Web-based Presentation: On the Conduct of the Understanding

Introduction

I have chosen to present on John Locke’s On the Conduct of the Understanding as part of this week’s focus on modern philosophy of education. I will first explain some critical ideas from the reading, which helped me frame my critical question/thesis. I will then consider a recent controversy within education; specifically, I will be looking at the debates around the common core standards relating to mathematics in the United States. Lastly, I will be concluding with my argument based on Locke’s ideas and my research.

 

Critical Ideas From The Reading

In On the Conduct of the Understanding Locke examines how humans should reason and rationalize to form understanding and come to well-founded conclusions. According to Locke, in order to gain the truth, one shouldn’t blind his/her own view (Locke, p. 3). In order to understand something well, one must be exposed to more than one sort of notion or viewpoint and consider these viewpoints in their reasoning. Good reasoning and understanding happen when all arguments, for and against, are examined against one another and a conclusion is formed based upon the whole picture (Locke, p. 9).

Locke argues that the process of reasoning must be taught early if one wants to reason well (Locke, p. 8). The goal is to teach and practice reasoning so that it becomes habit. Locke argues that although we may be born with the ability to be rational, we only develop the necessary skills to act rationally and reasonably if we use, exercise, and apply these skills and work hard to become “rational creatures” by teaching the mind to look for connections and different ideas. Thus, adapting minds to this process early allows them to use these skills to consider various sides to an argument rather than being limited by their one single view when making judgments.

Lastly, Locke argues that teaching mathematics is essential for making students “reasonable creatures”. He believes that the reasoning skills learned in the study of mathematics can be transferred to other parts of knowledge (Locke, p. 8).

 

Critical Question/Thesis:

In order for one to understand and reason well, he or she must be willing to consider more than one viewpoint and take these into consideration before making judgments or forming conclusions. The skills necessary to reason and rationalize well must be taught early so that they are practiced regularly and become habit. The question becomes: do reasoning skills need to be exercised and applied early in order for one to reason well? If so, how should educators teach reasoning, critical thinking and alternative problem solving skills to children early on in their schooling?

 

Debate or Controversy Within Education

Since Locke speaks about teaching mathematics as the essential way to fostering “reasonable creatures,” this made me think of the debates around the common core standards approach to mathematics. This approach was developed in the United States due to the understanding that students could learn mathematics by rote memorization of particular algorithms, but had difficulty applying that same knowledge to more advanced mathematics causing many students to not pursue advanced math classes (Kruger, 2018). After analyzing the processes of professional mathematicians, researchers found that these professionals had a deeper understanding of numbers and the relationships between numbers and they often did not follow the formulas and algorithms that are taught to elementary school students and, if they did, they understood the reasoning behind those formulas (Kruger, 2018).

 

The common core required students to learn different ways to attempt to solve math problems and tried to encourage strategic critical thinking and creative problem solving skills. For example, students may solve a problem using one method and then check their work by trying another method to see if they come to the same solution. The goal of common core math is to help students understand the relationship between numbers. 

The debate arose when parents found themselves not being able to help their children with their math homework, as they were unfamiliar with the approaches being taught in school and the math problems looked too complicated. There are some people who believe that this approach to mathematics is problematic as it takes more time and steps to arrive at the correct answer (Phillips, 2017). Questions arise as to whether “replacing rapid recall of arithmetic facts and memorization of multiplication tables” with this method is actually helpful to students when they apply to college or who wish to enter STEM fields (Phillips, 2017). According to Kruger (2018), “the Common Core State Standards initiative…did not, in and of itself, change the approach to teaching mathematics. It simply encouraged more widespread adoption of new methodologies that are rooted in well-researched, evidence-based best practice.”

 

Argument

I agree that critical thinking and problem solving skills need to be practiced in order to be developed well and in order to be transferred to different subjects of knowledge. According to psychologists, children learn to reason around the age of 7 and educational content should reflect their development (Sher et al., 2014). With regards to the controversy on the common core mathematics approach, after reading Locke, I think that teaching different approaches to solving math problems is beneficial to students. Providing students with options is important in adapting to different learning styles and offering students an option to learn about the relationships and connections between numbers is crucial for students who have difficulty grasping concepts using the traditional approach to teaching math. However, I do think that alternative methods to teaching should be taught in combination with traditional and standard methods so that students are not at a disadvantage when they enter advanced math or higher education.

In a more broad educational sense, I would argue that providing alternative instruction helps students understand that there is more than one way to tackle a problem and that they should explore different perspectives before coming to conclusions. Fostering critical thinking skills in any subject is beneficial to developing student’s reasoning abilities and the goal of making reasoning habitual means that it can be transferrable to other knowledge areas. Students will learn that considering alternatives is a necessary step in coming to educated conclusions and they will seek different ways of thinking to solve problems.

 

 

References:

Kruger, P. (2018, September 05). Why Did The Approach To Teaching Math Change With Common Core? Retrieved from https://www.forbes.com/sites/quora/2018/09/05/why-did-the-approach-to-teaching-math-change-with-common-core/#53c08ab69ff2

 

Locke, John. On the Conduct of the Understanding, ed. Jonathan Bennett, 2017, pp. 1- 10. www.earlymoderntexts.com/assets/pdfs/locke1706.pdf


Phillips, C. J. (2017). Knowing by number: Learning math for thinking well. Endeavour, 41(1), 8-11. doi:10.1016/j.endeavour.2016.11.001


Sher, I., Koenig, M., & Rustichini, A. (2014). Children's strategic theory of mind. Proceedings of the National Academy of Sciences of the United States of America, 111(37), 13307-13312. Retrieved from http://www.jstor.org/stable/43043466



Comments

KM
Kelly Milliken mai 31, 2019, 01:36

Excellent post and overview of some key points in Locke’s work. For the purpose of my commentary paper I would like to focus on your argument that teaching different approaches to solving math problems would be beneficial to students. I think that an argument can be made to say that it is not beneficial when looking at Locke’s views. When I read Locke’s thoughts on mathematics, he emphasizes that the process one goes through to gain an understanding of a math concept is what lends itself to helping individuals become “reasonable creatures”. Locke (2017) focuses on seeing the problem through from start to finish, on the sequence and dependence of ideas that math equations rely on (p. 9). He talks about how students will get stuck but that this is part of making the connection. When linking this to alternative methods of teaching math strategies, if we allow students to switch the method when they come up against a problem the are stuck on, then aren’t we taking away the struggle and sequencing, hence the opportunity to build reasoning skills. A key component of Locke’s belief is that we need to see different perspectives and move away from our own. Yes, teaching alternative methods is showing that there are different ways to solve a problem, different perspectives; however, if the alternative method that a student always chooses to use plays to their strengths and the way that the concept make sense to them, then are they really gaining a deeper understanding and different viewpoint or are they just sticking with what they know, what makes sense to them? Isn’t this just teaching them to find a method that fits how they see the problem verse working hard to see it another way. This would be in conflict with Locke’s beliefs on reasoning. They are making the math problem suit their style of learning, instead of working to improve and grow their understanding. Just knowing that there are other perspectives, other ways of solving the problem isn’t enough. To move towards being a “reasonable creature”, a student must apply themselves to work through the perspective/strategy even when it does not match their learning style and is difficult for them. Locke’s (2017) idea that, “ there is often a complaint of lack of basic abilities when the fault lies in the lack of a proper improvement of them” (p. 6) stood out to me when thinking about the concerns regarding students’ math skills. In the past few years in Alberta, where I teach, there has been growing concern regarding the Math curriculum and declining student performance. There is a push to return to traditional math strategies as there is worry that teaching multiple strategies causes confusion and does not allow students to truly master any strategy at all (Staples, 2018). To give our students the best chance of gaining understanding of math concepts, should we not narrow the focus which would allow them to put forth the hard work and practice of a specific strategy/skill that Locke notes is vital to moving someone closer to perfection of a skill (p. 5)? With this being said, wouldn’t teaching a combination of traditional and alternative strategies still allow room for a split in focus, improvement and practice, not allowing them to reach full potential on a specific competency? References: Locke, John. (2017). On the Conduct of Understanding. Jonathan Bennett (Ed). p. 1-10. Staples, D. (2018, September 12). Progress in Alberta math education threatened by new K-4 curriculum. Edmonton Journal. Retrieved from http://edmontonjournal.com/news/local-news/david-staples-new-k-4-math-curriculum-promising-but-has-some-defiencies

Bethany Pilgrim juin 1, 2019, 01:36
Replying to Kelly Milliken

Kelly, I agree with your argument that teaching a variety of strategies can make it more difficult to master a specific individual skill. There is not enough time in the school year to thoroughly teach each strategy, so it is inevitable that something suffers. My close friend teaches junior high mathematics in Nova Scotia, and she feels more and more overwhelmed every year with the new strategies that come down from the school board. I also agree with you Mila when you say that teaching how to tackle a problem from multiple perspectives can be beneficial. However, it is easy to see the other side of that argument, that such teaching can make it easy to give up and move on when one encounters difficulty. I think it is important to teach multiple strategies in order to help students better understand the numbers and the problem. My colleague always argues that the point of showing your work in mathematics is to ensure that you didn’t just luck out this time and can replicate the equation. I think this argument can be applied to teaching multiple mathematics methods. However, perhaps we need to re-evaluate how we teach those methods. I had an excellent math teacher in grade nine. He taught everyone the traditional methods, and for those of us who finished early or really needed more of a challenge, he gave us further instruction on other methods so we could experiment and deepen our understanding. That was the most fulfilling, enriching math class I ever had. Maybe he was on the right track.

Jesse Cardin juin 4, 2019, 01:36
Replying to Bethany Pilgrim

The idea of whether or not to teach a curriculum that is more established or to modernise or replace a curriculum seems to be something that is always at the forefront of systems of education. I find it intriguing and well analysed in your investigation of maths and the introduction to a standard curriculum being used in the U.S. One of the notions of maths that I have held is that it can be categorised as a traditional science and I have noticed that because of that many of the ideas, theories, equations, etc. that have existed and been taught for centuries or even millennia. I wonder if we were to take a look at a field that has more of a modern focus and that is changing at a fast pace, like computer science, how traditional forms of education would compare to the more strategic individualised form that comes from developing critical thinking skills?

Steve Hawkins teacher juin 2, 2019, 20:54

Hi Mila (and Kelly and Bethany), This is already shaping up to be a very interesting discussion, and I'll have more to say on it in a couple of days, once everyone has had their chance to weigh in. I did want to draw attention to an interesting point of contact between Mila's discussion and the discussion of the *other* reading, which deals with parenting. Mila drew attention to the impact on the educational relationship between parent and child at home when there are dramatic changes in content and teaching methods at school. If it is a strain on *teachers* (as Bethany suggests) to adapt to the latest paradigm, it would be asking a lot of parents to do the same. It would be an anxiety-provoking experience for parents to find themselves at a loss to help their 9-year-old with her homework. Those of you who are teaching smaller children: do parents complain that they do not understand (or see the point of) some of the material you teach, or your methods? Do these sorts of concerns affect your choices about whether to assign homework? In general, how good does an innovation in method need to be to counterbalance the costs (strain on teachers, disruption of learning at home) of such changes? Do regular changes in methods improve or undermine general public confidence in schooling (in the way that, say, flip-flopping about the health benefits or risks of eating eggs generates cynicism about the reliability of advice about healthy eating)?

Nihal Jamal juin 10, 2019, 20:54
Replying to Steve Hawkins

Hello all! I actually conducted a survey regarding homework for one of my other courses during my Masters. The theme of my research was measuring the affects of obstacles parents face when helping their children with homework. The survey asked parents to rate from 1-5 (5 being a major obstacle) on some of the more common obstacles suggested by my research. They were the following: 1. Don't have time or too busy with work 2. Too much energy/time spent helping another family member who needs extra support (senior, person with disability) 3. English as a second language 4. Don't have a high school degree 5. don't understand the strategies and methods being taught in school today Since my school is small, I was only able to survey about 30 parents so please keep the small sample size in mind. Only 10% of parents suggested that the methods and strategies being taught in school was the top reason for not being able to help their children with homework. The majority of parent split between being too busy and English being their 2nd language. As a result of this survey, I decided to initiate an ESL course for parents at our school. I'm hoping this will help parents gain the skills to read and understand the homework which will lead to students completing their homework on a more regular basis. Yes, the course will cost the school more money but the parents will pay a tuition fee which will cover majority of the cost. Thank you Nihal Jamal

Steve Hawkins teacher juin 11, 2019, 20:54
Replying to Nihal Jamal

This is a great post, Nihal. Even keeping in mind the small sample size, it's intriguing. In the neighbourhood I used to live in back in Montreal, there was a French-language primary school where the policy was to assign no homework, in part for one of the reasons you give: many of the parents were unable to read French. Thanks for posting this, Nihal. Really interesting.

TS
Terry Stevens juin 3, 2019, 16:50

Hello Mila, Very interesting post and I really like that you brought in the Common Core discussion. I have very little experience with it except for the few topics I have reviewed out of interest. What I found interesting was the points you brought up from Kruger about professional mathematicians and how they do not necessarily follow formulas. I’ve just about finished my first year teaching Euclidean geometry to Grade 9 students which seemed to follow this approach of a deeper understanding of the reasoning behind formulas, rather than just rote memorization. Locke mentions that in studying mathematics a person will see “how necessary it is in reasoning to separate all the different ideas, to see how all the relevant ones are related to one another, and to set aside those that are not relevant to the proposition in hand” (p.10). In this regard Euclid’s Elements is a great resource. Reasoning should be predicated on previous understanding and related and relevant information. Locke advocates for people to undertake learning from various sources in order to broaden their views. In mathematics this can be achieved through the use of a variety of methodologies for solving problems. From what I have seen it can become convoluted if too much is introduced too quickly, without adequate support and revision. From the perspective of parent as the first teacher, this can be difficult if they do not understand. I guess I would have to echo the question that Dr Hawkins asked, are the costs associated worth the potential gains? I do not have an answer. Initially when I started writing this I would have said no but I have reflected on my teaching of Elements and realized that was exactly what I was doing. If I reference back to Plato’s cave, then I would have to said yes. I know the results that can come from this methodology and I feel like I should be helping others to get there as well. Thank you for the great discussion.

Louis-Marc Robitaille juin 4, 2019, 21:38

Hello Mila, I think reasoning skills need to be exercised and applied early for one to reason well. Students in Quebec public schools start learning mathematics in kindergarten which I think is beneficial for them. However, I believe our children need to learn how to socialize at a young age as well. If they focus only on academics, they might have trouble making friends later in their lives. A certain balance between academics and socialization is important for the well being of our children at school. To your argument, I totally agree that teaching different approaches to solving math problems is positive for students. It is central for teachers to diversify their teaching approaches to make sure every student can use their critical thinking skills in the way they are the most comfortable. Have you ever taught your content using games in your classroom?

DC
Dan C. juin 5, 2019, 02:11

Hi Mila! Thanks for your detailed structured and informative post! Locke’s goal is to determine what it is to ‘understand’ something really well. Locke accurately describes the process of logical reasoning leading to ‘cognitive’ understanding. For Locke the judgement is always through our ‘cognitive’ domain. However, we need to unpack what it means to understand and ask if there are different ways of understanding? It is clear that the term understanding has other meanings. For example, it can mean to be ‘sympathetic’ as in - ‘she is a very understanding friend’(Cambridge dictionary). This usage can be explained by what has been identified as our ‘affective’ domain of learning (Bloom et al, 1956). Does Locke fail to recognize our feeling-toned values that are rational and need to be considered and included in many of our judgments? In addition, embodied cognition theory (EBL) postulates that our bodies can shape many of our cognitive experiences (Abrahamson and Lindgren, 2014). For instance, take a class involved in a creative challenge to try and build a bridge structure in the classroom. EBL asserts that the he students are also forming an “embodied sense” of connection; the human value and skill of interpersonal bridge-building that leads to a deeper understanding (Abrahamson and Lindgren, 2014). Likewise a student involved in gymnastics can be seen to also be learning and developing the cognitive idea of ‘balance’ that can then be demonstrated in math education with the study of algebraic equations. In this way, EBL can be seen to include the third domain of learning our Psychomotor domain (Bloom et al, 1956). It’s important to note that Locke is astute to assert the general necessity to consider opposing arguments and their viewpoints when coming to understanding. In philosophy there is the principal of charity which I often refer to which, in summary, states one ought to make the opposing argument as strong and a s rational as possible before evaluating it’s validity (Wilson N.,1959). To address your stated issue in current math education, I am keenly aware of this debate as both a math teacher/specialist and a tutor. I do agree with Locke in the sense that math calculations does build a rigor or what I call ‘the math muscle’ I call that provides students the endurance and concentration needed in higher level problem solving tasks. This is much like an athlete who trains and develops certain muscle strength and memory through drills that can lead to improved performance. However, I propose a third option in the debate. Lambert’s (1990) reconceptualizes and redefines the learning process of ‘coming to know’ math - it involves a ‘zig-zag- approach that gives students the opportunity to “move around in their thinking from observations to generalizations and back to observations and to refute their own ideas and those of their classmates.” In doing so, Lampert (1990) reimagines what counts as knowledge in the math class. Student’s answers to problems are no longer the primary indicator of whether they acquired math knowledge. In Lambert’s (1990) classroom students answer questions about mathematical assumptions, develop logical arguments, remain “open to revision” and engage in conversations and debate as members of a mathematical “community of discourse” (Lampert p.30 1990). For me, this approach introduces and brings a significant degree philosophical thinking and inquiry into the math education that is currently lacking. Lampert provides a detailed example of her approach while teaching a grade six class exponents. It is an excellent and inspiring read. And, I intend on implementing this approach into my classroom next year. Thanks for your work! References: Bloom, B.S. (Ed.). Engelhart, M.D., Furst, E.J., Hill, W.H., Krathwohl, D.R. (1956). Taxonomy of Educational Objectives, Handbook I: The Cognitive Domain. New York: David McKay Co Inc. http://www.nwlink.com/~donclark/hrd/bloom.html Lindgren, R., Abrahamson, D. (2014) Embodiment and Embodied Design Ch. 6 (Saywer, K) The Cambridge Handbook of the Learning Sciences (Cambridge Handbooks in Psychology, Cambridge: Cambridge University Press Lampert M. (1990) When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching American Educational Research Journal, Vol. 27, No. 1, pp. 29-63 https://doi.org/10.3102/00028312027001029 Wilson N. (1959)The Review of Metaphysics Vol. 12, No. 4 (Jun., 1959), pp. 521-539 https://www.jstor.org/stable/20123725?read- now=1&seq=2#page_scan_tab_contents Cambridge Dictionary (n.d.) webpage https://dictionary.cambridge.org/dictionary/english/understanding

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