Using Language Learning Methods to Learn Math, Introduction
An estimated 1.5 billion people are actively learning a foreign language. What can we learn from what they're trying to help kids who love math learn math?
Many people are learning foreign languages
In their 2020 report, Duolingo, one of many language learning companies, noted that they had achieved “enormous scale with over 500 million total users and around 40 million monthly active users, representing every country in the world”.1
Math is a language
Galileo Galilei, an Italian physicist and astronomer, published a book called The Assayer (Italian: Il Saggiatore) in October 1623. This book is famous for being considered one of the first works of the scientific method. Galileo explores the idea that nature should be studied with mathematical tools rather than those of scholastic philosophy, which had been happening until then.
Quoting from the book2:
Philosophy [i.e. natural philosophy] is written in this grand book — I mean the Universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.
Mathnasium, a Math tutoring company for kids, goes even further and, in a 2020 blog post3, breaks it down even more:
A language contains the following components:
There must be a vocabulary of words or symbols.
Meaning must be attached to the words or symbols.
A language employs grammar, which is a set of rules that outline how vocabulary is used.
A syntax organizes symbols into linear structures or propositions.
A narrative or discourse consists of strings of syntactic propositions.
There must be (or have been) a group of people who use and understand the symbols.
Mathematics meets all of these requirements. The symbols, their meanings, syntax, and grammar are the same throughout the world. Mathematicians, scientists, and others use math to communicate concepts. Mathematics describes itself (a field called meta-mathematics), real-world phenomena, and abstract concepts.
Learning languages is hard
The area of academic research that looks at learning a second language (assuming you have managed to learn your first) is called “second language acquisition”4. The research generally views the three key elements of learning the language5 as
Comprehensible input - being exposed to a “thing” in the new language and learning to understand it
Comprehensible output - learning to produce a “thing” in the new language.
Review and feedback - identifying errors and making changes in response
Each area requires a large amount of time and effort to get it right.
The Foreign Service Institute (FSI) of the U.S. Department of State, the area of the U.S. government responsible for training and helping U.S. diplomats, has compiled data gathered over the years on how long6, on average, a person should expect to take to learn a language to a passable level.
Category I Languages: ~24-30 weeks (600-750 class hours)
Category II Languages: ~36 weeks (900 class hours)
Category III Languages: ~44 weeks (1100 class hours)
Category IV Languages: ~88 weeks (2200 class hours)
If we spend an hour a day working on a language during the school week, at the bare minimum (600 hours), learning a language would take 120 weeks (600 ÷ 5) or a little over 2.3 years.
Learning languages methods
Learning a language is a hard and long process, so you can imagine that people have been trying various methods to accomplish their goals. In this section, we will examine a few language learning methods to see what they try to do.
Direct method (Oral or Natural method)
Learning is based on the student's active involvement in speaking and listening to the new language in realistic everyday situations. The learner is encouraged to think in the target language rather than translate. They hear and use the language before seeing it written. This is how babies learn their native language.Grammar Translation (traditional method of teaching Latin and Greek)
Learning is based on analysis of the written language. Uses translation exercises, reading comprehension, and written imitation of texts. The focus is on mastery of grammatical rules and memorization of vocabulary lists.Audio-Lingual (Aural-Oral method)
Learning is based on repeating dialogues and phrases. These everyday situations phrases are imitated, repeated, and drilled to make the response automatic. Reading and writing are both reinforcements of the phrases.Communicative (What most US classrooms use)
Learning is based on stressing the importance of communication. Schools have found that most students think of grammar exercises as boring, so students spend their time making requests, talking about past events, listening to native speakers, or describing images.Total Physical Response (Learning involves the body)
Learning is based on combining language and physical movements. Three ideas in this method are that grasping the language comes before speaking, comprehension is developed through body movement, and listening prepares one to speak so the student doesn’t have to speak while learning.
Language Immersion (use the language to learn the language)
Learning is based on using the target language as the means of instruction. Rather than the language being the subject material, the whole class is surrounded/immersed in the language. “The best way to learn the language is to move to the country/region where the language is spoken and just live your life”The Silent Way (developed by an Egyptian mathematician, Dr. Caleb Gattegno)
Learning is based on viewing the classroom as a space with a teacher and learning. The learners are more important than the teacher in language learning, so the teacher’s role is to direct students’ focus, facilitate self-reflection, and provide verbal and nonverbal feedback when necessary. On the other hand, students are encouraged to speak as much as possible.Whole language (everything matters)
Learning is based on seeing the language as a whole entity. This means writing, speaking, reading, and listening are integrated in every lesson and are seen to complement each other.Suggestopedia / Desuggestopedia (relax, you’ll learn better)
Learning is based on the power of suggestion for acquiring language knowledge. According to the method, students will be more receptive to learning new information if they feel relaxed and comfortable. Students listen to the language in a _very_relaxed environment (think over-stuff couches, dim soft-lights, no noise, etc)Shadowing Technique (don’t think and repeat)
Learning is based on the student pacing back and forth rapidly while listening to the target language. Students repeat out loud what they hear quickly and without hesitation. This combines Aural-Oral with physical movement.
And many, many more. As with the Shadowing Technique, you can start to combine methods to produce a new method, so many people have done this and named their method after themselves.
Using these methods to learn math
Over the next few weeks, we’ll look at what we can take away from each of these methods to help our kids who love math. We’ll even look at some unique ones like AJATT (All Japanese All The Time - a method reaching fluency in Japanese in a relatively short period written by someone who learned Japanese well enough to get a “Salary-man Office Job” in Japan in 18 months). We’ll also look at tools, like Anki and the spaced repetition world, to help us implement these methods. So stay tuned!
That’s all for today :) For more Kids Who Love Math treats, check out our archives.
Stay Mathy!
All the best,
Sebastian
https://blog.duolingo.com/global-language-report-2020/
Galilei, G. (1957) [1623]. "The Assayer". In Drake, S. (ed.). Discoveries and Opinions of Galileo (PDF). Doubleday. (Originally found through Wikipedia at https://en.wikipedia.org/wiki/The_Assayer)