# Learning theories in practice/Constructivism/Math
### Constructivism… in a Math classroom instruction?

Garelick (2005) provides a brief synopsis of the state of math education in the US in K-12. In his article, he characterizes typical but certainly not all practices in math teaching, focusing on constructivist learning theory and how it is incorporated in a series of textbooks that have far reaching influence.
The objective of this monograph is to contrast my personal teaching practice in a Houston Independent School District (HISD) elementary school against the major contributions of constructivist theorists, their epistemological considerations, and the practice derived from its framework, which are deeply connected and provide structuring forces upon each other (Ainley & Pratt, 2001).
Furthermore, it attempts to demystify the general presumption that constructivism is the law of the day in mathematics teaching in elementary school settings. Such myths seem to be common in the general public beliefs, as well as educators, in regards to the nature of constructivism and its application in US schools’ instruction (Clements, 1997).
As part of my practice as a Math teacher in an elementary school within HISD, I consciously attempt to incorporate some strategies that would be definitively characterized as a constructivist approach to teaching. Following in line with this paper’s objective I will attempt to shed some light on whether or nor it is at all possible to call my teaching constructivist and, by extension, to critique the idea that there is a constructivist reform movement (Andrew, 2007) going on in the current educational system in United States in general, and, in particular, in Houston, Texas.

### Constructivism

Many scholars are considered to be contributors to the development of the constructivist approach. Constructivism, though, is not an instructional theory but rather a learning approach. In this way, it is considered a philosophy frame describing the way humans learn more than a theory in education. Consequently, the descriptions of learning and cognition posited by constructivism do not prescribe a set of applicable teaching methods (Skott, 2004).
Constructivism views learning as an experiential process. It assumes humans can’t know the world as it is because there can’t be proved there is such a thing as objective cogniscence of the outside world. So, there is not outside truth that can be apprehended and, therefore, taught. In contrast, humans compare their learning and arrive to a canonical knowledge via intersubjectivity. Thus, there is not such a thing as shared knowledge. From this concept, constructivism is highly critical of the way children are taught truths fixed to a bank, or the “banking system” in Freire’s words, where teachers impart instruction and children passively absorb knowledge and skills. The typical picture of students seated and a teacher standing lecturing comes to mind immediately.
A central idea within constructivism is that learners play an active role in their own learning process so their interests, prior knowledge and past experiences should be taken into account when deciding the curriculum, the content, the methods and the evaluation of the learning process. They are active learners and there is an organic connection between education and personal experience (Dewey 1938, p. 25, cited by Edna, 1999). As consequence, students make concepts their own by manipulating concrete objects, through hands on activities, by playing with realia and having concrete relationship with their environment that provides the contextual fabric to their learning. In Mathematics, this means solving real life problems by manipulating concrete models that represent a real context to the problem. A typical example within this purpose would be the extensive use of multibase arithmetic blocks (MAB) created by Denis, now known as base 10 blocks (Edna, 1999).
The learning process is focused more on the process itself than on the outcome. Answers are derived as consequence of the process that makes the learner develop knowledge and construct meaning from within. So making sense of the learning object, usually through real life problems, take a central stage in the teaching-learning environment as that of a school classroom.
It is mandatory that verbal interactions be part of the process in which groups of students collaboratively search for solutions to those problems. Speech mediates conceptual development. Students are encouraged to communicate their thinking in this way, combining and sharing their learning via verbal exchanges with colleagues. This tenet of constructivism, central to Vygostky’s idea of learning, states that we learn in contact with others and we never learn in isolation. However, must not be thought that this is the so called discovery learning, in which students take their own way, develop their own thinking and knowledge and arrive to any conclusion they come up with. On the contrary, since constructivism is a learning approach, not an instructional strategy, there is no such a prescription in it (Anderson & Piazza, 1996), and teachers - or facilitators, or guides - play a fundamental role in having learners develop the conceptual framework that is conventionally accepted (Ward, 2001). If 5 groups of 5 could be represented by 5 baskets containing 5 eggs, then the result students must arrive at is 25 and the operation applicable to it could be either 5 added 5 times (5+5+5+5+5+5=25) or 5 times 5 (5 x 5 = 25). This is one of the major misconceptions of constructivism in which it is associated with laizze faire in the classroom and that everybody can do whatever they want and arrive at any conclusion that it seems plausible. This characteristic of the learning process is possible due to the emergent discourse (Leonard, 2000), which occurs when a relaxed from social norms communication happens, in contrast to the rigid top-down institutionalized communication typical of a traditional teaching-student pattern of interaction.
One of the consequences of taking this approach in to a learning setting is that the environment where it is being implemented becomes a dynamic cultural system. Teachers guide the learners’ inquiry to the accepted conventional knowledge within a constructivist cultural system. This is in contrast with the traditional classroom where curriculum, content, methods and strategies are set in advance tightly correlated - and usually within a non-flexible pre-established timeframe.
Students then, become curriculum builders while they work collaboratively with colleagues and teachers to develop their interest, elicit prior knowledge, decide on how to attack the content, and how they are expected to show that they got the content. As Windschitl (1999) puts it, if a class must learn symmetry, one student could choose learn it via symmetry in architecture; another student may choose learn it thorough music and the symmetry manifested in music writing or finding patterns while listening; other student could choose visual arts and patterns in paintings or sculptures while other learner would be learning about symmetry using numbers. Thus, students have content structured only to a moderate degree, and they can contextualize the content in ways that stimulates them. They create the powerful collateral or incidental learning that adds personal meaning to the original theme selected by the teacher. So while everybody’s objective is to make sense of symmetry, each student could take a different process through which make sense of the content, using different themes and concrete realia, and making possible that their performance evaluation be individually suited using authentic assessment.
Another characteristic that would emerge from such a cultural system within this learning environment is that engagements are long-term in the search of, as repeatedly stated above, solutions of real problems. Further, students engage in long-term projects and the development of metacognitive strategies while they reflect on their own learning and the cognitive strategies developed to solve the specific real life problems they help to decide to set up. Efficiency is not a part of this approach. Time management then becomes a secondary issue when it comes to obey a strict relationship between content mastered and time frame. This is in contrast with the standardized curriculum in our current educational systems, which is increasingly being associated with a strict timeframe per topic and subject and standardized tests measuring quantitatively students’ performance.

### Standards for Mathematics teachers and instruction

Even Constructivism is not a new idea, in recent years many different professional organizations have embraced the constructivist philosophy of learning and have recommended changes in the way Mathematics is taught, according to Inch (2002). The author, who is not alone in affirming propositions like this one, cites the National Research Council 1989, Grennon Brooks and Brooks 1993, and the National Council of Teachers of Mathematics (NCTM) 2000. Accordingly, we can assume following Inch, that the recommendations posted by these organizations would assemble a constructivist learning environment in Mathematics. We will focus on NCTM since I am currently implementing many of their propositions in my teaching and I have attended professional development courses that directed me in that direction.
The National Council of Teachers of Mathematics (NCTM) released three documents with the objective of give focus, coherence, and new ideas to efforts to improve Mathematics education in the United States and Canada. Their release of Curriculum and Evaluation Standards for School Mathematics in 1989 articulated goals for teachers and policymakers in Mathematics. In 1991 the NCTM published Professional Standards for Teaching Mathematics, which described the elements of effective mathematics teaching. Assessment Standards for School Mathematics, which appeared in 1995, established objectives against which assessment practices can be measured.
Aiming at revising and evaluating the standards to maintain their relevance NCTM appointed a commission to recommend how they could proceed to keep the standards current and relevant. As a result, the Standards 2000 project was begun in 1997, with the appointment of a Writing Group to produce an updated Standards document and an Electronic Format Group to produce an electronically enhanced version of that document. The exchanges between these two groups and other outside inputs resulted in a book, Principles and Standards for School Mathematics, which is recommended by NCTM as a source to improve mathematics curricula, teaching, and assessment (NCTM, 2007).
A good summarization of the NCTM perspective about how Math teaching should look like and be portrayed is in Lawson (1997). We synthesized her summary in Table 1 and Table 2.
Table 1. NCTM perspective on Curriculum and Evaluation for high quality Math teaching
INSERT TABLE 1
Note: Based on Lawson (1997)
Table 2. NCTM perspective on Professional Standards for high quality Math teaching
INSERT TABLE 2
Note: Based on Lawson (1997)
Additionally, the NCTM's Principles and Standards for School Mathematics (NCTM 2000) included a new process standard that addresses representations (emphasis ours) as a key element in developing conceptual mathematical understanding (Schultz & Water, 2000).
These guidelines have become lighthouses in the direction to which Math teaching has been going in the last decades. Scholars and researchers mention the ongoing revolution in Math teaching (Arthurs, 1999) that follows these precepts, and many people call themselves constructivist teachers, in one hand, without fully understanding its implications and, on the other, they want to be perceived as doing the right thing (Brewer & Daane, 2002). Now in spite of the above Inch’s affirmation that the NCTM recommendations imply a return to constructivism in Mathematics teaching, in fact, such recommendations could be part of any educational theory or pedagogical approach to instruction.
In order to evaluate these propositions as pertaining intrinsically to a constructivist approach to education of Mathematics, I will describe my elementary Math teaching practice in light of the ongoing 'constructivist revolution.'

### My teaching practice

Like any other teacher in this profession, I deal with intrinsic and extrinsic aspects in terms of the professional activity I perform daily in the school setting. Due to time constrains related to any human activity, one must consider the amount of time allocated to any task within his work environment so one can account for the real dedication of the actual professional activity in question. For example, if an accounting employee must answer two hundred emails a day, his accounting tasks will be hinder by the allocation of resources to answering emails.
Likewise, teachers must allocate time and resources to activities extrinsic to the actual learning process to which schools should be fully dedicated to. With the risk of oversimplification, I present figure 1 in which the analogy of a funnel is presented with the intention of graphing the point of contact between teachers and learners and how that point is the funnel tip of many other activities completely extrinsic to the learning process. Form collecting pictures’ money and forms and attending software vendors’ presentations to evaluate the “suitability” of the software for learners’ needs, to allocating time for meetings and paperwork dealing with anything but the learning process of the students. So the time limitations to effectively dedicate all energy and resources to the learning process is undermined by all extrinsic activities teachers are subjected to, and therefore decreasing the concentration of the teacher on planning and leading the learning process.
Let the representation of figure 1 be a contextual frame for the alleged constructivist environment under which Mathematics is currently taught in the United States.
Figure 1. Point of contact between learner and teacher as the vehicle for extrinsic and intrinsic educational activities
INSERT FIGURE 1
In regards to the intrinsic aspects of my professional practice I divide it in planning and instruction.
Planning relates to all activities related to the preparation of lesson plans, content, materials and activities for a particular theme or topic. The Texas law entitles every teacher to planning and preparation through the Planning and preparation time – Texas Education Code Sec. 21.404. During the teacher’s planning periods the district, or the school administration, can require the teacher to engage in no activity other than parent-teacher conferences, evaluating student work, and planning. Teachers must have at least 450 minutes of planning time every two weeks in increments of not less than 45 minutes within the instructional day (Texas Classroom Teachers Association, 2007). I have a hard time recollecting myself having 450 minutes of uninterrupted planning in any given 2-week period in my almost 5 years of teaching. An informal assessment on my colleagues would probably reflect the same impression, with those with over 15 years of experience stating how much the situation has worsen in the last decade. Thus if in order to prepare relevant concrete experiences a teacher must have plenty of time available to actually find the material relevant to the content, my current situation in this regard is far from the ideal. Moreover, if it is desirable to involve other persons that may be part of the real life situation demanded by true constructivism (such as government agency, a company’s personnel, any other organization, etc), we can conclude that the situation obstructs the realization of a constructivist learning experience.
In relation to the instruction, its source is the CLEAR curriculum which follows the mandated TEKS (Texas Essential Knowledge and Skills) established by the TEA (Texas Education Agency). The curriculum is the document where the content and its sequence, the suggested fragmented activities and materials for instruction are placed for the teacher as a “guide” and for the rest of the learning community as a reference. My curriculum is mandated by HISD. Not only the topics and their sequence but also the time allocated for each topic. In figure 2 the reader can see the current 4th grade curriculum for Mathematics for the beginning of the second nine weeks period in the 2007-8 school year. For fractions, the reader can see under the header Time Allocation, there are 13 classes of 90 minutes.
Figure 2. Snapshot of HISD curriculum for fractions in 4th grade
INSERT FIGURE 2
If your students get the learning objectives in that time frame you then move on; if the students don’t get it, you move on and other avenues kick in to have them acquire the knowledge and skills. The administration explicitly directs teachers to follow the scope and sequence of the CLEAR curriculum. Otherwise we could arrive to the standardized testing date without having touched a portion of the curriculum. Here we have the first wall in terms of applying anything constructivist in terms if learning experiences for the children.
This happens in the context of having to comply with a daily schedule in which all the subjects have a time slot which terms must be met. As part of these mandatory time slots, my grade level is departmentalized which means that students must rotate at a pre-established time. Therefore, if we are not done with the instruction at let’s say, 10:00AM, when my class needs to rotate for their Language Arts, Reading and Social Studies blocks, there is nothing I can do to flexibly accommodate my timing to make sure the instruction has make it to completion and, hopefully, to fruition. Meaning, the student, the teacher and the learning experience accommodate themselves to the daily schedule, and not the other way around. This is another point in which constructivism becomes an impossible practice.
The time required by a student to make sense of a topic or an activity is dependent on his or her prior knowledge, interests, and cognitive ability, and not to an obligatory rotation set by a clock, written in a document mandated by the school administration.
In addition to the time and the curriculum, there are the activities I plan for my students. I intent to have them speak to each other as much as possible and have them solve problems that can resemble to real life situations using manipulatives that help them construct models that represent reality. For example, the following recount is how I can introduce the concept of multiplication.
I place boxes with plastic manipulatives on the desks of groups. They are seated in groups of four with mixed levels of ability (Usually I place one “low” student, two “average” students, and one “high” student per group, following instructions from an external consultant who, according to their research, is the better way to have collaboratively groups). In addition to the manipulatives I give them a word problem and instruct them to create a visual model that can be used to explain to the rest of the class the solution to the problem. A typical multiplication problem would be like this:
Mrs. Smith has a garden to plan lettuce. She has 9 rows available to plant her lettuce. After planting all the seeds, she noticed that there are thirteen seeds in each row. How many lettuce plants will she be able to eat if all her seeds sprout and grow in to adults lettuce plants?
So the students create a model of the garden using the manipulatives and then all the class moves around the classroom. We review all different strategies and the thinking behind each model and its respective solution to the problem. Any student can ask questions. I usually guide them to come to the conventional or canonical knowledge. For example that we can add up repeatedly the same number of seeds (columns) as many time as the number of rows. So once we create the notion that we can use repeated addition to solve the problem, we move onto the more abstract aspect of it and introduce multiplication. We will repeat this problem solving activity for several days until the majority of the students understood what we did. I then may introduce multiplication related vocabulary and then the algorithm to multiply rows and columns. I will then provide the students with different color grid paper to create arrays to produce their own multiplication tables. We will use the arrays, shown in Picture 1, to help develop models of multiplication, relate it to polygons’ area, for computation purposes, and to memorize the multiplication tables.
Picture 1. Arrays constructed by students using color grid paper
INSERT PICTURE 1
This account is an oversimplification of the instructional strategy I use. Usually some, or even the majority of the students, have heard of multiplication and some even know a procedure to multiply two numbers (usually a two digit number multiplied by a one digit number). But the main point in me doing this is to have them understand and make sense of that when we have groups of equal quantities the most efficient way to find the answer is by multiplying. Also, to understand that even though we can solve it by addition (repeated addition), multiplication or even division (as a family fact of the multiplication expression) they need to develop the concept that what we did was to multiply groups of equal quantities. So I take my students from a very concrete, concept development activity (solving a problem by means of creating a model using manipulatives), to the most abstract use of an algorithm to perform a calculation. Further I attempt this via activity, reflection, and social interaction what would indicate one of the constructivist’s key elements (Bischoff & Golden, 2003).
One can contrast this strategy with the constructivist approach of real life problems derived from the students’ interests and prior knowledge. The fabrication of a real situation in which someone is planting a garden and needs to know how many lettuce plants she will be able to eat at the end of the season, is completely artificial. Even though I set up the experience based on my knowledge of the students (basically on their prior knowledge of addition, multiplication, solving problems strategy, etc), at no point the students are able to interject their input in the design of the learning experience as a whole. So the idea that students become curriculum builders, such as under the assumption of a constructivist approach, it is non-existent in the example provided above. Let us not forget the contextualization of time constrains and mandated curriculum to put things in perspective.
Another characteristic of my teaching is that I provide the students with different approximations to a unit theme. I usually begin a 10 minute Mental Math session in which I give a simple operation (such as 45 – 18 =) and the students must find in their head the answer and then orally explain to the rest of the class how they found it. I follow their oral explanation by posting on the board what they retell, using Mathematical conventional writing. Usually five or six students get to explain their thinking about how they find their answers to the incomplete number sentence provided. After that, I have them solve other simpler problems that can lead them to make connections between the mental math exercise, the simpler problem, and the word problem or problems that they solve by building a model and then explain to their classmates. One of such simpler problems is The Number of the Day in which they can “build” the specific number (or incomplete number sentence) by drawing, making groups, using words, using any of the four operations we use in 4th grade, number patterns, picture patterns, fractions, liquid containers and their capacity, money bills and coins, etc. By doing this I try to have my students make sense that numbers can be chunked and manipulated in any way we want to, with the purpose of giving real life use to such mathematical manipulations, like the operations we use in the classroom all the time.
One could argue that the discourse I am trying to encourage is emergent as opposite to the top down institutionalized discourse. However, there is no relaxed from social norms communication even though everybody participates in the problem solving and all the oral activities I plan as part of my Math lessons. I plan and mandate the activities and at end I will finally evaluate the students’ performance (whether informally or though a project or using a standardized test-like procedure is, relatively speaking, irrelevant; I will have the final word on the grade they’ll get).
The efficiency logic runs supreme when it comes to making any decision in the classroom. All the activities I planned are subject to my immediate reorientation according to my personal evaluation of the time constrains we have in order to complete the review of specific learning objective. So the moral of this story could be: “you participate as much as you want, but when time presses us, I tell you what to do and how so we can finish up and move on.”
When we have to move on because the curriculum suggested time frame points us to do so (the administration directs teachers to follow the time frame of HISD mandated curriculum), those students that did not get the content will receive remediation. How is it that they are selected to received assistance? At the end of the unit (i.e., multiplication), a common assessment (meaning the same test for the seven 4th grade classes in the school) is given. The test is TAKS like, with 4 answer options per word problem (standardized-objective-like examination).
The common assessment results are read by a scanner and fed into a district computer database. Those students who do not pass the common assessment will be required to be tutored by either the classroom teacher in small groups, or an outside tutor, or a software program, or stay in extended day after class. The decisions regarding the type of help made available to this group of students are put together by the school administration.
Here we can see how the students’ performances are evaluated through means other than authentic assessments that are provided in a constructivist environment. In a real life situation, the students’ performances could be assessed based on their production generated in solving the problems presented by the learning experience, in contrast to the objective type test.

### Discussion

I strived to synthesize the constructivist approach to learning and the standards and suggested approach to teaching Mathematics by NCTM through my personal experience as an elementary school Math teacher.
In lieu of the preceding statement I conclude that being a constructivist is an ideal impossible to put into practice under the current educational system in which I practice my trade. The most important aspects of constructivism cannot take effect because they would go against the most important characteristics of the current educational system. These characteristics relate to a mandated curriculum, and the need of putting efficiency as the motor that runs the system “forward”. Tight time frames linked to topics in the curriculum, suggested segmented activities and their correlation with high stake standardized tests result in a rigid top-down controlled system under which constructivist principles cannot be applied.
This is in spite of all the propaganda advertising the coming of a constructivist wave in which children will choose what to do and will ruin the achievements in accountability registered by the implementation of education law and its programs.