What I learned from 20 Years of Student Journals
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NCTM 2018 Annual Conference Washington DC Presentation about Student Journal writing in mathematics education
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What I learned from 20 Years of Student Journals
- 1. inspiration I remember having to slow down and show my thought process. Sometimes it was a chore and took more time than I allotted for it, however there was a sense that I was going beyond a typical experience with math. Writing about math made it more approachable. It was one of the first times that I found myself loving math. “Instead of cycling through math assignments, I feel the journal gave deeper meaning to my studies. I was never left with that all-too-common feeling of "I will never use this in real life" because I WAS using it in real life! ..and in combination with all of the other skills I was learning in school! Amazing, really. I struggled a lot with English and writing in high school, and I believe keeping a math journal helped me to be more confident when tackling writing assignments in general. It really helped to further my writing skills more than other methods, which is pretty outstanding for a math class.” “I honestly wish I could track down this journal and re-read my thought process and compare it to my current self. I think I might be surprised.” “I remember at the time I found it to be a stressful task. Which is interesting because I definitely excelled in my humanities classes more than math/science, so in hindsight I feel like I would a have appreciated the opportunity to use my language skills, but I think it was such a leap from anything I had been asked to do in math before that point that I found it foreign and intimidating.” “My math journal was something I really took pride in. To this day, I still have them. Keeping a math journal gave me the opportunity to exercise my skills in writing, math, and art/design all in one place. I will forever be grateful for your math classes and teaching methods. I never tested well as a student, but the first and only time I took the ACT I scored in the 99th percentile in Geometry..! It got me a scholarship for college. There really isn't much I remember from other classes or teaching methods throughout my time as a student, except for my math journals. The impact it made on my life is beyond words.” “I remember my improvement in the class. The first few entries were reminiscent of what I learned in math before. As someone who struggled with math but excelled with writing, I was soon able to write out my reasoning which strengthened both my understanding of and in interest in Carmel Schettino, 2018 “I am just about to enter graduate school for Art History. Writing in math class began the trend of deliberate, methodical, and passionate thinking to an extent. Of course other classes … aided with this. However, there is something to be said about writing about what would otherwise be considered a mechanical process. I feel as if I am a more careful curator now. I methodically work through my process with bursts of creativity and inspiration.” Alumni Journal Survey Responses
- 2. Metacognitive journaling in math: lessons from 20 years of student writing CARMEL SCHETTINO, PH.D. AVENUES: THE WORLD SCHOOL NCTM ANNUAL CONFERENCE APRIL 2018
- 5. inspiration
- 6. What I do - PBL An approach to curriculum and pedagogy where student learning and content material are (co)-constructed by students and teachers through mostly contextually-based problems in a discussion-based classroom where student voice, experience, and prior knowledge are valued in a non- hierarchical environment. (Schettino, 2010, 2013)
- 9. 1995-2000, 2001-2010, 2011-2017 Clear expectations and samples
- 10. empowering students to see themselves as capable of participating in and being doers of mathematics Showing Evidence of students engaging in active participation in reasoning and sense making Showing Evidence of students striving to make Their own mathematical thinking Visible and intelligible to others Using Multiple forms of discourse implementing equitable instructional practices : develop Positive Mathematical Identity NCTM, Catalyzing Change in High School Mathematics
- 11. implementing equitable instructional practices : Develop Mathematical Agency Telling others through words and actions who they are and what their purpose is Showing Evidence of taking risks and engaging in productive struggle Understanding that learning results when they successfully leverage an approach that works for THEM NCTM, Catalyzing Change in High School Mathematics
- 12. inspiration lesson #1: grading writing is hard
- 13. inspiration Find 2 lattice points that are sqrt(13) units apart. So you have to find two perfect squares that add up to 13 1 12 2 11 3 10 4 9. [two perfect squares] 5 8 3^2 + 2^2 = 9 + 4 = 13 Your two numbers are 2 and 3 Up 2 over 3 Up 3 over 2 Is it possible to find lattice points that are sqrt(15) units apart. No there are no perfect squares that add up to sqrt(15)
- 14. inspiration
- 15. inspiration
- 16. inspiration lesson #2: kids are funny
- 17. Some kids name their journals
- 19. Oops! So enlightening when you can help a student with a misunderstanding
- 22. “I’m Positive it’s real” Student’s description Of the Pythagorean Theorem
- 23. inspiration lesson #3: they write what they think
- 24. inspiration
- 25. inspiration When I originally looked at this question, I did not understand anything about how the question was related. I understood the individual parts. I knew that I had to use the distance formula because that is how you find points that are equidistant. The distance formula is…. As we went over it in class, my partner explained to me the process to find the answers.
- 26. “ ” I remember in class when we were going over this one, how Pete said, ‘but how were we supposed to know how to plot the points?’ I totally felt the same way because we were both thinking about it specifically. We were waiting for the book to spoon- feed us the answer. ~Student #20 about a problem deriving the distance formula with general points (a,b) and (c,d)
- 27. inspiration lesson #4: they don’t always get out of it what you want
- 29. By focusing on personal pronouns and modality, the researchers were able to see who the text recognizes as the people associated with the mathematics
- 30. “Voice” of Mathematics Textbook . Herbel-Eisenman and Wagner (2007) First person pronouns, like I and we, indicate an author’s personal involvement with the mathematics. The use of the second person pronoun you, also connects the reader to the mathematics because the textbook author is speaking to the reader directly. One way the pronoun you was used suggests an “absolutist image” of mathematics, portraying mathematical activity as something that can occur on its own, without humans.
- 31. “ ” First-person pronouns indicate the author’s “personal involvement with the activity portrayed in the text” . For example, the use of “we” could indicate that an author is speaking with the authority of the mathematical community; it may also be used in an inclusive way so as to involve the reader in the mathematics. HERBEL-EISENMANN (2007) Textbook analysis of first person plural – does it translate to student writing?
- 32. Advice on writing a math paper Unacceptable to use the plural pronoun “their” to refer to the singular “reader” or “you.” use the pronoun “we” to refer to the mathematical community in General “we” should not be used as a formal equivalent of “I,” and “I” should be used rarely, if at all. Kleiman (2004) http://www1.mat.uniroma1.it/people/manetti/tec2.pdf
- 33. Research study – Did journaling increase agency, ownership for students? 31 journals from 3 different schools – my students Page length, number of appearances of I, we, you Used average page length Calculated slope of regression line for each student for I, we, you over the course of their journal Looked for trend in growth or decline Difficult to see patterns
- 34. I Slope -0.04975369458 WE Slope 0.008957307061 YOU Slope -0.00632183908
- 35. I Slope 0.04824561404 WE Slope -0.1040935673 YOU Slope -0.03684210526
- 36. Student Number Difference of Last Entry –First Entry per student
- 37. Difference of Last Entry –First Entry per student Student Number
- 38. Student Number Difference of Last Entry –First Entry per student
- 39. One-Way ANOVA Test on 3 (I, We, You) different pronoun differences p = 8.585 𝑥 10−5 Hypothesis test says that The results that I had could not have happened by chance
- 40. inspiration lesson #5: evidence that in PBL students lose their sense of “I learned” and grow in “we learned”
- 42. inspiration
- 43. inspiration
- 44. inspiration Since we had done a lot of problems using the pythagorean theorem in the past I thought that I could use it in this problem. As my shows this method did not help me get any closer to t finding the height of the street lamp. I decided to wait until we discussed this problem in class to earn a more effective way to solve this problem. After our discussion in class, I now know two different way to find the solution to this problem. Will presented the first method. This method involves putting the entire diagram on a coordinate grid and using the equation of a line to find the lampost’s height. Here was the illustration of this method : [See handout]
- 45. inspiration lesson #6: all students are capable of deep reflection
- 47. Alumni survey on journals Question Average Response Helpful for knowledge of content matter 3.875 Helpful at improving your written communication skills in mathematics 3.875 Helpful for sorting through your thought processes & problem solving skills 4.25 Helpful for allowing to show a strength in alternate form of assessment 4.75 Helpful at organizing your thoughts in a PBL Classroom 3.125 As you remember it, how helpful was keeping a journal for…
- 48. Anonymous Alum, on writing Instead of cycling through math assignments, I feel the journal gave deeper meaning to my studies. I was never left with that all- too-common feeling of "I will never use this in real life" because I WAS using it in real life! ..and in combination with all of the other skills I was learning in school! Amazing, really. I struggled a lot with English and writing in high school, and I believe keeping a math journal helped me to be more confident when tackling writing assignments in general. It really helped to further my writing skills more than other methods, which is pretty outstanding for a math class. “ “
- 50. inspiration lesson #7: we do not learn from experience, we learn from reflecting on experience.
Notas del editor
- thank you all for coming – I’m assuming that you are here because somehow you are interested in writing in math class How many people have tried using journals? How many would like to try? Interest in writing, curiosity,
- Intro of myself started teaching in 1992 using journals in 1995 live in western massachusetts, travel to school, especially in new york city to do consulting in the secondary math classroom speciically with respect to PBL
- I’m assuming that you are here because somehow you are interested in writing in math class How many people have tried using journals? How many would like to try? Interest in writing, curiosity,
- How did I start? 1995 Klingenstein Summer Institute for New Teachers Columbia University We did a lot of journaling in our math cohort, can’t remember who the actual speakers was that said that writing in mathematics was a good thing, but as I wanted to learn and change one thing about my teaching I decided it would be that. Bought Joan Countryman book “Writing to Learn Mathematics” Started doing something that resembled journaling. I knew reflection was a huge part of learning process I was also teaching at a school that used PBL and I found students resistant to PBL without a textbook or resource, but I had also read some Dewey during this institute for the first time
- I new that in order to have truly curious, engaged and self-motivated students I needed to create a learning community where reflection was a regular part of the classroom practice. I tried other things – like responses to rubric, self- assessments, written feedback, etc. but found that their own writing was so personal and expressive that I could not find something that created the reflection the way journals did.
- In my research on the PBL Classroom and what students appreciated/stood out about their experience . The attributes of the PBL classroom that also fostered empowerment, agency, interest and enjoyment were also these four pillars. the most interesting thing about the use of journals is that they are interconnected in all four of these pillars.
- So where did my journey go from having the idea and the inspiration?
- Voluntary for all students Two entries a week Pick a problem Difficult to grade Could use on assessments Required for all students Twice a week Told them to work more on explaining their thought process Got easier to grade tried to make a rubric Could use on assessments All students required One entry every two weeks Focus was more on what did you not understand, how did you overcome it, do the problem, explain any moment of clarity Used rubric consistently, made more edits, easier to help students So through all this, I learned many lessons from my students that I would like to share with you all - as time went by I realized improvements that could be made to the way that I described the assignments and what I asked of them Created the explanation info sheet Could use on assessments LESSON #! Next slide: grading writing is hard but: gets easier the more you do it
- My realizations
- Read this out loud. - S rewrote problem wrote out some tries showed how the sum of the 2 square =13 describe direction drawing then described next question very little complete sentences, no connection of why something was done, what's the point, why are they looking for 2 perfect squares?
- Again, I'll read it for you. Notice this is all one sentence, it basically a run on sentence of stream of consciousness. Not very good writing but the ideas are there - which takes priority? how do you discuss this with students? What did I do? Look on handout Talk about the ones one the hand-out, Sample 3, Sample 4 – have them discuss the differences – what do you notice and wonder?
- Notice this is all one sentence, it basically a run on sentence of stream of consciousness. Not very good writing but the ideas are there Look on handout Talk about the ones on the hand-out, Sample 3, Sample 4 – have them discuss the differences – what do you notice and wonder?
- some kids like puns
- some kids make mistakes that are kind of cute
- they can be creative this was a problem that was about where the altitude of parallelograms were and how to visualize the altitude if it was 18 inches long - seeing the difference between the side length and the height (altitude) click -Next slide is Mathematical writing - intro
- so one of the hardest things for students to do is to figure out what to write about. For a long time I was just looking for the problem, explanation, etc. the concept of mathematical writing is foreign to most students – they don’t necessarily understand what they are supposed to do, write, think about and discuss, their experience with mathematical text is mostly reading – text books, or listening to teachers When posed a question they sometime don’t even know where to start:
- Sample 5 – what they are used to doing – explaining their understanding not what they learned, just method Sample 6 – Explanation of methods – sin and tan entry Sample 7 & 8 – found a way to reflect but in an organized way Second lesson
- And it doesn’t necessarily come out the way you and I might want to undersatnd it. Leads me to my next lesson -
- Finding the distance formula lesson#4
- Kids who voluntarily signed up to do their journals did not always do a good job – eventually made it required – evened the playing field. Part of my goal was to increase ownership, authorship, agency, not just create a place for them to take notes, cheat sheet Even towards the end of the year some kids were still just writing how to do the problems In the beginning I didn’t have a rubric so it wasn’t good – kids did not have a way of knowing how to get closer to what I wanted (actually I don’t think I knew what I wanted) By 2012 I had read a lot about writing - Candia Morgan, Writing Mathematically. read a lot of research on Textbooks
- what are kids used to "reading" or "seeing as text" in math? first person pronouns were entirely absent in textbooks includes such phrases as 'you find', 'you know' and 'you think', Depersonalization In the second-person pronouns, the authors obscured agency by having inanimate objects apparently perform animate activities. “
- The only research that is out there right now is on pronoun use and voice in textbook writing. Very little on student writing and agency with pronoun use. So if we want to see how the student recognizes themselves as the people associated with the mathematics it would make sense to look at the pronouns in their own writing.
- Depersonalization with "you" “textbook is devoid of first person pronouns, indicating that the authorial presence of human beings is concealed. In the second-person pronouns, the authors obscured agency by having inanimate objects apparently perform animate activities. “
- The second-person pronoun, “you,” were also examined to understand the construction of roles and relationships between reader and author. In one text researched the word you occurred 263 times. So the question posed is really "does this type of analysis translate to student writing about mathematics?"
- What are students told about mathematical writing - Does not foster ownership of the material – general math knowledge for the community If this is what they are told about mathematical writing – how can we get them to grow in their own ownership, authorship and agency of the mathematics they are learning? Froget the students of color or those who are disenfranchised. So this led me to want to do a study of all of the journals that students have left with me.
- So how could I answer that question that I wanted answered?
- I noticed that so many students’ data were different
- another student
- So instead I thought, OK if I can’t look at each kid individually or them as a whole, I wonder if I could look at my effect on the group of students over time. I decided to take the average amount of use of I, we and you in their very first entry in their journal and the average in their very last entry in their journal. Difference between average # of I per page for first entry (for all students) vs. Difference between average # of I per page for last entry (for all students) If d>0, pronoun use increased If d<0, pronoun use decreased This might give insight about the way that I was giving feedback on their work and helping them to grow in their ownership and agency in their writing because it was about ALL students.
- Difference between average # of I per page for first entry (for all students) vs. Difference between average # of I per page for last entry (for all students) If d>0, pronoun use increased If d<0, pronoun use decreased This might give insight about the way that I was giving feedback on their work and helping them to grow in their ownership and agency in their writing because it was about ALL students.
- Difference between average # of I per page for first entry (for all students) vs. Difference between average # of I per page for last entry (for all students) If d>0, pronoun use increased If d<0, pronoun use decreased This might give insight about the way that I was giving feedback on their work and helping them to grow in their ownership and agency in their writing because it was about ALL students. PBL is collaborative spirit – come in with independent view of math, spirit of learning in isolation, as the year progresses, they identify the knowledge more and more of the collaborative nature of PBL – what the teacher is subliminally trying to help them towards – they sense that it’s really working and they loose the sense that “I learned” this but that “we Learned “
- There are so many theories about reflection in learning – about it’s such an important part of the learning process. But when kids are just writing and not really reflecting is it helping? Is it making them
- physical representations of reflection
- verbal
- exlanation of their ideas, decisions, prior and after class discussion came to the conclusion lesson #6
- asked alumni to reflect on using journals
- This has all brought me to the larger goal of getting students to be aware of their learning process through writing, owning their part in that process and becoming stronger mathematicians because of it. Metacognition is the process of thinking and learning about your own thinking and learning. On Wednesday night, The two lifetime achievement award winners both mentioned writing and being aware of your learning as skills that they thought were most important in mathematics learning - If a student even practices this just a little bit and can improve just a little bit, they are learning in ways they could not imagine. Which leads me to my next and last lesson.