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.

Great quote.

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.

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.

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.

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.

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.

> 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.

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?

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.

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.

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.

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.

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.

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

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

In fact Japan teaches through problem solving as an Ontario math leader I have attended wksps by Japanese teacher/leaders. I can put you in contact with s brilliant woman who is in Toronto who speaks Japanese and wrote her doctorate on Bansho or the Japanese lesson structure including trips to Japan. Singapore brought in Japanese teachers to transform their instruction. Read Liping Ma’s work that shows Chinese educators have always taught more conceptually than N. American teacher. Read the Teacing Gap. PISA is just a pointer, you have to read the follow up ethnographic research to find out why these countries do well. Anyone who switches to a problem solving approach never goes back because the kids learn both the conceptual understanding which cannot be taught by direct instruction and the old goals of math of doing long division are pre-digital world just ask Siri literally. I have been teaching thru problem solving since 1988. Over many EQAOs in 3 and 6 my students laugh at EQAO. Data means nothing if the research uses just computation to judge the strategies used is at best highly incomplete because it is arithmetic not mathematics and at worst is preparing students to be successful in the 20th century.