IS CONCEPTUAL MATH, THE MOST IMPORTANT PART OF MATH INSTRUCTION
In recent years one of the most debated topics in math tends to be regarding the idea of what type of instructional focus is better for math instruction: conceptual or procedural. In this article, I will try to answer this question as best as possible, from an evidence-based perspective. However, before we attempt to answer the question at hand, it is probably best to define the terms.
Conceptual Math: Involves the core concepts behind math, that allow it to function and make sense. For example, in order for a student to have a conceptual understanding of fractions, they need to understand that a fraction is a part of a whole.
Procedural Math: Involves the processes and formulas that make calculations possible. For example, looking at fractions from a procedural perspective would mean understanding how to add, subtract, multiply, and divide fractions.
Computational Math: Involves the knowledge of basic math facts that makes it possible to use procedural math. Computational math is generally seen as less important than the above two types of math; however, it is still necessary for students to be successful. For example, a student could know what a fraction is and they could know how to add fractions, but they still need to be able to calculate the correct solution to a problem using their computational skills.
At the end of the day, we want students to be able to do all three of these types of math. However, there is a lot of debate over which type of math holds the greatest overall value and over whether or not a focus on one of these types of math, can lead to greater mathematical understanding overall for students. There are four main theories regarding what type of math instruction leans to the greatest improvements in student understandings of math.
The first theory is called the procedural view: Procedrualist believe that mathematical understanding is generally speaking driven by an increase in procedural knowledge and that if a student’s procedural understanding increases, so should their conceptual and computational skills. In some ways, we could define the Proceduralist view as the traditional view on mathematical education. The second theory is called the Conceptual View: Conceptualists believe that mathematical understanding is generally speaking driven by an increase in conceptual knowledge and that if a student’s conceptual understanding increases, so should their procedural and computational skills. The Conceptualist viewpoint has become extremely popular in modern education communities. The third theory is called the Inactivation View: Inactivationist believe that all three skills are developed largely separately. Well our understanding of the principle of teaching specificity, suggests that there has to be at least a kernel of truth to this idea, even a quick glance at meta-studies on the topic seems to disprove the core principle of this perspective. Lastly, the Iterative View: suggests that conceptual and procedural learning is interrelated and that both conceptual and procedural learning helps develop both sets of skills. While the Conceptualist view is the most commonly preferred theory by the educational community at large, the Iterative view is largely agreed upon within the academic community.
In order to establish efficacy for any of the above theories. We need to look towards meta-analysis and the impact of having a teaching perspective weighted towards one of these instructional types. If the Conceptualist Theory were correct, most case studies into the impact on overall mathematical learning should show a greater increase in learning, when teachers use a Conceptualist approach. However, in 2011 Durkin, Rittle-Johnson, & Star completed a meta-study on the topic and found that teachers who included both conceptual and procedural teaching in their classrooms had a mean effect size of .54, compared to teachers who did not. This would suggest pretty clearly, that the evidence suggests an Iterative view is more correct than a Conceptualist view.
Some Conceptualists argue that conceptual math education needs to be a greater focus because students are more likely to have a conceptual knowledge deficit than a procedural one. Rittle-Johnson et-al conducted a study in both 2007 and again in 2009 on whether or not students tended to have knowledge sets that were skewed towards conceptual knowledge or procedural knowledge. Their study showed that student knowledge in both the conceptual and procedural domain was largely symmetrical, meaning that if a student had a high degree of skill in one area, they typically had a high degree of skill in the other. Similarly, a study by Cowan et al in 2011, showed that variances between conceptual and procedural skill levels in individual students typically were less than 5%.
When comparing an Iterative view, versus a Conceptualist view, it becomes quite clear, quite quickly, that the Iterative view is more supported by the evidence. However, perhaps the reason that the Conceptualist view has become so prevalent in recent years has to do with the results of recent studies on Procederualist vs Conceptualist focuses on math instruction. There have been multiple studies comparing purely procedural teaching to conceptual teaching within the last 20 years, most of which have shown a slightly higher benefit to a conceptual approach when compared to a procedural approach. This has led many education professionals to conclude that a Conceptualist approach is superior. The problem with this conclusion is that it does not account for an Iterative approach. To put it more simply, just because a Conceptualist approach is better than a Proceduralist one, does not mean that a Conceptualist approach is better than all other approaches nor does it mean that teachers should not cover Procedural teaching through direct instruction.
So the question that remains, is how do we best teach both conceptually and procedurally at the same time? Some helpful suggestions from papers written by Rittle-Johnson et al in 2007 and again in 2009, include: encouraging students to find multiple procedural ways to solve the same problem and collectively assessing math work as a class to find errors. Both of these strategies are according to Rittle et al valid Iterative approaches, but also can be independently verified by the literature as high yield instructional approaches. A paper written by Schwartz and et al, in 2011, suggested that teachers begin lessons with a conceptual approach and transition to a procedural one. Essentially suggesting that conceptual instruction should form the ‘minds on’ section of the math lesson. While Schwartz et al, did not provide any evidence for the efficacy of this idea. There does seem to be little potential opportunity loss risk in trying the strategy, as we know already that an Iterative approach is better than a Conceptualist or Proceduralist approach. Similarly, a paper by Hiebert & Grouws, written in 2009 suggests, that teachers should start their math lessons with an inquiry based conceptual math question and follow up that math with actual direct instruction.
Personally, I think it is important for math teachers to be able to identify individual weaknesses, in individual students, whether they be conceptual, procedural, or computational. In my experience student learning can be hampered, when we mark incorrect math solutions, as simply wrong, without exploring why they are wrong with the student. This is especially true, when a student struggles consistently with the same type of math problem for an extended period of time. In my personal experience, when I conference with individual students regarding their math struggles, I often find they have a specific glaring weakness. That is not to say that I commonly find that students have specifically a procedural or conceptual weakness, but rather their understanding of a specific type of math problem can be hampered by a weakness in either domain. If we cannot identify the specific weakness of the student, we run the risk of accidentally teaching them over and over again, the same content, without actually helping them.
For example, let's say we want a student to be able to apply their knowledge of fractions to situational problems, but they do not know what a fraction actually is. It will be very hard for that student to identify when to use a fractional procedure to solve a math problem, no matter how many times, we show them how to solve a fractional equation. Inversely, if we have a student who is struggling with the procedure behind finding a common denominator and consequently keeps getting fractional math equations wrong, it will not help them to further reinforce their conceptual understanding of what a fraction is. Finally, even if a student does have a strong procedural and conceptual understanding, if they lack basic computational skills, they might still fail to answer basic questions correctly. This does not mean, we need to constantly teach all three types of math in every math lesson, nor does it mean we need to conference with every student every day. However, I would argue that there is a strong benefit to conferencing with struggling students, when you have the time, in order to identify their specific math needs.
One final consideration at the end of this discussion, is the actual efficiency of implementing an Iterative approach. While a Conceptualist approach is slightly superior to a Proceduralist approach, and an Iterative approach is significantly superior to a Conceptualist one, none of these approaches are what we would call high yield strategies, according to John Hattie’s research. That is not to say that it will not have a meaningful impact in your classroom teaching to switch to an Iterative approach, but rather, if the concept seems overwhelming to the average teacher, there are other higher yield strategies they could implement that are less challenging. Ultimately my goal in writing this article is not to entrench the reader into a new camp of math instruction, but rather to add nuance to a preexisting debate that has become quite inexplicably polarizing.
Written by Nate Joseph,
Last edited: 5/12/2020
Interested in learning more about this topic check out our interview with Dr. Jon Star on the topic:
References:
1. Bethany Rittle-Johnson, and Michael Schneider. (2011). Developing Conceptual and Procedural Knowledge of Mathematics. Oxford Press. Page 9.
2. Ibid
3. Rittle, et al. (2007). Common and Flexible Use of Mathematical Non Routine Problem Solving Strategies. Science and Education.
4. Cowan, R., Donlan, C., Shepherd, D.-L., Cole-Fletcher, R., Saxton, M., & Hurry, J. (2011). Basic calculation proficiency and mathematics achievement in elementary school children. Journal of Educational Psychology, 103, 786–803. doi: 10.1037/a0024556.
5. Schwartz, D. L., Chase, C. C., Chin, D. B., & Oppezzo, M. (2011). Practicing versus inventing
with contrasting cases: the effects of telling first on learning and transfer. Journal of
Educational Psychology, 103, 759–775. doi: 10.1037/a0025140
6. Hiebert, J. & Grouws, D. (2009). Which teaching methods are most effective for maths? Better:
Evidence-Based Education, 2, 10–11.
7. J, Hatie. (2017). Hattie Ranking: 252 Influences And Effect Sizes Related To Student Achievement. Visible Learning. Corwin. Retrieved from <https://visible-learning.org/hattie-ranking-influences-effect-sizes-learning-achievement/>.
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