Sometimes we do something just because that’s the way it’s always been done. This happens in sports just like in education. For example, consider the high jump. You’re probably familiar with the basic idea of seeing how high someone can jump over a bar.
What you may not know though is that athletes have not always participated in the high jump in the same way. It used to be that people would jump over the bar like a hurdle by using a scissor kick method. People always did it this way and people continued to do it this way. At least it was that way until 1968 Mexico City Olympics.
There, American Dick Fosbury introduced the world to a new technique by going over backwards. He utilized what would eventually be called the “Fosbury flop.”
Now it’s important to understand how strange this first appeared. His method was laughed by the fans. They would even chant, “Ole!” as he ran towards his jump.
His fellow athletes and coaches looked down on him like he disrespected the sport. The judges were even unsure if he was breaking the rules because they had not seen it done this way before. Ultimately, they agreed that it was ok and Fosbury used the technique to win the gold medal and set a new Olympic record. Today, virtually everyone uses this technique.
I share this story as an example of something called status quo bias. This happens when we do something in a particular way, not because it’s the best way but because that’s the way it’s always been done.
Dick Fosbury challenged the status quo with his new technique. It made people uncomfortable because it violated their expectations. However, once they got past that, they realized that his method was truly better and necessitated change.
I believe the same thing is happening in education, and specifically in regards to the mathematics standards we teach. Are our mathematics standards chosen because they are still the most important ones for students to learn or because that’s what we’ve always taught in math class?
I think that we need to take a step back and realize that so much of what we teach students is instantly useless because of technology. I’ll say it again to be clear, we spend days and weeks teaching students content that they’ll likely never use again because of the pervasive existence of technology like calculators.
Obviously, I’m not the first person to think of this. I love how Conrad Wolfram articulates it in his TED Talk. Just watch the first 30 seconds of the clip below (from 4:10 to 4:40).
Here’s what he said in those 30 seconds:
Now here’s the crazy thing right now. In math education, we’re spending about perhaps 80 percent of the time teaching people to do [computation] by hand. Yet, that’s the one step computers can do better than any human after years of practice. Instead, we ought to be using computers to do step three and using the students to spend much more effort on learning how to do steps one, two and four — conceptualizing problems, applying them, getting the teacher to run them through how to do that.
The first time I heard that part of the clip, it blew my mind. I had never thought of it that way.
So here are my thoughts:
- When I surveyed 383 K-12 mathematics teachers in the United States, 85% of them stated that they did not have enough time to teach one year of grade level standards in one year.
- Many of the math standards we teach students are computation based and can be done with technology.
- Many of the non-computation based math standards are ones that computers (currently) cannot do such as complex mathematical modeling. For example, computers alone still can’t tell if you’re pregnant, but people with computers can.
- As of 2015, 92% of American adults own cell phones they have with them constantly and most of those phones have calculators.
Based on this information, I reach the conclusion that we are spending too much time teaching procedures that will likely be replaced by technology. For example, how often are you multiplying multi-digit numbers, dividing decimals, or figuring out sales tax by hand?
I wonder what would happen if we greatly deemphasized teaching these procedures or cut them out entirely. What could we do with that time we freed up? Like with Dick Fosbury challenging the status quo with his new technique, I’m sure the idea of changing how these standards are prioritized will certainly make people uncomfortable because it violates their expectations.
What do you think? Please let me know in the comments.
Robert, you are right on the money with this post. I have said this repeatedly to all those around me. Why do we demand that denominators be rationalized? Because slide rules couldn’t handle them. This is just one example of the stranglehold ‘tradition’ has on our curriculum.
It is time to change this. We must push hard on this topic, as professionals.
Agreed Glenn. I’m not saying that everything has to go. I just think we need to step back and look at why we teach each standard as there is just too much that is never going to be used again which is taking time away from things that we should be spending more time on.
As a math and special education teacher I literally spend ALL MY TIME teaching kids stuff a calculator can do and then we hold them back from years of other content. Some of these kids are really great thinkers but cannot remember the steps of algorithms very well if at all. I’m currently homeschooling two of my own kids and I am spending almost all my time teaching them stuff they will use a calculator for and I want to just be sure they understand it. (We are in an o line charter so state testing it is )
I completely agree, but how do we get the “powers who be” to change the status quo? I would much rather have my students hammering out the critical thinking skills to figure out project-based lessons.
Who are ‘powers who be”? You are!!! You are the one who is in contact with your children day in and day out and have the power to influence their minds in ways that not many people can. Hammer out the critical thinking skills. You will be happy you did and your students will be so much better prepared for what lies ahead of them.
I think I understand your question Laura. It feels challenging and overwhelming when you have to balance what you feel is right for students with how you feel like you are being judged and evaluated by others.
What I’d say is along the lines of what Satinder shared: focus on the parts you do have control over. Here are two examples (of which I don’t know how many apply to you).
First, some teachers spend so much time on things like going over homework at the beginning of class. I go into this in more detail here (https://robertkaplinsky.com/heres-can-extra-month-teach-students/) but you might revisit how you spend time in class.
Second, consider things like classwork where students do worksheets or many of the same kinds of problems. I know I did that for years. But now I realize that after kids could do the first 3 to 5 problems correctly, they didn’t gain a whole lot from the other problems that followed. So, I think that you might replace some of those additional problems with ones from, say, openmiddle.com where they can do some of that problem solving.
Hope that helps.
Robert, love the post. I also enjoyed that TED talk. It makes so much sense. We are having those debates right now as many of our teachers have not been PLC trained in our department and we are going through the standards and trying to prioritize and vertical align. Yet some of the conversation revolves around what has always been done. As for the standards, although I agree with many of the changes there still needs to be some rethinking. Keep up the good work.
Hey Tina! Long time no talk. Yes, it’s not my intention to say that I know exactly which standards to keep or toss, but that something needs to change and that we should be having these tough conversations instead of perpetually avoiding them.
I love this idea. I’ve been admitting to my students that while the concepts we learn in Algebra 2 do apply to biomedical science, chemistry, engineering, and robotics, those who actually use it have computer do that part for them.
Then I try to spin it that we’re developing problem-solving skills by trying to apply formulas in different ways, but it’s still just an algorithm. They sit there, waiting for me to show them the steps, and then they copy them down.
I just watched Conrad Wolfram’s full TED talk yesterday (a little behind them the times, I know), and now I’m more inspired than ever to design my classes based around solving real problems, and fill in the state math standards where they apply.
It’s going to take some work, but I know it will be worth it.
Wonderful Kimberly. You wrote that in such a way that everyone can identify with what you are experiencing. It’s an awful feeling trying to balance the what we feel is important with the standards we have to teach.
What I especially like is how you are interpreting this. It’s not that we don’t ever use some of these skills, but that we shift it from the calculation part to applying the skill in problem solving.
I wish you luck because, as you said, it is going to take some work but it will surely be worth it.
I find the status quo versus best teaching tension to be a tough one. Also, the assessments. I so hate testing now. If you want an algebra 2 revamp partner, contact me!! @quantgal67 on twitter or email [email protected]. I wrote my thesis on Algebra 2 as a gatekeeper course. I’ve been teaching it for years and try new things all the time. Making a,change like this is tough to do alone. A potentially easy starting point is graphing. Why would anyone graph anything by hand in a work setting?
Hi Laurie,
I currently teach Algebra 2 and Algebra 2 honors. I too would like the opportunity to email you to discuss what Imyou currently do with teaching Algebra 2 and share with you what I do to get your feedback.
May I email you as well?
Thanks.
Scott
The problem I still have is when my students use a calculator for 3 x 5. Or when the hit a wrong button on the calculator and have no idea that their answer is wrong. I do think math education needs to change but i. worry about getting away from computational and procedure all together. There still has to be a benefit to know basic math facts.
Let me try and do a better job explaining myself. I definitely want students to understand WHY 3 x 5 = 15. I think that developing that kind of conceptual understanding is CRITICAL. However, fast forward to middle school. Kids are learning about things like sales tax and how to find out how much sales tax you have to pay if it is 9.375% and costs $84.56.
So, they make their kids multiplying $84.56 x 0.09375. My gosh! That is crazy. No one does that by hand in real life. It’s just not necessary. I want kids to have conceptual understanding and know that it should be less than $84.56 and something that is about $8. I don’t want them to have to do busy work like this.
In terms of “there still has to be a benefit to know basic math facts,” I think that is worth analyzing in more detail. How much math do you know HOW to do but not WHY it works? For example, many kids can divide fractions but don’t know WHY invert and multiply works. Is that really what we want?
Wow Robert, Couldn’t agree more with you here. The problem I believe lies in assessment practices and standardized assessments as they exist today. These focus almost exclusively on procedural fluency and either eliminate the use of a calculator or limit the use of a calculator to ensure students can do it by hand. The same is done with the recall of formulas and algorithms that are required in order to score well even if a calculator is available. This is an antiquated practice that must change. Any ideas for how we alter assessments to focus more on steps 1, 2, and 4? This nonsense trickles down from ACT / SAT college entrance exams to state standardized assessments and then to teacher’s classroom assessments and practices. Teachers will continue to teach this way because “it is on the test”. What can we do here?
Amen!!! And have you ever seen the math on a woodcock Johnson academic used to qualify kids for special education? It’s how fast they do math facts and no calculator allowed.
Suppose I was teaching a student to figure 9.375% tax on an $84.56 purchase. It would be bad practice to have the student BEGIN by lining up the digits vertically and starting the calculation procedure. Instead, it’s my duty to have the student reason through estimation. “The tax should be a little less than 10% of the total – a little less than 1/10 of the total. It should be a little less than $8.46.” Like you said, we want conceptual understanding. Once the conceptual understanding is there, the verification by calculation is important. But it becomes MUCH less important to do it by hand.
I guess I’d say that I’d like my students to KNOW that they can multiply something like $84.56 x 0.09375 without a computer. I’d like them to experience it, and to understand why their algorithm works. But to practice it? To spend lots of time rehearsing and repeating the procedure? Nah, life is short.
Thanks Cory and Tim. To be clear, I didn’t know all of these things while I was in the classroom. I definitely made kids multiply decimals out the long way (though not always) and I didn’t do a good enough job building conceptual understand. But now i have a decade more experience and have better perspective. There has to be a balance, and as Cory said, it’s too overloaded with what is best for standardized assessments.
As an upper middle, algebra 1 teacher I can say, I’d rather teachers let student put 3X5 in a calculator or better yet have a chart taped on each desk. It will stick and free up time for what’s important. Plus, if we are being honest most come to middle school not knowing them fluently. Talk to them about the why’s with authentic mathematical language. Send us thinkers with a rich mathematical vocabulary.
Well said, Penny.
I completely agree that the two examples you give are situations that would cause concern. That’s exactly why we need to hammer in number sense, problem solving, critical thinking, etc. instead of computation skills. If they have number sense, they will be able to check for reasonableness
Agreed Joel. I’m not saying “no computation ever” but something that’s much more balanced would be wonderful.
Rick. I agree. There is a balance. Most of my class either doesn’t know or needs to think for a minute to know what is 7 x 6. Calculators have their place, of course, but I believe teachers allow students in elementary and middle school to use them as a crutch rather than actually teaching so that students can know basic computation.
While I agree to a point, I also think it’s important for students to understand the reasoning behind the calculations done in a calculator. So many kids lack conceptual understanding because of “tricks” they learn in math class. What more gaps in understanding will be opened if we don’t teach how to do basic computation and *why* the answer in the calculator is correct?
I agree!
Yes, I am completely on the same page with you. I think kids definitely need to have conceptual understanding of why it works. You have to know WHY 3 x 5 = 15. However, once you have that conceptual understanding of multiplication, I don’t think you need to memorize all your math facts.
Consider also that many times we teach kids HOW to do math but not WHY it works. As an example, kids can divide fractions but have no idea why they invert and multiply. That’s no good either.
Robert, you’re absolutely right. It’s like not using the spell checker because we don’t know why it’s correcting us and having to memorize all the words in the dictionary. We can do spell checkers and calculators do their job and let people do the thinking/creating/analyzing/inventing etc.
I love this analogy. That’s a great way of thinking about it. Yes, spelling is valuable, but why would we spend so much time on spelling when helping people articulate their thoughts is so much more valuable?
Students also need to know THAT 3 x 5 = 15 without using a calculator. I teach 7th grade math, and it is astonishing/ alarming/ frightening how many kids don’t know their multiplication facts. It sure does make it hard to teach the concept of proportionality when kids see the numbers 4 and 32 next to each other in a table and don’t know they are related by multiplication by 8. Calculators, yes, to a point, but kids MUST be fluent with basic math facts.
I am in complete agreement with you. Well said!
I agree. Understanding and knowing the basic math facts are basic to doing and understanding anything in math, and math is everywhere in our lives.
Ok. I have read this post several times now. The first time I read it, I felt an odd/unfamiliar sense of disagreeing with you. I also admit that I did not read any comments on here or Twitter. I just walked away agreeing to disagree.
Now I have reread this post, these comments/replies and several more on Twitter. Here are some thoughts I’ve been hashing through:
-My initial feelings were that you felt that Ss do not need to spend time learning about basic operations at all; whole numbers, fractions, decimals, etc. When I read the post alone, it still reads that way to me. To me, the post sends a different message than your replies to comments do.
– Now I am feeling that you are more concerned with the time spent practicing/extent to which they are practiced in ways that we do not actually use them without calculators as adults. For example, Ss should understand the meaning of addition, but could spend their time in better ways than blindly adding 6 digit numbers using algorithms
-I also feel that Ss spend far too much time trying to perfect steps in algorithms. I would much rather have them spend their time on conceptual understanding over memorizing
– I wonder if your experiences teaching on the secondary level and mine in the elementary level are creating a gap in my understanding of this post. I find that at times, when I talk with secondary math teachers I feel misunderstood for what I do in elementary. I put in time on number sense and basic operations so you can help them apply them. Maybe I am being overly sensitive. Is the topic of this post more related to secondary math than elementary?
Hi. Thank you for taking the time to share this Lori. In short, I think we are more on the same page than not. I agree with your second and third points. Sometimes it’s hard to articulate all your thoughts in a single post, so I am glad I have the chance to expand upon my thinking in the comments.
I don’t think that this is so much of an elementary / secondary thing. I am using elementary examples so everyone understands, but the same stuff happens in secondary. As Glenn mentions in the first comment (https://robertkaplinsky.com/didnt-teach-calculator-can/#comment-300001) we do this kind of stuff in secondary too.
In general, I just want us to take a step back and think, “Is this really still a priority for our students?” Status quo bias makes that really hard though.
Status Quo Bias – indeed. Fosbury looked at the problem and worked out a way to make a better jump. But he didn’t remove the bar.
I’m from the UK, and help out an inventor from San Diego (if you ever used Tommee Tippee with your children, well, he was responsible for those – but not the one that didn’t work recently!) who thought long and hard about how calculators are used. He came up with the QAMAcalculator, which requires users to enter a reasonable estimate first. There is lots of research that says that estimation helps problem solving, helps understanding and sense of number and yet the presence of calculators removes the need.
If schools used these routinely in every subject, not just mathematics, then the discussion above would probably not be needed. The positive effect on numeracy of children (and probably teachers!) could be huge. indeed Jon Allen Paulos said
“A wonderfully appealing all-purpose calculator that encourages thought before it supplies an answer. Part Socrates, part sphinx, it forces students to really consider what they’re doing before it responds. It’s fun, addictive even. Try it.”
But… status quo bias!
As a secondary school teacher I have had a lot of high school students pull out a calculator to do 7+0… this is a huge problem when you are trying to teach them the concept of variables. A student needs to learn to crawl before they can walk, and walk before they can run. I agree that technology is a wonderful tool and I use it extensively in my class and when teaching students how to use a mathematical concept in the real world problems they will face, but we are facing a generation of young people who have been taught how to use a calculator in elementary school and so have never developed the number sense required to build higher level thinking.
I agree we should be doing more to teach students real world skills and how technology changes those skills they will need, but giving them a crutch too early in their development is causing us to have an entire generation that can not function with out it, and so we are trying to build these students education on a foundation of skills that the student does not have. Just as you can not build a house on shifting sand, if the students do not have a good foundation in understanding what a concrete number will do, how will they understand what variables are?
Thanks for sharing your perspective Berry. I think that we are having the same conversation but drawing our lines in the sand at different points. For example, when a kid can multiply 473 x 29 and they “bring down the zero”, is that what number sense looks like? If not, then how deep should we go with number sense because essentially that was a trick that felt ok then, so why is using a calculator so different?
Hi Robert! I always appreciate your thinking.
I’m deep in helping some 9th grade teachers teach quadratic expressions/functions. It’s by far my least favorite topic to teach because much of the grunt/algebra work is so unnecessary. We had students working on “Will it hit the hoop?” activity in Desmos…full engagement, full curiosity, full wonder…and full appreciation for the shape of parabolas…but now we “have” to be back in our books to make sure that we’re teaching a coherent set of skills that align to CCSS, vertically align Alg1 with Alg2, and make sure we’re “covering” what’s on the SBAC. So much of the work is by hand.
It seems like such a waste of our instructional time for us and learning time for our students. I really struggle with it professionally. Are we doing students harm?
What do we do when the needs of our students conflict with the mandates of our work? Your post inspired me to have the courage to write this: http://undercovercalculus.com/hector-dilemma/
I appreciate this work and curious to see what you learn as you move this argument forward. I’m sure there’s some pushback from the status quo. Here’s hoping we can be Fosburys and invite other math teachers to be so too!
Thanks Chase. I loved your post. I really think that the Hectors of the world would benefit from having a more thoughtful rethinking of how and what we teach.
I agree that we spend too much time on procedures that a calculator can do, but if we take away teaching the procedure we take away teaching the concept. If we only teach using a calculator the student will not care about the conceptual learning. I think we lose more than time.
I’d like to challenge your statement of “if we take away teaching the procedure we take away teaching the concept.” That has not been my experience at all. From my perspective, teaching the procedure does not lead to conceptual understanding while teaching conceptually often leads to procedural skills.
First off, I am in agreement with a lot of Wolfram’s perspective. My degree and first career was in engineering but now I have gravitated to teaching math. My issue, like many teacher’s, is I’m given a list of standards with no course syllabus, text book that goes along with the standards, etc. while at the same time, the standards are just a list that takes me to a while to decipher what the creator of the groupings might have been thinking to make my lessons flow. In other words, there is a lot of work for new teachers, no matter their comfort and knowledge in their subject matter to get to a point to just having something to do each day that flows with the last class and the next one, fills in learning gaps and challenges advanced students ideas (often I find that students who finish first have learning gaps that aren’t as easy to spot because they are good at following a procedure). So, while my intent is to shift my classroom to a lot of the ideas you, Wolfram, and many others bring up, please realize adding this on the teacher means it becomes a hurdle rather than an exciting time to be teaching math (which it should be!). Why aren’t these ideas being pushed onto those creating the high school courses and standards? I already spend why too much free time to be effective, as do most teachers. I’m all for getting a consensus but I think we are there on this topic. I hope I’m emphasizing that I’m resistant to change in the way the courses I teach are taught, I’m resistant to teachers redisgning the courses and all duplicating work within each state, district, or school when we could all be contributing far less time with a collaboration model at a much higher level. I am still blown away how much teachers hoard their work rather than making it publicly available at the district and state level (we do all right a the school level).
Well, I gotta tell you, I am 100% with you my friend. I do do computation….lol…but only new and fun strategies where kids can make some basic connections, and then they practice, if they want at home. Or…they teach their parents the new strategy. Other than that….2 digit multiplication is owed the calculator. I am not very well understood at my school. We play games, we create games and we make “connections”. I have yet to use a textbook in the last 3 years. The irony, I helped write the current approved text for here in Ontario. 🙂
Thanks Leanna and Lynne. I believe I feel your pain. There are so many emotions around this topic between the reality that we want to do it but the frustrations that we don’t have the time or energy to do it and are hampered by impractical resources and poor communication and sharing.
I get you. My hope for this post is more to get some controversy stirred up. For everyone who posts on this blog post in agreement, I get many more on social media who disagree with me. So, it’s not a done deal by any means.
I just wanted to say “thanks” for validating so much of what I’ve been feeling lately. If we spend more time exploring number concepts in a concrete and discussion-based way in lower elementary, we could build a better foundation for the future expansion into other domains where calculators would serve as a tool- not a crutch- to aid in real-world problem solving. We are pushing kids into abstract procedures too quickly and on top of that, not giving them enough time to master those procedures (which I’d agree are irrelevant in today’s world). *That’s* why you have high schoolers using calculators for 7+0. They have no confidence in their ability to think! Elementary teachers need a better set of standards that allow time to cover the most important underlying math concepts more deeply. When this happens, we can finally stop squashing our students’ number sense with procedures, and foster their understanding instead.
You are totally getting the message I intend Kelly. Perhaps this wasn’t my clearest blog post ever, but I’m not against building number sense. In fact, I think that is critical. However, there are some aspects that are just repetitive number crunching. For example, teaching dividing fractions as purely invert and multiply with no understanding of why that works is not particularly useful. So, I think we need to step back and re-evaluate why we do what we do.
Both of my daughters, now grade 6 & 8, used go math in a California school. I would regularly look through the curriculum and their work. I think that the curriculum pushes conceptuall understanding.
I found that it did an excellent job and teaching multiplication using several different conceptual models, and later arithmetic with fractions on a very good conceptual level.
I see nothing in the go math curriculum that requires overbearing computational practice.
Robert, your blog posts always push me to reflect on my own practices. I am usually left challenged and inspired; both leading to my personal growth. Thank you for continuing to share your own thoughts with everyone.
Thanks Brad. Not everyone will agree with me, but if they did, it might be a sign that I am not pushing hard enough. Glad you find them worth pondering.
Keep pushing. I know many of my colleagues in Arcadia benefit and ultimately so do our students. I suspect you will look back on this blog post in the near future and be surprised that this was even a debatable topic; at least I hope this to be the case.
A million times thank you! I work with math teachers K-12. My mantra for elementary teachers is that most of what they teach (algorithms, etc) becomes obsolete at 6th grade (when a calculator is introduced) if they don’t teach students to think. It obviously makes me really popular. How much more would students understand if they knew the why’s! For example, I ask students what mean or average “means”. The response is always the same which is a regurgitation of the algorithm. That’s why they can get calculator responses that are not reasonable, but they don’t know it. I break out the linking cubes and talk about “leveling off”. The next response is, “Ohhhhhh.” I appreciate this blog more than you know.
Thanks Stephanie. Yeah, I feel that we’ve lost our way in terms of balancing what we’ve always done with what students really need going forward. Hope this helps.
I have been telling my High school students the last few years if the phone can do it ( even photo math) why would someone hire you? Thanks for your support!
EXACTLY. We should be helping students focus on what calculators can’t do. The scary reality is that it’s a moving target. Once calculators evolve, what we teach will have to shift as well.
I absolutely agree with this article. I allow my students calculator use virtually every day, and I tell them that I don’t expect them to ever have to work without one. I also teach them how to use the calculator and make sure they can tell me what the calculator is telling them. I struggle, however, because of the expectation to teach skills that standardized tests assess. How can we convince the testing companies (or our school districts) to put less emphasis on computation skills on these assessments? I feel as though educators are beginning to shift the tide, but we are having to fight against standardized tests that don’t assess the things that we value.
There isn’t a way to convince testing companies or school district to put less emphasis on computation. The people to convince are the standards writers as they rule everyone.
Here’s my attempt to do so: https://robertkaplinsky.com/open-letter-writers-future-math-standards/
The big question is how can you move away from teaching students how to compute these problems that computers can do nowadays when this is how they are being assessed on state tests. Big stakes tests. Tests that determine whether I am “a good teacher or not” tests. As many have stated, it is difficult to get students to buy in to the teachings, when they are addicted to their phones AND they know that their phone can solve most all the computations they are working on. The way we are “forced” or cornered into teaching students is ultimately hurting them (and us), they are our future.
I really do wish that technology could make major leaps in assessment. Too often our assessments are driven by “what is technology capable of assessing” and not “what should we be assessing.”
I’m with you in spirit, but I have a potential problem with the theory; I’ve never known anyone to have well-developed number sense without strong computation skills. Is it possible to develop strong number sense without strong calculation skills?
How’s this for an example… briefly skim this biography: https://en.wikipedia.org/wiki/Roger_Penrose. Then watch this two minute video: https://twitter.com/msbjacobs/status/1059201146642849793.
Same!
Thank you, Robert, for bringing such a “controversial” topic of discussion to light with applicable examples. One “controversial” topic of discussion I have been having with my teachers lately deals with the “traditional” hundreds and 120s number charts used in elementary math classrooms. For years and years and years teachers have used number charts that start with 1 at the top left corner and end with 100, 120, or other variations at the bottom right corner. When asking teachers why the numbers are in this order, it’s because “this is how we teach students to read,” but mathematically and scientifically it doesn’t make any sense in terms of directionality. The bottoms up number charts flip the rows of 10 so that the number with smaller values are below numbers with larger values, which follows directionality of increase, decrease, more and less, place value, vertical number lines, and aligns to how scientific and mathematical tools are labeled (graduated cylinders, rain gauges,height charts, thermometers, y-axis/vertical axis for graphing, etc). The first mention (that I have found so far) in published documents about this whole concept is from NCTM 1977, but yet the traditional number charts are still be used today! In order to help our students develop mathematical number sense and fact fluency/fluidity, we need to help them build and identify where patterns emerge on number charts because the power they can have is immeasurable! I don’t understand why/how students are truly supposed to understand what happens to place value when looking at 10 more and 10 less if we are showing them a resource that has 10 less going up on a traditional 100s chart and 10 more goes down on a traditional 100s chart. If you asked a student to “count down from 20” why would you have them “count down” verbally but “count up” directionally on a number chart? Also if you have a student show what zero looks like or count down with physical movement, the child will start tall and shrink closer to the floor. I’m hopeful now that other educators and publications are beginning to see that there’s not a better way but actually a mathematically and scientifically accurate representation for number chart patterns and movement, shifts in thinking, teaching, building, observing, and developing will begin to take hold! Not because it’s “how it’s always been done” but more rather about what’s developmentally appropriate and aligned to all maths and science. Food for thought! Next phase…encourage companies to offer bottoms up number charts for sell! We won’t ever grow our students in the direction they need to be successful if we as educators continue to think as we “have always done it.”
https://www.nctm.org/Publications/Teaching-Children-Mathematics/2017/Vol24/Issue3/A-Bottom-Up-Hundred-Chart_/
Thanks for this. I haven’t thought about this much and will simmer on it more.
YES I totally agree with you and with Conrad. I agree that until those school comparing summative yearly assessments go away or better yet get changed to actually test the intent of problem solving , nothing will change in math class! I worked with my state dept to create a new set of standards long ago. We had fantastic math thinkers in charge and they knew our standards needed to go deeper. We even designed a test- The WESTEST ! But it required hand scoring so it did not last long! As a teacher trainer I once did an experiment as I traveled around our state conducting math PD for teachers. I figured I had reached around 500 teachers and I asked them all to do this task: Draw a picture that represents 1 3/4 divided by 1/2. Back then – around 2006 – I only got like 4 correct drawings and when I asked them to write the word problem the majority wrote a problem of sharing with 2 people! What I am trying to say is that until we change teacher development , math will never change! Today I work to convince elementary teachers that those nontraditional algorithms make sense. I tell them to do a test- go to your class and ask the kids to solve 199 + 198 and see how many look for pencil and paper! These summative tests will never encourage teachers to go beyond procedures. Our standards are written for deep understanding yet the tests allow the deep part of the standards to be ignored. So yes, I do think our standards are good yet the main intent – reasoning and conceptual understanding needs to be the highlight not an aside.
It’s certainly a complex conversation with a lot of competing interests. I hope that years from now people read this blog post and struggle to comprehend how the standards were ever not relevant.
Great post! I don’t want to completely downplay the benefits of understanding computation for number sense – but I always tell the teachers I support that the last time any of their students will ever do long division by hand is in 6th grade math (7th if there are negatives). And, if students understand the concept and number sense, there’s a good chance they wouldn’t use the common algorithm to do it in 6th grade even! Even for things as simple as addition and subtraction algorithms…when 90% of us do it as adults (usually in our heads), we don’t “borrow and carry” or any of that. Thanks for putting this out there!!!
Yeah, it’s definitely not my intention to say that there’s no value to teaching computation, but that we place far too great an emphasis on it and not nearly enough time on conceptual understanding.
Hi Robert, I teach secondary at an international school. We follow the British curriculum. I have taught the IB for many years (MYP and DP). I so much agree with what you are saying. I believe basic number sense is crucial, and most of this is taught in elementary schools. To me, it would be preferable if elementary students really learn the why of addition, subtraction, multiplication and division. Multiplication and division initially with one-digit numbers. To understand multiplication for two-digit numbers using the area model is also important. I want students to be able to show that they understand why this is. In my opinion, the only division that must be really learnt (what is division?) is division by a one-digit number. Using partial products.
Estimating the answers of simple operations is truly important, and hopefully students will always intuitively try to estimate. After that, I am fine that people use calculators. I so much agree that time should be spent on open problem solving, especially with not precise or complete info. Textbook questions often tend to have all whole numbers, which actually do not happen so often in the real world – so students should be familiar and confident with decimal numbers, which requires a calculator.
I also feel that multiplication and division of more than one-digit numbers and including decimals is often taught to students at a too young age to actually being able to understand the why. I would prefer that students learn the why of the basics and then do the more-digit numbers on a calculator, and that really small numbers and really big numbers are taught at a later age when students can understand these kind of numbers.
Yes, you absolutely get my point. I’m not saying to never teach this but rather to focus MORE on why it works and far less on practicing until you’re a robot.
This! I know that change is difficult, but we need to spend much more time examining the “why” and less time practicing the “how”. If students have number sense and are able to reason mathematically, they will not be so easily swayed by “but that’s the answer I got in my calculator.”
Keep fighting the good fight!
Thank you Leslie.