I love that you said this! Error analysis problems are my favorite type of problems and now they will be built in naturally. I am excited.
When helping students transition from concrete arithmetic to abstract algebra, I try to do things many of you have mentioned: manipulative, drawing pictures, relating new learning to old learning and/or use stories to help students imagine the situation, etc. Manipulatives can be limited at times so I often have students work together or I model to the class. This can be somewhat limiting in a large class as sometimes the learner who needs the kinesthetic option the most, might not get ample time or his/her hands-on the materials to create a more concrete experience. Some students are not quite developmentally ready for this transition, especially if they are in an advanced math class when they shouldn’t be. Our district believes that all students must take Algebra by 8th grade so the need for this jump from concrete to abstract happens way too quickly. Couple this with the pacing, I don’t feel like I get to spend even a fraction of the time needed to make this transition successful.
I’m hoping students will find programming to be another way to see the relationship between different representations of the same thing. My goal is for students to make the connection between the real world and mathematics; programming will hopefully be the tool to do this!
I have tried attaching meaning to “real Life” things and used images, tables, graphs to model abstract concepts. I’m hopeful that programming will help students see the relationships between a rule—and what the rule can do for you in a concrete way–it either works or it doesn’t and they have to try to figure out how to “fix” the issue themselves.
Letting students work in groups to be able to talk about their ideas seems to be the most helpful. That allows them to make conjectures and test their ideas before they share with a larger group. I think programming will also give them another avenue to test their conjectures in a non-threatening manner.
By starting with tables and graphing, we tend to look for patterns and then introduce variables later.
i don’t teach algebra. i teach technology. when teaching abstract concepts, i use analogies and everyday experiences to help students make connections.
my students find remember procedural steps hard to remember.
programming can help them by not requiring them to remember a single step! they can just program the computer to help them.
When making the transition for the 6th grade students that I work with I try to relate everything back to the concrete arithmetic that they have learned in elementary school. This seems to work for most of the students but some students that have gaps in their learning struggle even more to make these connections. I think that programming will open the doors for more students and show them the true life connection between mathematics and computer science
In the past for me having my students be able to use multiple representations has helped. Especially if they are able to explain the connections between each representation. I think programming will help the transition because they will be able to see a connection to real life. It will give them an avenue to apply mathematics.
I have had some success in moving from concrete arithmetic to abstract algebra using algebra tiles, or problems where someone is paid a certain total amount for all their lawn mowing jobs, say, and some of the money comes in envelopes and some in loose cash. Students need to figure out how much is in the envelopes. I have also had success using balance problems, where students need to make the two sides of a balance work and need to figure out the weight of variously shaped pieces using one know weight.
My students struggle with persevering with problems - going back and checking their work to find their error.
Programming forces students to do this, because the program doesn’t run unless you find all the errors.
It’s definitely true that having students do algebra when they are younger can be a struggle. I do think that kids who have been using common core standards for longer have a little easier time with this, since they introduce bits of algebra very early.
I’ve never taught algebra before so I’m looking forward to exploring all the ways to solidify these concepts in my middle school students.
Providing many visuals help students transition into different concepts. Students still have difficulty explaining their thinking and writing it down. I think programming can help transition by making connections and explaining the program they are working on to improve it or make it functional.
I have not taught much of algebra either but look forward to learning more and teach middle school students.
Programming will allow my students to access the material in a completely different way which may help some students understand the material better than they would have without programming. It allows for almost a hands-on approach to something that can at times be viewed as very abstract.
I think it will also help to gain student interest. Many times there are students who are not interested in mathematics but are interested in technology and this is a way to blend those two things and increase interest for all students who have at least a passing interest in one of these topics.
The best thing for my students is to see the same concept in different contexts. Application of real world examples helps the abstract become a little more concrete. Developmentally it is difficult to push students into this way of thinking if they are not cognitively ready for it. Knowing your students zone of proximal development, and their learning style helps determine what teaching strategy is best. However, you must have those tools as a teacher in order to help the students.
Things that have worked: using real-life examples that require algebra, using more kid-friendly words like having them think of variables as “mystery numbers”, repetition of key ideas.
Difficulties: deriving equations with variables from word problems.
I think programming can help by providing another way for them to understand these concepts. Also, they can create programs that contain algebraic equations.
Students to best when I can link abstract ideas to something concrete. It is critical that I find hands on activities that help build the connection between the concrete and abstract. Many concepts are hard to find real life examples/strategies so I am hoping programming with help fill in the gaps in my lessons for where these holes are most prevalent.
Students have difficulty working with unknowns. When letters are introduced into their math vocabulary, they struggle to understand what it all means. Visual representations of this aspect help in making sense for students.
I think this video described the problem a lot of my students have. I fear even bigger problems with my group next year so I’m excited to add this as a tool to help them connect on the idea of functions. Other groups can describe all the ways to describe a function but I’m never completely certain they understand exactly what it is.
I first transition students by acknowledging possible misconceptions, difficulties understanding vocabulary or concepts that may have been a staple in earlier mathematics classes, that could be used differently now. For example, x=1, x=2, x=3. Students may struggle with the notion that x can change.