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Lesson 42: Automaticity vs. Memorization

Please post your thoughts and questions on the 42 Fuelling Sense Making lesson here.

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…


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 1 year, 4 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 1 year, 4 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 1 year, 3 months ago by Anisha Baboota.
 This reply was modified 1 year, 3 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 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 1 year, 3 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!


Before speaking about this I was unaware about what an emotive issue this is. I spoke to a group at a Maths JAWs (Job alike workshop) on discovery learning vs explicit instruction and the room split, both metaphorically and physically (people actually moved chairs) into two distinct groups both with very strong views on the subject. Both groups found it hard to understand the other group’s view point.
From my own perspective (currently), i’m in the rare group seeing advantages and disadvantages of both ideologies.
I believe in introducing concepts moving from the concrete to the more abstract, not just memorising but developing a good understanding of a topic. I believe we should find out what students have retained in their long term memory, as this can be very different from short term memory. Students should be also given opportunities to apply their knowledge to unfamiliar scenarios.
I also believe that memorisation of key facts has a use. Cognitive load theory states that amount of information that working memory can hold at one time. Since working memory has a limited capacity, instructional methods should avoid overloading it with additional activities that don’t directly contribute to new learning. If students have to draw an array every time they need to multiply two numbers together, it may be impossible for new learning to occur.
Discovery learning is not new as is something which can be very effective with some topics. Criticising explicit instruction and memorisation is as old as the USA itself. <i style=”fontfamily: inherit; fontsize: inherit; color: inherit;”>“Learning facts is ineffective, pupils may be able to repeat them but not use them or understand how those facts can be deployed in different ways.” JeanJacques Rousseau (17121778).

What I’m hearing from your very well thought through response is the message of “balance”.
Around here, we strive to strike a balance because any extreme is likely not helpful.
Although you already said it pretty well, I like to say that we build conceptual understanding to build procedural fluency over time. The same is true for math facts; let’s explore them, conceptualize them, and always use them through emerging strategies and models. As we do this, a “memorization” is taking place. So I guess the key is how we try to get there… straight repetition without productive struggle or thinking is not as efficient as grappling with problems and using prior knowledge to reason and prove.
Great response here. Thanks for sharing!


I completed my elementary education in the 70s and 80s and did quite well in math by learning my math facts and algorithms taught by my teachers. I appreciated teachers that could teach it well and thought those in my class that didn’t do well were not adept at math. It wasn’t until university that I came to realize the importance of conceptual understanding. I was enrolled in an engineering program and it was clear that the conceptual understanding of algorithms and their usefulness was more important than application of algorithms. It was a struggle and I mean a struggle to build conceptual understanding. Even though I didn’t complete my degree in engineering, the active daily struggle with many math concepts has allowed me to really appreciate the struggle my students go through and it’s importance to their conceptual understanding.
I see the importance of automaticity, having worked with my father, a tailor for many years. Having immigrated from Italy and started his own business, it always amazed me how he easily worked with numbers to run the business and to tailor garments. He added totals, subtracted discounts, multiplied multiple purchases and divided measurements, often without the use of a calculator. I had such an appreciation for how he worked with numbers, I too would often work without a calculator and through automaticity did much of the day to day calculation in my head. There was always a purpose for working with the numbers and automaticity not memorization came of it.

This makes so much sense to me. There is such a difference between memorization and automaticity. I want my students to have automaticity. That is all 🙂

Love the reflections and how the idea of automaticity is resonating. Ironically, I think so many “back to basics” advocates would probably also agree that automaticity is way more rich than rote memorization and I think everyone wins when we try to help our students build that automaticity!

I think a lot of the three act math problems that Dan Meyer and Making Math Moments use model this thinking. That is why so many of the problems fit under the umbrella of proportional reasoning. So, in summary, working with these concepts to promote the fluency is critical. I am also going to completely agree with those pushing the balanced approach with the idea that those students who don’t know their multiplication facts can often become completely bogged down cognitively with equivalencies before they get to the actual operations in many problems.

Balance is definitely key. Unfortunately, we can often get too focused on rote memorization as a means to help students not have to think about certain “basics” that we can cause them to stop thinking at all. So just be cognizant that everything should be done through making connections in order to build automaticity.


This one is a complicated discussion for me to have. Overall, I am closer to the discovery/inquiry camp when it comes to approaching math. But I feel that there is a group of students that benefit from the backtobasics memorization. I look at the kids who have problems with dyscalculia, whose best chance to succeed in math is through memorization.
I feel there are many more resources available to students struggling to read (because it is not socially acceptable not to read as it is not to be able to do the math). My daughter, who suffers from dyslexia, has to depend on her sight words to get through an unfamiliar text, where she does not phonetically read the word but states the word because she recognizes it and its meaning.
I feel that memorization of math facts can help students get through complex ideas in math as well. Where they do not have to struggle with the concept of multiplication to understand proportional reasoning, granted, the automaticity of multiplication will allow students to understand proportionality a lot easier. Still, some students will need memorization to have a chance.
This lesson was amazing and I am entirely on board with the automaticity is the best way to learn math facts because it will help translate connections to other big math concepts. However, I think it is dangerous to state that it must fit everybody if the shoe fits some.

Hi Scott,
I’ll push back a bit here for a couple reasons.
First one is that our message here is that balance is important. However, rote memorization without understanding is not helpful – even for some students who you might feel it could be helpful for. Learning mathematics without meaning is not going to be useful even if one might be able to memorize it. What will they do when they attempt solving problems in everyday life if they didn’t learn it through problem solving?
Often times, special education classes are the biggest culprit for pushing outdated rote strategies without understanding on the students who require the experience learning through context the most. The thought by teachers is that because the process is slow that it isn’t working – yet the students tend to walk away without the content they tried forcing memorization on them with…
What is dangerous is believing that some students are incapable of understanding and are left to memorize information in meaningless ways.


I think about 3+2 – I do not think I memorize that; I see a hand; three fingers and 2. That becomes automatic after repetition but the visual remains. It reminds me of the developing visuals for systesm of equations with the shotput task. Even when the shotput task goes away, we still have the visual to go back to to calculate new problems. So 6 is 3 fingers and two fingers and one more. We need the visual to build off of. If that thing becomes automatic after a while, that becomes a new benchmark fact. So 2 things: (1) I do not think my students got these visuals when they were younger so they are working form a defecit. They can get by with facts that our algebra is based off but they are not automatic. Do I take the time to rebuild that? I defintely want our youngest students (and my own children to pracitce intential skip counting more!).
(2) Start with the visuals; let them devlop that for a concept so that it can form the foundation not only for that concewpt but future mathematical concepts then move to automaticy not memorization.

I think the answers are yes and yes. However, you don’t “go back” and redo earlier grades, you simply start with a low enough floor with the end goal being the current grade.
Take shot put for example. That is a systems of equations unit but starts with simply substituting values into an equation and evaluating.
This can be done by rolling back / lowering the floor but keeping a connected context.


I think the points made in this lesson make total sense and it is much better to aim for automaticity than memorisation. The other term that gets thrown around a lot, and misunderstood just as often, is fluency. To me, fluency aligns with automaticity. It means having the number sense and flexibility to be able to make sense of problems and to be able to quickly figure out 7 x 8 (for example), even if I haven’t actually memorised that fact. However, many people (including those who wrote the Eureka curriculum) seem to equate fluency with memorisation. I abhor math facts sprints and won’t do them. I have seen too many students with a crippling anxiety around math, and an overemphasis on memorisation does not help with that.

So glad that you’re aligning with our perspectives on this. So often curriculum writers have good intentions, but things get miscommunicated during the writing process. Also, there can be leads who are more comfortable in one grade band than another which can also cause issues with how the mathematics develops. Keep pushing for that automaticity and students will eventually get the side benefit of what those who “rote memorize” are aiming for.


The importance of building automaticity through inquiry and discovery before introducing algorithms is such a great concept. However, depending on what grade we teach, the algorithm approach is already engrained. I think it is still possible through problems like the 3act tasks to have students use visuals, manipulatives, number lines, or other means to solve with understanding.

So true. The older students are, the more exposure they’ve likely had to “plug and play” math.
By withholding information in a lesson, you are increasing the chance that students won’t be able to jump to an algorithm.


With the 14 to 16 years old students they have to learn algorithms, i mean remember them. But to use them they have to understand deeply and automate concepts like the order of operations, the distributive property of the multiplication, the operations with fractions, negative numbers… If they have just memorized, if they are not at least unconsciously competent with this skills then is really difficult to help them to use complex algorithms like the general formula to solve second degree equations, and it’s even more difficult for them to use different strategies. In this ages we have a really sad division of students and in some skills like geometry they are like more than a half of students that they don’t understand whats an area they just know the formula or they don’t remember. These students had pas all the primary school without understand anything, they are tired and frustrated. And in secondary school we have 30 students each class, we use to be people that we was suppose to be good of math and we have problems to understand why knowledge and skills that we consider completely obvious, this students haven’t understand. That’s why I’m doing this course I need more skills, strategies…

These are all great points and common struggles. Glad you’re on this journey with us!
For us (and many research papers), the key is building conceptual understanding and over time that procedural fluency will come. When we come at it from a “show and tell” then rote memorization standpoint, we are really just hoping that students will “eventually get it”.


The most important indicator of student success is them having math fluency. In a best case scenario students develop fluency by learning about math while doing interesting things in everyday life. But regardless of how they acquire it, attaining automaticity will impact long term math learning. Not having math fluency can cause anxiety and negative math experiences which students carry with them as they continue on through school.

The conceptual understanding is so critical! I’m contemplating ways to tackle the multiplication/math fact challenge in my classroom now. Most come to me as 7th graders without knowing their math facts, and it really hurts them. They are unable to grasp the concepts being taught because they’re bogged down in the arithmetic. My goal this summer is to arrive at Day 1 next year with a bunch of tools and strategies to help students create the conceptual understanding behind multiplying and dividing.

I want this same goal for myself. We have some englishlanguage learners who often lack automaticity with numbers. They struggle to perform the calculations and rely on memorization or procedures. I hope my tool box is better developed to offer then more concrete examples and tools to solve problems.

This will take time, but if your goal is as you’ve mentioned, you will help them to build that automaticity!



I teach at the high school level. Many students have been able to advance grades without really understanding the meaning of decimals and fractions. As you demonstrated unitizing with whole numbers, I was thinking, “Wouldin’t it help students to see groups of 1/2’s or 1/10’s so that they could visualize multiplication and division of fractions.
I constantly hear my colleagues say that they don’t have enough time, but they have to go back and reteach procedures and rules for multiplying, rounding, and division. Perhaps if students truly understood, the teacher would not have to create these side lessons so often.

You bring up a huge hurdle for educators – especially in the middle and high school grades. While it would be awesome if all students came with a strong conceptual understanding and procedural fluency of the concepts introduced in earlier years, we know that just typically isn’t the case much like how students “forget” factoring or trig. The key to developing this understanding is to use it frequently – which we tend to avoid so we don’t go “off track” during a lesson. This is the opposite of what we should be doing to build fluency.
The question we must ask ourselves is “how useful is ________ if students don’t understand ________”.
You can fill in “trig” and “fractions”
“Factoring quadratics” and “multiplication”
Or any other advanced vs. Rudimentary idea.


This is an area of great debate for sure. I understand the importance of learning math facts but I am definitely more from the camp of if they know how to get the answer does it need to be memorized. I dreaded math in 5th grade because we had to take the 1 minute tests. You had to get 100% to move to the next test. I began to hate math and going to math class. I was a much more conceptual learner and memorizing facts never made sense to me. I spend more time on the conceptual math in my class than the memorization of facts and formulas. We are required to do Sprints each day with our students. I use the Sprints for students to show personal growth, we would do the same Sprint twice and look for how many more than got correct the second time. It was challenge to beat your first score. But I still find them to be abstract and of not much value. I would much prefer for my students to explain their thinking of how they got their answer than to just quickly write a quantity on the paper. I also teach my students that being fast at math does not always mean that you are good at math. You need to get the correct answer and be able to explain why you got the answer. I have actually given my students credit for wrong answers because their explanation and thinking was so strong but they made a small mistake in computation.

I’m surprised you’re required to Sprint with your students with all of the research behind mindset and speed in math class from Jo Boaler and others. I’m wondering if you can morph your sprints so that after students can reflect on a sprint question and show more thinking to you.


I suppose memorization seem to be based on showing the learner the one and only way of finding the answer. Automaticity on the other hand seem to be based on showing the learner various ways of finding the answer.
Clearly, memorization is quick and dirty, straight to the point. While automaticity, I think, engages the learner in various ways. Case in point is the 5 + 3 example. Memorization calls for the answer, 8, case closed. Automaticity on the other hand calls for contextualization, I think, of the addends. It is the interaction that builds familiarity and deep undertsanding of the concepts. Simply put, I think, automaticity encourages the learner to learn math facts through the use of story telling. 5 red delicious apples from my garden and 3 red, equally delicious apples from the community garden…you get the point.

My story is similar to most of those you read here, so I’ll omit it. The struggle is even when your district adopts a discoverybased curriculum and encourages its use, a new teacher who never taught through discovery doesn’t know what do to with it. Even when the campus specialists are on board with discovery, how often can you see them? Unless a teacher actively seeks out discoverybased instruction they tend toward the traditional classroom because it is easier for the teacher. I wanted to teach discovery, but I didn’t know where to go or what it was meant to look like in the classroom. As teachers abandoned discovery, it made it harder for teachers that were still trying to learn to do it to be successful because the students didn’t come in with it and there was no professional support on the campus for it.
 This reply was modified 3 months, 2 weeks ago by Anthony Waslaske.
 This reply was modified 3 months, 2 weeks ago by Anthony Waslaske.


This is where I need some help. I am moving districts this year, but when they hired me I mentioned what I was doing with Making Math Moments and he was very interested. He wants me to share my ideas with the 8thgrade team, but it sounded like I might be the only one teaching projectbased lessons. Beyond the Facebook group in the next few modules, any ideas on how to collaborate? I almost wish there was a Stack Overflow for posting lesson plans for criticism.
 This reply was modified 3 months, 2 weeks ago by Anthony Waslaske.

Congrats on the new role! Glad to hear that you might be taking on a bit of a mentor ship role.
This is definitely something where you’ll want to stay curious a little longer and hold off on action taking / advice giving a little longer as well. Definitely don’t want to come in too hot especially when you’re still learning (never stops) and experimenting. I might recommend bringing your team on board to the academy so you can all do the learning together?


Having taught kindergarten using multiple methods of counting, I lost sight of continuing to teach and allow students to explore these concept using these multiple ways of counting in the older grades.

As a middle school teacher, I never really thought much about the different ways to look at counting, but even just listening to three of the counting principles in the video, I can now see just how important they are to consider. Plus, I think the whole idea could help explain a lot of the difficulties some kids have with the whole memorization piece. Looking forward to learning more about this…

Glad you’re finding these ideas helpful and implementable!


I agree that it is important for students to have an understanding of what multiplication actually is in order for them to be more successful in 8th grade, especially with understanding the Distributive Property. The example of “skip counting” by 2s actually being the counting of 2 items at a time relates well.
I could envision using the short video clip of counting by 2s in this lesson with my class and having them do a Notice and Wonder, before moving into the Distributive Property. I think using the vocabulary that you have _____ groups of ______ would help students to be able to draw out what they have and help them understand why all terms inside the parentheses need to be multiplied (changed) when you simplify the problem, instead of just the first term or just the constant term.

Like how you are applying the curiosity path and also fuelling sense making here! Keep it up!


Thinking of my own experiences in understanding operations, for a long time I was better at multiplying than adding. I attribute this to the face that my mother used to drill me on multiplication tables. I got really good at them and was one of the few students who really liked the timed tests. Adding and subtracting, however. . . They were not as automatic for me. I did come to recognize the combinations that add up to 10, which really helped. It was probably not until I started doing Kakuro puzzles that I got really good at adding numbers. I have found other techniques for subtracting, mostly that involve finding what you need to add to one number to get the other one. 148, for example means that 8+2=10 and 10+4=14. That gives you 148=4+2=6.
With my students, I encourage them to look for the patterns in operations instead of just memorizing them.

I think I can relate to your experience. I don’t think I was super flexible with composing and decomposing numbers which also limited my mental math skills for multiplication outside of the 12×12 tables.
Helping to make connections and build flexibility and fluency is so key to building those mental math skills!


This may be the key to why so many students have such a terribly difficult time learning their basic addition and / or multiplication facts. Most of my teaching years have been in 3rd through 5th grades, and there are always several kids who just cannot get to a point where they have fact automaticity on either one. They get so caught up in the addition / subtraction / multiplication / division facts that they can’t think about the problem itself. It’s almost like I can feel them thinking, “Oh no! This problem wants me to add and multiply!! I’m not good at either one!”
Rather than continuing to ask them to practice facts on this or that program or game, I need to go back and make sure they have the foundations like onetoone correspondence and hierarchical inclusion. Those two are so inherent in mathematical thinking that anyone would be lost without them. (Though of course they don’t know the names for either.) Lack of understanding unitizing explains why some kid just cannot seem to grasp the idea of place value, without which everything past the ones place is difficult. Lots to think about here! I never thought to go back and make sure that a fourth grader has onetoone correspondence, though now thinking back, I can name one or two who truly did not know this concept.

During my first year of teaching, I noticed that almost all my students struggled with basic multiplication. This prevented them from developing confidence in their ability to complete more complex problems. I found myself assigning multiplication drills at the start of class and when that produced more anxiety I turned to computer games. I eventually learned that I had to teach them how to find patterns and other relationships to make connections. I encouraged them to draw pictures, use counting blocks, base ten rods, and anything else we could get our hands on. Not all my students were on board with using manipulatives, but those that were began to show growth in problemsolving.

I love playing games with 10 sided dice in my secondary classroom. I learn so much more about how students see problems using these tools. Some days we add, some days we multiply, sometimes they are integers, so much more to learn when they discuss with classmates.

I used to think that if kids would just memorize their multiplication tables then factoring expressions would be easier. I know think that if they have a better understanding conceptually of what it means to multiply that they would have a much easier time with factoring. I often am breaking things down in my head as I teach to multiply things together quickly…. I only realized I did this as I started to learn more about the importance of letting kids make these connections before we just told them this is how it is.

As a middle school teacher I often see students who lack both the memorization and automaticity of basic math facts. I be noticed it plays a big part in confidence in further math. What do you do to foster growth in automaticity for later/different bloomers?

I think of two students in my class who came in “knowing” their multiplication facts to 12×12. One saw 13×13 as a learning goal. The other saw division as a great mystery. Both were bright students but did not have a conception of these facts they had memorized. Since, in our grade, we were working with place value, repeated addition, and arrays to 5×5, there wouldn’t be specific lessons for what they wanted to know. However, I could build growth in understanding through discovery and inquiry to allow them to move forward at their own pace. For both, I had to deconstruct their memorization to help them use numbers in authentic problemsolving. For both, what seemed a mystery at the beginning of the year became something they grasped without me needing to teach the specific fact they wanted.

I teach 6th and 7th grade. In 6th grade unit rates, I see students use a variety of methods from skip counting to adding on, to build their tables and graphs. But when we get to building proportions in 7th grade, it is quickly obvious that students who know their math facts see proportions and develop an understanding faster than the skip counters/addons, who need more work to develop the proportional reasoning understanding.
I am considering spending more time on equivalent fractions and emphasizing the identity property (for 6th grade) this year in an attempt to help to students develop the connections and to “level the playing ground.” Thoughts???

Teaching Advanced Functions and Calculus the past several years, I have noticed students only wanted to memorize and move on. When I pushed the understanding of the math, students would ask more questions. I am so looking forward to going back and teaching grade 9 math in September.

I teach the middle grades but I can tell some of my students struggle with some of these concepts. They do come out especially with algebra but at other times as well. Perhaps by doing more visual things for algebra I will also be addressing some other issues with understanding of numbers as well.

I think you’re right. The more students can see the thinking and make connections, the deeper the understanding will become.


Wow! My focus on counting and quantity will significantly increase going forward. The counting principles highlighted (hierarchical inclusion and unitizing) helped me grasp why some students have struggled with place value and representing quantities in a variety of ways. I have downloaded the recommended guide to share and to explore these 10 counting principles with the K2 teachers in my PLC.
It is a little frightening to realize that my “teaching” of these math concepts has been incomplete for so many years. I must be gracious with myself and remember…”know better, do better”.

As students enter sixth grade there is a big focus on “is the student understanding standard (blah), why or why not?” and “how can we reteach this to make them understand?” Often times it is a lacking in their background knowledge that is preventing this new understanding. For example, students struggle with rates, ratios, and proportions because they cannot simplify a fraction; but students struggle with simplifying fractions because they don’t understand their math facts; and some of my students are struggling with their math facts because they can’t skip count so they write out 2+2=4, 4+2=6, 6+2=8, 8+2=10, in order to multiply 2×5. Often times when completing simple operations such as addition or multiplication of single digit or even two digit numbers I flash back to my fifth and sixth grade school years when we would sit in the gymnasium and complete worksheets before we could go outside and play. First you had to complete addition, then subtraction, followed by multiplication, and finally division. These were timed activities and in both of those years, I never made it past the addition group. And yet my major at university was mathematics while I minored in education. I have even coauthored a paper in an Ohio mathematics journal, but I could not get past the addition timed activity. It wasn’t until I was attending a college class on how to teach elementary mathematics that I actively learned number sense and my passion for teaching math was awakened. I knew that I wanted to make a difference in children’s lives who think they are bad at math, and it all comes back to this; the difference between memorization and automaticity.

I can relate to much of what you wrote. I am a 5th grade teacher and I see students, more each year, coming so unprepared for 5th grade standards. They are often a few years behind with their math understanding and here we are trying to teach them fractions concepts, heavy concepts at that, and they really don’t have any idea. They are truly at the unconsciously and consciously competent levels. Yet, with the amount of new standards we are expected to teach, reteaching and going back to unpack earlier skills is difficult. However, students simply cannot move forward if they don’t have certain skills and understandings. Ugh!


By the time most students get to my class, they should know their facts. But many do not know them or how to get to the answer. The teachers in the grade before us say they knew them at the end of the year. I believe that we spend more time having them memorize them rather than building the skills needed to solve them. We need to work more on the meaning so that students can find the answer when they cannot recall the fact.

This year in my class I will try to add more activities that encourage Automaticity vs memorization of multiplication facts. In my 8th Grade class not only do students not know their facts but they are also grade levels behind on critical math concepts. Finding the time to balance encouraging automaticity building activities and helping students with grade level content is very difficult. Planning prior to the beginning of the new school year will be critical to being successful with this new focus.

Interestingly enough, I had a conversation with my principal in June, discussing the importance of math facts and that when students become more automatic they will struggle much less with some of the more “difficult” concepts allowing them to further dive into problemsolving. This course to date has expanded my thinking and made some areas questioned. Once again the use of automaticity and knowledge continues my thought process.

Great to hear! Of course building that automaticity will be extremely helpful, but as you’ve learned that can take significantly more time for some than others. Using strategies and models will be crucial to help all students build that fact fluency.


Automaticity vs Memorization
Thought: Automaticity leads more to being Consciously Masterful because they “Understand” the concept and would be able to see the use of the skill outside of the limits of school. Memorization leads more to reproducing by rote and not necessarily understanding the purpose of the skill being presented to them in school. (I like your Shot Put Unit posted and mentioned in an above comment: https://learn.makemathmoments.com/task/shotput/)
Question: At some point (having had conversations with people in various social groups) shouldn’t 3(5)=15 be expected to be done without the need of a calculator? Grade Level would expect it to be. I might accept that they used a calculator to get the right answer, as a teacher in the course, because it may not be the skill I’m measuring, I’m looking for deeper understanding in something else – but someone in the community might not be a fan of “Discovery and Inquiry” style of teaching if the student does not know their math facts (i.e. Multiplication Table) without the calculator by a certain Grade Level (e.g. having now reached High School).
 This reply was modified 2 months, 2 weeks ago by Velia Kearns.

Great thoughts and wonders.
The goal is that students build automaticity through the doing of mathematics. I’d argue that if we are learning about more complex mathematics and students need a calculator for 3(5) = 15, then the question becomes: how worthwhile is the time we are spending on complex math if they aren’t confident / automatic with much more simple math?
I’d be wanting to get students building their number sense and fluency through strategies and models like we share in many of our earlier problem based units like sowing seeds (learn.makemathmoments.com/task/sowingseeds) vs trying to somehow make sense of much more complex math.
Does that make sense?
It all comes down to helping them build the strategies and models that will get them to automatize math facts and extend to more complex concepts.

I really appreciate how this debate is summed up to memorization versus automaticity. The exploration for understanding really stands out to me as in my current role I am trying to have the teachers I work with move away from drills and superficial work to focusing on understanding through productive struggle with low floor/high ceiling tasks and facilitating high quality discussion. Encouraging depth over breadth.

I loved this lesson! I find it really hard to explain the need for multiple ways of doing and representing math to people who feel it should all be memorized. I was blown away at all the different ways we can break down something “so simple” as counting!

This makes me wonder if I should go back to teaching 7th grade. I feel like we were so worried that students didn’t know their math facts and we wanted them to memorize them but maybe we need them to just build their number sense and become more flexible with numbers in general. I am going to definitely share these ideas with the middle school teachers at my school and hopefully we can get students more confident and comfortable with numbers, through math talks, math fights, and WODBs.
My favorite part is how there are so many different ways to approach this problem that I saw in only one way a few years ago. Seeing so many different perspectives and approaches is so exciting and I think it will help my students become more confident when they recognize that their way is reasonable and can work even if it’s different.
It is understandable that we all want students to know math facts. Sadly, our go to strategy is typically just rote memorization without necessarily building understanding. If we leverage concrete and/or visuals, we can have a true understanding and automaticity!


Throughout much of my educational life, I have used memorization as a tool to learn but it was always backed up by visual images that I had in my head. I realize this is not a method that works for everyone. Understanding number relationships is key to working with numbers whether it be for multiplication, addition, algebra, integers, etc. Postsecondary education programs and school districts would benefit from reexamining the key principles required for teaching number sense that Makes sense!

I had always thought that ‘memorisation’ and ‘automaticity’ were synonyms, so an interesting point of view which I will give more thought to. I do encourage my students to try to learn their multiplication facts as single pieces of information, because even as an adult, I often have to start at “once __ is __” and go up to the one I want rather than just knowing it. But I also encourage them to develop strategies for working out ones they can’t remember.