mathematics

Improving Math Problem Solving Skills: Anchored Instruction

ANCHORED INSTRUCTION

Anchored instruction is an idea that purportes the necessity to “anchor” learned ideas, especially in the realm of mathematics, to real-world ideas. Students and teachers “engage [in] problem-rich environments that allow sustained exploration” (Cognition and Technology Group at Vanderbilt, 1992). The idea behind anchored instruction stems from constructivist pedagogy, and is intended not to increase computational skills in students but rather to improve problem solving skills in real world situations. It’s the difference between

25 x 3

and

Frederick has 25 potatoes on his farm. He grows his potatoes for 4 more months and triples his number of potatoes. How many potatoes does he have now?

The first is computational, and the second is problem solving.




Is Anchored Instruction a Good Thing?

Anchored instruction necessitates a change in pedagogical style, and with it, a change in the way that we assess and teach. Anchored instruction is a part of the “Guide on the Side” vs. “Sage on the Stage” movement popularized by Alison King (1993). Rather than instructing up front in a lecture style, anchored instruction puts students into “real world”, or at the least certainly more authentic, problems than if students were to complete worksheet upon worksheet of math problems. Complex, real world problems are not overly difficult to create for students either. Anchored instruction methods are highly collaborative, problem-solving based, and have more than one “right” answer.

Jasper Adventures: An Anchored Instruction Example

Up until now, we have only talked about how amazing anchored instruction can be, without much evidence to back up these claims. Let’s take a look at Hickey, Moore, and Pellegrino’s study of the “Jasper Adventures”, a video anchored instructional mathematics series from the 80’s (more information here).

The long and short, Jasper had 12 video “adventures” that students would watch. While watching, students would hear facts and figures as a part of the story that they would need to use to solve a final problem posed at the end of the video. Jasper was based on the idea that students needed to become independent thinkers, rather than only being able to regurgitate mathematics proofs and formulas (Cognition and Technology Group at Vanderbilt, 1992). Jasper seeks to make learning relevant, instead of creating inert knowledge.

Inert Knowledge: Knowledge that is not used spontaneously, even though it is relevant
— Bransford et. al., 1986; Gick & Holyoak, 1980, 1983; Scardamalia & Bereiter, 1985)
 

Overview of Hickey, Moore, and Pellegrino

  • A quasi-experimental design in which the authors studied 19, fifth grade classrooms from two well-matched schools.

  • One school was higher in socioeconomic status (SES), the other in low.

  • Half the classes used Jasper materials, half did not.

  • Classes were split into “more consistent” use of new reform mathematics curriculum encouraging research based practices, or “less consistent” to the new reformed curriculum. (USA National Council of Teachers of Mathematics (NCTM) curricular standards)

  • Had 4 groupings to study:

    1. High SES and more consistent classrooms using Jasper

    2. High SES and less consistent classrooms

    3. Low SES and more consistent classrooms using Jasper

    4. Low SES and less consistent classrooms

  • I found their research parameters and practices sufficiently rigorous in its methods. However, as always, be sure to check out the research for yourself. The research generalizability should be relatively high for other high and low SES classrooms in the united states and similar countries, and provides strong support for using constructivist style practices in mathematics education.

Research Goals

(a) consider student subjective motivational experiences,

(b) study a large-scale implementation that was initiated and carried out by the school system

(c) using newer ostensibly more appropriate standardized achievement measures

d) comparing consequences in classrooms that are more consistent and less consistent with the broader curricular reforms [NCTM & Jasper]

(e) comparing consequences in higher-achieving, high-socioeconomic status (SES) classrooms and lower achieving, low-SES classrooms (2001. pp. 615)

Main Findings

  • Teachers using Jasper materials had goals to use mathematics to solve real world problems, but did not define using collaborative methods as a goal (despite allowing for more collaboration in their classes).

“ All six of these Jasper teachers listed their first (or only) goal for the activities as something like "showing students how math problem solving is useful in the real world." Meanwhile, none alluded to the broader goal of supporting extended collaborative investigation around complex problems” (p. 634)

  • Increased Problem solving skills in students using anchored instructional methods.

“The mathematical achievement results in the area of problem solving and data interpretation clearly showed that the Jasper instructional implementation had very desirable consequences, with no evidence of negative consequences.” (2001, p. 648)

“ In other words, the scores in every Jasper classroom increased while the scores in every non-Jasper classroom stayed the same or went down slightly” (p. 638)

  • Improved conceptual knowledge and estimation skills were limited to high SES Jasper classrooms, not low SES classrooms.

  • High SES students using Jasper report lower subjective competence in mathematics than non-Jasper students. The researchers note that this is most likely this is due to the highly complex, challenging, and novel ideas of the Jasper activities. Low SES students reported increased subjective competence in mathematics.

  • Low SES students had more positive outlooks on mathematics education.

  • High SES students perceived the Jasper activities as “effectively delay[ing] their progress through [the] levels of curriculum” (p. 637).

  • Support against the idea that lower achieving students will not be able to handle high levels of complexity in problem solving style mathematics problems and are better served with traditional mathematics teaching methods. Using Jasper materials with lower SES students:

“supports the argument that academically disadvantaged students can profit from the complex problem-solving activities associated with the Jasper materials and that such students do not suffer negative academic or motivational consequence".” (2001, p. 648)

Conclusion

Given the above evidence, it seems clear, at least in the continental USA, that there is strong support for the idea that anchored instruction improves mathematics abilities in grade 5 students. The question now becomes, how can we, as educators, best incorporate anchored styles of instruction into our own practice, and how much time should we spend teaching a skill before sending students off to problem solve? Should we teach the skill parallel as students need for the skill arises, or should we front load this instruction?

REFERENCES

Cognition and Technology Group at Vanderbilt, (1992). The Jasper Experiment: An Exploration of Issues in Learning and Instructional Design, Educational Technology Research and Development, Vol. 40, No. 1, pp. 65-80. Retrieved from: http://www.jstor.org/stable/30219998

Daniel T. Hickey, Allison L. Moore and James W. Pellegrino, (2001). The Motivational and Academic Consequences of Elementary Mathematics Environments: Do Constructivist Innovations and Reforms Make a Difference? American Educational Research Journal, Vol. 38, No. 3 (Autumn, 2001), pp. 611-652. Retrieved from: http://www.jstor.org/stable/3202494

King, A., (1993). From Sage on the Stage to Guide on the Side, college Teaching, Vol 41, No. 1 (Winter, 1993). pp. 30-35. Retrieved from: http://www.jstor.org/stable/27558571?origin=JSTOR-pdf






ETEC 533: Interview with a Mathematics Teacher Veteran

Below are several excerpts from an interview that I had with a colleague, “M.”. Our conversation concerns the benefits and drawbacks of technology inside of a Middle School mathematics classroom. Below, M’s words are italicized while my own are in regular font. M.’s name has been written as a changed initial for privacy reasons.

About M.

M. has been a teacher for 12 years and previously worked as an engineer in a municipal role, designing builds for bridges and roads. He has a passion for teaching students applied mathematics, and has taught in Kuwait and South Korea during his teaching career. Presently, he is a Middle School Mathematics teacher, as well as an instructional coach for teachers K-12 in our school.


Real World Applicability of Mathematics

“For math and using tech, the simplest tech is the best and im trying to get my students to be ready for the real world; because my background is in engineering. To your question specifically... most applicable technology is excel. Just getting them to use the program...getting the students to be comfortable to be using excel in a practical application.

Working in an engineering firm, the type of program that we use...is basically excel or a spreadsheet on steroids...so i try to get student comfortable with area and volume...and make the worksheet look like a form... and get them to use excel in real life because that's the one that they;ll use the most often as engineers.

This was surprising to me, that M. believed the best technology that he has used in his classroom was simply using an excel program. However, the above transcript points out a few important things.

Many teachers often have not had real world experience in their field before they go in and teach the content. M., having been a city engineer before, has knowledge of what content he can include in his classes that will have the most carry-over to the real world. This means that he can look past much of the glamour of new technologies and design lessons that would transfer well to the real world of engineering for his students.

I realize that it is not possible for all teachers to be able to have had real world experience in their field, and in this case, having a course on real world applicability of technology in various careers, watching recorded interviews, or speaking with employees from a variety of careers would help teachers assess what technologies would be more applicable for use in the classroom.  

With technology, there is a growing notion that “newer is better” as the technology is the newest, most flashy piece of information on the market and this gets kids engaged, and teachers, as it looks cool. However, teachers need to take a step back and think pedagogically why we include the technology that we do in a classroom from the basis of an entire curricula. Excel, working off real world experience, seems to be an incredibly robust tool to use in a math classroom as it is a piece of software that students, should they decide to become engineers, will use every day in their lives. Not only do teachers need to stay up to date with technology, they must also wrestle with what technology is most applicable as well as fits in with the rest of their curriculum as well.





The Four Uses of Technology in Classrooms


One of the more interesting offshoots of our conversation was M’s notions of various uses of technology in the Classroom, and how he categorized the use of technology in his Math classroom.

1.Student interacting with technology

This is the Direct Application use of technology.

The most obvious category is students interacting with digital technology in the classroom, using applications to solve problems, and working with digital technology throughout various projects.

 

2. Formal & Informal assessments

“Here you can use [technology] as an exit ticket, informal assessment, formative assessment; You know, let’s check in and see [how you are doing]... let’s practice what we just talked about it.”

Using technology as a “quick and dirty” exit slip to check in on understanding of a concept for the lesson, was a low pressure way for students to engage and get involved in the content. Rather than using it as a formal assessment, it allows a quick check in with students to see if they understood the lesson content. As most digital forms can mark automatically, it can serve as a quick and efficient formative assessment option for teachers.

Using tech as a formal assessment method has been difficult for M. so as to not let students open another program in order to cheat on the exam. NWEA Lockdown browser may be a good tool for his formal assessments with technology as it locks students into an application and a single website/tab for the completion of a formalized test.

 

3.Presentation

“Not student[s] engaged [with technology], but presentation style. Using an ipad for drawing a picture and projecting that up on the board... there are a lot of different ways to present material [with technology]”

When designing curriculum, a teacher needs to decide what concepts should receive priority, and with this, what concepts can be taught in a more inquiry fashion versus a “typical” lecture style format. Depending on whatever structure chosen for the lesson content, technology can assist with the presentation of information. Obviously, apps like powerpoint, keynote, videos, and even video interviews come to mind when thinking about various presentation styles. However, technology can also assist with showing off one’s learning on an individualized basis, such as 3D printed models, laser cut objects, or even an interactive digital textbook.

 

4.Backend Assessment & Grading

“[This is] the backend students don't see. Putting together ways that we can directly input in grades and calculate grades using spreadsheets.”

The last use of technology in classrooms was on the backend where teachers can more efficiently calculate grades (or standards tracking if grades are not used). Spreadsheet programs allow for instant calculation, as well as visual representations of data that would take far longer to complete by hand. This gives a teacher more time to be able to focus on other tasks at hand in their teaching profession, than needing to spend time calculating marks.