Questioning in Mathematics Reading

Mason's ideas on questioning fit very well into an inquiry model. Inquiry is a student-led, challenge first/lecture later (if at all) style, and Mason's ideas about integrating difficult questions to lessons is directly tied to that. When we make questions rigorous for students, we teach students to not be afraid of getting stuck, and they learn to be more resilient to tough problems. If we model how to cope when a problem stumps you, we show we are empathetic (we also can be stumped!) and they can tackle problems on their own, without the constant guidance of a teacher, which inquiry requires.

In my unit planning, I want to make notes of specific questions I want to base my lessons around. I think by having a guiding question, lessons can have more of a natural flow to them, and the lesson will naturally become more inquiry based.

Lesson Plan for Micro Teaching: Linear Relations

Linear Relations Lesson Plan

Subject: Mathematics Grade:    8	Lesson Number: 7  of  8 Time: 15 minutes
Big Idea or Question for the Lesson: What is an ordered pair? What is a linear relation? How do you graph it?
PLO foci for this lesson: B1 graph and analyse two-variable linear relations [C, ME, PS, R, T, V]
Objectives: Students will be able to  (SWBATs) determine the missing value in an ordered pair for a given equation create a table of values by substituting values for a variable in the equation of a given linear relation describe the relationship between the variables of a given graph construct a graph from the equation of a given linear relation (limited to discrete data)
Content and Language Objectives: -   understand what an ordered pair is (verbal) -  determine what a linear relation is (verbal/kinesthetic)		Skills/Strategies required: -  understand what a variable represents -  be able to solve for a variable in a given equation
Materials/resources: masking tape sharpie/thick pen graph paper whiteboard/markers
Assessment Plan: formative assessment: visually see if the students graphed their ordered pair correctly
Adaptations: [ for EALs] just make sure when you say, “variable”, you point specifically to the variable, and same goes for any integers skeleton note sheet that they fill in at the end of the lesson Modifications: skeleton note sheet that they fill in at the end of the lesson Extensions: graphing negative slopes, or more complicated functions (individually on their own graph paper)
Lesson Plan - Teacher-led, Class/Group Activity (10 min) Ian: Today we are going to look at linear relations (draw on board). Everyone should pair up and tell everyone to pick an integer from -3 to 3 but you cannot repeat a number that someone else chose (throw the stuffy at them as they choose). Alison: An evil wizard kidnaps your stuffy. He locks your animal in a dungeon. You need to rescue it! But you don’t know where they are. You only are given two clues: (1) your stuffy’s cell number (x-coordinate), which is the number they chose, and (2) a cipher that represents a map of where all the cells are. Find your animal by finding the y-coordinate of its cell. Ian: This wizard has bamboozled us! Let’s try and figure out how to use this cypher. Take a look at this grid below us. This line is labelled x. Let’s all find our places on the x number line and place your stuffys there for now. Because this dungeon is this whole grid, we have this up down or y direction. In order to find out where your cell and your animal is in the y direction, we must find our y values! How can we do this?! Questions: how can you find the y coordinate using your x value? How can you show this visually on the map/grid of the evil wizard’s lair? What order should we rescue the animals so we’re the fastest? Now give students time to solve for their y value, and place their stuffy on the grid taped to the floor where they think their cell is Alison: open up a discussion of what the shape of the graph looks like, because it’s actually continuous between each person (object). Hint: It’s a straight line. It’s a straight line! So! What does the class think Linear means? If this equation is a two variable LINEAR equation, and it comes up with a straight line, thennnn!’ So each of you has an x value that you chose, and a y value that you figured out using the equation we gave you. What you did when you got those two values is figured out an ordered pair. Conventionally, we write it as such (x,y). This is arbitrary.Get everyone to represent their coordinates as ordered pairs, and then collect them all on the board. Now… level two! - Independent Work (5 min) Ian: Move to independent work, now LEVEL 2! You are now all kidnapped and placed in new cells!  But now you have a different cipher to figure out where everyone’s animals are (cell numbers are -3 to 3)! (Hand out the ciphers as scrolls) Create a list of ordered pairs (like we did on the board) and then plot them on graph paper (individually). cipher: y= -3x+2 Hand out graph paper, and have students create a list of ordered pairs to figure out where everyone else’s cells are (with x values -3 to 3) and then plot them on the graph paper (figure out their y values).   - Closing (1 min) Now we all know what linear means and how to graph when given an x value! Could you figure the graph out if given a y value?

Two Column Math Problem Approach

 Hopefully these pictures are clear!

Dave Hewitt Video Reflection

Watching Hewitt teach was inspiring. His use of the space, of the students, of choral speaking - all of it worked together to make an engaging and thought-provoking lesson. I've been thinking a lot about questioning and thinking skills lately, and being mindful of the way we frame questions, especially in mathematical contexts, is crucial for student learning. The way Hewitt phrased his lesson on algebra as a "thinking of a number" question provoked the kind of curiosity and puzzle solving that I want to be able to foster. I definitely want to test out his teaching strategies, such as the choral speaking, the long wait times, the repetition, etc., in my classes on practicum.

Snap Math Fair Reflection

The SNAP Math Fair was a fantastic experience. The kids were so excited to show us the puzzles they had been working on and I was very excited to see them excited! It was a very comfortable environment, and I didn't feel like the students were nervous or under pressure, just very willing to help others solve their problem. I think this is directly the result of the clear guidelines of SNAP, and the fair definitely got me on board with its approach. I thought it was great to have the students recreate the problem to suit an exhibit of the choice, and show their project in that context. You could tell it really personalized the question for the students. I wish I could have seen all of them - the time was too short and the discussions too good!

Arbitrary and Necessary

The arbitrary part of mathematics is the language and definitions we use. It is the stuff we invent for the purposes of communication. For example, Hewitt says that we choose to write thirteen as "13," but there are other equally valid ways to write it if you wanted to. 13 is just the widely recognized and accepted form. Arbitrary means it is something that must be memorized in order to be known. Necessary mathematics, on the other hand, can be worked out independently. Students need to be aware enough of to accomplish it though. To decide if something is necessary or arbitrary, we have the think about whether or not you could figure it out yourself, or represent the exact same thing in a different way.
This article made me think about what questions I ask. Am I asking students to regurgitate a arbitrary fact, or am I asking them to work something out logically? When I think about that, then I can provide students with the appropriate question. Hewett's example was a teacher getting a student to "think about" the name of the number that occurs most often, when that question had no thinking involved - either the student remembered it or not.
I really want to implement a lot of inquiry based learning in my math classes, and I think focusing on the necessary questions, where students get to work things out themselves, is definitely my aim.

SNAP Math Fair

In my elementary school, I remember there being a science fair that would happen every year, and each year I would get excited about it - thinking about fun ideas of what I could do. But since the fair was always done outside of class time, I would always end up being busy with extracurriculars, like dance and theatre, as well as regular homework. I love the idea that the work for the SNAP Math Fair is done entirely in class and that everyone participates.

I also love that it isn't competitive. I was very intimidated by math contests as a kid and an adolescent, so I never participated, despite being a fairly strong math student. I think the Math Fair creates an alternate challenging opportunity for students, where, instead of everyone receiving the same questions and ranking themselves against their peers' scores, students work on different puzzling problems, and can teach each other. 

I would love to implement a SNAP Math Fair in my practicum high school, but I've already heard from my SAs how densely packed the curriculum content is. I am worried that I won't be able to afford the time required for students to work on their projects during class time. I will do my best to make it happen though, because I think it would be really rewarding for the students! 

BC Math 8 and 9 Curriculum Documents

Nadereh and I looked at the BC Math 8 and 9 Curriculum documents - her blog post will focus on the grade 8 material and I will focus on the grade 9 material in this blog post.

I was impressed with the introduction to the curriculum documents, especially its inclusion of the Aboriginal perspective section. The introduction really set a tone that, though the material and curriculum are important, it is vital for teachers to recognize learners' backgrounds, attitudes, skills, and learning styles. I liked as well the 'Goals for Mathematics 8 and 9,' because it showed the big picture of what the curriculum is trying to do.

I liked how they included the key concepts for the grade 7 math curriculum to show how that leads into math 8 and 9 (because students might not have got that foundation in elementary school). I think they could've included the general grade 10 topics as well, just to give further perspective of the curriculum continuum, but I understand that once grade 10 hits, the curriculum diversifies, so that makes it complicated.

As for the actual curriculum itself, Math 9 seems pretty similar to what I did when I was in high school, except for the addition of probability. I appreciate that they have started to include probability earlier than grade 12, because I think it is an important, relevant part of mathematics.

I also think that the emphasis of different kinds of mathematical processes was interesting - I think it will help me focus on what kinds of activities I should do in class to encourage these ways of thinking. They even included various suggestions for assessment, which I think is a great way to encourage teachers to break free from the standard unit test.

Battleground Schools Reflection

I agree with the article that there is a prevalence of math phobic attitudes in our current North American society. It seems normal for people to assume that they lack the 'math gene' and are completely comfortable publicly sharing that opinion. When I would tell people that I was a Math major, people were very shocked, and I would reply with some joke about how "we aren't all hermits living in math caves."

The article points out that sometimes it is the case where those who succeed in the traditional, conservative math classroom, will pursue mathematics, and from that group, some will become teachers. As teachers, they will educate the same way they were taught, perpetuating the environment that only allows a few to succeed. As someone who did well in their math classes throughout high school and university, in which classes were for the most part very traditional, I worry that I will translate this style of teaching forward. One of my biggest goals as a math teacher is to break the stigma that math is dry, dull, and only for a select few. But I know that sometimes, in periods of great stress, teachers will revert back to how they were taught as a student. I want to avoid this! I want to use inquiry, despite its inherent messiness, to create an environment of experimentation, creativity, and student discovery, similar to Dewey. 

I found it very interesting to read about the crazy history of math education in North America. I think that we are still very much in a period of conflict between traditional and progressive approaches, and as a teacher, I will experience this conflict first hand. Parents, students, or colleagues may not appreciate my style of teaching and I may experience some backlash.  

Reflection on Microteaching

For my microteaching lesson, I taught the basic 6 count step for swing dancing. From the feedback I've received and my own reflection on it, I think it went well! My classmates picked up the "rock step, triple step, triple step" fairly quickly, and so we had time to partner it and dance to music. When we switched to the music, we initially tried it full speed, but when no one was following, I changed it to half speed, which really helped students. I think it was a good decision to be flexible with what I had planned and change it to the level of the students.
I really enjoyed teaching the mini lesson and I'm looking forward to getting to teach another!

Microteaching Lesson Plan

LESSON PLAN

Subject: Swing Dancing Grade: N/A

Lesson Number in Unit: 1 Time: 10 minutes

Objectives: Students will be able to perform either the lead or follow part of a 6-count Swing Basic, including rock step and triple step. Students will partner with each other and practice leading/following, and learn couple dancing techniques.

Strategies: ·         Learning by doing – full class participation ·         Formative assessment by playing music and walking around and observing students while they practice

Materials/equipment needed: Open space Music (preferably in 6 counts) and music player

Adaptations/Modifications: Include visual diagrams, or written instructions for those who have trouble remembering verbal instructions If students are very uncomfortable with dancing, lesson can be simplified to focus on non-partnered dancing (just learn the rock step and triple step) Students in wheelchairs, or those that are unable to participate by standing, can pretend their hands are feet and dance the steps on a table while seated If students are uncomfortable with being close to each other at the beginning, play a game where they run around and find someone to pair up with, assume the dance position, and randomly blurt out the name of a fruit or vegetable. This silly activity gets students feeling goofy and more comfortable! Extensions: ·         Students can learn basic turns, such as a tuck turn ·         Students can learn a double arm slide

LESSON COMPONENTS

Triple step (3 min)

Have follows stand on one side of the room and leads on the other, facing each other. Have the students gallop sideways from one side of the room to the other, then switch direction. Eventually, bring it down to only one gallop in each direction. Explain that this is all a triple step is!

Rock step (2 min)

Have leads step or “rock” back on their left foot, follows rock back on their right foot, then rock forward on to the other foot. Have students practice the pattern “rock step, triple step, triple step”. Make sure students aren’t looking at their feet the whole time!

Partnering (5 min)

Have follows and leads pair, and introduce the correct hand placement. Have students practice the steps together, leads practice guiding the follow and the follows practice going with the mistakes of the lead (ex. if the lead does and extra triple step, just go with it!). Try it with music! Switch partners and try with new people.

Soup Can Problem

The width of the bike on my computer screen is 2 of my fingers. The whole water tank is about 5 fingers long, so the water tank is 2.5 bike widths. Googling "bike dimensions," I found that a standard bike is about 1.8m or 180cm wide, so the water tank is 2.5*180cm = 450cm long. If we assume that the water tank has the same proportions to a regular soup can, we can find the ratio of heights to find the diameter of the tank. From google, I found that the normal dimensions of a Campbell's soup can is 10.16cm x 3.33cm. So the ratio of heights 450/10.16 = 44.29. So to find the diameter of the water tank we multiply this ratio by the diameter of a regular can: 44.29*3.33 = 147.5. So the radius is 147.5/2 = 73.75. The volume of a cylinder is the area of the base times the height, so we get:

pi*r^2*h = pi * (73.75)^2 * (450) = 7,689,293 cm^3 (approximately) 

1cm^3 = 0.001 L

So 7,689,293 cm^3 = 7689.293 L 

So the water tank holds about 7,700 L of water. 

Letters after 10 Years

LETTER ONE:

 

Dear Ms Sturrock

                I just wanted to let you know that I really struggled in your class in grade 9.

                You obviously really like math, but sometimes I just want the simple version of how to do something. Your explanations were too long and complicated when I usually just wanted a concise answer. I didn't usually follow your explanations, and I wish you'd have explained slower and more precisely. You made math seem way too complicated. I stopped asking questions after a while.

                I also found your classroom too loud to actually focus. I feel like you didn't have any control over the class volume when we were working through activities. I wish we could have had more quiet work time so I could process what we were learning. 

                Hope this feedback helps you -- Student X

Reflection: I sometimes feel I get carried away in my explanations of concepts, and make them overly complicated. I think this is because I'm trying to explain my thought process and scaffold relational understanding, but I worry that it might result in students being overwhelmed by the amount of information. I also am worried about my classroom management skills. I want to include many hands on activities and have a lot of student participation, but I worry that I won't be able to control the class volume and environment as well, and this will negatively affect certain students' focus.

LETTER TWO:

 

Dear Ms. Sturrock,

                I wanted to send you a letter to let you know how much I appreciated you as my grade 8 teacher. First of all, I felt like you really cared about us and our learning, and I always felt welcome to talk to you or ask questions. I also really liked how much activity you put into your lessons. We were never sitting for too long and I enjoyed getting to use lots of different materials. I liked how much we got to participate each class, instead of you just talking at us from the board the whole time. I also appreciated how we got to do projects that used math to understand real world issues - like my ecological footprint!

                Thank you so much -- Student Y

               

Reflection: My number one goal as a teacher is that students feel safe and valued in my class. After that, I hope that I can engage students in math by using fun, diverse, multimodal activities. I want students to be active, and give them ample opportunities to be collaborative. I hope to incorporate real world relevance as well.

 

 

 

Math Dance Project Reflection

I believe dancing is something we all are born to love, because dance is all about figuring out what we can do with our bodies, and expressing how we feel by moving. But once puberty hits, it seems people become uncomfortable with the idea of dancing, whether out of fear of judgement or because they feel awkward or silly. This is a shame, because dancing is so expressive, creative, and natural. If you didn’t judge yourself, there is no wrong way to dance.

I always have loved to dance. I also, of course, love math! So I was so excited by the opportunity to combine these two passions, though I wasn’t quite sure how they would connect. In my first quick google search, however, I discovered Dr. Schaffer and Mr. Stern’s TED talk, and immediately was on board with their ideas.

The first chapter of their book seemed like a great place to start with math and dance – shaking hands. Getting students, especially high schoolers, willing to participate when the word ‘dance’ is mentioned seems a bit daunting. But creating handshakes is a comfortable, low-risk of embarrassment way to start. The physicality of handshaking then lends itself nicely to a discussion of combinatorics, providing a memorable way to explore counting problems. I also enjoyed reading their other lesson plans, from clapping rhythms for LCM, to human tangrams for geometry.

After working on this project, I’ve looked over some of my favourite dance videos on youtube, and I am now more aware of the symmetry and geometrical patterns that are made. Choreographers are always looking for new combinations of movements and shapes to create, and I think that is fascinating to look at from a math perspective. I think dance is a great way to physically engage students and allows for their creativity to shine in math, which traditionally can be very “right or wrong.” I definitely want to incorporate movement into my class, and will continue looking for ways dance and math connect.

Dishes Puzzle

To solve this puzzle, I first drew a picture to visually see how the different dishes would be shared over a group of people. From the drawing, I could see that, with four people, they could split rice and meat evenly but not broth. In order to find the minimum number of people that could split rice, broth, and meat with whole numbers of dishes, I looked at the lowest common multiple of 2, 3, and 4, which is 12. (Since 2 divides 4, the LCM is just dependent on 3 and 4 - which have no common factors so the solution is 3*4=12).

To find out how many dishes of each I would need for 12 people, I used ratios (see above).
6+4+3=13, so for every 12 people we would have 13 dishes in total (13 dishes/12 people). To find out how many people use up 65 dishes, I divided 65 by 13 to find that I would need 5 groups. Therefore, I got the equivalent ratios 13 dishes/12 ppl = 65 dishes/60 ppl (12*5=60). So there were 60 guests that used 65 dishes!

I don't think the cultural context has much effect of the problem for me. I can still understand it and solve it without understanding the culture behind the puzzle. I think it is interesting that I know similar problems to this one from my high school days, so people from around the world have been solving this problem for such a long time!

Response to David Stocker Reading

When we did the TPI, I scored lower on the social reform perspective compared to the others. I am passionate about social reform, and really believe that social justice issues need to be involved in schools, but I always imagined it as a priority for other subjects, not in math class. After seeing my results, I started to question that assumption, and have been looking into different ways teachers have been incorporating the two together. This is why I really appreciate the reading we got this week. Stocker's approach to teaching mathematics seems like something I could really get behind. I agree with his assertion that numbers are everywhere, and in order to not be manipulated by them, students need to understand their meaning. As a teacher, I want to change students' perspectives towards math, teach something that will have a lasting impact on my students, and nurture a curious, supportive, and active class environment. I think a social justice perspective could provide an engaging lens for achieving these goals.
I did have some hesitations towards using social justice in math, but Stocker addressed them in his introduction, which put me slightly more at ease. I know it won't be easy, but I definitely agree it is something worth trying. Something I would like to practice in our class would be taking some higher level curriculum (grade 11 or 12), and brainstorming ideas for lesson plans that have a focus on social issues.

My plans for the Oct. 23 Pro-D conferences

I am part of the IBDP cohort, so I will be attending a conference Oct 23-24 for professional development in that area.

My Math Experiences: The Extremes

My best Math teacher:

In grade 6 and 7, I had wonderful teacher, Mr. K. You could tell he really enjoyed teaching, and he would even give us candy on a test day! (He called them "brain vitamins"). During math class, he would make us stand up and learn "math dances," which were silly sayings combined with movements that helped us remember certain mathematical rules. I still remember some of them today! He got us using using tactile tools as well, providing us with blocks and measuring instruments to allow for a more hands-on approach. He also allowed for differentiated learning within the class, grouping different students together and giving them the kinds of questions that would challenge their problem areas.



My worst Math teacher:

I've had quite a few unfortunately bad professors during my undergraduate degree, and they all had a two things in common. One - they would talk very fast and write messily, and I'd always be racing to write down everything (let alone understand any of the lecture). Two - it was obvious that they didn't care about teaching us. The professors would get frustrated at questions, and had no patience for students who asked for things to be repeated or explained again.
That being said, I have had some really engaging math professors, who welcome questions. I just wish there were more of them!

TPI Reflection

Completing the TPI and looking at the results was very interesting and informative for me. I didn't realize my priorities and opinions on quite a few of the topics until I was actually confronted with questions about them. For example, I believe strongly in social reform, but I don't prioritize it in my Math teaching. My main concern is students' social-emotional learning, and their views towards themselves as a self-motivated, confident student. This is also why I believe I scored slightly lower in transmission.
I found it interesting after completing the TPI to think about how I could incorporate more social reform or social justice ideas into my Math lessons. While I still believe focusing on nurture is my main priority, involving social issues could result in projects that involve more relevant, real life problems students can actually care about.
Some ideas I've been thinking about:

• Volume questions related to creating houses with water bottles in Nigeria 
• http://phys.org/news/2011-11-plastic-bottles-nigeria-housing-problem.html
•  Graphing questions related to minimum wage
• Dean, J.  (2007).  "Living algebra, living wage:  8th graders learn from real-world math lessons."  Rethinking Schools, 21(4), 31-35. 
• http://itec-ubc.ca/wordpress/mackowetsky/wp-content/uploads/sites/54/2014/02/Living-Algebra-1.pdf
• Understanding statistics, perhaps related to:
• From http://www.radicalmath.org/main.php?id=SocialJusticeMath#3
• Public Health: AIDS, asthma, health insurance, diabetes, smoking
• Educational access, funding, testing, achievement gaps
• Environment: pollution, hunger, food and water resources
• Welfare
• Immigration 
• Debt, credit cards, minimum payments, interest, loans, etc.
• Lotteries

 

Chessboard Problem

When I first considered this chessboard problem, I immediately jumped to 64 as the number of squares. But once I realized I could also consider squares of bigger sizes, the problem got a bit more interesting! I am a very visual person and I always jump at a chance to draw a picture, so first thing I did was draw a chessboard in my notebook. Next, I wrote down the number of small squares (64) because that was something I already knew. Then, I tackled the next smallest square, 2x2, so I could start systematically working my way through all potential square sizes, smallest to largest, not missing any along the way. I started counting 2x2 squares along the bottom row, and part way along, I realized I was counting the centers of each 2x2 square. So I decided to count how many centers along and how many up there would be in the chessboard, and multiply them together, which happened to be 7(across)x7(up)=49. I did the same method with the 3x3 squares, which I found was 6 middles across and 6 middles up (6x6=36). At this point, I realized the pattern would continue all the way to the 8x8 square, and just wrote down the answers without counting centers.
Finally, it was time to add up all the squares I'd found! To make the mental math a bit easier for me, I added first the numbers that make multiples of five (so 1+9, 4+16, etc.) and then added up the rest. My solution, as you can see from the picture, was 204 squares!

From the above, you can see there were a few changes in my thinking:
- Considering only small, 1x1 squares, to considering many sizes
- Counting individual squares, to counting unique centers
- Counting centers all the centers, to multiplying the number of centers across and up together
- Multiplying centers, to just assuming the continuation of the pattern

Tools I used:
- Drawing a picture
- Use of the center of the squares
- Multiplication and addition

How to extend the puzzle:
- How would the solution change if it was a bigger board? Or if the board was not a square itself (ex. 8x9 board)?
- How would the puzzle change if we were instead looking for rectangles of the form Ax2A? Would there still be a pattern in the answer?
- What if we instead had a board of triangles? (below)

Reflection on Our Instrumental vs. Relational Class Discussion

I was surprised at the evenness in how our class split on the discussion of instrumental vs. relational understanding in a math context. I much prefered relational, though I did see some of the benefits of instrumental, and I found it interesting how some of the points from the opposing side I felt completely different about. For example, when they raised the topic of technology, I thought that was a point in favour for relational understanding. Now, with computers to do most of the grunt work of calculations, I think it is more important for us to know how and why things work instead of the number-crunching, 'rules without reasons.' I also think, for certain concepts at least, learning relationally first, not instrumentally, is much more intuitive, not the other way around.
However, I do agree that, in terms of time-saving, instrumental understanding seems like the clear winner. It is much faster to teach, and when it comes to doing practice problems, it is much quicker. Understanding what multiplication and division is relationally is very important, but eventually you just need to memorize shortcuts to get things done in time! That speed and comfortableness is the "fluency" students need to move on to new topics. And though relational understanding is ideal, with the given constraints of class time, making sure every student can abstractly consider each topic isn't feasible.
I think both understandings are inherently linked to the way with must teach math. If I was teaching logarithmic rules, for example, students would understand how logs relate to exponents, and we would use that relational understanding to explain why log(xy)=log(x)+log(y). This meaning-making is important, but eventually for ease and quickness (i.e. fluency), students will memorize the rule.

Instrumental vs. Relational Understanding

                Skemp’s article solidified some of the ideas that have been vaguely floating around in my brain. While doing my undergrad, some of my favourite math moments came from finally learning where certain formulas came from – for example, when the mystery origins of the ‘area of a circle’ formula were finally proven in calculus class. In fact, proofs were often my favourite parts of my courses, and I didn’t really enjoy ‘rule’ heavy courses (ex. differential equations: “just ‘guess’ this solution!”). When I was in elementary school, I actually hated math because all it was to me was memorizing a bunch of times tables and facts. It wasn’t until I got to high school when I got to do some logical problem solving did I start to enjoy it. Because of this, I related a lot to Skemp’s arguments for relational understanding as a basis for mathematical instruction. I agree that to fully comprehend and appreciate math, having that foundation is crucial. I also believe that, at least in my case, it makes the subject more engaging, making students feel more like a detective and less like a machine. However, I really appreciated his devil’s advocate section, because in some cases, it is still appropriate to use instrumental approaches. And with the time given, it is extremely difficult for a teacher to make sure every student has a relational understanding of all the topics in the jam-packed curriculum.

                Three specific lines stood out to me. One, “well is enemy of better.” If our only focus is for students to perform well on tests, then teaching beyond simple rules seems unnecessary. But if this way of thinking is what is taught, it inhibits further development in the subject, because math builds on itself grade after grade. Without a good foundation, ‘better’ is much more difficult. Two, there can be “a less obvious mismatch … between teacher and text.” I hadn’t thought of this conflict before, but it makes sense. When there is disconnect between a teacher’s style and the text, or other teachers’ styles, this can make it a lot harder for students to understand the material in either relationally or instrumentally! I appreciated this acknowledgment of difficulties of introducing more relational based approaches into instrumental oriented schools. Three, “there are two effectively different subjects being taught under the same name, ‘mathematics.’” This sentence took me a while to muddle through - it is an unusual statement. But it really ties in with the earlier statement about mismatching the two approaches to understanding.

                

First Blog Post (Sept 14th)

Hello internet!