Mathematixed: Freedom and Power through Understanding Math

In this episode, we continue exploring multiplication by using what we learned in Episode 5 about the five meanings of multiplication. We consider how each meaning might help us understand how multiplication with negative numbers works, and by doing so, we develop the negative number multiplication "rules" for ourselves.

Timestamps for each "multiplication meaning" that we explore:

7:50 Equal groups

20:25 Scaling

32:50 Area

44:30 Rates

48:55 Combinations

And at 49:49, we touch on how the multiplication "rules" for negative numbers extend to division.

Next episode will delve much further specifically into division as a part of multiplicative reasoning.

Thanks for listening. Here's to freedom and power through understanding math!

Hello and welcome!

You're in for another "teacher-y" episode today (sorry, not sorry... I'm on a roll and I can't stop!) By teacher-y I just mean I'll be focusing on our topic (the nature of multiplication) through the lens of the language of the standards teachers use, the grade levels we introduce things, etc...

This episode is dedicated to Keith Devlin, whose articles (links below) helped me re-think both what multiplication *is* and how we should teach it:

It Ain't No Repeated Addition

It's Still Not Repeated Addition

This episode discusses five different conceptions of multiplication, in the chronological (grade-level) order they are introduced in the Common Core State Standards. Here are some time stamps you may find helpful:

4:10 - Order of Operations ("PEMDAS") & "Is Multiplication Repeated Addition?"

13.33 - Introducing #1: Equal Groups (2nd Grade)

17.08 - Introducing #2: Array/Area (2nd Grade)

20.50 - Equal Groups in 3rd Grade

25.57 - Array/Area in 3rd Grade

26.37 - Introducing #3: Multiplicative Comparison (4th Grade)

33.00 - Scaling/Multiplicative Comparison (5th Grade)

42.33 - (Should we teach #3: Scaling earlier?!?!)

46.21 - Introducing #4: Rates (6th Grade)

49.16 - Rates cont. ("Constant of Proportionality") in 7th Grade

50.48 - Rates cont. ("Slope of a line") in 8th Grade

54.18 - Introducing #5: Combinations (7th Grade)

Stay tuned over the next several weeks for two more episodes that continue to develop multiplicative reasoning as we discuss the nature of rational numbers as well as how multiplication and division behave when using negative numbers. 

Here's the conclusion to our mini-series on additive reasoning!

This episode is dedicated to the ancient Indian mathematician Brahmagupta, one of the first (that we know about) to not only treat zero as a number but to work extensively with negative numbers as well.

If you're interested in receiving a free guide to learning about (or teaching) additive reasoning with integers, sign up here!

It's the longest episode yet, so here are some time stamps in case you want to jump around:

• 4:54 Number categories
• 8:59 The story of integers
• 20:38 Addition: Static thinking (protons/electrons; money/debt)
• 32:22 Addition: Kinetic thinking (up and down number lines)
• 38:47 Subtraction as removal: Kinetic thinking
• 47:50 Subtraction as difference: Static thinking
• 55:01 Bringing it all together

We'll be back in a few weeks to discuss multiplicative reasoning, so in the meantime I'll leave you with this question: Is multiplication repeated addition? Find out how I ended up answering that question on our next episode!

In this episode we dive into the operation of subtraction and explore how intimately it is connected with addition. Both addition and subtraction are additive ways of reasoning, and it's generally not beneficial to try to consider one operation completely in isolation from the other!

There are so many great math educators out there providing training and resources that help build understanding around these operations, but the two specifically mentioned in today's episode are:

Christina Tondevold (specific vlog mentioned: How To Make Subtraction Not So Hard)

Steve Wyborney (activity mentioned: Splat)

Be sure to keep an eye out for next week when we allow ourselves to consider the entire realm of numbers that opens for us when we subtract a larger number from a smaller number, and how all these ways of thinking we've developed so far will enable us to navigate this uncharted territory with relative ease. :)

And if you'd like to stay informed of new episodes and everything else we're up to, be sure to sign up for our email list.

Happy listening!

In this episode we're going to start our deep dive into the type of thinking we do when we combine numbers additively. I'll discuss the two words I use to describe what's going on in my head both when considering contexts and performing calculations, and we'll get a peek into where additive thinking fits in the larger framework of mathematical reasoning.

Shout out to Pam Harris for her extremely useful Development of Mathematical Reasoning graphic. Check out her workshops, (including a free one that expounds on the DMR graphic!)

And thank you, twitter user @Wparks91 (and perhaps many others) for generating widespread awareness of different mental math processes for 27 + 48 through this tweet.

You can find my video regarding the mental development of "conservation of number" here.

Also, as for the math joke at the very end...

There are 10 kinds of people in the world: those who understand binary, and those who don't.

...Binary refers to a base-two number system. It compares with our decimal (base-ten) system in these ways: while base-ten has ten digits (zero through nine), base-two has two digits (zero and one). Also, as you shift each place to the left when writing digits in a number, when in base-ten, the value of your digit is increasing by a factor of 10, while in binary, it is increasing by a factor of 2. So the binary system has the ones place, then the twos place, then the fours place, etc.; and the numbers we consider 1, 2, and 4 in our decimal system would be considered 1, 10, and 100, respectively, in the binary system. Hence, referring to "10 kinds of people", while using the binary system, is referring to precisely the same amount of people that our familiar digit of 2 is referring to.

As always, my hope is that your day is filled with freedom and power, and if understanding math better can contribute to that, may it be so!

Welcome to our very first episode! We're going to start with the most basic of math concepts: counting. But we'll take it to a place beyond even the largest of numbers - and where the podcast leaves off, the exploration continues at mathematixed.com.

This episode is dedicated to the mathematician whose ideas we'll be exploring today, the one and only "Father of Set Theory" Georg Cantor (1845-1918). As discussed in the podcast, here are just a few of the many wonderful things he said:

• "Great innovation only happens when people aren't afraid to do things differently."
• "Don't always blindly follow guidance and step-by-step instructions - you might find something interesting."

And my personal favorite:

• "The essence of mathematics is in its freedom."

A few other things referred to on this podcast:

Peter Liljedahl and his outstanding book, Building Thinking Classrooms

Land of Oden

Amazing Grace

If you'd like to support the podcast by becoming a member of Mathematixed, you can find out more about that here.

Or feel free to just listen and think along with me... let's spread the love of math around!

Enjoy!