Make Math Moments Academy › Forums › Full Workshop Reflections › Module 1: Introduction To Making Math Moments That Matter › Lesson 15: Building Conceptual Understanding in A Problem Based Lesson › Lesson 15: Question & Discussion
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Lesson 15: Question & Discussion
Kyle Pearce updated 2 weeks, 5 days ago 82 Members · 135 Posts 
How does the lesson style you witnessed in the Lesson 15 video compare with traditional math lessons? What are the benefits of this style of lesson? What are your reservations about this style of lesson?

The lesson style witnessed may lead to more open ended questions and expectations that students will have to think for themselves rather than being led by similar examples demonstrated by a class teacher.
The fear of using this approach is students may get off task, not tackle the tasks in the most efficient and accurate way or learn new ideas.
If this lesson style is a success students will become more adept at problem solving and communicating their ideas mathematically.

Thanks for sharing!
I wonder what the result would be if a student didn’t tackle the problem in the most accurate or efficient way? Why might this be a negative thing? Is there any positive that could come of it?


Traditional math lessons focus on the teacher showing you what to do, students, you mimic what teacher did.
This lesson style is more focused on engagement and understanding. There is a sense of discovery for the student and there is very much the answering of the question of why/where do these formulas come from.
My reservations for this style of math is the pacing, and my own personal ability to flexibly sequence the strategies students are sharing and working with.

Thanks for sharing!
Pacing is certainly a concern for those who are new to problem based learning and understandably so. However, while pacing feels slower approaching lessons in this way, my wonder is how many students were keeping up in a traditional gradual release of responsibility lesson? I find we tend to keep a pace whether students are keeping up or not. Problem based lessons allow us to go deeper and meet more students where they are.


I like the definition by Peter Liljedahl that traditional teaching originates from assuming kids do not know anything, while problem based learning looks to build knowledge off of what the students already know. So, if you assume kids do not know anything, then you need to tell them everything. If you assume they have a base, then all you need to do is push them when they need help.

Great take away. The idea of “low floor, high ceiling” is so important to leverage this as well.


There are so many benefits to this type of lesson. Kids are intersted, they see the value in their indivudul style of solving the problem, they are differentiatable, and there is “Real Math” — not just learning algorighims — going on. My only hesitation is how to fit this into my curriculum. What can I do to make what I am teaching fit into this model — I work with a PLC, and the expectaion is that we are moving together and teaching the same things.

When you use the discovery method; students tend to remember the work they put into it and what they got out of it. Typically they can then apply that learning later. In a regular system, they try to remember a formula or steps to follow for the test and then forget about it. With this method shown, they can take and use what they learned in other areas and subjects and it doesn’t just stay an isolated case of learning rules to remember for a test.

I really enjoyed watching how the inquiry framework has been applied in a math learning context. Students are pushed to do the thinking/heavy lifting here, while the “fueling” portion of the threepart framework keeps students on track with the lesson goals/objectives. I teach math in a dual language context, where students are also receiving direct math instruction in their mothertongue: Mandarin. Math instruction in China is very direct and teaches skills in a very deliberate and targeted progression. My students have a very solid foundation, but sometimes do not have the opportunity to apply their thinking. I really look forward to using this framework in my math classes, because I think it will add depth of understanding to their strong foundation, and also give opportunity for students who see themselves as weak in math to gain some confidence.

This is very interesting to me. When you say “direct and targeted”, would you say that the learning is more procedural, conceptual, both?


This lesson style is very different than the traditional math lesson format. The accessible entry point, use of video, and withheld information, all make this lesson very engaging. I’m sure the idea of having your teacher and his friend star in the video is an added appeal! This brings every learner to the problem to share notices and wonders without performing mathematically yet.
The part of the lesson where students get to solve a question is where the productive struggle begins. I can see (and feel for) some students not knowing quite where to begin. I assume we as teachers can ask them some questions about the task to steer them in the right direction. This would be the main monitoring stage for the teacher, I assume.
One area I would like to become stronger in during this course and beyond is the ability to anticipate more than 2 ways to approach the problem. From there the selecting and sequencing skills would need to develop with the ultimate aim of connecting to the learning goal.

I love that these problem based lessons are accessible to all students, regardless of their “math level.” This framework gives all students the chance to be engaged (rather than leaving some behind in a traditional math class). By being accessible to everyone, this style of teaching encourages creativity, community, confidence and perseverance. It allows students to create their own models rather than simply memorizing a formula.

Great reflections here!
It sounds like the overwhelming response is that a problem based lesson is more engaging and has many benefits.
A couple struggles we are hearing include how do we help students when they don’t know where to start and how do I get through my curriculum.
For what to do when students don’t know where to start… this is common especially when math has been delivered in a direct instruction, procedures first approach. Building a culture of problem solving will take time and effort. We will also need to develop our questioning skills to ask students a question that will keep them thinking without robbing their thinking.
As for curriculum, teaching students to become resilient problem solvers will help you with pacing rather than slow you down. It seems counter intuitive, but it is true. Build that culture by investing in this process and you will save time and students will retain more.

This lesson was much more engaging then the traditional way of say teaching a lesson out of a math textbook. It was visually appealing with the video and it sparks curiosity and leads students to want to explore of what’s happening using numbers and then try to make sense of it all. It is great because it is a low floor, high ceiling task which allows for group work where everyone can contribute. I have tried similar tasks from your website and really appreciate the suggestion of having extensions ready for that randomly created group that is done ahead of everyone else. In the past, I would just give an extension off the cuff but planning for individual students that are scattered amongst the groups makes a lot sense.

Great words from Tom Schimmer: “plan with precision, so you can proceed with flexibility”


A traditional math class is a group of disconnected rules that we teach students to perform on homework and the unit test. The opportunities to generalize their knowledge into unfamiliar areas are minimal. It is significantly dependent on the teacher to be the carrier of the experience that shoves as many rules and algorithms as possible in a year.
The lesson style presented in the first five lessons will put the student in the driver’s seat of their understanding. They get to help create the questions being asked and find multiple ways to work their questions. I feel the video said it better, but seeing the various representations, in the end, will take away some of the fear of unfamiliar problems. The students will see that they can create a strategy to get to the end solution.
My fear of this is that I was not taught this way, and I haven’t been teaching this way, so am I going to be a firstyear teacher again to change my approach to a Problem Based system.

@scott.mcnutt I like your vulnerability here! We have to be ready to become that first year teacher again to change our practice! We’re here for you every step of the way.


The lesson style is different because it is like a movie. You are dropped into this world and you do not really know what is going on. You have to pay attention to the clues in the the exposition to figure out where you are, who is that guy (“oh, I see he is the bad guy because …”), where you are going. Therefore the story starts to belong to the student and not the teacher. Sometimes I spend so many hours/days/months/years developing a problem or something and then it is my baby/I know it inside and out and I drop it on my students who are thinking about the girl they will call that night or the fight thye had with their mom. This approach brings the student in and makes it more likely the student will make it his/hers and therefore they start to do some of the heavy lifting. Then the problem becomes their baby – the results theirs. The benefits are memorableness – which is important for how they see themselves as mathematicians/students/problem solvers but also in the concept they get that opens doors to other concepts. The limitation are yes time, but I think also mood. It is a lot of energy and sometimes I think my studnets (at least I do) need a day to sit and do some repetative equation solving. It is just good for some students mental health on some days to get away fro mthe engaging conversations – especially the introverts wherre that sucks up energy. Certainly not a reason to not do this approach but to consider some students and break it up with a little practice here and there. I think sometimes repetiion (once they get it and are doing it right can lead to pattern discovery). So the only limitation is it cannot be every class/every concept I think.

@david.diehl Definitely! We need a good balance so we can help all students achieve. The balance can also happen in the same lesson. Time for quiet thinking, time for active discussions, time for quiet practice, ect.


Perfect world math classes would all look like this! Students working to solve the problem rather than me telling them how. But many of the concepts in higher math (grade 11/12) will be the foundation for more “applicable” math (calculus) and I struggle with finding the best way to set kids up to come across the patterns on their own – what accessible problem could they solve but not be totally lost? For example, tried to have students graph x^2 vs. x^2 – 3 vs. (x – 3)^2 and have them come up with the rules, but found that the time it took them to graph even, let alone notice the pattern, was way longer than I had room for in the schedule. Working on a quad system, time is a real concern. I know this course is more applicable to younger grades, but wouldn’t it be perfect if all math classes followed this structure? My struggle is coming up with the “notice/wonder” investigations and still covering all of the outcomes.

Time is always a concern and on the minds of educators – especially in the senior grades. While this should always be on our minds, never forget to ask yourself whether flying through material at a surface level is getting the results you’re after. Covering vs conquering content are two very different things.


It is almost as though I have been given the freedom to conduct class outside of the box. I have found more success with delaying the traditional notes until the end of the lesson instead of the beginning to help solidify the thinking.

So glad to hear it! Yes, save the consolidating and “notes to your future self” creating to the end.


I thought that I was far from classic lessons: teaching rules and doing exercises. In my math classes I always try to ask students about their own way, and strategies to solve problems, then they show their calculations or strategies in a plastic board. Then I put the different ways that I think is more interesting in the black board even if they aren’t correct.
The new and very interesting things for me, and that they are also a great difference with classical lessons, are:
• The amazing and recycling context.
If for example I am teaching equations, I will accept and show students strategies even if they aren’t equations. But I will not do it myself, and now I think it is a huge handicap. Because if I do it, I could help students to connect with this past knowledge and do understand more students, and help others to make new math connections.
• I don’t usually use estimation in my class, I think is a good way to spark curiosity.
I hope that incorporating this 3 parts framework in my classes students will be more open minded to include different ways to thing and calculate and that they will not be lasy trying to understand the strategies of their partners.

It will certainly take time to build that culture to get students more actively and openly sharing their notices, wonders and strategies, but it will come with time if you keep at it! Thanks for the great reflection!


This lesson assumes that everyone has an entry point to solve it in some way. There’s no assumption that it’s too difficult or too easy because a teacher can guide students to stretch what they have solved. My reservations are what if I’m not sure how to stretch their thinking and how would I create this kind of task myself? I already feel overwhelmed with tasks so how do I build this (necessary) structure and keep up with all of my other responsibilities?

@kathleen.bourne these are definitely natural reservations and the beauty part is that we help you with these things in the rest of the course!


These types of lessons build the capacity to think through problems. I hope that it helps them build confidence in their ability to do math.

This type of problem is based on inquiry versus a traditional plug and solve problem. Benefit is that if the concept is taught with excellence then the concept allows for deep understanding especially if mistakes were made along the way. It is a safe way to learn. Traditional math can leave students behind quickly if they do not understand the concept because the learning is shallow. Traditional learning is based on mimicking what the textbook or teacher does and replicate it. This type of learning is forgettable. Reservations of inquiry learning is finding the right task and being a skilled enough teacher to direct the learning with what you are given. Being new to teaching math this year, I am finding that I struggle with connecting the learning. I am not afraid of the inquiry portions of the lesson from all my years teaching science or the discussion or not answering the student’s questions, but it is that magically step of pulling out that learning for everyone which I am struggling with. Some of my students get the connection easily and they are willing to complete all the thinking for everyone in their group, but others want the plug and chug which is frustrating because they do not make the connections.
 This reply was modified 8 months ago by Kay Walder.

This style of teaching employs conceptual understanding and thinking. Many students with disabilities have memory delays and are not able to memorize rote problems. This teaching allows students to visualize and draw conclusions on real life examples. It creates a meaning and purpose that allows for an increase in engagement.
As a resource math teacher, the struggle comes with students generating ideas and sharing them. They are usually silent in a lesson like this especially in middle school since there are developmentally insecure. They really have to be coached into this type of lesson. They generally sit and wait to be told how to do it. Therefore, a lot of prompting and scaffolding happens. Also, many general ed. teachers don’t do this because of the amount of time one lesson or “problem” takes. They have pacing guides to follow and struggle with doing both.

This style of instruction is not how most of the members of my department teach–and with a state assessment at the end of the course, the use of time is paramount to my thinking at all times. I can see how familiarity with the routines and protocols will speed things up as time goes on but I am still finding it difficult to trust the process and that we will be able to accomplish all that we must.

It is so hard to shift when you’ve been trained to think that “using time effectively” equates to teachers talking and kids copying.
I know it is hard to shift, but you have to just do it. It will feel like time wasted initially because your kids will probably sit and do nothing. They are trained to wait for the teacher and copy what they do. Not good at all. Rip the bandaid off! 🙂


This lesson style is much more engaging and respectful of students’ prior knowledge and individual abilities. I have tried to incorporate some of this kind of teaching and I know that it definitely increases the enjoyment factor for students. This might seem trivial, but spending some time addressing students’ prior baggage and potential math phobias will pay dividends down the line. However, it really is a tension between teaching this way which I can see is better for students, and keeping up with the fastpaced demands of the Eureka math curriculum that my school uses. There is so much ‘stuff’ to get through! And the pacing! Still, following the Eureka program leaves a lot of students behind because they’re not ready for it or they need more time to process than Eureka allows, while simultaneously leaving the advanced learners bored and frustrated. So I know I need to find a better way, which is why I’m taking this course!

Over the past few responses we hear that there is clearly a positive response to these types of lessons and folks even argue that they are more accessible and equitable for all learners! Fantastic!
However, the idea of “getting through” curriculum is still a challenge which is more of a mindset challenge than anything.
As I’ve said many times on the podcast “I used to be awesome at getting through the curriculum, but did my students learn it?”
The answer for me was – only the memorizers and the lucky ones.
This shift takes time but it is so worth it.

There are so many benefits to doing a lesson like this one! Obviously, engagement is a part of it – the intriguing way the problem is presented is much more likely to hook students than a traditional textbook problem. While not all students will race to gobble a bunch of chocolates, many will find this more relevant than the “real world” problems found in a textbook.
In addition, the format leaves things wide open for a variety of approaches. In the quest to nurture flexibility in problem solving, rather than compliance with rules, the ability to consider alternative viewpoints is key and is built into this lesson format. The lesson is inherently differentiated based on the approach the students will take, and everyone moves forward when the representations are connected in the consolidation phase.
Furthermore, there is opportunity for students to build their math identity. When we teach students the algorithm, we (inadvertently, perhaps) send the message that there’s “their way” and “the real way” to solve these types of problems. The consolidation phase validates a variety of approaches and reinforces the idea of multiple ways of knowing.
Lastly, as mentioned in the video, this is a context that can be used to develop a wide range of mathematical ideas, reinforcing the connectedness of mathematical concepts that might otherwise be taught in isolation.

What a thoughtful response.
I agree in all areas but one that pops out as so important is that differentiation piece. Often we think differentiation means completely different for all learners, when in fact we can lower the floor significantly simply by withholding information.
Thanks for sharing your thinking with us!


1. The lessons presented in the videos so far have been way more interesting than any lessons I had in school or that I have taught in my teaching career so far. The lessons involve way more thinking and inquiry and relate to real life in some cases.
2. The lessons increase problem solving, let students know it is ok to have unanswered questions, and to have fun teaching math.
3. I worry about the time the lessons may take, and whether or not I could teach lessons this way, since I haven’t. I try to come up with creative stuff and sometimes it doesn’t work, and but I am willing to try new things. I will definitely try to incorporate more inquiry and curiosity into my lessons.

I have designed a few of my own explorations based on the MMMtM philosophy and the students are really responding well so far. For some, they prefer the “traditional” style of “just give me the examples and let me do” but we are talking about how learning to do things multiple ways is valuable and that really understanding why we are doing something is more important than memorising the method. Some students are now really looking forward to Mondays now. As a teacher who lacks confidence in my own Maths ability, it is challenging for me and I am still scared of making mistakes (rather than following the textbook method) but we are all doing it together!

This lesson allows the students to do the thinking and the comprehension first before any algorithm is introduced. They get to activate prior knowledge and collaborate with classmates on ways to figure it out. The benefits are the learning will stick better because the students did the work! My reservations all stem from my fear of letting go or not being able to think fast enough to help my students extend their learning. I worry that I wouldn’t be able to make the connections and miss out on the learning.

We all struggle with that control and with the idea that “if I don’t say it, how will they learn it?”
This can be hard to grapple with, but it is so freeing once you get it in motion.
Remember that you are still teaching, but more so during the consolidation of the lesson. What are you hoping students learned? Get explicit at that point and use student approaches and strategies as your means to make connections. You’ve got this!


The main benefit is a more engaged group of students, where their voices are heard and legitimized. I also enjoyed the way you extended the learning for the “advanced” or quick students.
I am concerned that when students are working in groups, one will figure it out, write it all out verbally answer the questions and the other students just copy, agree, etc. without thinking it through for themselves, actually discussing answers, coming to the same conclusion together.

Your concern is valid and very probable when you are starting this. The culture you want to build in your classroom will take time and requires a lot of community building. Focusing on the idea of students working to avoid robbing the thinking of others will be important and your tool to tackle this is questioning students about why / how things work. If you’re just looking for a final answer, then there will be no thinking and no way to get students to think more deeply.


I am very excited about teaching in this way I want my students to be engaged, thinkers. I want to give them time to explore and learn from themselves and each other.
I am worried about the planning involved as I teach all subject areas not just math. This looks like it could be very time consuming to get started.
I am nervous about how to approach the group or students who don’t know where to start and get them started without directing them to the answer
I also worry about the group that has 1 student who takes over and solves the problem but does not include others in the process.

These are all valid concerns!
Note that the best thing you can do is to begin thinking of how you can transform your typical lesson into a similar experience – but don’t worry about the bells and whistles. There are many lessons already “good to go” out there including our units here learn.makemathmoments.com/tasks
But the key is starting with a question for students to estimate and then solve using their prior knowledge of strategies and models and then to nudge them further down that line. This will take practice and it is never an “all or nothing” experience. You’ve got this!


I’ve done a few of these things in class before, but I am really looking forward to intentionally making my entire room like this notice and wonder atmosphere with more curiosity and less stand and deliver. My biggest hurdle is figuring out these curious tasks to create…hopefully going further in these modules will help me!

We have a great lesson in Module 2 that helps to transform regular textbook problems into
Curious tasks that students stick with!


This lesson allows for students inquiry and also facilitates for students to have multiple pathways towards success. Students are encouraged to problem solve, and will naturally develop an interest in finding the answer. It creates a great Buy In to the topic.
My fear is that some students will not be able to get started on their own. I also would be worried about time. What happens if we aren’t even close to a solution at the end of class? Does this mean I can continue for another day?

All common struggles and concerns.
As we work through the online workshop we will address some of these hurdles. In the meantime, check out some of our problem based units to get you a sense of what a week might look like / sound like:
learn.makemathmoments.com/tasks


This style of lesson requires the students to do the thinking and the teacher to take note of/be aware of what thinking the students are doing. Then the Ts drive their discussions based on the moments that were created. Of course the teacher needs to have thought about the reason they are giving this problem in the first place and where the discussion is likely to go/what elements might arise to focus their light on to push the thinking further.

@jeff.harvey The teacher has to do a lot of thinking for sure. Like you said, the teacher has to match the problem to the learning goals and anticipate student strategies and solutions and be ready to address and connect those strategies to the learning goal! Not a small feat! We’ll be unpacking all of those moving pieces in this course!

Mr. Orr,
Looking ahead and anticipating the wealth of information I will be gathering, I am truly full of gratitude. And thanks for hosting this workshop.



I love that everyone can contribute something, especially in the Notice and Wonder sections. There’s an entry point for every student and a way to make their input valuable for class discussion. I also love that it models reallife math: solving a problem when you don’t know all the parameters. It’s forcing thinking and problem solving over rote repitition, which fosters true understanding over memorizing.

This type of lesson requires students to think because you are not teaching a concept and then giving them an assignment to apply it. I have had students get frustrated with lessons like this because they are not used to this kind of teaching. They want to be shown the formula or technique and then plug in the numbers to get the right answer.

I have tried the 3 act math lessons in my grade nine applied classes. I like them because students do engage in the lesson and students who normally shut down, are contributing to their groups and participating and getting some understanding from the activity.
What I struggle with, with these type of lessons, is how to take the activity to the math lesson I want them to get from it without doing a traditional lesson afterwards. I guess I worry about the curriculum too much, but I like the way you mention to build the activities so the students can just go with it…. and the pacing takes care of itself. I just am not sure of how to get there. I am missing a link I think. Which is why I signed up, so thank you! 🙂

Awesome to hear that you’ve been experimenting with these problem based lessons. You’re not alone when it comes to wondering how to go from the sparking curiosity part to the Fuelling sense making piece. Key is all about how to consolidate and the intentionality. We will dive into this deeply as we move through the workshop. In the meantime, be sure to explore these problem based units to assist: learn.makemathmoments.com/tasks


The lesson style was more open and had mutliple entry points. The chance to scaffold it at different levels thus bringing in the differentiation that is so important in order to feel successful in the subject.
My reservation is to think of such problems that facilitate this kind of thinking and exploration of concepts.
 This reply was modified 4 months, 2 weeks ago by Sakina Gadiwala.

This is a super common reservation. In module 2 we will explore the elements that are important to developing these types of experiences and how we can transform our resources that we are currently using without a huge amount of work.

I can’t wait. You’ve been awesome so far, and yes Jon also.
Thanks


This method encourages discussion and openended questioning. It allows multiple access points for students of all ability levels, whereas a traditional math lesson tends to favor the fast learners.

I agree that to a large extent, this approach provides an even playing field, without fear or favor.
Thanks for your post.


It is much more accessible for students of any ability. Some teachers will be apprehensive that they have to get student to talk to their partners. We have been encouraging partner work for a few years and some teachers will not direct students to talk to each other, but make it voluntary so some students are not engaged.

I also agree with K. Keifer. I have tried 3Act math tasks and really liked the level of engagement. I have not been able to find enough ways to apply that across the curriculum. I am excited to learn how.

The biggest way that this type of lesson is different then the traditional one, is the student’s responsibility for the learning. I think that ownership is powerful, and will lead to better math “by in” and problem solving skills.
My major reservation, is how to do this with my curriculum. I teach high school Algebra 1, Geometry and Algebra 2 CC Regent courses. The curriculum is wide and varied. I can’t seem to picture being able to do this type of lesson for such a varied course, as well as when teaching something pretty new and unrelated (like the quadratic formula, or trig).

Many traditional math lessons do not allow the students to discover the mathematical concepts, but instead are focused on teachers demonstrating the concepts repeatedly for students until they are able to mimic the teacher’s moves. In the lesson style demonstrated in lessons 15 we see how students can be introduced to problems that allow them to draw on their prior knowledge to start solving a problem. As they progress through the problem the role of the teacher is to help identify where each student needs support, and what type of learning gaps they may have. For those students who are able to use prior knowledge to solve, the role of the teacher is to know how to push them to dig deeper.
I can see several clear benefits of this type of lesson. Students must engage in thinking about a problem. The visual nature of the problem allows all students easy access to begin. By working in groups students will have the opportunity to collaborate as they are making sense of the math and attempting to solve a problem. The teacher is able to assess the knowledge of students as they attempt to solve, and can provide supports as needed.
I do not have reservations about the efficacy of this sort of approach, but I know that I need to be sure that I have a clear grasp of the progression of the mathematics. This year I will be changing schools and grade levels, from 5th to 7th grade, and I know that I have some work to do to become much more familiar with the grade level content over the summer.

TC,
I suppose this approach will provide the opportunity to unlearn some pedagogical tendencies.
Thanks for your post.


I find the ideas behind this type of lesson quite freeing. I used to work in a school system, where following the desired steps seemed to be required as it was the only way to get full marks in the standardized tests that ended high school. I always felt like I had to spend enough time teaching in the traditional approach or students weren’t successful in these exams, not because their answer was incorrect, but because their process didn’t match the answer key. The traditional approach seemed like it removed any sort of understanding math from the classroom, and just focused on mimicking procedures.
Since switching systems last year, I have experimented more with this type of lesson and found that there are numerous benefits in that it requires the students to understand more fully why they are doing what they are doing, and causes them to be more fully engaged in their learning.
Reservations that I have, based on things that I’ve experienced, is that I find it harder to introduce in older classes if that isn’t what they’ve experienced in younger grades. My grade 9s are all for it, whereas my grade 11s who’ve been in traditional classrooms up to this point were very resistant. As well, I work in a school where it seems like every student has a math tutor. I’ve noticed that the math tutors have started to “preteach” content, and so students come in with their failsafe procedures and aren’t willing to engage in the task the same way.

Traditionally, a lesson is introduced as it has been presented for years: mostly by telling the students what they’re going to see and do. However, with the lesson style witnessed, the students are challenged to engage with the material and then determine what they saw and then wonder what is coming up next.
I suppose the benefit of this approach is that the students take ownership of their discovery and can reference that knowledge for future encounters.
The challenge is that teachers will have to find a way to keep coming up with hooks for other concepts.

While it might seem hard to “find hooks”, the key is understanding the elements that matter. We will dive into this more during module 2, so get ready!


This lesson seems to be much more fun than traditional lessons. The lesson opens up more questions. I love that we could expand the lesson later in the curriculum…you can’t do that with worksheets. This helps with making connections between all the different concepts. My big concern (as with all pbls) is time. When we are expected to “stay together” with the other classes and those teachers are not using pbls or explorations, I will get behind. But I know and have seen the importance of these types of lessons.

I wonder why you will feel “behind”?
One big question we all need to ask ourselves is which method will help students better understand the concepts and skills? If other teachers are just “presenting” material, are they really “ahead” or have they just checked off a list regardless of whether students have conceptualized the learning?
I totally understand where Kristina is coming from. This year my board set up a calendar for us: we all needed to be doing topics in exact sequence, by an exact date in the calendar year. I felt that added a lot of pressure on teachers to “keep up” with the long range plans, to not “fall behind”. My question this year was: do I just move on through math for the sake of keeping up with the plans or do I teach my students and meet them where they are (but then not cover all topics)?



This style in comparison to traditional math lessons is definitely more thought provoking, interactive, and more enjoyable. The best part of this style is that it nurtures critical thinkers who learn to embrace challenges and eventually become resilient to obstacles. These are valuable skills that are transferable to all aspects of life, math or otherwise.
Benefits:
Students are more engaged.
Students cannot fall back on rules and plugin numbers without understanding.
Students are asked to take action in their own learning, instead of being passive.
Students are encouraged to think outside the box, get information, be creative.
Lessons meets groups where they’re at.
Lessons reach different types of learners, and lessons are not language heavy.
(I could go on)
Reservations:
Students who hide behind their peers or are afraid to voice out opinions, how do they show their learning?
How much time does it take to create such lessons (video, power point, images, variety of questions/representations/entry points)?
How would I, someone who did not specialize in math, come up with ideas such as the video for every lesson/concept/unit? For example, I can’t even think of a situation where I had to use order of operation, so how would I create a realistic problem where my students could relate to it?
How do you show evidence of learning of an individual when students work in groups?
Would students sneak a peak of other groups of help? What if a group has NO IDEA, would seeing others write on their boards put unnecessary pressure? Is comparing work always good or bad?

This type of lesson is different than traditional lessons because it doesn’t initially provide the problem students are trying to solve. It wants students to analyze the bigger picture without there being any correct answer at this point.
As many already said, a benefit of this type of lesson is that it allows all students to access the problems. Based on their responses, the teacher is able to get an idea how the students are thinking and seeing the problem. It also provides students an opportunity to solve problems in a way that makes sense to them as opposed to using standard algorithms.
The last benefit mentioned is also part of my reservation with this type of lesson. I’m torn between having students show their thinking for solving problems using any method and moving them to using methods which help build the foundation for solving more complex problems in higher levels of math.
Another reservation is getting students to do anything with the problems. Most of my students refuse to write anything down or attempt anything if they are not sure about a problem. Part of my focus this last year was to get students to take risks by asking them to write down what information they thought they might need from a problem, explaining that sometimes by doing that, they will see things differently. Very few did this. Many also refused to write down information given to them if they didn’t know how to do it. I also did some small notice and wonder problems or which one doesn’t belong and many struggled with it because they thought there had to be a correct answer, even after I explained that there wasn’t, this was just getting them to see what was there.

This is so different than a traditional math lesson. In a traditional lesson, all students work through (or don’t… 😯) the same 3 examples with the teacher. They are all held to the same standard – some of the students don’t learn anything new (because they already knew it, or because they checked out mentally or weren’t ready for the lesson) and some do learn. The traditional lesson is kind of like a bulldozer in that it prepares the class for the next lesson in the book (or does it…).
This meets the students where they are individually and so it makes it much more of a learning experience for everyone.

Great points here.
With a problem based lesson, the goal is that all students take away something – possibly something different than others – based on where they are in their mathematical journey. The “bulldozer” style approach definitely doesn’t help all students unless they are all at that same place. I used to try dragging students by the collar to “where we should be”, but it just left them and I frustrated when it would cause overwhelm and/or students shutting down.
Glad you’re on this journey with us!


The thing I love most about this type of math lesson is that students see, right from the get go, that they can solve a problem in multiple ways. Often in traditional math classes, students who struggle are so fearful of making the wrong initial step that they don’t even begin. I also love that this context can be used with multiple problems at various points throughout the year. While traditional math classes feel more like a checklist of skills/tasks to accomplish (that may or may not relate to things students are curious about), this model fits better with natural human curiosity.

The lesson began by hooking the student with a great problem that they could relate to. Also it asked for the thoughts and opinions from students about what “really” is going on. This increase in interest and involvement builds student buy in and ownership of the problems. In a traditional lesson notes would be given, example problems worked with the teacher and then students would practice the problems. There is a BIG Difference!

I love how this style of lesson completely smashes the tired “I do, we do, you do” shtick. It reminds me of a science teaching methodology called the “Five E’s.” (Engage, Explore, Explain, Elaborate, and Evaluate) This style, which essentially starts with a “hook” not only provides a reason to engage, but it also builds a sense of community where all students can participate and feel successful! I’m also really digging that the context can be used for multiple math concepts, which can all be extended to challenge kiddos that “get” the foundational ideas…

So glad this is resonating. We taught for many years using “I do, we do, you do” and it just wasn’t cutting it from an engagement perspective and from a constructivist perspective. Glad to hear you’re on board for doing this work with us!


This way of teaching I feel allows more students to be curious and feel like they are capable of solving the math. Sometimes the typical approach shows students one way, they don’t get that one way, they think it is the only way, so they check out. I feel like the 3 step framework here allows them to solve a problem (without them really even realizing they are using a bunch of math). I also think it is beneficial even for students who solve the problem in a really long and unpractical way, because hopefully they will see through other student strategies or the lesson part of the day a more efficient way to solve the problem. I think having the students already invested in the problem when they learn a new strategy will have them be more engaged/wanting to learn.
My wonder is how do you pull in the teaching a strategy part/learning target, without feeling like you are diminishing the student work they just did? I don’t care if my students solve it a different way but I know I am supposed to teach them a certain way. And I don’t want them to show their strategies and then feel like “no, this is the best way to do it”.

@tabitha.price You’ve listed many great benefits here of teaching through tasks! We’re excited to help you on this journey.
We’ll be unpacking how to create these tasks in upcoming modules/lessons as well as how to facilitate those discussions. Stay tuned!!!


My big takaway is the idea of meeting kids where they are. I think traditional teaching does that, but at a snail’s pace that keeps everyone together, even if there are those in the room that could move faster.
In this model, you meet kids where they are, but you leverage where they are to propel them forward with carefully designed tasks and logical extensions at hand that you can employ strategically.

Love this take away and couldn’t agree more. Nice work!


No answers, no obvious help. I can imagine a struggling student being intrigued, and being emboldened to share wonderings, especially if they see all answers being accepted. This is a great way to teach estimating – I can’t believe I’ve never seen the too low, too high question. I’m still worried that struggling students will not fully engage, just from too many years of discouragement.

Will kids remember the math? I know they’ll remember the video and the fun, but how to make the learning permanent seems the most important to me.

That is where purposeful practice still plays a big role as you’ll notice throughout our problem based units:
learn.makemathmoments.com/tasks


Compare: Students are more involved and engage with this lesson style
Benefits: strategies are not just given to students, inquiry, openended questions, can be built upon
Reservations: time involved to learn this lesson style and actively implement in a classroom

It will definitely take time to learn, but it is something you learn as you do along the way.


I have begun moving my lessons to more of a notice, wonder model. I have struggled with those who solve it quickly. Having task cards premade is a wonderful solution. I also struggle with engaging material for hooks/tasks at a third grade level that I am not creating myself. It will be interesting to see how I can fit this/merge this into our new math curriculum purchased by the district for next fall.

Third grade is great because students are pretty easily intrigued/made curious. I always think to myself “what is my learning goal and how can I withhold some information ?”
Lesson 2.4 will dive into this more deeply.


I think the biggest differences come from students being given the opportunity to problemsolve and discover math for themselves, instead of just being shown how to do something and then copying it.
The benefits are myriad: students are more engaged, they learn how to problemsolve, and they get to have more ownership in their learning than with traditional lessons.
My hesitation is mainly wondering if my students (who are used to being told how to do things and then having to copy them) will make real attempts to solve this, or will just say “I don’t know” and not try.

Foremost the biggest difference is who is doing the heavy lifting. Teaching this traditionally I’m talking 20 maybe 30 minutes and the kids are looking at the clock wondering not about the lesson but when the class will be over. The person doing the talking is doing the learning. There then lies the answer to the question, the traditional style lesson the teacher does most of the learning. Over the years, I would break the lesson into interactive components but without connections students still don’t get it. When the lesson flops your response the goto is more direct teaching.
I guess I’m still wondering about practice, is there anything students do at home. I read the 5 Practice and they suggest questions related to the task. Are is there anything more, I’m thinking parents might ask for it so they have something to do with them. I also wonder if I consolidated under the document camera if I could post that for the students so they had the notes at home or would that deter them from taking those notes in class.
 This reply was modified 3 months, 3 weeks ago by Anthony Waslaske. Reason: Break up the block of text

Great reflections here and wonders.
As for practice, we tend to set up a few consolidation prompts related to the problem based lesson like you’ll see on day 1 of each task:
learn.makemathmoments.com/tasks
Check out the reflection tab for those consolidation prompts.
On day 2, we go into a math talk and purposeful practice.

I just completed my first year of teaching so my experience is limited. As a student, I remember the old “sit and get” (teacher centered). Now its moved to the “I do, We do, You do” (teacher is in full control for the a third of the lesson). Y’alls style is truly student centered! While watching the video scenario, my mind kicked into gear and opened my mind to multiple solutions. All of this without the teacher intervening! This is what we are aiming for in our classrooms.

Love it and yes, our experiences as students were very similar to yours. We taught the I do we do you do model for a long time and just couldn’t get students to engage like we are now. So happy that you’re seeing the value snd are committed to making your math classroom a student centred experience!


There is no comparison to what I consider a traditional lesson. Students are actively engaged, not passively writing down everything in their notebooks.
The biggest benefit I see is students taking ownership of their work. I already do a lot of collaboration. This is definitely the next level that I need to take them. I don’t have any reservations, I simply need to make the investment in time.

This type of lesson is much more engaging for the students and allows all students to dive into the task. It takes the pressure off finding the ‘right answer’ since they don’t even know what the question is in the beginning.
I stress about sequencing student solution paths on the fly (so to speak) but now that with adequate prep and practice, this will get easier over time. Additionally, as many have already stated, time is always an issue (or a pressure). It’s difficult to spend what seems like such a large amount of time building classroom climate and going through this process – even though we know it will work itself in the end. Trusting the process and myself to stick with this and make it work are things I need to force myself to do.

This style of lesson is one that makes kids WANT to explore the question.
It allows them to move from an interesting, concrete or realistic problem, to discussion about how they would solve it, and finally to applying mathematical abstract concepts.
The benefit is that everyone’s ideas are honored, shared, and built upon. Kids are able to use their own type or level of mathematical reasoning first, and can then connect it to the abstract, or at least move up to another level of understanding as they listen to other people’s explanations. In other words, this type of lesson makes it possible for each kid to start “somewhere” wherever that may be, and either move forward, or at least see that their way is just as valid as that of others.
In contrast, the “traditional” way of teaching involves giving kids the abstract first, such as “Today we are going to learn how to find a ratio. Here’s how to do it. Let’s practice my way. Now you do it like I did.” The traditional method assures that at least several students will be lost from the beginning because they didn’t have a chance to explore their own math thinking first.
My only reservation for doing this type of lesson is that the person that I am matched with on my gradelevel insists that we do the same lessons at the same pace. We each teach two classes of math. If I am selfcontained, it is much easier to do this type of lesson because I can pace for my own kids and do lessons my own way. It takes a lot of confidence to say, “Yeah….. No. I’m doing it this other way because it’s best for kids.” The comeback is , “Well we have to get through all these concepts by (date) and if we take the time to do this lesson, we won’t get there.”
 This reply was modified 3 months, 3 weeks ago by Terri Bond.

I feel like so often students are just wanting to “get it right.” They just care about doing what is supposed to happen next. With this type of lesson, students are engaged in a way they weren’t before. They are able to push their minds into areas they might not be aware of. This is similar to science when we do inquiry based lessons. I LOVE having a situation that we can base future lessons around. The “buyin” is already there as is a solid foundation for the understanding of said foundation. The potential for challenge questions and future lessons are endless.

Glad you’re seeing the value not to mention how much more enjoyable learning and teaching math is through this approach!


I of course LOVE the idea of the lesson as a vehicle for teaching unit rate and know the kids will connect to the candies, but the video does not give me a clear idea of how to use the tools provided. I downloaded the zip drive and while there are many files, there is no suggestion for how to use and sequence them. The two entitled .DS_Store and Chocolate Mani.key cannot be opened so in essence, I don’t feel empowered. I don’t expect a script, that’s not what this is about…but perhaps a suggested flow chart.

Hi there!
The lesson is modelled through this lesson in the online workshop and a breakdown / summary can be found here:While we do not have that particular lesson crafted into a full unit like we have been building out in our PBL lesson area (https://learn.makemathmoments.com/tasks) checking out the flow of our 30+ problem based units and the teacher guides will help you put the curiosity path into play with those units as well as build the confidence / skills to apply them to other lessons as well.
Have a look and let us know where you are struggling.


In lessons like this, the students are engaged in the problem at the start of the lesson and excited to figure them out, whereas in traditional math lessons it can become mechanical, especially when students are not actively engaged until the practice which typically occurs at the end of the lesson. In the latter the teacher is doing most of the thinking.
Benefits: My students enjoy notice and wonder questions because they are all capable of noticing and wondering. I believe this aids in building the confidence of low performing students. I especially enjoy how problems like the ones in the video have a low floor and many different ways of approaching the problem which make it accessible for all learners. I love how Kyle mentioned coming back to a task that was previously used. I did the postit note and camera case lesson for slope this year and referred back to it informally last year. Now I am excited to think of questions I can use when reintroducing the problem for writing equations of lines and systems of equations.
Reservations: The challenging part for me is feeling overwhelmed, like I need to scrap everything I used to do and dive into this 150%. I’m not sure if I need to spend a year going fully problembased (try the illustrative mathematics curriculum), pull problems from all of the resources I’ve found through MMM Podcast, or figure out how to spiral my curriculum.
Thankfully I used the pandemic to test out problembased lessons and did some exploring and tried many new activities, so I have a place to start.

Traditional Math Lessons are more “Organized” and Compartmental.
The lesson here was more Organic, and could lead to connections with larger contexts like the math was intended for. Being able to pull the idea from a previous lesson was easier because the “moment” was closer to the memory. It existed in more than just rote copying of an example – but had handson, visual, friends talking about it, AND notes.
In this lesson, the notes weren’t the goal – the expansion of the idea was more the goal.
Questions:
When will notes ever be the goal?
I like to show students “Structure” in writing the answer; there is less structure for this lesson – for example in High School writing the x for multiplication wouldn’t be a good thing since it’s a variable – so brackets are advertised/used
After you do this type of learning – for how many days? – does the traditional math class ever come back?
This lesson style is there to spark and fuel, but then part of the connection is the rote copying of notes and examples, isn’t it?
I can use this for the concept, but I also need to “Show” them how to write the answer when dealing with the problem – after they learn how to identify and think through what’s being asked. This reply was modified 3 months, 2 weeks ago by Velia Kearns.

Great questions here.
In short, the goal is never to copy notes despite the fact that for about 10 years in my classroom, that was pretty much all students did. A question to ask about note taking is what skills are students developing and are those skills what our goal is for students in math class?
The structure can come during / after consolidation as we make connections and summarize our work/ findings. This should be coconstructed by the facilitator and the students (not a copy job).
Check out our problem based units to get a sense of what a 5 to 8 day “unit” would look like / sound like: learn.makemathmoments.com/tasks

Traditional lessons are planned and structured. The lesson starts with the teacher. The teacher tells the students exactly how to do the math for the lessons. Most of the time we are providing them with one way to solve the problem. We provide notes as a roadmap for later. The students copy the work of the teacher. When they get the right answer, we say they have mastered the skill. For students that are below level, they will need some intervention because the lesson started above their level. The enrichment for the students that are successful will happen once they have shown mastery of the lesson content. Most lessons are independent of each other. The traditional lesson seems to take away the student’s ability to think for themselves.
This lesson is also planned but is structured off of the students. The lesson starts with the students. The notice and wonder engages the students. As the teacher presents the problem for the students to solve, it allows students of all levels to start at their level of knowledge. The intervention or enrichment for all students can happen in the lesson. The lessons connect content.
My reservation comes from it feeling like you are giving up control. What if we don’t match up to the pacing guide? How do you build the culture in your classroom, if the students don’t know where to start?

These are great reflections and wonders.
Pacing guides are always tricky since traditional lessons didn’t help you keep pace either – not unless you were willing to move on without students understanding. The same could be true here: I can keep moving along despite students not being ready or I can meet students where they are and help them progress accordingly.
Building the culture takes time and we will dive into that more throughout the online workshop. When students are stuck, we use purposeful questioning to help get them unstuck.


An interesting result, of focusing in this direction, is watching the students who are bored because math is so “easy” becoming engaged in stretching their mathematical thinking. Making room for multiple ways to solve problems also validates the different thinking of less traditional learners. A third result is that everyone seems to be more engaged in what is learning. Many students who come in hating math begin calling math their “favourite subject” because their minds are engaged and their confidence has grown. I am on the journey of learning this style of teaching and have already seen these results.

Instead of the teacher being the one in action, the students are the ones in action. The teacher is a moderator instead of the center of learning. Many different models are used to solve the problem. It is problem based. The students have to think not mimic!

First of all this one situation can be used in many different ways. It is possible to come back to it later. It is also very visual. It avoids going to numbers and symbols and instead uses pictures and reasoning.

The benefits is how much the students are doing the thinking. They are the ones making the connections. You don’t have to do that for them. Kids will be way more engaged in these lessons then they ever could be in the traditional type lesson. Students also have multiple entry points in a lesson of this type. I loved it, especially the connection to the derivation of the volume of a sphere formula!

I feel like this really breaks it down into tactile pieces rather than a lecture style. I think this type of lesson would make it easier to creating grading based on the TEKS rather than trying to rush through a lesson and then fight with students to complete their work. I have seen so many students who have failed classes because they did not connect with the work or format of the lesson even though they completely understood the concept. I feel this could also be completed online with students who may be out sick and need to make up this lesson. We could also utilize feedback sites such as NearPod, Padlet, FlipGrid, and so many others.

This lesson style is more student guided instead of teacher led. The teacher still has a leadership role, but the students help lead the class as well with their ideas. This would give the students a feeling of control, and pride when their ideas are shared, heard, and encouraged. The reservations of this style of lesson would be the time it takes to get to the final product, as well as the nervousness of the teacher with what can be perceived as lack of control in the classroom while the Sense Making is taking place.

@vanessa.watt These concerns are definitely valid! Luckily being in this course should help you with both! Looking forward to helping you along the way.


The lesson gives students a place to start, no matter where they are mathematically. There is less pressure to have the math concepts thrown out right away. It gets students to start thinking and making guesses, whether right or wrong. This usually gets students stuck because they won’t move on unless they feel or are told that they are right. The lesson demonstrates math in its essence…which is what I have loved about the subject. Taking a very simple exploration and show how it can be used to demonstrate very high level of mathematical thinking while giving all students an entry point and now feel as intimidated.

I am looking for the Chocolate Mania Lesson under Tasks as you directed and I don’t see it. Does it go by another name?

The lesson style I saw in the 15 video feels less structured and not as predictable as what I have traditionally planned for. It feels much more engaging for me as the teacher and for the students in the classroom, I see this as a benefit for all. I also see benefits for my students because they are the ones doing the thinking and sharing their ideas with one another. I like the way all students have access to doing the problem the way they see it. Some reservations I have are about my lack of confidence in trying to carry it out since it is so new for me. I also worry about how one student’s idea may sidetrack what I think the main goal for the day is. I think that is what the consolidation process will help with.? My other reservation is about other “type A math teacher planners” out there. How can I convince them that this is worth trying?

Great reflection and great wonder.
You can still be a type A planner – and probably should be – as there is much to think about upfront. See our teacher guides here https://learn.makemathmoments.com/tasks and check the guide tab to see all the thinking and planning that goes into it.


This lesson is the complete opposite of the way I have been teaching. As per school board directed instructions, I have always posted and discussed the learning goal prior to any lesson. This lesson makes me realize that I have been “robbing” my students of their curiosity path. This lesson also validates what I tell my students. I am always saying that they don’t need to use the method of solving that I show them. My direct teaching method, however, really doesn’t promote that…now wondering how many students are actually thinking “yeah right!”, when I tell them this. I love how we can highlight all the different methods that students may use to come up with the correct answer…maybe now students will believe me:)

There is much more room for student discussion and for students to approach the problem in different ways.

This type of lesson is so engaging and fun! I wonder if kids who are typically “good” at math in a traditional classroom would feel more reserved about engaging since there is no memorized formula and kids who struggle would feel more comfortable since they are invited to engage with creativity. Really cool way to open up math to everyone and encourage reflective thought and discussion. Love it!

We definitely get some of our memorizers or traditionally “strong” students pushing back on this approach as they aren’t comfortable not knowing the answer or the procedure right away. They are put into a productive struggle which is uncomfortable when they’ve never been pushed to truly think in the past. They totally get over it though! 🙂


It is SO different from the way that I currently teach. I definitely use the traditional model (aka I do, we do, you do) and it becomes so monotonous as the year goes on. This activity invited the students to take ownership of their learning and it make them realize what skills they needed to solve the problem. I loved it!

Super cool! If you’re loving it, be sure to explore the problem based units which follow the same framework:
Learn.makemathmoments.com/tasks


I think with this style of lesson it does a lot of beneficial things. It encourages depth of learning over breadth. It connects learning and builds experiences that students can continue to reference. It encourages problemsolving and builds in group work and discourse (and vocab by extension). It is lowfloor/highceiling so students can enter the task in multiple ways and the discussion style honors individual, creative thinking.

I love how this type of lesson builds on students questions and offers multiple entryways to a singular problem/ concept!
What I find difficult is being present to help the different students where they may need while also anticipating the extensions needed. I have students who will do all the work/thinking for their peers without letting them attempt the problem(s) at hand.

The 3part framework values skills students currently have to solve contextual problems; there is a gradual progression to the instructor sharing the algorithm after students have chosen, shared, and discussed their own path to solve a problem. Everyone can find some success. The traditional math lessons remind me of a recipe that varies very little, and students are expected to follow it in order to get the same outcome.
The nontraditional math lessons provide students with more opportunity to explore different ways to solve problems as they build towards a conceptual understanding of the ‘new’ strategy. I love how there is more freedom for students to use the knowledge they have and then to build on it. Hopefully, fewer students would be left behind in the dust. Additionally, the 3part framework encourages those students who are very inquisitive to explore outside the box.
My only reservation: Can I get through the curriculum? I usually have at least one class where students are highly motivated to learn and work, so this style of lesson is fantastic to use. The other classes are a mix. It’s there where I worry about progressing too slowly to get to the end point; however, I will definitely give it a shot.😀

Great reflection here and great question. We often ask educators whether “getting through” means students learned the material? We used to be masters of “getting through”, but often times, our students didn’t make meaning of what they saw. So while there might be a fear/risk of a time crunch, we believe it is worth the risk.


The interaction with lessons like this, let’s us see how the students were solving the problems. We can hear what strategies they use, their way of thinking and also areas of development.
I think we always say one reservation will be time and the age of students.

I think this lesson style is beneficial if students have confidence already and they are willing to try looking at things another way. They will find it satisfying to defend their answer. Students who are not confident and don’t have strong number sense will still feel lost without adequate support. I, too, worry about getting through the curriculum I have to get through.

This is an interesting reflection. Have you tried problem based lessons enough to know this is reality or is this an assumption you are making?


The lesson style enables teachers to utilize more openended questions. This heightens the expectation of students having to think and solve problems using their problemsolving skills instead of waiting for the answer. This also does not limit students to think that there is only one way to solve problems. Versus traditional lessons that simply teach students to copy the teacher and not think for themselves.
The reservations about using this style is less structure in the way things are taught, task could take longer than expected and not allow for all task to be met in one day, how to know if students know how to accurately solve the problem, and is this the best way to present new ideas to all levels of learners.
Utilizing this type of style allows students to become more vocal in their understanding, problem solving and communication skills about math specifically. With this style students might be more willing to make mistakes and try again. This type of lesson engages the learners and develop their own style of learning why we solve problems a particular way.
Thanks for sharing! Great points and some common reservations. These are all ideas that one must grapple with as they begin a problem based lesson journey which will vary from grade to grade, concept to concept.
