If I understand correctly, there may be three separate concerns you've articulated:
1) Testing (school won't test, school won't accept outside results)
2) Acceleration (school won't accelerate, possibly due to concerns of
mixed age classes)
3) Math (student solves algebraically, school seeks visual-spatial solutions)
There is an art to staying in a conversation long enough to obtain the answers you are seeking, including clarifications to your understanding of what the teacher/school/district has said. Taking notes can help. For example, note taking can keep things moving, focused, and on topic. At the same time, note taking can also slow things down enough to "buy time" for a person to count to 10 (thereby avoiding a flash of anger in a response which may guess at motivations or sound accusatory or emotional); This may afford the opportunity to formulate a clarifying question or allow a person to rephrase what they understood.
I think the underlying message is that they just don’t want to accelerate this child.
This child? Or any child? Or... not accelerate any child more than 1 year? Etc? The more clarification a parent has, and the better an objection is understood, the more prepared a parent may be to advocate.
In all seriousness, how can you deny or even have a discussion about acceleration, when the child’s school won’t test him and ignores the results of independently administered test?
Not every battle is worth fighting. For example, when school district's testing practice does not meet a parent's expectations, only that family can decide whether the disappointment about testing is worth investing effort to hopefully bring about a change. If the family perceives a large potential impact on their child (or many children), and believes the issue is worth putting forth effort to make a change, the family may wish to prepare for individual advocacy efforts and/or work to amend school district policy.
If a family is determined to proceed, they may first wish to consider what managed their expectations about in-school testing and/or the acceptance of outside test results. For example, was there something in State law? ...in school district policy? ...mentioned by school district personnel in a meeting? Or possibly something posted to a forum indicating how another school district handled testing?
To prepare for individual advocacy efforts, parents may wish to:
1) become familiar with any applicable State laws and determine whether current school district actions follow applicable State laws
and/or 2) become familiar with school district policies and prepare to discuss whether current school district actions are in line with published district policies; parents can hold schools accountable to abide by their current published policy statements (for example, policies regarding identification/testing)
and/or 3) become familiar with the school district objections to the acceleration or placement you are seeking and determine whether you have sufficient material (research studies, statistics, facts, personal anecdotes) to overcome objections
Parents may also gather with others for strength in numbers and submit written suggestions for policy changes (often based upon what other districts may be doing). Finding and reading the policy and practice statements of other public school districts may be a good starting point for gathering ideas as to how a local school district's policy statements may be improved.