Inquiring Minds.....

I have recorded my thoughts and ideas related to invented strategies in mathematics throughout my inquiry. I used a variety of methods to record these thoughts, ideas, ponderings and queries. I made notes to myself on the articles as I read them, typed some of my ideas as soon as they occurred and others were recorded using whatever was available to me at the time that the idea popped in my head, a post-it note, a loose leaf....or the envelope that the phone bill came in.....join me now as I share my musings, my A-ha moments, my discoveries and my questions.



I tried several methods of organizing my ideas for the purpose of blogging, but did not like the idea of having my first post being the last that was read and I also didn’t like ordering them in reverse for the purpose of the blog, posting my first post last and vice versa, as it did not allow me to continue my blog in a way that I thought would make sense at a later time.....it is my hope that by organizing it in this way that you are able to follow me on my inquiry through the world of invented strategies as I explore what the experts have to say and add my own thoughts on their findings.



Inquiring...

Not really sure where I am going with this yet, but I am narrowing my focus on the use of invented strategies in primary/elementary mathematics....I know I need to narrow my topic and develop a specific question in relation to my inquiry topic, I am hoping that as I delve further into the literature that I will come up with the question that gives further direction to my inquiry....

Inspired to Inquire....

My inspiration comes from the dedication at the beginning of “Mathematics Their Way” by Mary Baratta-Lorton (1995). The dedication reads as follows:

TO CHILDREN, lost in a world of adult symbols which they cannot begin to fathom

and

TO TEACHERS, lost in a world of methods and materials they did not create and in which they no longer have faith.

No magic. No easy answers. But joy and growth and meaningful exhaustion, yes, a plenty!

Here’s to that better way that evolves when we look at learning in their way.

Good luck in your teaching!

- Mary Baratta-Lorton

(Baratta-Lorton, 1995, p. xiii)

I was introduced to this program in 2007 after spending a year struggling with “somebody else’s way” of teaching mathematics, a way that didn’t make sense to me and I feared made even less sense to my students.

With the rollout of the Western and Northern Canadian Protocol (WNCP) mathematics curriculum in 2007 came a series of week-long workshops designed to familiarize teachers with the changes to the curriculum and the changes to the “way” of teaching and thinking about mathematics in Alberta classrooms. At this workshop I was lucky enough to meet a group of primary teachers from Fort McMurray who introduced me to “Mathematics Their Way”(MTW) and invited me to participate in another week-long workshop later that summer, a work-shop that would open my eyes to what had been missing from my mathematics classroom; student’s ideas and voices. MTW helped me see the understanding that can emerge when students are given the time to think about mathematics and explore the connections between mathematics and their world, when students are permitted and encouraged to construct knowledge on their own, when students become active participants in the learning process.

While MTW provided me with a means of structuring my classroom learning experiences so as to promote student construction of meaning and understanding I am by no means intending this blog to promote this resource, I am merely hoping to provide you with insight into where my journey into invented strategies began. This same approach could be implemented using a variety of other materials, activities and resources. The idea which I am trying to promote is that of allowing students to become active participants in the creation of mathematical knowledge and understanding, the development and use of invented strategies, through whatever means.

In future posts I will explore the research relating to the use of invented strategies in mathematics.

Melanie

Baratta-Lorton, M. (1995) Mathematics Their Way – 20th Anniversary Edition. Parsippany, NJ: Dale Seymour Publications.





What is an invented strategy?

Chambers (1996) in his article “Direct Modeling and Invented Strategies: Building on Students’ Informal Strategies” presents several interesting examples of strategies that students used other than standard or traditional algorithms when presented with mathematical problems to solve. The strategies used by students are an indication that they were not bound to rules or procedures but experienced the freedom to solve the problems through the use of whatever materials and strategies they felt necessary or appropriate.

The examples below are all taken directly from the article cited above:

Example 1

The following problem was presented to a group of third grade students

On our hospital field trip we saw 12 emergency rooms.


Each room had 9 beds in it.


How many beds were there?

The standard multiplication algorithm would have been useful in this situation, if the student was familiar with and experienced in its use. Given a lack of exposure to the standard algorithm, the student was able to find an answer (the correct answer) in the following way;

- Pamela used base ten materials and counted out twelve groups of nine blocks. She then regrouped the blocks into groups of ten, replacing each group of ten with a tens block, and replaced the ten tens blocks with a hundred block. In doing this she found the answer to be 108.

Example 2

This problem and solution were taken from a kindergarten class.

A bee has 6 legs.


How many legs do 5 bees have?

- Sean solved this problem by using cubes to represent the bees and their legs. He set out 5 cubes, one to represent each bee and places 6 more cubes with each “bee” to represent the number of legs on each. He then counted only the blocks that represented legs and found the answer to be 30.

- Jeffrey used a number line, starting at 6 because that was the number of legs on one bee, he counted 6 more spaces on the number line and stopped at 12 for two bees, and continued to count by ones using the number line, stopping as he counted 6 legs to keep track of how many bees until he got to 30 legs for 5 bees

- Brianna knew the doubles fact 6+6=12, using her fingers she counted on 6 more to get 18 legs on 3 bees, she repeated this counting on process using her fingers until she reached 30 as the number of legs on 5 bees

The three students used different solutions to arrive at the same answer, yet each found the answer in a way that made sense to them, whether it was through the use of direct-modelling strategies using manipulatives such as blocks or counters or through the use of tools such as number lines of the less sophisticated strategy of counting on one-s fingers. Regardless of the method or strategy used, each student was able to solve the problem, it is unlikely that the same kindergarten students would have applied the standard multiplication algorithm correctly, having not been taught it, they used a method that was reliant more on self-constructed knowledge than teacher directed knowledge.

A shift in strategies from direct-modeling to invented strategies often occurs as numbers become larger (Chambers, 1996) as invented strategies do not involve the use of manipulatives or drawings (Carpenter, Franke, Jacobs, Fennema, & Empson, 1998).

The following examples of invented strategies appear in Chamber’s article.

Max had 46 comic books. For his birthday his father


gave him 37 more comic books. How many comic books


does Max have now?


Solutions:

- Lauren (Grade 2) started at 46 and counted by tens and then by ones 46, 56, 66, 76, 77,78,79,80,81, 82, 83

- Laurel (Grade 2) rounded 37 to 40 and then subtracted the 3 that she added in this step from her final answer, 46+40=86; 86-3=83

- Joel (Grade 1) added the tens and then the ones, 40+30=70, 6+7=13, 70+13=83

All three solutions were developed mentally, students did not use blocks or other manipulatives nor did they record their answers as they worked through the question. Although Joel’s strategy most closely resembled the standard algorithm for addition he began as many students do in invented strategies with the tens digits first.



Chambers (1996) along with pointing out the understanding and construction of knowledge that can occur through the use of invented strategies also points out the confusion that can occur through the use of standard algorithms that unlike invented strategies, do not build on student’s natural thinking. Chambers provides the following example in which a student uses first uses the standard subtraction algorithm to solve the problem presented to her and then uses an invented strategy based on direct-modeling.

Example

How would you do this problem?

 70


-23

Gretchen replies that the question is easy and completes it as follows:

 70

-23

 53

When asked to show the same problem using base-ten blocks she easily finds the correct answer of 47 and expresses confusion over the conflicting answers. When asked which she thinks is correct she selects the written answer, rejecting her own invented strategy using the base-ten blocks. She rejects an answer she has shown to be correct using direct-modeling in favour of an answer she is confused by, this is a remarkable example of the power of standard algorithms (Chambers, 1996), they are in fact so powerful that once this particular student was taught the standard algorithms she believed it to be the “right way” to do subtraction, despite not using it correctly or understanding how she arrived at two different answers.

This example provides some insight into the problems that arise when teaching methods favour the use of standard algorithms. Chambers (1996) suggests that standard algorithms are favoured by teachers as one of their key features is efficiency, but at what cost does this efficiency come? And can they really be considered efficient or effective if students do not use them effectively....more questions to guide my inquiry.



Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in children's multidigit addition and subtraction. Journal for Research in Mathematics Education , 29 (1), 3-20.

Chambers, D. L. (1996). Direct modeling and invented procedures: Building on students' informal strategies. Teaching Children Mathematics , 3, 92-95.





Making Math Make Sense

Carroll and Porter (1997) in their article “Invented strategies can develop meaningful mathematical procedures” point to the understanding and knowledge that can be developed as we encourage students to invent their own strategies as opposed to relying on the use of traditional algorithms. They present the notion that invented strategies are not only beneficial to students but that teachers also benefit from a way of teaching where “more emphasis is placed on discovering and sharing procedures than on memorizing and practicing traditional algorithms” (Carroll & Porter, 1997 p. 370).

It is easy to see how such an approach is more exciting for students and teachers alike. I must admit, my own willingness to embrace and encourage the use of invented strategies stems from my own boredom with textbook instruction and practice of traditional algorithms. I recognize that my boredom and lack of interest transfers to my teaching and my students are unlikely to develop much interest in the topics either. Those students who are able to follow the steps in the algorithm will do so and develop speed and accuracy, and others will continue to struggle with the use of a method that does not make sense. This is because most of us as teachers do not truly understand “why” the algorithms work, having learned these procedures because they produce the correct answers as opposed to because they make sense (Carroll & Porter, 1997). The traditional algorithm for subtraction is presented to illustrate this point. As we work through the calculation of:

112


-89

What does it truly mean to borrow a 1? We go through the motions we have been taught, follow the steps we have been taught and arrive at the answer of 23, but do we really understand how the answer was arrived at?

Students on the other hand when allowed to invent their own strategies arrive at an answer in a way that makes much more sense,

Add 1 to 89 to make 90, add 10 to 90 to get 100 and 12 more to make 112

+1 +10 +12 =23

Students answers tend to focus on the size of the numbers and the relationship between the numbers as opposed to decomposing the number into its individual digits, as is done in the traditional algorithm, (take 9 from 2, can’t do it, borrow a 1 and subtract from 12......doesn’t sound very sensible when you think out loud does it?) (Carroll & Porter, 1997).

Strategies invented by the student on the other hand promote mathematics as being meaningful, as making sense (Carroll & Porter, 1997). Unlike traditional algorithms, the reasons these strategies work can be clearly explained and understood by most students and gives them a variety of methods of solving problems as opposed to using the one size fits all approach provided by traditional algorithms. “Different problems are best solved by different methods. The power of standard written algorithms is that they can be applied to all problems. However, they are not always the best method.” (Carroll & Porter, 1997 p. 371) This is a powerful message in that if we choose to teach only the standard algorithms and ignore the strategies that children invent we may be in fact encouraging them to use inefficient strategies, strategies that they may in fact use incorrectly. To illustrate this idea, Carroll and Porter (1997) provide insight into Fuson’s investigation into how children solved the calculation of 7000-2. Fuson conducted his research in the late 1970’s and found that in a group of above average third and fourth graders none of them were able to calculate the answer correctly because they struggled with the use of the standard algorithm (Carroll & Porter, 1997). Carroll and Porter (1997) found in their interviews that 67 percent of fourth graders in classrooms that encouraged invented strategies were able to correctly calculate the answer by counting down; when they presented the same calculation later in the year the number of students able to correctly calculate the answer grew to 80%.

The strategies presented by students often differ from the standard algorithms, using methods that are more meaningful for the students. I will speak more to this as I share my thoughts on work by Constance Kamii in a later post. Kamii is a name that is encountered often in the study of invented strategies.

How as teachers can we encourage and support our students in the use of invented strategies? Carroll and Porter (1997) suggest the following as essential to this approach:

• Time – students need time to explore their own methods and those of others, time to experiment in a risk-free environment, one that does not emphasize speed or accuracy on the first attempt.

• Materials – manipulatives should be made available for those students who are still at the direct-modelling stage of invented strategies. Though the purchase of commercially available mathematics manipulatives may not be possible for all schools there are many other readily available, inexpensive materials that can be substituted for many of the purchased manipulatives, items such as bread tags, broken golf tees, small pebbles, craft sticks, etc

• Fact Strategies – invented strategies will help students build on the strategies necessary for multi-digit computations, they will not rely on knowledge on facts such as 9x12, but will have efficient strategies for finding the answer, such as decomposing the 12 into 10+2 and finding the answer for 9x10 + 9x2

• Meaningful contexts – problems presented in contexts that are meaningful to children will encourage them to invent strategies that are meaningful. Contexts related to real world application of numbers or situations derived from children’s literature can be used to provide meaningful context for mathematics.

• Sharing – children should be encouraged to share their strategies with others in a variety of ways, allowing them to learn from one another. Those students who struggle to invent their own strategies might find success with the strategy invented by another student.

Carroll and Porter’s (1997) suggestions are not beyond the reach of the primary teacher, they do not require any formal training, expensive materials or expert knowledge, they simply require a willingness on the part of the teacher to shift their instruction from a method of instruction where the emphasis is on skill in computation to one where the emphasis is on inventive ways to compute, where the role of the teacher and students are not so much reversed, but that they are equal, where student ideas and strategies are valued and encouraged.



Carroll, W. M., & Porter, D. (1997). Invented strategies can develop meaningful mathematical procedures. Teaching Children Mathematics , 3, 370-4.





Kamii

As I skimmed the articles I gathered in my research I came across this name quite often, so I realized that it would only make sense for me to read more about her work, about her studies in findings into children’s use of invented strategies.

The article “The Harmful Effects of Algorithms in Grades 1-4” (Kamii & Dominick, 1998) outlines Kamii’s work in schools from 1989-91. Kamii’s research was based on Piaget’s theory of Logico-mathematical knowledge, and that such knowledge is constructed through one’s own actions and mental processes, that is, while we can teach children the steps in the standard algorithms for computations, we cannot ensure that the child understands, that the relationship between the ideas is formed without mental or physical experimentation on the part of the child.

Kamii’s research involved four grades of students as follows:

• Grade 1

o Algorithms were not taught by any of the teachers

• Grade 2

o One of the three teachers involved taught algorithms

o One of the teachers did not teach algorithms, but parents did at home

o One of the three did not teach any algorithms and had convinced parents not to teach them at home

• Grade 3

o Two of the three teachers taught algorithms

• Grade 4

o All four teachers taught algorithms



Kamii termed those classes in which no algorithms were taught “constructivist” classes as opposed to traditional classrooms in which algorithms were taught. The results of this study provided overwhelming evidence in support of not teaching algorithms, in support of allowing students to construct their own strategies to use instead. The number of students who were able to correctly find the correct answers on each of Kamii’s performance tasks was much higher in constructivist classes, and those that did not find the correct answer found incorrect answers that were much more reasonable than the incorrect answers computed by those students using algorithms. Other interesting findings by Kamii included that as students were exposed to algorithms their willingness to attempt an answer decreased, their way of writing answers also differed from those who had not been taught algorithms, they wrote answers such as “8,3,7”, suggesting that students were concerned with the digits in each column as opposed to the number as a whole (Kamii & Dominick, 1998).

The harmful effects of algorithms identified in the article are: “ (1) They encourage children to give up their own thinking, and (2) they “unteach” place value, thereby preventing children from developing number sense” (Kamii & Dominick, 1998, p. 135).

Like other researchers they point to children’s natural tendency to add numbers from left to right as opposed to right to left in traditional algorithms. If children naturally have ways to compute then why is it that we are so concerned with teaching them the traditional algorithms, the algorithms that interfere with their natural sense making, their natural abilities. It seems that again, the idea of teaching algorithms relates to “that’s the way it’s always been done”, in that it is a part of our culture of mathematics and that these methods must be transmitted to our children (Kamii & Dominick, 1998).

Kamii’s research hypothesis was stated as “Children in the primary grades should be able to invent their own arithmetic without the instruction they are now receiving from textbooks and workbooks” (Kamii & Dominick, 1998, p. 132) and the evidence she collected through her research leads to a validation of this hypothesis, she provides evidence that points to the confidence students develop in their mathematical ability and their willingness to attempt computations even when they are initially unsure how to proceed, if confidence and a willingness to take risks in learning can be achieved in a “constructivist” classroom, why then would we not want to encourage this in every classroom, why then do we stick to the textbooks instead of providing students with the valuable practice that is not found in a workbook, but instead found in the creation and use of their own invented strategies.





Kamii, C., & Dominick, A. (1998). The Harmful Effects of Algorithms in Grades 1-4. In The Teaching and Learning of Algorithms in School Mathematics (pp. 130-140). Reston, VA: The National Council of Teachers of Mathematics, Inc.



Unteaching?

In exploring the potential harm caused by the use of standard algorithms before students are ready to understand and make sense of them I came across a paper presented by Doug M. Clarke at the 20th Biennial Conference of the Australian Association of Mathematics Teachers titled “Written algorithms in the primary years: Undoing the ‘good work’?”

Clarke (2005) begins by discussing the widespread teaching and use of written algorithms in primary classrooms and presents several reasons identified by researchers Plunkett, Thompson and Uiskin for the acceptance of written algorithms as necessary content in primary classrooms, I have included four of these reasons that I am most concerned with and my observations related to each.

• Algorithms have been traditional primary mathematics content around the world for many years

o This speaks volumes to me in that we are doing something simply because “that’s the way it’s always been done” rather than examining “WHY” algorithms have been part of primary math content for many years

• Algorithms are automatic, being able to be taught to, and carried out by, someone without having to analyse the underlying basis of the algorithm;

o For those students who are able to remember the steps in the algorithm they are provided with an “easy” way to do things, those who have trouble remembering have no way of deriving the algorithm on their own if they do not fully understand how it works.

• Algorithms are fast, with a direct route to the answer

o Speed becomes the emphasis of mathematics here, as opposed to understanding. In emphasizing speed in computation and mathematics students might come to believe that if they cannot quickly arrive at an answer then such an answer either doesn’t exist or they don’t have the necessary skills or knowledge to come up with the answer on their own.

• For the teacher, algorithms are easy to manage and assess

o This brings me back to our study of ability grouping in Boaler’s (2002) text in which it appeared that ability grouping was used mainly to benefit the teachers. Decisions regarding mathematics instruction should, in my opinion, be made in the best interest of the students, not for the sake of what is easiest for the teacher.

To contrast the reasons given in support of the inclusion of standard written algorithms in primary mathematics are the numerous reasons presented that call for not teaching written algorithms at the primary level, reasons that illustrate the dangers of teaching standard algorithms at this stage; numerous reasons that have been supported by researchers with evidence that clearly define the harmful effects of algorithm instruction in early grades, as opposed to many of the reasons in support of teaching algorithms which appear to be arbitrary.

Identified dangers in teaching algorithms in primary grades are summarised as follows.

• The discrepancy between the way algorithms encourage us to think about numbers as opposed to the way we normally think about them

o In the standard algorithm for addition the ‘4’ in 547 is thought of as ‘4’ and not ‘40’ which it actually represents

• The encouragement to abandon understanding

o Students move from a way of doing mathematics that makes sense to them to a way of following the rules

• The algorithms taught may not be as efficient as those constructed by students

• Students tend to accept incorrect answers that are not reasonable and use algorithms when not necessary

(Clarke, 2005)

If we are to accept student invented strategies instead of teacher taught algorithms in the primary grades, when then, should the standard or traditional algorithms be taught? This is a question that plagues me as I wonder if I do not teach the algorithms will my students miss out on something. Will other teachers assume I simply made the choice to leave out what is often thought to be an “important” part of the curriculum? Will my students ever learn the standard algorithms if I do not teach them?

Clarke (2005) suggests that while the algorithms should not be taught directly in the first five years of school that they may naturally arise during classroom problem solving, as parents and older siblings are likely to teach these methods at home. In this way, the standard algorithms are simply presented as another way of doing things, as opposed to being the only way. This works for me, for my quest to find a better way of doing things, of thinking about math “their” way, if students understand the standard algorithms and are able to use them efficiently then I would certainly allow them to do so, but I think it is unreasonable to expect that standard algorithms be the only way presented to students when they have so many more creative and sensible ways of approaching the mathematics, ways that they naturally develop and use.



Clarke, D. M. (2005). Written algorithms in the primary years: Undoing the 'good work'? In M. Coupland, J. Anderson, & T. Spencer (Ed.), Making Mathematics Vital ( Proceedings of the Twentieth Biennial Conference of the Australian Association of Mathematics Teachers) (pp. 93-98). Adelaide: Australian Association of Mathematics Teachers.



Thoughts from long ago

“The learner should never be told directly how to perform any operation in arithmetic...Nothing gives scholars so much confidence in their own powers and stimulates them so much to use their own efforts as to allow them to pursue their own methods and to encourage them in them.” (Colburn, 1912, p. 463)

The above quote is the concluding thought in Clarke’s (2005) presentation, it speaks volumes to me about the quest to allow students to become effective and efficient users of their own strategies, to have them make sense of mathematics in their own way. I was so taken with this quote that I sought out the original article in which it appeared. I was surprised to find that the quote was taken from a speech delivered by Warren Colburn to the American Institute of Instruction in 1830. I was shocked to read that the ideas presented by Colburn almost 200 years ago are remarkably similar to the ideas being promoted by educational reform today, that the “new” math Colburn speaks of is the same “new” math that we hear of today. I wonder now after reading this article how far we have really come in our instruction of mathematics, how much have we progressed? Why are we still discussing ideas that were brought up 200 years ago, ideas that still are not in full implementation in our classrooms?

Join me as I journey to the past, to unlock the secrets to “new math.”

I bring to your attention the “old system” of teaching mathematics as defined by Colburn, a system in which students are presented with rules to follow, procedures to perform that result in correct answers, procedures and rules that produce answers which the student has no way of knowing are correct or reasonable, and no idea as to why they have followed a particular rule or procedure, other than because the book or teacher directed them to do so (Monroe & Colburn, 1912). In contrast to the old system is the “new system”, a system that encourages students to think and reason for themselves, to work with numbers in a way that makes sense beginning with simpler concepts and using this knowledge to form the basis for more complex computations, to use one’s own knowledge and abilities to develop understanding; understanding that once realized, will make the learner unwilling to settle for anything which they do not understand, for ideas that are not their own (Monroe & Colburn, 1912).

Nearly 200 years ago Colburn was proposing that we allow students the opportunity to develop their own methods and strategies in mathematics, yet our classrooms today are still dominated by textbooks and repeated practice of teacher taught algorithms. I hope that this in an indication of teachers at a loss for a better way, of teachers unaware of the “harmful” effects of such algorithms, unaware of the understanding and methods that develop when students are given the chance to make sense of mathematics in their way.





Government of Newfoundland & Labrador - Department of Education. (2008). Grade 1 Mathematics - Curriculum Guide.

Monroe, W. S., & Colburn, W. (1912). Warren Colburn on the teaching of arithmetic together with an analysis of his arithmetic texts. The Elementary School Teacher , 463-480.

Final Thoughts
As my time for inquiring draws to a close I am left with questions, yet I am also left with hope, hope that others will see the benefits of invented strategies and allow their students to experiment and create methods in mathematics, hope that it won’t take 200 more years for us to implement a “new” way of doing things, hope that students voices will be heard and ideas valued, that they will see mathematics as making sense and being useful, hope.....

Melanie