49

It's purely a matter of how we choose to define the notation. The main reason for it is that it lets us write polynomial expressions (which are extremely common) without parentheses, e.g., $x^3 + 3x^2 y - 41x + 2z$ rather than $(x^3) + (3(x^2)y) - (41x) + (2z)$. However, what really matters is that the notation is clear and unambiguous, so expressions like $...


48

I couldn't agree more with @Steve's comment. The following response is written with elementary-to-high-school mathematics in mind. A lack of a decent number sense really does encumber making sense of and parsing word problems, as well as the process of exploring solution strategies. It is akin to interpreting a passage written in a not-so-familiar dialect: ...


45

https://www.theguardian.com/news/datablog/2013/may/31/times-tables-hardest-easiest-children There are links to a dataset in the article. As far as I can tell, this isn't a formal study: But some new data generated by pupils at Caddington Village School in Bedford sheds light on which multiplications are actually the hardest – and how kids do overall. The ...


36

You asked: "How do/would you explain why division by zero does not produce a result." Any such explanation that is not rooted in student understanding would be talking to ourselves, not to students. Therefore both meaning and student understanding are important. Otherwise, what's the point? So I have grounded my response there. Young students (...


28

Have the students tell you that division by zero is a non-sequitur. This is possible at any age where division is understood at all. Teacher: If there are eight cookies and four children, how many cookies does each child get? Student: Uh, two. Teacher: Yes! This is a division problem. $\frac{8}{4} = 2$. Now, if there are 8 cookies shared by only two ...


27

It's pretty rare these days for anyone to actually do division by hand. Most people reach for a calculator. Given those realities, I would question whether it even makes sense that we spend such a vast amount of time teaching young children even one algorithm for division. Maybe we should postpone it, deemphasize it, or replace long division with a slower or ...


23

I find the ability to estimate calculations quite useful and I think you need to be able do do calculations to estimate them. If you are keeping a grocery budget, I would suggest you should know what your groceries will cost within $10\%$ before they are rung up. To know that, you need to be able to add and multiply in your head. You don't need many ...


21

Yes! But the virtue doesn't lie in being able to do the calculation but in gaining a feel for numbers as well as algorithmic thinking. I teach Computer Science freshmen and one of the first things we need to do is introducing base 2 as well as number systems working with modulo (two's complement). We also introduce basic circuitry to do addition/...


20

The issue is the implied multiplication in 8÷2(2𝑥). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous. If implied multiplication works the same way explicit multiplication does, then we do the division first (left to right) and get 4(2𝑥). If implicit multiplication has a higher priority than ...


19

In third grade we taught division using repeated subtraction. To divide 6 by 2, subtract 2 until you get to 0. 6-2=4, 4-2=2, 2-2=0. It took 3 steps so 6÷2=3. This can also be shown on a number line, where it takes 3 steps of 2 units to go from 6 to 0. Teaching the concept of division this way is just the inverse of what we have done for multiplication. ...


16

If the expression were, say, $48\div 4\times 12$, there would not be much disagreement (multiplication and division are performed from left to right). But an expression such as $48\div 4(12)$ results in disagreement because the parentheses could mean one of two different things: a way of grouping or a way of multiplying. If one interprets the parentheses ...


16

I taught at the elementary and high school levels. At times we used calculators and at times we didn't. Students benefit from experience both ways. Students need to learn that calculators are only a tool and they still have to think. Students also need to learn that having a calculator doesn't guarantee that their computation will be correct. Finally ...


15

While there may be legitimate reasons behind the convention In $a \times b $ the $a$ denotes the number of terms and the $b$ denotes the individual terms the larger issue is the mismatch between the teacher's enforcement of that convention and the expressly stated purpose of the formative assessment, which is written at the top of the very same page: ...


14

You can use (basically) Newton's method. Take $x_0>0$, and define $$x_{k+1} = \frac{1}{2} (x_k + \frac{n}{x_k}).$$ The seqeunce converges to $\sqrt{n}$. And if one starts with $x_0= n$ one has, as soon as $|x_{k+1} - x_k|< 1$, that $\lfloor x_{k+1} \rfloor = \lfloor\sqrt{n} \rfloor$. But the above is not very feasible for computing by hand, and ...


14

My preferred model of multiplication with integers involves motion. Imagine you are recording a video of a car driving at a certain velocity. If the car is going forward, the velocity is positive; if the car is going backward, the velocity is backward. Now when you play the video, you can choose whether to play it at normal speed, or at 2x, 3x or 4x ...


13

Dividing $1$ disk into $\frac{1}{n}$-ths ($\frac{1}{3}, \ldots$), leads to $\frac{1}{1/n} = n$ pieces ($3,\ldots$): As $\frac{1}{n}$ approaches $0$, the number of pieces $n$ grows without bound. The result upon division by $0$, $1/0$, should be this limit. But there is no limit.


13

A reason why this form might be preferred is the way one says it: $5 \times 3$ is read out "five times three" so it says take $3$ five times, hence it "is" $3+ 3+ 3 + 3 + 3$. However I doubt there is any real standard. For what it's worth Wikipedia disagrees with itself. On the page on Multiplication it has $a \times b$ as $b + \dots + b$. On the page ...


12

My immediate response is 'wait a few years'. I've spent a fair amount of time with 3 year olds, and most of them are busy learning how to be a person in their own right, how to have a conversation, what the difference is between real and make-believe, and (often) how to tell when they need the toilet. I've read that they can't understand metaphors by that ...


11

One place in math where this issue actually does come up is in defining ordinal multiplication. From an ordinal perspective, the ordinal $5$ is the order type $a<b<c<d<e$, the ordinal $3$ is the order type $x<y<z$, and $5 \times 3$ is $$a_x < b_x < c_x < d_x < e_x < a_y < b_y < c_y < d_y < e_y < a_z < b_z &...


11

This answer is intended for the second category of students: How do/would you explain why division by zero is undefined for Algebra students? Begin by introducing the reciprocal of a real number as that number that satisfies the property of reciprocals: $$a⋅\dfrac 1{a} = 1$$ Note here that $\dfrac1a$ must be thought of as the result of dividing $1$ by $a$ ...


11

Perhaps it is worth pointing out that every programming language defines an operator precedence structure to avoid ambiguities. An example table for C and C++ can be found here. Ambiguities must be avoided in order for the language parser to create the correct compiled (or interpreted) machine code to implement the expression. For example, the expression $4+...


11

One way to see if the student understands the commutative property of addition is to have "fill-in-the-blank" questions such as $$2+3=3+\_\_$$ $$2+3=\_\_+2$$ $$2+\_\_=3+2$$ $$\_\_+3=3+2$$


10

There's a fun method which I've seen referred to as Russian peasant multiplication or ancient Egyptian multiplication. (I don't know if these names have a historical basis.) If you think about how the algorithm works behind the scenes, it's essentially multiplying the two numbers in base 2, so it could have more than one pedagogical use. This technique is ...


10

Generating systems. The same method that works for linear equations works also for polynomial equations. Starting with a solution in mind (in mathematics and computer science, we call this a planted solution), generate in some way left-hand sides of equations, and then compute the corresponding right-hand sides. If you use more than one equation per variable ...


9

There is an algorithm to calculate square roots with paper and pencil (like the long division algorithm). See How to calculate a square root without a calculator.


9

it sounds like the textbook has some poorly written and vague questions. there is no reason why either of the answers would be "better" for the first example. It is simply dependent on how you interpret the words. You would be surprised how many textbooks actually include similar vagueness in some questions and it is concerning to say the least.... keep up ...


9

You can find some further studies that cited the original writing on google scholar: link Mahajan (PDF) remarks that: Etta Berman, one of the teachers in the program, studied it for a masters degree (1935); she gave a battery of quantitative tests to 106 control and 82 experimental students in the sixth grade. ...


9

Here are the two previous pages from those materials (a pre-publication version found with a Google search): And here is the page containing the homework problem in question: The intent now seems pretty clear. Students know that you can join groups by adding them, but in the case where the groups are equal in size (e.g. five bags, each with nine goldfish), ...


9

The problem is due to imprecise specification of the intended result. Here's a more precise way. $\text{Recall that the }{\bf commutative\ law}\ \color{#c00}X+ \color{#0a0}Y = Y + X\ \text{ is true for all reals } X,Y$. $\text{Use the above law to $ $ simplify }\ 2\, +\, \color{#c00}{\pi}\, +\, \color{#0a0}3\ \text{ to the form }\, n + \pi\,\text{ for ...


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