Wednesday, May 10, 2017

Finding a Balance in Math

For the past month or so, I’ve been participating in the Not A Book Study look at Cathy Fosnot’s work on constructing multiplication and division.

This has been a tremendous opportunity to make great connections with other math educators across Northern Ontario, and really deepen my knowledge of how students construct mathematical concepts. I’ve been searching for something deep to sink my teeth into for a while, and I find this book - and all the discussion surrounding it in the #notabookstudy - has been rich and extremely thought-provoking.

As a secondary math teacher, I learned a lot in teachers’ college about tips and tricks to teach math through the intermediate and senior years. But we didn’t learn very much about how students learn math. How do they make the jump from thinking additively (counting 3 + 3 + 3 + 3 + 3 + 3 + 3) to thinking multiplicatively (7 x 3)? How do they approach the concept of division? How do they go from being able to problem solve in a specific context, to a generalized one, eventually using variables and equations?

Needless to say, I’m learning a lot about learning by reading this book, and this will certainly enrich how I approach teaching mathematical concepts in the future.

But I find I’m struggling with another aspect of teaching math as I continue learning about how we learn - finding the balance between discovering the concepts and practicing the skills.

In secondary, there is very little discovery (I’m generalizing here, based on my experience teaching). Most of what the students learn is taught to them in a very procedural way, with perhaps a bit of inquiry or hands-on activities, but almost never where the students are discovering the math themselves. And the students certainly aren’t owning the math.

On page 48, Fosnot says “Mathematics cannot be learned through transmission…” and I agree - students need to understand what the mathematics is telling them in order for them to remember what they learn. And yet we, as teachers, are still merely transmitting information.

However, in elementary (again, a generalization from what I’ve seen, mostly in primary and junior), there is a lot more discovery of the math, with lots of emphasis on concept construction, but not as much procedural learning. While this can promote a better understanding of what the math means, and a better growth mindset and approach to problem solving, the drilling of facts or repetition of procedure doesn’t seem to be, pardon the pun, part of the equation.

Partially because of this, in secondary we are seeing students who struggle more and more with basic math facts. That struggle leads to frustration, cancelling out any gains that may have been made from understanding the math initially discovered in earlier years.

Can students understand where the math comes from AND become procedurally efficient?

I’d like to think so, but we as teachers need to work on that balance. With respect to automaticity, Fosnot says “The issue here is not whether facts should eventually be memorized, but how this memorization is achieved: by rote drill and practice, or by fostering on relationships?” (page 86).

The former is shallow but “quick;” the latter is deep but takes time. And if “It is not up to us to decide which pathways out students will use [as they move toward constructing understanding]” (page 18), how can we ensure each student has the time to properly develop their thinking AND their automaticity?

Wherein lies the balance? My journey continues...


  1. Thanks for this Heather. As a principal, I attended a fair number of workshops where math inquiry was seemingly the only way. We would sit as a group and spend 45 minutes trying to figure out all the ways a math problem could be figured out. This seemed to me a real waste of time. This seems to be the way in education, and I think you are asking a fundamental question - where is the balance? My wife teaches grade 7 math and she would agree that a certain amount of drill is necessary to reinforce basic knowledge necessary for inquiry. Students do need to know their basic math facts. I believe the pendulum will swing back, but we waste so much time waiting for a certain amount of sanity to return to the system. The same can be said about language arts. Before becoming an elementary principal, I had the wonderful opportunity to teach grade 6 language arts for most of a year. I couldn't understand why we were not supposed to teach spelling! My kids had no idea how to spell even though they were in grade six. I started explicitly teaching spelling, looking at word origins, roots, suffixes etc to build some understanding of how words are constructed. Again, the basics to me seemed really important and were missing. Thanks for your article, I think a lot of math teachers will be on the same plane as you!

    1. Hi Paul - thanks for your comment! I really appreciate your views both through a teacher lens and an administrator lens. I'm torn about spending time figuring out all the ways a math problem can be solved. A lot of what we're looking at in the Not A Book Study is about recognizing (and valuing) all the ways a student can approach and construct their learning. To do that, it would help to know just how open some of these math tasks actually are.

      But to increase the value of the time spent looking at that question, I would want to link it back to the curriculum - recognize what earlier-grade knowledge the student is drawing from - as well as look at how to best help a student make connections/jumps from one concept to another (or from a specific case to a generalized case). And then how do we best reinforce those newly-attained concepts through purposeful practice? I see myself paying a LOT more attention to the strings of questions on practice worksheets than before, guiding the students toward more "efficient" means of solving problems. There's so much to think about!

  2. I wonder how many of those procedures are the standard algorithm. If a person had never learned that algorithm, but showed up in high school class really good at splitting the numbers to arrive at an answer, would that be OK? I often wonder this - how committed to the "old school" ways are many secondary teachers? I can remember the last time I sat in a high school math class. I wonder how much as changed in 25+ years (OK...28!! ugh.) I'd really love to see what a math inquiry would look like with a group of secondary students. Is there such a thing?

    1. Hi Lisa! Thanks for your comments :)
      I'm not sure that TRUE inquiry happens very much at the secondary level. A lot of it seems very procedure-specific, but I don't know if that's just my much-engrained way of approaching secondary math, or whether math/concepts eventually become too complex for true inquiry?

      re: your point about a high schooler being really good at splitting numbers but not jumping to the algorithm, I had that same discussion with colleagues a little while ago. If a student could DRAW the solution, but not work through the algebra, would that be okay? If a student could complete the square by literally completing a tile square, but not work only from the equation to a solution, has he/she still mastered the curriculum expectation? I'd like to say yes, but having never seen it through myself, I'm not sure... Definitely things to try when I'm back in the classroom!

  3. And I also want to say that this difference exists even in elementary schools. Some teachers are very inquiry driven, and others act like they are ("Here...let me show you what you are going to discover!") and still others are committed to a textbook more often than not.

  4. Kaarina McLaughlinMay 13, 2017 at 6:18 PM

    Heather, I have been living exactly what you are talking about the past two years. I taught grade four for many years and was very "inquiry driven". I quickly realized that students would understand the math much better than I ever did by teaching this way. Moving to grade six last year truly highlighted the need for more practice. I have struggled with the need to allow time for rich inquiries and the need for increased practice to understand the concepts.
    That being said, I was recently planning for the introduction of order of operations and it was blatantly obvious that the students now want to figure it out on their own and not by direct teaching. Last year, I had a strong group and many of them already knew about the order of operations and could explain to their classmates without much direction from me. This year, when I asked them to figure out 3x2+5 and 5+3x2, it was obvious that the students did not have prior knowledge of the order of operations. So, I used direct teaching and there was practically a revolt! Needless to say, I came home and had to do a total overhaul on my lessons to help them understand this concept by having them discover the "how" and "why" on their own. The students did not want me to teach them, they wanted to figure it out. So, as the inquiry model continues to flourish in much of the elementary panel, maybe more students will not be satisfied with a secondary teacher using direct teaching. I am hoping that they will demand time figure out the "how" and "why" on their own!

    1. Wow, Kaarina - that's so interesting! Amazing to hear that the students WANT to discover the math! This gives me a lot of hope :) I would love to hear more about how you find that balance between discovery and practice in your grade 6 class.

      I'm new to the idea of strings designed to link together ideas and then build on those ideas to increase automaticity. I'm starting to wonder how I can use those to build on more complex math concepts in secondary...

  5. If we are able to successfully implement the learning we are doing in all elementary classrooms (re Cathy Fosnot's research, guides and curriculum) do you think we will see the same challenges you speak to re Secondary? As we read Fosnot, I see balance... opportunities for constructing meaning and understanding and practicing coming nicely together just in diff ways than traditional instruction would approach it? When I read Fosnot, I see the two working beautifully together.

    1. That's an interesting question... if every child was able to traverse the landscape of learning on his/her own path, developing the skills & making connections in their own way and in their own time, would we need as much practice/drill as we currently perceive we need? Can this be done in a system where curriculum expectations are laid out by year/age? Can it be done when subjects are still taught in isolation of each other? I'm interested in seeing how other provinces and countries approach math learning to see how they address these learner needs.

  6. Hi Heather,
    I am a parent who finds the arguments at compelling on this. Anne Stokke in answer to the question of balance gave her recommendation at 80/20 in favor of instruction and practice over discovery or inquiry methods.

    And a teacher from Australia has an excellent blog where he cites the evidence to support a balance well in favor of teacher led instruction and practice based methods.

    This post highlights the measured benefit of teacher led instruction from the pisa results.

    One of the concerns I have is how much the alternative view sounds like wanting a free lunch. People are looking at ways to avoid the unpleasantness of practicing and memorizing in favor of pleasant activities such as discovering some mathematical fact. Most kids and parents understand not much works this way.
    Ask a child who is in A grade hockey or progressing through piano exams whether they could do well without drills and practice and they will have no doubt you need to do the practice to enjoy being good at something.

    I met two elementary math teachers who flat out refused to get kids fluent in the times tables. In their opinion this was material from earlier grades and if the kids needed practice that was up to the parents. These teachers had taken the idea that math lessons should contain nothing that feels like work to such an extreme that they would rather leave kids less able than tackle a memorization task.

    I do get the concern that a focus on memorization without insight would be bad too. I loved Lockhart's A Mathematicians Lament. But I think there are two flaws with his solution. One well written here

    is that it doesn't address the needs of students to get through the material in time for it to be available knowledge and skills for them for future work or learning. The second is that it requires a lot of expertise from the teacher.

  7. I agree that balance is the critical piece when thinking about conceptual and procedural understanding in mathematics. Kids need to own their learning and I believe the best way for this to happen is through inquiry, asking questions, and linking to previous understandings. However, there is a very important time for purposeful practice to help build in fluency and efficiency, but this can be and needs to be done without compromising conceptual understanding.
    I find the example above that mentions the revolt with learning the order of operations interesting. I believe that there are some things that cannot be learned well through inquiry, and I'm wondering if order of operations is one of these things. This is an agreed-upon structure in the world of mathematics that allows us to interpret arithmetic statements in a common way. I think the role that inquiry could play with this would be why there is a need for a rule like BEDMAS. I would be very interested in hearing more about the inquiry lesson on this topic!
    I think it's important that there is time spent in learning mathematics in a way that values and develops both the conceptual and procedural, without compromising either important way of thinking.

  8. Hi Heather;

    What a great post. I'm so thankful for educators who let us into their heads and allow is to "swim in the river you are swimming in" as Dai Barnes often says.

    Lots has been said here already, so I want to just comment on another similar issue I see in high school - the child who is not ready for the "course" being delivered to them.

    In elementary, it seems to me that we are pretty good at diagnosing where a student is in understanding and using progressively more efficient and eloquent strategies. Yet in high school, we deliver a course regardless of where that readiness is.

    And if they aren't ready, one of two things happens:
    a) the student is put into a "locally developed" course. Call me crazy but I have never seen manipulatives or problem solving, including diagnosing learning needs in one of these courses. I do see simple worksheets and assignment booklets
    b) the student fails the course and gets put into credit recovery with more booklets and worksheets.

    I don't see the diagnosis work, the "doing" of math outside of written worksheet-style and textbook-style exercises. Only once have a seen a student move from Locally Developed, to Applied, to Academic math based on a complete lack of opportunity in elementary school to develop number fluency and efficient strategies.

    How does our work with Cathy Fosnot help us to rethink how we address students who are not thriving in secondary school courses?

    How do we ensure that every single learner has the opportunity to excel in mathematics?

    1. This is an excellent point! I often think high school students who are struggling could use some "elementary" approaches. Maybe the second time around they'd be more ready for it.

  9. Dear Heather,

    Your reflection has provoked me to think about the wide-scale change we are asking all educators to embrace over the next couple of years specific to the renewed mathematics strategy.

    Through my work over the last couple years, I have come to believe that to enhance and support improved mathematics instruction in ALL of our classrooms and subsequently to realize significant levels of improved student achievement in mathematics, the collective ‘we’ as educators need to find the time to ‘re-learn’ or perhaps I should use the term un-learn. Ontario educators are ‘best in the world’ and there is no doubt in my mind we have the collective capacity to make these important changes to how mathematics is taught. It will require a continued collective responsibility by those teaching and leading the mathematics work in schools/systems to dig into the current research behind the Ontario curriculum and the accompanying Guides to Effective Instruction. Researchers like Cathy Fosnot, who we are currently studying in the notabookstudy, (and others such as J.Boaler, A.Lawson, C.Bruce, M. Small) are helping us to understand the current research that supports the why and how. As a province, I believe we are building a common understanding of what effective mathematics instruction looks like (see Anthony and Walshaw, 2009), sounds like, feels like when it is alive and well in our K-12 classrooms - such important work when we think about the connection between high levels of mathematics achievement and lifelong outcomes for students. The moral imperative of the work calls for us to embrace this challenge. There is still much work and learning to do....

    To me, this work is not about grades or divisions but more about building our collective understanding to establish the strong contexts Cathy Fosnot speaks to; contexts that allow our students to actually experience what it is to mathematize (K-12) - where ALL educators have the confidence, expertise and understanding to make math meaningful for students, helping it to come alive in our classrooms - mathematizing vs what Fosnot refers to as teaching ‘dead math,’ numbers on a page that hold little relevance/meaning for students. I look forward to the day where ALL students who are working on functions and relations or algebra or geometry see the context and relevancy in the mathematics…..and that all students have confidence because they have a strong foundation and a strong understanding of numbers (fluency and automaticity included) from which to draw upon and all students experience the joy of being a ‘mathematician’ and the power of mathematizing; where numbers, shapes and formulas on a page hold meaning and understanding and are not merely algorithms to be memorized….and later forgotten.

    We are all on this learning journey together; it is an incredible one that holds so much hope and promise for all of our students. Your blog is inspiring. I hope you will continue to share openly about the changes you are making in your classroom based on this new learning and the impact it is having for your students. Thank you for the opportunity to respond and reflect on your post, I look forward to our continued conversation over the next several years! There is no doubt in my mind we are going to see incredible stories come from your classroom.


  10. There's a lot in this post and in the comments that resonated with me. Here are a couple of things that I might add to the conversation.

    1) I don't think that exploratory or inquiry learning was ever intended to happen without direct instruction. The key, to me, is when we intervene. And in the case of math facts, when do we start teaching them? How do we review them? I think that too often than not, we swing the pendulum to extremes, so people embrace exploration without this instructional piece, and then when students struggle, they make it all about facts without exploration. Fosnot shows us well how to combine both, and I think her work is worth exploring.

    2) I'd also add a question that we are asked in our new K Program Document to always consider when planning for students: "Why this learning for this child at this time?" I think this is worth considering for all grades. I think about our K students. There is a group of students that have a strong number sense and are starting to combine groups of items and "add" them together. We've taught some math facts around doubles. We've helped them understand them, but we also work on the facts themselves. This learning is right for these students, but not others. We are working on some subitizing skills (some to 5 and some to 10) for another group of students. Again, this is almost drill-like -- still within play, but still drilled memorization (with a little explanation). This is needed for some kids, but not others. Could these same K considerations be valuable in other grades as well? I think it's worth talking about when it comes to the "math facts" conversation.

    Thanks for such a thought-provoking post!


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