**Teaching Multiplication and Division Facts**

When I reached middle school, the headmaster welcomed us and gave a little “talk” on what was expected of us in the middle school. He talked about forming the habit of reading everyday for pleasure and for school, importance of doing homework every night, selecting a sport that we could enjoy even after we left school, keeping a good notebook for classwork, writing everyday something of interest, become proud of our school, and then he said: “Those of you who have mastered your multiplication and division facts, you will be finishing eighth grade with a rigorous algebra course and then finish high school with a strong calculus course.” After laughter subsided, we realized the importance of the statement our elementary mathematics teacher–Sister Perpetua used to make as she was making sure that we had mastered our multiplication tables by the end of third grade and division facts by the end of fourth grade. We had heard about headmaster’s welcoming speech from her and the students who had gone before us. As headmaster and a demanding math teacher, he was very popular and respected by teachers, parents, and most students. He would repeat the ideas many times after that. It was more than sixty-five years ago, but his words are still fresh in my mind.

In my more than fifty-five years of teaching mathematics from number concept to Kindergarteners to pure and applied mathematics to graduate students (in mathematics, engineering, technology, and liberal arts), and preparing and training teachers for elementary grades to college/university, I am strongly convinced that no student should leave the fourth grade without mastering multiplicative reasoning—its language, conceptual schemas/ models, multiplication and division facts, and its procedures—including the standard algorithms.

1. After*Concept, Role and Place of Multiplication in the Mathematics Curriculum:*, additive reasoning, and place value, the next important developmental concept in mathematics is multiplicative reasoning.*number concept*^{[1]}**Multiplicative reasoning**is an example of quantitative thinking that recognizes and uses repetition of groups to understand the underlying pattern and structure of our number system. Multiplicative reasoning is the key concept^{[2]}in the mathematics curriculum and instruction in grades 3-4. Multiplication and division are generalizations and abstractions of addition and subtraction, respectively, and contribute to the understanding of place value, and, in turn, its understanding is aided by mastering place value. It helps students to see further relationships between different types and categories of numbers and it helps in the understanding the number itself.- Whereas, in the context of addition and subtraction, we could express and understand numbers in terms of comparions of smaller, greater and equal, with multiplication and division, numbers can be expressed in terms of each other and we begin to see the underlying structures and patterns in the number system. Multiplicative reasoning provides the basis of measurment systems and their interrelationships (converting from larger unit to smaller unit (you multiply by the conversion factor and vice-versa. It is the foundation of understanding the concepts in number theory and representations and properties of numbers (even and odd numbers; prime and composite numbers, laws of exponents, etc.), proportional reasoning (fractions, decimals, percent, ratio, and proportion) and their applications.
is not always smooth. Many children by sheer counting can achieve a great deal of accuracy and fluency in learning addition and subtraction facts, and at least for some multiplicaion facts. However, it is not possible to acquire full conceptual understanding (the models of multiplication and division), accuracy (how to derive them efficiently, effectively, and elgantly), fluency (answering correctly, contextually, in prescribed and acceptable time period), and mastery of multiplicative reasoning by just counting.*The move from additive to multiplicative thinking and reasoning***Definition**: Qualitatively and cognitively, for children,in their mathematical development. It is a higher order abstraction: addition and subtraction are abstractions of number concept and number concept is an abstraction of coutning. Addition and subtraction are one-dimensional cocnepts and are represented on a number line. Multiplication and division, as abstractions of addiotn and subtraction, start out as one-dimensional (as repeated addition and groups of), but they become two-dimensional concepts/ operations (i.e., as an array and area of a rectangle representations). Lack of complete understanding and mastery of multiplicative reasoning can be a real and persistent barrier to mathematical progress for students in the middle years of elementary school and later. Compared with the relatively short time needed to develop additive thinking (from Kindergarten through second grade), the introduction, exploration, and application of ideas involved in multiplication may take longer. Understanding of multiplicative reasoning (i.e, the four models–repeated addition, groups of, an array, and area of a rectangle) is truly a higher order thinking as the basis of higher mathematics.*multiplicative reasoning is a key milestone*of the*The main objective*, particularly in*mathematics curriculum and instruction*, for K through grade 4, is to*quantitative domain*. Numeracy means: A child’s*master numeracy*and*ability*in executing, standard and non-standard, arithmetic procedures (addition, subtraction, multiplication, and division), correctly, consistently and fluently with understanding in order to apply them problem solving in mathematics, other disciplines, and real-life situations. To achieve this: children by the end of fourth grade, should master multiplicative reasoning. They should master multiplication concept, facts, and procedure by the end of third grade and by the end of fourth grade, they should master concept of division, division facts, and division procedure. Mastering multiplicative reasoning means mastering multiplication and division and understanding that multiplication and division are inverse operations. They should be able to convert a multiplication problem into a division problem and vice-versa.*facility*: The first real hurdle many children encounter in their school experience is mastering multiplication tables with fluency. Even many adults will say: “I never was able to memorize my tables. I still have difficulty recalling my multiplication facts.” It is a worldwide phenomenon. Everyone agrees that chidren should master multiplication tables, but there is disagreement in opinions about*The reasons for difficuties in mastering multiplication and multiplication tables*Mathematics educators, teachers, and parents have formed opposing camps about it. One group believes in achieving understanding of the concept and believe that fluency will be reached with usage, whereas the other group believes in memorizing the tables and insist that conceptual understanding will come with use.*what it means to master multiplication tables and how to achieve this mastery.*Both work for some children, but not for all.*Both of these extreme approaches are inadequate for mastering mutiplication tables for all children.*

At the time of evaluation for a student’s learning difficulties/disabilities/ problems, when I ask him/her, ‘Which multiplication tables do you know well?’ Inevitably, the reply is ‘The 2’s, 5’s and 10’s.’ Some of them would add on the tables of 1’s, 0’s and 11’s to their repertoire. If I follow this up by ever so gently asking the answer for 6 × 2, then the response is: “I do not know the table of 6.” On further probing, I get the answer. Most frequently, the student finds the answer by counting on fingers 1-2, 3-4, 5-6, 7-8, 9-10, 11-12. 6 × 2 is 12. Some will say: 6 and then 7, 8, 9, 10, 11, and 12. 6 × 2 is 12. All along, the student has been keeping track of this counting on his/her fingers. Another way the answer is obtained by reciting the sequence: 2, 4, 6, 8, and 12. Here also the record of this counting is kept on his/her fingers. Both of these behaviors are indicative of lack of mastery of multiplication facts. They are also indications of the child having ** inefficient strategies **for arriving at multiplication facts. Skip counting forward on a number line or counting on fingers is not an efficient answer to masering multiplication facts.

Similarly, during my workshops for teachers, when I ask them to define “multiplication.” Most people will define multiplication as “repeated addition,” which is something that most of them know about multiplication from their school experience. Then I ask, according to your definition, what do you think the child would do to find 3 × 4? What does that mean to the child? The answer is almost immediate. “It means that child thinks 3 groups of four. He would count 4 three times.” As we can see, the person is mixing the two models of multiplication: “*repeated addition*” (3 repeated 4 times: 3 + 3 + 3 + 3 and “*groups of*” (4 + 4 + 4). Their definition and the action for getting the answer do not match. There is incongruence between their *conceptual* *schema* for the concept of multiplication and the *procedure for developing a fact*. Many children when deriving multiplication facts have the same confusion. To derive 6 × 8, A child would say (if he knows the table of 5–a very good sign): “I know 8 × 5 is 40 and then I add 6 so the answer for 6 × 8 = 46. The reason for wrong answer is this confusion in mixing the definitions. Children should understand** different definitions of multiplication. **The concept and problems resulting in multiplication emerge in several forms

On the other hand, repeated adddition and array model are limited to whole number multiplication. And, groups of model is helpful in conceptualizing the concept of multiplication of fractions and decimals. Children also acquire the misconception that “multiplication makes more” when they are exposed to only repeated addition and the array model. In such a situation, I say to them: “you are right. But what happens when you have to find the product of two fractions ½ × ⅓? What do you repeat how many times? The answer, invariably is: “You cannot. You multiply numerator times numerator divided by denominator times denominator.” Or, “what do I repeat when I want to find 1.2 × 1.3?” At this time, most teachers will give me the procedure of multiplying decimals. “Multiply 12 and 13 and then count the number of digits after the decimal.” If I pursue this further by asking: “How do we find the product (a + 3) (a + 2)?’ I begin to loose many in my audience. If, a person has complete understanding of the concept of multiplication, they can easily extend the concept of multiplication from whole numbers to fractions, decimals, and algebraic expressions. Only, the models “groups of” and the “area of a rectangle” models help us conceptualize the multiplication of fractions, decimals, integers, and algebraic expressions. And, only the area of a rectangle model helps us to derive the standard procedure for: multiplication of fractions/decimals, binomilas, distributive property of multiplication of arithmetic and algebraic expressions.

As one can see from this exchange, according to most teachers, the model or definition for conceptualizing multiplication changes from grade to grade from person to person. Rather than understanding the general principle/concept of multiplication, students try to solve problems by specific or ideosyncratic methods. Later, they find it difficult to conceptualize schemas/models/procedures for different examples of multiplication problems (with different types of numbers) and they give up. For example, they have difficulty reconciling the multiplication of fractions and decimals with their intial schema for multiplication (repeated addition or array andd even groups of, in some situations). We beleive, they should be exposed to and should be thoroughly familiar to the four models of multiplication before we introduce them to procedures. They should practice mastering multiplication tables when they have learned and applied these four models of multiplication. Then, they can accomodate different situations of multiplication into their schema of multiplication and create generalized schema for multiplication. The most generalized model for multiolicaiton is the area of rectangle.

Some of the difficulties children have in learning the concept of multiplication are the result of the lack of understnading of these different schemas and the emphasis on sequential counting in teaching multiplication in most classrooms. Students are not able to organize them in their heads, see the connections between them, and the importance of learning these models. They also think that different number types (whole numbers, fractions, decimals, integers, rational/irrational, algebraic expressions, etc.) have different definitions of multiplications. They do not see that the definitions and models should be generalizable.

is the teaching of multiplication: Children*Another reason for the difficulty*-a collection of sequential steps, sometimes the facts are derived just with the help of mnemonic devices, songs, and rote memorization as ‘a job to be done.’ This means: give a cursory definition of the term (e.g., multiplication is easy way of doing addition), give the procedure (e.g., this is how you do/find it), practice the procedure (do these problems now), and then apply the procedure (let us do some word problems on multiplication). It is a little exposure and then practice of the narrowly understood procedure. It is not mastery with rigor.*learn the tables and multiplication procedure in mathematics curriculum as mere procedures-*means, the student has the*Mastering a concept*, the*language*(effective and efficient strategies),*conceptual schema(s)*in skills and procedure, and*accuracy and fluency*it to other mathematics concepts and problem solving. The procedure of mastering multiplication tables should be based on solid understanding of the language and the concept. Students and the teacher should arrive at strategies and procedures by exploring and using the language, the conceptual schemas, and efficient and effective models. And then from several of these procedures should arrive at those that are efficient and generalizable (the standard algorithms). Students should develop, with the teacher, the criteria for efficient and effective conceptual schemas for deriving facts and procedures for multi-digit multiplication. The teacher should also help develop an efficient script for students to follow the steps needed to executeprocedure. Once children have arrived at an efficient procedure or procedures, they should practice it to achieve fluency and automatization. The fluency should be achieved by applying it in diverse situations. It means, ultimately,*can apply*,*they have understanding*, and*fluency*. Children learn tables successfully when teachers give them efficient strategies, enough practice in doing so and make it important to do so. They understand and are able to apply them according to how well they are taught.*applicability*

From the outset, we want to emphasize that it is important for children to learn (understand, have efficient strategies for arriving at the facts, accuracy, fluency, and then automatization) their multiplication tables. Eventually, by deriving the facts using efficient strategies and applying them to problems, they will be able to recall multiplication facts rapidly (*8 times 3*? *Twenty-four*!), and then use this knowledge to give answers to division questions (24 ÷ 3? Eight!); use these multiplication and division facts to do long multiplications and divisions; and use them appropriately in solving problems. When the concept of multiplication is understood, then one should introduce division concept and help them see that multiplication and division are inverse operations. Cyisenaire rods are the best material for making this relationship clear. (See *How to Teach Multiplication and Division*, Sharma 2018).

**Transition from Addition to Multiplication**:1.*Pre-requisite Skills for Multiplication and Multiplication Tables:*The instructional practice of having students count groups—skip counting—is an essential transition between additive and multiplicative reasoning. This counting should be limited to counting by 1, 2, 10, 5, and possibly 9. All other groups, when being added should be done by decomposition (adding 6 to 36 should be accomplished by asking: What is the next 10s? “40” How do I get there? “add 4” Where did the 4 come from? “from 6” What is left in 6? “2” What is 40 + 2? “42” So, what is 36 + 6? “42” Encouraging to count after 36 to add 6 does not amke the child acquire a robust numbersense. Just like visual clustering or representation of number as a group is a generalization and abstraction of discrete counting, skip counting, emphasizes the structure and efficiency that grouping gives to counting and, therefore, to addition. For example, counting by fives (using the fingers on hands as a starting model, then moving to TenFrame, Visual cluster cards representing 5, and then the 5-rod (yellow) of the Cuisenaire rods is the right progression for learning to count by 5. or twos (using eyes, or stacks of cubes, Visual Cluster Card representing 2, then the 2-rod (red) of the Cuisenaire rods) is very productive. Similarly, counting tens rods (in base 10 blocks or the 10-rod in Cuisenaire rods, however, using the Cuisenaire rods is better) as: 10, 20, 30, 40, and so on, emphasizes the concept of repeated addition and grouping. However, if these counting sequences are learned by discrete counting (Unifix cubes, fingers, number line, etc.) or without models to support the grouping and repeated counting activity then the order and the outcome will be learned without the concept and significant meaning about multiplicaiton or division.*Counting by 1, 2, 10, and 5.***Additive Reasoning pre-requisite Skills for learning and masrering Multiplication Tables:**(a) 45 sight facts of adddition, (b) Making ten, (c) Making Teens’ numbers, (d) What is the next tens, (e) Adding multiples of Tens to a two-digit number (e.g., 27 + 30 = ? 59 + 50 = ? 40 + 10 =?), (e) Commutative property of addition, (f) Counting forward and backward by 1, 5, 10, and 2 from any number.: Derivation of multiplication facts/tables is easier when the four models:*The Order of Teaching Multiplication Tables**repeated addition, groups of, an array, and the area of a rectangle*; commutative and associative properties of multiplication; and*distributive property of multiplication over addition and subtraction*: a(b + c) = ab + ac and a(b – c) = ab – ac have been mastered. Multiplication tables should be mastered only after theand*groups of*is clearly understood. If we use Cuisenaire rods for modeling multiplication, particularly for showing it as area of a rectangle, then the repeated addition and groups are already embedded in it and children can see the commutative, associative, and distributive properties also. Using these propeties, the teacher should derive multiplication tables up to 10 (i.e., 10 × 10 = 100 facts), in the following order (I cannot oveemphasize this order).*area of a rectangle*

(i) ** Commutative property of multiplication**: This reduces the work of deriving 100 facts to only 55, an easier task.

(ii) ** Table of 1 **(19 facts), (iii)

(vi) ** table of 9** [11 new facts] The table of 9 has several clear patterns hidden in it. Children need to see them. For example, (a) the sum of the digits in the table of 9, from the facts we already know (from tables of 1, 10, 5, and 2) is always 9: 9 × 1 = 9 = 09, 0 + 9 = 9; 9 × 2 = 18, 1 + 8 = 9; 9 × 5 = 45, 4 + 5 = 9; 9 × 10 = 90; 9 + 0 = 9; (b) the tens’ digit in the table of 9 is 1 less than the number being multiplied with 9, 9 × 1 = 9 = 09, 1 – 1= 0; 9 × 2 = 18, 2 – 1 = 1; 9 × 5 = 45, 5 – 1 = 4; 9 × 10 = 90, 10 – 1 = 9. Let us, therefore, apply these two patterns to derive 9 × 7 = ? We use the two patterns: here in the ten’s place will be 7 – 1 = 6, and, then to make the sum of the two digits as 9, we know that 6 + 3 = 9, thus, 9 × 7 = 63, and by commutative property of multiplication, we have 9 × 7 = 7 × 9 = 63. This process helps children to easily memorize the table of 9. We can also derive the fact 9 × 7 in several other ways: (a) by using the distributuve property of mulitplication over subtraction: we already know that , 10 × 7 = 70 ; 9 × 7 = (10 – 1) × 7 = 10 × 7 – 1 × 7 = 70 – 7 = 63; (b) using distributive property of multiplication over addition, 9 × 7 = 9 × (5 + 2) = 9 × 5 + 9 × 2 = 45 + 18 = 45 + 20 – 2 = 65 – 2 = 63, Or, 9 × 7 = 9 × 5 + 9 × 2 = 45 + 18 = 45 + 10 + 8 = 55 + 5 + 3 = 60 + 3 = 63.]

(vii) ** Table of 4 **(9 new facts). Since 4 is double of 2, the entries ib the table of 4 are double of table of the corresponding entries in the table of 2. For example, 4 × 7 = 2(2 × 7) = 2 × 14 = 2 × 10 + 2 × 4 = 20 + 8 = 28; Or, 4 × 7 = 4 (5 + 2) = 4 × 5 + 4 × 2 = 20 + 8 = 28. Or, 4 × 7 = (2 + 2)7 = 2 × 7 + 2 × 7 = 14 + 14 = 28.

(viii) ** Remaining facts**: The total number of multiplication facts derived so far: 19 + 17 + 15 + 13 + 11 + 9 = 84. The remaining 16 facts are: 3 × 3; 3 × 6, 6 × 3; 3 × 7, 7 × 3; 3 × 8, 8 × 3; 6 × 6; 6 × 7, 7 × 6; 6 × 8, 8 × 6; 7 × 7; 7 × 8, 8 × 7; and 8 × 8. And, because of the commutative proeprty of multiplication, the number is reduced to 10. These 10 facts can be mastered by children in a week. These remaining facts should be derived by decompositon/ recomposition. For example, let us consider: 8 × 6 = ?.

:*Improving Times Table Fluency**The Institute for Effective Education*(IEE) in the UK has published a new report on improving times table fluency, as a result of study of 876 children in 34 Year 4 (grade 3 in the U.S.) classes. All groups had similar pre-test scores and similar groups of children–same distribution of children with similar abilities. Each class used a different balance of conceptual nad procedural activities during times tables lessons. Conceptual activiities were games that focused on the conncetions and patterns in table facts, while procedural activiities were games in which students practiced multilication facts. All grous had same pre- and post tests. The report concluded that times tables may be best taught by using a balanced approach–teaching both the concepts behind them and practicing them in a range of ways with low-stakes testing.

In the light of many similar studies, concept-based instruction involving efficient and effective methods that can be generalized and uses pattern-based continuos materials (Cuisenaire rods, Visual Cluster cards, etc.) that help in developing the script are better. Once children know the tables of 1, 2, 10, and 5 and can derive the other facts by using effective scripts, they should paractice the tables with games. We have found the following games using Visual Cluster cards to be very effective.

**Game Four: Mastering Multiplication Facts**

*Materials:** *A deck of Visual Cluster Cards (Playing cards without numbers) without face cards or with face cards. Each face card is, intially, given a fixed value (Jack = 2, Queen = 5, and King =10), later they are given values as: Jack = 11, Queen = 12, and King =15).

*How to Play**: *

- The whole deck is divided into two to four equal piles (depending the number of players).
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays two cards face up. Each one finds the product of the numbers on the two cards. The bigger product wins. For example, one has the three of hearts and a king of hearts (value 10), the product is 30. The other has the seven of diamonds and the seven of hearts, the product is 49. The second player wins. The winner collects all cards.
- If both players have the same product, they declare war. Each one puts down three cards face down. Then each one turns two cards face up. The bigger product of the two displayed cards wins. The winner collects all cards.
- The first person with an empty hand loses.
- Initially, the teacher or the parent should be a player in these games. Their role is not only to observe the progress, mediate the disputes, keeping pace of the game and encouragement, but also to help them in deriving the fact when it is known to a child. For example, if the child gets the cards: 8 of diamond and 7 of spade. Teacher asks: What is the multiplication problem here? “8 × 7” The teacher asks: Do you know the answer? “No” Which is the bigger number? “8” Can you break the 7 into two numbers (point ot the clusters of 5 and 2 on the 7-card)? “5 and 2” If the 7-card was 5-card, then the problem would be 8 × 5. If the 7-card was 2-card, then the problem would be 8 × 2. Now, 7-card has 5 and 2, so the problem is: Is 8 × (5 + 2). Is 8 × 7 is same as 8 × (5 + 2). “Yes!” So, 8 × 7 = Is 8 × 7 = 8 × (5 + 2) and is made up of two problems: 8 × 5 and 8 × 2. What is 8 × 5? “40” What is 8 × 2? “16” Now, What is 8 × 5 and 8 × 2 together? “40 + 16” What is 40 + 16? “56” Good! What is, then, 8 × 7? “56.” All this should be done orally.

In one game, children will derive, use, and compare more than five hundred multiplication facts. Within a few weeks, they can master multiplication facts. Once a while, as a starting step, I may allow children to use the calculator to check their answers as long as they give the product before they find it by using the calculator.

*Game Five: Division War*

** Objective: **To master division facts

*Materials:** *Same as above

** How to Play: **Mostly, same as above.

- The whole deck is divided into two to 4 equal piles (depending on the players.
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays two cards face up. Each one finds the quotient of the numbers on the two cards. The bigger quotient wins. For example, one has the three of hearts and a king of hearts (value 10). When 10 is divided by 3, the quotient then is 3 and 1/3. The other has the seven of diamonds and the seven of hearts, the quotient is 1. The first player wins. The winner collects all cards.
- If both players have the same quotient, they declare war. Each one puts down three cards face down. Then each one turns two cards face up. The bigger quotient on the two displayed cards wins. The winner collects all cards.
- The first person with an empty hand loses.

In one game, children will use more than five hundred division facts. Within a few weeks, they can master simple division facts. I allow children to use the calculator to check their answers as long as they give the quotient before they find it by using the calculator.

*Game Six: Multiplication/Division War*

** Objectives: **To master multiplication and division facts

*Materials:** *Same as above

** How to Play: **Almost same as the other games

- The whole deck is divided into two to four equal piles (depending on the number of players.
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays three cards face up. Each one selects two cards from the three, multiplies them, and divides the product by the third number (finds the quotient of the numbers). The bigger quotient wins. For example, one has the three of hearts, the seven of diamonds, and a king of hearts (value 10). To make the quotient a big number, the player multiplies 10 and 7, gets 70, and divides 70 by 3. The quotient is 23 1/3. The other player has the seven of diamonds, the seven of hearts, and the five of diamonds. He/she decides to multiply 7 and 7, gets 49, divides 49 by 5, and gets a quotient of 9 4/5. The first player wins. The winner collects all cards.
- If both players have the same quotient, they declare war. Each one puts down three cards face down. Then each one turns three cards face up. The bigger quotient on the three displayed cards wins. The winner collects all cards.
- The first person with an empty hand loses.

In one game, children will use more than five hundred multiplication and division facts. They also try several choices in each display as they want to maximize the outcome. This teaches them problem solving and flexibility of thought. Within a few weeks, they can master simple division facts. I allow children to use the calculator to check their answers as long as they give the quotient before they find it by using the calculator.

^{[1]}See previous posts on *Numbersense*; *Sight Facts and Sight Words*; *What does it Mean to Master Arithmetic Facts?, etc. *

^{[2]}See previous posts on *Non-Negotiable Skills at the Elementary Level*. For a fuller treatment on the topic see: ** How to Teach Multiplicative Reasoning** by Sharma (2019).

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*Game Nine: Visual Clustering and Comparison of Integers*

This game can be played with two or three players but is most effective between two players.

** Objectives: **To learn the concept of integers and comparing fractions

** Materials:** Take an ordinary deck of playing cards including jokers and face cards. The game is more effective if cards are without numbers at the corners. But even if you cannot get such cards, you can use an ordinary deck of cards. Each card is identified as follows: The black cards have positive values. For example, the three of spades and clubs will be denoted as +3. The red cards have negative values. For example, the three of diamonds and three of hearts will be given a value of -3. The face cards of jack, queen, and king have a numeral value of ten. The sign of the numeral is determined by the color of the card. The joker can assume any different value.

*How to Play: *

- The whole deck is divided into two equal piles.
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays a card face up. The bigger card wins. For example, one has the three of hearts (-3) and the other person has the seven of diamonds (-7). The three of hearts (-3 is bigger than -7) wins. Similarly, if the first person has the two of clubs (+2) and the other person has the eight of diamonds (-8), the two of clubs wins (+2 > -8). The winner collects all cards.
- If both players have the same value cards, they declare war and each one places three cards face down. Then each one turns a fourth card face up. The bigger fourth card wins. The winner collects all cards.
- The first person with an empty hand loses.

This is a very appropriate game when the concept of integers is being introduced and for children who have not mastered number conceptualization. The concept of integers, just like number conceptualization, is dependent on three interconnected skills: one-to-one correspondence, visual clustering, and ordering. This game develops all three of these prerequisite skills. Children with a lack of understanding of integers have a great deal of difficulty learning operations on integers. They continue to count on fingers and derive integer relations by sequential counting on the number line or other sequential counting materials. Students who have not mastered operations with integers make many more errors in pre-algebraic and algebraic operations. This game and the next games help students master integer operations.

*Game Ten: Combining Integers*

** Objectives: **To master adding and subtracting fractions

*Materials:** *Same as above

*How to Play: *

- The whole deck is divided into two equal piles.
- Each child gets a pile of cards. The cards are kept face down.
- Each person turns two cards face up. The two cards represent two integers. Each one finds the result from combining the two integers. In combining the two integers one uses the following patterns.
- Same signs (same colors), you add and keep the common sign. For example, the four of clubs (+4) and five of spades (+5). Their sum will be written as + 4 + 5 = +9. Similarly, -4 -6 = -10 have numerals 4 and 6 with the same sign -, so they become numbers -4 and -6, and their sum will be written as -4 -6 = -10.
- Opposite signs (different colors), you subtract and keep the sign of the larger numeral. For example, -5 + 10 = + 5. Here the numerals 5 and 10 have opposite signs. We subtract 5 from 10 and keep the sign + of the larger numeral 10. Similarly, -7 + 4 = -3. The numerals 7 and 3 have opposite signs, -and +. We subtract 4 from 7 and keep the -sign of numeral 7.

Once again, the bigger sum wins. For example, one has the three of hearts (-3) and a king of hearts (-10), the sum is -13. The other has the seven of diamonds (-7) and the seven of hearts (-7), the sum is -14. The first person wins (-13 > -14).

Let us take another example. One player has the three of clubs (+3) and the three of diamonds (-3). The sum is 0. The other has the four of clubs (+4) and the two of diamonds (-2). The sum is +2. The second person wins. The winner collects all cards.

- If players have the same sum, they declare war, and each one puts down three cards face down. Then each one turns two cards face up. The bigger sum wins. The winner collects all cards.
- The first person with an empty hand loses.

This is an appropriate game for students who have not mastered/automatized addition of integers. This game teaches, reinforces, and helps them automatize integer combinations.

Initially, children can count the objects on the cards. For example, if one has the three of clubs and four of hearts, each black icon cancels each red icon. In this case, one club will nullify one heart. Three hearts will nullify three clubs. There is one heart extra. It will be left out. Therefore, we have +3 -4 = -1.

Although children may initially count objects, fairly soon they begin to rely on visual clusters to recognize and find sums. In one game, children will use more than five hundred sums. Within a few weeks, they can master addition and subtraction of integers.

*Game Eleven: Multiplying Integers*

** Objective:**To master multiplication of integers

*Materials:** *Same as above

*How to Play: *

- The whole deck is divided into two equal piles.
- Each player gets a pile of cards. The cards are kept face down.
- Each person displays two cards face up. The two cards represent two integers. Each one finds the result from multiplying the two integers. In multiplying the two integers one uses the following patterns about multiplying:
- + × + = +
- -× -= +
- + × -= –
- -× += -.

For example, +4 × +5 = + 20, -4 × -5 = + 20, +4 × -5 = -20, and -4 × +5 = -20.

Once again, when playing the game, the same rules apply: the bigger product wins. For example, one has the three of hearts (-3) and a king of hearts (-10), the product (-3 × -10) is + 30. The other has the seven of diamonds (-7) and seven of hearts (-7), the product (-7 × -7) is + 49. The second person wins (+ 40 > +30).

Let us take another example. One person has the three of clubs (+3) and three of diamonds (-3). The product (+3 × -3) is -9. The other has the four of clubs (+4) and two of diamonds (-2). The product (+4 ×-2) is -8. The second person wins (-8 > -9). The winner collects all cards.

- If both players have the same product, they declare war and each one places three cards face down. Then each turns two cards face up. The bigger product wins. The winner collects all cards.
- The first person with an empty hand loses.

This game is appropriate for students who have not mastered/automatized the multiplication of integers. This game teaches, reinforces, and helps them automatize integer multiplication.

Initially, students can recall facts sequentially. However, soon they begin to rely on visual clusters to recognize and find the products with sequentially recalling the multiplication tables. In one game, children will use more than five hundred products. Within a few weeks, they can master the multiplication of integers.

*Game Twelve: Dividing Integers*

** Objectives: **Mastering division of integers

*Materials:** *Same as above

*How to Play: *

- The whole deck is divided into two equal piles.
- Each student gets a pile of cards. The cards are kept face down.
- Each person displays two cards face up. The two cards represent two integers. Each one finds the result from dividing the two integers. In dividing the two integers one uses the following patterns about division:
- + ¸+ = +
- -¸-= +
- + ¸-= –
- -¸+= -.

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**What is Needed to Fight Stereotype?
**In order to effectively minimize the effects of stereotype, eradicate the conditions that foster stereotype in institutions, and create environments where our children do not encounter these conditions will take time, will, and effort. For the moment, we need to focus on a few key factors. Here, our focus is only on the mathematics education related factors:

(physical, affective, cognitive, mathematical),*Classroom environment*(training as a teacher, subject matter mastery, attitude—toward the discipline and learner differences, usage of language in communicating mathematics, questioning and assessment techniques, mastery of teaching and learning tools and their effective and flexible use, collaboration with colleagues and students, interest in learning, etc.),*Teacher characteristics*(knowledge of pedagogy, choice of instructional materials/models, selection and sequencing of introductory and practice exercises, amount of time devoted on mathematics instruction—tool/skill building, main concept, collaboration, practice, problem solving, etc.).*Instructional strategies*

As educators and policy planners we react to situations where stereotypes are manifested. Then we seek easy solutions: we increase the number of female and minority faculty, provide mentors, and actively recruit students in STEM programs. These changes provide only opportunity for positive impact, but to have long lasting effect, they need to accompany significant changes in pedagogy, understanding of learning issues and the aspirations, assets—strengths and weaknesses of these students, the nature of classroom interactions, type of assessments, and the nature of feedback.

For example, even when the number of females and minorities increases in STEM programs, not enough students remain in the programs. It is because they may not have the information about what kinds of pre-requisite skills they need to succeed in these programs. How to acquire these pre-skills? What efforts should they make to succeed? They may not know how to be effective and successful learners. They may not know what types of jobs they can get if they succeed in STEM programs. When they perceive that they are not succeeding, they change course, programs, and aspirations for careers.

Many students change majors during their undergraduate years. The rate at which students change their major varies by field of study. Whereas 35 percent of students who originally declare a STEM major change their field of study within 3 years, 29 percent of those who originally declare a non-STEM major do so. However, about half (52 percent) of students who originally chose math major switch major within 3 years. This change of major is much higher than that of students in all other fields, both STEM and non-STEM, except the natural sciences. [1]

The challenge of keeping students—especially women and underrepresented minorities—is on the agenda of every policy decision at every level of government—local to federal and education—from early childhood to graduate school. According to studies, among the culprits of attrition in STEM programs are uninspiring introductory courses, a culture that can be unwelcoming, and, and lack of adequate preparation of students, specifically in mathematics. [2]

**Learning Strategies and Stereotype
**Differences in students’ familiarity with mathematics concepts explain a substantial share of performance disparities between socio-economically advantaged and disadvantaged students and males and females. Many children, particularly girls and minorities, do not get exposure to quality mathematics content and effective pedagogy. Access to proper and rich mathematics language, transparent and effective conceptual schemas, and efficient and generalizable procedures is the answer to higher mathematics achievement for all students and fewer inequalities in mathematics education and in society.

When gender differences in math confidence, interest, performance and relations among these variables are studied over time, results indicate that *gender differences in math confidence are larger than disparities in interest and achievement in elementary school*. Research shows that confidence in math has become a major problem for girls and many minority children. It is one of the reasons women are vastly outnumbered by men in STEM professions later in life. Differences in math confidence between boys and girls show up as young as grade 2 and 3, despite girls and boys scoring similar marks. That trend continues through high school.

According to several studies, about half of third grade girls agree with the statement that they are good at math compared to two-thirds of boys. The difference widens in grade 6, where about 45 per cent of girls say that they are good at math compared to about 60 per cent of boys. This information is important for teachers as these attitudes are significant predictors of math-related career choices.

Early gender differences in math interest drive disparities in later math outcomes. At the same time, math performance in elementary school is a consistent predictor of later confidence and interest. There is a reciprocal relation between confidence and performance in middle school. Thus, math interventions for girls should begin as early as first grade and should include attention to developing math confidence, in addition to achievement. Even in preschools, the kind of games and toys children play with can determine the development of prerequisite skills for mathematics learning.[3] Confidence in learning is a function of metacognition and that in turn develops executive function and cognitive flexibility.

In elementary school, boys often utilize rote memory when learning math facts whereas girls rely on concrete manipulatives such as counting on fingers, number line, etc. And they use them longer than they should. Their arithmetic fact mastery puts boys in a better preparation for related arithmetic concepts (e.g., multiplication, division, etc.). These differences in strategies result in girls demonstrating slower math fluency (i.e. the ability to solve math problems related to arithmetic facts quickly) than boys. Both these methods (mastery by rote memorization and prolonged use of counting materials) are inefficient for arithmetic fact mastery for anyone. But, these inefficient strategies reinforce the gender stereotype for girls. Girls may blame their slower mastery of facts on being girls rather than the inefficient methods and strategies.

These inefficient models, methods, and strategies might bring higher achievement in elementary mathematics up to grade 3 and 4 (one can answer addition, subtraction, multiplication, and division problems by sheer counting), but they do not develop skills that are important for later concepts (e.g., difficulty dealing with fractions, ratio and proportion, and algebra as they are not amenable to counting). As a result, the average mathematics achievement of an average American is fifth-to-sixth grade level.

Clear understanding of concepts, fluency in procedures, and application of proportional reasoning (e.g., fraction concepts and procedures, ratio, proportion, etc.) are the gateway to algebra. Algebra is the main door to STEM fields. Students who do not opt for STEM related topics are the ones who have experienced difficulty understanding and operating on fractions. To become competent in understanding and fluently operating on fractions, students need:

(e.g., repeated addition, groups of, an array, and the area of a rectangle*Mastery of multiple models of multiplication*,*)*(tables of 1 to 10),*Mastery of multiplication tables*(for 2, 3, 4, 5, 6, 8, 9, and 10),*Divisibility rules*(computing division with one digit divisor and multi-digit dividend without long-division procedure),*Short-division*, and*Prime factorization**Knowing that a/b = 1,for any a and b and b ≠0, and that multiplying by a/a (by a fraction whose value is 1) gives an equivalent fraction.*

Skills listed above cannot be achieved by being proficient only in counting, memorizing and using manipulatives. These skills are acquired with having strong numbersense (e.g., mastery of number concept, arithmetic facts, and place value). Students have mastery of an arithmetic fact if they demonstrate:

- Solid understanding of number concept and sight facts,[4]
- Efficient strategies for arriving at the fact (e.g., using decomposition/ recomposition of number),
- Fluency (giving a fact in 2 seconds or less orally, 3 seconds or less in writing), and
- Applying it to other facts, mathematics concepts, and/or to problems.

** Developmental Trajectory of the Competence in Mathematics** (Additive and multiplicative reasoning and facts, place value, decomposition/recomposition of number)

Strong numbersense

to STEM fields.

To be prepared for higher mathematics, all children should learn, master and apply strategies that are based on

*decomposition/ recomposition*of numbers (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14; 8 + 6 = 4 + 4 + 6 = 4 + 10 = 14; 8 + 6 = 2 + 6 + 6 = 2 + 12 = 14; 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14; 8 + 6 = 7 + 1 + 6 = 7 + 7 = 14),*flexibility of thought*(able to arrive an answer to a problem in more than one way),*generalizable skills*(moving from strategies that give the exact answer to efficient strategies that give the accurate answer easier and quicker with less effort and then move to strategies that are elegant—that can be abstracted into formal systems, and can be extrapolated), etc.

Inefficient strategies and simplistic definitions and models such as ** addition** is counting up/forward,

Teachers and parents need to emphasize that ** mastery** of math facts and concepts is not just memorizing or arriving at the answer by counting. It is:

- Deriving facts with efficient strategies, strong conceptual schemas, precise language, and elegant procedures,
- Accuracy and fluency, and
- Ability to apply this information in diverse situations (e.g., intra-mathematical, interdisciplinary, and extra-curricular applications).

When girls are encouraged to continue counting to find answers, they become self-conscious of their strategies and give up easily. This happens to boys too. But, in most cultures boys are given more support and encouragement. In addition, the stereotype that “boys are good in math and girls are good in reading” gives boys the benefit of doubt—they will ultimately outgrow inefficient strategies.

All students must understand that ** mastery of certain math concepts** is important for any quantitative problem solving in most professional fields. To acquire efficient and effective strategies for number relationships and gain confidence in their usage is even more important. For example, students in early grades show high interest in STEM. But in later grades, without fluency in basic skills, lack of flexibility of thought and poor/or no conceptual schemas for key mathematics concepts, they tend to lose that interest. They also have difficulty connecting the diverse strategies and experiences in problem settings to related disciplines. For example, they have difficulty in applying conversion of units and dimensional analysis algebraic manipulations from mathematical setting to physics and chemistry. As a result, the sheer size of numbers and complexity of concepts and procedures they encounter in the STEM fields overwhelms them. As another example, a calculus course requires students to have an in-depth understanding of rates of change (e.g. proportional reasoning and its applications). The foundation of the concept should be introduced to students early in their mathematics education, and their understanding of it should evolve from middle school up to and including calculus. They should explore rates of change using numbers, tables, graphs and equations:

- Investigate and model applications of rates of change, and
- Explore how integrating concepts and technology appropriately enhances student understanding across grades.

When students are exposed to interesting and challenging problems from early grades and are shown clear developmental trajectory of each concept and procedure, they see connections between concepts. When students see the relationships between mathematics tools—strategies, skills and procedures and problems and where do these problems come from they remain engaged. This is particularly true about many female students as they are not sure of their competence. When problems are selected and their relationship with the STEM fields is made transparent, students get interested in these fields. Many students do not know what types of problems are solved in different fields. For example, in surveys 34 percent more female students than male students say that STEM jobs are hard to understand, and only 22 percent of the female respondents name technology as one of their favorite subjects in school, compared to 46 percent of boys.

**Increasing STEM Participation is a Whole School Activity
**Turning students’ interest toward mathematics and then STEM has to be a school-wide effort. Every educator (teachers—regular and special education, administrators, guidance counselors, coaches, para-professionals, etc.) should be aware of their own beliefs about math, gender/minorities, and their biases. For example, I have observed interactions between many adults (including principals) openly admitting their incompetence in mathematics to students. Here is a sample of interaction between a guidance counselor and two ninth grade students.

**Female Student**: Mr. Wilson, I am having very difficult time in my algebra I class. It looks like I do not have what it takes to be successful in Algebra I. I guess, I need to be taking the simpler algebra course or pre-algebra again. Could you please sign this paper for change of course?

**Guidance Counselor**: Let me see! Do you have a note from the teacher or your parents? Yes, algebra is kind of difficult. I have seen, over the years, more girls changing from this Algebra I class to easier courses. Have you tried getting some help from your algebra I teacher?

**Female Student**: I tried. I went to her a couple of times. It did not work. I will get a note from my father. He did warn me that algebra might be difficult. I will see you tomorrow.

**GC**: OK!

Another day:

**Male Student**: Mr. Wilson, I am having great deal of difficulty in my algebra I class. It looks like I do not have what it takes to be successful in Algebra I. I guess, I need to be taking the simpler algebra course or pre-algebra again. Could you please sign this paper for change of course?

**Guidance Counselor**: Let me see! Did you do poorly on the first test? You know the first test in a course is not really an indication of poor preparation for a course. One has to get used to the new material and the teacher—her style of teaching and her expectations. Now do you know what the teacher wants? Have you tried getting some help from your algebra teacher? You know she is one of the best teachers in our school. I know she is a little demanding, but she is an excellent teacher.

**Male Student**: Yes, she is demanding. Not a little, but a lot.

**GC**: You should join a study group. David, your friend on your soccer team, he is very good at math. Have you asked him for help? He even lives near you. Why don’t you try the course for few more weeks, maybe till the next test and then you still have difficulty come see me. Meanwhile, I will talk to your teacher. By the way, before you come see me next time, get a note from the teacher explaining that you did try. And, I also need a note from your parents so that they know about your changing the course? I know, algebra is kind of difficult, but trying is even more important.

**Male Student**: I guess, I will give it another try. If it doesn’t work, I will come to you, again. Yes, I will get a note from my mother. My father wants me to have algebra on my transcript. He says: “It looks good for college applications to have algebra in eighth grade or latest in ninth grade. I will see you later.

**GC**: OK!

**Quality of Concepts and Quality of Instruction
**Quality instruction has the greatest impact on student achievement and the development of a positive attitude about a subject matter. The major changes in student outcomes are obtained by teachers’ instructional actions. Generally, the premise is that teachers who implement effective instructional strategies will, in turn, help students use mental processes that enhance their learning. However, it is not enough to merely use an instructional strategy; it is more important is to ensure that it has the desired effect on student learning.

The opportunity to learn and the time students spend learning quality mathematics content and practicing meaningful and rigorous mathematics tasks assure higher mathematics achievement. Differences in students’ familiarity with mathematics concepts explain many performance disparities between socio-economically advantaged and disadvantaged students and between females and males. Widening access to meaningful mathematics content—proper mathematics language, efficient, effective, and generalizable conceptual schemas, and efficient and elegant procedures—are the answers raising levels of mathematics achievement and, at the same time, reduce inequalities in mathematics education.

Poor learning environments and poor mathematics teaching create gender, race, and class disparities in quantitative fields, and the gaps begin to develop as early elementary school. Initially small and subtle, they grow into causative factors for low achievement in and avoidance of mathematics in high school, college, and even graduate school. They become most pronounced in quantitative professions such as university-based mathematics research and STEM fields. It is worth noting that women who drop out of quantitative majors do not tend to have lower scores on college entrance exams or lower freshman grades than their male peers. Females are leaving math fields when they are performing just fine; it is therefore worth considering that the reasons hardly lie in them, but in our educational environments that might induce them to leave.

**Attitudes and Values
**Various explanations exist for gender differences, beginning with small differences in elementary school to consequential differences in high school, undergraduate mathematics classes, advanced mathematics, and in math-related career choices. Below we summarize some of the factors that contribute to gender, race, and ethnic differences in mathematics and math-related courses and career choices.

*Attitudes toward numeracy
*Even from a young age, girls are less confident and more anxious about math than boys. These differences in confidence and anxiety are larger than actual gender differences in math achievement. The differences are shaped by the social attitudes of adults toward work, careers, education, and achievement. These attitudes are important predictors of later math performance and math-related career choices.

Early perceptions and attitudes formed at home and in early childhood classrooms form the basis of future attitudes toward learning. For example, many parents read to their children regularly and with interest. This instills the love for literacy and learning. During this reading, some parents do discuss the quantitative and spatial relations—ideas about number, number relationships, and numeracy, in the reading material. Many families play board and number games and with toys that develop prerequisite skills for mathematics learning [5].

From early childhood, traditionally, boys and girls play with toys and engage in games that are responsible for the development of different types of skills. Boys engage in activities that develop spatial skills and girls participate in games and toys that develop sequencing skills. For example, boys tend to be stronger in the ability to mentally represent and manipulate objects in space, and these skills (ability to follow sequential directions to manipulate objects mentally, spatial orientation/space organization, visuo-spatial representation, rotations, transformations, pattern recognition and extensions, visualization, inductive reasoning, etc.) predict better math performance and STEM career choices. In teaching mathematics, it is important to use those models—concrete materials, visual representations (diagrams, figures, tables, graphs, etc.) that develop these skills and to make up earlier deficits in these skills and aid in the development of numeracy skills effectively.

Attitudes thus formed about quantitative and spatial relations become the basis of later interest and competence in numeracy and its applications. By high school, these skills and related attitudes are well established. Personal aptitudes and motivational beliefs in the middle and high school have profound impact on individuals’ interest in science, technology, engineering, and mathematics in college and later in choice of occupations and professions.

*Attitudes toward work and professions
*Occupational and lifestyle values, math ability, self-concept, family demographics[6] (particularly, financial and educational status of family), and high school course-taking more strongly predict both individual and gender differences in STEM careers than math courses and test scores in undergraduate years.

People’s life styles, values, attitudes, and interests and that of those around them influence gender and class differences about occupational and career choices and the role of work in their lives. For example, women tend to care more about working with people, and men tend to be more interested in working with things. This difference, in turn, relates to gender-gaps in selection of math-related careers and even within STEM disciplines—health, biological, environmental and medical sciences (HBEMS) versus mathematics, physical, engineering, economics, accounting, and computer sciences (MPEEACS).

Women’s preferences for work that is people oriented and altruistic predict their entrance into HBEMS instead of MPEEACS careers. For example, for the first time ever, women make up a majority (50.7 percent) of those enrolling in medical school, according to the Association of American Medical Colleges. In fall 2017, the number of new female medical students increased by 3.2 percent, while the number of new male students declined by 0.3 percent.[7]

Women prefer biological sciences, where they represent 40% of the workforce, with smaller percentages found in mathematics or computer science (33%), the physical sciences (22%), and engineering (9%). To change this phenomenon active intervention and education are needed.

Even in medical sciences, number of women can be increased, if we pay attention to the attitudes and beleif systems can be intervened. For example, as girls leave high school, they are much more likely than boys to express interest in a medical career. And, they have the entrance requirements. They have higher grades and attend college at higher rates.

In college and univeristy, however, women leave the pre-med courses, a pre-requiste for medical degrees and careers. But, the recent research points out that they leave this pathway at a higher rate than men do.

They do not leave for lack of competence. It’s not that women perform worse academically, according to the research points instead to the *competency beliefs* of high-achieving women.

That finding, the authors write, suggests that “motivational interventions in pre-med science courses will be critical for retaining high-performing women in pre-med, an important outcome with implications for equity and women’s health.”

**Role of Problem-solving Strategies
**Mathematics is the study of patterns in quantity, space, and their integration. This means mathematics is thinking quantitatively and spatially. For example, in elementary school, understanding the concept of place value in representing large numbers is the integration of quantity and space. We are interested in a digit’s quantitative value in the number by its location in relation to other digits in the number. Similarly, coordinate geometry is a good example of this integration: each algebraic equation and inequality represents a curve in space and every curve can be represented by a system of equations/inequalities. Thus, to do better in mathematics and in subjects dependent on mathematics, one needs to have strong visual/spatial integrative skills: the ability to visualize and see spatial organization and spatial orientation relationships. Students who are poor in these skills, generally, have difficulty in mathematics. Those students who can do well in arithmetic up to fourth grade by just sequential counting begin to have difficulty later when concepts become complex (e.g., fractions, ratio, proportion, algebraic thinking, geometry, etc.). Then they blame themselves for their failures in mathematics – “I cannot learn mathematics.” “Mathematics is so difficult.” However, to a great extent, the reality lies in lack of these prerequisite skills and inefficient strategies.

The prerequisite skills for mathematics learning can be improved through training and intervention. Gender differences in spatial abilities and visual-spatial skills can be reduced and/or stronger compensatory strategies can be developed with effective interventions. The pattern of differences in the prerequisite skills for mathematics learning can be broken through these intervention programs. The types of games and toys children play, in early childhood, determine the fluency in these skills[8]. This means that one way forward is to ensure that all students spend more “engaged” time learning core mathematics concepts, solving challenging mathematics tasks, acquire prerequisite skills for learning mathematics.

Success in STEM fields depends on a person’s ability to apply efficient math strategies and exposure to diversity of problem solving strategies. For that, students need to engage in thinking both quantitatively—analyze ideas and strategies (deductive thinking), and qualitatively—synthesize ideas (inductive thinking). Thinking quantitatively means developing a strong numbersense and its applications. Thinking spatially/qualitatively means seeing patterns in numbers, shapes, objects, and seeing connections amongst ideas.

Research and observations show that in our schools boys tend to and are encouraged to use novel problem-solving strategies whereas girls are likely to follow school-taught procedures. In general, girls more often follow teacher-given rules in the classroom. It could be that girls are trying to fit in the class and they learn that these behaviors are rewarded. This tendency inhibits their math explorations, innovations, and the development of bold, efficient, and effective problem-solving skills. They need to explore and acquire strategies that can be generalized to multiple situations than just solving specific problems.

Such differences in learning approach and types of experiences contribute to gender related achievement gaps in mathematics as content becomes more complex and problem-solving situations call for novel approaches rather than just learned procedures. The rigorous use of the Standards of Mathematics Practices (CCSS-SMP)[9] in instruction at K-12 level and student engagement in collaborative and interdisciplinary research and internships at the undergraduate level can better prepare our students for higher mathematics and problem solving. Then, they will stay longer in STEM fields.

To be attracted to and stay in mathematics, students need to engage from a very early age with appropriate and challenging mathematical concepts. That means to experiment more and experience widely. This happens when they collect, classify, organize, and display information (quantitative and spatial); analyze, see patterns and relationships, arrive at and make conjectures; and communicate these observations using mathematics language, symbols, and models. These skills are central to a person’s preparedness to tackle problems that arise at work and in life beyond the classroom. Unfortunately, many students do not have a rigorous understanding of basic mathematics concepts (integration of language, concepts, procedures, and skills) and are not required to master these skills. In school, they practice only routine tasks procedurally that do not improve their ability to think quantitatively and qualitatively and solve real-life, complex problems—involving multiple concepts, operations, and meaningful ideas.

Mathematics also means communicating mathematical thinking using mathematics language, symbols, diagrams, models, mathematical systems: expressions, systems of equations, inequalities, etc. All these skills are central to a person’s preparedness to tackle problems that arise at work and in life beyond the classroom. The best approach to keeping students in the STEM fields is not only to give them skills but also to give them the “taste” of success in applications of mathematics in problem solving.

**Collective Course Design
**In 2011 the Association of American Universities started a project to improve the quality of STEM teaching at the undergraduate level. Among the conclusions of this project are:

Success is more likely when interdisciplinary departments take collective responsibility for introductory course curricula in STEM fields. For example, mathematics teachers should select application problems from the STEM disciplines so that students see connections between use of mathematics tools and concepts and the nature of the problems they can solve from other disciplines.

Along with interdisciplinary integration, there should be active collaboration between K-12 and undergraduate curricula, pedagogy, instructional strategies, and teachers. To improve the situation, colleges and universities should collaborate more, with K-12 schools, industry, and one another.

Applications of mathematics should involve its language, concepts, skills and procedures, not just formulas and procedures using calculators, computer packages, and apps. The objective of applications, therefore, is to see the utility, the power, and the beauty of mathematics. The application of mathematics fall in three categories:

: Applying a concept, method, procedure, or strategy from one part of mathematics to solve a problem in another part of mathematics. For example, solving a geometry problem using algebraic equations; seeing the study of coordinate geometry as the integration of algebra and geometry; understanding statistics as the integration of algebraic concepts (e.g., permutation/combination, binomial theorem), geometry (e.g., representation and presentation, graphing, and displaying of data, etc.), and calculus (differentiation and integration of probability functions), etc.*Intra-mathematical applications*: Applying mathematics modeling to problems in other disciplines (e.g., understanding and explaining concepts and principles in physics, chemistry, economics, psychology, etc. using mathematical models and systems). Students should understand and realize that mathematics is used first for understanding and explaining physical phenomenon and then mathematics modeling for solving problems in natural, physical, biological, and social sciences. For example, understanding the airline routing problem using mathematics (linear and non-linear programming, and permutation/combination, etc.) and then extend the approach to modeling similar problems.*Interdisciplinary applications*: Applying mathematics in solving problems in real life through group projects, independent and small group research, etc. This involves applying combination of skills from different branches of mathematics in solving real life problems. The skills involved are: identifying a problem; asking right questions about the problem and solution requirements and constraints on solutions; defining knowns and unknowns; identifying unknowns as variables; identifying and articulating relationships between knowns and unknowns—functions, equations, inequalities, etc.; identifying already known facts relating to the problem and the variables involved, postulates, assumptions, results that apply in this situation; developing strategies for solving the problem; collecting data, classifying, organizing, displaying the data; analyzing data; observing patterns in the data; developing conjectures/hypotheses, and results; solving the problem; relating the solution to the original problem; conclusion(s); if needed rethinking/redefining the problem with modified conditions and restraints; etc.*Extra-curricular applications*

The course planning, design and course delivery are, therefore, more than a sole faculty member’s task. Colleges and schools showing the most improvements in attracting and retaining students in STEM use teams of faculty, instructional-design experts, data analytics on student learning, administrative supports like teaching-and-learning centers, creative-learning spaces, mentoring and tutoring, and multiple means of delivery, and meaningful and timely feedback.

An approach that is attracting more students into mathematics is undergraduate research, where students engage in independent individual or small group research projects for a sustained period of time under the supervision of a faculty member.

To keep many more students in STEM fields cannot be the activity of an isolated individual or an office. To address the issue of female and racial achievement gaps university and school reforms must be campus-wide and embraced by all faculty members in order for women, black and Latino students to truly thrive.

Schools must also move away from forcing students of color into remedial programs before their participation into proper programs. Those students need to learn how to navigate the boundaries of the different social worlds that make up higher education. They have to learn how to “try on” the identities of the professions, to feel that they own them and have the right to play in them. This approach is in opposition to remedial programs, which lead people to think of themselves as outsiders, inferior, and not worthy of achievements. With remedial programs, they learn math as a compliance activity.

Achieving change takes total campus/school commitment, with the most powerful and knowledgeable people involved. Diversity offices can be supportive, but the power of the academy is in the hands of faculty. They can either motivate or demotivate a student from a course, program or degree. When everybody – faculty, staff, and administration, makes the success of students from different backgrounds a top priority on campus, only then can we make a difference.

**Minimizing the Effect of Stereotype
**Our main goal should be to create conditions so that stereotype does not exist in our schools, but that will take time and a great deal of effort. Concurrently, we also need to minimize the effect of current conditions that have been affected by stereotype. Here are some strategies for improving mathematics instruction for all and for making sure that children do not adopt the cultural stereotype that math is for a select few.

**Tracking and Its Role
**Placing students in ability groups, particularly minority and low performing female students, is the beginning of closing doors for meaningful mathematics and STEM. Some instruction grouping may be considered at sixth grade and beyond. But these groupings should be to accommodate interventions—both for gifted and talented and those who struggle, not in place of regular classes. From seventh grade on there may be two levels: honors and regular. However, each group should be provided challenging and accessible instruction that is grade appropriate and rigorous in content. The difference should be on time on task rather than in the quality of content or nature of instruction.

Teachers should ensure that each and every student has access to meaningful curriculum and effective instruction that is balanced with respect to rich language of mathematics, strong conceptual understanding using multiple models and representations, efficient procedures and fluency, diverse and flexible problem solving, and the development of a productive disposition for mathematics.

Each teacher should provide every student the opportunity to learn grade-level or above mathematics using efficient strategies and provide the differentiated and targeted instructional support necessary for every student to successfully attain this goal. However, some may need more and others may need less practice to reach proficiency. For example, to differentiate, all students should be asked questions appropriate to their level but on the same concept or procedure being taught in the class.

Differentiation does not mean making groups and teaching them lower or higher level mathematics. Differentiation should offer children exercises and problems at different levels but on the same grade level concept or procedure. Small groups and individualization can be organized for brief but frequent periods of practice, reinforcement, and deepening their understanding, but not for initial teaching and for long intervals. Students learn more from other students than from most teachers. Teachers can make this happen.

Teachers should affirm and help students develop their mathematical identities by respecting their mathematics learning personalities.[10] For example, each student falls on the ** mathematics learning personality** continuum of learning mathematics processes. On one end of this continuum are students who process mathematics information parts-to-whole. They process information sequentially, deductively, and procedurally. They are known as

To engage all students, it is important to be cognizant of different ways people learn mathematics. Teachers should view students as individuals with strengths, not deficits. This manifests when they value multiple contributions and student participation and recognize and build upon students’ realities and strengths.

**Multiple Models
**Teachers should provide students multiple opportunities to grow mathematically by providing:

- multiple entry points and multiple models for the same concept (e.g., for multiplication—
*repeated addition*,*groups of, an array, area of a rectangle*; for division—*repeated subtraction, groups of, an array, area of a rectangle*), - multiple procedures for the same problem (e.g., the quadratic equation:
*2x*can be solved by using^{2}– 5x – 7 = 0*algebra tiles*,__by graphing__,*by factoring*,*by completing the square*, or*by quadratic formula*; - multiple strategies for deriving a result (i.e., the sum 8 + 6 can be derived as 8 + 2+ 4 = 10 + 4 = 14, 2 + 6 + 6 = 2 + 12 =14, 8 + 6 = 4 + 4 + 6 = 4 + 10 = 14, 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14, 8 + 6 = 7 + 1 + 6 = 7 + 7 = 14, etc.), and
- multiple expressions for and demonstrate their knowledge in multiple ways—
*models*(*in words, symbols, tables, graphical, equations*),*forms*, and*levels*(concrete, pictorial, abstract/symbolic, etc.).

**Intervention and Remedial Instruction
**We should provide additional targeted instructional time as necessary and based on the results of common formative assessments—make instructional time variable, not student learning. A teacher has four opportunities for remedial instruction:

*Tool building*—identifying the tools necessary for students to be successful in the main lesson and quickly reviewing them (orally using the Socratic method) before the main lesson (e.g., commutative property of addition, N+ 1, making tens, and what two numbers make a teens’ numbers before beginning addition strategies; rules of combining integers before solving equations; prime factorization before reducing fractions to lowest terms; differential coefficients of important functions before starting integration of functions, etc.).*During the main lesson*—if during the lesson a concept is found to depend on a previous concept, briefly review that concept in summary form and write important formulas to be used in the new concept (e.g., divisibility rules during fraction operations; laws of exponents during combining polynomials; addition strategies during teaching subtraction strategies; important algebraic expressions before factoring: (a + b)^{2}= a^{2}+ 2ab + b^{2}, (a – b)^{2}= a^{2}– 2ab + b^{2}, (a + b) (a – b) = a^{2}– b^{2}, etc.).*Individual and small group practice*—at the end of the group lesson on a major concept, making small groups and helping each group of students to practice skills, concepts at different levels and helping them to practice previous concepts.*Intensive intervention*—organizing and providing intensive intervention for select students (e.g., those with dyscalculia, learning problems in mathematics, gaps in previous concepts and procedures, etc.). This type of intervention should be provided by specialists after or before class and should be in addition to regular math education instruction (math specialists with mastery of mathematics concepts and understanding of learning problems in mathematics using efficient and effective methods, not by a special educator who is weak in mathematics).

**Instructors and Pedagogy
**Quality instruction goes a long way toward keeping students — especially underrepresented minorities and women in the STEM fields. But measuring educational quality is not easy. Assessing the quality and impact in STEM at the national level will require the collection of new data on changing student demographics, instructors’ use of evidence-based teaching approaches, deeper and meaningful student engagement, student transfer patterns and more.

Most experienced, effective teachers (who have a clear understanding of the trajectory of the development of mathematics concepts and procedures and how children learn) should provide instruction to students who need more support rather than the least trained and least effective teachers and paraprofessionals. Highly effective teachers have the skills to support students who may not have previously been successful in mathematics. Effective teachers can make up almost three years of the result of poor teaching. Similarly, a poor teacher can nullify the gains of three years of effective teaching and even turn students off from mathematics.

**Keeping Diversity in Math to Fight Stereotype Threat
**An important way to address underrepresentation of minorities and women in mathematical pursuits is to create environments without stereotype threat (gender, race, ethnicity)—environments in which these groups are not concerned about being judged according to negative stereotypes. This can be done by mentoring where students are assured that:

**They are respected***.*Assigning simplistic work just to keep students in a program is not respecting learners of any kind and is insulting to their intelligence. A mathematics teacher should have fidelity only to (a) students and (b) to mathematics; teach meaningful math to all children in meaningful ways.**The work should be challenging.**Students should realize that although the work they are asked to do is challenging, it is accessible to them, they do indeed have the ability to succeed at it, and they believe that the teacher is there to help them succeed. The role of the teacher is to help them develop self-advocacy of their abilities, their strengths, and their usage.**They trust the mentor and her intent and capacity to help achieve the mentee’s success.**This means that they should feel secure that someone is there to help them to reach their goals; they believe in the process that they will have a higher level of skill set as the outcome and they will increase their potential to learn.**Importance of learning skills**Student work should focus not just on the content of mathematics but on how to learn—planning, goal setting, organizing, gaining learning skills—developing executive functions, marshaling resources, self-assessment, self-advocacy, self-regulation, self-reflection, and seeking and using feedback properly.*.*

The emphasis in this kind of mentoring is on stressing the importance of the expandability of one’s learning potential and realization of one’s goals—in a sense intelligence itself—students should grasp and internalize the idea of the role of the plasticity of the brain in learning potential and learning.

The intent of this mentorship is that at the end of these experiences (math courses, STEM program, etc.), mentees realize the idea that intellectual ability is not something that one has a finite amount of, and that it can be increased with genuine effort, experience and training. The role of the mentor here is more than cheerleading; it is helping the mentees to reach new personal heights in success and the attitude that they are capable of learning mathematics.

The increase in female representation in faculty and mentor roles, for example, has positive effects as long as they do not emphasize the uniqueness of their achievements and do not place undue pressure and importance on their mentees to view themselves as female mathematicians or scientists.

The more our female math students are exposed to women role models who can show them that not only can women “do math” but also that their feminine identities need not be viewed as a liability, the more they are likely to view math environments as places where they can belong and succeed. The same applies to other underserved groups.

**Identity and Mathematics
**The effects of stereotypes are far reaching. Research shows that stereotype threats induce undue pressures on women in the quantitative fields who are in the process of shaping their identities. Do women in mathematical arenas bifurcate their identities in response to prolonged exposure to threatening stereotype in these environments, or are women with bifurcated identities simply more likely to study mathematics? Research shows that because of the threat of negative stereotypes about female math ability many female math students (but not their peers in other fields) bifurcate their feminine identities. For example, in responding to stereotype threats, women in mathematics related fields often

They report keeping fewer feminine traits (e.g., sensitivity, nurturance, and even fashion-consciousness) in order to avoid negative judgments in math environments. They foster fewer feminine traits even though these may not lead to negative judgment. Thus, the stress of engaging in such adaptation could constitute yet another deterrent to women’s persistence in quantitative fields. Although sacrificing fashion-consciousness as an aspect of one’s identity may seem trivial, sacrificing an interest in having children does not.

Similar pressures in identity formation are present on students from minority groups. Many minority students try to adapt to situations by sacrificing some of their positive traits. We need to understand that stereotypes affect not only others’ judgments but also people’s own judgments of their own competence. For example, new immigrants to the country respond to discrimination and stereotype by blaming themselves. They blame their lack of knowledge, skills—language, experience, and knowhow, and as a result harshly judge themselves. Some of them are negatively affected and others may succeed by paying a price in both cases. For example, first- and second-generation children of immigrants respond to these situations with skills and knowhow (e.g., Chinese, Indian, and East European immigrants children flock to STEM majors and want to succeed to fight discrimination); they do not fight against stereotype and place higher expectations on themselves. Many of these children feel a great deal of pressure to succeed.

To reduce and nullify the effect of stereotype organized response is needed on several fronts. There is need for both systemic and tactical changes; change in systems and people that inhabit these organizations as these effects are situational. For this reason, the focus should be on both the people who are vulnerable to stereotypes and the organizations where the stereotypes exist (which applies to almost all organizations).

First, the change has to come in these organizations to reduce and then to eradicate these conditions.

Second, the affected groups need self-advocacy skills in responding to these situations. They need skills to achieve. They need skills in organizing and taking advantage of the positive situations. A good example of this is the METCO program in the Metropolitan Boston area. In this program, volunteer minority students (both male and female) from Boston public schools are bused to suburban schools instead of Boston Public Schools. These students get the opportunity to get a first-rate education in their suburban host schools. Moreover, the suburban students who realize the importance of this program and the schools who treat METCO students as their own also derive benefits. Many METCO students have gained skills from this program and many of them have gone on to become leaders in STEM fields.

As the METCO example shows, along with systemic change, there is need to work at the individual level. There is need for a mindset change on the part of individuals to actively gain skills to minimize the negative effects of stereotypes. For example, we can *a priory* determine a group’s needs and aspirations. Women’s ideas about themselves, their academic and career needs, and aspirations cannot remain fixed. Therefore, just focusing on organizations may not be enough – we also need to focus on the individuals.

Organizations—schools/colleges, social organizations, work places, and parents can do three key things to effect change.

** First**, they can control the messages they are sending by making sure there are no negative beliefs about any group in the organization. For instance, an experimental study on the evaluation of engineering internship applicants found that the same resume was judged by a harsher standard if it had a female versus a male name. Applications should be judged by the same standard.

** Second**, they can make performance standards unambiguous and communicate them clearly because when people don’t know what the standards are, stereotypes fill in the gaps.

** Last,** organizations can hold gatekeepers in senior management accountable for reporting on gender, race, and ethnic disparities in hiring, retention and promotion of employees.

[1] National Center for Education Statistics (NCES). (2017). Percentage of 2011 – 12 First Time Postsecondary Students Who Had Ever Declared a Major in an Associate’s or Bachelor’s Degree Program Within 3 Years of Enrollment, by Type of Degree Program and Control of First Institution: 2014. Institute of Education Sciences, U.S. Department of Education. Washington, DC. https://nces.ed.gov/datalab/tableslibrary/viewtable.aspx?tableid=11764

[2] President’s Council of Advisors (2012).

[3] Pre-requisite Skills and Mathematics Learning (Sharma, 2008, 2016).

[4] Sharma (2015). *Numbersense a Window to Understanding to Dyscalculia *in* An International Handbook of Dyscalculia (Steve Chinn, Editor).*

[5] For the role of prerequisite skills in mathematics learning see *Games and Their Uses in Mathematics* *Learning* by Sharma. A list of games that develop prerequisite skills can be requested (Mahesh@mathematicsforall.org) from the Center free of cost.

[6] Lost Einsteins: The Innovations We’re Missing by David Leonhardt, New York Times, Dec 3, 2017.

[7] Scott Jaschik (December 19, 2017), *Women are Majority of New Medical Students.* *Inside Higher Ed*.

[8] See *Games and Their Uses* by Sharma (2008). A shorter version of this book, *Pre-requisite Skills and Mathematics Learning*, in electronic form, is available free of cost from the Center. This document includes a list of games to develop these pre-requisite skills.

[9] See www.CCSS.org/math for the eight *Standards of Mathematics Practices*.

[10] See **The Math Notebook** on *Mathematics Learning Personalities* (Sharma, 198?)

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The third of the Standards of Mathematics Practice (SMP) is mathematicians’ key occupation: construct viable arguments and critique the reasoning of others in a mathematical discourse. They discover, invent, and develop mathematics knowledge by constantly engaging in this process.

**Nature of Mathematics Knowing**

Mathematics is learned and generated by observing concrete situations and models, identifying and extending patterns, using analogous situations, and applying formal logic and reasoning to new and old situations. Developing formal reasoning provides a stronger base for learning and the development of mathematical ideas. In two previous Standards of Mathematics Practice (SMP), the emphasis was on understanding the problem—the language and concepts involved in the problem and then taking the specific concept to a general situation.

Developing reasoning, supporting one’s argument, critiquing another’s approach should not be reserved for high school geometry or advanced calculus; they should be part of all mathematics learning from Kindergarten on. Kindergarteners and first graders should be as familiar as high achieving high school students with the appropriate language (vocabulary, syntax, and mathematics sentence structure) and the development and practice of reasoning and logic (deductive and inductive; direct and indirect) such as: “prove it” “how did you know?” “how did you find out?” “defend your answer” “how can you be sure?” They should know answers to these questions and many others such as: “What definition or result did you use in this approach?” “What is wrong with this answer?” “Do you agree with …?” “why do you agree with … reasoning?” “What conclusions can you make from this?” “Is this a correct inference?” “Do you agree with that person’s reasoning?” “Why?” “Why not?” Development of and insistence on providing reasoning for their statements is not to make mathematics difficult; it is to understand mathematics better, deeper, and with understanding. Such mathematical thinking offers students the choice whether they want to be generators of mathematics knowledge or its users.

The origin of reasoning is intuition. When children’s intuitive answers are encouraged, they feel confident and are ready for formal reasoning. Mathematics is about removing obstacles to intuition and keeping simple things simple. Doing good mathematics is the interplay between intuition and reasoning—making things simple.

**Viable Arguments and Critique of Others’ Reasoning**

Mathematically proficient students understand and use stated assumptions, definitions, derived formulas, proven theorems, and established results in constructing arguments in the process of forming equations, relationships, and representations.

They can give examples for terms and definitions. They make conjectures and build a logical progression of statements to explore the truth of their conjectures and ideas.

They are able to analyze situations by breaking them into cases and recognize and use counter examples. They justify their conclusions, communicate them to others using mathematical language, and respond to questions and the arguments of others using appropriate reasoning.

*Mnemonic Devices and Mathematical Reasoning*

One hallmark of mathematical understanding is the ability to justify, in ways appropriate to the student’s mathematical maturity, why a particular mathematical statement is true, where a mathematical rule comes from, and how and when that can be applied.

There is real difference between students who can give the sum 8 + 6 as 14 by counting up or by rote memorization and the difference 17 – 9 by counting down or rote memorization and those who can find the sum by using strategies: decomposition/ recomposition of numbers, making ten, and knowing teens numbers. They see sum 8 + 6 as the outcome of strategies: 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14, or 4 + 4 + 6 = 4 + 10 = 14, or 2 + 6 + 6 = 2 + 12 = 14, or 8 + 8 – 2 = 16 – 2 = 14, or 7 + 1 + 6 = 7 + 7 = 14. They develop mastery (understanding, fluency and applicability) and develop efficient procedures.

There is a world of difference between a student who can summon a mnemonic device (**DMSB **= **D**oes **M**y **S**ister **B**ite or **D**ead **M**ice **S**mell **B**ad or **D**oes **M**cDonald **S**ell **B**urgers) to conduct the long division procedure: divide, multiply, subtract, and bring down and the student who knows why particular digits in the quotient are in a certain place or what will be the probable size (estimate in the correct order of magnitude) of the quotient before and when he completes the division procedure. Learning and applying procedures by just memorizing mnemonic is not mathematics.

Similarly, using the mnemonic device **PEMDAS (= Pl**ease** E**xcuse** M**y** D**ear** A**unt** S**ally to implement the order of operations:** P**arentheses, **E**xponents, **M**ultiplication, **D**ivision, **A**ddition, and **S**ubtraction) is purely procedural and shows a lack of understanding. It is important to know the reasons behind this order of operations (for details see the post on *Order of Operations*).

- Addition and subtraction are one-dimensional operations (linear—for example joining two Cuisenaire rods or skip counting on a number line); they are at the same and the lowest level of operations. If both operations appear in the same expression, they are executed in order of appearance, first come first serve ();

- Multiplication and division are two-dimensional operations (as represented by an array or the area of a rectangle), therefore, are at a higher level than addition and subtraction, they must be performed before addition and subtraction and if they both appear in an expression should be treated as first come first serve ();

- If all the four operations: addition, subtraction, multiplication, and division appear in a mathematical expression, the order should be: Two-dimensional operations first and then the one-dimensional operation in order of their appearance ().

- Exponential expressions are multi-dimensional (depending on the size of the exponent, e.g., a 10-cube = 10
^{3}is a 3-dimensional expression with an exponent of 3 and a base of 10; therefore, exponentiation operation is more important than multiplication (and division) and definitely higher than addition and subtraction, therefore, must be performed before all of them. Therefore, the order of operations so far is: ();

- Grouping operations are expressions included in groups such as brackets, braces, parentheses either transparent and/or hidden (compound expressions in the numerator and denominator of a fraction, function and radical operations are hidden operations. They may involve some or all of the above operations in multiple forms, therefore, are or higher preference than all of the above operations. In transparent grouping operations, the order is parentheses, braces, and brackets. The hidden grouping operations are performed in the context. Inside a grouping operation, the same order as in the above operations is kept.

Hidden operations such as: fraction and radical operations need to be brought to students’ attention. For example, the fraction expression has hidden groupings as the numerator and denominator involve extra operations, even though there is no transparent grouping operation. In order to simplify the fraction read it as: (3 + 5) ÷ (3 －1). Therefore, before we simplify the fraction (performing the division operation), we simplify the hidden operations in the numerator and the denominator. Similarly, function and radical operations are hidden: e.g., *f(a)* = 3a where *a = *and *x=2*).

Therefore, the grouping operations (parentheses, braces, and brackets in this order) are performed first. The hidden grouping is contextual. Then exponential operations need to be performed. After that multiplication and division are in order of their appearance. The last operations to be performed are addition and subtraction in order of their appearance. Therefore, the grouping operations should be performed before all of the other operations. The order of operations, therefore, should be: (). Here, G represents grouping operations—transparent, hidden, and both.

The mnemonic devices are important for remembering the sequence of activities in a multiple step procedure or operation; however, the use of acronyms and memory reminders should be only after students have understood the concepts and procedures and the reason for a particular order of operations. They do not take the place of conceptual understanding and derivation of procedures.

Similarly, there is a difference between a high school student who uses the mnemonic (FOIL) to expand a product such as (a + *b)(x *+ *y)= ab +ay +bx +by *and a student who can explain where the mnemonic comes from (application of the distributive property of multiplication over addition, applied twice: (a + *b)(x *+ *y)= (a + b)x +(a + b)y = ax + bx + ay +by*. The student who can explain the rule understands the mathematics and can use the mnemonic device productively as he may succeed at a less familiar task such as expanding (a + *b *+ *c)(x *+ *y +z) or (2x + 3y)(-2x ^{2}+6xy -5y^{2}). *

Another practice that does not develop mathematical reasoning in students is the emphasis and introduction of procedures before the appropriate conceptual schemas are developed. It is important to develop the language containers and the conceptual understanding before a procedure is introduced. Fluency of a procedure or skill without conceptual strategies robs students of applying mathematics with understanding and reasoning. It is, therefore, important to assess them both for understanding before students are asked to apply them. Both conceptual and procedural understanding can be assessed by teachers by using mathematical tasks of sufficient richness and constantly asking the question: how do you know it?

The student should first have the conceptual understanding and then use it to acquire the procedure and only then should mnemonic devices be introduced to remember and automatize the steps.

When mnemonic devices and algorithms/procedures are introduced before conceptual understanding and the development of language containers (vocabulary, terminology, language expressions), students do not show interest in conceptual understanding and apply these without knowing the reasons behind them.

When students are given mnemonic devices before they understand the concept and procedure and the related reasoning, it may be difficult for them to apply the concept, defend their work and reasoning, and communicate their results and understanding. The classrooms where use of these mnemonic devices as a proxy for mathematics is paramount, real interest and passion for mathematics are absent and difficult to achieve.

*Deductive and Inductive Reasoning*

Many in the general public and non-mathematicians and even some teachers have the misconception that mathematics is a collection of sequential procedures, and the only justification for their actions is the sequence of steps and best case the use of deductive logic. It is true that the foundations of mathematics including arithmetic are established by formal deductive logic. However, in learning school mathematics and even in some higher mathematics, there is an interplay of deductive and inductive logic. In inductive logic, one moves from many specific examples to a pattern, that helps develop conjectures and then we arrive at a general principle—theorem, formula, and procedure. It is a right hemispheric activity—looking for patterns.

Deductive logic, on the other hand, starts from the general principle—formula, theorem, definition, etc. and proceeds to its application to specific situation. It is a left-hemispheric activity—engaging in sequential reasoning. Mathematics reasoning is the interaction of these opposite but complementary activities; it is similar to the corpus callosum integrating the flow of these activities from one side of the brain to the other. In that sense mathematics reasoning is a whole brain activity—integration of thinking originating from kinesthetic to linguistic to spatial orientation/spatial organization to inter- and intra-personal to logico-mathematical intelligence. The integration of inductive and deductive reasoning spans from seeing a concept geometrically/spatially to following step-by-step procedures using sequential procedural logic.

Mathematically proficient students are able to reason deductively and inductively about data, concept, and procedures, making plausible arguments that take into account the context from which the data arose and understanding the nature and quality of the concepts and procedures.

**Applying Mathematical Reasoning as Communication**

Just like any communication, mathematics communication has two parts: expressing one’s ideas succinctly with reason and understanding others’ ideas correctly. It is defending one’s ideas but also understanding others’ ideas and identifying strengths and finding fallacies in arguments from both sides. It is not enough to defend one’s argument, it is equally important to:

(a) understand and identify others’ reasoning and its validity and effectiveness

(b) recognize the fallacies in one’s own and other’s reasoning and arguments, and

(c) correct the fallacies in one’s own and others’ arguments and approaches in a mathematics context, e.g., problem solving.

This means students are able to compare the soundness, effectiveness, and efficiency of the two (or many) plausible arguments and approaches, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is and how to fix it.

Mathematical reasoning is developmental and contextual. Children are capable of developing reasoning according to their age and mathematics concepts. For example, elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. The concrete materials they use to show their reasoning should be (a) effective, (b) efficient, and (c) elegant. Such concrete arguments should make sense and be correct even though they may not be generalizable or made formal until later grades. For example, even Kindergarten students can easily see the commutative property of addition using Cuisenaire rods and then generalized to numbers and even variables.

Later, students learn to determine domains to which an argument/reasoning based on language, diagrams or formal logic applies. At the same time, students, in all grades, can observe, listen or read the arguments of others, decide whether they make sense, and ask questions and add to clarify or improve the arguments.

**Developing Mathematics Reasoning**

Mathematical reasoning develops when we provide students experiences that help them acquire the component skills of such reasoning. Teachers’ questions aid the development of mathematical reasoning:

(a) What strategy can be used to find the sum 9 + 7?

- I can count 7 after 9.
- Can you give more efficient strategy?
- I can use blue and black Cuisenaire rods.
- Can you give a strategy without concrete materials?
- I can use Empty Number Line.
- Can you give any of the addition strategies?
- Making ten: (9 + 1 + 6 = 10 + 6 = 16)
- Making ten: (6 + 3 + 7 = 6 + 10 = 16)
- Using doubles: (2 + 7 + 7 = 2 + 14 = 16)
- Using doubles: (9 + 9 – 2 = 18 – 2 = 16)
- Using missing double: ( 8 + 1 + 7 = 8 + 8 = 16)

(b) What strategy can be used to find the difference 17 – 9?

- Using teens number: 17 – 9 = 7 + 10 – 9 = 7 + 1 = 8
- Using making ten: 17 – 9 =10 + 7 – 7 – 2 = 10 – 2 = 8
- Using doubles’ strategy 17 – 9 =18 – 1 – 9 = 18 – 10 = 8
- What to add to 9 to get to 17: 9 + 1 + 7 = 17 = 9 + 8 = 17, so 17 – 9 = 8.

(c) What is the nature of the figure formed by joining the consecutive mid points of a quadrilateral?

- To get a sense of the outcome of this construction, I will first consider a special case of quadrilateral: a square or a rectangle.
- What does the constructed figure look like in such a special case?
- What if the quadrilateral is concave? Is this assumption correct? What is your answer in this case? Why?
- What if it is convex quadrilateral? What is your answer in this case?
- Is it true in both cases?
- Is it true for any quadrilateral?
- Can you prove it by geometrical approach?
- Can you prove it by algebraic approach?

(d) How many prime numbers are even?

What is the definition of prime numbers?

How many factors does an even number have?

- 2 has 2 factors, namely, 1 and 2
- 4 has 3 factors, namely, 1, 2, and 4.
- 6 has 4 factors, namely, 1, 2, 3, and 6.
- All even numbers, except 2 have more than 3 factors.

What conjecture can you form?

For upper grades:

Can you predict the nature of any even number?

Can you prove that ___ is the only even prime number?

Is a square number a prime number?

Why is a square number not a prime number?

If *n* is a prime number, what can you say about *n + 1*?

(e) Is the product of two irrational numbers always an irrational number?

- What is the definition of an irrational number?
- Is every number an irrational number?
- Why? Can you prove it?
- If not, why?

Can you give a counter example to justify your answer?

(f) Will the range of the data change if every piece of data is increased by 5 points?

David says: It will increase by 5. Is he right?

Why? Can you prove it?

Melanie says: It will not change. Is she right?

If not, why? Can you prove it?

Can you give a counter example to justify your answer?

(g) What other central tendencies are affected by such a change? Why? Explain.

One of your classmates just stated: Such a change will not change the median of the data, is this true? Why?

When students are given opportunities to make conjectures and build a logical progression of statements to explore the truth of their conjectures, they learn the role of reasoning and constructing arguments. The teacher should constantly ask questions such as: “How did you get it?” “What did you do to get this?” “Can you explain your work?” When teachers ask children to explain their approach to finding solutions and the reasons for selecting the particular approach, children develop the ability to communicate their understanding of concepts and procedures and the ability to trust their thinking. Some questions are applicable to all grade levels:

- What mathematical evidence would support your assumption/ approach/strategy/solution?
- How can we be sure of that ….?
- How could you prove that …?
- Will it work if …?

However, some questions should be at grade level. For example, at the high school level the questions can be more content specific.

** Question:** Your classmate claims that the quadratic equation:

- What is a solution to an equation?
- What is a real solution?
- Do you agree with this claim?
- Why? Why not?
- What information in the equation assures you that it does not have any real solutions?
- How did you determine that this does not have a real solution?
- Can you change the constants in this equation so that it will have two real solutions?
- Only one real solution.

Teachers should analyze general situations by breaking them into special cases and ask students to recognize, use and supply examples, counter examples, and non-examples. This can be exemplified by questions such as:

- What were you considering when …?
- Why isn’t every fraction a rational number?
- Is every rational number a fraction?
- Is every fraction a ratio?
- Is every ratio a fraction?

To help children how to learn to justify their conclusions, communicate them to others, and respond to the arguments of others, teachers can ask questions such as:

- How did you decide to try that strategy?
- Do you agree with David’s statement? “Between two rational numbers, there is always a rational number.”
- Why do you agree?
- How will you find it?
- Why don’t you agree?
- Do you have a counter example?
- Is this statement true for all real numbers?
- Why?
- Is every square a rectangle?
- Why?

**Analogies and Metaphors as Aids to Mathematical Reasoning**

Students’ reading comprehension is improved when their thinking involves the understanding of analogy, metaphor, and simile. Similarly, the use of analogies and metaphors is an example of reasoning in mathematics, particularly in the initial stages of learning a concept. Students need to learn to reason by using analogies and reason inductively about data, making plausible arguments that take into account the context from which the data arose. For that teachers need to follow with questions such as:

- What is different and what is same about this problem and the other you solved before?
- Did you try the method of the previous problem?
- Did it work?
- If it did not work, how did you know it did not work?
- Why did it not work?
- Could it work with some changes in your approach? Why or why not? What changes would you make?
- How did you decide to test whether your approach worked?

One of the important aspects of thinking children need to develop is to know the conditions under which a particular definition, formula, or procedure applies and the parameters of its limitations. To develop this ability, teachers could ask questions about the content under discussion. For example:

- Is 3,468 divisible by 4?
- Yes or no?
- Why? Justify your answer without actually dividing the number by 4.
- Is this number divisible by 12? Why? Justify your answer without actually dividing the number by 4.
- In a fraction , if a = b, and b ≠ 0, then the fraction is equal to 1.

Do you agree wit this statement? If so, can you prove it? If not, can you give or construct a counter example to this situation?

Students need focused training and support in comparing the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is. For this a teacher may articulate questions that focus on:

- How to differentiate between inefficient and efficient lines of reasoning?
- How to focus and listen to the arguments of others and ask questions to determine if the reasoning and the direction of the argument make sense?

Finally, teachers should ask clarifying questions or suggest ideas to improve/revise student arguments.

All skills, from cognitive to affective to psychomotoric, can be improved by efficient and constant practice. In classrooms where expectations of high levels of rigor are standard, students develop proper mathematics reasoning and are keen to identify others’ reasoning and critique it.

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