Providing for a Range of Learners

One underlying aspect of the philosophy of Investigations is that all students learn by different methods, think differently, are motivated by different learning situations, and progress through levels of understanding in different ways for different concepts. This is because each student has to make sense out of each new idea, and no two students will do this in exactly the same way. So, the authors of Investigations strongly recommend that students be grouped heterogeneously in classes so that they can learn from each other. One word that is often used to describe the classroom environment is safe. The teacher has the responsibility to promote a classroom community of learners so that everyone can share equally in learning and no student feels that his/her thinking is not valued.

Enrichment

How will this program benefit a child who is able to solve problems quickly and who masters concepts quickly?

(This question is also addressed on another page on this site that specifically explains the connection between Investigations, Algebra I, and higher mathematics.)

First, we must consider the question of whether we truly understand an idea if we only have really thought about it in one way (what if you only knew one way to drive to work from home, or one way to make soup?). Along with this, we must consider the question of whether it is possible to know everything there is to know about a given idea (especially as a child) and whether it is worth examining the idea from another angle. The more ways in which we think about an idea, the more likely we are to be able apply it in other situations (which, incidentally, is a major stumbling block for many students in higher level math courses who have been able to get by until that point with only one way of understanding an idea).

Very often, the students who have been the high achievers in math are those who have the most trouble adapting to this classroom culture because they are probably good at memorizing and reproducing facts and skills, but perhaps not so experienced at justifying their thinking (orally or in writing) and listening to the thinking of others, which are abilities that we want to develop in students for life, not just math class. (It should also be noted that students will need to write on the state achievement tests and the Ohio Graduation Test. See the page on helping your child at home for some suggestions related to this.) For these students, it will be important to build their confidence in their ability to solve non-routine problems since they may not be used to doing this. Also, the tasks in Investigations are open-ended enough to be easily adapted, often just by changing a number or the wording of the question, to really create a challenge for those students who need it. These students are often at their best when given a unique challenge – a What if? question that still relates to the activity at hand. We as adults need to show students that we are lifelong learners and can learn right along with them. Example questions that are great to ask highly motivated students are:

Is that true all the time? How do you know?

Can you think of a counterexample (that does not fit the rule)?

How would you prove that?

What assumptions are you making?

What would happen if ( )? What if not?

Do you see a pattern? Can you explain the pattern?

Can you predict the next one? What about the last one?

How does this relate to ( )?

What if you had started with ( ) instead of ( )?

What if you could only use ( )?

What are the key points or big ideas in this problem?

Very important is the idea that we cannot assume that if students are able to get right answers quickly, they should move on to more complex ideas. It is very often the case that these students have simply memorized facts or skills and do not understand them. An analogy: if an adult could very quickly read a list of words in French, would it necessarily be the case that she could speak French? One easy way to check to see if a student truly understands a solution to a problem is to ask how he/she knows the method used makes sense. If the answer does not involve mathematical thinking but instead involves a restatement of the procedure, comments about the positions of numbers in the problem, or the infamous That is just what we learned in class, chances are that the child does not understand and is not ready to move on. Another way to confirm this is to ask if he/she has another way of solving the problem (if not, the student probably does not understand). We do a disservice to children by assuming they understand and pushing them on toward skills for which they actually have no conceptual foundation. Eventually, they may bottom out when they get to a concept that they cannot memorize or simply reproduce.

Investigations provides extension activities with many lessons. Providing challenges within the context of the activities, as described above, can help promising students to explore at their own levels while still being involved in the learning of the class as a whole. To help the typically high-achieving students to feel comfortable explaining strategies and listening to those of other students, we need to remind them that ideas are much more useful when they are shared with others, and we need to let them know that we are truly interested in their unique ideas (and their ideas are often very unique!). Also, we need to remind them that they can always learn if they listen to the ideas of others. The idea is not at all to hold these students back, it is just the opposite: to encourage them to think and reason and to provide mathematical challenges through which they can truly blossom and reach their personal best in learning.

Click here for a Microsoft Word document that contains quick suggestions for creating challenges within the homework assignments in the second unit at each grade level. Kindergarten is not included here because the minimal homework is usually very open-ended. Similar adaptations can be made for the assignments in any unit.

Click here for a document called Creating Mathematical Challenges by Jan Mokros, one of the primary authors of Investigations.

Click here for an article (PDF file) called Providing Mathematical Challenges by Marlene Kliman, a member of TERC (the group that authored Investigations).

Click here for a document about ways teachers may enrich within the classroom to challenge students.

Remediation

Because this program was designed to be more developmentally appropriate for students than any traditional program, teachers often find that students who once needed some type of remediation do not need as much, if any. The hands-on and pictorial learning that occurs is often just what these students need in order to eventually reach that symbolic level of learning. However, there will be some students who will still struggle with concepts, and as described above, Investigations tasks can be easily altered for these students.

Often, there is a question about students with language disabilities and how these students can succeed in this program. Essentially, there are two ideas to consider: 1) In any content area, these students can best develop their language skills by using them; 2) As needed, teachers can make modifications as they would for any assignment. Particularly if a student is on an IEP for language, a teacher can account for this in grading responses that involve writing or verbally explaining, as the teacher would do in any content area.

Click here for a document called Supporting Math Learning in Inclusive Settings.

Click here for a document called Strategies for Special Needs Students.

Click here for a document called English Language Learners.

Click here for a document called Students with Special Needs.

Click here for a document called Specific Inclusion Strategies. This specifically addresses issues related to ADD and Aspergers Syndrome.

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