Ben Hartley - Math 8 Squares, Cubes, Roots, Pythagorean Thm - Unit Plan (Draft)

Disclaimer: Some of the of formatting is a bit funky after pasting from my other document. Sorry about that.

 

 

EDCP 342A Unit planning: Rationale and overview for planning a unit of work in secondary school mathematics

Your name: Ben Hartley
School, grade & course: J.N. Burnett, Math 8

Topic of unit: Squares, Square Roots, Pythagorean Theorem

Preplanning questions:

(1) Why do we teach this unit to secondary school students? Research and talk about the following: Why is this topic included in the curriculum? Why is it important that students learn it? What learning do you hope they will take with them from this? What is intrinsically interesting, useful, beautiful about this topic? (150 words) This unit builds some background geometry skills and introduces the concept of radicals and square roots with integers. The first reason why this topic is taught in secondary school is that these are mathematical skills that will be used regularly in subsequent math classes. The second reason is that these skills are generally useful in many real work applications. Many people will, at various points in their life, want to build something with their hands. They will likely need to construct a right angle or calculate an area. These skills are relevant for tasks like that. The third reason to learn about the Pythagorean Theorem is that it is a valuable bit of our collective human history. The fourth reason is that the Pythagorean Theorem is one of the first theorems that students might come across. Teaching it introduces students to the concept of theorems and potentially to the concept of proofs!

(2) A mathematics project connected to this unit: Plan and describe a student mathematics project that will form part of this unit. Describe the topic, aims, process and timing, and what the students will be asked to produce, and how you will assess the project. (250 words) Topic: Mathematical Poetry Aims: Motivate students to reiterate and recall how to solve a specific kind of mathematical problem (Pythagorean geometry problems), Give students opportunities to associate math and art as related fields, Students have fun! Process: At the beginning of the unit students are presented with the project: For one type of problem that we learn about during this unit students will have to write ‘instructions’ on how to solve such a problem in poem form. (The instructions can be fairly vague, especially for poem forms like haiku that are very short) However, the instructions do need to be accurate! This should be challenging linguistically! When writing the poem is challenging students will want to try to write a poem for each new topic when they think that it might be easier. Writing poetry is also meditative. Giving students time to clear their desks and work quietly in their own mind is a chance to step away from any other worries. Students should NOT use any AI in order to write their poems. As the teacher, I will spend some time producing as many distinct examples out of chat gpt as possible so that if students produce something too similar then I will see that they used AI tools. The other method to prevent AI usage is to have students work on this entirely during class, without their phones. Optionally, students will have a chance to share their poems aloud. Timing: Small amounts of work time spread out throughout the entire unit. Product: Students will produce a poem that captures how to solve a particular problem that they will want to solve. Assessment: The poems will be assessed based on the accuracy of the student’s poems (correctly describes the process of solving their chosen problem) and the inclusion of poetic features.

(3) Assessment and evaluation: How will you build a fair and well-rounded assessment and evaluation plan for this unit? Include formative and summative, informal/ observational and more formal assessment modes. (100 words) Though assessment methods have to be different for different learning standards, there are few different categories that are fairly similar. Formative assessments: Students are assessed daily in the form of “thumbs up / thumbs down” feedback when discussing new topics. Class engagement, and small quizzes, also factor into this kind of assessment. This data helps the teacher identify students that might struggle with particular concepts or to identify topics that need additional review as a class. Summative assessments: The students will have to option to take a written paper exam or to do an oral exam. In either case each student will demonstrate whether or not they meet the learning standards specified in the BC Curriculum. Some of the learning standards cannot be evaluated in these ways (learning standards related to collaborating in mathematics or to social skills) and instead are evaluated based on the products of group activities and projects and their behavior during.

Elements of your unit plan:

a) Give a numbered list of the topics of the 10-12 lessons in this unit in the order you would teach them.

Lesson	Topic
1	Perfect Squares
2	Perfect Cubes
3	Square Roots of Perfect Squares / Cube Roots of Perfect Cubes
4	Quiz 1
5	Squares (Fractions and Decimals)
6	Estimating Square Roots of Non-Perfect Squares
7	Estimating Cube Roots of Non-Perfect Cubes
8	Quiz 2 & Open Cubes Math Art
9	The Pythagorean Theorem I
10	The Pythagorean Theorem II
(11)	Quiz 3
(12)	Exam

b) Write a detailed lesson plan for three of the lessons which will not be in a traditional lecture/ exercise/ homework format. These three lessons should include at least three of the following six elements related to your mathematical topic. (And of course, you could include more than three!)

These elements should be thoroughly integrated into the lessons (i.e. not an add-on that the teacher just tells!)

a) History of this mathematics

b) Arts and mathematics

c) Indigenous perspectives and cultures

d) Social/environmental justice

e) Open-ended problem solving in groups at vertical erasable surfaces (“thinking classroom”)

f) Telling only what is arbitrary, and having students work on what is logically ‘necessary’

Be sure to include your pedagogical goals, topic of the lesson, preparation and materials, approximate timings, an account of what the students and teacher will be doing throughout the lesson, and ways that you will assess students’ background knowledge, student learning and the overall effectiveness of the lesson. Please use a template that you find helpful, and that includes all these elements.

Lesson Plan 1: History of Mathematics – Pythagorean Theorem

Lesson Plan 2: Open-ended Problem Solving – Estimating Cube Roots

Lesson Plan 3: Sol Lewitt’s Variations on Open Cubes – Arts and Mathematics

Lesson 1

Title	History of Mathematics – Pythagorean Theorem
Duration	1.5 classes – project
Description	This lesson introduces the Pythagorean Theorem by embedding it in the history of its use and discovery. Each student will practice their research skills by investigating how ancient people used the Pythagorean Theorem, and the different ways that it was discovered or expressed in different cultures. The product of the presentation can be a short presentation, a written report, or an art piece that summarizes their findings.

Prerequisite Learning Standards

Learning Standard	Relevance	Assessed by...
Two-Step equations with integer coefficients, constants, and solutions.	Solving an equation using the Pythagorean Theorem requires some algebraic manipulation skills.	Outcomes in prior units
Square and cube roots.	Taking square roots is necessary to solve using the Pythagorean Theorem.	Outcomes in prior units
Perfect squares and cubes.	Squaring numbers is necessary to understand Pythagorean Triples.	Outcomes in prior units

Learning Standards

Learning Standard	Relevance	Assessed by...
Model mathematics in contextualized experiences.	Students must investigate how ancient people’s used the Pythagorean theorem.	Analysis of project product and the work that went into it.
Visualize to explore mathematical concepts.	The Pythagorean theorem is related to geometry, which is very visual.	Analysis of project product and the work that went into it
Engage in problem-solving experiences that are connected to place, story, cultural practices relevant to local First Peoples Communities, the local community and other cultures.	The whole point of the project is to have students engage with mathematical history.	Analysis of project product and the work that went into it
Explain and justify mathematical ideas and decisions.	The students will have to consider how the ancient people’s discovered these equations, which involved justification.	Analysis of project product and the work that went into it
Communicate mathematical thinking in many ways.	The various allowed forms of the final product give students a chance to communicate their thinking in many ways.	Analysis of project product and the work that went into it
Represent mathematical ideas in concrete, pictorial, and symbolic forms.	Geometry and algebra are both very relevant to the Pythagorean Theorem.	Analysis of project product and the work that went into it
Pythagorean Theorem	Of course, the students will be engaging with the Pythagorean Theorem and investigating how it is used.	Analysis of project product and the work that went into it

Lesson 2

Title	Estimating Cube Roots
Duration	0.5 classes – instructional
Description	This lesson is a chance for students to try to work out a solution to a tough problem without being presented with the answer. In prior lessons the students will have learned a method for estimating square roots (what amounts to a linear interpolation at each of the perfect squares). The students will then be given time to come up with a strategy for estimating cube roots of non-perfect cubes. There are many strategies they could come up with.

Prerequisite Learning Standards

Learning Standard	Relevance	Assessed by...
Square roots and cube roots	The students need to understand what it means to take a cube root before they can try to estimate it.	Outcomes in prior units
Perfect squares and cubes	In order to understand the concept of cube roots it is helpful to understand cubing.	Outcomes in prior units

Learning Standards

Learning Standard	Relevance	Assessed by...
Use reasoning and logic to explore, analyze, and apply mathematical ideas.	The students need to use reasoning in order to connect this estimation problem to other concepts they’ve learned or to come up with new ideas.	Observation during the activity work time along with the final product.
Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving.	This is an inquiry and problem solving task.	Observation during the activity work time along with the final product.
Visualize to explore mathematical concepts.	Students will likely want to graph these functions in order to get a sense of what is happening.	Observation during the activity work time along with the final product.

Lesson 3

Title	Sol Lewitt’s Variations on Open Cubes – Arts and Mathematics
Duration	3 classes – project (distributed in ½ classes throughout semester)
Description	This lesson is an ongoing project that students can work on throughout the semester. It is not explicitly related to any of the content learning standards, however, it does provide practice in creative thinking, spatial reasoning, and logic applied to mathematics. The project is modeled on an art piece by Sol Lewitt’s called “Variations on Open Cubes”. The project asks, what are all of the possible ways that you could make a 3D wire-frame cube with some edges missing (requiring that it remain 3D and that the remaining edges are connected.) Answering this question requires lots of experimentation, spatial reasoning, and discovery. Plus, it is something that is very easy for students to actually create with their hands. The final produce is a set of as many distinct cubes as the students can find along with some justification that those cubes are distinct.

Prerequisite Learning Standards

Learning Standard	Relevance	Assessed by...
n/a

Learning Standards

Learning Standard	Relevance	Assessed by...
Use logic and patterns to solve puzzles and play games.	This is a mathematical puzzle.	Observation during project work time and final product.
Use reasoning and logic to explore, analyze, and apply mathematical ideas.	The students need to use a lot of reasoning if they are going to find all of the incomplete cubes.	Observation during project work time and final product.
Apply multiple strategies to solve problems in both abstract and contextualized situations.	The students will likely have to use multiple strategies to work through this problem.	Observation during project work time and final product.
Visualize to explore mathematical concepts.	This is a very visual problem at its core	Observation during project work time and final product.
Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving.	The students will demonstrate their understanding of rotation and symmetry.	Observation during project work time and final product.
Use mathematical vocabulary and language to contribute to mathematical discussions.	Students will need to use some mathematical terminology in order to justify their work.	Observation during project work time and final product.
Represent mathematical ideas in concrete, pictorial, and symbolic forms.	The students will be encouraged to actually build models of the cubes they discover.	Observation during project work time and final product.
Construction, views, and nets of 3D objects.	The students will be working with cubes and thinking about the various views that you can look at a cube with.	Observation during project work time and final product.

Curricular Micro-Teach Reflection

 Based on the feedback slips that we got after our curricular micro-teach, there are a few things that we could improve on. Here is a summary of some common points that people brought up

• The Desmos activity was difficult to navigate to. we should've had a QR code or something easier to help us get there
• The pace was a little bit fast and we ended up having time at the end so we could've slowed down.
• The transitions felt slightly clunky going from one teacher to the next. 

I generally agree with all of these points and they are things to keep in mind for future teaching endeavors. I think it is also difficult to put together a group presentation in a short amount of time and have it run perfectly smoothly, especially when we have other classes and work to get done. For me personally, getting the timing right is the main area that I want to improve in. This is something that I also encountered during my short practicum and beyond.

 Some things that I think went well were the way that our group split up the work evenly and checked in with each other to make sure that our topics were aligned. We were able to play into each of our preferred styles in order to make the whole presentation more enjoyable and engaging.

Non-Curricular Micro-teach reflection

 This was a blog post that I forgot to put together earlier and then it fell by the wayside as other assignments have come up. I've refreshed myself by re-reading the feedback sheets that my group members wrote.

 I did my non-curricular micro-teach on whistling and humming. There were a few things that went well:

• Everybody was very engaged and actually genuinely wanted to be able to make the whistling sound.
• Most of the group was able to make the whistling sound by the end of the 10 minute micro-teach
• My group members related their new skill to topics in the international phonetic alphabet

Based on these I would summarize that I picked a good topic and (with the help of shifting groups around) was able to make it so that the students in my group were prepared to learn this new skill. I can imagine that it might've gone differently if most of my group was not able to whistle.

Some things that could've gone better were the timing of the lesson. I did not anticipate that we would get through it so quickly! I definitely had time to maybe throw in a quick quiz on how the concepts relate to the IPA or extend the activity by asking the group what other sounds we make could be voiced or unvoiced. However, I don't think that this is so bad because we did have a fruitful discussion about how this skill relates to other musical skills in singing and in instruments (its a lot like how a trumpet is played, which I did not know).

 Overall I am very happy with it because even I learned something new! 

Textbook Blog Post

"Think about your answers to these questions, discuss
your ideas with other students and your teacher, and
then write a summary of your findings in your journal.
(e.g., p. 25)
From this, the model student might see himself or herself
doing mathematics independently, but reflecting on his or
her mathematics in community. " (p. 12)

I recall that as a student I definitely treated mathematics as something that I would do in a solitary way and then (when forced to) I would communicate it with others. However, I was a particularly introverted student in primary and secondary school. A passage like this in a text book would be a big relief to me and make me more comfortable during the activity.

As a teacher I want to respect students that may have similar temperaments to my past self while also helping students to meet learning standards that include social and collaborative competencies.

I don't think that it is accurate to say that a passage like this will cause students to take up a particular view of mathematics as being more individual than collaborative. Students are not that small-minded.

"Morgan (1996) notes that the absence of first person pro-
nouns obscures the presence of human beings in a text and

affects
not only [...] the picture of the nature of mathematical
activity but also distances the author from the reader,
setting up a formal relationship between them. (p. 6)
In both example textbooks, first person pronouns were
entirely absent. Two forms of the second person pronoun
‘you’ are common in mathematics textbooks:
1. ‘you’ + a verb
2. an inanimate object + an animate verb + ‘you’ (as
direct object)." (p. 10) 

 I think that this statement is absurd. The crux of the argument here is clear in the statement: "The absence of first person pronouns obscures the presence of human beings in a text." But this can be equally applied to Pride & Prejudice by Jane Austen. Pride and Prejudice is written in the third person so it contains very little first person pronouns. Does that mean that the presence of human beings is obscured? No. The whole book is entirely focused on people and their relationships.

"1. How might a textbook position students in relation
to mathematics?
2. How might a textbook position students in relation
to their peers?
3. How might a textbook position students in relation
to their teacher?
4. How might a textbook position students in relation
to other people?
5. How might a textbook position students in relation
to their own experiences?" (p. 9)

In response to these questions I will try to answer them with respect to some of my favorite textbooks. One of my favorite textbooks (from a student perspective) is Landau Lifshitz's Introductory Mechanics Textbook. It is a physics mechanics textbook, not a math textbook, but I think the principles are generally applicable.

This textbook positions the reader as somebody that has a very strong mathematical background but who maybe does not yet know how to apply that knowledge to physics. Moreover, it has a lot of confidence in the reader to be able to figure out tough questions.

The textbook does not position the reader with respect to peers in any way. It presents exercises but it give no explicit instruction on how to use them. However, there is an implicit positioning in that the exercises are typically *very hard* this means that students need to talk to each other or to other sources in order to succeed efficiently.

The position in relation to the teacher is very similar to the position in relation to peers. Part of this is because the exercises are unique and tough enough that even many professors and teachers will struggle for a time to solve them. This deconstructs the differentiation between student and teacher. All of the above are peers for some time.

The textbook presents no explicit or implicit positioning for students in relation to the broader population.

The position of the student with respect to their own experiences is very important in this text because one of the aspects of this book that I love the most is how it forces you to struggle and then succeed. The experience of struggling and then ultimately succeeding based on my own thinking and engaging with the text is a very empowering feeling. 

"The equations below express the relationship between
a person’s height, H, and femur length, F:
Males: H = 27.5 + 2.24F
Females: H = 24 + 2.32F
[...] If a man’s femur is 20 inches long, about how tall
is he? (Lappan, Fey, Fitzgerald, Friel and Phillips,
1998, p. 21)" (p. 8) 

This excerpt is clearly a strange example to have in a textbook for a wide variety of reasons. I think that it exemplifies a lot of lazy ways in which mathematics is applied to the lives of humans. Our examples should take into account cultural background, avoid generalizations, and respect individual diversity. The concept of 'normal' and 'average', though it has a "neutral" meaning in mathematics, is not neutral in broader society and thus we should always be careful about how it is applied. 

Flow State Response

 Thoughts on the Flow State Talk:

I have a handful of thoughts on the idea of flow. 

 The first thought that comes to mind is skepticism over the concept itself. Is 'flow' something that actually happens in the brain or is it an illusion that we attribute to other actual phenomena? Take, for example, the diagram that we have seen associated with flow:

 

I want to compare this diagram and its subtext with the proposed mechanism behind 'flow'. Specifically, how does the idea of flow being caused by the rate limits on the brain's ability to process information. Based on this theory the vertical axis of this diagram should be interpreted as the rate of information that you need to process in order to succeed at a task while the horizontal axis is the rate of information that you can process for that task. Then flow should roughly be anything along the upward diagonal. It is unclear to my where apathy comes from in this model and why this diagram makes statements about happiness, sadness, depression, and contentment. In my personal experience, I have experienced many instances in which I am in what this model would describe as a flow state, but it is not a happy state. Moreover, I would not say that the difference between boredom and relaxation is primarily a difference in ability at a task. These emotions seem much more complicated to me. The extent to which this diagram seems to extrapolate far beyond its foundations makes me skeptical of its scientific and explanatory merit.

 
My next thought is less of a skeptical perspective. Assuming that flow is a good state of mind for students to reach in the classroom, I'm interested in ways in which flow can be cultivated in a classroom. Based on this chart, I am imagining the path of a student w.r.t. a particular topic as they go from a beginner to a more advanced level. They necessarily must start on the left and hand side and move to the right hand side. As such, it is then our choice as teachers to deter mine what challenge level they should be presented with. Thought none of the left hand side emotions have particularly positive connotations, I think that there can be positive kinds of stress so I would prefer to initially present students with medium to highly challenging material. Then, as they improve, I would keep the level consistent or let it drop a bit so that they enter the area of flow, control, or relaxation.

I think that this recommendation that I've extrapolated from this diagram is interesting because of how counter-intuitive it is. Coming into this program, I would've imagined that we should be scaling up the challenge of material along with student ability. However, this chart suggests that material should be consistently challenging, if not decreasing in its level, as students improve. 

 

My last thought on this topic is another separate critique of the idea from another perspective. Flow, as described, seems to fall in line with a very old-fashioned view of 'genius' in which a solitary person produced their work in something like a state of ecstasy. This is intrinsically a non-social process. It is almost* by definition impossible to be in a flow state while communicating with other people, while collaborating. In other words, for the same reason that a person in a flow state is 'unaware of their own existence' they are also 'unaware of others' existences', and thus cannot collaborate with them as people. I see this as another reason to hesitate before embracing this concept in classroom design. 

Soup Can Problem

 Estimating the size of the soup can:

The soup can seems to be 3 bikes wide and 2 bikes tall. 

 My estimate for the height of the bike is roughly 70cm and the length of the bike is probably pretty close to 140cm. This would make the radius of the soup can's circle 70 cm and the width of the soup can 420 cm. So the volume would be given by V=pi R^2 W. Plugging in we get V=3.14 * 4900 cm^2 * 420 cm. This comes out to approximately 6.46 x 10^6 cm^3 which in units of m^3 is 6.46 m^3.

 Is this enough to put out an average house fire? I'm not sure but I would guess that a small house fire would be able to be put out with the water from a single firetruck. I think a firetruck could probably carry 1000 Gallons of water (more than 100, less than 10,000). This corresponds to about 3.75 m^3 of water. So by this estimate the can has enough water for a small house fire and some extra.

  

 Extension of this puzzle:

 I would like to extend this puzzle into something creative that also touches on the topic of scale factors. I would have students calculate the scale factor between the large can and an actual Camble's soup can. Then I would have the students create a model of the bicycle scaled to the smaller soup can out of some classroom materials (toothpicks, cardboard, card-stock).

Non-Curricular Micro-Teach Edits

 CHANGES HIGHLIGHTED IN BLUE

Lesson Plan

Duration: 10 minutes

Topic: whistling and humming at the same time

Goal: To teach the group how to whistle and hum at the same time and understand the connection to different phonetic categories.

Required Background: The students must already know how to whistle.

Lesson Breakdown:

Introduction to IPA and the difference between voiced and unvoiced consonants (2 minutes): Go over some basic sounds that all of the group members can make such as 'f' and 'v' and point out how when you make these sounds your mouth is in the same shape but your vocal chords are doing something different. The 'f' sound is 'unvoiced' while the 'v' sound is 'voiced.

Practice going back and forth between voiced and unvoiced consonants (3 minutes): Practice going back and forth between making the 'f' sound and the 'v' sound or the 'b' sound and the 'p' sound. Do this for a bit so that the group members get a feel for the difference, something they probably haven't paid attention to before.

Move on to considering whistling in this same context. Demonstrate whistling and humming and have students try (3 minutes): Work with students to try to replicate the whistling and humming sound. Some students will get it right away and other students will struggle for a bit longer. At this point students that get it right away can help out students that are struggling.

Trouble shooting and playing with the new skill (2 minutes): Give students time to play around with their new skill or to troubleshoot if they still haven't figured out how to make the noise. Assess at this point whether or not each student can make the sound.

Lesson Play:

Teacher: "Ok, let's begin. How many of you are familiar with the IPA?"

        Student: "It sounds familiar but I don't quite remember what it is."

Teacher: "The IPA stands for the International Phonetic Alphabet. It is an alphabet designed specifically to capture all of the vowels and consonants that humans use across all spoken languages. Each character in the language can be sorted out into various categories that determine how that sound is made. For example, the sound that 'f' makes is called an unvoiced labiodental fricative. Let's make the 'f' sound now."

        All: "fffffff" 

Teacher: "Unvoiced means that your vocal chords are not vibrating (i.e. it is a sound that you can whisper). Labiodental means that the sound is made with your teeth against your lip. Fricative means that the sound is made by friction when area is forced through a small-ish opening. For today we will focus just on whether or not a sound is voiced. The voiced version of 'f' is 'v', so let's practice making the 'v' sound."

        All: "vvvvv"

Teacher: "Great, now let's go back and forth a few times, 'f' then 'v', then 'f', then 'v'."

        All: "ffff - vvvv - ffff - vvvv"

Teacher: "Ok great. Now can anybody tell me what the voiced version of 'p' is?"

        Student: "p - p - b - b... I think the voiced version of 'p' is 'b'."

Teacher: "Amazing, you're getting it now. Let's get to the fun part. Can everybody whistle for me, just an easy steady note."

        All: *whistle*

Teacher: "Whistling, even though it is a sound that we can make, is not a part of the IPA because it is not a phoneme in any known natural human language. People clearly use it to communicate but you'll never find it as a part of a word with other sounds. However, we can still wonder: What if we tried to do a voiced whistle? That's what we are going to do now. I'll demonstrate." *hums and whistles at the same time* "Who would like to give it a try?"

        Student: "I'll try, I think I can do it." *struggles to hum and whistle, alternates back and forth.*

Teacher: "If you are struggling to get it try practicing switching back and forth between voiced and unvoiced consonants that you know and really pay attention to what your body is doing. How does an 'f' feel different from a 'v'? How does the 'th' sound in 'math' feel different from the 'th' sound in 'the'? How does the "sh" sound compare to the 'z' sound in 'azure'? Go back and forth a bunch and then try it with whistling."

         Student A: "sshh, zzhh - p, p, b, b - s, s, z, z - f, f, z, z..." *whistles* - *whistles and hums at the same time*

        Student B: "... I'm still struggling. I'm not quite getting it."

Teacher: "Let's try another way of getting there. First whistle. Then stop whistling and hum. Notice that you can 'hum' (humming is not just making the 'm' sound) with your mouth in any variety of positions: open, closed, pursed, smiling, tongue out, etc.. Now try doing the following. Whistle, stop whistling, hum, stop humming, and then start doing both simultaneously. Once it clicks you'll be able to do it forever."

        Student B: *whistles and hums at the same time* "Woah that is so weird!" 

Assessment: Did they successfully whistle and hum at the same time? 

Arbitrary and Necessary Response

 I like this article because it highlights the importance of language within mathematics. I have a few thoughts.

 First, I think it might be compelling to show students examples of instances in which different fields of study use different terminology or symbols for the same concept and how that occasionally leads to confusion. For example the use of 'i' for the imaginary unit in mathematics and the use of 'j' for the imaginary unit in engineering. The specific symbol that is used is almost entirely arbitrary and we can see that in the simple fact that different groups use different symbols. This is similar to the language analogy given in the article.

 

Another thought I had was about how it might also be interesting to let students create their own terminology during exploration. I believe that though the terms themselves that get created during exploration are largely arbitrary, the content that gets encoded does significantly affect the direction of exploration. In that way it is maybe much less arbitrary?

 

In my own lesson/unit planning I will definitely be working with these ideas. Getting students to use the vocabulary of mathematics properly is extremely important to me. In my own experiences tutoring students in mathematics and physics, there is a stark correlation between using terminology correctly and successfully performing the tasks presented. I cannot think of a single instance in which students used all of the terminology correctly but could not perform the tasks. There are instances in which students can do the tasks but cannot use terminology correctly, however, I think that that is more indicative of them having learned to imitate the mechanical problem solving steps.

 

I would be interested to see how research on teaching languages can be applied to teaching mathematics terminology in light of this work.