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Lesson 42: Automaticity vs. Memorization
Posted by Jon on May 1, 2019 at 11:55 amPlease post your thoughts and questions on the 42 Fuelling Sense Making lesson here.
Mary Olsen replied 2 months, 1 week ago 101 Members · 131 Replies 
131 Replies

It is amazing that everyone agrees that we need to learn these ideas in our teacher training. I was trained to teach Primary/Junior (grade k6) in teachers college, but then took my intermediate and senior (712) additional qualification courses. Not once can I remember discussing teaching students different ways of counting. Now that I am teaching secondary I have had a number of discussions about the value of teaching conceptionally vs. memorizing. My department members feel that the students need to memorize and then will later come to understand the ideas rather than learning the concepts and automatizing the memory. I am not sure why but I know that I did not really memorize my math facts, but had developed a mental method of computation that worked. Sometimes slower, but then sometimes I would also get lost in thought about the connections that were being made with the numbers. I always thought this was due to my visual nature of learning and visualizing the concept or idea in my head assisted me to understand the concept. I am still struggling with a way to convince my fellow teachers that conceptual learning is better in the long run.

Since most teachers were taught math through memorization, and most students are still taught math through memorization, it is very important that I, as a middle school math teacher, really try to spark student’s curiousity into WHY these memorized math things work.
I really want them to have a deeper understanding of math instead of just seeing a few numbers in a story problem and thinking: should I add them, subtract them or multiply them?
It’s interesting when you ask students to “restate the problem in your own words”, they can’t. And when I question: why would you multiply those two values? they have NO idea.
I can’t wait to use the notice/wonder concept with rewritten problems in my classes. I think that is a gamechanger, to get them interested and asking why instead of just expecting memorization of facts & formulas.

Could not have said it better myself…I have 2 children and 1 memorized and one never really needed to. Maybe it is not all one way or the other…

Well said, especially with what you said about the story problems, just because they’re in math class they might go to that surface level thinking of “do I add/subtract, etc.?”, but almost having them consistently ask “What is the goal of what I’m trying to figure out? How can I get there? What strategies could help me approach that?” That is what leads into the multiple strategies, and then deciding what will bolster effectiveness and efficiency… they need to see the purpose and conceptual nature behind having fluency first.


For most of us, we grew up learning most of our math through rote memorization. This is often what many parents today want and expect for their children as well. They don’t understand that the idea of common core math or developing a mathematical foundation is really going to take their children so much further and make them so much more confident in their understandings of math. So many of the students today who say they are “bad at math” usually feel this way because they don’t have a deep conceptual understanding of math, but rather have just been taught some math facts that have no relevance to anything else for them.

I definitely learned to read and do math through memorization. When I was in fourth grade, I was pretty far behind my peers in most in reading, spelling, and math. My mom took me to a tutor where I memorized my math facts, worked on spelling, and developed my reading skills. I really took to memorization and my confidence grew. However, I notice that I tend to fall back on what I know and sometimes struggle with tackling something unfamiliar. A very good example of this is when reading, I tend to have difficulty with sounding out names of people or places that are not common.
As a teacher, I notice that many of my students do not have their math facts memorized or have automaticity with them. Even more important that knowing math facts is making meaning out of the concepts we are learning. I would much rather give my student a multiplication chart and watch them solve a problem than worry about their memorization of math facts.

It is important to lead students to understanding of base concepts before moving on. Even though I did well in math, I did not really understand until began teaching math to my students. I teach the way I was taught. It will take real effort to overcome this.

I was 25 or 27 years old before I realized there is MORE than one 1 to subtract! I relied on memorization my whole life. I want my students to see a problem and have stratigies to solve it. Automization works because it’s basis is the strategy, it is founded in the connections made.

I always found the progression from little understanding to some understanding to mastery is very interesting with students. Finding out what they actually know and see what gaps they have with number sense, for instance, can dramatically affect a teacher’s approach to further experiences the student is given in class.
I really appreciate the document “Principles of Counting and Quantity” as I have used the concept of subitizing with my middle schoolers who have difficulty with math. It seems like an easy task for older students, but it is surprising how many need experience and further practice to master this concept. I will be checking out these other ideas and see what I can find out with my other students.
Having to use Distance School as a method of teaching, I may not be able to check all this out during this school year, but will definitely have it handy when we are in a classroom and I have my students’ attention daily.
Thanks !

When is comes to students learning their math facts I think it is important to give them a strategy to first solve the problem. This way they have something tangable to refer to. Then from this point I think it is important for some memorization to happen.
With addition I fee as though kids need something to count at first then they can move to the dot counting on the numbers. When adding I try and make a ten first then remember what I have left over for the ones. This is for the more advanced math fact learners.
With multiplication I was just expected to memorize the facts. I memorized them then later I learned some strategies when it came to learning them. If you learn the startegy first then memorize them it can work as well. Either way I feel they are both sufficient in learning/knowing them.

I often struggle with the need for students to know their math facts but knowing that just have them memorize them is not the best way for that to happen. I really like the idea that students learn them through really knowing what the concept of multiplication means and then using it in real world context.

This is a conversation my department is getting ready to have, so the timing of my watching this lesson is uncannily relevant. I am definitely on the discovery and inquiry side of this argument. The school is very progressive as well, but the Maths department is full of traditionalists. The administrative level is changing next year and they are going to start requiring more inquiry in the Maths department. I am excited that maybe the leadership in the department will also change and I won’t be so lonely in my thinking anymore. 🙂

It is not uncommon for high school math departments to be more on the traditional side. I think one of the keys is ensuring that they know that what they are doing isnt “wrong”, but rather maybe it is just too quick. Let’s toss that inquiry / investigative experience in there first and allow students to do more of the thinking! 🙂
Good luck as you move forward with this discussion!


I still remember how I was taught multiplication through memorisation without understanding anything!! Before my exams I had to memorise the whole multiplication table from 2 to 12. But I understand that I cannot teach the way I was taught!!
I need to help students understand problems and help them to make sense of it.

Hi Premila, I keep seeing your posts and I wonder if you would share what grade(s) or courses you teach and where (what country) you were educated? Would you be up for being part of a sharing group? Stephanie


I agree that teachers are ill prepared for teaching multiplication facts and that we rely on memorization which is how we were taught. It wasn’t until I started my masters classes that I even knew there were three phases students go through when learning multiplication facts. We discovered in our masters classes that many students don’t get enough practice at the concrete phase and are forced to moved to the fact strategies phase or the mastery/ automaticity phase.
From what I have seen in third grade math, students don’t spend much time in the concrete phase, at least in our curriculum. As a 4th/5th grade teacher I’m always frustrated when students are having to use their fingers for math facts. It seems like they become reliant on that strategy and have a hard time getting to the mastery stage. Does anyone else feel this way? Should i be more accepting of this? It just seems like they need more concrete practice.
 This reply was modified 2 years, 9 months ago by Claire Beck.

I just want to say that I am proficient with math and sometimes, when my brain is tired or I just don’t really want to think, I also use my fingers. I feel that using their fingers will help them build automaticity.

Claire, I think that I have good mental computation skills, however, when my memory is engaged in other robust activity like teaching a class, monitoring behaviour etc, I often “loose my automaticity of facts”. Over the years I have discovered that using my fingers or dice dots on the board or a quick decomposition of a multiplication fact, not only gives me a correct fact answer, but it reinforces for my students that tools are useful. Fingers are always available for skip counting or adding up or down. I remember getting in trouble in grade 2 for the dots on my math pages – quick tool for adding on. I actively encourage the use of tools. When working memory is available and facts are automatic, students extinguish their use.

This is a great point! Often times when we are teaching a lesson we have taught many times before, we almost know the answers without even thinking of facts. I try to remind myself to slow down and “play the part” (if need be) to show students that being a math person doesn’t mean you’re fast or that you have a lot of information memorized. Can be easy to forget though, so be intentional about it!


Every year I have so many students entering 5th grade not knowing their math fats even though there is a huge emphasis on memorization and drill in 3rd grade. It just doesn’t stick. Why? Because they gained a temporary fluency without having the conceptual learning down, and haven’t been given enough time to interact with the concept before moving on to automaticity. I have to work into my class opportunities for students to interact conceptually to support them and help them get to that automatic level of dealing with numbers. Christina Tondevold over at BuildMathMinds.com has some wonderful resources and ideas here for elementary teachers.
 This reply was modified 2 years, 9 months ago by Nancy Van Hall.

I think had I been taught by discovery, subtracting would have made a lot more sense in 1st grade and multiplication more sense in 2nd, but discovery was virtually unheard of back in the ’60s (lol).
Since I don’t have a teaching certificate, just a masters in math, I really haven’t been exposed to how math is introduced and built up in the primary levels (from an intentional teaching perspective and built on how humans learn) at all, so I’m really excited to learn more!

This idea of automaticity vs memorization has been a struggle for me. On one hand I understood that allowing students to explore and discover rather than force them to memorize makes sense. However, as a middle school math teacher I see those students who never were given ample opportunities to explore and thus realize the importance of memorizing or learning the facts struggle. To me it’s like expecting a middle school student to read grade level text when they have never been taught to automaticity the letter sounds and their relationships. It’s not that the student can’t read, it’s that they spend so much of their working brain on figuring out the word that they can’t understand the meaning of the text. Based on this, it will be important for me to bridge this he gap by providing them appropriate experiences/tasks that will encourage their thinking, push them to strengthen their math facts while making sense of the 7th grade standards. I must be more focused on the purpose of my lesson and the outcome I want the students to walk away with. In addition, you have taught that I need to anticipate the thinking of students before the lesson so that I can guide them to deepen their thinking or understanding.

It is certainly difficult trying to give students the necessary experience to build fluency, flexibility and then automaticity when they have large gaps. However, if we can start with those low floor, high ceiling tasks, we can help to “nudge” them along. Keep at it!


I’m a middle school teacher, so I love hearing about what is so essential in the early grades. I do often see kids who just can’t seem to latch on to the ideas and concepts we study in middle school, and I do think that lack of understanding about things like counting, making groups, multiplication are likely the root of these difficulties. I do try to include some of this as it comes up naturally in class, but wondered if there are any resources out there that might fit with a middle school classroom to reinforce these ideas?

Like Ericka in the last post, I too am a middle school teacher and I also encounter a range of misunderstandings regarding math. I feel I have a duty to head off as many of them as I can before they get into high school where things really begin to count for graduation credit, etc. I never really thought about how many ways there are to count – exciting!! That just blows my mind because there again something I think of as very dry and easy, and I get a small glimpse into how complicated it can be. I need as much of that as possible to be able to better understand my students and become a dramatically better teacher.
Thank you!

This is SO important!! So many times students get answers that are nonsensical and they don’t have the number sense to know that their answers are not reasonable!

I totally agree! I would MUCH rather a student get a crazy answer and stop dead in their tracks to question its plausibility than have a student punch numbers into a calculator (based on an equation from the book) and have no concept of whether or not it makes sense.


One of the things that I firmly believe is that automaticity is stronger and more valuable when learning mathematics than memorization. I think, unfortunately, that as a specialized math teacher — rather than being a general ed elementary teacher — allows me a little more privilege when it comes to having this belief. In the US, so much of our elementary education programs that teachers complete focus on literacy, and not so much on mathematics and math development. Therefore, if most elementary teachers are not strong with their foundational mathematics and the conceptual knowledge behind such math, they are more likely to default to memorization. Most likely it’s how they learned and it’s a lot easier to facilitate than having to develop lessons that are deep and conceptual that lend towards automaticity. I believe that we need a change in the system of teacher education if we really want to train our elementary teachers (and many secondary teachers) to focus on automaticity instead of memorization. I think that math illiteracy and fear is the biggest obstacle when it comes to this debate. And it’s frustrating as a middle school teacher to get students who are comfortable with memorization battling the types of lessons that are more conceptual and build automaticity — they feel unsuccessful and get upset!

I don’t think I’ve ever focused on the difference between memorization and automaticity. It makes a lot of sense to me, and I’d like to focus on the latter. When students simply memorize facts or numbers, there is a disconnect, and more often than not it’s for the shortterm, not the longterm. I’m hoping I can create more situations that build automaticity for my 3 three year old and my future fourth grade students.

I also think we often say we wish this or that was part of our undergraduate program, but honestly, so many things are taught when you’re learning to become a teacher and without real classroom experience, it lacks the amount of relevance I get from continuing to do professional development as a teacher. So, instead, I think every teacher should continually enroll in courses, seminars/webinars, and online opportunities like this one so we are connecting it to what we do/what we should do in our classrooms.

I could not agree with you more. What we were taught to become a teacher is nothing compared to what we learn in the classroom, from colleagues, and from PD.



In the long run yes, I want students to memorize certain things. I’m always torn when a middle or high schooler does not know them and they are letting it hold themselves back mentally. I think it is a great point that we memorize things that have no reference to them. I never thought about the alphabet being an order that has no reason to be that order, but numbers are in that order for a reason.

I used to think I was bad at math because I couldn’t state my multiplication facts quickly enough (playing around the world game). Instead, I would break numbers apart, multiply the pieces, and then put them back together to get the correct answer. Now I know that I was actually thinking more deeply about number sense than some of my classmates who could jump to the product within a couple of seconds. I often see numbers as dots on dice. So 8 to me looks like the 5 die and the 3 die. Or sometimes, when I think of 8, I see 2 less than 10.

This lesson was enlightening because I was in the memorization of multiplication facts groups because I believed that if students had that foundation, it would make everything else we were learning easy. I now see the difference between memorization and automaticity and how this will enable students to apply their knowledge beyond knowing math facts up to twelve.

The differentiation between memorization and automaticity is super helpful for me to learn more about since I was a student who was very quick at memorization and later, figured out the meaning of what I had memorized. For a long time, I thought that was just the way math was — that children were ready to memorize but needed more time to understand rather than building understanding and then that leading to automaticity. It’s hard to convince parents, most of whom learned math through memorization, that reciting memorized math facts is not the gold standard of elementary school. Our elementary teachers often have trouble with parents teaching the algorithms to their children when they are working on visual models.

I am a big believer in automaticity vs memorization. Most of us grew up memorizing which doesn’t stick if there is no visual that pops into your brain to connect math facts to. Getting to that automaticity of math facts seems to be an area where my students have fallen short and ultimately where I need to reflect on how my math program needs to evolve. There is such a wide range of learners coming into grade 3 and beyond, some who have a solid understanding of place value and others who are still striving to move beyond onetoone counting and everything in between. Have you found that following the Curiosity Path has helped create a more level playing field and move everyone forward? I’m thinking building in time for math stations and guided math groups alongside may give opportunities to help the concepts that come out during the Curiosity Path stick and to build in some gaps in counting strategies. Would love to hear your thoughts.

Being a middle school math teacher I struggle with the results of the lack of memorization or automaticity. I know in the school I work at multiple of the teachers in the younger grades consider math their weakest and least favorite subject. I don’t think this is uncommon unfortunately. Sometimes I think I should move to the lower grades to start automaticity at the beginning rather than so many years into a child’s education.

I have taught for 12 years. During my first handful of years teaching 4th grade, I was caught up in the “traditional” ways of teaching math facts by having students take timed facts practice EVERY. SINGLE. DAY. It took a few years, but I soon realized what I already knew, which is that some kids memorized them and did these easily, while others struggled to memorize them. Many even had anxiety over these timed facts. I don’t really remember too much about my math facts instruction that I received as a student in elementary school, but I know that I did not have an understanding of why the multiplication worked. I started realizing that my students deserved better from me. However, every year, I still hear fellow teachers blame so much on students not “memorizing” or “knowing” their math facts. Now that I teach 5th grade, I do not emphasize memorizing math facts. However, I do find that students who struggle to understand their facts often do struggle with other skills/concepts. I love the idea of giving students opportunities to learn math facts in a conceptual way. My concern with this is that students are not usually receiving this type of instruction prior to entering 5th grade. They do not have anything to build on. I like the idea of conducting number talks, and would love to find out more resources on how to do so. We also cannot focus on rewarding students for memorizing their math facts without having a deeper understanding of how they work. I’d love to learn some ideas on how to approach this topic in 5th grade, since by that level students do need a solid understanding of their facts.

Sooooo many students don’t know their math facts. Where I see this being an issue is when we are engaging in a complex theory; those that don’t know their facts are working on facts computation rather the theory at hand. I do believe understanding is important but having this basic tool of knowing math facts is critical in higher learning.

As a fourth grade teacher, I have tried songs, using arrays, multiplication charts, and baseten blocks to try to help students visualize what is happening when we multiply. This works for some students. What other ways can I use to help students build automaticity? I learned by the drill method with flash cards and I remember the pain of it to this day!

Representations using models is key… both visual and concrete… if the end goal is just to get to the algorithm, then often times students see through that. If you are pushing them to solve problems without the use of a calculator and to use models and strategies to get there, you’ll see more growth in this area.


How can I help high school students develop automaticity when the embarrassment of not knowing their math facts causes them a great deal of stress and anxiety?

This is a great question and it is very important that we think about it as to not turn students off …
I’d say introducing math talks if you haven’t is so important and so easy to do… if you’ve never done math talks, check out the Shot Put problem based unit and on day 2 onwards we have math talks framed out…


I have been doing learning math facts through automaticity for the last couple of years, but it does bring up some problems. At my school there are still teachers who still do rote memorization of multiplication facts, including Mad Minute. How do you bring the other teachers around to my view point or do I sort of “undo’ what the teacher did last year?
The other issue is parents. I have a number of students who know their times tables among all sorts of things, but have little idea how it works. Do you know of a good article that I could send to parents explaining this approach?

<div>
</div>Growing up I learned Math through rote memorization and we were taught that in order to move to the next level or stage it was important to fully understand the foundation because every concept was a building block for the next step. This is why I often struggled in Math because when we were give the textbook example to follow and understand during a teacherdirected lesson if you didn’t understand why the answer was what it was and how it came to be, well then you were going to struggle in some aspect. Now when I teach Math to my students I do remember and use the methods using the way I was taught but I am also remaining receptive and open to showing students that there is not just one way to learn this “old school style” because it may be an unfamiliar means of learning for them (especially since nowadays textbooks are hard to come by, and/or aren’t even used!)
 This reply was modified 2 years, 8 months ago by Anisha Baboota.
 This reply was modified 2 years, 8 months ago by Anisha Baboota.

I’m glad these topics are being discussed! Lately, I’ve been feeling like I wish I had some insight on how my high school students were taught basic math skills like addition and multiplication. Some of my students get to our high school classes and they still add on their fingers and require a multiplication chart to do any of the other work we are doing. I feel like making meaningful connections with addition and multiplication would greatly help them succeed with the more complex tasks we are doing. I also want to utilize these strategies in remediating the learning for students who are struggling with these things.

These are huge concerns especially for secondary math teachers. When older students are using fingers often or multiplication charts, that is an immediate red flag that they do not have the conceptual understanding AND haven’t developed any fluency/flexibility with the operations they are being asked to use. This makes the high school math curriculum pretty ineffective and not very useful since the concepts are much more complex, yet students haven’t built a true understanding of much more simple (but still complex) concepts. It is also even more difficult when a high school teacher isn’t really sure how to nudge a student from where they are “at” to the next place in their journey aside from trying to get them to rote memorize facts. It really is a challenging circumstance that I believe only building our own teacher content knowledge AND pedogogical practices can help to address!


I am still torn on this issue. As I stated in another reply I wonder if different kids need different approaches and I have watched a lot of kids fall behind in math because they felt like they could not keep up with the kids who (however they got there) were able to compute efficiently without a calculator. I wonder if we should let them use calculators sooner so more of their brain ppower can be used for the creativity of problem solving…

I feel the same in some ways. SOme kids also need more repetition


I have heard a lot of people asking for supports and strategies for building automaticity of math facts. Greg Tang picture books give some great strategies. Here is his website https://gregtangmath.com/ along with a picture of some of his books. Graham Fletchy has a video on counting progression. https://vimeo.com/210115211 Berkeley Everett has a website with counting videos. Making it visual https://mathvisuals.wordpress.com/counting/ Here is an example of one. https://vimeo.com/235431163

What great comments above. I was a visual learner like Mark and as I have always had short term memory issues my ability to visualize and use patterns helped greatly in math as a student. In my intermediate math classes students were only using traditional algorithm in division and multiplication. Only 1 student used halving and doubling and actually knew what it was. Counting stages were not mentioned at all and neither was mental math strategies in our teaching training. A lot of this information has really only been around for a relatively short time. This means that unless teachers are reading about the latest theories and development in teaching then they will not be teaching the students this information either. Adult trainers need to be masters trainers. Lack of time to teach, constant changes, increasing behaviour issues in classes and ever increasing rotary schedules at the younger grades add to this problem. Don’t forget only 3% of English graduates were hired at one point upon graduation at the elementary panel. Only recently has this changed. This all impacts math instruction for sure.

I am a proponent for board games for meaningful practice of math facts, especially games with dice and or that utilize a deck of cards. Rolling dice to move around a gameboard (Monopoly, Parcheesi, etc) give students a chance practice math where math problems are not the focus…it is a skill/component that is a means to achieving a goal (aka beating my sister finally in this game!).
Repeated exposure to simple addition problems builds memorization, fluency, and flexibility. Rolling a 6 and a 1, I can see that I will move 7 spaces, and it doesn’t matter if I move 6 spaces then 1, or 1 space then 6 (WOW! The commutative property). My sister rolls a 4 and a 3, and I can see this is just another decomposition of 7.
Black Jack is another fun game (with or without the gambling!) for students to practice their basic addition and subtraction skills.
“This Game Goes to 11” is a more kidfriendly games that allows students to practice their addition skills.

I agree! I think math games are a great way for students to build their understanding of the basics.


I’ve always struggled with the concepts of memorization vs. automaticity. I love the idea of providing students with more time to use visuals, problem solving situations, and manipulatives. So often, my students ask if they can just use a calculator or refer to a multiplication table. Sometimes I’ve actually caved and felt pure defeat and said yes, just go ahead!! I’m motivated to now put the calculators away, avoid Mad minutes, and spend more time on number talks, rich discussion, and 3 act math tasks.

I taught 4th grade before becoming a 7th grade math teacher. The district made it a goal that all students would learn their addition and subtraction facts by the end of 2nd grade and their multiplication and division facts by the end of 4th grade. We did timed tests once or twice a week. They were rocking those x and / facts. Well I has two groups of those students again as 7th graders and said, what the heck happened; they know these facts 3 years ago. Point proven to me. My question would be, what is the feel on the use of calculators in grade 7?

I feel that automaticity and learning how to become an automatic thinker in math is crucial. I can only speak for myself but I don’t feel I really understand beginning math concepts such as the Counting Principles or the difference between automaticity and memorization before taking this (and other math PD). I don’t remember learning this in teacher’s college (I don’t remember learning the fundamentals either). We discuss the fundamentals in PD sessions at school but we don’t break them down and explicitly talk about them either. I don’t know if if no one wants to imply that teachers don’t know something but I’m kind of under the impression that we’re missing the boat because no one has said “Hey, by the way, these ideas about math are super important and this is what we need to be teaching and observing in our students”. We use Lawson’s continuum which has been helpful but I feel that only scratches the surface. Maybe the answer is to have specialized teachers teaching core subjects such as reading, writing, and math. In elementary we often describe ourselves at the “jacks of all trades” but maybe it’s time to make the shift so that we have the time to deeply learn and understand specific areas and do that most of the time, rather than one hour a day?

Memorization seems to be a lost art these days because every answer to any question we have is at our fingertips. We don’t ask kids to use that part of their brain anymore, just like Jon mentioned about phone numbers (we have to stop and think about any phone number besides our own – when back in the day I knew LOTS of phone numbers). I think memorizing with understanding is important…if I know my 2 times tables can I easily use those to get to 4 times and 8 times? Can I get from 3 times to 6 times? How are they related and how can I use what I have memorized to get an answer to something I don’t have memorized? Maybe that is the Consciously Masterful?

This lesson aligned with my belief that automaticity is more important than memorization of facts. For example, a student doesn’t have 8 x 6 memorized but understands that 8 x 5 = 40 so 8 x 6 is one more group of 8. That student can arrive at 8 x 6 = 48 quickly but does not have it memorized. Understanding multiplication becomes more important than having all of the multiplication tables memorized.
I have taught grades 5 through 8 math in my teaching career. It’s always the students who have developed good number sense and automaticity with numbers that have the most success in these middle school grades. Over the last two years, I’ve taught 5th grade math classes where I incorporate number talks regularly. These number talks improved students understanding of the basic operations.
 This reply was modified 2 years, 8 months ago by Amy Rensko.

Memorization vs. Automaticity – having come from third grade and having worked very hard to get my students “fluent” with their multiplication facts, only to have the fourth grade teachers complain that “they don’t know their facts” three months later, I wish I would have developed this mindset earlier. Nevertheless, I can see that having students develop a deep conceptual understanding of multiplication AND work with the relationship of number would have helped them to become more automatic, ie, fluent, but that that automaticity would have stayed with them better. I’m reminded of many lessons at the board trying to shove the idea of the distributive property into their brains, if only I would have guided them to construct their own strategy. I was Consciously Incompetent – intuitively, I knew there was a better way, I just hadn’t found it yet.

I really appreciate your discussion of memorization versus automaticity. I had heard for a long time of the different methods students were learning in elementary school for addition, multiplication, etc., and had only heard how foreign they were. I’d also heard from parents who were frustrated at their inability to help their kids. Only just last year did they start showing us at the high school level these new conceptual ways of understanding the principles, and I really like them! I wish I knew more about them.

It is very difficult to build automaticity unless students have baseline facts that they know. They need handson experience with counting and quantity in the early years, especially using visual structures like 10 frames and rekenreks, and then drawing models and labeling them with numbers to move to representational and abstract. Games can provide great reinforcement of addition facts. Until kids can make 10, know (or readily derive) the sums of singledigit doubles, and can mentally juggle 2 more/less than those, there’s not a lot to build on to derive unknown facts from known ones. That, and being able to add/subtract 10 and multiples of 10 are the basic skills for which I wish every child had at least conscious competence when they come to me in 3rd grade!

I teach middle schoolers and often wonder why some of the students know their multiplication facts and some do not. I also find it interesting that some of the students know their multiplication facts but do have automaticity. I agree that automaticity is a better, more useful way to learn your math facts. It seems to me that students who learned their facts through automaticity have more number sense than students who learned through memorization . I will say that I had a student this year who knew his multiplication facts (by memorization) but needed to work on number sense. This student was very proud of himself because he was always the first person to answer when we needed to know the product of 2 numbers. The fact that he knew his multiplication facts gave him great confidence which helped him when he so often felt like he was not so good at math. So although I recommend automaticity for learning multiplication facts, I still feel that it is better to learn through memorization, then not to learn the multiplication facts at all.

I agree with this model. A great way to build automaticity is through games that build fluency. My state has a document for each grade level that has building fluency through games. The one below is 5th grade.
https://drive.google.com/file/d/0Bze4eU0rInwwRExXZHI4RUVkMGM/view

I remember all to well sitting dumbfounded as to how my teacher was able to manipulate a strategy by just looking at the question. We were given a lesson and somehow when the word problems came along, the strategy in its taught form did not work. I thought that whether we are old or new school we desire the same goals. However, I realize the old way doesn’t really lead to autonomy. I wonder how different my experience would have been if I truly understood how numbers worked. I’ve always loved math and even with grit could not see the various ways around solving different problems. I definitely have a different outlook on my teaching pedagogy around facilitator.

I never knew that there were so many different counting principles. I was trained in secondary math, so a lot of the counting and basic math principles were not taught to me. I find it fascinating how much math students truly learn during their first years in school!

I feel it’s important that the students can recall multiplication facts automatically . I think using calculators stunts the need or importance. But if a student knows that multiplying by 8 is the same as double, double, double, then there’s some strategy to go to.

I actually think that memorization of facts, esp. 112 in multiplication, is very helpful. Having automaticity with multiplication (be it using known facts, breaking a factor apart, etc.) makes it easier for students to conquer larger or unknown facts. (Sorry my talk is all multiplication. It’s pretty much life in 3rd grade.)

I think I go back and forth between thinking students should memorize facts and believing that they should form understanding and automaticity with numbers. While I do think it is important for students to understand the meaning of 3×7, in 4th grade as we move into multiplication of larger numbers and multistep problems involving multiplication, it makes it very difficult for students to complete when they have errors or are still skip counting or drawing pictures.

I now realize why so many of my students lack number sense. They don’t view counting as qualitative. This lessons video has given me some ideas on how to help them gain a better understanding.
I believe as students first encounter the concept of multiplication, it is more important for them to gain an understanding of what is happening when they “multiply”. They should be encouraged to make representations and connections between the facts to grow their understanding. And this process should continue as the equations get larger. But here is my concern, due to required curriculum coverage, we have a tight timeline that must be followed. Natural automaticity comes because of the amount of time students spend working on a concept. For many students that doesn’t happen within the time frame. Is asking them to do activities to help them memorize their facts a negative approach if the student can articulate what “to multiply” means?

I think activities that build fluency are important especially after the foundations are laid. I’d just ensure that the activities are “worthwhile” and build on top of that conceptualizing instead of just rote memorization.


I found the lesson to be very interesting. I have been questioned by some of the teachers and parents that I have worked with on this backtobasics approach verses discovery approach, which is sometimes hard to explain to someone. I like the explanation used here. Going back to the beginning of concepts, I think is huge in order to see exactly what and where the students are at. I found myself doing this this year multiple times in order to make sure the students had a full understanding. I also think that part of the struggle is that not all teachers think in this discovery way, so when you get students who have been in a “backtobasics” classroom for a couple of years, it takes a while to have be comfortable in the more discovery type of classroom. It takes them a while to feel like they are capable in math class and know that they can do the work it was just never delivered to them this way before. I think that knowing and having a recollection of multiplication facts does help in the older grades, I don’t think that they have to be “memorized”. Students can have an understanding of multiplication that will enable them to solve any type of multiplication; including the basics facts.

As an elementary teacher, I have found there are students who can’t memorize their multiplication facts. Now I am wondering if that is because they can’t unitize or never grasped it in earlier grades. Same with place value. I’ve always trusted younger grades to build this base, but maybe not enough time is spent on unitizing? I’m curious to find out.

This is such a big discussion that has been had over and over again. I teach 5th grade and my thoughts on this have changed over the years.
I do agree that students need to know their basic facts. However, my view on that has changed. I use to be the automatic you need to know this and above all else DON”T USE YOUR FINGERS. After many different trainings, including these, I am much more in favor of teaching automaticity. Students still come up with the answer, but I feel that this way they better understand how they got to the answer. It’s kind of like the challenge of Old School vs. New School thinking.

Last school year I was in 3rd grade. I was trying to have my students “memorize’ their multiplication facts because I thought it would help them become fluent! Whoops!! I will say that we also spent a lot of time digging deeper into what those facts really meant by using manipulative, arrays, and other hands on/visuals. The Do/Don’t really hit home with me “Don’t: Memorize math facts in order to engage in interesting math. Do: Explore interesting math to develop knowledge and understand math facts!”

I am in total agreement with your reasoning. This talk brought be back to a grad class I took and the teacher asked use to solve the question. We all did it while using our graphing calculator. He then said, why did you solve it that way and we all said that was how it was shown in the text book. What we couldn’t provide to the teacher is why did we use the number and symbols to get the answer and the reason was we just mesmerized what was done in the book and repeated it. Basically, we were not able to understand why the problem needed to be solve this way. This discussion opened my eyes and showed that memorizing facts is not the route to go. I know that we could eventually memorize facts but at by teaching the why of the concept will help the students understand the concept fully.

What a huge take away and your reflection will hopefully continue to remind you as we tend to have to fight the urge to “show” kids how to do math vs. letting them investigate and explore first.


I sometimes feel like the media’s presentation of the 2 distinct groups of math teachers helps fuel teachers that are mostly focused on memorization that a deep understanding of concepts. At the beginning of my career I was more focused on memorization I came to discover. Many of my students performed well because they answered only questions that were familiar to them. Questions that were unfamiliar were often not tried by students.

In my experience I feel that there is a big divide between the “back to basic” and “discovery” groups of teachers. When you have this at a school it really hurts students from year to year when teachers are not on the same page. Students are often unsure of what to do when they don’t have a fact memorized and often resort back to strategies that are inefficient, but comfortable with (like counting on, or skip counting).

You’re so right! And, the toughest part is that it is never a complete “either/or” situation which makes it worse when we flip flop from one extreme to the other each year.
We want to build conceptual understanding AND procedural fluency – just not through rote memorization without understanding.


I always thought the problem with memorization of multiplication facts shows up when a child’s memory fails them.
Students often resort to guessing when they lack a conceptual understanding of what multiplication means.
That was a brilliant point made in this video connecting memorization to information difficult to link to other relevant memories.
Counting to multiplying, seems to me to be a continuum. The gradual changes that occur along the way offer so many connecting points that students that simply memorize are being cheated.

I really appreciate learning about how to break down what I have always thought of as basic knowledge, multiplication tables. I have never understood math in this depth and I am excited to see how everything is coming together in my own understanding.

Fantastic to hear! Don’t worry… many of us “think” we understand the math, but there is always something there that we are unaware of. The more we understand it, the better we can teach it.


Understanding the base ten system is a must for students. As a math coach and an interventionist, I work with students every day who do not understand our base ten system. They have no idea, “why a teacher told them to carry a one” in addition. They are totally confused. I love the concept of unitizing, it is essential. I love showing them how everything is interrelated in counting.

So true! Without building a solid conceptual understanding of unitizing, students will certainly continue struggling until they decide that they “aren’t a math person”. Thank you for the work you do!


I do think it’s a both/and rather than a one or the other. I like the word, “automaticity,” better than “memorization.” Students need to realize the relevance of numbers and understand what is happening with numbers then apply that understanding to addition, subtraction, multiplication, and division. Then, they should build upon that understanding to be quick to know that 3 X 2 = 6 without going through the processing.

Agreed. Giving students repeated opportunities to think and then develop that automaticity is so important.


I was taught to memorize math facts along with other things in school. When I am in a classroom with kids that don’t know math facts I find it hard for myself to find ways to help them remember without memorization. Sometimes i learn new ways from the students. Songs are a good way to learn things, like the alphabet, etc. This is an area I am trying to improve on to teach math conceptually.
I am also curious if there will be more talk in this module of the Counting Principles.

I teach students that have IEP’s in middle school. My take is that if they haven’t memorized them by now , then they certainly are not going to memorize them in middle school. I have been using the “Math in Bed” notice and wonder prompts to build counting principles and automaticity. The more my students are exposed to different ways of counting the more connections they make to to multiplication and concept based number sense.

While I agree, we certainly want to continue providing them with opportunities to build fluency and flexibility – hence why so many of our problem based units focus on working without the calculator and developing strategies and models. Keep up the great work!


This really made me think about my math “moment”! When I was in the third grade I could not pass my timed multiplication table tests because I just didn’t get what multiplication was! It was just a memorization exercise and I HATED math at that point. It wasn’t until later in high school that I started loving it because I was understanding what I was doing! I like how you give credit to both parts though because the automaticity is what gives students flexibility to do problems in different ways.

I had a frustration recently with my 11th grade Higher Level IB class the other day that brought to mind the idea of memorization vs automaticity. It was basically solving complicated algebraic expressions (involving radicals, pi, fractions, etc), and they couldn’t see the connections between all the algebra that had gotten them to that point. You don’t get to HL math without being a traditionally “good” algebra student, and probably even someone who enjoys algebraic manipulation. My students had so much experience and success solving algebraic equations with whole numbers, but they just weren’t making the connection between, say, 3x=6 and 5pi/sqrt2=6.
They kept asking for help, and I just kept on writing simpler algebra problems on the board. Eventually, we all came to an AHA moment, but it felt to me as if I was in front of a room of students who had memorized, rather than really understood, algebraic manipulation.

Just last week, my director and I were working on an informational sheet to send out to teachers depicting just this: memorization vs. automaticity! I really like that you said that the goal for both is to know the answer quickly, however, the difference is how you get there. Memorization is almost like “numbers working in silos” – there’s no correlation to anything else.
The question then is, how do we know when students have achieved this? As a 3rd grade teacher, we taught the concepts of multiplication, the “groups of”, we did explorations, etc. but it got to a point where students then needed to show that automaticity, and if it wasn’t there, then rote memorization would be the way to learn them. How do we prevent this from happening? How do we know students have automaticity or they have just memorized the facts?

Glad your perspective aligns and you both see the value in automaticity!
I find that digging deeper helps us to quickly determine whether just memorized or automaticity had set in. When they tell you quickly, asking them how they know is a quick tell.


I loved this lesson. It helps prioritize the concepts to work on before multiplications and to build a solid foundation of counting for kids. Loved the examples! I will use it in grades 3 and 4 with the teachers that I work with.

As a kid, my dad was a math teacher and he always said, “if you understand the fundamental concepts, you’ll never need to think about memorizing anything.” I think that this is so true but many kids miss the bus at some point and then they are forced to play catch up by memorizing. Sometimes, I think that these kids see math as something that requires memorization.
 This reply was modified 1 year ago by David McKnight.

so true! Our task now is how can we provide our students who are playing catch up a chance to “rumble” with those concepts while allowing others to stretch forward.

I agree with the post about “now put this in your own words” Students cannot because they have just memorized everything.

I used to skip over communication type prompts with my students because it was such a roadblock. Now I realize that prompts like that are the real work / learning.


As a high school math teacher I have always struggled to understand why the students struggle so much with the basics, but this has been quite enlightening. The part about unitizing was really helpful because so many of my students have no idea about place value, and this helps explain why. Just this morning, one of my better students was asking for help with a problem, and the only reason she needed help was because she had no idea what place value any number was even in, much less an understanding of what it actually means. I am fortunate to be working in a school that understands that I have to take my students where they are and build from there, so I will probably end up using a lot of these ideas which are really aimed at younger students, and hopefully see some amazing growth.

Glad this helped to shed some light. I was in a calculus class yesterday and similar struggles were evident. It isn’t the new concepts holding students back – it is the lack of any actual understanding. Let’s help to address this!


This is a huge conversation in my school right now for students transitioning from elementary to middle school, along with conversation of what to do when student don’t have automaticity by middle school. One of things I appreciate being mentioned is the order of things. Teachers often feel they can not do thought provoking math until after students are fluent. But sometimes it’s the context that gets students interested in the math, asking to learn new efficient methods, or recognizing patterns.

Exactly! It also offers another opportunity to practice those skills in context while extending to more complex ideas at the same time.


This is really a hot topic for me. I teach several classes a year of intervention math for 8th graders. I believe being able to multiply is the key to everything for these struggling students. So many of their problems in math can be traced back to their inability to multiply.
I am eager to see what these guys have to say. I am willing to try anything.

As a high school teacher I would be nervous to be given a child and told to teach them number sense with onetoone, hierarchical inclusion and unitizing. I wish my education would have included more elementary level math so that I understand how to get students thinking in this way. On the plus side I have three kids I can now work with and learn with them as the oldest starts kindergarten next year. My husband and I have been working with him already in Mommy and Daddy Preschool and I will be sure to focus on these three ideas this summer. I also have prepared to use the MMM videos that were given in the previous modules so that I know he is being taught to think and explore.

One thing that I have noticed over the past three years is that my students who have a handle on their math facts are far more confident as they are introduced to more complex math concepts. It is heartbreaking to watch most of my students bog down while trying to solve a one step equation. We struggle with this every day and I don’t have a good answer.

This makes perfect sense and it often comes down to not having a real understanding of what they are doing. We have lots of great one and two step problem based units that can help you and them along in this journey in the tasks section of the website. Have a peek!


I have been told that I need to make sure that I provide a lot of practice problems and give students calculators because kids can’t calculate in their heads. This bothers me. I don’t want kids to memorize and yet I don’t think that giving a student a calculator is a bad thing.
Understanding what each computation will do to a number is important and I want kids to get the basics of that. I want kids to know how the computations work and what the numbers mean.

You’ll see that with our problem based units in our tasks section of the website that we are constantly promoting models instead of calculators for students to build automaticity. Check them out!


It’s interesting to hear that the “Back to Basics” vs. “Discovery and Inquiry” argument has been going on so long. When I started teaching, Common Core was in full force and had been for several years. It’s frustrating to listen to nonteachers complain about “Common Core math” when they have never even looked at the standards or practices. Sure, they have their faults, but in a perfect world, they do point students toward a deeper understanding than they would gain through pure memorization. I do still think there’s value in “knowing” math facts, but it truly is useless if students don’t know what those facts mean. I’ll be curious to see what else you have to say about exploring interesting math to build understanding of math facts.

Isn’t it bonkers? I think folks still believe it is an either or when in reality we’d like them to understand AND know. That is what developing true automaticity is all about!


My 5 year old son can count up to 100 and he can also skip count by 5. He finds it fun and I think I should ask him tomorrow if he can represent the counting with concrete materials and what it means to him.
With my students, some of them struggled with multiplication. My 12 year old daughter had a very difficult time with multiplying numbers since grade 2. We didn’t know how she developed her way of multiplying numbers (especially multidigit numbers) until this year. Embarrassed to say, my husband and I (both math teachers) finally sat her down and worked with her. She understood the area model and she could even work on multiplying fractions but she was just not able to multiply numbers “efficiently”. Yes, if she has all the time in the world, she can finish all the questions on a test. Unfortunately, our world is not a perfect world! I thought she would somehow advance to the automaticity level but she didn’t. I turned back to the “old way”. I wonder how to encourage students to move from consciously incompetent to consciously competent/ unconsciously competent before moving to consciously expertise!

Modelling is key, so stick with it. Using partial products is a huge strategy also and the model makes it visible enough to see those connections and “chunk” it down into smaller, more manageable pieces.

Would you give me lesson plans to strengthen multiplying numbers from MMM site? Also, what is your opinion on mobymax and IXL? Would you use them at all?

Btw, I checked with my 5 year old son. We had a number of candies in the bag. I asked him to put them in groups. He first put them in 3 groups of 6 candies and (found out there were 19 candies), then 6 groups of 3 candies then he also put them into 9 groups of 2 candies. He asked if he could keep that one candy since it cannot be placed in any group. LOL. I’m blown away by how he can show his thinking.



I really want to incorporate more number talks in my class (6th grade) to help with automaticity vs. memorization. The understanding part is so important. I find students coming from elementary who believe they are strong in math rely on their memory. I want to deepen the understanding and have them know why.

I think about watching kids “skip count” by fives on their fingers (yes, I am in middle school and yes, this is common…), but they don’t do it for other numbers. We have the “nine’s trick”, but when I see kids using it, they aren’t thinking about 4 groups of 9, just putting down their 4th finger so they have 3 and 6 for 36. I feel like this is an area that would help older kids make better connections and be less of a “trick”. Also I like the idea of group counting up and down to prebuild the multiplication/division connection. Very often I will hear kids say they can multiply but not divide.
The concept of unitizing makes so much sense for how it ties in to fractions and can be used (hopefully) to help demystify them even more.

As others have posted number sense is huge in Higher level math course, if a student does not have a good number sense they tend to struggle with the why of some of these concepts.

The topic of memorization vs automaticity is huge in my district right now. We have teachers and parents on both sides of this that are passionate about their view. It is interesting as the district math coach that because of my age (let’s just say I look like a veteran teacher) that many parents assume I am on the memorize bandwagon. But I would much rather have a student who can think and reason their way to a computation because they see the relationships between numbers in order to do so. So I am in this workshop to learn ways to help teachers help students become mathematicians and not rote memorizers!
 This reply was modified 8 months, 1 week ago by Julie Gonzales.

(This is more of reflection as we are currently in summer and it is hard to do all the call to action pieces without students)
I don’t usually think about all the principles of counting, yet in college one of my favorite classes (and toughest) was combinatorics. It all came down to a deep understanding of how to count and quantities of numbers relate to one another. In a high school setting, teaching juniors, we often see this lack of number sense. I really enjoyed using some of the tools presented here with my Freshmen, but it was a lot tougher to bring it back down to basics with the juniors taking Algebra 2 when their “math facts” foundation was so fragile. I am excited to build these connections in the coming year from the first week of school.

I agree with what many have said about this topic above. My perspective is being a 4th grade teacher. I often find myself frustrated when teaching 6 digit addition and subtraction and 4 x 1 multiplication or division when students are not yet reaching automaticity with their facts. I do provide multiplication charts to students who need them so they can learn the grade level content. I wonder what we should be doing in our elementary classrooms to ensure that students have ample opportunities to become automatic in basic facts so that they are able to solve middle elementary and middle school standards with less difficulty. It seems to be a challenge everywhere (Canada and the US at least…) I wonder what we all need to do differently…

I have struggled immensely over the past couple year with my own beliefs about inquiry based learning and factbased memorization, essentially jumping back and forth over the fence many times. It is not until recently through my own exploration of numeracy strategies that I have realized how understanding how numbers can work can lead to quick fact recall. While I used memorization throughout most of my life for math fact recall, new learning of various math strategies has help my brain when the fact recall is just not coming. Here I can use my own reasoning strategies to help. A specific example is the 9 x table facts. I used to use the finger trick all the time, sharing this trick with many students, until I was taught the strategy of compensation where I could use my understanding of 10 x table facts in order to help me calculate 9 x facts.
Ex 9 x 7 = 10 x 7 = 70
70 minus one group of 7 is 63

I really appreciated the discussion around memorization vs. automaticity. In the seventh grade I work with a very inquiry based learning style in my math classroom, but given that my students have a lot of prior mathematical knowledge it means that they have more strategies in their toolbox so to speak to use to help them work through things. Those students who have their times tables memorized tend to be the same students who are not sure how to connect what they know to a new situation and ask me for the formula or steps to follow so that they can just do it. I would have to say that although their facts are well established in their memory bank, they do not have that mathematical fluidity one would have if their facts were understood on a more conceptual level. Hoping this makes sense as my long covid brain fog seems to be hampering my word retrieval at the moment…

Thanks for sharing. This isn’t the first time we’ve heard a perspective like this… knowing facts doesn’t make problem solvers and hopefully throughout this Online Workshop, you’re seeing ways to put students in a position to do more problem solving!


It’s fascinating and enlightening to spend time in a K class. Especially after teaching middle and high school math for a while. Seeing these fundamental numeracy concepts just beginning to develop gives a great perspective on the overall progression that students go through during elementary years… and the challenges that elementary teachers face to try and help students discover their math facts through numeracy exploration on the path toward automaticity.

I have always felt that math facts need to be more readily available to kids and definitely in my earlier years of teaching I worked on the memorization of facts. “Around the World”, anyone? But what I have found is that many students are not flexible with their math facts, so even with my 7th graders, while I may not do a whole section of class of “what do I need?”, I do find myself when certain numbers come up asking questions like, “huh, the answer is 57, what do I need to get to 100? how about 150?” (an activity from Pamela Harris) Or .. when we have worked on finding the mean or multiplying larger numbers, I have asked about different strategies they used to add those numbers or getting their facts that they need. I find I am embedding the fluencytype questions within my lessons more than I would have years ago.

I teach 8th graders so I am past the point of memorizing math facts even if students don’t know them well. Yes we could have many other things to memorize in 8th Grade but I choose not to. I tell my students the only thing they have to memorize in my class is perfect squares up to 20. What I want them to do is discover things like volume formulas, angle relationships, components of linear functions, etc. Once they are discovered, we work and manipulate concepts in so many different ways that the automaticity comes. I know they will forget parts of what they learn but my hope is that I have built a foundation that they can either reason their way back to understanding or with a little information, I can get them back to a mastery level.

It is my 3rdgrade year and 1979. Math class is an “Around the World” competition. Fortunately for me, I was pretty darn good, either that or the rest of my classmates were just bad. I am going with the former. I was a lucky one and memorization came easily for me but I really don’t think it was all just memorization. I think I always understood that 4 groups of 5 is 20 and could see that in my mind’s eye. Too many students cannot make sense of repeated addition and it makes math so frustrating for them (and me).
It doesn’t even have to be multiplication facts. So many students struggle with simple addition and subtraction. My 7thgrade and 8thgrade math classes consist of many students who still count on their fingers. For instance, 12 – 8 equals 4. A possible strategy is to subtract 2 two times. The same applies to 17 + 8. Add three to get to 20 and then add five more.
Our best friends love to play the card game Euchre. The scores get into the hundreds. I literally watched my friend, who has an advanced degree in the medical field, add two numbers in the hundreds using the traditional algorithm – add the ones, carry, add the tens, carry, etc. I actually was a little shocked. I would have added the hundreds, then tens, and ones being sure to adjust accordingly.

The idea that counting is so much more complex and different than the ABC’s is certainly an interesting perspective that I haven’t thought about before. I also think that students finding their own multiplication “rules” is important and strengthens their number sense. The harder part is going to be how we incorporate this in higher grades to students who are unwilling to let go of some of the rules.

I was playing Monopoly with my grandkids over the break and my 7th grade grand daughter had to count the pips on the dice to add them up. I look at the dice and do not even see an addition problem anymore, I just see the sum of the dice as a number. I played a lot of board games growing up and this basic addition became automatic. But not because I sat down and practiced addition over and over. It just happened as I was playing board games. As I noticed this I decided I wanted to play more board games with my grandkids that have 2 number dies in them. So that they will see the addition automatically some day. But it also helped me realize what you are saying about automaticity. I take away two main things. As teachers we need to give opportunities for students to make sense of numbers, addition, multiplication etc.. through planned activities that are designed to help students see the math and experience the numbers. Secondly they need to have multiple opportunities for these experiences. Automaticity does not come from one experience. It comes from seeing the numbers in multiple ways over multiple experiences. Its making sense of the different ways to see them. These opportunities can come as a warmup, a math game near the end of the bell. As an planned activity once a week where students work with manipulatives, draw models, show ways to make sense of what ever math they need to work on. I think every teacher can build these into their weeks. I think if we did we would see deeper understanding of student over time. But it will take time…and we must be committed to the process.