Drill-Based Math Instruction Diminishes the Math Teacher as Well

Emma Gargroetzi posts an astounding rebuttal to Barbara Oakley’s New York Times op-ed encouraging drill-based math instruction. Gargroetzi highlights two valid points from Oakley and then takes a blowtorch to the rest of them.

I haven’t been able to stop thinking about her last sentence since I read it yesterday.

Anyone who teaches children that they need to silently comply through painful experiences before they will be allowed to let their brilliance shine has no intention of ever allowing that brilliance to shine, and will not be able to see it when it does.

I’m perhaps more hesitant than Gagroetzi to judge intent. Lots of teachers were, themselves, victimized by drill-based instruction as students and may lack an imagination for anything different. But I’m absolutely convinced that a) we act ourselves into belief rather than believing our way into acting, and b) actions and beliefs will accumulate over a career like rust and either inhibit or enhance our potential as teachers.

A math program that endorses drills and pain as the foundational element of math instruction (rather than a supporting element) and as a prerequisite for creative mathematical thought (rather than a co-requisite) inhibits the student and the teacher both, diminishing the student’s interest in producing that creativity and the teacher’s ability to notice it.

Teachers need to disrupt the harmful messages their students have internalized about mathematics. But we also need to disrupt the harmful messages that teachers have internalized as well.

What experiences can disrupt the harmful messages teachers have internalized about math instruction? Name some in the comments. I’ll add my own suggestions later tomorrow.

2018 Aug 25. I added my own suggestion here.

Featured Comments

Faye calls out the process of learning content and pedagogy simultaneously:

Many mathematics teachers do not have the mathematics content knowledge that they need themselves. The Greater Birmingham Mathematics Partnership has found that teaching teachers mathematics using inquiry based instruction results in increased content knowledge for the teachers and a change in their beliefs about how and what all children can learn, i.e., acting themselves into changed beliefs.

Chris:

Math teachers circles (www.mathteacherscircle.org/). They provide the space for math teachers to be mathematicians (in the same way a lot of the arts teachers I know are still practicing artists).

Another Chris echoes:

It wasn’t until I was asked to think about mathematical tasks and ideas for my own understanding that I could ask the same of my students. And then, it was unavoidable…there was no going back.

William Thill elaborates:

But when I can tap into the emotional and intellectual highs that emerge from playing with cherished colleagues, I am more likely to “set the buffet” for my students with more open-ended exploration times.

Martha Mulligan:

… watching yourself teach on video is a great experience to disrupt harmful messages about math instruction, like talking too much as the teacher. I know that many math teachers feel the need to provide the most perfect, refined, rehearsed explanation so that students can see what they are supposed to see in the way they are supposed to see it. I certainly felt (at time still feel?) that way. That practice diminishes the students’ roles of sense-making on their own. But watching a video of myself teaching was one of the most humbling things I’ve done and it changed my practice so much. I also watched them among other trusted teachers from whom I learned so much. Having time to stop a video, talk about, reflect on it, etc is very powerful. Even seemingly simple things like wait time and teacher movement/positioning can look very different than what we imagine we look like.

Alexandra Martinez calls out the limitation of reading narratives and watching videos of innovative teaching:

I think the most powerful way to disrupt teacher’s own experiences and expectations is new creative experiences with their own students. The evidence and reflection can support teachers in seeing what is possible. If we ask teachers to imagine what is possible through narrative, they won’t always believe it. But when they see their own students speaking and thinking as mathematicians, that evidence disrupts their established belief systems. So I’d say observations, modeling, Coteaching, pushing in, PLC planning with lesson study can all potentially do this.

Be sure, also, to check into Chris Heddles’ a/k/a Third Chris’s dissent:

I’m going to go against the grain and admit that I use drill as a prerequisite (or at least an opening activity) with many of my students.

About 
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

84 Comments

  1. Reply

    A disruptive message which I subscribed to early in my career was “if students aren’t doing problems the way I taught them, I must {choose 1} correct them / downgrade them / take away points”. I have long since cancelled my subscription to this and now treasure and highlight student thinking which runs counter or differently that how the group established the mathematics during class.

  2. Reply

    A disruptive message I subscribed to was that I must immediately know whether a student’s answer is right or wrong and explain why. I combat that by choosing to ask my students spontaneous questions that I don’t know the answer to. A bonus is that the answers are often messy or ambiguous, just like in real life. We do the sense-making together, and sometimes the high school freshman gets to school the old woman with the PhD.

    Another disruptive message was that early math, like addition, multiplication, division, etc., are simple and uninteresting. I regularly get goosebumps in discussions about that early math with my high-schoolers, now that I give myself a chance to really think about it.

  3. Reply

    A lousy message I was supposed to believe was that if they got it right on the drill they were ready to move on because they understood that math. Related is the idea that if they got it wrong they need mpre practice or easier math, not … instruction and learning.

  4. Reply

    I’m not sure this fits into what you’ve asked us to comment about, but your piece reminded me of the advice from a friend who helped me a great deal in my early years of teaching. She said, “Hold your beliefs as opinions.” In my early years, I had students chant, “Divide, multiply, subtract, bring down.” i thought I was helping.

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    Well, I was helping them if my goal was for them to repeat what I showed, whether or not they understood. And now? I tell students what Phil Daro told me, “Do only what makes sense to you, and persist until it does.”
    • That line from Phil is a nice way to distribute obligation between the teacher and student. The teacher is obliged to help the student make sense and the student is obliged to persist through the nonsense.

  5. Reply

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    Math teachers circles (www.mathteacherscircle.org/). They provide the space for math teachers to be mathematicians (in the same way a lot of the arts teachers I know are still practicing artists).
  6. Reply

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    I think the ability to sit in someone else’s classroom and listen to student’s thinking is an experience that would potentially disrupt multiple harmful messages.

    Max talks about this in 2 > 4. When you are not the person driving instruction and trying to get from point A to point B in a lesson you are free to pick up on so much more. You have the ability to walk the room and pick up on student’s mathematical ideas either because they have shared them out loud or because you can see what they have written. As an Instructional Coach I have that opportunity every day and it has been eye opening. When you are at the fron of the room trying to advance the lesson it’s easy to miss just how MANY differet ideas your students have. One of my goals this year is to create the opportunity to visit other classrooms for my teachers. You see so much more. You hear so much more.

  7. Chris Heddles

    August 24, 2018 - 4:47 pm -
    Reply

    I’m going to go against the grain and admit that I use drill as a prerequisite (or at least an opening activity) with many of my students.

    The reason for this is student confidence. Big, open tasks are fantastic for students with the confidence and self-efficacy to tackle the unknown. Unfortunately, by the time students arrive in my classroom (aged 16+) they have the firmly-entrenched belief that they are simply unable to do maths. This means that they will refuse to engage at all with the rich puzzle on offer because they are convinced that nothing they try will make any difference anyway so why bother trying. I can’t give them guidance based on their thinking because they are too terrified to even articulate any of their thinking. Even Dan’s excellent idea of “every student can guess” is only partially-effective because they then sit back and wait for someone else to tell them if their guess is correct.

    In contrast, drill-based activities are simple, predictable and private. This provides a safer-feeling option for students to start doing something. From here, I can use their written working to start figuring out their thinking and what they need to learn next. Their progress through a sequence of drill-based activities then serves as evidence of ability that they can use for confidence in tackling a more complex problem that uses those skills practiced in the drills.

    I would love nothing more than to set a big, gnarly problem for my students and then feed them the skills as those skills might be useful for the part of the problem that they are facing right then. When I started teaching, I did just that but found that the confident students threw themselves into the problem while the majority sat back feeling silly and helpless so they waited for their “smarter” colleagues to give them “the answer”. For these students, rote drills are an extremely useful on-ramp.

    • There is nothing wrong with starting where the student’s are. How sad it is that the longer students are in the “system” the more damaged they become! When I taught grade 8 it often took until Halloween or later before students trusted me to take the risks necessary. I am in Ontario, Canada where the new government is trying to move back to the “tried and true” i.e. drill and kill. If you want to try and mess with their heads to change their mindset, a powerpoint on learning the multiplication tables through the properties of multiplication (commutative, distributive) that of course extend to algebra if you want it.

    • I understand where you are coming from. I think the answer to that problem is to purposefully teach problem solving strategies through carefully selected and sequenced big problems. The aim in this sequence is not primarily getting the right answer. It’s giving students specific strategies that confident problem solvers use instinctively and providing feedback along the way on the growth they demonstrate in using those strategies. It’s ok – expected even – that they will make mistakes as part of their process and that they can learn from those mistakes and adjust their solution process along the way. Good problem solvers tinker with the problem. They draw pictures or diagrams. They make tables and look for patterns. They are persistent.

      It’s the “I have to get the right answer right away” mindset that has been drilled into students that must be overcome. We can make headway on that front by not being answer driven ourselves as we guide students through these experiences. We need to value process at least as much as we do the product because the learning objective is that students become more confident problem solvers.

      It won’t happen overnight and we can’t completely overcome their past experience but we can make progress.

    • Maria Marquis

      August 26, 2018 - 4:10 am -

      Love this; completely relevant at secondary level. With their deeply entrenched fear/dislike of math at this age, simple DO-able drills starts building their confidence (or helps the teacher to discern gaps) at the start of class.
      Math is one of the only subjects that it is socially acceptable to admit that one “hates”. This message comes from parents, teachers, and all manner of media and social circles. When students reach their teens, it’s a Sisyphean task that teachers face to overcome the deeply ingrained belief, “I can’t do math.”
      Foundational drills allows students to dare to think otherwise.

    • Great quote.

      I’m reminded of this statement in NCTM’s Catalyzing Change in HS Mathematics (p27), “Our math identity should be based on reasoning and sense making, not correctness – getting the answer right or wrong. If mathematical identities are based only on correctness, it is fairly likely that students will be unwilling to contribute their mathematical ideas or reasoning when they are uncertain whether it is correct.”
    • This is a reply to the comment about it being ok to “hate maths”.

      I’m rethinking my thinking on this, due to recent conversations with fellow teachers, student responses to the survey question – “how does maths make you feel?” and just general observation.

      So much of the “I hate Maths”, too cool for school vibe seems to be a mask for students whose previous maths experiences have made them feel scared, steamrollered, horrified. It takes a lot of time, care and patience for that mask to drop and for the students to move beyond their fears.

  8. Reply

    I notice that math teachers often draw a dichotomy between rich, open tasks and drill-oriented practice. I wonder if it would be helpful to try and articulate some of that middle ground, the rich tasks that also act as practice, practice that one can look back on and draw new connections, and any number of other places to bridge the gap and help teachers move more fluidly between open tasks and practice.

    • Yeah, I think that dichotomy between open tasks and drill practice is enormously counter-productive. I’m pretty enthusiastic about purposeful practice like the Open Middle tasks (which no one should confuse as rich or fully open). I’m also very enamored of routines as a medium-level grain size between the small size of drills and the large size of tasks. The OpenUp curriculum has several routines around mathematical language and warmups that recur throughout multiple years of math. Productive for the student and practical for the teacher.

      Dylan, are there any other “bridges” between open tasks and practice that seem reliable to you?

    • Connor Wagner

      August 27, 2018 - 5:37 am -

      Dylan,

      I couldn’t agree more. I believe we have set up a dialogue in the math community and within school systems that creates a pendulum. We spend years drilling, and then realize student’s don’t understand the structure or “whys” of math, so we swing back the other way, until we realize they lack the foundations to access these “higher-level” tasks.

      I have come to believe learning mathematics has more similarities to reading than we want to admit. The “drilling” being the equivalent to “phonics,” while “tasks” are more akin to reading comprehension. A child must learn to read before he or she can read to learn–though a teacher can read to the child and promote comprehension prior to the child’s ability to read.

      In one of my favorite math books, “Here’s Looking at Euclid,” Alex Bellos interviews a Japanese teacher who states, “First we get them to recite it [their times tables], and only sometime later do they come to understand the real meaning.” Why are we so scared of this idea in the U.S.? The structure needs a foundation before you can build.

    • The structure needs a foundation before you can build.

      What is the foundation and what is the structure in this metaphor?

  9. Reply

    +1 on Chris’ above call for teachers’ opportunity to engage in the doing of mathematics as a positively disruptive experience.

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    It wasn’t until I was asked to think about mathematical tasks and ideas for my own understanding that I could ask the same of my students. And then, it was unavoidable…there was no going back.
  10. Martha Mulligan

    August 25, 2018 - 5:06 am -
    Reply

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    Similar to Scott’s answer above, watching yourself teach on video is a great experience to disrupt harmful messages about math instruction, like talking too much as the teacher. I know that many math teachers feel the need to provide the most perfect, refined, rehearsed explanation so that students can *see* what they are supposed to see in the way they are supposed to see it. I certainly felt (at time still feel?) that way. That practice diminishes the students’ roles of sense-making on their own. But watching a video of myself teaching was one of the most humbling things I’ve done and it changed my practice so much. I also watched them among other trusted teachers from whom I learned so much. Having time to stop a video, talk about, reflect on it, etc is very powerful. Even seemingly simple things like wait time and teacher movement/positioning can look very different than what we imagine we look like. I learned better questioning and discussion techniques that I probably would not have learned without using videos of my (and colleagues’) teaching. I also learned to just talk less and pause more after students talk. Watching other teachers teach live is great, too. It doesn’t provide the reflection time that video clubs do.

    (Also! I know Emma Gargroetzi! She’s my cousin-in-law!)

  11. Reply

    In the past year I have introduced “choice” as a means to expect students to see/assess their own skills and own the learning process in my Math 8 classes. It takes patience and guidance, but I’m seeing advantages. Choice is usually in the form of the number of practice problems completed, the variety of skills chosen for practice in a class period, and alternative or extension practice available through the Chromebook. Our results indicate that most students are more engaged daily, collaborate with peers more, and see improvements on assessments.

    Admins are reluctant to embrace these strategies; preferring that we do only one skill each day and that we only teach standards one unit at a time. Not what I consider best practice, but my admin sees best practice through her own lens.

  12. Reply

    I found standards based grading an important way to break the habit of creating assessments designed to let the weak students succeed. More specifically, when colleagues and I collaborated on common assessments, 15-20 years ago, we often padded the assessment with low level skill problems to ensure the student could “at least earn a 60%”. Since then, as we have incorperated a standards based approach to assessment and teaching. we have a better way of capturing where kids are in their understanding of the standards, which in turn we can build upon. We find more students are willing to attempt richer problems and are less fearful to at least try something, instead of leaving papers blank, as they would have in the past. Their grades are a better reflection of their mathematical knowledge rather than, in some part, a measurement of their work ethic.

  13. Reply

    Excellent. Congratulations. I hope to use your experience in my on going discussions with my admins who are reluctant to embrace SBG. They’d prefer what you describe as your past. It seems that students failing assessments is hard for them to deal with when Mom calls. Oddly, they’d rather have the student pass the course, then fail the state test. I’d rather they do both and see SBG, retakes for learning, and rigorous practice and assessment as the best route.

  14. Reply

    When I first started teaching I followed the lead of another teacher who took points away on quizzes and tests for things that had nothing to do with mathematical computation. It was her way or no way. (Including giving 0’s for using pen) It took me two weeks to realize that was not how I wanted to teach. In my classroom, I don’t ever “teach” a topic until my students have been allowed to explore it first – in any way that they are comfortable. I have amazing brainstorming going on and students like math again. If they are using crayon and paper – it makes it colorful. Don’t lose sight of the math being done just because it’s not the way you want to see it. Be flexible.

  15. Reply

    I need to take off the “teacher hat” and put my student hat back on – or perhaps, my “player-explorer” hat on. When I play with math (through shared problem sessions, Daily Desmos Challenges, Park City Math Institute problem sets, Math for America Problem Sets), I experience the joy, wonder and options available to me. By playing near other cool folks, I also have developed cherished friendships and professional relationships with people like Peg Cagle, Lani Horn, Chris Luzniak, and others. We talk about life, share silly of inappropriate humor, and appreciate how we each think about a puzzle of challenge. This lays the environment for a garden of ideas and creative exploration. This disrupts the default assumptions about what kind of setting I need to create for my students. Experiencing mathematics as an exploratory playful activity, rather than a performative activity (for a grade/judgement about my value as a math person) helps disrupt the harmful perspective that students are, ultimately expected to somehow “prove themselves.” Math seems to be the place where we accept that kids should be judged, reduced to a number, ranked, and sorted.

    Featured Comment

    But when I can tap into the emotional and intellectual highs that emerge from playing with cherished colleagues, I am more likely to “set the buffet” for my students with more open-ended exploration times.
  16. Kathrin McGregor

    August 25, 2018 - 7:03 am -
    Reply

    I too was drilled to NEVER use pen in math. It was unheard of and always questioned by the few kids who jump back and forth over the line. Last year I started encouraging pen when I saw a clear value in seeing mistakes they made as a way to move the lesson deeper. I use the strategy ” my favorite no” and my students are learning natural mistakes are valuable!! There are still a few line dancers that are not mature enough to value the strategy, but they are just 10/11 yrs old! :p

  17. Alexandra Martinez

    August 25, 2018 - 7:13 am -
    Reply

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    I think the most powerful way to disrupt teacher’s own experiences and expectations is new creative experiences with their own students. The evidence and reflection can support teachers in seeing what is possible. If we ask teachers to imagine what is possible through narrative, they won’t always believe it. But when they see their own students speaking and thinking as mathematicians, that evidence disrupts their established belief systems. So I’d say observations, modeling, Coteaching, pushing in, PLC planning with lesson study can all potentially do this.
  18. Kristen Propst

    August 25, 2018 - 7:13 am -
    Reply

    As a first grade teacher, all mathematical concepts must begin with “real life” and proceed to mathematical representation. Perhaps that is the real disruption to teacher and learner… keeping it real. Ruth Parker’s Algebraic Reasoning class was the paradigm for me.

  19. Reply

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    Many mathematics teachers do not have the mathematics content knowledge that they need themselves. The Greater Birmingham Mathematics Partnership has found that teaching teachers mathematics using inquiry based instruction results in increased content knowledge for the teachers and a change in their beliefs about how and what all children can learn, i.e., acting themselves into changed beliefs.
    • Thanks, Faye. I love the idea that adult teachers would learn content and pedagogy simultaneously. Certainly, whether we knew it or not, we were all learning content and pedagogy as young students.

  20. Reply

    The idea that maths homework must be graded for accuracy, accompanied by the other fallacy that if you don’t grade homework, students won’t do homework.
    There are a number of problems with this.
    1. It stops students seeing that mistakes are important, as they spend all their time trying to ensure perfection.
    2. It stops students practicing maths without fear.
    3. Homework that is graded in this way is rarely differentiated.
    I’ve never believed that grading homework for accuracy was very useful, but the practicing without fear came alive for me as an adult learner of a foreign and I realized how impactful it was for me to be able to try something and get it wrong sometimes in order to understand how I needed to change it.
    I also believe that marking work that is practice for accuracy also puts up barriers to some students taking risks.

    • I have pretty much reduced HW to once a week, hand out on Friday and collect the next Friday, usually only five problems, and the opportunity to make corrections for full credit.

      Most kids don’t have the time to complete daily work outside of class and what they resort to is getting a pic of one student’s work and then sharing it via social media. Not what I hope for when giving assignments. And as I teach Algebra 2, it usually is technique review that will be needed in the upcoming week (proportionality, number sense, factoring, solving equations with fractions). Most kids will do their own work, ask questions, and help one another.

    • I set skills practice homework in the following way:
      * the amount is set as a minimum time doing the practice
      * they have the answers available to check their own work
      * they practice skills until they are satisfied with their own fluency and confidence

      Khan Academy is my favourite tool for this sort of practice but some students prefer to work from textbooks. In any case, I have no requirement to mark their work as they can check it themselves. I only ask them to show me their work if they get stuck and can’t work out why.

      The only check I do of this type of homework is to ask them to do another question of the same type that I give them in class. It literally doesn’t matter how many mistakes they made in practice, only that they now have the skill.

      I set the amount of homework by time rather than number of questions for two main reasons:
      1. the students are encouraged to continually reflect on their own progress in deciding how much practice they really need for each skill and what ‘done’ feels and looks like. There is no temptation to rush through X questions because it’s not a metric that is considered.
      2. the less-experienced students don’t feel that they spend hours doing what takes the confident kids only a few minutes.
      3. Each student practices the skills that they individually need to practice. For some that is more advanced than what we are doing in class. For many, it is prerequisite skills that they missed many years prior and doing them is the most learning and success that they have ever experienced in maths class.

    • Christine Thereault

      September 1, 2018 - 7:31 am -

      I love your homework policies. They require a high level of student agency and self-assessment! It is my goal to create a learning environment with my students where they are able to do this independently. Thanks for the great idea!

    • Totally with you on your homework practice. I want students to be able to make decisions about what is right for them and what will move their learning on.
      I also emphasize and praise constantly error analysis. I encourage students to do work, check it and most importantly look at what went wrong, before moving on.

  21. Reply

    One harmful message that teachers often get trapped in is often created by tracking. When they say, “in my honors class.” Somehow expectations are different for their students based on which class they are enrolled, even though the standards and assessments are exactly the same. Guess which students are not in the “honors” class . . . this is a problem that is sadly created by policy makers who are afraid to stand up and do what is good for ALL learners!

  22. Reply

    “Teachers need to disrupt the harmful messages their students have internalized about mathematics.”

    The insight I gained from the NYT article is that students turn relative competencies into pejorative terms. It makes sense to me that a student who thinks they are better at literature will eventually assign the word “bad” to their math competence. I’ve never thought of it in this way. I suppose I will try to have conversations with my students this year (seniors) about the development of their attitudes towards math.

  23. Charlene O'Brien

    August 25, 2018 - 8:16 am -
    Reply

    One way to disrupt; certainly not new. Do less to do more. Find rich tasks, more open ended problems and then, try them alongside other teachers before you give them to students. Have a conversation about the work you did. At the least, have a conversation with other teachers about drill based problems. Thanks for the start, here! Our learning and thinking becomes what we can anticipate from students. That thinking will transfer to other problems.

  24. Reply

    I’m not sure what the specific experience was that changed my thinking, it was probably cumulative over time, but when I started seeing myself as the formalizer of informal thinking as opposed to the bestower of knowledge it was a game changer. I think it’s extremely important for students to play with an idea and informally see what’s going on. They will come up with their own words and their own “rules” then it’s my job to transition that to the formal vocabulary and connect the formal way of doing things to their own thinking. That way I am building on their own understanding and connecting things they already know. I have found that they internalize things much better this way than if I had simply given them the formal version up front.

    • You’re describing a pretty large pedagogical shift for a lot of teachers here. Perhaps it really was just a gradual accumulation of small insights and experiences. If you can recall anything particularly disruptive, let us know.

  25. Margaret Taranto

    August 25, 2018 - 8:37 am -
    Reply

    Teachers need to remember to listen to their students. Listen to the way they perceive math learning. Listen to the way they are experiencing math. Listen to the way they are understanding math. Teacher’s need to be open-minded and realize that they can learn a lot from their students.

    • I’ve been giving what I described (and your response to it) a lot of thought over the last few days trying to remember if there was a single point in time where I shifted my teaching in a big way. I still come up empty in terms of a significant event. I think it really was gradual. I’ve been teaching for 20+ years (and am now an instructional coach) and I didn’t always think of it in these terms. I started out teaching from Saxon and it could not have been more different.

      I also wanted to clarify that I don’t think every lesson or topic fits into such a nice facilitator of informal to formal box, but I do think my best lessons do. I absolutely love it when I can put students in a sand box, let them play with a concept and decide how they think it works. Letting them share there mathematical thoughts as is without worrying if the language is precise (yet) or whether an idea is completely formed. It’s exciting then to meet them in that place and layer in the formal vocabulary and complete the ideas.

      Like I said, some lessons lend themselves to this better than others. Sometimes you’re extending prior knowledge and you’re just trying to help them see what comes next and sometimes you’re operating from a blank canvas. Sometimes you’re the eighth grade teacher going over a concept for the 3rd time. Sometimes you’re reviewing old concepts to lay the groundwork for an upcoming lesson that DOES lend itself to guiding them from informal to formal.

      But then again maybe I’m just using different vocabulary and it sounds more dramatic than it really is :) It is a shift in perspective and it’s certainly different than being the person at the front of the room explaining that slope is rise over run. I do think it just involves a lot of noticing and wondering, meeting them where they are at and understanding that the path from “not understanding” to “understanding” may not be as clear as you would like it to be for a formal lesson plan from your college teacher education days :)

  26. Reply

    Really interesting commentary, particularly in the varied interpretations of the word “disruptive.” My meaning in the post wasn’t pejorative at all. We need experiences that disrupt negative ideas about students and math, particularly the ideas that have calcified over time.

    My own ideas will sound a lot like two of the three Chris’s above.

    Back when I was offering workshops to teachers more or less full time, individual teachers would attend and ask, “How do I bring this back to my department?” (Meaning the pedagogical ideas.) My answer was always the same.

    Do math of a disruptive sort together. (I used “weird” instead of “disruptive” but the intended effect is the same.) Then discuss together what made it different. Was it different-good or different-bad? Then discuss how you as a faculty could become more different like this together? What are small, practical steps we can take, even if it’s as small as committing five class minutes each week to an estimation challenge?

    I find it helpful to assume good faith for as long as possible. Teachers aren’t embracing harmful pedagogies deliberately. They need an image of the alternatives. Then they need support in realizing them.

    You’ll see lots of highlighted comments if you re-visit them here. Thanks for those.

    I think Dylan Kane and Chris Heddles form a really interesting juxtaposition: one convinced that drills are a necessary (or at least useful) prerequisite for larger tasks. The other wondering if there is a third category in between drills and larger tasks that would be useful to explore.

  27. Reply

    I am a physics teacher, not a math teacher, but obviously due to the nature of my subject have an interest in math education as well. I subscribe to a couple of math blogs, and what is interesting is this strong dichotemy between “skill and drill”, and “inquiry” in maths. Whenever it is argued, it is always through opinion pieces and anecdotes. Are there any good studies that look at children’s attitudes, retention rates, college preparedness, or general academic effectiveness between the two? Both sides are passionate, but I need more to convince me!

    • Christine Thereault

      August 25, 2018 - 12:50 pm -

      Have you read the book, What’s Math Got to do With it? by Jo Boaler? That includes the results of how attitudes and performance changed as a result of an inquiry approach to mathematics.

    • Matthew McGovern

      August 25, 2018 - 1:38 pm -

      Great thanks. Ive only read “The elephant in the room”, so will have to find a copy

  28. Travis Logslett

    August 25, 2018 - 1:18 pm -
    Reply

    I often ask students what it takes to be successful in math class and routinely I hear – pay attention to the teacher, take notes, get problems right. Never do I hear – think creatively, question others ideas, learn from other students. Admittedly I was taught like the experience my students describe and I started my career teaching like that as well. But it has been experiences in my teaching career stemming from listening to students and digging into their thinking that has made me a better teacher. Drills don’t do that for me

  29. Reply

    I’ve taught Algebra 1 to 8th graders for 15 years. All things in moderation. Frill and practice is as essential in mathematics as it is in learning to play basketball. Practice reinforces consepts and procedures.
    Drill and practice comes AFTER on target, interesting, active, demanding instruction.

  30. William Carey

    August 28, 2018 - 3:51 am -
    Reply

    Featured Comment

    I’m not surprised that Professor Oakley’s response begins with a language analogy. Having taught both Latin and Math, I’ve seen the results of abandoning fluency in the grammar and vocabulary of a subject. It’s no fun reading Latin if you have to look up every word. That’s why, in an age of Whitaker’s Words, students still memorize their Latin vocabulary. Similarly, it’s no fun wrestling with a good problem in math if the avenues forward are obscured by a lack of procedural fluency.

    If you took the quotation you highlighted and substituted “coach” for “teacher”, I think most people would dismiss it — to be an excellent pitcher, you must silently (or with loud complaint) comply with the literally painful experience of practicing throwing tens of thousands of pitches. But no one would say that coaches who compel their players to take batting practice or to run footwork drills have no intention of allowing their players’ brilliance to shine.

    Removing drill and practice also removes one of the **best** avenues for inquiry in mathematics teaching: allowing the general to emerge from the particular.

    When I teach the zeroth power to Algebra students, I teach it by a rote drill we do together. The students say aloud and together something like “two cubed”. I have a hand signal that means “multiply by two” and a hand signal that means “divide by two”. I make those, and the students say together, “two squared” or “two to the first”, and so on. I have another hand signal that means switch to actual numbers, so from “two cubed” to “eight”, then “sixteen”. We practice with that. Eventually, I give the “divide by two” signal when we’re at “two to the first”. Perplexity. Eventually someone ventures, “two to the zeroth?”. Then I give the hand signal for switch to actual numbers. And we have a great conversation. Because the students are fluent at that drill, the general rules of exponents can emerge from particular mathematical facts that they know. Giving up on drill means that you have to go from the general to the particular all the time, and that’s a disaster in the making.

  31. Marcia Weinhold

    August 29, 2018 - 9:09 am -
    Reply

    My own “disrupter” was being hired at a school where I was told I would use a graphing calculator every day. I was handed a Casio (I think it was 7000 – the first hand-held grapher) and told that all my students would have one. I would be teaching precalculus for the first time, but also covered some algebra 2 concepts. All I had for ‘curriculum’ was a list of topics as would appear in the first NCTM standards (yeah, I’m old). At some point we had to dig deeper into parabolas to get into polynomials, so I set up y1 = x+2 and y2 = x – 1 (or something similar – two linear equations). The students were well versed in linear equations and their graphs. The previous teacher had used the graphing calculator to develop the concepts of slope and x and y intercepts and what affected them by using the graphing calculator. Then I wondered if I could get the calculator to show (x+2) and (x – 1) as factors. I tried y3 = y1*y2 and watched (I was glad that the Casio graphed slowly enough to watch what was happening). Yes, two straight lines with slope of 1, and then a parabola! It was such a surprise as well as an “Ah ha” moment.

    I never went back to tables of values to plot parabolas (my previous students had had a terrible time making a correct table of values when there were negative values anyway) or higher order polynomials! We built polynomials out of factors, checked a few choice points as a product of the points on all the linear factors having the same x-values, and convinced ourselves that the resulting curve was indeed the product of all those factors. At the same time, we were having to continually re-adjust the y-scale on the calculator in order to see the max and min points. When we could see them, the x-intercepts often disappeared because they couldn’t be distinguished from the x-axis. We learned a lot about the difficulties of technology and how to display very large numbers.

    We worked entirely with factored numerator and denominator when we studied rational functions. We saw what happened if we then added a constant or multiplied by a constant — using the rational function as a base function and looking at “families of functions” as we had with linear and quadratic functions. How could we use these multipliers and addends to control the function and put the zeros or asymptotes where we wanted them? This desire for control was motivated by looking at some data sets that we wanted to describe using functions… When we finally looked at un-factored polynomials or rational functions, the need to factor was immediately evident and no one asked “when will I ever use this?”

    Eventually we got to trigonometric functions. The text I was using for that unit set up the functions in the form y = C + A sin[B(x-D)]. Using their graphing calculators and some suggestions, students worked in groups to find the results of changing each of the parameters. Note that A and C act as the multiplier and addend used with other function types. Students learned to control these functions also, and could use them to model tides and other periodic motion. Finally, we talked about polar coordinates, figured out how r and theta were related to x and y, and then explored the families of functions produced by r = C + A sin[B(theta-D)]. I asked the students to come up with a definition of a “polar rose” and then market them to the Rose Bowl parade by determining the conditions on A, B, C, and D that would produce a rose of a specific size… Then I challenged them to create a 6-petaled rose for extra credit.

    These methods of teaching were so thoroughly disrupted that when I tried to use them in a university precalculus course they were met with some dismay…

  32. Alfonso Garcia

    August 30, 2018 - 6:25 am -
    Reply

    Thanks everyone for the insightful comments, it is inspiring to see a discussion like this, and how much we are challenged in this profession.
    I love the idea of the “group disruption” to begin with a group of teachers. To the extend that we continue to explore our own mathematical understanding, we will be able to better support students.
    At the core of our behaviours in the classroom are our beliefs, so if we are able to have a set of beliefs that can be flexible, we are better equipped to support kids. I continue to reflect on what I see works with students in the short term …and wonder how to bridge their learning for the long term.

  33. Reply

    Dan,

    I want to comment on these two points in your blog:

    1)Emma Gargroetzi’s line:
    Anyone who teaches children that they need to silently comply through painful experiences before they will be allowed to let their brilliance shine has no intention of ever allowing that brilliance to shine, and will not be able to see it when it does.

    2) From your blog above: A math program that endorses drills and pain as the foundational element of math instruction (rather than a supporting element) and as a prerequisite for creative mathematical thought (rather than a co-requisite) inhibits the student and the teacher both, diminishing the student’s interest in producing that creativity and the teacher’s ability to notice it.

    I am a trained electrical engineer and a math professor at a community college. As my kids were proceeding through math in elementary school, I saw the train wreck of Everyday Math. All three of my kids had extra math on the weekends through at least 6th grade. I knew at that time that if they did not master the basics that they would be unable to do Algebra and other higher level mathematics. You talk about killing the creativity. Not mastering the basics hobbles students in Algebra and beyond. Creativity abounds when students are not bogged down in the basics because mastery of the four operations including fractions and decimals, and order of operations were required.

    I have two children who chose to pursue an education in engineering (one in graduate school and the other a freshman). They are excited to solve real world problems and are be creative.

    Their creativity can be unleashed because they have a solid foundation in the basics.

    I would like to point out that creativity and brilliance often come from practice which may not be fun all of the time. Great musicians practice scales which unleashes their creativity. To make that incredible soccer goal, a soccer player practices drills. To solve the world’s engineering problems, a student needs to have mastery of math facts. Math is no different than any other field where people aspire to do great things. If students cannot be successful in Algebra because they did not master the basics, then they will not be able to be that engineer, mathematician, or scientist.

    • I don’t think anyone here has said that practice is not important in learning mathematics. It is an important part of achieving fluency. Likewise, I believe all math educators agree that having quick recall of basic facts is essential for subsequent learning and success in higher level mathematics and science. Efficient use of “algorithms” or strategies for computation are essential as well. The difference is where the practice is placed in the learning sequence in order for students to understand why they are doing what they are doing. Additionally, we cannot allow a student’s learning trajectory to influenced by his/her not knowing basic facts. Skilled educators can support both needs at the same time.

    • Amy Flax:

      Their creativity can be unleashed because they have a solid foundation in the basics.

      Creativity is possible even without a solid foundation in the basics. That’s the point here, not that we should avoid procedural fluency. (I don’t know who is arguing any such thing.)

    • > Creativity is possible even without a solid foundation in the basics.

      Dan – I’ll confess to not understanding that at all, and I’d love for you to write a full blog post about what this looks like in, say, calculus, or linear algebra, or even something like graph theory.

    • Here’s a canonical example from research (p. 135). Students can make creative arguments for which pitching machine is most reliable without having procedural fluency in calculating variance from a set of numbers or procedural fluency in calculating the distance formula given two points.

    • Can you connect the dots for me on that one? I think I’m misunderstanding your argument. It looks like creativity on that task required a really strong foundation in the basics. Per the note on p.139, it seems like the solutions the students created relied on:

      1. Finding the area of a polygon whose vertices are cartesian coordinate.
      2. Using the pythagorean theorem (or distance formula?) to find the length of a line segment.
      3. Averaging real numbers.
      4. Determining which quadrant a cartesian point occupies.
      5. Measuring distance precisely with a ruler.

      If you gave that exercise to students who hadn’t built up some confidence or expertise in those mathematical tools, would they be able to generate as many and as creative solutions?

    • Yeah – I’m definitely not saying students can lack any procedural fluency in any mathematical concept. I’m saying (and pointing to evidence) that students can understand a given conceptual domain in ways that yield creativity even if they haven’t mastered the operations. In the paper, students are able to create definitions of variation without having solved twenty problems where they calculate (by hand, naturally) the variance of a set of numbers.

    • Ah – that’s a much more constrained and limited argument. So, if I’m reading you right, you’re arguing that students can exhibit creativity in thing x without explicit previous drill in thing x. I think much of the heat and noise here is that it seems like you’re arguing a more general case: that students can exhibit creativity in thing x without explicit previous drill period. I think most people would be fine with the more constrained version of your argument.

      I think what Professor Oakley (and I, though I’m far less eminent and scholarly than either of you – I just teach math) are arguing is that *some* drill in *some* things a, b, and c is necessary to allow students to exhibit creativity in thing x.

  34. Reply

    What I appreciate about this blog is that Dan allows comments that may, at least at a surface level, appear to contravene some of Dan’s own ideas and approaches.

    NYT-published author complaining about censorship.

    Dan may like Emma Gargroetzi’s blog posting, but she apparently got so many pointed criticisms of the inaccuracies in her posting, (which she refused to post–she only posted my rebuttal after it first appeared elsewhere), that she turned off comments, leaving us all hanging.

    The way to reconciliation in disagreements in teaching math is through dialogue, not one-sided pronouncements. I can see through much of the discussion in the comments section here that all of us, whether reform or traditional, agree on far more than we disagree.

    • I feel that basic skills need to be mastered before moving on. I also believe that the most efficient algorithms are important as well. Too many loosely defined algorithms paralyze students later on. I am believer of mastery of skills. I see far too many students dependent on a calculator and do not have a deep understanding of what they are doing because they did not master the basics long ago. Mastery of fractions makes Algebra easier to learn and solve as well.

    • Great question.

      So what is the most efficient way to do 2013-1985? Clearly someone who actually thinks mathematically does something like add 13 + 15 or use a quick adding up 5 (1990)+ 10 (2000) + 10 (2010) + 3.(remember counting back change?) The most efficient way to subtract these numbers is to add.

      Students who are taught algorithms as THE method first seldom become mathematical thinkers. And in the 21st century we want citizens who are flexible thinkers not “computers” as we have lots of those – just ask Siri or Alexa (both will even do calculus for you). In fact when they asked several dozen PhD’s from a range of math specialties to do traditional sheets of computation, they only used traditional algorithms less than 5% of the time. (http://www.psy.cmu.edu/~siegler/418-Dowker.pdf)

      They spent time proving to each other whether the methods they chose were generalizable. This is also the best way for students to prepare for dealing with rational numbers – proportional reasoning. What the PhD’s continuously said was: look to the numbers. Luckily, if you actually ask young children – they use the same thinking as the PhD’s. Rote mastery destroys this thinking – i.e. mathematical thinking. As someone who has been teaching this way since 1988 – (after spending the first 7 years of my career teaching rote math) – it is incredibly more fun, interesting, for me and the students – it requires my professional judgement and observational skills – not at all demeaning to me or to my student’s natural mathematical skills and understanding they have been developing since birth. They learn more math – and are equally as effective at arithmetic. After the first few lessons I did this way way back when, one of the students said: “You’re messing with my mind sir!”. Exactly what education should do.

    • Daniel – I think you’re missing the point of stasis with Barbara here, and your linked source actually argues *in favor* of the sort of mathematical practice she advocates.

      Dowker’s paper is an argument that if you compel mathematicians to do a routine problem repeatedly, they will become creative and develop many different general techniques for that problem. It’s an argument that the general can emerge from the particular. If you don’t do lots of the particular, you’re passing on a wonderful technique for encouraging general reasoning. There has to be grist for the mill.

    • “She only posted my rebuttal after it appeared first elsewhere … she turned off comments, leaving us all hanging” writes Oakley.

      While I was initially excited to engage in a meaningful dialogue about math education, especially with the person who wrote such a widely read NYT Op-ED, there were some serious red flags that led me not to engage the conversation more fully, and subsequently to close the comments.

      The number one red flag arrived at 1:24am on the morning of Sunday, August 23 when Professor Oakley posted her first comment on my blog and her friend Barry Garelick posted that same comment on his own blog, along with suggestion that I was unwilling to post it. Yes, these two posts were posted AT EXACTLY THE SAME TIME.

    • Emma, I had sent that comment to your blog several days before YOU finally decided to approve it for posting–presumably, when you saw that Barry Garlick had posted it.

      Your portrayal of what actually happened is utterly deceptive.

    • Emma Gargroetzi

      September 6, 2018 - 1:08 pm -

      Barbara – you are correct that my dates were off as Sunday was Aug 27, not Aug 24. I posted the blog on Thursday Aug 23, Dan Meyer posted a blog that linked it on Aug 24. My original post received large amounts of traffic on Saturday Aug 26 – presumably the day that you heard about it and composed a reply. Sunday morning when I woke your comment and a link to Barry’s blog were in my email. It was not until yesterday that I looked and saw that they were posted within the same minute.

    • All I know is that I posted my rebuttal and waited and waited for days, and you didn’t approve it to appear, even while your posting was obviously getting much traffic.

  35. Reply

    I very much agree with these comments. Learning math shares many commonalities with language learning. Let’s say a 15-year-old child started learning Russian. The child learns that the word “понимать” means “to understand.” Great–conceptual understanding has arrived! But just because a child has a conceptual understanding of the meaning of the word понимать in no way means she can use it, recall it, or apply it correctly. Quibbling about whether practice and drill is foundational versus whether it is supporting, or whether or not a child can be creative unless she learns the rules in part (perhaps large part) through practice and drill, is just that–quibbling. The child needs to understand that “понимать” means “understand”–AND she needs to practice with that concept through a variety of drills, (some of which can be very creative, but some, like practicing piano scales, just foundational).

    The problem with many thought leaders in mathematics education is that they are highly intelligent and have excellent memories. They can retain information relatively easily. They think their own way of learning, which doesn’t demand as much practice, repetition, or attention to the memorization process, is the best method for everyone to learn. But, as Morgan’s work amongst that of others shows, it isn’t.

    No one wants to make learning Russian–or math–tedious. But there are some aspects of language learning, or learning in math, or learning anything, that involve getting in the trenches, practicing, and developing procedural fluency. Thus, it is deeply misleading to say ‘Anyone who teaches children that they need to silently comply through painful experiences before they will be allowed to let their brilliance shine has no intention of ever allowing that brilliance to shine, and will not be able to see it when it does.’ That’s like saying any practice and drill in learning a foreign language is unfair, because it doesn’t let a child’s brilliance shine. The same applies to learning math. Great teachers provide the passion and exuberance and great methodologies that can get kids past some of those more tedious aspects of really mastering whatever they are trying to master.

    • The child learns that the word “понимать” means “to understand.” Great–conceptual understanding has arrived!

      We both use the same words but we aren’t talking about the same thing when we talk about conceptual understanding.

      Your metaphor here is like saying that a student has conceptual understanding of a triangle because she knows its definition. That is emphatically not how mathematicians and math education researchers define conceptual understanding.

      You also quote Paul Morgan’s study approvingly but he’s even even further from the mark, equating conceptual understanding with dance and music activities (p. 186). In his Psychology Review piece he links the word “understanding” to nothing more precise than NCTM’s Principles to Action volume.

      If you folks are going to comment on serious issues in national publications and tier one journals, you need to actually understand the arguments. Ask any math ed researcher for a good published definition of conceptual understanding and they’ll all put Adding It Up in the top five.

    • The definition of conceptual understanding you are bringing forth here simply cannot be obtained without substantive procedural fluency. You note “Creativity is possible even without a solid foundation in the basics. That’s the point here, not that we should avoid procedural fluency. (I don’t know who is arguing any such thing.)” Yet you wrote this entire blog post to denigrate an op-ed that was written in support of the value of the development of procedural fluency, an op-ed written to say, not that one should make the study of math painful, but rather that it simply can’t always be made fun.

    • Hi Dan, again I think that your comments misrepresent my work. I am not “equating conceptual understanding with dance and music activities.” Instead I reported on an empirical evaluation that contrasted various types of math instructional practices that 1st grade teachers reported using and their relations with math achievement gains by students.

      We found that teacher-directed instruction predicted achievement gains by both students with and without prior histories of struggles in math. In contrast, student-centered instruction predicted achievement gains only by those who had not previously struggled in math.

      One type of instructional practices that we examined is the use of movement and music, which others have described as facilitating conceptual understanding. For example, the title of the cited Wood (2008) article is “Mathematics through movement: An investigation of the links between kinaesthetic and conceptual learning.”

      We found no evidence indicating that use of movement and music were related to math achievement gains. However, teachers reported using these same practices as they taught classrooms with greater shares of students who had struggled in math. This suggested an instructional mismatch. I think we want to provide effective instruction to all students including those who are struggling.

      My write up appeared in Psychology Today, not Psychology Review. I link to the NCTM’s Principles of Action because of its emphasis on students first attaining conceptual understanding before being provided with practice opportunities to attain procedural fluency. You can find further discussion of the relevant empirical work in this review (https://link.springer.com/article/10.1007/s10648-015-9302-x). The review concludes that it is a “myth” that instruction should be ordered so that students first acquire conceptual understanding before working to acquire procedural fluency through practice. Instead, conceptual understanding and procedural fluency influence each other bidirectionally. Effective math instruction should be emphasizing both.

      This suggests, to me, that providing more opportunities to practice math will help students acquire both procedural fluency and conceptual understanding in math, thereby helping the students to become more proficient. Explicit instruction and practice opportunities may be especially beneficial to students who are struggling.

      Helping all students to do better in math is a goal that I believe we both share. And I do not doubt your sincerity in trying to help students learn math.

      I value constructive dialogue. However, relying on arguments that mischaracterize another’s position while making disparaging comments about professional standing to speak on the field’s evidence base only undermines one’s own credibility.

      Paul

    • I reported on an empirical evaluation that contrasted various types of math instructional practices that 1st grade teachers reported using and their relations with math achievement gains by students.

      The names you assign the responses on that survey instrument matter. They’re what gets picked up by the popular press. Decisions you made that have varying levels of justification: (1) naming a cluster of survey responses “student-centered”; (2) linking a cluster of survey responses to “conceptual understanding”.

      We found no evidence indicating that use of movement and music were related to math achievement gains. However, teachers reported using these same practices as they taught classrooms with greater shares of students who had struggled in math.

      Great! That’s interesting. Report that. But as I’ve mentioned here and on Twitter, your warrants for turning that finding into a referendum on conceptual understanding are extremely weak. Even if I felt great about your citation of a teacher writing her opinion in a non-research publication, I am very curious how you justify the Huntley et al. citation, which does nothing to link conceptual understanding to movement and music. Nothing. What’s the deal? You haven’t addressed that citation here or on Twitter.

      I value constructive dialogue. However, relying on arguments that mischaracterize another’s position while making disparaging comments about professional standing to speak on the field’s evidence base only undermines one’s own credibility.

      Honestly, if I misrepresented fundamental concepts in policy or engineering that were then picked up by the popular press, I hope my professional standing in policy and engineering would get disparaged.

    • Hi Dan,

      We describe in detail (pp. 8-10) the study’s factor analyses (http://journals.sagepub.com/stoken/rbtfl/J2BxFXoAWRPSo/full). Our terminology is consistent with other empirical work examining instructional practices including as cited in the study. Our findings are also consistent with other empirical work. We did not link a cluster of survey responses to “conceptual understanding.” Yet we did report on exactly what you say we did not—the lack of evidence of relations between movement and music with math achievement, as well as that 1st grade teachers were more likely to be using these practices in classrooms with greater shares of students who had struggled in math. Unfortunately, I do not see you making any serious attempt to engage with the study’s empirical findings. Despite my repeated attempts to clarify, you continually misread the text where we cited Huntley et al. and Woods. You insist that Huntley et al. says nothing about use of music, yet we cited this study in reference to calculator use. You insist that Woods is not a research journal but instead a blog, and then insist that “Paul’s out there HOPING nobody here knows the difference between a professional journal and a RESEARCH journal” after I clarified to you that is was published in a professional journal and not on a blog. I then provided multiple examples, including from both the NEA and NCTM (http://www.nea.org/tools/lessons/music-and-math.html; https://www.nctm.org/Grants-and-Awards/Grants/Using-Music-to-Teach-Mathematics-Grants), where use of music is suggested as a way to teach math concepts including to students who might find the concepts “difficult to grasp.” And you continually misread my position as a “referendum on conceptual understanding” when I make no such claims and instead present an empirically-based argument for why frequent math practice might help increase math achievement, particularly for students who struggle (https://www.psychologytoday.com/us/blog/children-who-struggle/201808/should-us-students-do-more-math-practice-and-drilling). You then end your reply by again insisting on the need to disparage one’s professional standing when the views “misrepresent fundamental concepts,” yet you have not shown this to be the case in any credible way that I can see. Instead you rely on red herring attacks to which I have repeatedly responded. I value your efforts to help students learn math. Yet reasoned exchange over peer-reviewed empirical findings that run contrary to your own views looks to be, well, not occurring at present. I do hope we are able to engage in more productive dialogue elsewhere in shared–but evidence-based–efforts to help students do better in math.

      Paul

  36. Reply

    My teaching changed when I realized that the drills I had done to learn weren’t the reason why I knew what I knew. Drills were an opportunity but I had given myself permission to mess around with the boredom. There’s a conflict in the minds or the teenagers I teach. They want the freedom but they still will do just what you told them unless they understand that freedom is what is expected of them.

    I changed the way I taught when I stopped asking myself “what do I know?” (which is easy to model) and, instead, “why do I know that?” (which can be a tough question). I go under the assumption that we all change what we’re told to do into what we think we should do. We’re not always successful but when we are, that’s when we learn . . . when we’ve made the task our own.
    The message that a student has permission to find the answer their way is vital.

    It’s the same principle behind a number talk. You can ask students what 2013 – 1985 is, but the answer isn’t as interesting as how different students approached it. Drill (repeating the same types of problems) can be fine but the value is in accepting that there are multiple ways to solve the problem. If students understand that knowing one way to solve a problem is acceptable but knowing five ways is phenomenal, they get closer to knowing mathematics.

  37. Reply

    I’ve been pondering this whole issue since Dan posted it and and it seems to boil down to a quote from an earlier post by Dan about Graspable Math:

    “No tool is good. We can only hope to figure out when a tool is good and for whom and for what set of values.”

    If we consider repetitive skills practice as a learning tool (which it is) then we can stop arguing over its inherent “goodness” and start asking more useful questions such as those listed by Dan about tools in general. At different times, for different students, repetitive skills practice serves a useful purpose. In that context, it is a good tool.

    Where most educators get upset is when those in charge of policy (or with influential voices) assert that mindless drill is the entirety of maths learning and should be imposed upon all students by all teachers all the time. Of course this is a ridiculous proposition and not one that was put forward by Barbara Oakley in her original op-ed. Instead, she seemed to be arguing that we shouldn’t shy away from having our students do drill as part of their mathematical learning.

    I’ve never spoken with a maths teacher in person or online who thinks that drill should be all of mathematics learning nor any who think that it should be entirely absent. Instead, all think that it serves a useful purpose within a broader learning framework. Our remaining disagreement is about how much of a role the drill should take and what else can be done with the rest of the program.

    • I’ve never spoken with a maths teacher in person or online who thinks that drill should be all of mathematics learning nor any who think that it should be entirely absent.

      A view I find prevalent with many people who have never taught math but nonetheless think they should comment on it is that procedural fluency in a concept must precede any creative or inventive use of the concept.

      They aren’t suggesting balancing instruction more towards procedural fluency. They’re making an argument about what students can’t or shouldn’t do before submitting to pain. They’re suggesting a precedence and a prerequisite. That’s the argument that needs addressing and dismantling.

  38. Reply

    “They’re making an argument about what students can’t or shouldn’t do before submitting to pain. They’re suggesting a precedence and a prerequisite. That’s the argument that needs addressing and dismantling.”

    Can we ever get past this cherry-picked strawman? You know that’s not the position that many of your critics take. All traditional math textbooks and teaching methods introduce concepts first in a carefully-built scaffold. Then comes the homework to get individuals to better understand the subtle variations that are far more mathematically meaningful than basic concepts. Even basic skills require subtle understandings. Very few things are ever rote. This deeper level of understanding has to be carefully constructed on a unit-by-unit basis over years using individual problem sets. That’s what all STEM-prepared students get with AP and IB math sequences.

    I’m open to seeing other opt-in sequences for those who might have other beliefs, but drill-and-kill, if it ever existed, has been gone from K-6 for decades without any opt-out options. Where are the results? I had to help my son at home with math when his schools foisted MathLand, and then Everyday Math on him, but once he got to high school, I didn’t help one bit. He just got a degree in math. All of his STEM-prepared friends had to get help at home or with tutors in the early grades. When I grew up, I got to calculus in high school with absolutely no help from my parents. That’s no longer the case with full inclusion and curricula like “trust the spiral” Everyday Math.

    All properly-taught math sequences start with concepts, and engagement and curiosity are no magic wands for ensuring mastery. What works is what we see for AP and IB math. I don’t see STEM-prep success cases any other way. The problem now is that students in lower grades are stuck with a CCSS slope to no remediation in College Algebra and that the only ones who make the nonlinear transition to proper high school math have to get help at home and with tutors in the lower grades. If that doesn’t happen, it’s all over and no amount of engagement and concepts or “Pre-AP” math will fix it.

    This has never been a question about basic concepts. it’s a question about eliminating low expectations and ensuring proper mastery and mathematical understanding on grade-by-grade basis that keeps all math doors open for each individual student for as long as possible. With curricula like Everyday Math, schools trust the spiral and have abdicated all responsibility of mastery beyond the low CCSS slope. Just ask us parents of your best students what we had to do at home. Engagement and curiosity was not my main focus, but that didn’t stop him from playing with GeoGebra for hours at a time – something that was neither necessary or sufficient. Mastery increased his curiosity, not the other way around.

  39. Reply

    I have never taught school math. It’s pedagogy should be left to those who have. I did spend a long career using mathematics, and I’ve devoted the last few years trying to figure out why students have trouble with negative numbers, fractions, and real-world problem solving. It is a long standing condition unlikely to have a to have a pedagogy explanation, it would have been found by now.

    Is is pretentious, but I want to suggest the problem is in the content now taught. I have no illusions about changing the system of school math education. I would like to move discussion toward content. though. Pedagogy discussion grinds inexorably to another battle in the culture wars.

    I assert school math circumvents abstraction to race toward ” fluency” at he expense of building a foundation for learning the math to come, the math that will be critical to success in the world today’s students will see.
    Mathematics is the result of extracting its essence form the real world where it may have first emerged. This abstraction is what it is, and why it is powerful. Mathematics is self-contained.

    On the other hand school math takes mathematical meaning from connections made in the real world: “subtraction is removal” confuses a mathematical operation with the physical subtractive process leaving students at a loss when asked for meaning of subtraction of a negative. The meaning of negative is taken from real world as “down”, or “the other side of” which are asserted to be intrinsically negative. They are not. The mathematical meaning of signed numbers follows from the construction of the integer number system. They are constructed from . oriented pairs of natural numbers.
    A parallel argument holds for fractions and rational numbers. Fractions as parts of things is not a mathematical concept.

    • I agree with much of what you contend. I teach 7th and 8th grade math and Algebra and I try to avoid equating math with the real world examples. We can use the examples to show one of the ways math can be used but to infer that it is just a tool for “normal life” seriously detracts from the power of math and give students opportunities for artificially simplistic thought processes rather than driving harder toward the understanding the abstract nature of math, particularly algebra.

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