I agree with you Lauren.

Great points here, Lauren, Jian and Dan.

Reflecting on my personal mathematics journey, I was one who mostly memorized procedures and *sometimes* made connections along the way. I personally believe that there is a group of students, like myself, who can make it out just fine this way. However, there is another group of students who fail to make those connections, even when the teacher attempts to explain the “concept” behind the procedure.

During Chapter 4 in “Adding It Up”, the section that summarizes the importance of procedural fluency mentions a few points that I will paraphrase below:

1) Learning procedures without conceptual understanding presents the possibility of a student learning the procedure incorrectly and this can be difficult to reverse – especially since there is little or no conceptual understanding to help guide them.

2) Students may struggle to determine whether an answer is realistic/reasonable.

3) Learning a procedure without understanding requires extensive practice in order to remember the steps; much more practice than is required when students understand the inner workings of the concept and can “reconstruct” their understanding if difficulty should arise. The reasoning behind why it can be much more difficult to remember without conceptually understanding is that each step is stored as an individual piece of information as opposed to one “chunk” when the concept is connected to prior knowledge.

Something that also resonated with me was the line “once students have learned procedures without understanding, it can be difficult to get them to engage in activities to help them understand the reasons underlying the procedure.” I saw this in my own classroom when I first began my shift from teaching more procedurally to a more conceptual approach via inquiry. Once students “knew” the formula, there was an unwillingness (or maybe fear) to attempt unpacking the “why”.

My big takeaway is that math is an experience and if we want that experience to be memorable (and useful), we need to help students by seeing math class as differentiable experiences. When I think back to teaching via procedures primarily, I see the same experience. When I think back to teaching via 3 act math and inquiry approaches, I see separate chapters of the learning unfolding. Hopefully my students do as well.

Dimitra,

I completely agree about the necessity to help teacher candidates learn about mathematics concepts conceptually. I am amazed at how much I learn every day that I was unaware of when I learned as a student and taught as a teacher. I am still ignorant to many pieces of the mathematics puzzle, but I think as the math community continues to dig and share, we will make significant progress.

In our district, we are using a continuum model to attempt starting with a concept from the beginning and work with it as it develops into bigger ideas. Seems as though there is so much that we will never “get there”, however I think the more we do, the more others will continue exploring in other areas and share back to the community.

I’m convinced that for at least grades K-10 in Ontario, that there is a way we can make the learning concrete and visual for students. Sometimes we might not know what a specific topic “looks like”, but that just means we need to dig deeper.

Good thread.
Laurie,

I wonder if the question we need to ask first is “why do they need to knwo the differences of cubes? Early in my 25 year career I made that a big deal. I have come to realize all the drills and practice meant nothing long term as they did not know why they needed to factor. I failed to connect factoring deeply enough with prime factors and with other things. I also failed to accept when a student did long division to reduce (x^3-1)/(x-1) and other examples, including using synthetic division… Tools I also “taught without context”. I would suggest that factoring without context is useless. It is like learning vocabulary words and not grammar. I learned a new way to show a connection my kids never seemed to make. The Fund theorem of Algebra – factors make polynomials. – But I now show this by having students draw lines, create a table of values and find products and plot that. Ideally knowing that the factors make a polynomial, as the first step. Investigating and playing they see a lot. SO MAYBE that is the “repetition” needed? One sees patterns by REPEATED PLAY and not repeated algorithmic practice. We use this model in our JK-5 math and try to move it forward through the rest of the grades to 12.
“Conceptual Understanding can come from repeated Play”

I agree with Lauren’s opinion since that has been my experience in math. I remember doing a course called quantitative techniques, and we would learn math techniques, and then we would apply it to solve business problems. I knew all the math, but its application was slow to develop. In a revision class I explained my problem to the instructor and he started explaining how to move from concepts to application and a light went on. I remember we were given a sample to determine some fact about the population, and he said reflect on what you know about statistical analysis using samples; look at your sample size, and try and determine the significance of a sample size in relation to making predictions about the population. This explanation lead me to always relating what I knew to the problem at hand. I had a hilarious experience once in putting my deliberate practice to use. A colleague was doing a post graduate degree in accounting, and she had a problem, and when I saw the problem, I recognized it from my accounting studies, and I repeated what I remember doing to solve it. She told me that when the instructor showed them how to work the problem, I had used the correct procedure. I had left her with an equation, and the final step was to perform a differentiation the equation. At that point I then understood the procedure. Moral of the story procedural knowledge is a good foundation for gaining conceptual knowledge.
Sometimes people need modeling to initiate transfer, and this is particularly important in math. I know that I have practiced many procedures to automaticity, and after seeing how it is applied in practice, then I gained a complete understanding of it and could use it generally. I work with students receiving special education, and I am privy to many math lessons, and one of the things I noticed is that after one or two lessons, the students will be given problems to solve using the concepts they have just learned. They usually have problems, because the concepts are not yet cemented in their minds, and having not seen worked examples illustrating its practical use, they are unable to apply them to the problems.
I recently did an online course in trigonometry, and I kept noting all the topics one needed to know to do trig proofs using the different identities: you had to be fluent in the basic math operations-addition, multiplication and subtraction; you had to know exponents, radicals, working with complex fractions, factorization and the use of conjugates to rationalize square roots and some more. It involved a lot of memory to know which techniques were relevant. Knowing concepts are just one part of math, you need to remember them and know when they are to be used. The only way you can remember so many things is through practice, and when deliberate practice is removed from strategies to gain competence in math in favor of knowing concepts alone there is a problem. I am not a researcher, but I know from reflection on my past actions what strategies are necessary for me to gain competence in a subject.

I appreciate the described scenarios and the advantage of the creative approach. The reality has to include time constraints however, since a classroom includes a group of students and a set amount of time to complete the curriculum. I do agree that covering less curriculum can be a richer experience (“less is more”), however even within this parameter there will be students with different levels of comprehension. Some may need to resort to pattern recognition and then recall the algorithm at a later opportunity.

I agree with Kyle Pearce on the practice of interleaving -the use mixed sets as a way to boost understanding. However, that is not what is practiced. With the focus on assessment, and what I view as piecemeal assessment, students are tested after every topic; thus they only encounter mass practice. This leads to the inability of the students to discriminate where things are similar but a small difference requires a different treatment. When a varied set of problems are presented, they find it difficult to integrate different skills.I know the benefits of mixed or interleaved practice, but who is to institute it where the syllabus is a mile long and an inch deep, so everything receives a cursory hit. I was schooled on interleaved practice, since our end of year test covered all the topics we had learned during the school year. I also studied accounting for which interleaved practice is par for the course.

I like the analogy too, Erik. I believe being this “chef” is a worthy destination. But what about the chef’s journey?

I find that in most cases the only chef is the one standing in front of the room. So in this case, the question is whether it’s possible to train chefs by letting them work with recipes. Do we build sous-chefs first and assuming so, how do we get there? Frequently, there are instances when we need to just practice with a knife for a while. Should we develop these skills with the occasional recipe? Of course there are occasions for this; but you’re right, this alone will never help us grow into chefs.

Some of my classroom’s most successful days are more like the TV show “Chopped”: Here are a few ingredients that are required, the kitchen is stocked with utensils you might need, let’s explore this question, GO! But then there are days when we need to work on some basics. Today, my PreCalculus students were asked to draw their own graphs of functions with different types of discontinuities at given points. After seeing repeated examples of graphs that weren’t even functions, I knew we had some knife practice to do. Of course this means that the first time around with functions, they likely didn’t cook their own soup but it’s often impractical to start from scratch.

So we have a messy kitchen, and we need more thyme.

hmm.. we know one can “practice music” and play the notes as they are listed – but the “feel of the music” is very different. Procedure is the process/notes/recipe. But great musicians and chefs do not repeat – they envision, they inspire, they own it.
To get students to own it, means we need to tap into their needs and dreams. 10,000 repeats does not lead to mastery, it leats to repeats.
Students (and all people) need DEEP practice – not MORE practice. In The “talent code” this is discussed. Deep practice needs to be more than a procedure.

Well said Erik,
You articulated well my thoughts. I can cook fairly well and I know that feeling. But there are some times that you need to follow recipes. I am a good artisan baker, and I am always researching how to make the perfect bread. One day a friend saw my well-leavened golden loaf, and decided to come for some lessons on making bread, but after seeing the rigmarole that was required, she said it was too difficult. But becoming proficient in math is something like that chef, but the self discovery is really self realization. individuals must realize that they know all the techniques required for problem solving. Sometimes they know it without knowing that they know it. I have looked at some of Salman Khan’s videos, and I see the same problems that most teachers make -they assume that everyone could follow their applications. The videos are good for me, but not for my daughter. Sometimes you know something, but you do not know how to use it. And it would help if someone would remind you when solving a problem why they do what they do.
I think the worse situation is sometimes you get it and sometimes you do not. I remember sometimes getting all my factorization correct, and sometimes getting it wrong. Then, I realized I was patterning without understanding- although I always wondered why I sometimes seemed fluent and other time not. When I got it correct was when I regurgitated something I did before (standard recipe), when I saw a problem that I had never done I was stuck, but all along I had the requisite knowledge, but did not know the technique. The moment I realized how to work out the middle term ‘bingo’. I knew my tables very well, but I missed the explanation of the technique-find two factors of the last term that will return the middle term. The chef has flexibility, and a person who can only pattern has no flexibility. A chef has flexibility because he can bring to bear all his knowledge of cooking to create new dishes, our artisan cook may not have that flexibility. The chef easily transfers his skills to create new recipes. Students need to have skills, and to know how to transfer those skills, but to transfer those skills, they must have them. So it is in cooking so it is in math.

Micahel C, I agree with you. Living in two opposing math worlds is quite a challenge. The paradigm shift needs momentum, and quickly…

In response to Dick. Repetition is a part of the learning process. As someone who had learned math by the old method of repetition, I swear by it because, I do not think that concepts help children very much in understanding. Recently, I witnessed a lesson where children were shown the concept of area through counting squares, and the concept of the perimeter through counting the squares on the edge of the figure. They were then taught how to calculate area and perimeter using dimensions, and asked if they knew what they were doing with the expectation that they would associate this new way of computing area and perimeter with the concrete way they had previously learned. Unfortunately they did not.

I will now talk about my daughter who was educated in North America. She recently had to do a standardized test to obtain a job. The math questions were easier than the language questions, but she was unable to work them out mentally. she had to rely on the language problems. Let us hope they were enough to gain a pass. The questions were simple, any student with fair arithmetic skills should have been able to do it. But her problem is that the way she was taught in Canada, she never did repetition, she also never did interleaved practice, and interleaved practice is not new. i am 62, and that is what I was exposed to in my little third world country. I do not blame her for her situation, although she should not really be in the position she is now because I had always predicted that the way they were being taught in school, only the very best will be competent in math. I offered to teach her arithmetic, algebra and some geometry, but since that idid not correspond with the school syllabus it seemed like a different kind of math. What they did at school is work out problems without being taught the methods that will allow them to generalize or transfer their knowledge. They were not taught changing the subject of a formula. They applied it in solving an equation, but were not aware that if you have one unknown in any equation, you can solve it by changing the subject of the formula. Thy were not taught a general method for calculating the Lowest Common Multiple, so she is unable to do fractions fluently. They were taught to find out by doing iterations until you get the same product. I can name many more instances of what should have been taught but never was. She is now trying do undo the damage. Recently, she told me that she realized that you must know your tables to master factorization. I think that it is unfair to students when the authorities who control what they learn indulge in educational innovations for which there is no robust evidence that they will work. At the end of the day, the bells a whistles never stay, and what should have been learned to form the foundation for higher math was never learned. My daughter is a victim of the education system like so many other children are victims. I have done many math courses that involved application of math, and learning by repetition never hampered my conceptual understanding instead it fostered it. It provided me with a good foundation for higher math. By the way repetition alone is not responsible for making one remember multiplication tables; using them all the time is.

Why does knowing 7×8 in your head quickly a math requirement?

I have dyslexia and ADHD… and a masters in math, was on my way to a phd…. before I loved teaching high school math….. but I do not have that in my head… instead I think 7×4 =28 x2…. takes me a few extra seconds… but not memorized… visualized…

I think strategies beats memorizing….

I would suggest as a person who works as A jk-12 math chair… use the manipulative and multiple approaches… seeing 7×8 is believing….

Yes,it’s true

The use of visuals does not advance understanding in any way because most times the children to not transfer from the concrete to the abstract. And the problem that most students face in math is working in the abstract. I have received most of my education outside of North America and I am no researcher but early in my learning there was rote, and now I understand because of all the procedures that I have learned to automaticity. I remember before we were taught fractions we were taught to find the LCM of numbers by factoring out primes, and I was wondering why we were spending so much time learning that. When we started to do fractions, it was like drinking milk because I realized we were creating equivalent fractions all with a common denominator when we computed an LCM.
The point about this debate over rote or discovery, I view as mathematical innovation-consistent with views of Andrew Nikiforuk author of the book” If Learning so Natural, Why Am I Going to School”. Nikiforuk in his book, talks about Innovations versus Reforms” He defines innovations as ideas that administrators implement in the hope that they may do some good. He goes on to say that ” innovations are implemented with no foreknowledge or subsequent measurable proof of their effectiveness”. He posits that genuine educational reforms yields measurable improvement in student learning. There is some research by Jennifer Kaminski and Vladimir Sloutsky that says that manipulative or visuals impedes the ability of students to work in the abstract. The proponents of discovery math sell the idea that if people can visualize procedures they will understand them. I have a daughter who is the victim of discovery math. Every time I show her how to change the subject formula to solve an equation she is flummoxed. That is what you get from learning by problem solving rather than learning a defined technique that can be generalized. I remember reading that the late mathematician Herb Wilf said that after examining a slew of math research that they are not robust enough to be relied on. He described some of them as being simplistic.

Research is difficult to gather – without bias… I have looked at a lot of brain research and I am still confused. But I think it has a lot to do with how we “wire” our brains and that changes as kids mature. SO we have looked to ST Math (mindresearch.org) as a spatial temporal tool to enhance learning and “Investigations series” games as another tool when the students first learn.

The question to me is simple – why do I want to know anything? – facts or ideas or higher thoughts? My answer is – because I find a purpose in it. I do not know the capitals of the states, or the square numbers to 20, or the formulas I need for some math problems. Why? I can find those on a webs search easily, and I never “used them”. I do know the powers of 2 (computer programming) and the velocity formula for parabolic motion (as I can use calculus to get position/acceleration.) Yet we ask kids to “learn” things by doing them over and over again. This is not learning.

Now if we look at early child development – kids do things in their own ways – and can be correct. We make a MISTAKE when we say “do it this algorithmic way, because you will see it is easier in the future.” (essentially like watching a movie and being told the ending just when you got interested. WHY?) kids like to create – let them create an idea of addition, and multiplication etc… Then gradually share all strategies kids did. Finally show the algorithm.

It all starts with “trusting the kids”, “giving them time to learn” and “making it meaningful”. Hard to do with constraints we have, but worth the result.

Hi Chris,

I see where you’re coming from here. However, I think that it is not accurate that students could not even begin to play with the new concept. I think tackling it from a visual approach using square tiles and/or grid paper to introduce the relationship would be completely possible (and useful) without having to jump straight to the algebra.

If one approaches each concept directly from a symbolic approach using algebraic representations, then I think your argument holds up. However, if you were to begin from a concrete representation, move to a visual representation and then finally make connections to the symbolic (see “Concreteness Fading”), then I think the game changes significantly.

Is it possible that through some tinkering and exploration with concrete / visual representations that more deep connections could be made algebraically? Imagine how awesome it would be if a student saw that an array of 3 groups of 3 actually made a square and thus that is why 3 to the exponent 2 is called “squaring”. Once students “see” the Pythagorean relationship, students could then solve problems based on simple composing and decomposing skills they learned in primary/junior grades prior to reaching the abstract. However, none of this would be possible if we rush to the symbolic especially when we are well aware that many students are not ready for it.

Michael C,
I want to know if creative mathematics works in every situation if you are doing a trig proof, a differentiation or a composition of functions, I do not think that creative math can help you there. You must know the standard procedure for solving these problems. In my mind creative math is an attempt to create a way for everyone to be successful in math. A grade 3 girl in my spelling bee class challenged me to solve the problem 3953/5 i did the traditional division and I returned the answer 690 remainder 3. She told me that was incorrect and set about solving it by deducting 500 each time until she reached to 453 then she deducted 450 and ended up with the remainder 3. she divided each of the deductions by 5 and and added up the quotient of the division by 5. You can do a division like that, but using the traditional algorithm is a more efficient way, and where time is of the essence efficiency counts; it also help you to use the concept of division in practical applications. I remember when we had word problems my mother would demonstrate it to us. she would show me that if she had 30 oranges to share between my siblings and I she would divide by 5 to see how many each of us would receive. In those algebra questions where you divide both sides by the coefficient of x to find the value of x, I follow how she demonstrated it to me-if you bought 4 oranges and you spent 40 cents how much would each cost; she explained that you had to share the cost among the oranges by dividing by the number of oranges to find the cost of each just as she divided to find how to share the oranges. In math you need to know the traditional way of doing things, and learn how they are relevant to everyday problems.

“while it’s legitimate to ask if elementary school kids should learn rote arithmetic before conceptual understanding, most research done at high school level suggests the opposite.”

Most research at the elementary level agrees with what you say the High school research is… conceptual before wrote….

Your’s is a good perspective. Experiencing a part of mathematics from different perspectives provides the opportunity to make the connections with other parts, but you still have to make the connections. Experiencing the result of them is not sufficient. I experience understanding as an active component of experience, at the same time generated and generating.
This is a long and fascinating thread, but I still don’t think I would know how how it feels to understand. Seems to me this is the most valuable gift education can bestow. Here is my take: when I come to an understanding/insight now, I can look back and see more reason and more structure, and I can look forward to greater success with less stress.