We had some intense discussions last night for sure and
there exists confusion about the models we were using. This surprised me
somewhat since in our first session when we modeled the problems everyone
seemed to understand how the models worked. Then today I looked at samples of
student work to discover a serious misconception about bar or tape diagrams
(either term is acceptable but I will try to stick to TAPE diagram since that
is what is used in the frameworks).
If we think about a ratio of 2 reds : 3 blues we are
describing a multiplicative relationship that is represented in a comparative
model as shown below.
Reds
|
||||
Blues
|
Now if
we are also told there are 32 reds and some blues we DO NOT need to add
anything to the drawing BUT we need to show those 32 reds are evenly divided
into the 2 cells/blocks we ALREADY drew.
Reds
|
16
|
16
|
||
Blues
|
And each cell must have the exact same scale factor…each cell
represents a variable and the same variables must have the same value. So we
know that there were 3 x 16 blues or 48 blues.
The problem then stated an additional 10 red skittles were added to the jar and the students had
to find the new ratio of reds to blues.
The student misconceptions included adding 10 new cells for
a ratio of 12 : 3 rather than understanding that 10 reds is less than one new
cell…this indicates the students who underlined “key” words saw “added to” in
the problem and simply added…illustrating a serious misconception that ratios
are additive. Those students who thought about the problem correctly recorded
42 reds : 48 blues. So…what caused this widespread misconception???? A misguided
belief that underlying KEY WORDS is a helpful math strategy to work with “word
problems.” RATHER the students should be guided to restate what they think the
problem is asking and to focus on the meaning of the problem…this will
alleviate that serious issue of reading words outside of the context.
Next, I erred in handling the question about the Larry, Tom,
and Georgia money and percents. What I should have done was to put our thoughts
on the conjecture board and for us to rationally discuss what is going on in
the problem, what we were representing and that we were talking around each
other. MY FAULT! I apologize for not modeling what I preach in this regard. I
have given this problem and our discussion lots and lots of consideration since
Thursday…even losing sleep over it. Never very helpful. It turns out we were representing and solving
two different problems just as Helen and Rich pointed out. I will go over it at
our next meeting and we can examine the model and connect it to the algebraic solution.
We also need to complete the problem we left hanging after
our first class. Lots to do, so little time.
Anne
If only literacy and numeracy were a seamless, fluid whole... I think verbal language is very ambiguous. But we can't have the mathematical language without the verbal. Each informs the other. Still, literacy and numeracy don't always have a compatible relationship. The way we use the English language in the math class has to be very deliberate, purposeful, and as precise as possible. I am trying to persevere! --Helen Son
ReplyDeleteI think that overall the questions, models and discussions have been great and I’ve learned a lot about strategies that seem to work. They work for me and I think they would help many of my students who seem to struggle with the abstract and visually complex mathematics that they are given. It’s very frustrating to watch them try so hard and repeatedly fail at algebra. No matter what I do, it doesn’t make sense to some. Therefore, I’m not really that concerned over the issues we’ve talked of with interpretation of language. I think a few problems were just plain difficult and I’m not that worried about those few because I don’t plan to give those to my students. I think it’s interesting that some of the common core standards that I see when writing IEPs are completely unfamiliar. If I don’t know what certain concepts are, what is that saying for how well we are aligning instruction to the core? For example, what is bivariate data- think it must be data related to 2 different variables and some kind of relationship between the two-right? Also, when I had difficulty with the Georgia problem from homework one, I asked for help from an eighth grade math teacher. He set up and solved an equation to find the answer which was different from the one that was accepted as correct. When I went to tell him, he not only said that he had the right answer, but that he doesn’t agree that teaching with bar diagrams is a good idea. He thinks that it would be unfair to “make” students have to think this way. This is funny to me because he can’t see the other side-the students who struggle to logically think through and solve algebraic equations without a visual model. What about them? I am not saying that all kids have to be forced to think the same, but that this gives those that need it, the tools and steps to get to the abstract level of thinking they’re expected to reach. I knew that I wasn’t going to win that argument when he continued to disagree. My point is that because some teachers are unwilling to change and see the benefit in different kinds of thinking, tools and models, we continue to stay behind the expected standards. Another thing I really like about the problems we’ve done is that they are real world problems (also part of common core standard language) and the connection to algebra makes algebra meaningful to kids. At this time, they don’t see any reason for learning it. In terms of getting students used to this thinking, I do think that is going to be a difficult feat, but once practiced it will improve just like everything else. I’m looking forward to going through more problems in class. I will probably need help to model one of them-found the answer by solving an equation! I’m still proud of that even though I know I wasn’t supposed to!!! Hopefully someone can help me out with the bar diagram if I’m not able to figure this one out. See you soon, Andrea
ReplyDeleteAndrea,
DeleteI am glad you are recognizing the effectiveness of using models with your students. It is so frustrating to me to hear and see that so many math teachers will not consider including new models in their teaching. Only 3% of our population are successful at "school" mathematics because of the way it is taught. As one student told me...I can do all the procedures and get the right answer but I don't know how to think about the mathematics so when I get an unique problem I just don't know what to do...and I got a 5 on an AP calculus exam.
This student was reacting to the type of teaching your colleague does...and if he expects his students to be successful on the MCAS/PARCC will have to change his thinking. Interestingly Singapore, Hong Kong, and Japan all use the bar diagrams and call them model drawing. And their students are very successful.
Anne
Since last class I have also struggled with the language. In the case of the Tom, Larry, Georgia problem, the addition or subtraction of one word from the sentence changes the whole understanding of the problem. When I realized that my original answer was incorrect I looked at the problem for 30 minutes trying to figure out how I could change my reading of the text to have the correct understanding. This is something I think my students would struggle with given a similar problem.
ReplyDeleteThe modeling using the double number line is something that has already proven extremely valuable for both me and my student. I began talking about ratios this week in class and I saw some faces light up with understanding a way I have not seen for most of the year. The ability to visualize the ratios on the two lines makes it much easier to understand the concept of unit rate. My students who had started on the topic already (I teach students from 5 different math teachers) said that my explanation using the double number line made it much easier for them to grasp the concept of ratio.
Andrea, I think it is interesting that you spoke with another math teacher and he thought teaching this way was not a good thing. This being my first year of teaching, I am open to any and all new ways of thinking and being only 12 years removed from the grade I teach, I remember how difficult some topics were for my classmates who did not think the same way as the teacher. As Anne has said in class many times, we have to change our teaching because our students are struggling.
I agree a lot with what Helen is saying. The language in a math problem has got to be SO specific. Many times I have read a problem and not understand what they were really asking. "Well, do they want this, or do they want that"? It gets harder for second language learners to figure out an answer. Years ago a math guru explained in a class I was taking that a second language learner has to take the math language, translate into his native language, find an answer, translate it back into math language in the host country...no wonder they might get confused!
ReplyDeleteTom, Larry and Georgia, that one gave me fits. I wasn't really sure where I was supposed to go with that. Larry made 10% more than G. Tom made 10% more than L, At first I thought T made 20% more than G, but when I did the math, it turns out he made 21% more. Which went along nicely with Helen's graph of two 10 bars, and 1 single box.
I did try some number line problems with my fifth grade class. It was their first exposure to double number lines, and the problems were very easy. Still, a few didn't grasp the visual, and I kept thinking that only 1/3 of the class thinks like I do. Trying to figure out alternative explanations is much easier said than done. I feel I'm going to need a lot more experience in failure before I can become a better math teacher.
Larry,
DeleteOh my goodness, I am glad I cannot count the ways documented my unsuccessful lessons over the years...but I was up front with my students that we would work things out and by using the conjecture board I could buy myself time to find a better way of discussing the math. Thank goodness for the support of other teachers who were willing to work with me as I changed my practice for the better over the years. There are a lot of resources, ATMIM, NCTM, and other math associations are and have wonderful resources as you work through this journey.
Anne
Larry its not failure. Your learning both about yourself and your students. What works and what doesn't. Don't be so hard on yourself.
DeleteAs I missed last class, I hope to catch up at our next meeting. In the meantime, I have done my best to solve the homework problems. So far, I am concerned about students who cannot multiply and hate percents and fractions. Maybe my approach is different from last class. So, I will wait before any additional comments.
ReplyDeleteHopefully the work we will do with percents this evening will be helpful.
DeleteWe missed you at our last meeting.
Anne
Rich,
ReplyDeleteI think as you experience more and varied methods for introducing concepts not only will you develop a greater understanding of why the algorithms we learned worked but better still, as you are seeing, your students will avoid the misconceptions and frustrations that many of our classmates demonstrated if they were not good "rememberers." Also as you note these visual models help a lot of our students.
Anne
I found myself more successful with this weeks homework, or atleast I thinkI have been more successful. I found that the time we were able to spend in class clarifying misunderstandings about the bar/tape diagrams was very helpful, and that is why I felt better about my work out of class this week. I think it is important for us as educators to be able to see our students misunderstandings and be able to work with them to clarify and correct. This is something I struggle with greatly in my class, both because of time and student work output. So many of my students wont attempt practice on their own (homework) and it is hard to catch or clarify their mistakes when they aren't putting in the time/effort to practice and make those mistakes.
ReplyDelete