Chapter 10: Why do we Invert and Multiply?

When dividing two fractions, we are taught to "invert and multiply" (and don't ask why). It is the rule that all math students inevitably learn, relatively few challenge, and fewer can rationalize. It is like wetting your toothbrush before adding toothpaste and brushing...you don't quite know why you do it, you just know it works.

This week in class, we were challenged to rationalize the "invert and multiply" rule. Why does it work? How can we explain why it works? How can we show that it works? Below is my thought process on why we are able to "invert and multiply" two fractions that are being divided:

Let's use the following equation:

1/2 ÷ 3/4

Since these fractions are being divided, we could rewrite the equation as:

1÷3 
2÷4

Now we have a fraction in the numerator (1/3) and a fraction in the denominator (2/4). We can now multiply the entire equation by the fraction of 3/3 (to get rid of the 3 in the numerator fraction) and 4/4 (to get rid of the 4 in the denominator fraction)...excuse my terminology!:

[(1/3) ÷ (2/4)] x (3/3) x (4/4)

which can also look like:


(1÷3)(3)(4)
(2÷4)(3)(4)

As you can see, multiplying then dividing by 3 in the numerator and 4 in the denominator would cancel the respective terms, leaving:

(1) x (4)     
(2) x (3)

 If we separate the terms, we see our rule... invert and multiply!

(1/2) x (4/3)

https://www.pinterest.com/explore/dividing-fractions/

The mathematics applied to this solution is not necessarily complex, and I am sure there are a million other more effective and simple ways to solve it (see the picture to the left). The point I am trying to make, is that, using either numbers, visuals, or manipulatives, mathematics educators should work with students to facilitate the derivation of the "invert and multiply" (and other!) math rules. An unexplained/non-derived formula or rule is a difficult concept to grasp, and passing it along without question promotes a passive style of learning which resonates with primitive methods of teaching. I truly believe that something as simple as collaboratively deriving the "invert and multiply" rule could drastically affect the confidence, and therefore academic ability, of all learners within a mathematics classroom.




Thanks!


UPDATE: Here is a link to essentially the same proof as the one shown in the blog, but shown in a much clearer manner! It also shows other ways of proving the "invert and multiply rule"!

http://www.mikesmathclub.org/div_fractions.pdf

Chapter 9: Marking - Behind the Scenes

Marking a math solution is fairly straightforward, right? You take a student's response to a question, you compare it to the actual solution, and you give marks according to how well their response compared to the actual solution. But is it really just that straightforward? I would argue strongly that it is not. 

Take for instance, a mathematics problem that can be solved in more than one way (otherwise known as any math problem). If a student were to be evaluated against an "answer key" that differs in structure from their response, they would not receive full credit, despite getting to the correct answer. 

What about students who use manipulatives or geometric representations to discover a solution? If they solve a question using pictures, is their "full solution" any less valid? For example, lets examine the question below:

A online store called Amathon is running an "add-on" promotion. The promotion states that if you order an item that costs $25 or more, they will "add-on" a free gift. If you place an order with two items that cost $25 or more each, they will "add-on" three free gifts. If you place an order with three items that cost $25 or more each, they will "add-on" five free gifts, and so on.

1) If you placed an order with ten items that cost $25 each, how many free gifts will you receive?

2) Write a rule that could be used to determine the number of free gifts you would receive for any order containing items of $25 or more. 

This answer could be solved in a number of ways. Most academic math students would simply model the situation using listed values, or values in a table. However, some students may want to use of a graphical or geometric representation of the situation (see example below) in order to make conclusions. 

original image 

Are any of these methods more valuable than the others? Again, I would strongly argue that no, they are all equally valuable assuming they lead to the correct answer. There is no way that a "marking scheme" could prepare for all of these possible solutions, so how can we fairly and accurately mark this question? The situation becomes even more complicated when we consider that a student may get only some of the question right. 

I believe that we need to revise the way we think about marking. A solution to a problem is not a static set of steps, but rather a fluid group of mathematical processes that lead to a final conclusion. Although a "mark scheme" can help us determine whether a student was able to get the correct answer, it should not be used to dictate whether a students' thinking process is right or wrong. When marking math questions, educators should take more of a holistic (for lack of a better word) approach. The final solution is important, but the process a student takes to get there is just as, if not more, important. Sure, it might be hidden within a graph, a seemingly random group of numbers, or a drawing that might look like a doodle, but it is our job to extract a students thought process and evaluate it accordingly. 

Thanks for reading!

Chapter 8: Online Forum Reflections

Online Session One (Orchestrating Mathematical Discussions):

This session focused on ways that teachers can help their students move from a "show and tell" type dialogue, to building a mathematics community that shares rich, meaningful, and productive math conversation. The article provided highlighted five methods that mathematics teachers are able to do so. We then practiced one of these methods (anticipating student responses) through the "tiling task" activity. In this activity, we solved a question in our own method, as well as in a method we might anticipate a student would use. We then consolidated the session by performing a similar task, and interacting/reflecting on our peers work.

How important is the task when trying to stimulate the richness of discussion in the mathematics classroom?

I believe that the task is critical to the richness of discussion in the mathematics classroom. Think of a rich, deep, or just long conversation you have had in the past. I am confident that at least one of the parties involved in the conversation was very passionate about the topic. As mathematics educators, it is our job to make a math task something that students can "buy into", or be passionate about. Although there is certainly more at play than just the task (presentation of the task, materials provided, etc), its nature alone can stimulate rich conversation if students are able to invest in it.

Discuss how the use of different talk structures (whole-class, small-group, teacher-led, student-led, etc.) can affect mathematical discourse in the classroom.

I think that the way mathematics information is exchanged should vary throughout an instructional period. Relating the topic back to differentiated instruction, different students respond better or worse in different situations. For instance, some students may feel more comfortable speaking to a partner or small group rather than to the class, or to the teacher. The richness of dialogue will change under those different circumstances. Therefore, in order to optimize discussion, as well as reach the need of every child, different talk structures should always be used in the mathematics classroom.

Online Session Two: Formative Assessment

This session focused the impact that assessment can have, and how teachers can provide effective assessment to students. We began by watching and commenting on videos regarding the aforementioned topics. Next, we read about, and co-created learning goals and success criteria for a math problem. We also reflected on a time where assessment has really worked for us in the past, and discussed what it means for assessment to be "effective". Lastly, we consolidated the session by responding to students work. The goal for the consolidation piece was to provide effective assessment.

How do you decide as a teacher that you need to “re-teach” a lesson or part of a lesson?  How would you let the students know why you are re-teaching? 

Assessment is key to determining whether students are able to move past a topic, or whether they need to be "re-taught". If the assessment piece you are using for a topic indicates that some students have not mastered the subject, I think it is perfectly fine to let the class know. Saying something along the lines of "Okay folks, so I looked over our exit cards from last night, and I feel as if we could all benefit from a little clarification". The assessment piece should indicate specifically who needs help, and where they need it.

How important do you think timing of feedback is? Explain.

I believe that the quicker students are able to receive feedback, the quicker they are able remediate any issues they had. Therefore, by providing instant feedback, we can optimize the way students learn. Todd Malarczuk from the I/S Math Conference provided us a few different methods for providing instant feedback such as google docs or DESMOS. He stated that instant feedback has changed the way his students learn, and is something we will continue to investigate throughout his career.

Representation of how DESMOS can be used to provide students with instant feedback from https://teacher.desmos.com/marbleslides-parabolas#