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Published on March 18, 2014;
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Published on March 18, 2014 by Mario Pineda, Ph.D.
at http://www.drpineda.ca/challenges-in-mathematics-education.html

If I'm having to remember ..., then I'm not working on mathematics. - Hewitt (1999)

Someone once said that mathematics is not a spectator sport. Mathematics is unique among the subjects studied in school in that, other than its established conventions, mastery requires a thorough understanding of the topic while at the same time requiring a minimum of memorization (Hewitt, 1999). In the light of this it is curious that a common belief among the general public seems to be that the successful study of mathematics requires "old school" rote learning and the memorization of mathematical facts, e.g. the multiplication tables (Tran-Davies, 2014). The idea behind this approach rests on the assumption that it is beneficial to be able to rapidly recall mathematical facts when solving mathematical problems. While, in principle, this is correct the interpretation of what this means in terms of how mathematics should be taught is often erroneous. For example, currently there is a movement in Alberta (and elsewhere) towards discrediting 21st century learning strategies in mathematics educations, which emphasize understanding and hands-on problem solving, in favour of traditional rote learning and extensive memorization (Eberts, 2014; Tran-Davies, 2014).

The purpose of this post is to present and analyse a number of challenging scenarios from the mathematics classroom that would benefit from a 21st century approach for teaching mathematics. The scenarios are based on actual events I have encountered in the classrooms, either as a teacher or a a student. I have modified any identifying details, e.g. time and locations, grade, gender, names and the exact nature of the challenge.

- Scenario:
- As Ms. Fermat was introducing her grade 3 class to multiplication she described the following approach to her class. If you are multiplying five times an even number; halve the number you are multiplying by and place a zero after the number. For example, 5 x 6, half of 6 is 3, add a zero for an answer of 30. If you are multiplying five times an odd number; subtract one from the number you are multiplying, then halve that number and place five after the resulting number. For example, 5 x 7, subtract one from 7 to obtain 6, half of 6 is 3, add a five to obtain the answer of 35.
- Analysis:
- Whether this multiplication trick actually would make it simpler or faster to perform simple mental multiplications is dubious. A far more serious and insidious objection towards introducing mathematical methods such as this to students when teaching fundamental mathematical concepts is that it provides no understanding into how the concept (in this case multiplication) works nor does it convey useful insights that students could generalize when solving similar problems. While this algorithm may make sense for someone well versed in multiplication, for a grade 3 student recently introduced to multiplication, however, this trick will be indistinguishable from conjuring magic.
- Modifications:
- Students can be asked to use mental mental math skills to come up with their own tricks for multiplying numbers, similarly to the mental math strategies they learned when doing addition and subtraction. For example, the following mental math strategy can be used for solving 5 x 6:
- Double 5 to 10,
- we know that skip-counting 6 times by 10 gives us 60,
- since we doubled 5 we now need to halve the 60 to get the right answer,
- hence 30.

- Scenario:
- Mr. Fermat is introducing Ohms Law in his grade 9 science class. Ohms Law describes the relationship between voltage (\(V\)), current (\(I\)) and resistance (\(R\)) in DC electrical circuits and is given by \(V=I \times R\) . To help his students Mr. Fermat introduces them to the equation triangle for Ohms Law (aka Ohms Law Triangle) where the arrangement of the variables in the triangle provides a visual mnemonic for "solving" the Ohms Law formula for different unknown variables.
For example, to solve Ohms Law for the voltage (left most panel below) one would perform the mathematical operations involved between the remaining two variables to (i.e. multiplying I and R). - Analysis:
- By providing an equation triangle for Ohm's Law Mr. Fermat is not doing his students any favours and he is certainly not helping them to increase their understanding of physics, mathematics, or the close connection these two subjects have. In Alberta grade 8 students are introduced to solving simple linear equations (Education Alberta, Program of Study) that are identical to Ohms Law. Providing this type of visual mnemonic (there are others in both physics and chemistry, e.g. the density triangle, force and motion triangle, wave equation triangle, mole-concentration-volume triangle, etc.) misses an opportunity to connect new material covered in science class to mathematics that the students already are familiar with.
- Modifications:
- As a mnemonic representation of simple equations with only three variables, equation triangles provides no benefit over algebraic manipulation of the actual equation. These "aids" have no place in classrooms where learning, understanding and thinking is promoted.

These case studies illustrate the antithesis of a concept-rich instruction aimed at increasing the students' understanding and developing problem solving skills (Ben-Hur, 2006). Instead of having students develop useful and generalizable strategies and skills student are provided with "rules without reason" (Skemp, 2006). What is the nature of the teaching process and what are some characteristics that might make mathematics teachers' role more challenging in these scenarios? In the grade 3 class where the multiplication trick was introduced the Provincial Achievement Tests (PATs) were taking place later during the school year and, perhaps as a result of this, the focus of the mathematics instruction was on preparing the students for these high-stakes tests. These tests are timed (students have 60 minutes to answer 40 questions) and throughout the school year there was an emphasis on an instrumental understanding of mathematics often with problem solving under time constraints, exemplified by e.g. the introduction of the multiplication trick. As it turns out this particular school has historically performed exceptionally well on both the grade 3 and 6 PATs in mathematics suggesting that rote learning and use of algorithms for solving problem does indeed improve success on tests. While research confirms that an instrumental understanding of mathematical concepts is correlated with success on standardized test, e.g.

If what is wanted is a page of right answers, instrumental mathematics can provide this more quickly and easily. (Skemp, 2006)the problem is, however, that real life is not a standardized test and this approach does not provide students with a relational understanding of mathematics nor does it develop problem solving skills that are critical in today's world. As it turns out, both students in the student case studies (below) had been studying mathematics in elementary school under the "old" Program of Study that placed a stronger emphasis on instrumental instructional strategies. Both students also attended elementary schools with academically focused programs using traditional teacher-directed instruction with a strong emphasis on repetition of mathematical facts and memorization.

In the scenario of the equation triangles the situation was quite different. As this was a science class, the teacher, while well-qualified to teach the topic at hand, was not trained to teach mathematics. The teacher's approach towards presenting mathematics in his science class was likely an instructional strategy balancing the need for teaching students how to use Ohms Law and a trade-off due to limitations in the teacher's ability to solve equations algebraically. One problem with this approach is that, perhaps inadvertently, by not explicitly connecting mathematics to the science lesson an opportunity of illustrating how mathematics (in this case, solving equations algebraically) can be applied in a real world scenario has been missed. As a result there is a risk of introducing the misconception to students that it would be more difficult, complicated or time consuming to solve Ohms Law using simple algebra (Ben-Hur, 2006). The challenge for non-mathematics teachers, particularly in the sciences, is that mathematics is pervasive and, unless the teacher is comfortable with mathematics, it may be difficult for them to teach topics in their subject using mathematical concepts.

- Scenario:
A grade 10 student is working on problems involving simplification of radical expressions and encounters the following problem in her text book:

Simplify and provide an exact answer for \( \sqrt{50} \).

She proceeds by solving the problem as follows:

\( \sqrt{50} = \sqrt{25+25} = \sqrt{25} + \sqrt{25} = 5 + 5 = 10 \)

- Analysis:
- This example illustrates a a common mistake where students over-generalize mathematical rules. In this particular students over-generalize the rule for the product of radicals, i.e. \( \sqrt a \sqrt b = \sqrt{ab} \), believing that the rule is also applicable when adding and subtracting radicals, i.e. \( \sqrt a + \sqrt b = \sqrt{a+b} \) (while in reality \( \sqrt a + \sqrt b \neq \sqrt{a+b} \)).
- Modifications:
- When the rules for multiplication and division of radicals are introduced they typically are followed by a large number of staged practice problem where students are expected to apply only the newly introduced concepts. Often these types errors, however, occur out of context, i.e. in problems that may require different mathematical concepts altogether. One possible solution to this common mistake would be to make students explicitly aware when they can and and cannot apply these rules by interspersing the practice problems with problems where the rules are no applicable. This would force students to be more mindful when solving problems.

- Scenario:
- A grade 11 student has asked his math teacher for help to simplify \( \frac{\displaystyle 1}{\displaystyle x} \times x^3 \).
- [Teacher:]
- What seems to be the problem?
- [Student:]
- I don't know how to simplify this expression.
- [Teacher:]
- Do you remember the exponent rules?
- [Student:]
- I think so, but the \(x\) is below (pointing to the \(x\) in the denominator)
- [Teacher:]
- Remember that \( x^{-1} \) is the same as \( \frac{\displaystyle 1}{\displaystyle x} \). You just apply this rule backwards here.
- [Student:]
- Oh, ok. I see, well that is easy then. (student quickly jots down) \( x^{-1} x^{3} = x^{-4} \).
- [Teacher:]
- Almost correct, but what do you get when you add 3 to -1?
- [Student:]
- -4? (student is looking at the teacher with a puzzled expression)..., or 4? (student reaches for the calculator and punches in the numbers), oh, it's 2?

- Analysis:
- I always find it surprising how many students even in high school find it difficult to perform addition and subtraction involving negative numbers. This particular example also illustrates the common reliance on using a calculator for even the simplest mathematical operations.
- Modifications:
- Everyone gets stumped occasionally by a simple mathematical problems. The key for getting out of this type of pinch is to have good problem solving skills. The last course of action, particularly in a mathematics classroom, should be to use a calculator as this approach provides the least amount of insights nor an improved understanding of how to approach this type of problem next time around. In this particular case, the student can be asked to rephrase the problem using a different wording, e.g. if it is one degree below zero and it warms up three degrees, what is the final temperature? Another possible approach might be to ask the student to draw a number line and think about if "adding 3 to -1" will result in a smaller number (\( <-1 \)) or larger number (\(>-1\)) and by how much.

The over-generalization of mathematical concepts and a lack of understanding of basic concepts are two common challenges students encounter in their study of mathematics. While these challenges may appear very different, both have a common ground in that they are manifestations of a lack of understanding of the underlying mathematical principles. What is the nature of the student and what are some characteristics that might make their role more challenging in these scenarios? While it may be apparent that a lack of understanding of the underlying concepts contributed to the erroneous answers in both scenarios, this is not the only challenge students encountered. A far more insidious challenge these scenarios illustrate is a mindless and passive approach to learning mathematics and to problem solving. For example, applying the rule for radical multiplication by treating the rule as a formula where numbers are just to be plugged in or relying on calculators to solve simple arithmetic calculations requires the bare minimum of mathematical skills and is doomed to fail. What student's and teachers ultimately need to ask themselves is what the purpose is for solving mathematical problems. UCLA mathematician Terence Tao expressed this deftly on his blog:

The purpose of solving mathematical problems is to increase your understanding of mathematics, not just obtaining a solution (Terrence Tao)

It is always possible to correct misconceptions and misunderstanding students have by revisiting and reviewing material they find difficult, even if that means reteaching arithmetic operations involving negative numbers. It is far more difficult, however, to stem the tide of mindless and passive problem solving, e.g. blindly applying formulas or relying on calculators to provide answers. Hence the main challenge students are facing when studying mathematics is their own attitude. Over the last few years I have been tutoring junior high and high school students in mathematics and one of the most common excuses I hear from students for why they study curtains topics in their mathematics class superficially or not at all is that their teacher told them that that this particular material will not be on the next test. Because mathematics beyond the established conventions such as notation and terminology is necessary (necessary as opposed to arbitrary) the study of mathematics is an incremental enterprise where new concepts build on previous concept (Hewitt, 1999).

Do these four case studies tell us anything about the nature of the teacher and learner and the challenges they face in the mathematics classroom? I think they do. They all fundamentally lack the insights required for learners to master the fundamentals of mathematics. The quote by Hewitt (1999) (see the beginning of this post) could be paraphrased as "If I'm not understanding ..., then I'm not working on mathematics.". This stands in juxtaposition against the misguided ideas that repetition and rote learning is required for mastering the fundamentals of mathematics (Tran-Davies, 2014). This approach will, at best, create students that are passive learners of mathematics that may be able to quickly recall facts but will lack the understanding of how to create these facts on their own. If we relegate mathematics instruction to rote learning, memorization and "rules without reason" we rob our students of the opportunity for acquiring a deeper understanding of the logic, beauty and the vast applicability of mathematics in all aspects of our lives. Instead of reaching the awareness that mathematical concepts are, in Hewitt (1999) words, *necessary* students will come away with the belief that the concepts are arbitrary. Ultimately the lack of conceptual understanding and active learning among students reflect a failure by parents, teachers and the (old) Program of Study to engage students with meaningful and authentic material. The only way we will be able to educate students to master the basic of mathematics and have solid problem solving skills is if we teach mathematics using concept-rich instructions that include meaningful practice, decontextualization allowing students to generalize concepts and skills, allow students to reflect and verbalize mathematical concepts and their generalizations, recontextualization allowing students to generalize their knowledge in new contexts and integration of mathematical concepts and problem solving skills into other subjects (Ben-Hur, 2006).

Ben-Hur, M. 2006. Concept-rich Mathematics Instruction : Building a Strong Foundation for Reasoning and Problem Solving.

Eberts, R. 2014. Curriculum Redesign – The Backlash Begins… URL: http://eberts.ca/2014/02/28/curriculum-redesign-the-%20-begins/. Accessed on: March 18, 2014.

Education Alberta, 2014. Program of Study for Mathematics K-9. URL: https://education.alberta.ca/media/645594/kto9math.pdf. Accessed on: March 17, 2014.

Education Alberta, 2014. Provincial Achievement Test. http://education.alberta.ca/media/6738143/02%20math3%20bulletin%202013-14-20130826.pdf. Accessed on: March 17, 2014.

Hewitt, D. 1999. Arbitrary and Necessary Part 1: A Way of Viewing the Mathematics Curriculum. For the Learning of Mathematics, 19(3): 2-9

Skemp, R.R. 2006. Relational Understanding and Instrumental Understanding. Mathematics Teaching in the Middle School, 12(2):88-95

Tran-Davies, N. 2014. Back to Basics: Mastering the fundamentals of mathematics. URL: https://www.change.org/en-CA/petitions/back-to-basics-mastering-the-fundamentals-of-mathematics. Accessed on: March 17, 2014.

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