Today's Math I.D.s for Tomorrow

What struck me most about the math reading and class this week was the connection to that and my experiences in 7th grade math classes in the fall, and the deeper understanding I have now about how much of math teaching is not actually specifically teaching math facts.  On top of that, how much not only isn't math facts, but is actually life skills.  Additionally, all of it, the math facts, deeper mathematical reasoning skills, and life skills all are learned through every individual's self-identity and emotional relationships with the subject.  Through this "healthier" (Leatham & Hill, p. 228) math class consciousness I myself even felt more comfortable and empowered to do the geometry puzzles and measurement algebra work with my group. 

As a result of this, I will make self-reflection and discussions of what mathematics really is about, how it really is relevant in students' daily lives, and even personal emotions, a major part of any math instruction I will do.  

In thinking about what mathematics students actually need to know and how my mind has pretty much been blown away in class this quarter thinking about all the potential of the math tools that could easily be in everyone's hand soon,  like motion and temperature probes for smart phones, measuring devices, the WolframAlpha site that seems to have every-everything, I have some questions.  Mostly my questions have to do with, what really are students going to need to be taught in class, what do they need guided experiences with in class, what are they being taught that is a waste of their 21st Century time?  And not only what are they going to need to be taught that I don't even know I don't know, but what of the same stuff did I learn  that they should learn even better than me.  I guess this last question is, if I had grown up with the technology that exists today, would I be using the mathematical knowledge I did gain in school much more regularly or effectively?  What in my life would be different if I was using mathematical thinking more regularly or effectively?  I think there is a solid chance I would really understand my world at a deeper level than I do already.  As an educator, I will do my best through professional development and any other means I can find, keep my own learning progressing with an awareness of what is possible in the present looking towards the future.

Dynamic Mathematical Thinkers and Technology

After today's class I am really thinking about how I can't know what mathematics students are actually going to need to know in their lives.  But I do know I can help them practice mathematical thinking, that I can help this thinking be transferable across the great technologies that are constantly coming about, and help give them the self-confidence that they are mathematical thinkers that can and should use these new technologies for the betterment of their own lives and others'.  For example, if they can use a variety of programs to make a histogram, understand and analyze a histogram in meaningful ways, then making a perfect histogram by hand won't be of the utmost importance.

If in the not-so-distant future everyone has cell-phone probes to understand the world around them, will we be assessing if their understandings are logical?  With more technology, will mathematics assessments and materials be less hands-on with the real world or just more integrated?  For example, we worked with the shapes on a screen this afternoon, this is hands-on, right?  Collecting our own data and analyzing it with technology, that makes it all very "real-world" for students, right?   

As for implications for classroom practice, my thoughts from class today will help me keep my own reality-check for what students really could find useful in knowing and being able to do in their lives.  How will I keep perceived mathematical limitations off and away from every individual - probably by guiding them through mathematical practices with dynamic technology that keeps them as mathematically dynamic as the technology that will hopefully always be viewed as individuals' personal tools.

Mathematics with the Right, Slight Struggle

From class this past Monday, I learned that by doing something like posting a review sheet online for everyone to do two problems and comment on one another's answers, students are challenged in a way that makes them more responsible to one another and especially responsible for their own learning.  It also adds a slight element of struggle, which I've noticed the right amount of "slight" struggle can really help learning.  I learned there are great viable options for math journaling; there are super fun ways to incorporate learning about measurement, mean, and range; and, there can be quite high-order mathematical thinking involved when working with those elements and tanagrams.

A question I have right now is if the group task we did on Monday, making our tanagram giraffes twice as big, was a group-worthy task, right?  Because it helped a whole lot to have more than one of us working on it and because mathematical tasks involving changing size like that do come up in the real world? 

Implications for my classroom would be that I see the direct relevance of using math journaling, tanagrams, and fun activities, all in every grade, all advancing learning to deeper levels.  I think this seems so plausible because they all help me with my mathematical thinking as an adult and can see how they all help in the 1st/2nd grade classroom I'm in right now.  Even in this class of young students they have something called Math Notebooks where they respond reflectively to written mathematical prompts like journals where they explain their thinking.

Understanding Geometry Understanding

After our class this week focused on thinking about geometry, and then the reading, "The Rationale for Using Manipulatives in the Middle Grades", by Dana M. Freer Weiss, my understanding of geometry understanding has definitely increased. 

In class I learned about the Ban Hiele Levels of understanding geometry, and that visualization is the first.  This makes sense to me because anytime I am doing any kind of mathematical thinking I am visualizing.  I really start running into trouble when my visualizations start becoming just memorized number rules because then I've lost the meaning, I don't have those mathematical ideas connected to anything else, so when I try to build some new information in I don't have anything to connect that to either, it has to have its own isolated little spot and isn't very useful then. 

In the 1st/2nd grade class today the students were struggling to find the differences between 2-digit numbers and were getting frustrated because their fingers weren't getting them very far and drawings were taking forever.  As I moved through the students trying to help and wondering how to best suggest manipulative to my master teacher, I remembered the 7th grade math classes I was I was in this past fall and how the teacher always encouraged them to use manipulatives, even on tests, and the students at that school were some of the highest achieving math students in the district.  Probably not just because of the manipulatives, but I bet there were part of it.  I am fully convinced, whatever time I spend finding and learning to effectively use manipulatives with students will be worth it.

Finally, I am just wondering about geometry.  How can it make sense all the time?  Is it because it is all repeated patterns?  Is it because of the physical properties of the universe, like a plant growing straight up from flat ground makes a right angle and if another is close by and a leaf lays across the two you have a rectangle and it's properties just because that's how it is?  Is there any geometry that hasn't been explained yet or has it all made sense to somebody sometime?

Group Tasks

I had a super-great learning experience in class this week.  My group and I had felt our data story poster was near completion two weeks ago, but upon seeing it again yesterday with only a data table and no graphs, we decided graphing would help others understand the data and our story better.  It took some hard thinking and discussion to decide how we wanted to show phone subscriptions over time to GDP of different countries over time, but finally decided for a reason we couldn’t really put a finger on, that two graphs would work well: one showing phone subscriptions to time and the other showing GDP of the three countries we chose over time. 

Our brains warmed-up nicely as we traveled the room of posters, discussing our questions and thoughts of the data stories as we went.  At one point we reached a poster that was challenging to understand at first, comparing U.S. military and education expenditures to each other.  After looking and discussing further we did understand, and again, didn’t really put our finger on exactly why the graph had been tricky to understand.  Finally, back at our own poster, we saw a post-it question from another group asking why didn’t we put our two graphs together as one.  That’s when I really got it, that we didn’t put them together because if that meant comparing phone subscriptions to GDP, it might imply that one determined the other.  We didn’t necessarily conclude that GDP determined phone subscriptions, we just wanted to show the trends of each GDP and phone subscriptions over time.  We wanted both to be compared to time (as our x-axis) and would have had trouble combining GDP and phones for the y-axis in a meaningful way. 

This group task of the entire process of creating the data story, representing it in a poster, discussing other posters for understanding, and finally considering where others thought our poster could share understanding better, made me really understand an aspect of graphing quite consciously.  It was more meaningful for me to experience why my group and I felt more comfortable sharing our data in a certain way, considering other ways other groups shared theirs, and then thinking about ours again together that my understanding deepened. 

Now, not only do I have a deeper understanding of graphing, but I also have a deeper, more meaningful understanding of the value of group tasks in the learning process.  I will definitely take this awareness of the value of group tasks for understanding with me into classrooms.  I guess I am just left wondering when does the teacher need to step in if understanding is not coming through the group tasks?  How patient with the process should the teacher be?

Mathematics and Creative Writing

I just read the article about using creative writing and literature in mathematics classes, and what stood out most to me was the point that not only do students enjoy it while utilizing math concepts and vocabulary in personally authentic and meaningful ways, but it is also a really good way for the teacher to assess student knowledge.  The examples given made it clear that between the written story plots, descriptions, and graphics, students' mathematical misconceptions can be spotted to be addressed in instruction in ways just as effective as well-designed tests meant to identify misconceptions.

While thinking about mathematics and literature I was reminded of the mathematizing literature assignment from last quarter.  I worked with a small group of 7th and 8th grade girls, and I read a children's storybook to them while we stopped periodically to discuss what math ideas stood out to them and what math they thought younger kids might pay attention to in these stories.  It was quite an exciting endeavor as they considered so much of their own mathematical knowledge and fed off of one another's ideas.

I wonder how the 1st and 2nd graders I am working with in my main placement now might use creative writing in their mathematics studies.

Equity and Algebra Tiles

Using the grid frame with algebra tiles to work through multiplication in algebra blew-my-mind.  It made algebra so much more clear to my head, and I don't remember ever particularly struggling with it, I guess I just never understood there was a bigger concept to understand.  After that activity during our last class, where we experienced many different ways algebra can be thought about, the readings, Group Worthy Tasks (Lotan, 2003) and Orchestrating Discussions (Smith, Hughes, Engle, Stein, 2009), really hit home.  Just as described, I needed time to think about an algebraic equation myself, discuss with others, and work through with manipulatives a variety of thought processes to get to the point where the algebra tiles on the grid made sense to me.  And it was the connection I made myself, just as we talked about in class and was discussed in both readings, that x2 is an area and on the grid I could see an actual area.  If only I knew I was such a mathematical wizard when I was younger, well, I'd probably actually be in the same place, but I would have gotten a lot more of a thrill out of math classes.

I must admit, I am still working through the beans as negative and positive numbers.  Is it that you don't subtract because you can't really subtract in real life anyway, you can just move it or change it?  Money doesn't actually go away, it just leaves one pocket for another; apples that get eaten don't disappear, they go through the digestive tract.  Is all this why we don't subtract, we just add the opposite?

Some points in the readings remind me of the Culture of Power Delpit described, and that when we were talking in class and the readings explained having multiple entry points to the learning, it is about equity, so that complex instruction with group-worthy tasks and effectively organized discussions in mathematics isn't just a good suggestion, it is necessary for equity.

Math Questions

Allowing the time and space for students to work through contextualized math problems is what stood out most to me in our class this week as well as in the readings (Reinhart, 2000; Herbert & Brown, 2000).  Space is the multiple entry points good math problems can provide, and I think that by encouraging small groups to work together with a variety of materials even more space and entry points are provided.  One of the readings (Reinhart) emphasized the importance of allowing some time in the beginning for students to work independently and then work together.  I appreciated this point because I think it is an important part of providing multiple opportunities for students to participate.  The independent work at the beginning is more time to think about the math and provide the space for independent ideas to be generated that can be shared and tried out in the group discussion that follows, even encouraging some metacognition as the student thinks about what they know and what they want to know.

As I engage in a mental wrestling match with the iPod Touch, I wonder more about how it can be used to aid learning and I wonder if it can be used as a tool for this larger question I have had over this week's math class and readings: How do we document progress in learning using these mathematics techniques of group work, questioning and communication?  Maybe digital technology can help.  Because when a student is working as a small group discussing and working with manipulatives together, questioning their classmate's ideas as they deepen their own understandings, how do I document this?  Maybe I could record their communication of ideas to the group?  Take pictures of how they show their thinking?  Should I just go ahead and do written summative assessments along with this complex style of teaching?

I definitely appreciate how what we have been studying this week, especially the importance of student questions in their own learning has brought up many of my own questions, and I will absolutely utilize questions as a learning resource and tool in classrooms.