Memorizing math is so hard! Is there a better way?

Picture this: the finale of a school-wide Mathematics Competition. Both finalists are front and center in an auditorium, pencils poised over their blank notebooks, as the host asks the final question. What is the product of 16 times 7?

"112!" Both shout.

As the tie-breaker round is discussed, the judges notice something peculiar in the way the two finalists solved the multiplication question. Using two different approaches, they both arrived at the same correct answer simultaneously. One student, in a blue shirt, had solved using a standard multiplication algorithm, aligning the place value of the digits vertically before solving ones times ones (6 x 7) and carrying the ten to multiply (1 x 7) and add 4 tens. The result being a 2 in the ones place and 11 tens. 112.

The other student, in a yellow shirt, solved using an approach that seemed to rely on decomposition of the 16 into 10 and 6 so that they could first multiply (10 x 7) then (6 x 7), and then add those partial products (70 + 42) to find the final answer of 112.

The judges muttered and fussed over the "right way" to do math. So, audience, is there an ideal way that students should learn math?

Personally, I survived math classes in school by memorizing formulas and algorithms. I was never sure if I was putting the numbers in the right spots, but by trial and error I usually figured out a pattern. It wasn't until 2013 when I was a pre-service teacher I learned about “concept-based mathematics”.

I remember sitting with a young learner in a 2nd grade classroom, and they were showing me how they'd solved a subtraction problem given by their teacher. This student knew that they could use "friendly numbers" to jump back on a number line 1 spot, then by 5s until they had to jump 1 final place to find the difference of 17 for the problem 36-19=17. The anxiety-inducing challenge of 6-9 hadn't even entered their mind as they knew a strategy I'd never been taught.

Here's the point. Borrowing from the 3 tens to make 16 ones then subtracting 9 ones to find 7, then subtracting 1 ten from the remaining 2 tens to find 1 ten... is a perfectly reasonable strategy too. But knowing another way gives a learner options.

Concept-based math is about having options when solving problems, thinking about numbers, and doing math.

Concept-based learning focuses on making learning active, engaging, and transferable by teaching the concepts that build to understanding.

Conceptual math prioritizes helping students be able to explain why processes work, rather than automatically knowing how to complete a procedure.

In an elementary math class, this may look like students using blocks or objects to physically represent expressions, followed later by drawings and visual representations, and finally writing numerals with symbols - the Concrete-Pictorial-Abstract approach.

This progression helps learners see connections between representations. See more here: https://stonydean.bucks.sch.uk/maths-concrete-pictorial-abstract-cpa-model/

In traditional methods of mathematics instruction, which include rote memorization and timed practice, students are taught known formulas and algorithms, and practice the given rules for an operation through repetition.

Nod your head if you ever took a timed math test or memorized a math acronym like PEMDAS. Timed exercises are competitions, and if the stakes are high for learners (such as tied to grades) who are practicing these skills before fluency has been achieved, then the timed exercises are likely demoralizing and overall damaging to the learner in the long-run. Timed exercises are cited by many as a reason they have math anxiety as adults and don't consider themselves "math people".

Memory devices like PEMDAS on their own aren’t bad for learning, as our brain wants to make connections, but if learners rely on a phrase instead of knowing why the various operations should be conducted in that order, they don’t actually understand each operation's role.

I suggest introducing acronyms and visual posters charting a sequence of steps after learners have explored all representations of that math concept. Even better if the students create group posters and their own unique acronyms, as these will be formed from their understanding, not from what the teacher said was correct. See more here: https://singapore-math-blog.com/2020/01/19/teaching-long-division-so-it-makes-sense/

Conceptual math is available for all ages and abilities, as it is focused on delivering the why behind the mathematical formulas and algorithms. Many of us lose an interest in math as we enter more challenging topics, such as Algebra or Trigonometry. It is becoming more common that these topics are taught using concept-based approaches, such as using physical models, drawn diagrams, simulations, and of course concrete objects so that learners can manipulate the mathematical components and learn what happens if they change a variable, move a decimal point, or adjust an angle. See more here: https://www.cuemath.com/trigonometry/

Okay, conceptual math sounds cool, and we understand that is an approach to learning not a magic solution. So how might we help students both “do math” AND “understand math”?

Let's look first at traditional methods for teaching math, which often include rote memorization. These often get a bad reputation, with many articles, books, and professional educators saying that new ways are best.

I think timing may be most important.

If students are exposed first to meaningful learning approaches (concept-based) and second to routine practice (repetition, recall), they will be able to conceptually understand the steps that they are enacting in a given procedure or algorithm, and identify errors. Instead of being robotically on autopilot solving a formula by moving numbers around, these learners will ideally know how to visualize or even draw out the situation to solve fraction multiplication problems. (It's a bit challenging to write about, so I suggest checking out how Mix and Math solves fraction multiplication problems: https://www.mixandmath.com/blog/multiplying-fractions-with-pattern-blocks

Let’s use an analogy of driving a car being like solving math problems. There are tasks you want to be automatic at doing, like shifting gears and accelerating smoothly from a stop. When I say, “5 x 6” you think “30” without needing to make 5 groups of 6 objects and then count by 5s. 1-digit multiplication already has a place in YOUR long-term memory.

That’s an example of a possible benefit from rote memorization. This “skill and drill” approach can, over time and with correct practice, move new knowledge from short-term to long-term memory.

BUT a third grader who hasn’t been exposed to multiplication until now, needs that grouping objects experience and time to draw area models on graph paper to understand what multiplication IS. They need driving lessons. This is why if someone suggests flash cards, timed tests, or another quick-recall activity to use with learners who are NEW to a concept, I explain that it is a terrible decision. You wouldn't ask someone just learning to drive to go on the Autobahn, so why would ask a math kid to recall 7x9 after only a few weeks of learning multiplication?

Okay so when can we use flash cards, "around the world", etc? After SIGNIFICANT evidence has been gathered that the student is FLUENT. Recall activities are best used to maintain knowledge already secured conceptually and over time that knowledge is even more secured in the long-term memory since it's been accessed in various contexts. If a student doesn't know 7x9, asking them to spit it out on demand isn't going to work. They need to have learned 7x9 by using concrete tiles, drawing area models, separating 7x9 into 7x4 and 7x5 to solve for partial products, recognizing the connection to 7x10 as another group of 7. Then when this student can answer '63' when asked without stress or high stakes, they can now maintain that knowledge using flash cards, games, and recall activities. Not before. Ideally 1-digit x 1-digit product understanding can be maintained by memorization practices, such as like flashcard review, online math games, and, my students’ favorite, math bingo. Timing is most important.

Practicing math knowledge for recall can be engaging, but the student’s learning foundation must be made of solid conceptual understanding before repeating a skill becomes enjoyable and automatic. It needs to move to their long-term memory.

Next time you see a frustrated math student, imagine a car engine in their mind. They are grinding their math car’s gears because they don’t conceptually understand yet to solve ___ problems, with fluency, with automaticity. They are stalling, the clutch is stuck, they can't parallel park, and you are asking them to do so before they know how. Learning isn't linear, it's wibbly wobbly.

The book Concept-Based Mathematics (by Jennifer T.H. Wathall) explores so many of these ideas, and there are a lot of resources on the author's YouTube channel as well. See more here: https://youtu.be/Q4g3uiu5pdw?si=aMsCKa8ST5cgmgC4

So we agree that the goal is to have students who can first “understand math” and second “do math”. Along with the concrete-pictorial-abstract approach, educators can focus on Number Sense foundations. Especially for those learners in the very young grades or even still at home.

Jo Boaler’s YouCubed site hosts lots of ideas and math tasks based in conceptual math, and offers games and activities for classrooms. I’ve used Snap It and How Close to 100?, both of which are recommended in her article titled “Fluency without Fear”. https://www.youcubed.org/wp-content/uploads/2017/09/Fluency-Without-Fear-1.28.15.pdf Jo writes, “The activities given are games and tasks in which students learn math facts at the same time as working on something they enjoy.”

I can confirm that my students loved math games. So much so that they often asked to take the math game to recess. If they could could shout-out a few enjoyable options they’d likely suggest Math Bingo, Guess My Number, Dice Multiplication, Scoot, and Snake Puzzles. All of these are partner or group games that are perfect for shy students who are hoping to make a friend. I also saw my EAL students gravitate to the same math games, and I think this was because these games have instinctive instructions and visual appeal.

I’ll also suggest 3-Act Math tasks https://gfletchy.com/3-act-lessons/ by G Fletchy for discussing real-world situations that involve math with students. Number Talks are a great way to tip your toes into concept-based math, as you essentially allow the students to share theories about the number situation and facilitate with questions, instead of direct instruction.

Concept-based mathematics is about making math tangible and helping students connect new knowledge to prior knowledge through concepts, not repetition. Not yet.

When students use cognitive skills such as evaluating, analyzing, and comparison to discover the answer, they are constructing the mathematical foundations that they’ll use in their lives. Like building a deep and sturdy foundation for a skyscraper – so that your student feels that ‘the sky’s the limit’. When their understanding is secure, open up the trove of games, cards, bingo boards, and math challenges so that your learners can intentionally practice their understanding in new contexts (and without stress, always with enjoyment as the setting). This is intentionally-designed practiced repetition, not rote memorization, and it is important. It belongs in the classroom AFTER conceptual knowledge has been secured.

As stated, practicing recall through games, collaborative challenges, and/or real-world applications will be more engaging to a learner than worksheets or speed competitions. Your students might also have ideas for games and ways to practice, to cycle through their previously secured skills. Maybe you can make a class game! We want students to understand and feel capable in math, right? So leave the competition and speed out of it.

To sum it up, educational research has shown that students who possess deep understanding of concepts are more secure in their knowledge and able to transfer their learning into new contexts.

Consider the future, won’t we always have a calculator to find the quotient? And will an AI be able to generate a math lesson plan using knowledge freely available in published texts? If so, what will future human brains be asked to do regarding mathematical understanding?

Math education in 2023 should be about more than identifying the answer and practicing the algorithm. It should be about solidifying understanding of concepts so that learners can confidently critique, evaluate, defend, and explain their understanding of mathematical ideas. To simplify, concept-based learning accomplishes this goal better than traditional methods. Memorization can get you to the right answer, and if you’ve never considered yourself a “math person”, it could be because you were never introduced to the concepts and instead survived math class by plugging in numbers to a given formula. Like me. I believe we owe it to our learners to help them discover meaning in math well before they are leading their own adult lives. Hopefully future teachers won't be making a post like this about their math experience.

One final great resource: Anything Erma Anderson does it worth a peek as it is often math gold. https://www.taism.com/community/news-articles/post-details/~board/news-posts/post/what-is-a-rigorous-mathematics-program-highlights-of-erma-andersons-recent-visit-to-taism

Thanks for reading!

Christina Kottmann