Author Archives: vivien kamhoua

VISUALISING FRACTIONS OPERATIONS

Misconceptions in mathematics are common, and are usually misunderstanding and/or misinterpretation based on incorrect meanings. Knowing the nature of a misconception and its source would help to fathom ways of planning appropriate instruction that is beneficial to students. A likely misconception to see in my math class is   This misconception could be associated with students transitioning from operations with whole numbers to operations with fractions because, to them, the rules have changed. Because of previous “knowledge”, the students performed operations distinctly, considering the numerators and the denominators in separate operation. In order to teach additions and subtraction involving fractions, it is important to impress upon the students that the numerator indicates the number of parts and the denominator indicates the types of part (the whole). The computational rule usually don’t help students think about the operations and what they mean. When the students are only armed with rules, even when the rules are mastered, it is quickly lost in the short term as the myriad of rules soon become meaningless when mixed together (On complex operations).

I am using the 3-step T-GEM cycle to create activities that cover operations involving fractions.

To generate relationship: the students will begin with simple contextual tasks. The students will challenge themselves with an interactive activity on Phet Interactive Simulations website. The goal is to enable them to find matching fractions using numbers and pictures, to make the same fractions using different numbers, to match fractions in different picture patterns, to compare fractions using numbers and patterns. Here is a link to the activity.

To evaluate the relationship: at this point the students are expected to have a conceptual understanding of fractions that is the numerator indicates the number of parts and the denominator indicates the types of part (the whole). The students will engage in operations involving fractions. Because the students usually handle addition better than subtraction, they will start with addition, adding fraction with same denominator. The vast majority of my students can tell that a half plus a half is one or one whole. I will write down the question and apply their general strategy (the misconception stated earlier), then interesting questions will raised.

This example will shake the misconception. The students will be encouraged to connect the meaning of fraction computation with whole number computation such as ¼ is one part out of four equal parts of a whole. In order to practise addition of fractions with same and different denominators, the students will use the interactive activity “Fraction wall” . The activity illustrates the addition of fractions using array technique. The students will explore similar array techniques to perform subtraction. An illustration of using array technique is

The aim of this model is to help the students learn to think about fraction and the operation, to contribute to mental methods, and provide a useful background when they eventually learn the standard algorithms of LCM.

To modify the relationship: the students will reflect on the work by using an interactive activity that allow them to visualise an array of fractions, compare, add, subtract, and multiply two fractions with animations providing a conceptual understanding over the use of LCM to perform fractions operations.  Here is a link to the activity.

 

Reference:

Khan, S. (2011). New pedagogies on teaching science with computer simulations. Journal of Science Education and Technology, 20(3), 215-232.

 

 

Knowledge construction in mathematics

When it comes to knowledge construction, the theory of constructivism perfectly resonates to me. I share the view that knowledge is not a thing that can be simply given by the teacher at the front of the classroom to students in their desks. Rather, knowledge is constructed by learners through an active, mental process of development; learners are the builders and creators of meaning and knowledge. In a math classroom, in the most general sense, knowledge is constructed when students are encouraged to use active techniques, that is experiments, real-world situation, and problems solving, to create more knowledge and then to reflect on and talk about what they are doing and how their understanding is changing. David Tall (2004) introduced the notion of the three different worlds of mathematics, that is (1) conceptual – embodied world, (2) proceptual – symbolic world, and (3) axiomatic – formal world. Tall (2004) came to this conclusion after he found that there are three fundamentally different ways of operation in mathematics, one through physical and mental embodiment, including action and the use of visual and other senses, a second through the use of mathematical symbols that operates as process and concept in arithmetic, algebra and symbolic calculus, and a third using formal language in increasingly sophisticated formal mathematics in advanced mathematical thinking. This offers a useful categorization for different kinds of mathematical context encounter by the students in mathematics. Each category has its own individual style of cognitive process and together they cover a wide range of mathematical activity. Also, in each category we usually have abstract and concrete topics. The concept of abstract and concrete usually depends on the cognitive level of the students. The knowledge is constructed when the student’s understanding transitions from abstract to concrete objects. Sfard (1991) defines interiorization, condensation, and reification as the necessary phases to transition from abstract to concrete knowledge.

Scheiner (2016) argues that a concrete object is an object for which an individual has established rich representations and several ways of interacting with, as well as connections between it and other objects. He suggests two interesting ways through which the students can construct mathematical knowledge, (1) abstraction-from-action: that is the students first learn processes and procedures for solving problems in a particular domain and later extract domain-specific concepts through reflection on actions on known objects; (2) abstraction-from-object: that is the students are first faced with specific object that fall under a particular concept and acquire the meaningful components of the concept through studying the underlying mathematical structure of the objects. These ways of knowledge can be apply to Tall’s (2004) three different worlds of mathematics (conceptual, symbolic, and formal world).

In general, the students would construct knowledge in math when they actively engage in learning, practicing and reflecting on their understanding. They generate knowledge when they individually reflect upon their experiences and their ideas. The social context is also important for knowledge construction in math. The students by sharing their experiences and their ideas with others, modify their knowledge, altering concepts that need to be altered and rejecting concepts that are not correct. I agree with Scheiner (2016) when he says that the learner must make sense of the mathematical concept through restructuring knowledge structures that are built on previously constructed knowledge pieces. In fact, because the students usually make sense of the mathematics they learn by connecting them with the mathematics the already know. The mathematics curriculum is vertically aligned in such a way that knowledge is gradually added onto concepts as the students are progressing in their learning.

 

References

Scheiner, T. (2016). New light on old horizon: Constructing mathematical concepts, underlying abstraction processes, and sense making strategies. Educational Studies in Mathematics, 91(2), 165-183.

Bryant, P., & Nunes, T. (Eds.). (2016). Learning and teaching mathematics: An international perspective. Psychology Press.

Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5-24.

Embodied learning with interactive math games

(Merleau-Ponty, 2004) argues that our engagement with the world is not just cognitive or theoretical, but involves the emotional, practical, aesthetic and so on. He said that human experiences connect strongly with the notion that learning involves the exploration of the world from where one is and a clear understanding of how things relate to each other and to ourselves in the world. This resonate to me but does not give me answers on how we actually learn. I have read various theories of learning, such as behaviourism, constructivism, cognitivism and other teaching and learning empirical findings, all very interesting and describing different learning experiences. Also, the use of technology has been very helpful in designing a learning experience that the students can understand meaningfully.

However, regardless of the type of the instruction designed in classroom. In general, I always have different level and quality of understanding in class. Contemporary studies on education show that students have different and multiple intelligences, they perceive the world around them differently and therefore learn differently. Mathematics can be represented and perceived in various ways, and how the students come to learn math is equally varied and diverse. I use technology to provide the students with comprehensive activities through which they can experience learning and meaningfully grasp and understand the concepts being taught. It remains that some students learn better than others. Why do some learn better than others? Is this related to the instructional design or to an intrinsic inert ability? Check out this video.

I watched this video where students in a primary school are playing an interactive math game on the floor. I understand that students easily get and stay engaged with embodied learning. However, I am not seeing in this video any cognitive learning pertaining to the body taking place. Is embodied learning only about engaging students?

Reference

Stolz, S. A. (2015). Embodied learning. Educational philosophy and theory, 47(5), 474-487.

The deal between anchored instruction, WISE, LfU, and T-GEM

A goal of anchoring instruction is to foster students engagement in learning activities in which they are actively involved in the construction of their own knowledge through exploration. In the reading I have done on anchored instruction I noticed that its major goal is to support students in building relationships and connections with prior knowledge while constructing new knowledge. The students develop the ability and skills to identify, define, and solve their own problems. In regards to WISE, it is an online science inquiry learning environment that supports deep understanding with features such as (1) observation – where the students make observation of authentic artifacts anchored in authentic situations, (2) interpretation and construction – where the students construct interpretations of observations and construct arguments for the validity of their interpretations, (3) contextualisation – the students access background and contextual materials of various sorts to aid interpretation and argumentation, (4) cognitive apprenticeship – the students serve as apprentices to master observation, interpretation and contextualisation, (5) collaboration – the students collaborate in observation, interpretation and conceptualisation, (6) multiple interpretations – the students gain cognitive flexibility by being exposed to multiple interpretations, (7) multiple manifestations – the students gain transferability by seeing multiple manifestations of the same interpretations.

In regards to LfU, the goal of this model of instructional design is to embed instruction in activities that facilitate knowledge construction. The model is based on the principle that the structure of knowledge continuously changes while experiencing and exploring of new concepts. T-GEM has a straightforward framework where the students engage into a cyclical reflection process that is generate, evaluate, and modify knowledge. Over this process, they naturally acquire new knowledge that anchored their prior knowledge.

I have noticed that the principle into practice is different for each of these technology enhanced learning tools. However, the theory of learning elaborated in their framework revolves around constructivism. All these four tech enhanced learning tools help the student in constructing their own understanding and knowledge of the world, through experiencing things and reflecting on those experiences. They allow the students to reconcile everything new that they encounter with their previous ideas and experience, maybe changing what they believe, or maybe discarding the new information as irrelevant. In Mathematics, these instructional tools are useful in creating authentic tasks that engage students into active learning. I find anchored instruction, WISE, LfU, and T-GEM very helpful to design instruction in math in a way that address students’ preexisting conceptions and help them to build on them. There are also very helpful in creating activities that constantly assess students understanding in math and develop increasingly strong abilities to integrate new information.

References:

Petra, S. F., Jaidin, J. H., Perera, J. Q., & Linn, M. (2016). Supporting students to become autonomous learners: the role of web-based learning. The International Journal of Information and Learning Technology, 33(4), 263-275.

Edelson, D. C. (2001). Learning‐for‐use: A framework for the design of technology‐supported inquiry activities. Journal of Research in Science teaching, 38(3), 355-385.

Serafino, K., & Cicchelli, T. (2003). Cognitive theories, prior knowledge, and anchored instruction on mathematical problem solving and transfer. Education and Urban Society, 36(1), 79-93.

Misconceptions over combining non-like terms.

One of the most common misconceptions for students new to algebra occurs in simplifying and evaluating algebraic expressions, that is the idea that 5x+4 equals 9x. It is understandable how students might simplify this expression based on their prior knowledge of adding integers, but once again they fail to recognise x as a variable term. Since the second term only has a numeric part, 5x and 4 are not like terms, and so they cannot be combined into a single term. Even higher-level algebra students make this mistake, it is not uncommon to see the expression 3x2+4x3 simplified to 7x5 rather than recognising that these terms cannot be combined since they have different exponents. I came across a simulation on PhET that would help to addressing this issue. Following is a suggested pedagogical approach using a graphic that depict the cyclical nature of GEM and its phases,

Generate relationship – the students will generate pictorial representation of algebraic expressions using the simulation on PhET. They will use different representation such as circles, triangles, and coins to generate graphical expressions. The aim is to have them categorise items and generate expressions such as . Then generate the relationship between the variables in occurrence bananas and apples in this expression, the students should generate an idea on how to simplify it to .

Evaluate relationship – the students will replace variables in pictorial representations with variables such as x, y, or z. They will then evaluate the expression, comparing their results with the pictorial representation. They are expected to produce x+y+y+y+x+x+x equals 4x+3y and establish an analogy with what they have done in the previous step.

Modify relationships – the students are also encouraged to substitute a random value in for any variables. In this case, x will have a random value, and y will have a different random value. They will evaluate and see whether the algebraic expressions are equivalent or not.

Reference:

Khan, S. (2011). New pedagogies on teaching science with computer simulations. Journal of Science Education and Technology, 20(3), 215-232.

 

 

 

Breakthrough in LfU

The LfU principles of learning resonate to me and constitute another apparatus to anchor instructions. I particularly like the description of the three-steps process of the LfU model, that is motivation, knowledge construction, and knowledge refinement. I used these to create focus points of interest to designing an activity in math.

To foster motivation, the LfU approaches in creating demand and eliciting curiosity are interesting. My strategy at this step is that the activity should build upon students’ existent knowledge, and be just challenging enough to help the students to develop new knowledge by building on those that has already been established. Making sure to stay in the zone of proximal development (ZPD) as defined by Lev Vygotsky. The students will most likely remain motivated if the information in the activity is within their ZPD and represents the next logical step in their knowledge construction. Also, the activity should be constituted of authentic tasks that would create a natural demand of the knowledge to be used.

To foster knowledge construction, the LfU model that is observation through firsthand experience, and reception through communication with others once again resonates with Vygotsky’s approach of ZDP. My strategy here will be to design experience through which the students would collect relevant data for concept formation, identifying what they know, what they don’t and/or need to know. Also, open-ended discussions and research should guide the students in constructing and consolidating new knowledge. Open-ended because students have their unique experience to world. Some constructivist may argue that there could be no ultimate, shared reality (Duffy & Jonassen, 1992) and only has best “description”. And the best description is developed through collaboration and communication (Vygotsky, 1978). So, the objectives of the activity should not be predetermined and contents bounded.

To foster knowledge refinement, the students should reflect upon the inquiry process that contributed to their knowledge construction. During the activity, the students should acquire new knowledge, interact with their peers and face with ideas, explanations and information that are inconsistent with, or contradict, their prior knowledge and beliefs. Designing an activity that would confront these inconsistencies and contradictions will challenge students’ current cognition and reorganize their knowledge structure.

I would like to design such activity for each learning objective I teach in mathematics, but I don’t think that this is realistic. This brings me to the open issue “Although horserace comparative evaluation of instructional approaches is difficult in education, it is important to engage in summary evaluation that can start to quantify the effectiveness of LfU activities at achieving content and process objectives in terms of the time and resources used.” (Edelson, 2000). When I compare the mathematical contents I need to cover in one academic year for each Grade level, I personally don’t think that it possible to cover all contents in classroom with the allocated time. Technology could potentially help to do so if the students work off of class time. I am thinking of the pedagogical approaches of flipped learning.

 

References:

Edelson, D. C. (2001). Learning‐for‐use: A framework for the design of technology‐supported inquiry activities. Journal of Research in Science teaching, 38(3), 355-385.

Sparks, K. E., & Simonson, M. (1999). Proceedings of Selected Research and Development Papers Presented at the National Convention of the Association for Educational Communications and Technology [AECT](21st, Houston, Texas, February 10-14, 1999).

How wise are parabolas?

The scaffolded knowledge integration framework developed in WISE is a good guideline for designing a technology integrated instruction in math. (1) making thinking visible, (2) making science accessible, (3) helping students learn from each other, and (4) promoting lifelong learning are the general guidelines in creating a WISE. I like the website WISE of the University of California, Berkeley. It has good features, here is a link to the website. I haven’t played enough with this website to familiarise with all its features. However, I noticed that once a student starts a project, he or she cannot go back to make any change on an activity he or she already completed. This kind of features is suitable for assessing rather than learning.

I customised a project to help students in middle school appreciate the power, beauty and utility of their own knowledge of quadratic equations. From a constructivist approach to learning that is we care about learning those things that hold meaning for us, I chose to approach the concept of quadratic equations using some examples of parabola in real world. In the lesson, I aim to engage the students in structured experiences designed to support accurate and meaningful knowledge construction. The students misconception over this topic is that parabolas are confined to mathematics textbooks and have no real-life applications. This project will help the students to visually recognise parabolic forms in photographs and architectural landmarks, to practice and refine internet-based research methodologies, to develop an increased appreciation of the utility of algebra in the world around them.

Activity #1: what do you know about parabola?

All germane response that the students give about parabola will be write down.

Activity #2: Examples of parabolas in Architecture – student suggestions

Activity #3: I will guide the students’ research of some examples of parabolas in Architecture

Useful examples from modern architecture appear on Santiago Calatrava’s website.

Activity #4: research and identifying parabolas

After identifying parabolas and other geometrics forms in multiple examples of architecture, students are given the mission to search the internet for interesting examples of parabolic form, that are not “slanted”, as they will self-generate equations using Desmos.

Activity #5: Self-generate equations using Desmos.

In this activity, the students will use the transformation formula over quadratic functions (y = a(x-h)2+k) to self-generate quadratic equations. The student will be directed to a file on Desmos, here is a link Ski Jump. They will need to use the sliders to find the values of ‘a’, ‘h’, and ‘k’ to fit a quadratic equation onto the skiers. They will describe how they got their function to match the path of the athlete.

Activity #6: The students will self-generate equations of their own picture from activity #4

 

References

Linn, M. C., Clark, D., & Slotta, J. D. (2003). WISE design for knowledge integration. Science education, 87(4), 517-538.

Linn, M. C., Slotta, J. D., Terashima, H., Stone, E., & Madhok, J. (2010, December). Designing science instruction using the web-based inquiry science environment (WISE). In Asia-Pacific Forum on Science Learning and Teaching (Vol. 11, No. 2, pp. 1-23). The Education University of Hong Kong, Department of Science and Environmental Studies.

Creating a digital video that anchored instruction

The philosophy behind the anchored instruction is to contextualize instruction, creating an environment where learning happens through experiences, ones that connect and embed any information to a problem solving procedure. So that, students would remember the information when they encounter similar problems. The argument is that, when students learn new information to solve problems that are contextual and meaningful to them, they assimilate the information to the tools for problem solving rather than facts for problem solving. Dewey (1993) on the importance of viewing knowledge as tools, noted that when people learn about a tool, they learn what it is and when and how to use it. This analysis on cognition is new to me. The major goal of anchoring instruction is to facilitate a spontaneous recall of knowledge in a problem-solving context. Aside from remembering the information as tools to solve a problem, I wonder if  students could also effectively use these tools to solve newly identified problems. Although, two problems can be contextually similar in mathematics, the approach and information needed to solve them could be slightly different. Therefore, the problem solving techniques would be different, though the same tools could be considered when solving the problems.

Whitehead (1929) argued that there is a knowledge that can usually be recalled when people are explicitly asked to do so but is not used spontaneously in problem solving even though it is relevant. He called this inert knowledge. This has made me reflect on some mathematics I sometimes teach and that are mainly mechanical and lack contextual inputs. As teacher and learner, I recognise myself in Sherwood, Kinzer, Hasselbring, and Bransford’s (1987) illustration of inert knowledge. When asked questions related to their knowledge of logarithms and their understanding of the use of logarithms, entering college students responded that they remembered learning them in school but they thought of them only as math exercises performed to find answers to logarithm problems. Unfortunately, I still have some topics in math that I teach the mechanics only. I guess if the same question was asked of my students in college, they would probably have a similar answer.

Contemporary research on education shows that learning that is contextual and meaningful to a student can develop his or her critical thinking skills and increase his or her capacity of retention. The principles of anchored instruction will be helpful in designing an instruction that fosters students’ engagement through contextual and meaningful concepts. Recently, I experienced the positive impact that a video can have on my students’ learning experience. For the purpose of elaborating on possible features embedded in videos that would contribute in anchoring instruction, I will use the video I posted last week on ‘Design a TELE’.

This video helps to introduce the concept of exponential functions. I wanted my students to recognise the importance of rate factors such as growth and decay factors of exponential functions in real life situations. I wanted them to be able to describe something really important and contextual about exponential functions. I wanted them to be able to interpret the information given in the video for themselves without having to be told how to make the connection between exponential function features and the situation described in the video. I must admit that the outcome of my students’ learning experience from this video went beyond my expectations. I believe the reason for this is that the situation described in the video was part of my students’ everyday experience. The information in the video made sense to them and it was easily connected to a daily experience. Experiences that are lived through different perceptions have different connotation and interpretation. Because of this, the students explored the same video from multiple perspectives. I think without intending to do so, I designed an activity where the students learned to think mathematically without using a textbook’s guidelines or questions. This was an authentic activity through which the students developed knowledge and understanding over an everyday reality in their culture.

A reflection on mathematical cognition in everyday settings leads to think that an apprenticeship developing consistent critical thinking over problems that present an evolving level of challenge has the chance to significantly help students’ reflection on the types of skills and concepts necessary to deal with everyday problems. Digital videos that elaborate on authentic tasks with evolving levels of difficulty can have a positive impact on students’ knowledge and skills, and learning experience in mathematics. However, though digital videos coupled with anchored instruction facilitate retention, I don’t think it always facilitate learning transfer. Remembering tools to solve a mathematical problem do not mean that we can solve any problem involving similar mathematical concepts. It takes hours of practices and experience in a field to becoming expert. Digital videos that emphasis on analyzing similarities and differences among mathematical problems and on bridging new areas of math application will facilitate the degree to which transfer occurs. But The students do not have that much time to spend on a particular topic in order to build similar problem solving skills as experts.

 

Reference:

SITUATED, I. R. T. (2000). ANCHORED INSTRUCTION AND ITS RELATIONSHIP TO SITUATED COGNITION. Psychology of Education: Pupils and learning, 1(5), 231.

Barron, B., & Kantor, R. J. (1993). Tools to enhance math education: the Jasper series. Communications of the ACM, 36(5), 52-54.

Bransford, J. D., Sherwood, R. D., Hasselbring, T. S., Kinzer, C. K., & Williams, S. M. (1990). Anchored instruction: Why we need it and how technology can help. Cognition, education, and multimedia: Exploring ideas in high technology, 12, 1.

Shyu, H. Y. C. (2000). Using video‐based anchored instruction to enhance learning: Taiwan’s experience. British Journal of Educational Technology, 31(1), 57-69.

My experience of PCK and TPACK

PCK and TPACK are usually acquired through experience, teacher’s schools do not provide an extensive training on these. I think the reason for this is that we need a specialist in mathematics and pedagogy to deliver extensive PCK lessons in math, and we do not always have such teachers in teacher training schools. So, during their training, the teachers are usually limited to broad PCK concepts from specialist either in math or in pedagogy. The lack of specialists in both PK and CK, and in TK, PK and CK is the reason why we usually do not have extensive trainings in PCK and TPACK in teacher’s education. I have personally acquired substantial PCK and TPACK over years of experience in teaching. An exposure to students’ misconceptions and immersion into technology versed environments has been very helpful in this process.

Geogebra and Desmos are two software that changed the way I teach functions and transformations in mathematics. The transformation of functions and graphs has become easy to teach and easy for the students to understand because the dynamic character of the graphs allow to visualise the changes on the graphs as parameters in the functions change. The lessons over functions transformations and graphs became more accessible to the students because, when the parameters are introduced in a function, the students are able to see and test all the dynamic changes on the graph of the function. Before, I used to draw static graphs and show how a graph would transform or translate into another one, using two images, the initial graph and its image after the transformation or translation occurs. Whereas with Desmos or Geogebra, the initial graph will move till its image, and all the changes during the transformation or translation can be seen. This facilitates student’s conceptualisation and understanding of functions transformations and graphs.

 

Here below is an example with sinusoidal functions on Desmos.

Vygotsky Theory of Learning for math and science

I think of technology as a way to alter the ‘natural world’ with the goal of fulfilling ‘needs and desires.’ Something much bigger and complex than a single device or site. We use it in math and science classroom to understand principles and strategies needed to develop solutions and achieve goals. Designers of learning experience should strive to implement instructions that reach and challenge students’ socio-cultural and historical knowledge and understanding. Understanding occurs when students meaningfully contextualise and conceptualise what they are learning. According to Vygotsky (1978, 1986, 1997; Vygotsky &Luria, 1993) concept formation is an ongoing interaction between the concrete dimension (socio-cultural-historical experience) and the abstract dimension (math and science concepts in this context) that does not require that the child reinvent information, and does not expect the child to form abstract conceptualisations without first having engaged in concrete activities that support the formation of mental models.

I would design a technology-enhanced learning experience where technology would be used to meet goals around developing strategies to achieve what I think is important for concept formation in math and science, that are differentiated learning, personalisation and customisation, depth and complexity, digital citizenship, collaboration, flipped teaching research, and organisation.

 

Reference

Charnitski, C. W., & Harvey, F. A. (1999). Integrating Science and Mathematics Curricula Using Computer Mediated Communications: A Vygotskian Perspective.