mastery.
I'm open to being wrong about all this, and I'll have to see how my test scores look in a month. But I do firmly believe getting K-2 right is key for any approach to be successful down the line.
Daniel, thanks for this detailed and thoughtful response. There are so many things I want to say here.
First, your decimals lesson with Usain Bolt was rich, and for me it cut through a lot of the labels. You started with a problem, asking students to predict how many seconds it took, and this led to a discussion. Then you did some explicit work on procedures, which you had motivated the students to be interested in. You could label this as problem-based instruction or you could label it as explicit instruction or you could forget the labels and just call it good teaching.
I was struck by your "productive success" phrase. Engineering situations where kids succeed and build confidence is important. But I wonder how different this is from productive struggle. I think people sometimes overemphasize the "struggle" part and forget the "productive" part. You have flipped that. But I would say that a well designed task where students succeed *because* they figured something out is both productive success and productive struggle. Your decimals lesson is an example of that.
As to your point about high-SES schools, ouch, yes. Your survey results, that 88% of the teachers say the curriculum is failing struggling students, are striking. However I appreciate your care not to create a false dichotomy here. An inquiry-based approach has to take care to be building on what students have in long-term memory, and in a school where the students have better supports outside school teachers can probably rely on them having more fact fluency. Whereas in your case you have to build that fact fluency. It's truly impressive what you have done there. But what I see is that you don't stop there. You then bring that fact fluency to bear on rich conceptual development as you did with the decimals lesson.
Honestly, I think fact fluency should be part of any curriculum, whatever the approach. It's strange to me that somewhere along the line inquiry became a slogan that forbade memorization. It's so easy to fall into false dichotomies. What I see is that you are trying hard not to do that, and I acknowledge the difficulty of what you are doing. I'd love to hear how it goes.
Great way to wake up this morning. This is my area of passion. Becoming fluent with the necessary facts and foundations are essential for freeing up space - preventing cognitive overheating. - In exploring the research of Arthur Baroody- the initial learning of these basic facts can involve a Constructicist pedagogical approach in order to have them stored in long term memory
Bill, this is a synthesis I’ve been waiting for — and I find myself in genuine agreement. What strikes me most is how naturally you’ve sorted the guides by learning target, to illustrate how each one reaches for a different set of tools without any apparent anxiety about the label.
The way I hear you presenting this: when a lesson is aimed at conceptual understanding, the most effective path begins with why — why does this work mathematically — then builds toward how we generalize the procedure, and closes with when this technique is most useful or appropriate, perhaps set alongside another approach. The direction of thinking matters: student articulation first, not teacher performance first.
When the focus shifts to applications and messy real-world modeling, the teacher’s toolbox gets leveraged differently. Expert demonstration sets a frame of reference. Guided inquiry leads students down more productive paths than leaving them entirely to their own devices. And there has to be structured time for reflective iteration — refining and improving models through feedback. The right close for a modeling unit might be a gallery walk, a display of the various methods different groups employed, with a teacher-led comparison that helps students see the landscape.
For the youngest learners, developing fluency to offload working memory burden — and clear the way for conceptual understanding — is best accomplished through structured and semi-structured explorations intertwined with explicit instruction. The explicit interventions aren’t carrying the content alone; they’re there to emphasize, revoice, and sharpen the precision of patterns students have already been guided to notice.
And then there’s algebra — the symbolizing of patterns into a language of structure. A research-backed entry point here is the targeted examination of a solution as a lesson for learning. The analysis of reasoning builds deeper understanding of structure, and the close of a lesson in this territory ought to center on the strategic selection of strategies for mathematical problem solving.
None of these are in tension. They’re just different jobs, calling for different tools. I’m glad you’re building this thing in public — looking forward to the next Fridays.
Thank you for taking the time to write these posts and delve into the research. All of this is so important. Discussing the different types of mathematical learning and approaches that work in each case is extremely practical. In my 10th grade math class I use a wide variety of instructional approaches depending on the content, where we are in a unit, student mastery data, and the nature of the mathematics involved. Most of the time students are constructing meaning by working through problems and discussing them in a guided format. In my support block at the end of the school day where I work with 10-15 students who are struggling to access the curriculum, I lean heavily on explicit instruction. Teachers need to know how to leverage a variety of instructional approaches and when each one is appropriate but that will never happen if there are entrenched viewpoints that only one type of instruction is “right”.
Kristen, this is music to my ears. What you're describing is a thoughtful effort to choose the right tool for the job, exactly the thinking I'm looking for.
“Teachers need to know how to leverage a variety of instructional approaches and when each one is appropriate but that will never happen if there are entrenched viewpoints that only one type of instruction is “right”.” <— agreed! However, with certain ages and populations of students, I think we can say that some approaches are likely to be more beneficial as a default. I teach in a Title I elementary school with a high percentage of students experiencing poverty.
Most current curriculum mandate problem-solving first constructivist approaches. The problem is that these approaches assume a certain amount of background knowledge that many of our students don’t possess. The subsequent lack of systematic practice (the guided inquiry takes up the bulk of the lessons) means many of our students never reach automaticity or secure conceptual understanding. The kids who thrive are those who get extra support and practice outside of school. In my context, the research supports explicit instruction being the default, with inquiry added on as students progress and show mastery.
Explicit instruction, for most of the students I teach, is the path that leads to equitable inquiry later on.
I agree, Daniel, that many curriculum materials are lacking sufficient explicit instruction and practice resources. In my current school 50% of our 9th grade was level 1 on SBAC, and before this I taught 6th grade in a Title I school. What I have found works best is to actually front load the explicit instruction and practice of root skills in class (I use a progression of Do Now then Skill Builder for fluency followed by explicit instruction of any content skills that are needed for the inquiry task) that means by the time students get to engage with the inquiry task they have had 30 minutes of explicit practice and skill building first. It also ensures that they get that explicit practice because it happens at the beginning. I’ve found this also makes the inquiry task move more quickly because students can access it better. The key is that I don’t explicitly teach the concepts that students arrive at through the inquiry task — I only teach the skills they need to access it. Most curriculums don’t do enough to point out to teachers what the root skills are that are needed for access, and it has taken many years of work for me to modify my curriculum in this way.
That sounds like a solid approach, and I’m glad you found something that works. I love the idea of front loading the practice. Do you have any thoughts about K-2? I’ve been thinking a lot about this age and inquiry. Two things make me wary about curriculum that relies on problem-solving first approaches at this age. The first is that students working memory is still developing, so their working memory is easily overwhelmed. The second is that their executive functioning is also still developing. When you combine these two factors, I’ve seen that by the time many students reach 2nd grade, they are already years behind. When I taught 3rd grade, this meant only a portion of the class had enough secure knowledge to profit from all the rich tasks I was using.
I keep thinking about this, because if we get K-2 right, I think it opens a lot of doors for different approaches later. If we don’t, the task of catching them up can feel herculean.
In the real world what I see happening is this: problem-solving first curriculum is used K-2. This separates out kids who have support at home. Kids who don’t have this privilege never build the foundations necessary for inquiry in later grades. The rich get richer and so many kids get left behind.
I don’t have any experience with elementary and I do believe it is different. What you are describing sounds like a genuine problem. My thoughts based on what I’ve seen work in 6th grade would be again to front load explicit instruction and practice and then use the inquiry tasks to build conceptual understanding when there is already fluency with facts. I wonder if your school has any built-in support time? In middle school I have found that to be game-changing when you can pull a select group of students who need additional support with fluency outside of class time so that in class they can have a more level playing field.
Thank you for this—this is exactly the kind of concrete, experience-based perspective I was hoping to hear. And I think the research backs you up in important ways. The Gersten IES Practice Guide, which I cited in the post, recommends explicit and systematic instruction specifically for students who are struggling, with the strongest evidence rating of any recommendation in that guide.
Where I'd love to hear more from you is about what explicit instruction looks like in your classroom on a good day. When it's working for your students, what's happening? How do you sequence modeling, feedback, discussion, and so on?
Thanks for the question, Bill. It's a good one to reflect on. I especially like the focus on a "good day" because this is such a work in progress for me.
For context, I teach in a district where only 30% of students pass 5th-grade end-of-year tests. By eighth grade, only 17% are on grade level, and fully 50% of our current 8th-grade cohort is two or more grade levels behind. COVID plays a role, but the crisis is deeper than that.
More context: I've been a die-hard believer in inquiry-based teaching for most of my career. My critique of our curriculum was always that the inquiry tasks weren't sufficiently low-floor, high-ceiling. I used to substitute IM's material even though it's not our adopted curriculum because it was so much more teacher- and student-friendly.
But over the past year and a half, my professional practice and beliefs have shifted. What changed was watching my daughter, who was struggling to learn to read, finally start to make rapid progress as soon as she started receiving systematic instruction in foundational skills as a result of our district embracing the science of reading. The sudden growth and success that followed led me to interrogate my beliefs when it came to math. This all led to a deep dive into cognitive load theory, the instructional hierarchy, desirable difficulties, and a hard look at the evidence for the specific population I teach.
As a result, I shifted. Now I think of my teaching as a trajectory, with inquiry as the destination, not the starting point.
To answer your question more directly, I can share what I aim for based upon what we’re working on now. It's the end of 4th grade and my students are learning about decimals. I anchored the unit with a story about Olympic sprinters. I used Usain Bolt as an example and had students predict how many seconds it took him to run the 100-meter dash. We watched a clock tick upward, looked at a photo of a track, and estimated. From there, we discussed the problem with measuring a runner in whole seconds. I showed them a stopwatch, drew their attention to the hundredths place, and taught them directly and interactively why decimals are needed. Students made updated predictions, then watched the world record sprint to see how close they were. Lots of storytelling, explicit vocabulary, turn and talks, and conceptual engagement up front.
The next phase was more direct. I explicitly taught the procedures for working with decimals, such as converting tenths to hundredths. We practiced with paired problems, scaffolding, and frequent checks for understanding. Kids used mini-whiteboards constantly. Critically, I didn’t embed any of this learning in story problems or rich tasks at this point. I wanted students to become secure in their knowledge and vocabulary first. Once they show enough success, I will move on to a generalization/adaptation level inquiry task.
Am I pulling this off perfectly? Not even close! Some days I wing it. I have three young kids at home and am teaching all subjects. The struggle is real. The curriculum is imperfect. But following this approach, even on rough days, I see students experiencing far more success than I have in the past. And when I employ inquiry tasks now, they are more joyous and productive for everyone. It also has alleviated a lot of anxiety for everyone.
When I meet with families at the beginning of the year, the number one thing parents mention is how anxious their children are about math. I now directly attribute this to an overuse of inquiry. Our curriculum uses inquiry every day for nearly every lesson, which leaves many students engaging unproductively on a daily basis. By giving them more explicit instruction, with lots of opportunities to respond successfully, I've watched their attitudes toward math transform. They are more willing to engage and more confident. This isn't brought about through productive struggle. It's engineered through productive success. And once students experience that success and build confidence, I see them naturally desire more challenges and engage in deeper thinking.
Something I think gets overlooked in a lot of these conversations about what approaches are best is that in high-achieving districts, many students are getting explicit instruction outside of school, through family support, tutoring, and structured practice at home. What those students experience in the classroom looks like successful inquiry, but the foundation was often built somewhere else. My daughter is in the 99th percentile, but I’ve been teaching her explicitly since she was 3. Most of the math she’s learned has been from me, at home, and not in school. So the question I keep coming back to is: how do we build that foundation inside the school day for students who don't have that privilege?
It's telling that when I engage online with teachers experiencing strong success with highly inquiry-focused programs in elementary schools, and I look into their context, they almost invariably teach in a high-SES setting. My hunch is that those students would do well regardless of approach, because so much support exists outside of school.
I want to be careful here, because I know how easily this gets pigeonholed. I am not saying explicit instruction is for poor kids and inquiry is for rich kids. I want to be emphatic that this is not my argument. What I am saying is that I want all students to be able to access rich tasks, and I think there is sometimes a real blindness about what it takes to get there and about the outside influences that make inquiry appear to work seamlessly in certain settings.
I've also been able to collect a piece of data I think is pretty rare in these conversations. I got my union to survey all of the teachers in our district on our math curriculum. Overwhelmingly, teachers say there is not enough practice. The curriculum moves too fast, too many strategies are introduced at once, and students aren't retaining what they've learned. Roughly 9 out 10 of teachers say our curriculum is failing students who are below grade level or receiving special education services. I think if you were to ask teachers across Title I schools who are asked to teach with inquiry-first curriculum, you’d find similar responses across the nation.
So I’ve shifted. I used to believe in productive struggle as it’s popularly conceptualized as the best way to teach young students mathematics. I no longer think productive struggle, in the sense of having students grapple with novel problems they haven't been taught how to approach, is appropriate as a daily default for our youngest learners, particularly outside of highly privileged contexts. It eats up too much instructional time and leaves too many kids behind.
Two other shifts have made a difference. First, I now teach math facts explicitly, daily, and systematically using Facts on Fire, which is structured and incremental timed practice. In three months, I went from 1 out of 23 to 18 out of 23 students fully fluent in multiplication facts. Before, I had internalized that all timed practice was harmful and anxiety-producing. I actually see the opposite now. Once students achieve automaticity, everything else becomes so much easier for them. Second, I use SpringMath for high-quality fluency practice in the core knowledge and concepts students need to access on-level mathematics. Together, these two tools are building the foundation that makes everything else possible.
After all that, my clearest answer is that on a good day, explicit instruction looks like hooking them with a connection to lived experience, teaching a procedure or concept with lots of checks for understanding and interaction, some discussion, and finally independent practice. But it doesn't always follow that pattern. It depends on how much knowledge the majority of students have and their level of mastery.
I'm open to being wrong about all this, and I'll have to see how my test scores look in a month. But I do firmly believe getting K-2 right is key for any approach to be successful down the line.
Daniel, thanks for this detailed and thoughtful response. There are so many things I want to say here.
First, your decimals lesson with Usain Bolt was rich, and for me it cut through a lot of the labels. You started with a problem, asking students to predict how many seconds it took, and this led to a discussion. Then you did some explicit work on procedures, which you had motivated the students to be interested in. You could label this as problem-based instruction or you could label it as explicit instruction or you could forget the labels and just call it good teaching.
I was struck by your "productive success" phrase. Engineering situations where kids succeed and build confidence is important. But I wonder how different this is from productive struggle. I think people sometimes overemphasize the "struggle" part and forget the "productive" part. You have flipped that. But I would say that a well designed task where students succeed *because* they figured something out is both productive success and productive struggle. Your decimals lesson is an example of that.
As to your point about high-SES schools, ouch, yes. Your survey results, that 88% of the teachers say the curriculum is failing struggling students, are striking. However I appreciate your care not to create a false dichotomy here. An inquiry-based approach has to take care to be building on what students have in long-term memory, and in a school where the students have better supports outside school teachers can probably rely on them having more fact fluency. Whereas in your case you have to build that fact fluency. It's truly impressive what you have done there. But what I see is that you don't stop there. You then bring that fact fluency to bear on rich conceptual development as you did with the decimals lesson.
Honestly, I think fact fluency should be part of any curriculum, whatever the approach. It's strange to me that somewhere along the line inquiry became a slogan that forbade memorization. It's so easy to fall into false dichotomies. What I see is that you are trying hard not to do that, and I acknowledge the difficulty of what you are doing. I'd love to hear how it goes.
https://education.illinois.edu/profile/art-baroody
His research is listed on his site - I shared the article
https://mathematicalmusings.substack.com/p/strike-three?utm_campaign=comment-list-share-cta&utm_medium=web&comments=true&commentId=241431194
Great way to wake up this morning. This is my area of passion. Becoming fluent with the necessary facts and foundations are essential for freeing up space - preventing cognitive overheating. - In exploring the research of Arthur Baroody- the initial learning of these basic facts can involve a Constructicist pedagogical approach in order to have them stored in long term memory
to be accessed later. Thank you for this.
Thanks Matthew. Can you point me to an article by Baroody that you would recommend starting with?
Bill, this is a synthesis I’ve been waiting for — and I find myself in genuine agreement. What strikes me most is how naturally you’ve sorted the guides by learning target, to illustrate how each one reaches for a different set of tools without any apparent anxiety about the label.
The way I hear you presenting this: when a lesson is aimed at conceptual understanding, the most effective path begins with why — why does this work mathematically — then builds toward how we generalize the procedure, and closes with when this technique is most useful or appropriate, perhaps set alongside another approach. The direction of thinking matters: student articulation first, not teacher performance first.
When the focus shifts to applications and messy real-world modeling, the teacher’s toolbox gets leveraged differently. Expert demonstration sets a frame of reference. Guided inquiry leads students down more productive paths than leaving them entirely to their own devices. And there has to be structured time for reflective iteration — refining and improving models through feedback. The right close for a modeling unit might be a gallery walk, a display of the various methods different groups employed, with a teacher-led comparison that helps students see the landscape.
For the youngest learners, developing fluency to offload working memory burden — and clear the way for conceptual understanding — is best accomplished through structured and semi-structured explorations intertwined with explicit instruction. The explicit interventions aren’t carrying the content alone; they’re there to emphasize, revoice, and sharpen the precision of patterns students have already been guided to notice.
And then there’s algebra — the symbolizing of patterns into a language of structure. A research-backed entry point here is the targeted examination of a solution as a lesson for learning. The analysis of reasoning builds deeper understanding of structure, and the close of a lesson in this territory ought to center on the strategic selection of strategies for mathematical problem solving.
None of these are in tension. They’re just different jobs, calling for different tools. I’m glad you’re building this thing in public — looking forward to the next Fridays.
Casey, this is a great addition to the building! Thanks for joining.